diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzomrm" "b/data_all_eng_slimpj/shuffled/split2/finalzzomrm"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzomrm"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\nThe origin in nature of the neutron-rich isotopes \nheavier than the iron group remains uncertain,\nbut rapid progress in both theory and observation has been\nmade in recent years. For example, \nanalysis of the metal deficient star CS 22892-052 ([Fe\/H] $\\approx\n-3.1$) by Sneden et al. (1996) shows\nstriking evidence that the solar abundance\npattern for the $r$-process isotopes exists even at very early times\nin our Galaxy, and suggests an origin for these nuclei\nquite distinct from the $s$-process. The fact that the $r$-process\nabundance pattern in this very metal deficient star is so strikingly solar\nacross the entire range $56 \\le Z \\le 76$ also suggests that this\npattern is generic and reflects the conditions constantly obtained\nin a unique astrophysical environment responsible for the $r$-process.\nHowever, we note that the current observational data for $Z < 56$ \nare not so conclusive.\n\nOn the theoretical front, many recent calculations (e.g., \\cite{wh92};\n\\cite{how93}; \\cite{twj94}; \n\\cite{wwmhm94}) have shown that a promising site for $r$-process \nnucleosynthesis\nis the neutrino-driven wind blowing from a nascent neutron star following\na core-collapse-driven supernova. This $r$-process site has some\nattractive features. First of all, the $r$-process would be primary, \nin accordance\nwith the observation of \\cite{sneden96} (see also \\cite{cow96};\n\\cite{matcow90}; \\cite{ctt91}). \nFurthermore, the total mass loss in the wind\napproximately accounts for the amount of\n$r$-process material ejected per supernova, $\\sim 10^{-5}\\ M_{\\odot}$, \nas expected from the supernova rate and total $r$-process yields of the Galaxy.\nFinally, the conditions in the wind are determined by\nthe properties\nof the neutron star and the characteristics of its neutrino emission.\nTherefore, the $r$-process nucleosynthesis might be approximately constant\nfrom event to event, reflecting the near constancy of the neutron star\nmass and cooling history, if possible complications due to fallback \ncould be ignored. \n\nHowever, problems have emerged in the theoretical\nmodel. The successful $r$-process calculations, with the possible exception of\nWoosley et al. (1994),\nall utilize parametric modifications to the key parameters,\nespecially an artificial increase in\nthe entropy. The high entropies of Wilson's supernova model reported\nin Woosley et al. (1994) have not been replicated elsewhere.\nThis discrepancy has been\nemphasized by the analytic calculations of Qian \\& Woosley\n(1996, hereafter Paper I). In general, the deficiency in\nentropy is only a factor of two, but the gap is proving difficult to\nbridge. Paper I also suggested that other relevant parameters of the\nproblem, especially a short dynamic time scale or \nlarge neutron excess, might\ncompensate for the low entropy, or that there may emerge other sources\nof entropy hitherto neglected. But, for the time being, a potentially\nbeautiful solution to a classic problem --- the origin of \nthe $r$-process --- falters by a factor of two.\n\nWe will not resolve this quandary in the present paper. What\nwe shall do however, is: a) show that the standard models of\nthe neutrino-driven wind \nderived in Paper I give interesting nucleosynthesis above\nthe iron group, but with the electron fractions obtained in \nWoosley et al. (1994), do not give the classical $r$-process; and b) determine, both\nanalytically and numerically, the conditions that {\\sl are} required\nin this sort of model to produce the\n$r$-process, in particular the platinum peak.\n\n\\section{Nucleosynthesis In Neutrino-Driven Winds}\n\nAn $r$-process might occur for various combinations of\nentropy, electron fraction, and dynamic time scale.\nDifferent authors, using numerical supernova models, \nhave arrived at qualitatively different values for \nthese key parameters in the neutrino-driven wind. \nIn order to obtain a better understanding\nof the physical conditions,\nboth analytic and numerical studies of the wind were carried out in Paper I. \nThe primary goal of that paper was to\nexamine the dependencies of the physical parameters in the wind\non the neutron star mass ($M$) and radius ($R$), and the \nemergent neutrino luminosity ($L_{\\nu}$) and energy spectra. \nThe analytic study was in the same spirit as that of \nDuncan, Shapiro, \\& Wasserman (1986),\nbut was more extensive and directed towards nucleosynthetic issues. \nThe analytic results of Paper I were given by equations\n(48a), (48b), and (49) for the entropy ($S$), \nequations (58a) and (58b) for the mass outflow rate ($\\dot M$), \nequation (61) for the dynamic\ntime scale ($\\tau_{\\rm dyn}$), and equation (77) for \nthe electron fraction ($Y_e$).\n\nPhysically, the first three parameters ($S$, $\\dot M$, and $\\tau_{\\rm dyn}$)\nare determined by the sum of heating produced by all neutrinos,\nwhereas the evolution of $Y_e$ mostly reflects the difference between the\n$\\nu_e$ and $\\bar\\nu_e$ fluxes through the inter-conversion of\nfree nucleons by $\\nu_e$ and $\\bar\\nu_e$ captures. \nAs discussed in Paper I, the first three\nparameters are not sensitive to the exact values of $Y_e$, and \ntheir determination can be essentially\ndecoupled from the evolution of $Y_e$ in the wind.\nThe analytic results for these three parameters were tested by\na series of numerical calculations\nusing the one-dimensional implicit hydrodynamic code KEPLER.\nA total of nine models were studied. The neutron star mass and radius,\nand neutrino luminosity at the inner boundary of \nthese models were varied to observe the corresponding \neffects on these three parameters.\nA comparison between analytic and numerical results was summarized \nin Table 1 of Paper I. \n\nFrom these numerical models, we have extracted the velocity ($v$),\ndensity ($\\rho$), and temperature ($T$)\tof a mass element as functions \nof its position ($r$) in a steady-state wind. Using $dt=dr\/v(r)$,\nwe can obtain the evolution of $r$, $\\rho$, and $T$ with time $t$ for\nthe mass element. Starting at $T_9\\approx10$ ($T_9$ is the\ntemperature in units of $10^9$ K), when nuclear\nstatistical equilibrium (NSE) is assured, we then follow the time\nevolution of the nuclear composition in this mass element with \na reaction network (\\cite{wh92}), until\nthe composition freezes out at $T_9\\approx 1$.\nTo good accuracy, the mass element is initially composed of free nucleons, \nin proportions specified by the initial electron fraction\n$Y_{e,i}$. The initial electron fraction is essentially determined by\nthe equilibrium between $\\nu_e$ and $\\bar\\nu_e$ captures on free\nneutrons and protons, respectively (\\cite{qian93}; see also Paper I).\nIf we denote the rates for $\\nu_e$ and $\\bar\\nu_e$ captures on \nfree nucleons as $\\lambda_{\\nu_en}$ and $\\lambda_{\\bar\\nu_ep}$, \nrespectively, the initial electron fraction is given by\n\\begin{equation}\nY_{e,i} = { 1\\over{1+\\lambda_{\\bar\\nu_e p}\/\\lambda_{\\nu_e n} }}.\n\\end{equation}\nIn turn, the rate $\\lambda_{\\nu_e n}$ ($\\lambda_{\\bar\\nu_e p}$) is determined\nby the $\\nu_e$ ($\\bar\\nu_e$) luminosity and energy spectrum.\n\nIn realistic supernova models, the neutrino luminosity and energy spectra\nevolve with time. The individual wind models of Paper I represent the\nsteady-state configurations reached at different neutrino luminosities\nover time scales much shorter than the evolution time scales of the\nneutrino luminosity and energy spectra. Because the time evolution of\nthe neutrino energy spectra is much less pronounced than that of the\nneutrino luminosity, two generic sets of neutrino mean energies were\nassumed for these models. From the analytic results of Paper I, we\ncan see that the mass loss rate, the dynamic time scale, and especially\nthe entropy would not change significantly if a more precise prescription\nof the neutrino energy spectra were used. However, because the electron\nfraction is sensitive to the difference between the $\\nu_e$ and $\\bar\\nu_e$\nfluxes, and the nature of heavy element nucleosynthesis is extremely\nsensitive to $Y_e$, in this paper\nwe calculate the $\\nu_e$ and $\\bar\\nu_e$ reaction\nrates according to the time evolution of neutrino luminosity and \nenergy spectra in Wilson's 20 M$_\\odot$  supernova model used in\nWoosley et al. (1994). Specifically, we take \nfrom Wilson's supernova model the neutrino energy spectra\ncorresponding to the same neutrino luminosity as in\nthe wind model. Starting with the initial value in equation (1),\nwe then follow the evolution of $Y_e$ in the reaction network, taking into\naccount $\\nu_e$ and $\\bar\\nu_e$ captures on free nucleons and heavy nuclei\n(\\cite{macful95}),\nelectron and positron captures on free nucleons and heavy nuclei, \nand nuclear $\\beta$-decays.\n\nThe results of the nucleosynthesis calculations are presented in \nTables 1--3. \nThe entropy, the initial \nelectron fraction, \nand the dynamic time scale are given for each wind model.\nThe dynamic time scale roughly corresponds to the time over which\nthe temperature changes by one $e$-fold (Paper I). In order to\ncheck the influence of the neutrino flux on the nucleosynthesis,\nwe have carried out five different calculations.\nAll runs included electron and positron captures on free nucleons\nand nuclei, as well as nuclear $\\beta$-decays. The individual runs\ndiffer in the inclusion of various neutrino reactions. Respectively,\nthey cover the cases\nincluding (1) no neutrino reactions (column 2), (2) $\\nu_e$ and\n$\\bar\\nu_e$ captures on free nucleons only (column 3), \n(3) $\\nu_e$ and $\\bar\\nu_e$ captures on free nucleons, and \nneutral-current neutrino spallation on $\\alpha$-particles (column 4),\n(4) $\\nu_e$ and $\\bar\\nu_e$\ncaptures on free nucleons and nuclei (column 5), \nand (5) $\\nu_e$ and $\\bar\\nu_e$\ncaptures on free nucleons and nuclei, as well as neutral-current\nneutrino spallation on $\\alpha$-particles (column 6).\nFor all runs, we give the electron fraction ($Y_{e,f}$), the\naverage mass number of nuclei excluding free nucleons and \n$\\alpha$-particles ($\\bar A$), and the neutron and $\\alpha$-particle\nmass fractions ($X_{n,f}$ and $X_{\\alpha,f}$) at the freeze-out of\nthe charged-particle reactions ($T_9\\approx 2.5$). \n\nIn Figures 1--9, detailed nucleosynthesis results from the runs\nthat included $\\nu_e$ and $\\bar\\nu_e$ captures on free nucleons (column 3\nin Tables 1-3) are given \nin terms of the production factor, defined as the final mass\nfraction (after all weak decays) of a given stable nucleus divided by its\nsolar abundance (\\cite{ag89}). These results are also representative of the \nnucleosynthesis obtained in the other runs. \nIn these figures, the most abundant\nisotope in the solar abundance distribution for a given element is plotted as\nan asterisk. Isotopes of a given element are connected by solid lines. A\ndiamond around a data point indicates \nthat the isotope is produced chiefly as a\n(neutron-rich) radioactive progenitor. The dotted horizontal\nlines represent an approximate\n``normalization band,'' bounded from above by the largest\nproduction factor in the calculation and from below by a production \nfactor four times smaller. Nuclei\nthat fall within this band will be the dominant species produced.\nIn Figures 1--9, no re-normalization has been attempted. \n\nModels 10A--F produced interesting nucleosynthesis representative\nof the $\\alpha$-process (\\cite{wh92}; \\cite{wjt94}).\nThe most abundant nuclei produced\nhave mass numbers $90 \\leq A \\leq 110$.\nModel 10A ($Y_e\\approx 0.47$, $S\\sim 70$ per baryon) shows the \nproduction of the $N=50$ \nclosed-neutron-shell nuclei which were grossly overproduced in\nprevious studies. Models 10B and C, \nwith progressively lower neutrino luminosity\nand lower values of $Y_e$, made heavier nuclei. Production of Sn, Sb, and Te\nwas not accurately calculated, as the radioactive progenitors for these\nspecies were isotopes of Ru, the last element in our reaction network. \nModels 30A--C       \nproduced nuclei near the iron group. Model 30A exhibits interesting\nnucleosynthesis for $Y_e > 0.5$, while Model 30C shows the\nproduction of $^{64}$Zn (made as itself), the dominant isotope of\nthis element. This nucleus was not accounted for in the surveys of Galactic\nchemical evolution and nucleosynthesis in massive stars\n(\\cite{tww95}; \\cite{wowev95}), \nand appears to be made predominantly under conditions similar to those\nobtained in the neutrino-driven wind (\\cite{hwfm96}).\n\nFrom Tables 1--3, \nit is also clear that inclusion of the neutrino \nreactions made a difference. The $\\nu_e$ and $\\bar\\nu_e$ captures on\nfree nucleons have the largest effect. The electron fraction\nincreases appreciably due to these capture reactions when free nucleons\nare being assembled into $\\alpha$-particles.\nThis so-called ``$\\alpha$-effect'' (\\cite{fulmey95}; \\cite{mfw96})\nis evident when we compare the case including no neutrino reactions (column 2)\nwith the cases including various neutrino reactions (columns 3--6).\nThe inclusion of neutral-current neutrino spallation on $\\alpha$-particles \nand $\\nu_e$ and $\\bar\\nu_e$ captures on heavy nuclei did\nnot have a major effect on the nucleosynthesis, \nat least before the freeze-out of the charged-particle reactions.\nFor the relatively low entropies studied here,\nthe neutrino spallation on $\\alpha$-particles did not have \nany appreciable influence on the final $\\alpha$-particle mass fraction. \nThis is to be contrasted with the dramatic effect of these spallation\nreactions on the $r$-process in Wilson's high-entropy supernova model\n(\\cite{mey95}). The $\\nu_e$ captures on heavy nuclei\nmay have important consequences for the ensuing \nneutron-capture phase of the $r$-process (\\cite{nadpan93}).\nRegardless of these issues, it is clear that none of these wind\nmodels produced an $r$-process. The neutrons had been consumed\nby the time the charged-particle reactions froze out \n($T_9\\approx 2.5$),\nand the nuclear flow did not even reach the $A\\sim 130$ $r$-process peak.\n \nAnother important issue concerns the overall ejection\nof the synthesized material in the wind into the interstellar medium. \nAs the following simple argument will show,\nthe magnitudes of the production factors may preclude the\nejection of such material in all nine wind models. \nFrom the time evolution of neutrino luminosity in Wilson's\nsupernova model, we find that, for example, the neutrino luminosity\ndecreases from twice to half the value in Model 30B over a time of\n$\\tau_\\nu\\sim 1$ s. With a mass loss rate of $\\dot M\\approx 1.1\\times\n10^{-2} \\ M_\\odot \\ {\\rm s}^{-1}$ and the largest \nproduction factor of $(X_{w}\/X_{\\odot})_{\\rm max}\\approx 2.7\\times 10^4$\nin Model 30B (Figure 8), \nthe corresponding normalized production\nfactor is $\\sim(\\dot M\\tau_\\nu\/20\\ M_{\\odot})\n(X_{w}\/X_{\\odot})_{\\rm max}\\sim 15$, if the total amount of ejecta\nfrom the supernova is $20\\ M_{\\odot}$. Models 30A and C give normalized\nproduction factors of up to $\\sim 30$. \nFor Models 10A--F, the normalized \nproduction factors for the nuclei produced in the largest amount\nare typically\nof order 100. However, Woosley \\& Weaver (1993) find that the\nnormalized production factor should not be much above 10 in order\nfor supernovae to produce the observed solar abundance of oxygen.\nTherefore, these wind models cannot represent in detail what commonly\noccurs in supernovae. \n\nIt is quite possible that appropriate \nmodifications of these standard wind models\ncan lead to the physical conditions for acceptable nucleosynthesis.\nPaper I studied the effects of an additional energy\nsource on the entropy\nand dynamic time scale in the wind.\nModel 10F of Paper I was recalculated\nwith an additional\nenergy input of\n$5\\times 10^{47}\\ {\\rm erg\\ s}^{-1}$ distributed uniformly in volume between\n15 and 25 km. At these radii, the mass loss rate has been more or less\ndetermined. The additional energy input represents\na moderate perturbation to the total amount of heating provided by\nneutrinos ($1.2\\times 10^{48}\\ {\\rm erg}\\ {\\rm s}^{-1}$). As a result,\nthe entropy increased from 140 to 192 per baryon,\nand the dynamic time scale decreased from 0.11 to 0.022 s,\nwhile the mass loss rate slightly increased from \n$2.8\\times 10^{-6}$ to $3.7\\times 10^{-6}$\n$M_{\\odot}$ s$^{-1}$. \n\nThe effect on the nucleosynthesis was dramatic as shown in Table \n4. The neutron-to-seed ratio (cf. eq. [3]), \nless than 10 in all of the\nunmodified models, rose to $\\sim 166$. \nWith an average mass number of $\\bar A\\sim 90$ for the seed nuclei,\nuranium could be produced if all the neutrons were to be subsequently\ncaptured.\nThe $A\\sim 195$ $r$-process peak probably could have been produced\nwith less additional energy input, and hence a \nlower entropy and a longer dynamic time scale,\nthan assumed in Paper I. We conclude that lower values of $Y_e$ than\nthose calculated by Wilson in Woosley et al. (1994),\nor additional energy input\nlike that considered in Paper I, are necessary to produce an $r$-process\nin a spherically symmetric wind model.\n\n\\section{The Requisite Conditions For The $r$-Process}\n\nFrom our studies of nucleosynthesis in the neutrino-driven wind\nin the previous section, we learn that the standard wind models \nof Paper I fail to make an $r$-process. On the other hand, we also\nsee that reasonable modifications of these models can\nsignificantly change the physical conditions in the wind, \nand therefore give\nrise to a possible $r$-process. In order to better understand the\ndeficiencies of the standard wind models, and furthermore, to\nmotivate and direct physically plausible modifications of these\nmodels, we now survey the important physical parameters required\nfor a strong $r$-process. (See also \\cite{twj94}; \\cite{tak96};\n\\cite{fri96}; and \\cite{mey96}). \n\nWe consider the following generic model for the $r$-process.\nNeutron-rich material initially\ncomposed of free nucleons at high temperatures ($T_9\\approx9$)\nadiabatically expands and cools. After nearly all the protons\nare assembled into $\\alpha$-particles at $T_9\\approx5$, an \n$\\alpha$-process occurs to burn the $\\alpha$-particles into heavy\nnuclei. The $\\alpha$-process stops when charged-particle reactions\nfreeze out at $T_9\\approx2.5$. The heavy nuclei produced at the end\nof the $\\alpha$-process then become the seed nuclei for the subsequent\nrapid neutron capture process, or the $r$-process. We do not intend\nto account for the full detail of the $r$-process, such as the final\nabundance distribution, which becomes meaningful only in the context\nof a consistent astrophysical model. What we are most interested in\nis the physical conditions favorable for the production of the most\nabundant $r$-process nuclei, such as those in the platinum peak of\nthe solar $r$-process abundance distribution. For this purpose, we\ncan think of the $r$-process as the transformation of seed nuclei\ninto $r$-process nuclei through simple addition of the available\nneutrons. In this sense, the possibility of producing $r$-process\nnuclei around a certain mass number depends only on the relative\nabundances of seed nuclei and neutrons at the end of the \n$\\alpha$-process. \n\nIn general, the composition resulting from the $\\alpha$-process\nsatisfies \n\\begin{equation}\nX_{n,f}+X_{\\alpha,f}+X_s\\approx1,\n\\end{equation}\nwhere $X_{n,f}$ and $X_{\\alpha,f}$ are the final mass fractions of\nneutrons and $\\alpha$-particles, respectively, and $X_s$ is the total\nmass fraction of seed nuclei. If we represent the seed nuclei with a\nmean proton number $\\bar Z$ and a mean mass number $\\bar A$, we can\ndefine a neutron-to-seed ratio at the end of the $\\alpha$-process as\n\\begin{equation}\n{n\\over s}\\approx{X_{n,f}\\over X_s}\\bar A.\n\\end{equation}\nIn terms of this ratio, our simplified condition for making $r$-process\nnuclei with mass number $A$ becomes\n\\begin{equation}\n\\label{nesrp}\n{n\\over s}+\\bar A\\approx A.\n\\end{equation}\nWe are particularly interested in the production of the platinum peak,\nand will describe the numerical calculations for $A\\approx 200$ in\nthe following.\n\nTo follow the nucleosynthesis in the adiabatically cooling material\nwith a nuclear reaction network,\nwe need the initial composition and the time evolution of temperature\nand density. The initial composition at $T_9\\approx9$ can be simply \nspecified by the initial electron fraction $Y_{e,i}$, with the initial\nmass fractions of neutrons and protons given by \n$X_{n,i}\\approx 1-Y_{e,i}$ and $X_{p,i}\\approx Y_{e,i}$, respectively.\nBecause temperature and density are related through the constant\nentropy for the adiabatically cooling material, we only need to specify\nthe temperature as a function of time. For simplicity, we introduce\na dynamic time scale ($\\tau_{\\rm dyn}$) over which the temperature\nchanges by one $e$-fold, i.e.,\n\\begin{equation}\nT_9(t)\\approx T_9(0)\\exp(-t\/\\tau_{\\rm dyn}).\n\\end{equation}\nWith this time evolution of the temperature, the duration of the \nadiabatic expansion from $T_9\\approx 9$ to 2.5 is\n\\begin{equation}\n\\label{netaut}\nt_{\\rm exp}\\approx 1.28\\tau_{\\rm dyn}.\n\\end{equation}\nHereafter, we will refer to $t_{\\rm exp}$ as the expansion time.\nWe further assume that the entropy is dominated by contributions from\nradiation and relativistic electron-positron pairs, and is given by\n\\begin{equation}\n\\label{neent}\nS\\approx 3.33{T_9^3\\over\\rho_5},\n\\end{equation}\nwhere $\\rho_5$ is the density in units of \n$10^5\\ {\\rm g}\\ {\\rm cm}^{-3}$,\nand $S$ is in units of Boltzmann constant per baryon. For convenience,\nwe frequently refer to entropy without its unit throughout this paper.\n\nNow it is straightforward to determine the combinations of \ninitial electron fraction, entropy, and expansion time, for which an\n$\\alpha$-process can lead to a sufficient \nneutron-to-seed ratio for production of $r$-process nuclei\nwith $A\\approx 200$. We choose a range of expansion times\n($0.005\\leq t_{\\rm exp}\\leq 0.25$ s). For each $t_{\\rm exp}$, we survey\na broad range of initial electron fractions ($Y_{e,i}=0.20$, 0.25, 0.30,\n0.35, 0.40, 0.45, 0.46, 0.47, 0.48, 0.49, and 0.495). \nWith a particular set of $t_{\\rm exp}$\nand $Y_{e,i}$, we seek through iteration the appropriate entropy\nwhich enables the $\\alpha$-process to produce a final composition \nsatisfying equation (\\ref{nesrp}) for $A\\approx 200$. \nIn our calculations, we\ntake into account the effects of electron and positron captures and\nnuclear $\\beta$-decays on the evolution of the electron fraction.\n\nThe results are given in Table 5 and Figure 10. For each successful run, \nthe composition at the end of the \n$\\alpha$-process are given\nin terms of the final neutron and $\\alpha$-particle\nmass fractions and the mean proton and mass numbers for seed nuclei.\nThe mean proton and mass numbers were calculated\nfrom $\\bar Z=\\sum Y(Z_s,A_s)Z_s\/\\sum Y(Z_s,A_s)$ and $\\bar A=\\sum\nY(Z_s,A_s)A_s\/\\sum Y(Z_s,A_s)$, respectively, with $Y(Z_s,A_s)$ the\nnumber fraction of the seed nucleus with proton number $Z_s$ and\nmass number $A_s$. The final electron fraction and the neutron-to-seed\nratio are also given in Table 5. \nThe combinations of initial electron fraction\nand entropy in the successful runs for three specific expansion times\nare shown as filled circles connected by solid lines in Figure 10.\nAt a given $t_{\\rm exp}$, values of $Y_{e,i}$ and $S$\nto the left of the solid line will not give a sufficient neutron-to-seed\nratio for production of the platinum peak. In addition, the following\nfeatures of this figure are worth mentioning:\n\n(1) Depending on the expansion time, there exist many possible \ncombinations of initial electron fraction and entropy that can\nproduce nuclei with $A\\sim 195$. The\n``high entropy'' scenario ($S\\gtrsim 350$ and  \n$0.495\\gtrsim Y_{e,i}\\gtrsim 0.40$) \ncorresponds to longer expansion times ($t_{\\rm exp} \\gtrsim 0.1$ s). \nThe results for $t_{\\rm exp}=0.25$ s are consistent with the conditions \nseen at late times in Wilson's supernova model (\\cite{wwmhm94}),\nand those employed in the successful $r$-process calculations of\nWoosley et al. (1994) and Takahashi, Witti, \\& Janka (1994),\nalthough the later required an \nartificial increase in the entropy (by a factor of five) to produce \nnuclei with $A\\sim 195$.\n\n(2) Alternatively, there is a ``low entropy'' scenario ($S\\lesssim 200$ and\n$Y_{e,i}\\lesssim 0.4$) that requires shorter expansion times\n($t_{\\rm exp}\\lesssim\n0.025$ s). For the most extreme case shown ($t_{\\rm exp}=0.005$ s), the\nplatinum peak can be made for any $Y_{e,i}\\lesssim 0.495$ if $S\\sim 150$.\nSuch an expansion would correspond to very high velocities which\nmight be more appropriate to a jet than to a quasi-steady-state wind as\nconsidered in Paper I. However, these rapid expansions may not continue\nafter the $\\alpha$-process in order to allow enough time for the neutron\ncapture phase of the $r$-process. The slowing down of the expansion\ncould be facilitated by a massive overlying mantle in the case of a\nType II supernova.\n\n(3) For smaller values of $Y_{e,i}$, the required entropy \ndecreases regardless of the\nexpansion time. This merely reflects the very neutron-rich nature of the\ninitial composition. The results of Paper I suggest\nthat values of $Y_e$ below 0.3\nmight be very difficult to achieve in the neutrino-driven wind. \nMaterial with $Y_e<0.3$ would need to be ejected without any \nsignificant interaction with neutrinos. \n\n(4) For a fixed expansion time, the required value of entropy actually\n{\\sl decreases} as $Y_{e,i}$ increases from $\\sim 0.48$ to $\\sim 0.495$.  \nLower entropy translates to higher\ndensity (cf. eq. [7]), and would normally produce more seed nuclei.\nHowever, in these cases of high $Y_{e,i}$, the electron fraction\nhas even more leverage on the seed production. As $Y_{e,i}$ increases\ntowards 0.5, the neutron abundance decreases to vanishingly small values\nwhen the $\\alpha$-process begins\nat $T_9\\approx 5$. This in turn diminishes the efficiency of burning\n$\\alpha$-particles through the main \nreaction path bridging the unstable mass gaps at $A=5$ and 8, i.e., \n$^4{\\rm He}(\\alpha n,\\gamma)^9{\\rm Be}(\\alpha,n)^{12}{\\rm C}$.\nTable 5 shows that these high $Y_{e,i}$ values give extreme\n$\\alpha$-rich freeze-outs with final $\\alpha$-particle mass fractions\napproaching unity and comparable mass fractions for neutrons and\nheavy seed nuclei.\n\nIn deriving the above results, we have made two major assumptions:\n(1) the entropy is proportional to $T^3\/\\rho$ with the proportionality\nconstant calculated for a mixture of radiation and relativistic \nelectron-positron pairs, and (2) weak interactions other than electron\nand positron captures and nuclear $\\beta$-decays can be neglected in\nthe nuclear reaction network. We will discuss how our results are affected\nif we drop either assumption in turn.\n\nDue to its logarithmic dependence on the temperature and density, the\nentropy of non-relativistic particles roughly stays constant over the\ntemperature range in our calculations ($9\\gtrsim T_9\\gtrsim 2.5$).\nFor a total entropy of $S\\gtrsim 20$, the change in density essentially\nmaintains a constant entropy of relativistic particles as the material\nadiabatically cools. At high temperatures ($9\\gtrsim T_9\\gtrsim 5$),\nthe relativistic particles include photons and electron-positron pairs,\nand the corresponding entropy is given by equation (7). However, as\nthe temperature cools below $T_9\\sim 5$, electron-positron pairs begin\nto annihilate. The situation is much like the Big Bang. Eventually,\nat $T_9\\sim 1$, the only relativistic particles in the material are\nphotons, with the corresponding entropy given by \n$S\\approx 1.21T_9^3\/\\rho_5$. In general, we can write the \nentropy in relativistic particles as \n$S\\approx C(T_9)T_9^3\/\\rho_5$.\nBecause $C(T_9)$ decreases noticeably from $T_9\\approx 5$ to 2.5 when\nthe $\\alpha$-process is taking place in our calculations, we have\noverestimated the density by using $C(T_9)\\approx 3.33$ throughout \nthe adiabatic expansion of the material. Consequently, we have\noverestimated the seed production and underestimated the neutron-to-seed\nratio at the end of the $\\alpha$-process.\nThis is especially true for the cases of high entropies where more \ntime is available to produce seed nuclei.\n\nWith an accurate expression\nfor the entropy, the condition in equation (\\ref{nesrp})\nis satisfied at slightly\nlower entropies than those given in Table 5 for the same initial \nelectron fraction and expansion time. In fact, we have repeated our\ncalculations for $Y_{e,i}=0.30$, 0.40, 0.45, 0.47, and 0.49,\nusing an exact adiabatic equation of state to compute the density \ncorresponding to a specific temperature. This equation of state\ntakes into account the contributions\nto the entropy from radiation, electron-positron pairs, and ions.\nThe results are shown as filled squares in Figure 10 for three of the\nexpansion times explored in the initial numerical survey.\nAs expected, our previous results using equation (7) overestimated\nthe entropy required to produce the platinum peak.\nWhen we use the exact adiabatic equation of state, the required entropy \nis lower by $\\sim 10$\\% for the longest expansion time \n$t_{\\rm exp}=0.25$ s, \nwhereas for the shortest expansion time \n$t_{\\rm exp}=0.005$ s, the results are essentially unchanged.\n\nIf the $r$-process occurs in an environment with intense neutrino flux,\nperhaps we should also include various neutrino interactions in the nuclear\nreaction network. In fact, we could have specified a less\n``generic'' model by considering an $r$-process site similar to\nthe neutrino-driven wind. In this case, the adiabatically expanding\nmaterial is also moving away from the neutrino source. We can define\na constant dynamic time scale as $\\tau_{\\rm dyn}\\approx r\/v$, with\n$r$ the distance from the neutrino source and $v$ the expansion velocity.\nThe time evolution of temperature in equation (5) follows on assuming\n$T\\propto r^{-1}$. We can then introduce an additional parameter, e.g.,\nthe initial neutrino flux $\\Phi_{\\nu,i}$ at $T_9\\approx 9$, in our\ncalculations. As the material adiabatically expands, the neutrino flux\nit receives decreases as $r^{-2}\\propto\\exp(-2t\/\\tau_{\\rm dyn})$. In principle,\nusing the above prescription, we can repeat our calculations \nfor a range of $\\Phi_{\\nu,i}$\nwith various neutrino\ninteractions included in the nuclear reaction network.\n\nFrom our discussions in $\\S2$, we have seen that for relatively low\nentropies of $S\\lesssim 200$, the major role of the neutrino flux in\ndetermining the neutron-to-seed ratio is to increase the electron fraction\nby $\\nu_e$ and $\\bar\\nu_e$ captures on free nucleons through the \n$\\alpha$-effect. In addition, Meyer (1995) has shown that for high entropies\nof $S\\sim 400$, neutral-current neutrino\nspallations on $\\alpha$-particles during the\n$\\alpha$-process can increase the production of seed nuclei. In both cases,\ninclusion of neutrino interactions tends to reduce the neutron-to-seed\nratio at the end of the $\\alpha$-process, although the effects of these\nneutrino interactions are less important\nfor shorter expansion times. Therefore, we can interpret the entropies\nin Table 5 and Figure 10 as the {\\sl minimum} values\nrequired to produce the platinum peak for given sets of initial electron\nfraction and expansion time. With this interpretation, we can avoid\nrepeating our calculations and \ncomplicating our results with an additional parameter $\\Phi_{\\nu,i}$.\nOf course, in a consistent astrophysical model for the $r$-process where\nintense neutrino flux exists, the exact conditions for production of, e.g.,\nthe platinum peak, have to be determined with full consideration of\nvarious neutrino interactions.\n\n\\section{Analytic Treatment Of The $\\alpha$-Process}\n\nIn order to provide some physical insight into what determines\nthe neutron-to-seed ratio, and extend our results in the previous\nsection to production of $r$-process nuclei in general, we present\nan analytic treatment of the $\\alpha$-process in this section\nbased on our generic $r$-process model. If we ignore possible\nneutrino interactions, the electron fraction at the\nbeginning of the $\\alpha$-process ($T_9\\approx 5$) is about the\nsame as the initial electron fraction $Y_{e,i}$ at $T_9\\approx 9$.\nAt $T_9\\approx 5$, the material is essentially composed of free\nneutrons and $\\alpha$-particles for $Y_{e,i}<0.5$, with almost\nall the protons already assembled into the $\\alpha$-particles.\nThe mass fractions of $\\alpha$-particles and neutrons at the\nbeginning of the $\\alpha$-process are then approximately given by\n\\begin{mathletters}\n\\begin{eqnarray}\nX_{\\alpha,0}&\\approx& 2Y_{e,i},\\\\\nX_{n,0}&\\approx& 1-2Y_{e,i},\n\\end{eqnarray}\n\\end{mathletters}\nrespectively.\n\nThe composition at the end of the $\\alpha$-process ($T_9\\approx 2.5$)\nsatisfies\n\\begin{equation}\n{1\\over 2}X_{\\alpha,f}+{\\bar Z\\over\\bar A}X_s\\approx Y_{e,f},\n\\end{equation}\nwhere $Y_{e,f}$ is the final electron fraction at $T_9\\approx 2.5$.\nUsing equation (9) together with equations (2)--(4), we find that\nthe final mass fractions of $\\alpha$-particles and neutrons have to be\n\\begin{mathletters}\n\\begin{eqnarray}\nX_{\\alpha,f}&\\approx&{Y_{e,f}-\\bar Z\/A\\over 1\/2-\\bar Z\/A},\\\\\nX_{n,f}&\\approx&{1-\\bar A\/A\\over 1-2\\bar Z\/A}(1-2Y_{e,f}),\n\\end{eqnarray}\n\\end{mathletters}\nrespectively, in order to produce $r$-process nuclei with mass number $A$.\nComparing equations (8a) and (8b) with equations (10a) and (10b),\nwe see that the fractional change in the neutron mass fraction during\nthe $\\alpha$-process is less than that in the $\\alpha$-particle mass\nfraction for $Y_{e,f}\\approx Y_{e,i}<\\bar Z\/\\bar A$. Obviously,\nbecause neutrons carry no charge and the mean charge per nucleon for\n$\\alpha$-particles exceeds $\\bar Z\/\\bar A$ for the heavy seed nuclei,\nthe final composition favors the presence of neutrons for \n$Y_{e,f}<\\bar Z\/\\bar A$. Accordingly, we will present the analytic\ntreatment of the $\\alpha$-process for two different cases:\n$Y_{e,i}<\\bar Z\/\\bar A$ and $Y_{e,i}>\\bar Z\/\\bar A$. In both cases,\nwe will assume $Y_{e,f}\\approx Y_{e,i}$.\n\nAs the temperature declines from \n$T_9\\approx 5$ to 2.5, $\\alpha$-particles and neutrons \nare partially assembled \ninto heavy seed nuclei. We can describe the $\\alpha$-process in terms of\nthe time evolution of the $\\alpha$-particle and neutron abundances.\nDuring the $\\alpha$-process, the burning of $\\alpha$-particles\nmainly proceeds \nvia the reaction sequence \n$^4{\\rm He}(\\alpha n,\\gamma)^9{\\rm Be}(\\alpha,n)^{12}{\\rm C}$.\nThe production of seed nuclei occurs through the efficient $\\alpha$-capture\nreactions starting with $^9{\\rm Be}(\\alpha,n)^{12}{\\rm C}$. Consequently,\nthe rates of change in the $\\alpha$-particle and neutron number fractions \ncan be approximately written as\n\\begin{mathletters}\n\\begin{eqnarray}\n\\drvf{Y_\\alpha}{t}&\\approx&-FY_\\alpha Y_9\\rho \nN_A\\langle\\sigma v\\rangle_{\\alpha n},\\\\\n\\drvf{Y_n}{t}&\\approx&-GY_\\alpha Y_9\\rho\nN_A\\langle\\sigma v\\rangle_{\\alpha n},\n\\end{eqnarray}\n\\end{mathletters}\nrespectively, where $Y_9$ is the number fraction of $^9$Be, and\n$N_A\\langle\\sigma v\\rangle_{\\alpha n}$ is the reaction rate for \n$^9$Be($\\alpha,n$)$^{12}$C in units of cm$^3$ s$^{-1}$ g$^{-1}$.\nIn equations (11a) and (11b), \n$F$ and $G$ are the numbers of $\\alpha$-particles and neutrons that \nmake up a typical heavy seed nucleus. For a seed distribution with\nmean proton number $\\bar Z$ and mean mass number $\\bar A$, we have\n$F\\approx\\bar Z\/2$ and $G\\approx\\bar A-2\\bar Z$.   \n\nDue to its low $Q$-value of only 1.573 MeV, the reaction \n$^4{\\rm He}(\\alpha n,\\gamma)^9$Be is tightly balanced by its reverse reaction\nessentially over the entire temperature range $5\\gtrsim T_9\\gtrsim 2.5$.\nAccording to statistical equilibrium, the number fraction of $^9$Be during\nthe $\\alpha$-process is given by \n\\begin{equation}\nY_9\\approx 8.66\\times 10^{-11}\nY_\\alpha^2 Y_n \\rho_5^2T_9^{-3} \\exp(18.26\/T_9).\n\\end{equation}\nBecause the density $\\rho$ is related to the temperature $T_9$ through\nthe constant entropy $S$, and $N_A\\langle\\sigma v\\rangle_{\\alpha n}$ depends\non $T_9$ only, equations (11a) and (11b) can be expressed in a more\nconvenient form if we regard $Y_\\alpha$ and $Y_n$ as functions of\ntemperature. Using equation (5), we have\n\\begin{mathletters}\n\\begin{eqnarray}\n{dY_\\alpha\\over dT_9}&\\approx&FY_\\alpha^3Y_ng(T_9)\\tau_{\\rm dyn},\\\\\n{dY_n\\over dT_9}&\\approx&GY_\\alpha^3Y_ng(T_9)\\tau_{\\rm dyn},\n\\end{eqnarray}\n\\end{mathletters}\nwhere $g(T_9)$ has the unit of s$^{-1}$, and is given by\n\\begin{equation}\ng(T_9)\\approx 8.66\\times 10^{-6}\\rho_5^3T_9^{-4}\\exp(18.26\/T_9)\nN_A\\langle\\sigma v\\rangle_{\\alpha n}.\n\\end{equation} \n\nNow we can solve equations (13a) and (13b) for the two different cases\nmentioned previously. For $Y_{e,i}<\\bar Z\/\\bar A$, the final composition\nfavors the presence of neutrons. So we can approximately take\n$Y_n\\approx Y_{n,0}=X_{n,0}$ during the $\\alpha$-process. In this case,\nit is straightforward to solve equation (13a) and obtain\n\\begin{equation}\nY_{\\alpha,f}^{-2}-Y_{\\alpha,0}^{-2}\\approx 2FY_{n,0}\\tau_{\\rm dyn}\n\\int_{2.5}^5g(T_9)dT_9.\n\\end{equation}\nLikewise, for $Y_{e,i}>\\bar Z\/\\bar A$, we can approximately take\n$Y_\\alpha\\approx Y_{\\alpha,0}=X_{\\alpha,0}\/4$ during the $\\alpha$-process,\nand solve equation (13b) to obtain\n\\begin{equation}\nY_{n,f}\\approx Y_{n,0}\\exp\\left[-GY_{\\alpha,0}^3\\tau_{\\rm dyn}\n\\int_{2.5}^5g(T_9)dT_9\\right].\n\\end{equation}\nEquations (15) and (16) implicitly constrain the combinations of\n$Y_{e,i}$, $S$, and $\\tau_{\\rm dyn}$, for which the $\\alpha$-process\ncan give a sufficient neutron-to-seed ratio for production of $r$-process\nnuclei with mass number $A$.\n\nTo proceed further, we approximate the constant entropy during the\nadiabatic expansion as $S\\approx C(T_9)T_9^3\/\\rho_5$,\nwith $C(T_9)$ decreasing from 3.33 at\n$T_9\\gtrsim 5$ to 1.21 at $T_9\\lesssim 1$. So equation (14) can be \nrewritten as\n\\begin{equation}\n\\label{negt9}\ng(T_9)\\approx 8.66\\times 10^{-6}S^{-3}C(T_9)^3T_9^5\n\\exp(18.26\/T_9)N_A\\langle\\sigma v\\rangle_{\\alpha n}.\n\\end{equation}\nWe plot $g(T_9)S^3$ as a function of $T_9$ in Figure 11, assuming\n$C(T_9)\\approx 3.33$ and using the fitting formula for \n$N_A\\langle\\sigma v\\rangle_{\\alpha n}$ given by \nWrean, Brune, \\& Kavanagh (1994). \nAs we can see from this figure, $g(T_9)$ decreases\nmonotonically with temperature over $2.5\\lesssim T_9\\lesssim 5$. \nAt $T_9\\approx 4$, $g(T_9)$ has\nalready fallen below half its value at $T_9\\approx 5$. Clearly, the\nmain contribution to the integral $\\int_{2.5}^5g(T_9)dT_9$ comes from\n$4\\lesssim T_9\\lesssim 5$. This remains true even when the exact form\nof $C(T_9)$ is used. For our analytic estimates, we can approximately\nevaluate the integral with $C(T_9)\\approx 3.33$, and obtain\n\\begin{equation}\n\\label{negt9i}\n\\int_{2.5}^5g(T_9)dT_9\\approx 6.4\\times 10^8S^{-3}\\ {\\rm s}^{-1}.\n\\end{equation}\nUsing equation (\\ref{negt9i}) and assuming $Y_{e,f}\\approx Y_{e,i}$, we can\nrearrange equations (15) and (16) as\n\\begin{mathletters}\n\\begin{eqnarray}\nS&\\approx&\\left\\{{4\\times 10^7\\bar Z(1-2Y_{e,i})\\over\n\\left[{{1\/2-\\bar Z\/A}\\over{Y_{e,i}-\\bar Z\/A}}\\right]^2\n-\\left[{{1}\\over{2Y_{e,i}}}\\right]^2}\\left({\\tau_{\\rm dyn}\\over{\\rm s}}\n\\right)\\right\\}^{1\/3}, \\ {\\rm for} \\ Y_{e,i} < {\\bar Z\\over \\bar A},\\\\\nS&\\approx&Y_{e,i}\\left\\{{8\\times 10^7(\\bar A-2\\bar Z)\\over\n\\ln\\left[{{1-2\\bar Z\/A}\\over{1-\\bar A\/A}}\\right]}\\left({\\tau_{\\rm dyn}\n\\over{\\rm s}}\\right)\\right\\}^{1\/3},\\ {\\rm for}\\ Y_{e,i}>{\\bar Z\\over \\bar A},\n\\end{eqnarray}\n\\end{mathletters}\n\\noindent{where we have also used equations (8a), (8b), (10a) and (10b).}\n\nTo compare our analytic results in equations (19a) and (19b) with\nthe numerical results for $A\\approx 200$ in $\\S3$, we take \n$\\bar Z\\approx 34$ and $\\bar A\\approx 90$ from the numerical survey,\nand obtain\n\\begin{mathletters}\n\\begin{eqnarray}\n\\label{nesyet}\nS&\\approx&10^3\\left\\{{1-2Y_{e,i}\\over\\left[\n{{0.33}\\over{(Y_{e,i}-0.17)}}\\right]^2-\\left[{{1}\\over{2Y_{e,i}}}\\right]^2}\n\\left({t_{\\rm exp}\\over{\\rm s}}\\right)\\right\\}^{1\/3}, \\ {\\rm \nfor} \\ Y_{e,i} < 0.38,\\\\ \nS&\\approx&2\\times 10^3Y_{e,i}\\left({t_{\\rm exp}\n\\over{\\rm s}}\\right)^{1\/3}, \\ {\\rm for} \\ Y_{e,i}>0.38,\n\\end{eqnarray}\n\\end{mathletters}\n\\noindent{where we have replaced $\\tau_{\\rm dyn}$ with $t_{\\rm exp}\\approx\n1.28\\tau_{\\rm dyn}$. These analytic results are shown as open circles\nconnected by dotted lines in Figure 10. The general\nagreement between our analytic results and the numerical survey using\nequation (7) (filled circles connected by solid lines) is quite good.\nBecause $Y_\\alpha$ and $Y_n$ can decrease during the $\\alpha$-process\nand effective burning of $\\alpha$-particles may start at $T_9<5$,\nwe always tend to overestimate the entropy by holding either $Y_\\alpha$\nor $Y_n$ constant and doing the integral in equation (\\ref{negt9i}) over the\nentire temperature range $2.5\\lesssim T_9\\lesssim 5$ in our analytic\ntreatment. The largest discrepancies occur at $Y_e\\gtrsim 0.47$\nwhere we overestimate the entropy by $\\sim(15$--50)\\%.\nUsing the specific values of $\\bar Z$ and $\\bar A$ found in the \nnumerical survey would slightly improve the agreement. Further improvement\ncould be obtained by solving equations (11a) and (11b) together instead\nof approximately solving each for a specific case. However, these\nimprovements would add little to our understanding of the physics \ndetermining the neutron-to-seed ratio. We also notice that the same level\nof agreement with the numerical survey using an exact equation of state\n(filled squares) can be achieved if we use\n$C(T_9)\\approx 3$ instead of 3.33 to account for\nthe annihilation of electron-positron pairs into photons at $T_9<5$.}\n\nFrom our analytic treatment of the $\\alpha$-process, we can clearly see\nthe individual roles of the initial electron fraction, entropy, and\ndynamic time scale in determining the neutron-to-seed ratio. In addition\nto specifying the overall availability of neutrons (cf. eq. [8b]), \nthe initial electron\nfraction serves to direct the path of nuclear flow during the \n$\\alpha$-process. As our analytic treatment indicates, the comparison\nof $Y_{e,i}$ with the ratio $\\bar Z\/\\bar A$ for the typical seed \ndistribution reflects whether $\\alpha$-particles or neutrons are\nmainly consumed during the $\\alpha$-process. The influence of entropy\nis manifested through the density dependences of the equilibrium\nabundance of $^9$Be (cf. eq. [12]) and the rate for burning \n$\\alpha$-particles (cf. eq. [11a]). Physically, a high entropy means\nmany photons per baryon in radiation-dominated conditions. A significant\nfraction of these photons can be on \nthe high-energy tail of the Bose-Einstein\ndistribution, and therefore can maintain a low $^9$Be abundance through\nthe photo-disintegration reactions. In turn, this limits the overall\nefficiency of burning $\\alpha$-particles, and hence the production\nof seed nuclei. The dynamic time scale, or the expansion time,\nspecifies the duration of the $\\alpha$-process (cf. eq. [6]). Obviously,\nthe expansion time also acts to limit the production of seed nuclei. \nIn general, a lower initial electron fraction, a higher entropy, and\na shorter expansion time all give a larger neutron-to-seed ratio.\nIt is most interesting to notice that for a given $Y_{e,i}$, the\ncomposition resulting from the $\\alpha$-process essentially only depends\non the combination $S^3\/t_{\\rm exp}$ (cf. eqs. [20a] and [20b]).\nConsequently, the same neutron-to-seed ratio can be achieved for the same\ninitial electron fraction with an expansion time 8 times shorter\nif the entropy is reduced by a factor of 2.\n\n\n\\section{Conclusions}\n\nFor reasonable assumptions regarding neutrino luminosity, neutron star mass\nand radius, \nand the time history for $Y_e$, the nucleosynthesis resulting from the\nanalytic model developed in Paper I for the neutrino-driven wind does not\nresemble the solar $r$-process, although a number of interesting species\nin the mass range $90 \\leq A \\leq 120$ are produced. This failure may be a\nconsequence of important (but unknown) physics neglected in Paper I, or\nour results may reflect the true nucleosynthesis from typical \ncore-collapse-driven\nsupernovae (however, see the discussion concerning the ejection of the\nwind material in $\\S3$). \nExtra (but, so far, artificial) energy input to the wind beyond\nthe injection radius (where $\\dot M$ is determined) does give a successful\n$r$-process. Possible sources of this energy were discussed in Paper I.\n\nA numerical survey has delineated the necessary combinations of key\nparameters --- $Y_e$, entropy, and expansion time --- \nneeded to produce the third\n$r$-process peak (i.e., the platinum peak). \nHigh entropy is not a unique requirement. A shorter expansion\ntime also serves to limit the number of heavy seed nuclei produced and thus\nincrease the neutron-to-seed ratio. A lower $Y_e$ also leads\nto a larger neutron-to-seed ratio.\nThe sensitivity of the neutron-to-seed ratio to $Y_e$ diminishes \nas one proceeds to shorter expansion times,\nas does the sensitivity to the entropy. \nSpecific values of $Y_e$, entropy, and expansion time to produce\nthe third $r$-process peak are given in Table 5 and \nFigure 10.\n\nApproximate analytic formulae (eqs. [20a] and [20b]) were derived that give\nthe requisite entropy needed to \nproduce the heavy $r$-process nuclei as functions of\n$Y_e$ and expansion time. \nThese equations can be used to gauge whether other unstudied supernova\nmodels or other astrophysical environments are appropriate sites for \nmaking $r$-process nuclei.\n\nGiven that the standard wind models, without artificial modification, produce\na set of abundances distinct from the $r$-process, one must be concerned\nabout the observational consequences. One possibility already mentioned is\nthat important physics has still been omitted from the simple wind model ---\ne.g., neutrino flavor mixing, added energy input from shocks, rotation, or\nmagnetic fields, convection, etc.  --- and that the conditions required\nfor the $r$-process may still ultimately be achieved in common \ncore-collapse-driven\nsupernovae. Perhaps material having a much lower $Y_e$ than calculated by\nWilson in Woosley et al. (1994) is ejected (\\cite{bur95}). Another\npossibility which must be seriously considered however, is that the\n$r$-process has more than one important site and neutron exposure.\n\nSneden et al. (1996) and Cowan et al. (1996) have observed elements\nattributed to the third $r$-process peak in the metal poor halo giant stars\nHD 126238 ([Fe\/H]$=-1.7$) and CS 22892-052 ([Fe\/H]$=-3.1$). In HD 126238, the\nscaled solar abundances of both Os and Pt have\nbeen clearly observed. Both elements are made almost exclusively by the\n$r$-process and fit the solar $r$-process abundance pattern. \nSimilar results hold for the\nmore metal poor star CS 22892-052, although Pt was not observed and the\ndetection of Os is less certain. Coupled with previous data, the fit to the\nsolar $r$-process pattern for all elements between Ba and Os is striking,\nsuggesting that, for this range of nuclei, the solar $r$-process abundance\ndistribution appears to be made in its entirety in the progenitor(s) of these\nmetal poor stars. This result argues for a primary production scenario. Due to\nthe star's very low metallicity, especially so for CS 22892-052, the \nobserved $r$-process abundance pattern\nprobably arose from only a few supernovae.\n\nInterestingly, the abundance pattern for CS 22892-052 shows that elements in\nthe first neutron capture peak (Sr, Y, and Zr) are below their scaled solar\n$r$-process fractions relative to Pt, Os, and Th, yet well above the\nthe iron group. Thus locally the $r$-process abundances seem solar,\nbut globally they are not. Sneden et al. (1996) suggest that the bulk of the\nsolar abundance of Sr, Y, and Zr is due to the  $s$-process, but these \nnuclei are also easily produced in the neutrino-driven wind (Figure 1\nhere; \\cite{wh92}; \\cite{wjt94}; \\cite{wwmhm94}). This is consistent\nwith the existence of two sources for the $r$-process, one responsible for\nproduction of the $r$-isotopes for $Z < 56$, including those in the $N=50$\npeak, and the other for the heavier elements, possibly operating in another\nsite or a higher entropy version of the same site.\n\nA very different type of $r$-process would arise also for much shorter\nexpansion times, as might occur in accretion induced collapse where the\nwind is not slowed down by a massive overlying mantle (\\cite{wobar92};\n\\cite{wh92}). A high entropy $r$-process with short expansion time\ncould occur through ejection by relativistic jets in coalescing neutron\nstars (\\cite{ruf96}). With short expansion times, the duration of\nthe neutron capture phase of the $r$-process may require\nspecial consideration. If the material undergoing\nnucleosynthesis cannot be slowed down during the neutron\ncapture phase, the $r$-process may have to be accelerated\nby $\\nu_e$ captures on heavy nuclei (Nadyozhin \\& Panov 1993).\nUltimately, observational signatures of these\nvery distinct physical processes may be needed to resolve the nature and\nsite(s) of the $r$-process.\n\nWe would like to thank George Fuller and Gail McLaughlin for very\nhelpful discussions concerning the evolution of $Y_e$ in the\nneutrino-driven wind and also for providing the\nneutrino capture rates used in this work. We also gratefully acknowledge \nthe Institute for Nuclear Theory at the University of Washington and\nthe Max-Planck Institute for Astrophysik for their \ngenerous hospitality during completion of this paper. \nThis work was supported by NSF grant No. AST 94-17161 at UCSC. Woosley\nwas also supported by an Alexander von Humboldt Stiftung in Germany.\nY.-Z. Qian was supported by the D. W. Morrisroe Fellowship at Caltech.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro}\nDiscovered in the 90s, (65803) Didymos is a near-Earth asteroid (NEA) and target of the NASA's DART mission. Designed to be an astronomical-scale kinetic impact test, the mission was launched in November 2021 \\citep{cheng2018} and impacted Didymos' secondary, Dimorphos, in September 2022. The ESA mission HERA is scheduled to be launched in October 2024 and will also visit Didymos system. The goal of this mission is to do a post-impact characterization of the system \\citep{michel2018}. Over the years, Didymos has been observed through radar and light-curve observations \\citep{michel2016,naidu2016}, which allowed the characterization of shape and physical parameters of the object. Didymos is a S-type asteroid \\citep{deLeon2010,Pravec2012} and has a top-shape appearance, resembling a spherical body with an equatorial ridge. \\cite{naidu2020} evaluated the average object radius as $R_D=390$~m and spin period of $T_D=2.26$~h, while the bulk density is the least constrained physical parameter of the system ($\\rho_D=2170\\pm350~{\\rm kg\/m^3}$). The physical properties of Didymos and its secondary Dimorphos are summarised in Table~\\ref{tab:physical}\\footnote{We assume that Dimorphos is in a tidally locked state \\citep{agruda2022}}.\n\\begin{table}{}\n\\caption{Physical properties of Didymos and Dimorphos \\citep{Hirabayashi2017,naidu2020,Terik2022}\\label{tab:physical}}\n\\centering\n\\begin{tabular}{llclc}\n\\hline\\hline\nParameters & & Didymos  &  & Dimorphos \\\\ \\hline\nPrincipal semi-axes (km) & $\\alpha_D$ & $0.40$ & $\\alpha_d$ & $0.10$ \\\\\n & $\\beta_D$ & $0.39$ & $\\beta_d$ & $0.08$ \\\\\n & $\\gamma_D$ & $0.38$ & $\\gamma_d$ & $0.07$ \\\\\nMass (kg) & $M_D$ & $5.12\\times 10^{11}$ & $M_d$ & $4.92\\times 10^{9}$  \\\\\nBulk density (kg\/m$^3$) & $\\rho_D$ & $2170$ & $\\rho_d$ & $2170$  \\\\\nAverage radius (km) & $R_D$ & $0.39$ & $R_d$ & $0.08$  \\\\\nSpin period (h) & $T_D$ & $2.26$ & $T_d$ & $12.15$  \\\\\nOrbital distance (m) &  & -- & $a_d$ & $1183$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\nDidymos, such as 2001 SN263 \\citep{becker2015}, 1999 KW4 \\citep{Ostro2006}, and 1994 CC \\citep{Brozovic2011}, is in a class of binary asteroids in which a top-shaped primary rotates near its spin limit \\citep{Warner2009,margot2015,walsh2015}. When close to this limit, material on the surface of the object feels a centrifugal force that may overcome the gravitational attraction \\citep{Pravec2007}, resulting in geological phenomena such as landslides \\citep{walsh2008,Scheeres2015} or shape deformation \\citep{Hirabayashi2014}. The Yarkovsky\u2013O'Keefe\u2013Radzievskii\u2013Paddack (YORP) effect is a well-known effect responsible for accelerating the rotation of small bodies on a timescale of $\\sim10^4-10^6$~yr, due to the re-radiation of the sunlight absorbed by the asteroid \\citep{Bottke2006}. Given this, \\cite{Scheeres2006} propose that when accelerated by the YORP effect, surface material flows towards the equator, which would result in the top shape observed in the asteroids rotating near the spin limit. Given this, one might ask whether material is put in orbit due to this process, giving rise to moons.\n\nIt is shown by \\cite{walsh2008} that asteroids with high-friction angle may eject particles when rotating near the critical spin. Material dislodged in the ejection is believed to be deposited in low-eccentricity orbits around the object, giving rise to a satellite (secondary). However, the direct modelling of the friction forces was not realised and the effects of cohesion were not included by \\citeauthor{walsh2008} in their hard-sphere approach. \\cite{sugiura2021} goes a step further by adopting the effective bulk friction, which can effectively include the effect of cohesion. They demonstrate that, in fact, an initially spherical asteroid can achieve a top-shaped figure and form a transient ring in its vicinity. \n\n\\cite{hyodo2022} performed longer-term simulations of \\cite{sugiura2021}. They showed that a mass on the order of $10\\%$ of the initial spherical body is ejected in a single avalanche event around the top-shaped central rubble-pile body, forming a transient particle disk followed by formation of rubble-pile moons via disk spreading. This may be the origin of Didymos' moon Dimorphos, the focus of our work.\n\n\\cite{Zhang2017,Zhang2018,Zhang2021} use the gravitational N-body code PKDgrav to study Didymos' spin and shape evolution \\citep{Richardson2016}. Assuming Didymos as a rubble pile object, they found that a minimum cohesion of $10$~Pa is required for Didymos to maintain its structural stability and have no material detached, although its actual stability depends on the material arrangement and the density distribution within the body. The mass shedding phenomenon on Didymos was studied in \\cite{Yu2018}. The increase in the asteroid's spin rate increases the mass detachment on Didymos and also affects the evolution of the lofted material. A slowly rotating Didymos is capable of trapping material for at least a few months in its vicinity, possibly promoting the formation of large debris via accumulation (possible Dimorphos' building blocks). As the spin rate increases, the stable region in the vicinity of the object becomes smaller, releasing trapped material, which is in agreement with the results of \\cite{walsh2008} and \\cite{hyodo2022}.\n\nAnother proposed mechanism for the formation of binary systems that can give rise to Dimorphos is the primary fission due to the rapid spin, which occurs for sufficiently cohesive bodies \\citep{Pravec2010}. In such objects, the cohesion holds the object's shape until a break-up threshold energy is reached and a large fragment is detached from the primary, resulting in the satellite. If Didymos is cohesive enough, it may have fragmented into larger fragments in the past, meaning that Dimorphos possibly formed through this process. Now, if Didymos is a low cohesion object it is expected that material has been deposited at Didymos' equator due to avalanches on the object. In this case, Dimorphos can be formed by the accretion of the ejected material \\citep{walsh2008,hyodo2022}.\n\nIn this work, we study the formation of Dimorphos from the material ejected from Didymos due fast spinning \\citep{walsh2008,hyodo2022}. We study the viscous evolution of the ring of ejected material, with 1D numerical simulations using the \\texttt{HYDRORINGS} code \\citep{charnoz2010,salmon2010}. Here, our simulations are much longer timescale than previous studies, and we include the effect of tidal evolution of the formed moons. The effect of Didymos shape and the mass shedding process are not studied, which would require a more adequate tool. We assume that the mass deposition rate in the ring follows an e-folding temporal function. \n\nThe paper outline is as follows. In Section~\\ref{sec_dynamical}, we describe our dynamic model and numerical simulations. The orbital evolution of a system composed by Didymos and a ring of material is analysed in Section~\\ref{sec:ring}. In Section~\\ref{sec_parameters}, we analyse the effects of flow and tidal parameters and in Section~\\ref{sec:grow}, we focus on the formation pathways of Dimorphos. In Section~\\ref{sec_discussion}, we discuss the possible implications of our results on Dimorphos shape and the limitations of our model. We address our conclusions in Section~\\ref{sec_conclusion}.\n\n\\section{Dynamic Model} \\label{sec_dynamical}\nTop-shaped structures are thought to be formed through deformation due to rapid rotation \\citep{watanabe2019}, being the formation of ring the consequence of such a process \\citep{walsh2008,Yu2018}. The spin period of Didymos today is near the spin limit, suggesting a possible formation of its top shape through a fast spin-up \\citep{Zhang2017,Zhang2018,Zhang2021}, which could imply the formation of Dimorphos from a ring. Recently, \\cite{sugiura2021} performed Smoothed Particle Hydrodynamics (SPH) simulations of granular bodies to model a rotational deformation of rubble-pile bodies. They showed that the top shape formation occurs through an axisymmetric set of surface landslides of a primary, induced by a fast spin-up. This landslide, then, results in a mass ejection around the primary, forming a debris ring followed by a moon formation \\citep{hyodo2022}.\n\nBased on this, we envision the following scenario for the formation of Dimorphos: the fast rotation of Didymos caused by an external effect, e.g. the YORP effect, induces avalanches on the object \\citep{Yu2018}. Such an event is responsible for the asteroid reshaping and deposition of material in the equatorial plane (ring of material). Due to angular momentum conservation, Didymos spin rate decreases \\citep{sugiura2021}. The ring spreads viscously and satellites are formed while part of the ring material falls on the primary \\citep{charnoz2011,hyodo2015,madeira2022}, which increases its rotation. \n\n\\cite{Zhang2021} and \\cite{sugiura2021} find that Didymos reshaping would occur via a single avalanche, with material that falls back to the body being reassembled by it. However, due to computational limitations, these works only perform short-term simulations, and it is not clear how the history of ejections by Didymos would take place in a long-term period. Since the YORP effect is continuously increasing Didymos rotation, it is possible that new avalanches will occur on the object, possibly re-injecting into the ring part of the material fallen onto Didymos \\citep{trogolo2021}. In this case, we expect the total amount of material deposited in the ring to be greater than the amount of material actually detached from Didymos, since part of the material would be recycled. At the end of this process, Dimorphos and possible smaller objects are obtained, while the rest falls back onto Didymos, giving rise to its equatorial bulge \\citep{hyodo2022}.\n\nThe viscous spreading is the consequence of inter-particle interactions within the ring (viscosity), where mass in transferred radially in order to conserve the total angular momentum of the system. We can associate three different viscosities to the viscous spreading: translational, collisional, and gravitational viscosity. Translational viscosity is related to the transfer of particle and is the result of momentum transport due to the random motion of particles \\citep{Goldreich1978}. Collisional viscosity is an effect of momentum transfer due to impacts, where angular momentum is transported between the centre of the particles \\citep{Araki1986}. Finally, the gravitational viscosity results from the scattering of particles due to self-gravity wakes structures in the ring \\citep{Daisaka2001}. We include the three components of viscous spreading in our numerical simulations \\citep{salmon2010}.\n\nThe ring spreading timescale ($\\tau_{\\rm vis}$) is a key parameter of the model, as it defines the formation pathways for Dimorphos, discussed in Section~\\ref{sec:grow}, and the validity of the scenario envisioned by us. $\\tau_{\\rm YORP}$ gives the timescale for Didymos to reach the critical spin rate again after an avalanche, so we have that the material deposition will only be possible if $\\tau_{\\rm vis}$ be less than such a value. Otherwise, Didymos will push outward all the material overtime, preventing the formation of the bulge, which it is not in agreement with the numerical results obtained in \\cite{hyodo2022}.\n\nThe YORP timescale is given by \\citep{rubincam2000,Vokrouhlicky2002}\n\\begin{equation}\n\\tau_{\\rm YORP}\\sim\\frac{3\\pi c M_D}{5\\Phi R_D^3T_D}(\\alpha_D^2+\\beta_D^2), \n\\end{equation}\nwhere $c$ is the speed of light and $\\Phi$ is the solar flux the distance of Didymos distance from the Sun. We obtain $\\tau_{\\rm YORP}\\sim10^5$~yr for Didymos.\n\nIn turn, the viscous spreading timescale can be written as \\citep{Brahic1977,salmon2010}:\n\\begin{equation}\n\\tau_{\\rm vis}=\\frac{\\zeta\\Omega^3}{\\Sigma^2}(a_{\\rm FRL}-R_D)^2\n\\end{equation}\nwhere $\\Omega$ is the keplerian angular velocity, $a_{\\rm FRL}$ is the location of the fluid Roche limit (FRL), $\\Sigma$ the ring surface density, and $\\zeta$ a parameter that depends on the location and physical parameters of the ring particles \\citep{salmon2010}.\n\nThe surface density depends on the mass deposited in the ring, implying that the viscous spreading timescale will be intrinsically related to the timescale on which Didymos deposits material in the ring. For simplicity, we assume the instantaneous mass flux into the ring as \n\\begin{equation}\n\\dot{M}(t)=\\frac{M_T}{\\tau}{\\rm exp}\\left(-\\frac{t}{\\tau}\\right),\n\\end{equation}\nwhere $M_T$ is the total mass deposited in the ring, $t$ is the simulation time, and $\\tau$ is the deposition timescale. The total mass deposited in the ring at time $t$ is $M(t)=\\int_0^{t'} \\dot{M}(t')dt'$ ($M(\\infty)=M_t$). The ring mass at every instant is much smaller than $M_T$ for cases with $\\tau\\neq0$.\n\nWe vary $M_T$ and $\\tau$ in the ranges [$0.01-0.5$]~$M_D$ and [$0.1$-$10^4$]~yr, respectively. For such values, we obtain that the surface density vary in the range $\\Sigma\\sim10^2-10^4~kg\/m^2$ when the satellites are formed, implying in $\\tau_{\\rm vis}\\sim10-10^3$~yr, a value at least two orders of magnitude smaller than $\\tau_{\\rm YORP}$. A more robust approach to the mass deposited in the ring would require the study of avalanches at Didymos, depending on the its internal mass distribution and strength \\citep[see][]{Zhang2017,Zhang2018,Zhang2021}, which is beyond the scope of this work. Here, our intention is to focus on processes related to the ring evolution.\n\nFor this, we carried out numerical simulations using the one-dimensional hybrid code \\texttt{HYDRORINGS} \\citep{charnoz2010,salmon2010}. The code couples two distinct modules: a finite volume module that tracks the viscous evolution of the ring  and satellite formation and an analytical orbital module to track the satellite evolution due to tidal effects and ring-satellite torques \\citep[see][]{charnoz2011}. The code uses the formalism described in \\cite{salmon2010} for modelling the ring's effective viscosity. Material that spreads beyond the fluid Roche limit ($a_{\\rm FRL}=1.6926R_D$) is converted into one satellite per grid cell at each time-step. Collisions are assumed to happen when the distance between two satellites is less than twice their mutual Hill radius and are treated as perfect merge events. The formalism described in \\cite{meyer1987} is used to account for satellite migration due to ring interaction. The code does not take into account mutual gravitational perturbations between the satellites. \n\nThe ring is modelled as 50 cells distributed from the Didymos mean radius ($R_D$) until the FRL, composed of particles of size s=1~m with density $\\rho_D$. Such a radius is roughly consistent in order of magnitude with the size of the constituent particles of Dimorphos, recorded by the DART mission. We simulate a scenario where Didymos starts with a disk of material orbiting inside its Roche Limit (case $\\tau=0$), mimicking a scenario with a very fast deposition of material or with a debris disk due to an external impact \\citep{Michel2020}. We also simulate a scenario where the ring region is initially empty. In such, we assume that Didymos deposited mass appears in the first cells of the grid (i.e. close to Didymos' surface). We perform simulations assuming $M_T=0.01$, $0.1$, $0.25$, and $0.5~M_D$ and $\\tau=0.1$, $1$, $10^2$, and $10^4$~yr. \n\nWe consider cases with and without tidal migration ($k_2\/Q=0$, $\\tau_{\\rm tide}=0$). In the cases with tidal migration, we vary the tidal parameter $k_2\/Q$ for Didymos within the value range expected for rubble-pile asteroids \\citep{nimmo2019}: $k_2\/Q=10^{-6}$, $10^{-5}$, $10^{-4}$, and $10^{-3}$, which correspond to tidal migration timescales $\\tau_{\\rm tide}\\sim 10^3$, $10^2$, $10$, and $1$~yr, respectively. Such values of $\\tau_{\\rm tide}$ are at least an order of magnitude smaller than the BYORP timescale ($\\sim 10^5$~yr, see Section~\\ref{sec_parameters}) and as a simplification we do not include this effect in our simulations. Didymos is assumed to be a spherical object and we evolve the system until the ring surface density become less than the threshold value $10^{-1}$~kg\/m$^2$. For such a value, the mass in a bin becomes less than the mass of a ring particle, condition for which we consider the ring region to be empty.\n\n\\section{Didymos starting with a ring} \\label{sec:ring}\nOnce we described our dynamic model and numerical code, we will show the evolution of a case with tidal parameter $k_2\/Q=10^{-5}$, starting with an initial ring of mass $M_T=0.25~M_D$ but without considering deposition of material (case with $\\tau=0$). The different panels of Figure~\\ref{fig:standard} show for different times (at the top of each panel), the ring surface density ($\\Sigma$, solid line) and satellite radius (black dots) as a function of the distance to Didymos centre. The surface density is given on the left scale and the satellite radius on the right. \n\\begin{figure}[]\n\\subfloat[]{\\includegraphics[width=0.49\\columnwidth]{figures\/tau0\/Asigma0.png}}\n\\subfloat[]{\\includegraphics[width=0.49\\columnwidth]{figures\/tau0\/Asigma12.png}}\n\\\\\n\\subfloat[]{\\includegraphics[width=0.48\\columnwidth]{figures\/tau0\/Asigma19.png}}\n\\subfloat[]{\\includegraphics[width=0.48\\columnwidth]{figures\/tau0\/Asigma35.png}}\n\\\\\n\\subfloat[]{\\includegraphics[width=0.48\\columnwidth]{figures\/tau0\/Asigma75.png}}\n\\subfloat[]{\\includegraphics[width=0.48\\columnwidth]{figures\/tau0\/Asigma15200.png}}\n\\\\\n\\subfloat[]{\\includegraphics[width=0.48\\columnwidth]{figures\/tau0\/Bsigma14700.png}}\n\\subfloat[]{\\includegraphics[width=0.48\\columnwidth]{figures\/tau0\/Bsigma27737.png}}\n\\\\\n\\caption{Ring surface density (solid line, left scale) and satellite radius (black dots, right scale) as a function of distance to Didymos, for a system with $k_2\/Q=10^{-5}$, starting with a ring of mass $M_T=0.25~M_D$. Each panel corresponds to a different snapshot where the simulation time is given at the top of the panel. The dashed lines give the Fluid Roche limit (FRL) and the maximum distance that satellites can migrate due to ring torques ($a_{2:1}^{FRL}$). The dotted line places the 2:1 inner Lindblad resonance location with Dimorphos (2:1 ILR). \\label{fig:standard}}\n\\end{figure}\n\nThe ring material spreads radially due to viscous effects (Fig.~\\ref{fig:standard}a), with part of the material falling back onto Didymos and part flowing outwards, eventually crossing the FRL. Due to gravitational instabilities, material outside the FRL gives rise to a proto-Dimorphos (Fig.~\\ref{fig:standard}b) that accretes all the ring material inside its Hill sphere \\citep{karjalainen2007,charnoz2010,charnoz2011,hyodo2014}. When the satellite's mass increases, its Hill sphere grows, increasing the accretion. As a consequence, we observe a rapid growth of Dimorphos (Fig.~\\ref{fig:standard}c) that reaches 0.93 Dimorphos masses in 0.01~yr ($<40~T_D$) after its formation  (Fig.~\\ref{fig:standard}d).  This growth corresponds to a growth in the ``continuous regime'' according the classification of \\cite{crida2012}. This means that the satellite grows progressively, at the Roche Limit, by accreting small pieces of ring material that are captured at the satellite surface. The satellite's critical Hill density is $\\rho_c=M_D\/(1.59a^3)\\sim1.2~{\\rm g\/cm^3}$ \\citep{porco2007}, which corresponds to a minimum density for a rubble-pile formed by the accumulation of material. Taking this value as fiducial, we obtain that the object is expected to have a maximum porosity of $\\sim 50\\%$.\n\nWhile growing by accretion, Dimorphos migrates outward due to tidal migration and resonant interactions with the ring. When its Hill sphere no longer overlaps the ring, new moons can form from the ring, as can be seen in Fig.~\\ref{fig:standard}e. In $t=4$~yr, the satellite is massive enough to confine part of the ring due to 2:1 inner Lindblad resonance (ILR, dotted line), which can be noticed by the step structure at about $570$~m. From this time on (Fig.~\\ref{fig:standard}e-h), Dimorphos grows due to impacts with the newly formed moons \\citep[``discrete regime'',][]{crida2012}. In about 40 years, the satellite mass reaches $0.98$ Dimorphos masses. \n\nIn the following years of simulation, the almost formed Dimorphos confines part of the ring due to the 2:1 ILR (Fig.~\\ref{fig:standard}f,g). Ring material mostly falls back onto Didymos. The unconfined material gives rise to objects with radius of meters. These objects do no migrate due to the weak ring torque. They accumulate near the Roche Limit. After $\\sim 6400$~years, the ring is empty, leaving a population of debris around the FRL (Fig.~\\ref{fig:standard}h). Work on the dynamics around non-uniformly shaped asteroids \\citep{Scheeres2007,Madeira2022a,agruda2022,Ferrari2022} shows that the region near its surface is essentially unstable or chaotic. The irregularities in the asteroid induce variations in the eccentricity of nearby objects, that will have limited lifetimes, colliding or being ejected from the system. This could be the fate of the debris population, given its proximity to Didymos surface. In the next sections, we discuss how our results change when we consider the scenario with material deposition by Didymos.\n\n\\section{On the flow and tidal parameters} \\label{sec_parameters} \n\\begin{figure}[]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.7\\columnwidth]{figures\/mass.png}}\n\\\\\n\\subfloat[]{\\includegraphics[width=0.7\\columnwidth]{figures\/radius.png}}\n\\caption{a) Mass of the largest satellite (in Dimorphos masses) and b) mass ratio between second largest and largest object on the system as a function of the system parameters. The total mass deposited in the ring is given on the x-axis. The different colours stand to the different deposition timescales and the markers correspond to the tidal parameters. \\label{fig:parameters}}\n\\end{figure}\n\nFigure~\\ref{fig:parameters} summarises all our numerical simulations, showing the mass of the largest satellite and mass ratio between second largest and largest object formed in the simulation as a function of the total mass deposited in the ring (x-axis), deposition timescale (colours), and tidal parameter (markers). We find that the mass of the largest satellite is mostly controlled by the total mass deposited in the ring ($M_T$), regardless of whether the mass is delivered at the very beginning of the simulation or not (Fig~\\ref{fig:parameters}a), meaning that $\\tau$ and $k_2\/Q$ have only little influence on Dimorphos mass. The mass of the largest satellite is directly proportional to the total mass deposited in the ring. We obtain that a mass of $0.25$ Didymos masses is required to form a satellite with the mass of Dimorphos (smaller panel in Fig~\\ref{fig:parameters}a).\n\nWe clarify once again that $M_T$ corresponds to the total amount of material deposited in the ring over time and not to the mass of the ring at any instant \\citep[as in works like][]{crida2012,charnoz2011,madeira2022}. We get that the ring will have its maximum mass just before the formation of proto-Dimorphos, reaching a value up to $0.1~M_D$ in the simulations with $M_T=0.25~M_D$. If we assume that Didymos can re-inject part of the fallen material into the ring, we have that the value $0.1~M_D$ can be interpreted as a lower bound for the mass detached from Didymos in a single avalanche.\n\n\\cite{hyodo2022} find that a mass of the order of $10\\%$ of the primary is ejected via a single landslide when the resultant shape is a top-shape \\citep[cases with spin-up timescales $\\lesssim$ few days and effective friction angles $\\gtrsim 70^{\\circ}$,][]{sugiura2021}. They obtain that the timescale of the mass ejection (i.e., landslide) is comparable to the critical spin period, that is, $\\sim 2-3$h (i.e., $\\sim 10^4$ seconds). This corresponds to $\\tau \\sim 10^{-4}$~yr in our numerical description of mass deposition, implying that the formation of Dimorphos may be somewhat similar to the scenario shown in Figure~\\ref{fig:standard}. \n\nThe correlation between the ring mass and the largest satellite mass is quantitatively demonstrated by 3D direct $N$-body simulations of ring spreading \\citep{hyodo2015}, and was also found in \\cite{charnoz2010,charnoz2011}. We point out that the mass of the largest satellite depends on the initial angular momentum of the ring \\citep{hyodo2015} -- consequently, it depends on the initial width of the ring. If the ring is initially wider, as would be expected in a scenario with a ring formed due to an impact, the mass needed to form Dimorphos becomes smaller. Therefore, the numbers obtained by us should not be taken as engraved on marble, but as responsible for assigning orders of magnitude to the system quantities.\n\nWe also get a relation between $M_T$ and the mass ratio of the second largest satellite (Fig~\\ref{fig:parameters}b). When the largest satellite is very massive, it accretes a large amount of ring material. As a consequence, only small objects will form in the system. Systems with larger $M_T$ form a very massive satellite and some meter-sized bodies, while systems with smaller $M_T$ forms dozens of objects with similar mass ratio -- with radius of tens of meters. These objects are in the ''pyramidal regime'' defined by \\cite{crida2012}. The exceptions are the cases with $\\tau=10^4$~yr and $k_2\/Q=10^{-3}$, in which the most massive satellite migrates fast enough to allow other massive satellites to form \\citep{charnoz2011}.\n\nThe deposition timescale mainly controls the formation timescale. In general, the cases with $\\tau=10^4$~yr form Dimorphos that are slightly more massive than those obtained in simulations with $\\tau<10^4$~yr. Cases with $\\tau=10^2$~yr, however, form Dimorphos that are less massive than those obtained in simulations with $\\tau<1$~yr, for most cases. Despite the weak dependency of Dimorphos mass on deposition timescale, we get different formation pathways depending on $\\tau$, as will be discussed in Section~\\ref{sec:grow}.\n\nWe also find no clear relation between the tidal parameter and the mass of the largest satellite. However, we do find an effect of tidal parameter on the mass of the second largest object. The resonant interactions between satellite and ring is responsible for both the outward migration of satellite and the radial confinement of the ring, with the 2:1 ILR location with the FRL ($a_{2:1}^{FRL}=1049$~m) being the maximum location to which a satellite can migrate due to this effect \\citep{charnoz2011,madeira2022}. Upon reaching this position, the satellite confines most of the ring material and only tiny satellites can be formed, as observed in the cases with $k_2\/Q=0$. In such simulations, Dimorphos does not migrate after reaching $a_{2:1}^{FRL}$. Due to the tidal migration, however, the satellite eventually migrates out of the $a_{2:1}^{FRL}$ position, leaving the ring  unconfined. Thus, larger objects can form. A larger $k_2\/Q$ translates to shorter periods of ring confinement, which explains why larger objects are obtained in systems with larger $k_2\/Q$.\n\\begin{figure}[]\n\\centering\n\\includegraphics[width=0.7\\linewidth]{figures\/smamass.png}\n\\caption{Evolution of Dimorphos mass as a function of the semimajor axis for different simulations with $M_T=0.25~M_D$. Each colour corresponds to a different simulation. The solid lines correspond to cases with $k_2\/Q=10^{-3}$ and the markers to cases with $k_2\/Q=10^{-5}$. The vertical dashed line gives the current location of Dimorphos. \\label{fig:mass_sma}}\n\\end{figure}\n\nAs expected, the parameter $k_2\/Q$ will control the location of the satellite when the ring disappears. Figure~\\ref{fig:mass_sma} shows the evolution of mass and semimajor axis of Dimorphos in simulations with $M_T=0.25~M_D$. As can be seen in all the simulations with $k_2\/Q=10^{-3}$ (solid line), the ring disappears when the satellite is out of the current location of Dimorphos (vertical dashed line). In fact, we find that the ring is completely emptied before Dimorphos reaches its current location only for $k_2\/Q\\leq10^{-5}$. These results lead us to conclude that the Didymos tidal parameter must be in this range $k_2\/Q\\leq10^{-5}$. \n\nThe orbit of Dimorphos may be further affected after its formation by other physical processes, such as the binary YORP (BYORP) effect \\citep[e.g.,][]{cuk2005}. Such an effect is an extension of the YORP effect, affecting a binary system when the satellite is in a spin-orbit resonance (as in the case of Dimorphos). Due to the BYORP, the satellite's orbit can be expanded or contracted, depending on the physical properties of the satellite, which are expressed mathematically through the dimensionless coefficient $B_s$ \\citep{cuk2005}. \\cite{hyodo2022} propose that the top-shaped asteroids, such as Ryugu and Bennu, could have at least one satellite in the past, with the BYORP effect being the responsible for expanding the orbits of these objects on a relatively short timescale of $\\sim 10^5$~yr, until their eventual ejection from the system.\n\nBy analogy, we could speculate that a similar process is happening in the Didymos system, that is, Dimorphos is recently formed and is now on the way to being ejected. However, as BYORP and tidal effects can act in opposite directions, an equilibrium state can be reached for \\citep{walsh2015}:\n\\begin{equation}\nB_s=\\frac{4\\pi^2}{3}\\frac{k_2}{Q}\\left(\\frac{M_d}{M_D}\\right)^{4\/3}\\frac{\\rho_D^2R_D^2G}{\\Phi a_d^7}.    \n\\end{equation}\nFor the tidal parameter $k_2\/Q=10^{-5}$, we get $B_s=0.0064$ which is in line with the expected values for satellites of top-shaped asteroids \\citep[$B_s\\sim 0.01$,][]{McMahon2010,walsh2015}. Therefore, it is possible that Dimorphos formed a long time ago and has kept being at an equilibrium location around Didymos. Next, we explore the effects of $\\tau$ on the formation of Dimorphos.\n\n\\section{Formation pathways for Dimorphos} \\label{sec:grow}\n\\begin{figure}[]\n\\centering\n\\subfloat[]{\\includegraphics[width=0.8\\columnwidth]{figures\/sigma1e0.png}}\\\\\n\\subfloat[]{\\includegraphics[width=0.8\\columnwidth]{figures\/sigma1e4.png}}\n\\caption{Left scale of panels shows ring surface density (solid line) and right scale, the radius of the satellites (black dots) for systems with $M_T=0.25~M_D$ and $k_2\/Q=10^{-5}$. The top and bottom panel correspond to the cases with $\\tau=1$~yr and $10^4$~yr, respectively. The different colors correspond to different times (shown at the top of each panel). The Fluid Roche limit (FRL) and the maximum distance that satellites can migrate due to ring torques ($a_{2:1}^{FRL}$) are given by the dashed lines, while the 2:1 ILR location with Dimorphos is given by the dotted lines. \\label{fig:compare}}\n\\end{figure}\n\nDifferent formation pathways are obtained for Dimorphos, depending on the deposition timescale. For $\\tau=0$, the pathway is the one described in Section~\\ref{sec:ring}: a single satellite forms and reaches a mass $>0.9~M_d$ in the continuous regime. When this is far enough away from the FRL, smaller moons begin to form. The satellite then starts to grow through impacts with these moons (discrete regime).\n\nWhen we increase the deposition time, we decrease the mass flux and the amount of mass available to build the first satellite is smaller. So, we get a smaller initial satellite. The Hill radius of the satellite will also be smaller and, as a consequence, it will accrete less material directly from the ring. This can be seen in the difference of slope of the curves with $\\tau=0$ and $1$~yr in Figure~\\ref{fig:mass_sma}. Figure~\\ref{fig:compare} shows the ring surface density (solid line, left scale) and satellite radius (black dots, right scale) of simulations with $M_T=0.25~M_D$ and $k_2\/Q=10^{-5}$. The top panel is the case with $\\tau=1$~yr and the bottom one is the case with $\\tau=10^4$~yr. The different colours correspond to the system at the different times indicated at the top of each panel.\n\nFor $\\tau=1$~yr, the system starts without ring but receives material ejected by Didymos, which spreads out of the FRL in $1.09$~yr. A small satellite forms at the Roche limit. The ring is supplied by Didymos, which in turn feeds the satellite that reaches $0.3~M_d$ in 0.01~yr after its formation. Then, new satellites form and Dimorphos grows to its current mass by accreting meter-sized satellites in the discrete regime. For high deposition rates, we have a rapid formation of Dimorphos, first by accretion of ring material and then by impacts with smaller moons, and the parameter $\\tau$ is responsible for defining the amount of material accreted by Dimorphos at each stage. The accretion of Dimorphos directly from ring material is called ``continuous regime'' according the classification of \\cite{crida2012}.\n\nNow, for $\\tau=10^2$ and $10^4$~yr, satellite formation takes place in a different way. Due to the small mass flux in the ring, meter-sized moons form and migrate outwards due to ring torque, allowing the formation of new moons of tens of meters in radius (Fig.~\\ref{fig:compare}, right column). Therefore, the system evolves into a collection of moons as seen in the right column of Figure~\\ref{fig:compare}, with Dimorphos being formed by merge events between these bodies. According to the classification of \\cite{crida2012}, this corresponds to complete formation in the pyramidal regime, which would imply Dimorphos with different properties than those obtained in the case of the continuous accretion regime (see Section~\\ref{subsec:shape}).\n\n\\section{Discussion} \\label{sec_discussion}\n\\subsection{Dimorphos Shape} \\label{subsec:shape}\nWhereas we do not directly track the shape of Dimorphos in our simulation, previous studies have shown that different accreting regimes may lead to different shape. Here, in light of our results, we build assumptions for the possible shape of Dimorphos based on the different accretion regimes in which the satellite can be formed. The influence of the formation environment on the shape of a satellite is an intricate problem and it is not our intention to draw conclusions about Dimorphos physical properties, but only to raise hypotheses for future work.\n\nAlthough the Dimorphos mass is mostly controlled by the total mass deposited in the ring ($M_T$), we obtain completely different formation pathways depending on the deposition timescale $\\tau$. For short timescales ($\\tau\\lesssim$yr), Dimorphos grows in both continuous and discrete regimes, while for large timescales, all growth occurs in the pyramidal regime. These formation pathways are expected to affect the final shape and structure of Dimorphos.\n\nMaterial flowing out of the FRL collapses into seeds \\citep{karjalainen2007} which accrete the ring material inside their Hill sphere. They could form as lemon-shaped self-gravitating aggregates \\citep{porco2007,Tiscareno2013}. The maximum porosity expected for these objects in the Didymos system is $50\\%$. For high deposition rate, Dimorphos accretes a fraction of its mass in this way and then starts to grow by impacts with smaller satellites. Due to the low impact velocities, smaller satellites are expected to sediment onto Dimorphos, preserving its general porosity \\citep{benz2000,Jutzi2015}. Depending on the impact geometry (grazing impacts), a significant amount of debris can be released, causing deposition on the crater and also restructuring due to large deformations \\citep{Jutzi2008a,Jutzi2009b}. Such processes favour the formation of a smooth regions on Dimorphos surface, preserving (at least in part) the ellipsoidal shape of the object. This scenario seems to be compatible with the Dimorphos images obtained by DART mission.\n\nFor low deposition rates, Dimorphos would be the result of impacts between gravitational aggregates of similar sizes. As the objects are in nearly circular and coplanar orbits, impacts are expected to occur at low velocities, resulting in a merge or partial accretion event, depending on the impact geometry \\citep{Leinhardt2012}. \\cite{Leleu2018} show that, depending on the collisions conditions, the impact of two moons can also lead to hit and run events before they finally merge. \\citeauthor{Leleu2018} find that growth in the pyramidal regime can produce flattened satellites with ridges or objects with elongate irregular shapes. Impacts in the Didymos system are not expected to reduce porosity, however mechanisms such as shear dilatation or re-accumulation can increase porosity. Thus, Dimorphos may have higher porosity than their parent bodies. Regarding the composition, Dimorphos is expected to have a similar composition to the surface of Didymos, source of the ring material.\n\n\\subsection{Solar radiation pressure and modelling limitations}\nDue to their proximity to the Sun, NEAs are strongly affected by angular momentum exchanges with the solar radiation \\citep{walsh2015,margot2015}. A classic example of this is the YORP effect (Section~\\ref{sec_dynamical}), responsible for increasing the asteroid's spin. The particles around the NEAs are also expected to be affected by the different components of the solar radiation force, such as radiation pressure, Poynting-Robertson drag, and Yarkovsky effects \\citep{Burns1979}. The last two effects are responsible for variation in the semi-major axis and spin rate on timescales of $\\gtrsim10^6$~yr for our system \\citep{Mignard1984,rubincam2000}. Thus, they are not expected to significantly affect the formation of Dimorphos. However, it is possible that the ring particles are affected by radiation pressure.\n\nRadiation pressure is caused by the transfer of momentum due to impacts of solar radiation on the ring particles, being responsible for causing variations in eccentricity in periods of few orbits \\citep{Hamilton1993} and small periodic variations in the semi-major axis \\citep{Madeira2018,Madeira2020}. The variation in eccentricity due to radiation pressure can be estimated as \\citep{Hamilton1996} \n\\begin{equation}\ne_{\\rm RP}=\\frac{C}{\\sqrt{1+C^2}}    \n\\end{equation}\nwhere \\citep{Hamilton1996}\n\\begin{equation}\nC=\\frac{9}{8}\\frac{Q_{\\rm pr}\\Phi}{nac\\rho_Ds}    \n\\end{equation}\nin which $n$ is the orbital frequency and $Q_{\\rm pr}$ is the solar radiation efficiency, computed from Mie theory \\citep{Irvine1965}.\n\nFor one-meter particles made of ice we get $e_{\\rm RP}\\sim 10^{-3}$, a value small enough that the particles do not collide with the central primary at its pericenter distance, so in this case radiation pressure can be negligible. Nonetheless, such an increase in eccentricity is responsible for increasing the velocity of collisions between particles, which may result in fragmentation and grinding into smaller particles. This means that part of the ring material might be lost in some orbits due to this effect. Now, if we assume that the particles are a mixture of ice with a small fraction of silicates, we have $e_{\\rm RP}\\sim 10^{-4}$, dropping to values of the order of $10^{-5}$ for particles made of more resistant materials or ice particles with a silicate concentration of dozens of percent \\citep{Irvine1965,Artymowicz1988}, which will probably be the case for particles ejected by Didymos. For such cases, we expect the radiation pressure to have a minor effect on Dimorphos formation for the particle size assumed in our simulations (one meter).\n\nWe point out that particles with radii of the order of mm to cm must also compose the ring, as data on NEAs indicate the possible presence of material with this size range on the surface of Didymos \\citep{pajola2022}. Such particles will present eccentricities $\\gtrsim0.01$, therefore, it is expected that the evolution of the system will be more complex when considering their presence in the disk. Here, we consider the ring composed only of meter-sized particles due to a computational limitation.  \n\nMore sophisticated simulations of the ring evolution with solar radiation and fragmentation would require tracking the evolution of the particles as single entities and with more than one size in a same simulation, which is not possible with the current version of our code. The \\texttt{HYDRORINGS} code describes a ring as a 1D entity and using a hydrodynamic approach, for which there is still no formalism with multiple particle sizes in a closed form. Furthermore, a more realistic study of the formation of Dimorphos from a ring requires the investigation of different size distributions in the ring. We leave this investigation for future work.\n\nAnother effect that likely removes material from the ring is Didymos shape irregularities. As demonstrated by a set of papers \\citep[eg.][]{Scheeres2007,Madeira2022a,agruda2022,Ferrari2022}, the region in the vicinity of non-uniformly shaped objects is normally unstable (or chaotic) and the particles have limited lifetimes. These are two caveats of our model. Including the actual shape of Didymos in our simulations would make it impossible to track the ring's evolution over billions of orbits with current computer capacities. This is the same limitation presented by works that reproduce Didymos shape, such as \\cite{Yu2018,Zhang2017,Zhang2021} and \\cite{hyodo2022}, for which the system can be evolved only by a limited number of orbits. The results of this article should be treat as a first-order study (like other studies of this kind), and the ring masses as lower bounds, since material can be removed due to Didymos shape and solar radiation, among others unmentioned external effects.\n\n\\section{Conclusion} \\label{sec_conclusion}\nIn this work, we studied the formation of Dimorphos from material released from Didymos. The YORP effect increases the spinning of Didymos, detaching surface material that settles in a ring around the primary. Most of the material should falls back onto Didymos, possibly being re-injected in the ring due to Didymos fast spin, while the remainder would give rise to satellite seeds outside the Roche limit, and finally to Dimorphos. Through numerical simulations, we analysed the evolution of the ring material, assuming an e-folding temporal function for the mass deposited in the ring. We varied the total mass deposited in the ring, deposition timescale, and Didymos tidal parameter. \n\nDimorphos with its current mass is obtained for a total mass deposited in the ring equal to $25\\%$ of Didymos' mass, regardless of the timescale on which the material is delivered. The material falling on Didymos is possibly responsible for giving birth to the asteroid's equatorial ridge \\citep{hyodo2022}. Some of the material can also be re-injected into the ring due to Didymos spin-up. Given the simplicity of our model, we believe that our results can be applied qualitatively to all similar binary systems with a primary close to the critical rotation. We leave this work for a future publication.\n\nDimorphos is formed in its current position without a ring for the a Didymos tidal parameter of $k_2\/Q\\leq10^{-5}$, being obtained that the satellite can be kept in its position through a BYORP-tidal balance for $k_2\/Q\\sim10^{-5}$. Different deposition timescales can lead to different shapes of Dimorphos. For example, timescales $\\geq10^2$~yrs would give rise to Dimorphos with an irregular shape due to accretion in pyramidal regime. However, the deposition timescale is expected to be less than one year \\citep{sugiura2021}, which would give rise to a nearly lemon-shaped (ellipsoidal) Dimorphos composed of meter sized rocks, which seems to be in agreement with recent images obtained by DART mission. DART and HERA spacecraft will allow us to obtain details on the shapes and composition of Didymos and Dimorphos and improve models of formation of binary systems.\n\n\\section*{Acknowledgments}\nG.M. thanks FAPESP for financial support via grants 2018\/23568-6 and 2021\/07181-7. R.H. acknowledges the financial support of MEXT\/JSPS KAKENHI (Grant Number JP22K14091). R.H. also acknowledges JAXA's International Top Young program. We acknowledge Dr. Keisuke Sugiura for discussion. Thanks to the reviewers for the comments that helped us to improve the article.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{INTRODUCTION}\n\\label{sec:intro} \nThe \\gls{WFIRST}\\cite{spergel_wide-field_2015} \\gls{CGI} is a technology demonstration\\cite{noecker_coronagraph_2016,kasdin_wfirst_2018,douglas_wfirst_2018,bailey_wfirst_2019-1} that will use high-contrast imaging and spectroscopy (coronagraphy),  wavefront sensing, and wavefront control\\cite{shi_low_2016,sidick_wfirst_2018}, to image planets in reflected light \\cite{kasdin_wfirst_2018}.\nFormerly known as the Wide-Field Infrared Survey Telescope, \\gls{WFIRST} is orders of magnitude more sensitive than \\gls{HST} or ground-based observatories\\cite{bailey_wfirst_2019}.\nTwo primary coronagraph technologies will be tested, a \\gls{HLC} coronagraph\\cite{trauger_hybrid_2016} for high contrast small-\\gls{FOV} ($<$0.5 as) imaging, a bow-tie \\gls{SPC}\\cite{balasubramanian_wfirst-afta_2015} for spectroscopy and a wide-\\gls{FOV} \\gls{SPC}.\nThe design and  preparation for \\gls{CGI} has spawned many new modeling tools to accurately predict performance. \nThis work attempts to summarizes the most common of these tools, both to serve as a roadmap for future potential users of \\gls{CGI} and to aid other missions which may benefit from reuse of the many open source tools which have been shared by \\gls{CGI} science and engineering teams. \nThis review cannot completely capture the simulation work that has gone into developing Roman CGI but it is intended to serve as a starting reference for the community to engage with the technical and science capabilities of the instrument. \nAdditionally, to better understand the context of these tools, we suggest previous works, as well as several papers in these proceedings, that provide detailed descriptions of the design reference mission and typical observing modes\\cite{kasdin_wfirst_2018,bailey_wfirst_2019,poberezhskiy_cgi_2020,kasdin_wfirst_2020}.\n\n\\begin{table}[ht]\n\\renewcommand*{\\arraystretch}{1.5}\n    \\centering\n    \\caption{Partial Listing of Open Source CGI Software Packages and Libraries}\n    \\label{tab:my_label}\n    \\footnotesize\n    \\begin{tabular}{L{3.75cm}|L{3.75cm}|c|p{6.25cm}}\n    \\hline\n    Package & Application & References & URL\\\\\n    \\hline\n    Observing Scenarios    & Complete Observation Simulation  & \\citenum{krist_numerical_2015}& \\url{https:\/\/roman.ipac.caltech.edu\/sims\/Coronagraph_public_images.html}\\\\\n    PROPER & Diffraction Simulation & \\citenum{krist_proper:_2007,krist_wfirst_2017} & \\url{https:\/\/github.com\/ajeldorado\/proper-models\/tree\/master\/wfirst_cgi}\\\\\n     \\verb+FALCO+  & Coronagraph Simulation   & \\citenum{riggs2018falco1} & \\url{https:\/\/github.com\/ajeldorado\/falco-matlab},  \\url{https:\/\/github.com\/ajeldorado\/falco-python} \\\\\n     Lightweight Coronagraph Simulator & Coronagraph Simulation & \\citenum{pogorelyuk_effects_2020}& \\url{https:\/\/github.com\/leonidprinceton\/LightweightSpaceCoronagraphSimulator}\\\\\n     CZT-based Optical Propagation & Diffraction Simulation && \\url{https:\/\/github.com\/ARCExoplanetTechnologies\/ACED}\\\\\n           \\verb+WebbPSF+ & Diffraction Simulation & \\citenum{perrin_simulating_2012,perrin_updated_2014}  & \\url{https:\/\/github.com\/spacetelescope\/webbpsf}\\\\\n\n     MSWC & Binary Star Simulation & \\citenum{dsirbu2017mswc,dsirbu2018RomanMSWC} & \\url{https:\/\/github.com\/ARCExoplanetTechnologies\/MSWC}\\\\\n     \\verb+EXOSIMS+ & Mission Simulation & \\citenum{savransky_exosims_2017,savransky_wfirst-afta_2016} &\\url{https:\/\/github.com\/dsavransky\/EXOSIMS}\\\\\n     Imaging Mission Database&Mission Planning&\\citenum{savransky_exploration_2019} & \\url{https:\/\/plandb.sioslab.com}\\\\\n     Coronagraph convolved Debris Disks & Disk Simulation & \\citenum{mennesson_wfirst_2018}& \\url{https:\/\/roman.ipac.caltech.edu\/sims\/Circumstellar_Disk_Sims.html} \\\\\n     Known debris disk simulated scenes & Disk Simulation &\\citenum{chen_wfirst_nodate}&  \\url{https:\/\/roman.ipac.caltech.edu\/sims\/Chen_WPS.html}\\\\\n     direct-imaging-sims + model spectra & Target simulation &\\citenum{lacy_characterization_2019, lacy_prospects_2020}  & \\url{https:\/\/github.com\/blacy\/direct-imaging-sims},\n    \\url{https:\/\/www.astro.princeton.edu\/~burrows\/wfirst\/index.html}\\\\\n         Giant planet albedo spectra & Target simulation & \\citenum{cahoy_exoplanet_2010}  &  \\url{https:\/\/wfirst.ipac.caltech.edu\/sims\/Exoplanet_Characterization.html}\\\\\n     \\hline\n    \\end{tabular}\n\\end{table}\n\n\n\\section{Coronagraph Simulators}\n\\label{sec:packages} \n\\subsection{Observing Scenarios\nThe most detailed and physically realistic simulations of \\gls{CGI} observations released to date have taken the form of numbered observing scenarios. \nThese are referred to as OS$n$, e.g., OS9 for the ninth scenario.\nPhysical optics simulations of the instrument including optical surfaces, coronagraphs, and wavefront sensing and control are generated using PROPER\\cite{krist2007proper,krist_wfirst_2018}.\nThe scenarios include speckle time series, derived from wavefront maps produced using \\gls{STOP} modeling of the Roman observatory\\cite{saini_impipeline_2017}.\nThe OS model outputs include noisy and noiseless datacubes of coronagraphic intensity images versus time, with injected planets and realistic observing scenarios that include reference stars, and off-axis \\gls{PSF}s for injecting additional targets.\nThese scenario files are available to the public from IPAC\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics{os9_target_roll_-11deg_processed_image_with_planets.pdf}\n    \\caption{Images of 47 Uma c generated from publicly available OS9 simulations of the \\gls{HLC}  including time dependent speckle and detector effects. Left: simulated image with injected planets. Right: the same scene after reference subtraction, showing injected planets at a much higher \\gls{SNR}.}\n    \\label{fig:coronagraphs}\n\\end{figure}\n\n\\subsection{FALCO\nThe FALCO\\cite{riggs2018falco1} library provides a framework for running wavefront sensing and control algorithms in MATLAB and Python 3. FALCO can use a PROPER\\cite{krist2007proper} model as its truth model in simulations, and examples using the CGI PROPER model are included in the publicly available FALCO repository. Example status window updates for the hybrid Lyot coronagraph are shown in Fig.~\\ref{fig:FALCO}.\n\n\\begin{figure}[ht]\n\\centering\n\\subfigure[] { \n    \\label{fig:before_wfsc}\n    \\includegraphics[width = .4\\columnwidth,trim=.05in 0in .05in 0in]{fig_falco_before.png}}\n \\subfigure[] {\n    \\label{fig:after_wfsc}\n    \\includegraphics[width = .4\\columnwidth,trim=.05in 0in .05in 0in]{fig_falco_after.png}}   \n\\caption{Status report windows from FALCO before (a) and after (b) performing wavefront control using the CGI PROPER model as the truth model. The large starting shapes on the deformable mirrors are the HLC design settings combined with the settings for flattening the starting wavefront errors.}\n\\label{fig:FALCO}\n\\end{figure}\n\n\n\\subsection{Lightweight Space Coronagraph Simulator}\nBased on FALCO\\cite{riggs2018falco1}, the ``Lightweight Space Coronagraph Simulator'\ncomputes small linear perturbations about the nominal dark hole instead of propagating the full optical model.\nIt allows quickly simulating observation scenarios with time evolving \\gls{WFE}, \\gls{DM} drift, \\gls{LOWFSC} residual jitter and \\gls{HOWFSC} \\cite{pogorelyuk_effects_2020}. \nThe sensitivities to DM commands (the Jacobian) and to \\gls{WFE} were computed in 6 wavelengths using FALCO and remain valid in the linear regime (up to 10 nm phase perturbations). \nThis allows specification of DM voltages, \\gls{WFE} Zernikes, LOWFS residual jitter, detector noise and switching between broadband and narrowband modes.\nThe Python code is designed to run fast and only requires NumPy\\cite{numpy}. \nFig. \\ref{fig:leonid} shows an example dark hole time series (left) and dark hole image (right).\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{leonid_thumbnail_LSCS.png}\n    \\caption{An example of contrast evolution in presence of various wavefront instabilities in a linearized model of Roman-CGI. Left: Closing the loop allows maintaining the contrast throughout the observation. The drift rate is exaggerated to illustrate dark hole maintenance in the worst-case scenario of random walk of Zernike coefficients. Right: A broadband photon-counts image of a single exposure (used to close the contrast loop). See Pogorelyuk et al.\\cite{pogorelyuk_effects_2020} for details.}\n    \\label{fig:leonid}\n\\end{figure}\n\n\n\\subsection{CZT-based Optical Propagation}\n    The Chirp Z-Transform (CZT) based optical propagation library, developed at Princeton and NASA Ames, implements 2D Fraunhofer and Fresnel diffraction propagation on arbitrarily sampled input and output planes. It provides the same functionality and answer (to within numerical precision) as the more commonly used MFT technique (Matrix Fourier Transform), but is asymptotically faster. \n    \n\\subsection{MSWC}\n\nBinary stars represent a special challenge for a diffraction simulation due to the angular separation between the on-axis target and its off-axis companion. The off-axis companion can contribute stellar leakage due to optical aberrations at high-frequencies for every component in the optical train. The Multi-Star Wavefront Control (MSWC) package can be used to simulate binary stars, predict the expected leakage for different binary star imaging scenarios, and determine if the contrast leakage due to the off-axis companion introduces a background contrast floor for a particular Roman CGI imaging mode. This can flag known binaries as benign or requiring suppression. Depending on the Roman CGI imaging mode, wavefront control techniques may remove the binary companion's leakage \\cite{sthomas15snwc, dsirbu2017mswc,dsirbu2018RomanMSWC}. \n\n\n\\subsection{WebbPSF\nWebbPSF\\cite{perrin_simulating_2012,douglas_accelerated_2018-1} is a \\gls{PSF} simulation tool originally developed for \\gls{jwst} in Python. \nBasic \\gls{SPC} modes are currently included in WebbPSF\\cite{perrin_poppy_2016}\\footnote{\\url{https:\/\/www.stsci.edu\/jwst\/science-planning\/proposal-planning-toolbox\/psf-simulation-tool}} and are in the process of being updated to match current filters and mask designs.\n\n\n\\section{Yield}\n\\subsection{Yield Calculator}\nNemati\\cite{nemati_sensitivity_2017,nemati_method_2020} developed an analytic model of instrument sensitivity that calculates the time-to-\\gls{SNR} for known \\gls{RV} exoplanets. \nThis model has been widely used by the project team as an Excel spreadsheet and is now publicly available as part of EXOSIMS (see below).\n\n\n\\subsection{EXOSIMS\nEXOSIMS\\cite{savransky_exosims_2017} is an open source, full exoplanet imaging mission simulation tool,  which generates a survey ensemble of possible exoplanet detections given an underlying universe (e.g. exoplanet phase curves and occurrence rates) and observatory properties such as orbit, optical system performance, and background sources.\n\n\\subsection{Imaging Mission Database\nThe online Cornell Space Imaging and Optical Systems Lab \\textit{Imaging Mission Database\nuses stellar and \\gls{RV} exoplanet physical properties\\cite{batalha_color_2018}. As shown in Fig. \\ref{fig:plandb}, these properties can be combined in joint distributions and compared to the instrumental sensitivity floor (blue curve) to assess the frequency of  detection in an observation.\nSimilarly, the Imaging Mission Database generates depth of search maps\\cite{garrett_2017} using EXOSIMS for blind-search targets such as the  EXOCAT database\\cite{turnbull_exocat_2015}.\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{plandb_47umac_completeness.png}\n    \\caption{Map of the joint distribution of planet projected separation and $\\Delta$mag for 47 Uma c generated using the Imaging Mission Database.}\n    \\label{fig:plandb}\n\\end{figure}\n\n\\section{Science Targets}\n\n\\subsection{Light from Young Giant Exoplanets\nSome of the substellar companions and young giant planets discovered by ground-based direct imaging surveys will be viable targets for Roman-CGI spectroscopy and photometry. This will present the first opportunity to make observations of such objects at wavelengths shorter than 0.95 $\\mu$m. At the time of writing, we have identified HD 984 B, $\\beta$-Pic b, HD 206893b, HR 8799e, and 51 Eri b as possible targets for Roman-CGI bandpasses 1, 2, and 3 which include spectroscopy and photometry. HR 2562b, HR 8799d, HR 8799c, $\\kappa$-And b, HD 95086b, HR 3549b, HD 1160b, and HIP 65426b  are possible targets for photometry in bandpass 4. We used \\textit{coolTLUSTY} to generate self-consistent 1D radiative-convective equilibrium atmosphere models for each possible target, assuming values for effective temperature, radius, and surface gravity taken from the literature and that atmospheres are clear with solar abundances. For some targets, we also compare to models with higher metallicities and with forsterite clouds. These are discussed in detail in Lacy \\& Burrows 2020\\cite{lacy_prospects_2020} and are available publicly\nFor all these objects, the dominant spectral features are the pressure broadened Na and K resonant doublets around 0.59 $\\mu$m and 0.77 $\\mu$m respectively, especially for the cooler objects where there is stronger pressure broadening at the photosphere. For the hotter objects (above $\\sim$1300 K), metal hydrides like CaH and MgH have not yet condensed and rained out, so their absorption features are present in the spectra. For intermediate temperature objects and higher metallicity hot objects, absorption features from TiO and VO also appear in the spectra. When forsterite clouds are included they increase the continuum flux ratio in the optical range and weaken gaseous absorption features. Combining optical spectroscopy and photometry with NIR and MIR observations of these objects from other instruments will aid efforts to characterize these young planets since changes to cloud properties and metallicity have different effects in the optical than at longer wavelengths.  \n\nIdeally, Roman-CGI spectroscopy will measure the strength of the $\\sim$0.73 $\\mu$m methane absorption feature for a reflected-light target, and constrain parameterized models incorporating cloud properties, a temperature-pressure profile, and mixing ratios of major gas-phase absorbers\\cite{lupu_developing_2016,nayak_atmospheric_2017,Damiano2020}. Whether this task is achieved will depend on the nature of the planets observed, the quality of data Roman-CGI collects, and the quality of auxiliary measurements like mass and orbital separation that are available from other sources. In the event that Roman-CGI performance is insufficient to constrain detailed models, Lacy et al.\\cite{lacy_characterization_2019} put forward a set of models suitable for addressing a simpler task: assessing how cool giant exoplanets compare to the cool gas giants and ice giants in our own solar system. Saturn and Jupiter have higher cloud layers and lower metallicities than Uranus and Neptune. Lower metallicities decrease the amount of methane, and higher clouds make for a smaller amount of gas above the cloud layer. Together these effects weaken Jupiter and Saturn's methane absorption features compared to Neptune and Uranus. At shorter wavelengths, from 0.4-0.6 microns, Saturn and Jupiter exhibit absorption from an unidentified chromophore which reddens their appearance. This is similar to the mix of hydrocarbons, commonly termed Tholins, that give Titan its yellow-orange appearance. Retrievals on simulated observations showed that Roman-CGI observations should be able to fit a model consisting of a two-part linear combination of Jupiter and Neptune's reflective properties. One parameter sets the short wavelength weighting towards Jupiter versus Neptune and essentially depends on whether a chromophore is present or not. A second parameter sets the longer wavelength weighting towards Jupiter versus Neptune and mainly reflects the amount of methane present above the cloud layer. This approach supposes that Jupiter and Neptune represent two bounds of cool giant planet atmosphere behavior and that those observed will fall somewhere in between. Of course, this framework is over-simplified, but, in the absence of high resolution high SNR spectra, it could provide a useful starting point. A small grid of geometric albedo models in this form is also available\\footnote{\\url{https:\/\/www.astro.princeton.edu\/~burrows\/wfirst\/index.html}}, along with a GUI for calculating the light curve through out a planet's orbit and a set of models representing Jupiter's geometric albedo with constant cloud properties but varying metallicity. \n\n\n\\begin{figure}[ht]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{all_self_lum_targets.png}\n    \\caption{Model spectra for potential self-luminous targets including imaging and spectra. All models assume solar metallicity and an atmosphere free of clouds. Other model parameters are listed in table 2 of Lacy \\& Burrows 2020\\cite{lacy_prospects_2020}. Lines for different model spectra are colored in the order of the assumed effective temperature for each object. The shaded regions indicate the Roman-CGI bandpasses. The horizontal dashed lines mark the required contrast for technology demonstration success and the engineer's best estimate of the contrast that Roman-CGI can achieve. Note that the spectral resolution of model spectra shown here are higher than the Roman-CGI spectral resolution.}\n    \\label{fig:self_lum_models}\n\\end{figure}\n\n\\subsection{Reflected Light from Cold Gas Giants\nGiant Exoplanet Albedo Spectra and colors as a function of planet phase, separation, and metallicity from Cahoy et al\\cite{cahoy_exoplanet_2010} have been released along with a reference solar spectrum.\n\n\\subsection{Debris Disks}\nThe  unprecedented point source sensitivity of CGI is expected to also lead to many new scattered light detections of debris disks and exozodiacal dust.\nSimulations of dusty systems, particularly 1  Vir,  Eps  Eri,  HD  10647,  HD  69830,  HD 95086,  HR  8799,  and  Tau  Cet were prepared by a the Preparatory Science Project: The Circumstellar Environments of Exoplanet Host Stars by Chen et al\\cite{chen_wfirst_nodate,chen_circumstellar_2014} and have been publicly released and include injected giant planets.\nGeneral libraries of disks\\cite{mennesson_wfirst_2018}  with varied morphology and geometry have  convolved with coronagraph transmission functions have also been released publicly.\n\n\\subsection{Fast Extended Source Simulation with a PSF Library}\n\nThe effect of the spatially varying coronagraph transmission functions on complex scenes, such as exozodiacal dust models can be simulated using public \\glspl{PSF} libraries\\cite{douglas_simulating_2019}. These simulations are efficiently performed by generating an over-sampled scene model and applying a coronagraphic transfer function via matrix multiplication. Because of the field-dependent evolution of the PSF, a PSF must be generated for every angular offset of the pixel coordinates of the scene model. These PSFs are generated via interpolation of the PSFs available from IPAC and stored as matrices in memory. For example, an exozodiacal model can be flattened into a vector rather than a 2D array and multiplied by the matrix of interpolated PSFs since the coronagraph is still assumed to be a linear system. For details and examples of  disk generation, see Milani et al. \\cite{milani_faster_2020}. \n\n\\subsection{PSF subtraction}\nPost-processing via \\gls{KLIP} subtraction of residual speckles has been extensively explored for Roman\\cite{ygouf_data_2015-1,ygouf_data_2016} and exoplanet extraction has been one of the data challenge topics\\cite{mandell_wfirst_2019}.\npyKLIP\\cite{wang_pyklip:_2015} supports generic data and the documentation currently includes an example of \\gls{KLIP} run on an observing scenario. \n\n\n\n\n\n \n\n\n\\section{Conclusions}\n\nRoman CGI has has stimulated and nurtured a wide array of coronagraph and mission simulation tools. \nIn addition to the simulation software describe above, Data Challenges\\cite{hildebrandt_wfirst_2018,mandell_wfirst_2019,girard_2019_2020} have allowed the community to engage and develop additional tools\\footnote{\\url{https:\/\/roman.ipac.caltech.edu\/sims\/Exoplanet_Data_Challenges.html}}.\nIntegral field spectrograph modes were considered for Roman and simulated in Coronagraph and Rapid Imaging Spectrograph in Python (\\verb+crispy+)\\cite{rizzo_simulating_2017} which may also prove useful for other missions\\footnote{\\url{https:\/\/github.com\/mjrfringes\/crispy}}. The majority of the tools described above have been developed in Python; however, other tools have been developed for estimating coronagraph noise\\cite{robinson_characterizing_2015} in languages such as IDL\\footnote{\\url{https:\/\/github.com\/tdrobinson\/coronagraph_noise}}.\nIt is hoped that the open-source availability of all these tools will allow the community to more rapidly develop science questions with Roman CGI and provide springboards for future coronagraphic missions.\n\n\\acknowledgments\n This research has made use of the Imaging Mission Database, which is operated by the Space Imaging and Optical Systems Lab at Cornell University. The database includes content from the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with the National Aeronautics and Space Administration under the Exoplanet Exploration Program, and from the SIMBAD database, operated at CDS, Strasbourg, France\\cite{wenger_simbad_2000}.\n\n\n\nPortions of this work were supported by the Roman\/WFIRST Science Investigation team prime award \\#NNG16PJ24C.\nPortions of this work were supported by the Arizona Board of Regents Technology Research Initiative Fund (TRIF).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}}\n\nWe study the problem of when a pair of approximately central (AC) projections are Murray-von Neumann equivalent by means of a  partial isometry (or unitary) that is approximately central. In doing so, additional K-theoretic information on the AC projections is required.\nSuch information was found by Kishimoto \\ccite{AK}\\footnote{Kishimoto's main Theorem 2.1 in \\ccite{AK} is stated for separable, nuclear, purely infinite, simple C*-algebras satisfying UCT. In Remark 2.9 of \\ccite{AK}, he notes that it also applies to simple AT C*-algebras of real rank zero, which includes the irrational rotation C*-algebras, known to be AT from \\ccite{EE} and \\ccite{EL}, and also includes their canonical orbifolds under the canonical automorphisms of order 2, 3, 4, and 6 as they are known to be AF from \\ccite{BK} \\ccite{ELPW} \\ccite{SW-CMP} \\ccite{SWcrelles}. Recall that AF-algebras are AT-algebras (\\ccite{Rordam}, Corollary 3.2.17).} (Theorem 2.1) for certain classes of C*-algebras, which include algebras studied in this paper.  The objective of this paper is to formalize this information into a topological K-theory invariant for AC projections, and proceed with computing it explicitly for an AC Flip-invariant Powers-Rieffel projection in the irrational rotation C*-algebra $A_\\theta,$ which will be denoted throughout this paper by $e$ (see equation \\eqref{ezetaE}).  The projection $e$ is similar to one constructed by Elliott and Lin \\ccite{EL}, and is essentially the same as the unit projection in the Elliott-Evans tower construction \\ccite{EE}.\n\n\\medskip\n\nOur main results are stated in Theorems \\ref{MaintheoremA}, \\ref{Kmatrixthm}, and \\ref{S3imagesofe} of this section.\n\\medskip\n\nIn this paper we will be concerned with the Flip orbifold C*-algebra\n\\[\nA_\\theta^\\Phi = \\{x\\in A_\\theta: \\Phi(x)=x\\}\n\\]\nthe fixed point C*-subalgebra of the irrational rotation algebra $A_\\theta$ under the Flip automorphism $\\Phi$ defined by \n\\[\n\\Phi(U) = U^{-1}, \\qquad \\Phi(V) = V^{-1}\n\\]\nwhere $U,V$ are unitaries generating $A_\\theta$ (also called noncommutative torus)  satisfying the usual Heisenberg commutation relation\n\\begin{equation}\\label{VUUV}\nVU= e^{2\\pi i\\theta} UV.\n\\end{equation}\nThroughout the paper, $\\theta$ is a fixed irrational number, $0 < \\theta < 1$. Both $A_\\theta$ and its Flip orbifold have canonical bounded traces which are unique normalized traces denote by $\\tau$. \n\n\\medskip\n\n\nThe approximately central projections studied in this paper depend on integer parameters. In our case for example, $e = e_{q',q,p,\\theta}$ is a Powers-Rieffel projection that depends on a sequence of consecutive convergents $p\/q,\\ p'\/q'$ of $\\theta$. Since the rotation algebra $A_\\theta$ is generated by the unitaries $U,V$, a projection $e$ is AC in $A_\\theta$ if $\\|e U - U e\\|, \\|e V - V e\\| \\to0$  as $q\\to \\infty$. A projection $e$ is AC in the Flip orbifold $A_\\theta^\\Phi$ if\n\\[\n\\|e(U+U^*) - (U+U^*) e\\| \\to0,\\quad \\|e (V+V^*) - (V+V^*)e\\| \\to0\n\\]\nas $q\\to \\infty$ (since it is known from \\ccite{BEEKa} that $U+U^*, V+V^*$ generate $A_\\theta^\\Phi$). We do not know if AC in $A_\\theta^\\Phi$ implies AC in $A_\\theta$ in general (though for many projections in the C*-algebra generated by certain powers of $U,V$ this can be checked).\n\\bigskip\n\n\\begin{dfn} Two AC projections are {\\it centrally equivalent} in an algebra $A$ (or {\\it AC-equivalent} in $A$) if they are Murray-von Neumann equivalent by a partial isometry in $A$ that is approximately central in $A$ (for large enough parameter).\n\\end{dfn}\n\\medskip\n\nKishimoto's Theorem 2.1 in \\ccite{AK} (restated in Section 2.4 below),  \nas applied to the Flip orbifold $A_\\theta^\\Phi$ (which known to be an AF-algebra \\ccite{BK}, \\ccite{SW-CMP}), implies that two AC projections $e$ and $f$ in $A_\\theta^\\Phi$ are centrally equivalent in $A_\\theta^\\Phi$ if and only if the cutdown of a given finite generating set of projections $[P_j]$ for $K_0(A_\\theta^\\Phi)$ by $e$ and $f$ have the same $K_0$-class,\n\\begin{equation}\\label{efPj}\n[\\chi(eP_je)] = [\\chi(fP_jf)] \\ \\ \\in \\ K_0(A_\\theta^\\Phi)\n\\end{equation}\nfor each $j,$ where $\\chi$ is the characteristic function of the interval $[\\frac12,\\infty)$. Of course, equation \\eqref{efPj} is understood to hold for large enough integer parameters which $e$ and $f$ depend on.\nSince $A_\\theta^\\Phi$ is AF (so its $K_1 = 0$), the $K_1$ side of Kishimoto's conditions (see Theorem 2.2 below) are trivially satisfied. In our particular case, $P_j$ are projections in $A_\\theta^\\Phi$.\\footnote{It seems reasonable to expect that if $[\\chi(ege)] < [\\chi(fgf)]$ for each $g = P_j$, then there exist AC partial isometry $u$ such that $uu^* = e$ and $u^*u \\le f$; but the author has no proof. }\n\n\\medskip\n\nIn Section 2.3 we construct a specific basis $[P_1], \\dots, [P_6]$ for $K_0(A_\\theta^\\Phi) = \\mathbb Z^6$ (see \\eqref{basis} and \\eqref{Kbasis}), with specific projections $P_s$ in $A_\\theta^\\Phi,$ with respect to which we compute the classes $[\\chi(eP_se)]$ -- which would therefore determine the central equivalence class of the  projection $e$ in the Flip orbifold.  These $K_0$-classes will be identified explicitly by computing their Connes-Chern character  \n\\begin{equation} \n\\bold T : K_0(A_\\theta^\\Phi) \\to \\mathbb R^5, \\qquad\n\\bold T(x) = (\\uptau(x); \\phi_{00}(x), \\phi_{01}(x), \\phi_{10}(x), \\phi_{11}(x))\n\\end{equation}\nin terms of the canonical trace $\\tau,$ and four basic unbounded traces $\\phi_{jk}$ (defined in Section 2.1 below). The map $\\bold T$ is known to be a group monomorphism for irrational $\\theta$ (see \\ccite{SWa}, Proposition 3.2). Therefore, in terms of the Connes-Chern character we will calculate the numerical invariants for $e$\n\\[\n\\tau \\chi(eP_se), \\qquad \\phi_{jk} \\chi(eP_se) \n\\]\n($s=1,...,6,\\ jk=00,01,10,11$ ). For ease of notation, let us write\n\\[\n\\chi_s := \\chi(e P_s(\\theta) e)\n\\]\nfor the cutdown projections by $e$ (for large enough parameter).  We also find it convenient to organize these classes for $e$ into a vector\n\\[\n\\vec \\tau(e) =  \\begin{bmatrix} \n\\tau \\chi_1 & \\tau \\chi_2 & \\tau \\chi_3 & \\tau \\chi_4 & \\tau \\chi_5& \\tau \\chi_6\n\\end{bmatrix}  \n\\]\nconsisting of the canonical traces of the cutdowns, together with a topological K-matrix involving the unbounded traces which we denote by\n\\[\nK(e) = \n\\begin{bmatrix} \n\\phi_{00}(\\chi_1) & \\phi_{00}(\\chi_2) & \\phi_{00}(\\chi_3) & \\phi_{00}(\\chi_4) & \n\\phi_{00}(\\chi_5) & \\phi_{00}(\\chi_6)\n\\\\\n\\phi_{01}(\\chi_1) & \\phi_{01}(\\chi_2) & \\phi_{01}(\\chi_3) & \\phi_{01}(\\chi_4) & \n\\phi_{01}(\\chi_5) & \\phi_{01}(\\chi_6)\n\\\\\n\\phi_{10}(\\chi_1) & \\phi_{10}(\\chi_2) & \\phi_{10}(\\chi_3) & \\phi_{10}(\\chi_4) & \n\\phi_{10}(\\chi_5) & \\phi_{10}(\\chi_6)\n\\\\\n\\phi_{11}(\\chi_1) & \\phi_{11}(\\chi_2) & \\phi_{11}(\\chi_3) & \\phi_{11}(\\chi_4) & \n\\phi_{11}(\\chi_5) & \\phi_{11}(\\chi_6)\n\\end{bmatrix}\n\\]\nwherein the $(jk,s)$-entry consists of the unbounded trace $\\phi_{jk}(\\chi_s)$. Therefore, in the Flip orbifold, the central equivalence class of $e$ is fully determined by the pair consisting of the canonical trace vector $\\vec \\tau(e),$ and the $4\\times6$ topological matrix $K(e)$.\n\\medskip\n\n\\begin{ntn}\\label{notation}\nWe shall use the divisor delta function $\\delta_n^m = 1$ if $n$ divides $m$, and $\\delta_n^m = 0$ otherwise. We also use the notation $e(t) := e^{2\\pi it}$. Thus, $\\rmsumop_{j=0}^{n-1} e(\\tfrac{mj}n) = n \\delta_n^m$.\n\\end{ntn}\n\n\\begin{std}\\label{standing} Without loss of generality we can assume that there are infinitely many consecutive convergents $\\tfrac{p}q, \\tfrac{p'}{q'}$ of $\\theta$ such that\n\\[\n\\frac{p}q < \\theta < \\frac{p'}{q'}, \\qquad \\frac12 < q'(q\\theta - p)  < \\frac45.\n\\]\n(See Remark \\ref{tracecondition} for why.) This will be assumed in the hypotheses of Theorems \\ref{MaintheoremA}, \\ref{Kmatrixthm}, and \\ref{S3imagesofe}.\n\\end{std}\n\\medskip\n\nThe AC Flip-invariant Powers-Rieffel projection in $A_\\theta^\\Phi$ that will be studied throughout this paper is \n\\begin{align}\\label{ezetaE}\ne \\ := \\ e_{q',q,p,\\theta} \\ &=\\ G_\\tau(U^{q'}) V^{-q} + F_\\tau(U^{q'}) + V^q G_\\tau(U^{q'})\n\\\\\n&\\ = \\  \\zeta_{q',q,p,\\theta} \\ \\mathcal E(q'(q\\theta-p))\t\t\\label{ezeta}\n\\end{align}\ndepending on the convergent parameters $p,q,p',q',$ as stated in the Standing Condition, with trace $\\tau(e) = q'(q\\theta - p) \\in(\\tfrac12,\\tfrac45)$. The Rieffel functions $F_\\tau, G_\\tau$ in \\eqref{ezetaE} and the continuous field $\\mathcal E(t)$ in \\eqref{ezeta} are described in Section 2.2 (see \\eqref{rieffelproj}), and the C*-morphism $\\zeta$ is defined in \\eqref{zetamorphism}.  No confusion should arise with occasionally denoting the trace of $e$ simply by $\\tau := q'(q\\theta - p)$.\n\\medskip\n\nIn light of this background, our results can now be stated in terms of the following three theorems (some of the notation of which is explained later).\n\\medskip\n\n\\begin{thm}\\label{MaintheoremA}\nLet $\\theta$ be irrational with convergents satisfying the Standing Condition. Let $t \\to P(t)$ be a continuous section, defined in some neighborhood of $\\theta,$ of smooth projections of the continuous field $\\{ A_t^\\Phi \\}_{0<t<1}$ of Flip orbifolds.  Then the unbounded trace of the cutdown projection $\\chi(e P e),$ where $P=P(\\theta),$ is given by\n\\[\n\\phi_{jk} \\chi(e P e) = a_{jk}^- C_0(P) + a_{jk}^+ C_1(P)\n\\]\nfor $jk=00,01,10,11,$ where\n\\begin{equation}\\label{CP}\nC_0(P) = \\phi_{00}(P) + \\phi_{10}(P), \\qquad\nC_1(P) = \\phi_{0,q'}(P) + (-1)^{p'} \\phi_{1,q'}(P)\n\\end{equation}\nand\n\\[\na_{jk}^- =\n\\begin{cases} \\tfrac12 (-1)^{pjk},  &\\text{    for even } q' \\\\ \\\\\n\\updelta_2^{q} \\updelta_2^{j} \\updelta_2^{k}  + \\tfrac12 (-1)^{pjk}\\updelta_2^{q-1}   &\\text{     for odd } q'\n\\end{cases}\n\\]\n\\smallskip\n\\[\na_{jk}^+ =\n\\begin{cases} \n \\tfrac12 (-1)^{j+pjk} &\\text{    for even } q' \\\\ \\\\\n \\tfrac12 (-1)^{p'j} \\left[ \\updelta_2^{k-1} + (-1)^{pj} \\updelta_2^{q-k-1}  \\right]    &\\text{     for odd } q'.\n\\end{cases}\n\\]\nThe canonical trace of the cutdown is \n\\[\n\\tau \\chi(e Pe) = \\tau(e) \\tau_{p'\/q'}(P(\\tfrac{p'}{q'}))\n\\]\nwhere $\\tau_{p'\/q'}$ is the canonical normalized trace of the rational rotation algebra $A_{p'\/q'}$.\n\\end{thm}\n\n\\medskip\n\nWith this result established, we can write down the topological K-matrix of the projection $e$ according to the following theorem.\n\n\\medskip\n\n\\begin{thm}\\label{Kmatrixthm}\nLet $\\theta$ be an irrational number. The Powers-Rieffel projection $e$ has K-matrix given by the $4\\times 6$ matrices according to parities (as indicated by subscripts):\n\\begin{equation*} \nK(e_{q',q,p,\\theta}) \\ = \\ \\frac12 \n\\begin{bmatrix} \n2 & 0 & 1 & 1 & 1 & 1 \n\\\\ \n2 & 0 & 1 & 1 & 1 & 1 \n\\\\ \n0 & 0 & 1 & -1 & 1 & -1\n\\\\ \n0 & 0 & (-1)^p & (-1)^{p+1} & (-1)^p & (-1)^{p+1} \n\\end{bmatrix}_{q' \\text{ even}}\t\t\n\\end{equation*}\n\\begin{equation*}\nK(e_{q',q,p,\\theta}) = \n\\begin{bmatrix} \n1 & 0 & 1 & 0 & 1 & 0 \n\\\\ \n0 & 0 & \\updelta_2^{p'} & \\updelta_2^{p'-1} & -\\updelta_2^{p'} & -\\updelta_2^{p'-1}\n\\\\ \n0 & 0 & 0 & 0 & 0 & 0\n\\\\ \n0 & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}_{q \\text{ even}}\n\\end{equation*}\n\n\\begin{equation*}\nK(e_{q',q,p,\\theta}) =\n\\frac12 \\begin{bmatrix} \n1 & 0 & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1} & \\updelta_2^{p'-1} & -\\updelta_2^{p'-1} \n\\\\ \n1 & 0 & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1} & \\updelta_2^{p'-1} & -\\updelta_2^{p'-1} \n\\\\ \n1 & 0 & \\updelta_2^{p'-1} &  -\\updelta_2^{p'-1} & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1}\n\\\\ \n(-1)^p & 0 & \\updelta_2^{p'-1} & -\\updelta_2^{p'-1} & (-1)^p(1 + \\updelta_2^{p'}) & \n\\updelta_2^{p'-1} \n\\end{bmatrix}_{q,q' \\text{ both odd}}.\n\\end{equation*}\nIn addition, the canonical trace vector is\n\\[\n\\vec \\tau(e_{q',q,p,\\theta}) = \n(q\\theta-p) \\begin{bmatrix} q' & \\ p_2 & \\ p' & \\ p' & \\ p' & \\ p' \\end{bmatrix} \n\\]\nwhere\n\\[\np_2 := \n \\begin{cases} \n2p' &\\text{for } 0 < \\theta < \\tfrac12\n\\\\\n2(q' - p') &\\text{for } \\tfrac12 < \\theta < 1.\n\\end{cases}\n\\]\n \\end{thm}\n\n\\medskip\n\n\\noindent{\\bf Application.} One interesting application of the preceding theorem is determination of whether or not $\\alpha(e)$ and $\\beta(e)$ are centrally equivalent for smooth automorphisms $\\alpha, \\beta$ of the Flip orbifold. Or, equivalently, when is $\\alpha(e)$ centrally equivalent to $e$? We will answer this for the canonical automorphisms given by the Fourier $\\sigma$ and Cubic $\\kappa$ transforms (studied in \\ccite{FB} \\ccite{BW}  \\ccite{SWChern} \\ccite{SWcjm} \\ccite{SWcrelles}) defined by\n\\begin{align*}\n&  &  & & \\sigma(U) &= V^{-1},  &\t\\sigma(V) &= U\t\t&  &  & & \n\\\\\n&  &  & & \\kappa(U) &= e(-\\tfrac\\theta2)U^{-1}V,  & \\kappa(V) &= U^{-1}.\t&  &  & & \n\\end{align*}\nThese have order 2 and 3, respectively, on the Flip orbifold, and they have a simple relations with the unbounded traces: $\\sigma$ swaps $\\phi_{01}$ and $\\phi_{10},$ fixing $\\phi_{00}, \\phi_{11},$ while $\\kappa$ induces the cyclic permutation $\\phi_{11} \\to \\phi_{01} \\to \\phi_{10},$ and fixing $\\phi_{00}$. With this information at hand, one easily calculates the action of these automorphisms on the basis $[P_1], \\dots, [P_6]$ of $K_0(A_\\theta^\\Phi)$ mentioned earlier. This would then lead to the following result.\n\n\\medskip\n\n\\begin{thm}\\label{S3imagesofe}\nConsidering the Powers-Rieffel AC projection $e$ (given by \\eqref{ezetaE}) as running over parameters where $q$ is even and $p'$ is odd, one has its K-matrix and those of its Fourier and Cubic transforms\n\\[\nK(e) = \\begin{bmatrix}\n1 & 0 & 1 & 0 & 1  & 0\n\\\\\n0 & 0 & 0 & 1 & 0 &  -1\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix},\n\\qquad\nK(\\kappa(e)) = \\begin{bmatrix}\n1 & 0 & 1 & 1 & 0  & 0\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\\\\n0 & 0 & 0 & 0 & -1 & 1\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}\n\\]\n\\[\nK(\\kappa^2(e)) = \\begin{bmatrix}\n1 & 0 & 1 & 0 & 0  & 1\n\\\\\n0 & 0 & 0 & -1 & 1 & 0\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix},\n\\qquad\nK(\\sigma(e)) = \\begin{bmatrix}\n1 & 0 & 1 & 1 & 0  & 0\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\\\\n0 & 0 & 0 & 0 & 1 &  -1\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}\n\\]\n\\[\nK(\\sigma\\kappa(e)) = \\begin{bmatrix}\n1 & 0 & 1 & 0 & 1 & 0\n\\\\\n0 & 0 & 0 & -1 & 0 & 1\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix},\n\\qquad\nK(\\sigma\\kappa^2(e)) = \\begin{bmatrix}\n1 & 0 & 1 & 0 & 0  & 1\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\\\\n0 & 0 & 0 & 1 & -1 & 0\n\\\\\n0  & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}.\n\\]\nIn particular, as these matrices are pairwise distinct, the AC projections\n\\begin{equation}\\label{ecyclic}\ne, \\quad \\sigma(e), \\quad \\kappa(e), \\quad \\kappa^2(e), \\quad \\sigma\\kappa(e), \\quad  \\sigma\\kappa^2(e)\n\\end{equation}\nare pairwise not centrally equivalent. \n\\end{thm}\n\n\n\\medskip\n\nNote that the six projections in \\eqref{ecyclic} all have the same orbifold $K_0$ class and constitute the $S_3$ orbit of $e$ (where the symmetric group $S_3$ is generated by $\\sigma$ and $\\kappa$). Indeed, in the case considered in the preceding theorem where $q$ is even ($q',p$ odd), the Connes-Chern character of $e$ can be obtained from \\eqref{etraces} as\n\\[\n\\bold T(e) = (\\tau(e);\\ 1, 0, 0, 0)\n\\]\nwhere the trace of $e$ is also the trace of any automorphism of $e$ (by uniqueness of the normalised canonical trace). In view of what was just noted regarding how $\\sigma, \\kappa$ act on the $\\phi_{jk},$ we see that all the projections in \\eqref{ecyclic} have exactly the same Connes-Chern character as $e,$ so that they are all Murray-von Neumann equivalent (in the Flip orbifold). Theorem \\ref{S3imagesofe}, however, says that they are pairwise centrally inequivalent.\n\\medskip\n\nIn \\ccite{WaltersModular} we showed that in fact for any AC projection $e$ in the Flip orbifold, its $S_3$ orbit \\eqref{ecyclic} gives the only possible central classes, in the sense that for any canonical automorphism $\\alpha$ arising from SL$(2,\\mathbb Z),$ the projection $\\alpha(e)$ is centrally equivalent to one of the projections in \\eqref{ecyclic}.  The point of Theorem \\ref{S3imagesofe} is then that all six projections can be pairwise centrally distinct. It is possible also to give examples where some or all of the projections in \\eqref{ecyclic} are centrally equivalent - see Theorem 1.4 of  \\ccite{WaltersModular}.\n\\medskip\n\nWe have included many details in our calculations in this paper so as to save the reader from numerous and onerous checking (and hopefully to also be clear about our reasoning). We hope this may be of help. \n\n\\medskip\n\n\\noindent{\\bf Structure Of Paper.} Let's summarize what we do in this paper.\n\nIn Section 2 we gather the necessary background material, notation, results and introduce our notation for the K-matrix relative to a constructed basis for group $K_0(A_\\theta^\\Phi)$ required by our proofs.\n\nIn Section 3 the approximately central Powers-Rieffel projection $e$ is constructed from the continuous field constructed in Section 2 and its unbounded traces are calculated.\n\nIn Section 4, the projection is realized as a C*-inner product $e = \\Dinner{f}{f}$ using Riefflel's equivalence bimodule paradigm \\ccite{MRb}. This realization will facilitate computation of the topological invariants of the cutdown $\\chi(ePe)$ of any projection $P$ by $e,$ which is carried out in Section 5. This will prove Theorem \\ref{MaintheoremA}. \n\nIn Section 6, Theorem \\ref{MaintheoremA} is applied to the $K_0$-basis constructed in Section 2.3 in order to obtain the full K-matrix of $e$ (for various parity situations of integer parameters that $e$ depends on). This will then prove Theorem \\ref{Kmatrixthm}.\n\nIn Section 7, the unbounded traces of two technical C*-inner products required in Section 6  are computed (see Lemma \\ref{phiV1V3}).\n\nSections 8 and 9 are appendices involving some basic unbounded trace computations used in the paper. (The Acknowledgement paragraph is right before the References.)\n\n\\medskip\n\nIn a forthcoming paper \\ccite{WaltersKAC} we calculate the K-matrix of Fourier-invariant  projections in the Fourier orbifold $A_\\theta^\\sigma$ for the Fourier transform $\\sigma$. The greater complication in the Fourier case arises from the fact that Fourier-invariant projections do not have a Powers-Rieffel form, from additional unbounded traces on $K_0(A_\\theta^\\sigma) = \\mathbb Z^9,$ as well as additional $K_0$-basis elements, giving rise to a $9\\times6$ K-matrix. \n\n\n\\textcolor{blue}{\\Large \\section{Background Material}}\n\nIn this section we write down the relevant Connes-Chern character for the Flip orbifold; define a continuous field of Powers-Rieffel projections, which is a continuous section of the continuous field of Flip orbifolds $\\{A_t^\\Phi: 0<t<1\\};$ construct a canonical basis consisting of continuous fields of projections for $K_0(A_t^\\Phi);$ state Kishomoto's Theorem (\\ccite{AK}, Theorem 2.1) in the form we require; and state a couple of Poisson Summation formulas used in our calculations.\n\n\\bigskip\n\n\\subsection{The Connes-Chern Character} \nThe Flip automorphism $\\Phi$ of the rotation C*algebra $A_\\theta$ has four associated unbounded $\\Phi$-traces $\\phi_{jk}^\\theta$ defined on the basic unitaries $U^mV^n,$ satisfying \\eqref{VUUV}, by (see \\ccite{SWa} or \\ccite{SW-CMP})\n\\begin{equation}\n\\phi_{jk}^\\theta(U^mV^n)\\ =\\ e(-\\tfrac{\\theta}2 mn)\\,\\updelta_2^{m-j} \\updelta_2^{n-k}\n\\end{equation}\nfor $jk = 00, 01,10,11$,  $m,n\\in \\mathbb Z$ (and $\\updelta_a^{b}$ was defined in Notation \\ref{notation}). Invariably, we may write $\\phi_{jk}^\\theta$ simply as $\\phi_{jk}$ when $\\theta$ is understood. These are linear functionals defined on the canonical smooth dense *-subalgebra $A_\\theta^\\infty$ satisfying the $\\Phi$-trace property\n\\[\n\\phi_{jk}(xy) = \\phi_{jk}(\\Phi(y)x)\n\\]\nfor $x,y\\in A_\\theta^\\infty$. (Of course, such maps are $\\Phi$-invariant). In addition, they are Hermitian maps: they are real on Hermitian elements.  Clearly, on the smooth orbifold $A_\\theta^{\\Phi,\\infty}$ -- the fixed point *-subalgebra of $A_\\theta^\\infty$ under the Flip -- they give rise to trace functionals that are not continuous in the C*-norm. Together with the canonical trace $\\uptau$ one has the Connes-Chern character which we write as\n\\begin{equation} \n\\bold T : K_0(A_\\theta^\\Phi) \\to \\mathbb R^5, \\qquad\n\\bold T(x) = (\\uptau(x);\\, \\phi_{00}(x),\\, \\phi_{01}(x),\\, \\phi_{10}(x),\\, \\phi_{11}(x)).\n\\end{equation}\nIn \\ccite{SWa} (Proposition 3.2) this map is known to be injective for irrational $\\theta$.\\footnote{In \\cite{SWa} we worked with the crossed product algebra $A_\\theta \\times_\\Phi \\mathbb Z_2$, but since this algebra is strongly Morita equivalent to the fixed point algebra, the injectivity follows for the latter and is easy to see.} For the identity element one has $\\bold T(1) = (1; 1, 0, 0, 0)$. The ranges of the traces $\\phi_{jk}$ on projections in $A_\\theta^\\Phi$ are known to be half-integers, while the canonical trace has range $(\\mathbb Z + \\mathbb Z\\theta) \\cap [0,1]$ on projections.\n\n\\bigskip\n\n\\subsection{Continuous field of Rieffel projections}\n\nThere is a natural continuous (section) field $\\mathcal E: [\\frac12,1) \\to \\{A_t\\}$ of Flip-invariant Powers-Rieffel projections\n\\begin{equation}\\label{rieffelproj}\n\\mathcal E(t) = G_t(U_t) V_t^{-1} + F_t(U_t) + V_t G_t(U_t) \n\\end{equation}\nwhere \n\\[\nF_t(x) = \\rmsumop_{n \\in\\mathbb Z} f_t(x+n), \\qquad \nG_t(x) = \\rmsumop_{n \\in\\mathbb Z} g_t(x+n)\n\\]\nare periodizations of the $t$-parameterized family ($\\tfrac12 \\le t < 1$) of continuous (or smooth) functions $f_t(x), g_t(x)$ on $\\mathbb R$, compactly supported in $[-\\tfrac12,\\tfrac12]$, as graphed in Figure \\ref{figfg}. \n\n\n\\begin{figure}[H]\n\\includegraphics[width=3in,height=1.5in]\n{fandgplotspdf.pdf}\n\\caption{\\SMALL{Graphs of $f_t$ and $g_t$. }}\\label{figfg}\n\\end{figure}\nOne requires \n\\begin{equation}\\label{ftxt}\nf_t(x+t) = 1-f_t(x)\n\\end{equation}\nfor $-\\tfrac12 \\le x \\le \\tfrac12 - t,$ sets $f_t = 1$ on $\\tfrac12 - t \\le x \\le t - \\tfrac12$, and $f_t(-x)=f_t(x)$ for $-\\tfrac12 \\le x \\le \\tfrac12$.\\footnote{This is a slight modification of the usual constructions; e.g., see proof of Lemma 2.1 \\ccite{SWa}.} The condition \\eqref{ftxt} also holds for $-\\tfrac12 \\le x \\le 0$ (trivially for $\\tfrac12 - t < x \\le 0$). Since $f_t$ is even, we have\n\\begin{equation}\\label{ftx1}\nf_t(x) + f_t(t - x) = 1, \\quad \\text{for }  0 \\le x \\le \\tfrac12.\n\\end{equation}\nTaking $x = - \\tfrac{t}2$ in \\eqref{ftxt} and using the fact that $f_t$ is even, gives \n\\begin{equation}\\label{ftt2}\nf_t(\\tfrac{t}2 ) = \\tfrac12.\n\\end{equation}\n\nFurther, $g_t$ is defined by $g_t(x) = \\sqrt{f_t(x)(1-f_t(x))}$ for $t - \\tfrac12 \\le x \\le \\tfrac12,$ and $g_t(x) = 0$ elsewhere. It is easy to check that $g_t(t-x) = g_t(x)$ holds for all $x$ (since $x\\to t-x$ leaves its supporting interval \n$[t - \\tfrac12, \\tfrac12]$ invariant.\n\n\\medskip\n\nNote that the Flip invariance of $\\mathcal E(t)$ requires that $F_t(U_t) = F_t(U_t^{-1})$, which translates into saying $f_\\tau(x) = f_\\tau(-x)$, and $G_t(U_t) = G_t(e(t)U_t^{-1})$, i.e. $G_t(x) = G_t(t - x)$ for all $x$. These are easily checked as $G_t$ has period 1 and $x\\to t-x$ leaves the interval $[t-\\tfrac12,\\tfrac12]$ invariant.\n\n\\medskip\n\nTo give a specific smooth example of $f_t$ (where $t\\ge\\tfrac12$ is now fixed), choose any $C^\\infty$ function $h(x)$ on the closed interval $[-\\tfrac12, -\\tfrac{t}2]$  such that $h(-\\tfrac12) = 0,$ $h(-\\tfrac{t}2) = \\tfrac12,$ and all derivatives of $h$ vanish at the endpoints $-\\tfrac12, -\\tfrac{t}2.$ One can then define $f_t$ based on such $h$ by\n\\[\nf_t(x) = \n\\begin{cases}\nh(x) & -\\tfrac12 \\le x \\le -\\tfrac{t}2 \\\\\n1 - h(-x-t) & -\\tfrac{t}2 \\le x \\le \\tfrac12 - t \\\\\n\\ \\ 1 & \\tfrac12 - t \\le x \\le t - \\tfrac12  \\\\\n1 - h(x-t) & t - \\tfrac12  \\le x \\le \\tfrac{t}2 \\\\\nh(-x) & \\tfrac{t}2 \\le x \\le \\tfrac12\n\\end{cases}\n\\]\nIt is easy now to see that $f_t$ is a smooth even function, compactly supported on $[-\\tfrac12,\\tfrac12],$ and satisfies the condition \\eqref{ftx1}.  Of course, the corresponding function $g_t$ will also be $C^\\infty$ and compactly supported in the interval indicated by Figure \\ref{figfg}. These ensure that the projection $\\mathcal E(t)$ is smooth for each $t,$ which we emphasize is defined at $t=\\tfrac12$ as well.\n\n\n\\begin{rmk}\nIt is worthwhile remembering the conditions which ensure that $\\mathcal E(t)$ is a projection. Writing $F:=F_t, \\ G:=G_t$, those conditions are: both $F,G$ are non-negative, of period 1 (as functions of the real variable $x$), and satisfy\n\\begin{align*}\nF(x) &= F(x)^2 + G(x)^2 + G(x-t)^2 \\\\ \nG(x)&\\big[1 - F(x) - F(x+t)\\big] = 0 \\\\\nG(x)&G(x+t) = 0\n\\end{align*}\nfor all $x$. These conditions ensure that $\\mathcal E(t) = \\mathcal E(t)^2 = \\mathcal E(t)^*$ is a projection. \n\\end{rmk}\n\n\n\nThe field $\\mathcal E(t)$ has trace\n\\[\n\\tau(\\mathcal E(t)) = \\tau(F_t(U_t)) \n= \\rmintop_{-\\frac12}^{\\frac12} f_t(x) dx = t\n\\]\n(in view of Figure \\ref{figfg}). In Appendix A (Section 8), the Connes-Chern character of $\\mathcal E(t)$ is computed to be\n\\[\n\\bold T(\\mathcal E(t)) = (t; \\tfrac12, \\tfrac12, \\tfrac12, \\tfrac12)\n\\]\nfor $\\tfrac12 \\le t < 1$.\n\nNow we extend the field $\\mathcal E$ to the interval $0 < s < \\tfrac12$ by setting\n\\[\n\\mathcal F(s) = 1 - \\beta_s \\mathcal E(1-s)\n\\]\nwhere $\\beta_s: A_{1-s} \\to A_s$ is the canonical isomorphism\n\\[\n\\beta_s(U_{1-s}) = -U_s, \\qquad \\beta_s(V_{1-s}) = -V_s^{-1}\n\\]\nwhich commutes with the canonical Flip so $\\mathcal F(s) \\in A_s^\\Phi$ is also Flip invariant.\nWe have $\\tau(\\mathcal F(s)) = s$ and the unbounded topological invariants of $\\mathcal F(s)$ can be obtained from the (easy to check) relation\n\\begin{equation}\\label{phibeta}\n\\phi_{jk}^{1-s} = (-1)^{jk+j+k} \\phi_{jk}^{s} \\beta_s.\n\\end{equation}\nThis gives us the Connes-Chern character of $\\mathcal F$\n\\[\n\\bold T(\\mathcal F(s)) = (s; \\tfrac12, \\tfrac12, \\tfrac12, \\tfrac12)\n\\]\nfor $s\\in(0,\\tfrac12]$ - so it conveniently has the same unbounded traces as $\\mathcal E$.\n\n\n\\subsection{The $K_0$ basis}\nRecall that the torus $\\mathbb T^2$ acts canonically on the rotation algebra by mapping a pair $(a,b) \\in \\mathbb T^2$ to the automorphism $U \\to a U,\\ V \\to bV$. It is easy to check that the only such toral automorphisms that commute with the Flip are (aside from the identity)\n\\begin{align*}\n\\gamma_1(U) &= -U, \\qquad \\gamma_1(V)=V\n\\\\\n\\gamma_2(U) &= U, \\qquad \\ \\ \\ \\gamma_2(V)=-V\n\\\\\n\\gamma_3(U) &= -U, \\qquad \\gamma_3(V)=-V\n\\end{align*}\nwhere $\\gamma_3=\\gamma_1\\gamma_2$.  These give us Flip invariant projection fields $\\gamma_1\\mathcal E(t), \\gamma_2 \\mathcal E(t), \\gamma_3 \\mathcal E(t)$ defined for $t\\in[\\tfrac12,1)$\nas well as fields $\\gamma_1\\mathcal F(s), \\gamma_2 \\mathcal F(s), \\gamma_3 \\mathcal F(s)$ defined for $s\\in(0,\\tfrac12]$.  It's easy to check the following relations between the unbounded traces and $\\gamma_j$\n\\[\n\\phi_{jk} \\gamma_1 = (-1)^j \\phi_{jk}, \\qquad \n\\phi_{jk} \\gamma_2 = (-1)^k \\phi_{jk}, \\qquad\n\\phi_{jk} \\gamma_3 = (-1)^{j+k} \\phi_{jk}.\n\\]\nAs a result, we obtain the invariants for the $\\mathcal E$-fields\n\\begin{align*}\n\\bold T(\\mathcal E(\\theta)) & = (\\theta; \\tfrac12, \\tfrac12, \\tfrac12, \\tfrac12)\n\\\\\n\\bold T(\\gamma_1\\mathcal E(\\theta)) &= (\\theta; \\tfrac12,  \\tfrac12, -\\tfrac12, -\\tfrac12)\n\\\\\n\\bold T(\\gamma_2 \\mathcal E(\\theta)) &= (\\theta; \\tfrac12, - \\tfrac12, \\tfrac12, - \\tfrac12)\n\\\\\n\\bold T(\\gamma_3 \\mathcal E(\\theta)) &= (\\theta; \\tfrac12, - \\tfrac12, - \\tfrac12, \\tfrac12)\n\\end{align*}\nfor $\\theta \\in [\\frac12,1)$. Similarly, for the $\\mathcal F$-fields one has\n\\begin{align*}\n\\bold T(\\mathcal F(\\theta)) & = (\\theta; \\tfrac12, \\tfrac12, \\tfrac12, \\tfrac12)\n\\\\\n\\bold T(\\gamma_1\\mathcal F(\\theta)) &= (\\theta; \\tfrac12,  \\tfrac12, -\\tfrac12, -\\tfrac12)\n\\\\\n\\bold T(\\gamma_2 \\mathcal F(\\theta)) &= (\\theta; \\tfrac12, - \\tfrac12, \\tfrac12, - \\tfrac12)\n\\\\\n\\bold T(\\gamma_3 \\mathcal F(\\theta)) &= (\\theta; \\tfrac12, - \\tfrac12, - \\tfrac12, \\tfrac12) \n\\end{align*}\nfor $\\theta \\in (0,\\frac12]$.\n\n\\medskip\n\nWe shall let $P_1 = 1$ (the identity projection field), and write\n\\[\nP_3(\\theta) = \\begin{cases} \n\\mathcal F(\\theta) &\\text{for } 0 < \\theta < \\tfrac12\n\\\\\n \\mathcal E(\\theta) &\\text{for } \\tfrac12 < \\theta < 1\n\\end{cases}, \\qquad\nP_4(\\theta) = \\begin{cases} \n\\gamma_1 \\mathcal F(\\theta) &\\text{for } 0 < \\theta < \\tfrac12\n\\\\\n\\gamma_1 \\mathcal E(\\theta) &\\text{for } \\tfrac12 < \\theta < 1\n\\end{cases}\n\\]\n\\[\nP_5(\\theta) = \\begin{cases} \n\\gamma_2 \\mathcal F(\\theta) &\\text{for } 0 < \\theta < \\tfrac12\n\\\\\n\\gamma_2 \\mathcal E(\\theta) &\\text{for } \\tfrac12 < \\theta < 1\n\\end{cases}\n\\qquad\nP_6(\\theta) = \\begin{cases} \n\\gamma_3 \\mathcal F(\\theta) &\\text{for } 0 < \\theta < \\tfrac12\n\\\\\n\\gamma_3 \\mathcal E(\\theta) &\\text{for } \\tfrac12 < \\theta < 1.\n\\end{cases}\n\\]\nWe now construct $P_2(\\theta)$ as follows. If $0 < \\theta < \\frac12$, we set\n\\[\nP_2(\\theta) = \n\\begin{cases} \n\\eta \\mathcal F(2\\theta) &\\text{for } 0 < \\theta < \\tfrac14\n\\\\\n \\eta\\mathcal E(2\\theta) &\\text{for } \\tfrac14 < \\theta < \\tfrac12\n\\end{cases}\n\\]\nwhere $\\eta: A_{2\\theta} \\to A_\\theta$ is the C*-morphism $\\eta(U_{2\\theta}) = - U_\\theta^2, \\ \\eta(V_{2\\theta}) = V_\\theta$. From the relations\n\\begin{equation}\\label{phieta}\n\\phi_{0k}^\\theta \\eta = \\phi_{0k}^{2\\theta} - \\phi_{1k}^{2\\theta}, \\qquad\n\\phi_{1k}^\\theta \\eta = 0\n\\end{equation}\none obtains  \n\\[\n\\bold T(P_2(\\theta)) = (2\\theta; 0, 0, 0, 0).\n\\]\n\nNow suppose $\\frac12 < \\theta < 1$. Then $0 < 1-\\theta < \\frac12$ so that $P_2(1-\\theta) \\in A_{1-\\theta}$ (as in preceding case), which we compose with $\\beta_\\theta$ to obtain the projection\n\\[\nP_2(\\theta) = \\beta_\\theta P_2(1-\\theta) \\ \\in \\ A_\\theta,\t\\qquad (\\tfrac12 < \\theta < 1)\n\\]\nthus giving\n\\[\n\\bold T(P_2(\\theta)) = (2-2\\theta; 0, 0, 0, 0)\n\\]\nin view of equations \\eqref{phibeta} and the vanishing unbounded traces of $P_2(1-\\theta)$.\n\nFrom the above it is clear that regardless of whether $\\theta < \\tfrac12$ or $\\tfrac12 < \\theta$, the projections $P_j(\\theta)$ have the same unbounded traces. \n\nWe now claim that the group $K_0(A_\\theta^\\Phi) = \\mathbb Z^6$ has the following projections\n\\begin{equation}\\label{basis}\nP_1=1, \\quad P_2(\\theta), \\quad P_3(\\theta), \\quad P_4(\\theta), \\quad P_5(\\theta), \\quad P_6(\\theta)\n\\end{equation}\nas basis. As noted above, their Connes-Chern characters are\n\\begin{align}\\label{Kbasis}\n\\bold T(P_1) & = (1; 1, 0, 0, 0) \\notag\n \\\\\n \\bold T(P_2) &= \n \\begin{cases} \n(2\\theta; 0, 0, 0, 0) &\\text{for } 0 < \\theta < \\tfrac12\n\\\\\n(2-2\\theta; 0, 0, 0, 0)\t &\\text{for } \\tfrac12 < \\theta < 1\n\\end{cases}\n\t\\notag\n\\\\\n\\bold T(P_3) &= (\\theta; \\tfrac12, \\tfrac12, \\tfrac12, \\tfrac12)\t\n\\\\\n\\bold T(P_4) &= (\\theta; \\tfrac12,  \\tfrac12, -\\tfrac12, -\\tfrac12)\t\t\\notag\n\\\\\n\\bold T(P_5) &= (\\theta; \\tfrac12, - \\tfrac12, \\tfrac12, - \\tfrac12)\t\\notag\n\\\\\n\\bold T(P_6) &= (\\theta; \\tfrac12, - \\tfrac12,  -\\tfrac12,  \\tfrac12).   \\notag\n\\end{align}\nIt is known from Proposition 3.2 of \\ccite{SWa} that a basis for the range of the Connes-Chern character $\\bold T K_0(A_\\theta^\\Phi)$ consists of the six vectors\n\\begin{align*}\n&(2; 0, 0, 0, 0)  \\\\\n&(1; 1, 0, 0, 0) = \\bold T(1)  \\\\\n&(1; 0, 1, 0, 0) \t\\\\\n&(1; 0, 0, 1, 0)\t\\\\\n&(1; 0, 0, 0, 1)\t\\\\\n&\\begin{cases} \n(\\theta; \\tfrac12, -\\tfrac12, \\tfrac12, -\\tfrac12) = \\bold T(P_5(\\theta)) &\\text{for } 0 < \\theta < \\tfrac12\n\\\\\n(\\theta; \\tfrac12, \\tfrac12, -\\tfrac12, -\\tfrac12) = \\bold T(P_4(\\theta)) &\\text{for } \\tfrac12 < \\theta < 1.\n\\end{cases}\n\\end{align*}\nIt can be checked that these six vectors have the same integral span as the vectors in \\eqref{Kbasis}. Therefore, the $P_j(\\theta)$'s form a basis for $K_0(A_\\theta^\\Phi)$. \n\n\\medskip\n\nWe do not actually use the exact form of the basis projections $P_j(\\theta)$ in computing the traces of their cutdowns by the AC projection $e$. Their topological invariants, together with those of $e$, will be sufficient for that purpose. The interesting thing we learn here is the manner by which these invariants interact to giving invariants for cutdowns, as in Theorems \\ref{MaintheoremA}, \\ref{Kmatrixthm}.\n\n\\bigskip\n\n\\subsection{Kishimoto's Theorem} \nWe now state Kishimoto's Theorem 2.1 \\ccite{AK} in a form appropriate for our purposes and specifically for simple AT-algebras of real rank zero (satisfying UCT).\n\n\\begin{thm} (Kishimoto \\ccite{AK}, Theorem 2.1.)\nLet $A$ be a simple AT-C*-algebra $A$ of real rank zero or a separable simple nuclear purely infinite C*-algebra satisfying the Universal Coefficient Theorem. Assume that $K_0(A)$ is finitely generated by classes of projections $g_1, \\dots, g_k$ in $A$, and that $K_1(A)$ is finitely generated by classes of unitaries  $u_1, \\dots, u_\\ell$ in $A$. Then for each $\\epsilon > 0$ and each finite subset $F\\subset A$, there exists $\\updelta > 0$ and a finite subset $G\\subset A$ such that for any pair of projections $e_1, e_2$ in $A$ satisfying\n\\[\n\\|e_1 x - x e_1\\| < \\updelta, \\qquad \\|e_2 x - x e_2\\| < \\updelta\n\\]\nfor $x \\in \\{g_1, \\dots, g_k\\} \\cup \\{u_1, \\dots, u_\\ell\\} \\cup G$, and also satisfying\n\\begin{equation}\\label{ege}\n[\\chi(e_1g_i e_1)] = [\\chi(e_2g_i e_2)] \\ \\ \\in \\ K_0(A)\n\\end{equation}\nfor $i=1,\\dots,k$, and\n\\[\n[u_j e_1 + (1- e_1)] = [u_j e_2 + (1-e_2)] \\ \\ \\in \\ K_1(A)\n\\]\nfor $j=1,\\dots,\\ell$, there exists a partial isometry $v$ in $A$ such that \n\\[\ne_1 = v^*v, \\qquad e_2 = vv^*, \\qquad \\|vx - xv\\| < \\epsilon\n\\]\nfor each $x\\in F$.\n\\end{thm}\n\n\\medskip\n\nThe conditions \\eqref{ege} on the generators $g_j$'s imply that $e_1$ and $e_2$ have the same class in $K_0$. In our case, the algebras ($A_\\theta$ and $A_\\theta^\\Phi$) have the cancellation property so that the projections are Murray von-Neumann equivalent and in fact are unitarily equivalent via a unitary in the algebra.\n\\medskip\n\nIn our case, we apply this theorem to the orbifold $A_\\theta^\\Phi$ which has vanishing $K_1$ (since it is AF) so we are only concerned with the $K_0$ conditions \\eqref{ege} in classifying AC projections with respect to AC Murray von-Neumann equivalence.\n\nTo this end, we shall use the basis projections \\eqref{basis} for $K_0(A_\\theta^\\Phi),$ and the $K_0$ classes of their cutdown projections which will be convenient to write as\n\\[\n\\chi_i := \\chi(e P_i(\\theta) e)\n\\]\n(for large enough integer parameters in $e,$ of course).  Since the Connes-Chern character $\\bold T$ mentioned in Section 2.1 is injective on $K_0$, it follows that the central equivalence class of $e$ is fully determined by the canonical traces $\\tau(\\chi_i)$ and the unbounded traces $\\phi_{jk}(\\chi_i)$ for $i=1,\\dots,6$. We find it convenient to organize these numerical invariants for $e$ into a 6-dimensional trace vector \n\\[\n\\vec\\tau(e) =  \\begin{bmatrix} \n\\tau \\chi_1 & \\tau \\chi_2 & \\tau \\chi_3 & \\tau \\chi_4 & \\tau \\chi_5& \\tau \\chi_6\n\\end{bmatrix}  \n\\]\nconsisting of the canonical traces of the cutdowns, and a topological K-matrix involving the unbounded traces which we lay out as a $4\\times6$ matrix\n\\[\nK(e) = \n\\begin{bmatrix} \n\\phi_{00}(\\chi_1) & \\phi_{00}(\\chi_2) & \\phi_{00}(\\chi_3) & \\phi_{00}(\\chi_4) & \n\\phi_{00}(\\chi_5) & \\phi_{00}(\\chi_6)\n\\\\\n\\phi_{01}(\\chi_1) & \\phi_{01}(\\chi_2) & \\phi_{01}(\\chi_3) & \\phi_{01}(\\chi_4) & \n\\phi_{01}(\\chi_5) & \\phi_{01}(\\chi_6)\n\\\\\n\\phi_{10}(\\chi_1) & \\phi_{10}(\\chi_2) & \\phi_{10}(\\chi_3) & \\phi_{10}(\\chi_4) & \n\\phi_{10}(\\chi_5) & \\phi_{10}(\\chi_6)\n\\\\\n\\phi_{11}(\\chi_1) & \\phi_{11}(\\chi_2) & \\phi_{11}(\\chi_3) & \\phi_{11}(\\chi_4) & \n\\phi_{11}(\\chi_5) & \\phi_{11}(\\chi_6)\n\\end{bmatrix}\n\\]\nand we call it the K-matrix of $e$. Its entries all lie in $\\tfrac12\\mathbb Z$, where the $i$-th column consists of the unbounded traces of $\\chi_i$ (arranged from top to bottom in the same order they appear in the character map $\\bold T$). Therefore, in the Flip orbifold, the central equivalence class of $e$ is fully determined by the pair $\\vec \\tau(e)$ and $K(e)$. \n\\medskip\n\nFor instance, the K-matrix of the identity is\n\\begin{equation}\\label{Kidentity}\nK(1) = \\begin{bmatrix}\n1 & 0 & \\tfrac12 & \\tfrac12 & \\tfrac12  & \\tfrac12\n\\\\\n0 & 0 &\\tfrac12 & \\tfrac12 & -\\tfrac12 & -\\tfrac12\n\\\\\n0  & 0 & \\tfrac12 & -\\tfrac12 & \\tfrac12 & -\\tfrac12\n\\\\\n0  & 0 & \\tfrac12 & -\\tfrac12 & -\\tfrac12 & \\tfrac12\n\\end{bmatrix}\n\\end{equation}\n(since $\\chi_i = P_i(\\theta)$) where the columns are just the (unbounded) topological invariants of the basis projections $P_1, \\dots, P_6$. Its canonical trace vector is\n\\[\n\\vec\\tau(1) = \n\\begin{cases}\n\\begin{bmatrix} 1 & 2\\theta & \\theta & \\theta & \\theta & \\theta \\end{bmatrix} &\\text{for } 0 < \\theta < \\tfrac12, \n\\\\\n\\begin{bmatrix} 1 & 2-2\\theta & \\theta & \\theta & \\theta & \\theta \\end{bmatrix} &\\text{for }\n\\tfrac12 < \\theta < 1.\n\\end{cases}\n\\]\n\\medskip\n\n\nIf $\\alpha$ is a smooth automorphism, the K-invariant of the AC projection $\\alpha(e)$ is determined by the $K_0$ classes\n\\[\n[\\chi(\\alpha(e)P\\alpha(e))]  \n= \\alpha_* [\\chi(e \\alpha^{-1}(P) e)] \n\\]\nwhere $P = P_i$ as in \\eqref{basis}. These are determined by their canonical traces\n\\[\n\\tau [\\chi(\\alpha(e)P\\alpha(e))] = \n\\tau \\alpha_* [\\chi(e \\alpha^{-1}(P) e)] = \\tau [\\chi(e \\alpha^{-1}(P) e)]\n\\]\n(since the canonical trace is unique, $\\tau \\alpha = \\tau$), and the unbounded traces\n\\begin{equation}\\label{automP}\n\\phi_{jk} [\\chi(\\alpha(e)P\\alpha(e))] = \\phi_{jk} \\alpha_* [\\chi(e \\alpha^{-1}(P) e)].\n\\end{equation}\n\nFor the unbounded traces, one needs to calculate $\\phi_{jk} \\alpha$ in terms of $\\phi_{jk}$,  and determine the $K_0$ class of $[\\alpha^{-1}(P)]$ in terms of the basis.  This would then allow for the calculation of the K-matrix of $\\alpha(e)$ in terms of that of $e$. Here is a useful and relevant case, particularly for Theorem \\ref{S3imagesofe}.\n\n\n\\medskip\n\n\nFor canonical automorphisms, such as those arising from SL$(2,\\mathbb Z)$ (or even $\\gamma_1, \\gamma_2, \\gamma_3$), $\\alpha^{-1}$ permutes the basis $[P_i],$ and $\\phi_{jk} \\alpha$ is a permutation of the $\\phi_{jk}$. (The $\\gamma_j$'s would change the signs of some $\\phi_{jk}$'s.) Therefore, to obtain the K-matrix of $\\alpha(e)$ from the K-matrix of $e,$ one\n\\medskip\n\n(1) permutes the columns of $K(e)$ according to how $\\alpha^{-1}$ acts on $[P_i],$ \n\n(2) permutes the rows of the result in (1) according to how $\\alpha$ acts on $\\phi_{jk}$.\n\\medskip\n\nIt can be checked, almost by inspection, that applying this procedure to the identity element, where $\\alpha$ is the Fourier and Cubic transforms or $\\gamma_j$'s, leaves the K-matrix \\eqref{Kidentity} of the identity unchanged (as it should).\n\n\n\\subsection{Two Poisson Lemmas} We shall have need for the following two Poisson lemmas for our later computations.\n\n\\medskip\n\n\\begin{lem}\\label{poisson} (Poisson Summation.) Let $f(x)$ be a continuous function on $\\mathbb R$ that is compactly supported. Then for each $x$,\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft f(n)\\ e^{2\\pi inx} \\ =\\ \\rmsumop_{n=-\\infty}^\\infty f(x+n)\n\\]\nwhere $\\ft f(s) = \\rmintop_{\\mathbb R} f(t) e(-st) dt$ is the Fourier transform of $f$ over $\\mathbb R$.\n\\end{lem}\n\n\\begin{proof}  \nThe short proof below doesn't require $f$ to have compact support, only that $f$ is integrable and decays at $\\pm\\infty$ so that $h(x) := \\rmsumop_{n=-\\infty}^\\infty f(x+n)$ is a well-defined integrable periodic function (for instance, for Schwartz functions $f$). The Fourier transform of the 1-periodic function $h$ is\n\\begin{align*}\n\\ft h(m) &= \\rmintop_{0}^{1}  h(x) e(-mx) dx\n= \\rmintop_{0}^{1} \\rmsumop_{n=-\\infty}^\\infty f(x+n) e(-mx) dx\n= \\rmsumop_{n=-\\infty}^\\infty \\rmintop_{0}^{1}  f(x+n) e(-mx) dx\n\\\\\n&= \\rmsumop_{n=-\\infty}^\\infty \\rmintop_{n}^{n+1}  f(t) e(-mt) dt\n=  \\rmintop_{-\\infty}^{\\infty}  f(t) e(-mt) dt = \\ft f(m).\n\\end{align*}\nTherefore, by Fourier inversion for $h(x)$ we get\n\\[\n\\rmsumop_{n=-\\infty}^\\infty f(x+n) = h(x) = \\rmsumop_{m=-\\infty}^\\infty \\ft h(m)\\ e^{2\\pi imx}\n= \\rmsumop_{m=-\\infty}^\\infty \\ft f(m)\\ e^{2\\pi imx}\n\\]\nas required. \n\\end{proof}\n\n\\medskip\n\n\\begin{lem}\\label{poissonparity}\nFor any Schwartz function $H(x)$ on the real line we have the following forms of the Poisson Summation (for all real $x$),\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(2n)\\ e(nx) \\ \n=\\ \\frac12 \\rmsumop_{n=-\\infty}^\\infty H(\\tfrac{x}2+n) + H(\\tfrac{x}2+ \\tfrac12+n)\n\\]\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(2n+1)\\ e(nx) \\ \n=\\ \\frac12 e(-\\tfrac12x) \\rmsumop_{n=-\\infty}^\\infty H(\\tfrac{x}2+n) - H(\\tfrac{x}2+ \\tfrac12+n)\n\\]\n\\end{lem}\n\\begin{proof} From Poisson Summation for $H$,\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(n)\\ e(nx) \\ =\\ \\rmsumop_{n=-\\infty}^\\infty H(x+n)\n\\]\nreplace $x \\to x + \\tfrac12$ \n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(n)\\ (-1)^n e(nx)  \\ =\\ \\rmsumop_{n=-\\infty}^\\infty H(x+ \\tfrac12+n)\n\\]\nand add the preceding two equalities to get\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(n)\\ e(nx) [1 + (-1)^n] \\ \n=\\ \\rmsumop_{n=-\\infty}^\\infty H(x+n) + H(x+ \\tfrac12+n)\n\\]\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(2n)\\ e(2nx) \\ \n=\\ \\frac12 \\rmsumop_{n=-\\infty}^\\infty H(x+n) + H(x+ \\tfrac12+n)\n\\]\nreplace $x$ by $\\frac{x}2$,\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(2n)\\ e(nx) \\ \n=\\ \\frac12 \\rmsumop_{n=-\\infty}^\\infty H(\\tfrac{x}2+n) + H(\\tfrac{x}2+ \\tfrac12+n).\n\\]\nSubtracting we get\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(n)\\ e(nx) [ 1 - (-1)^n] \\ \n=\\ \\rmsumop_{n=-\\infty}^\\infty H(x+n) - H(x+ \\tfrac12+n)\n\\]\nwhich becomes\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(2n+1)\\ e(2nx) \\ \n=\\ \\frac12 e(-x) \\rmsumop_{n=-\\infty}^\\infty H(x+n) - H(x+ \\tfrac12+n)\n\\]\nor\n\\[\n\\rmsumop_{n=-\\infty}^\\infty \\ft H(2n+1)\\ e(nx) \\ \n=\\ \\frac12 e(-\\tfrac{x}2) \\rmsumop_{n=-\\infty}^\\infty H(\\tfrac{x}2+n) - H(\\tfrac{x}2+ \\tfrac12+n)\n\\]\nas desired.\n\\end{proof}\n\n\n\n\\textcolor{blue}{\\Large \\section{The AC Powers-Rieffel Projection}}\n\nIn this section we construct the Powers-Rieffel projection and compute its unbounded traces.\n\n\\medskip\n\nTo build approximately central projections from the continuous field $\\mathcal E$ of Section 2.2, we consider, for given integers $q',q,p$ and irrational $\\theta$, the natural C*-monomorphism $\\zeta = \\zeta_{q',q,p,\\theta}$ defined by  \n\\begin{equation}\\label{zetamorphism}\n\\zeta: A_\\tau \\to A_\\theta, \\qquad \n\\zeta(U_\\tau) = U_\\theta^{q'} = U^{q'}, \\qquad\n\\zeta(V_\\tau) = V_\\theta^q = V^q\n\\end{equation}\nwhere we will write $\\tau := q'(q\\theta-p)$ for brevity (which shan't be confused with the trace map $\\tau$!).  One now uses the field $\\mathcal E(t)$ to obtain the Powers-Rieffel projection \n\\begin{align}\\label{rieffelAC}\ne &= \\zeta_{q',q,p,\\theta} \\mathcal E(\\tau) \\\\ \n& = \\zeta (G_\\tau(U_\\tau) V_\\tau^{-1} + F_\\tau(U_\\tau) + V_\\tau G_\\tau(U_\\tau))\n\\notag\n\\\\\n&= G_\\tau(U^{q'}) V^{-q} + F_\\tau(U^{q'}) + V^q G_\\tau(U^{q'})\n\\end{align}\nthe K-matrix of which will be computed.\n\nIt is not hard to see that $e$ is approximately central in the rotation algebra (e.g., it's easy to see that it approximately commutes with $U$ since $\\|V^q U - U V^q\\| \\to 0$ easily follows).\n\\medskip\n\nFollowing \\ccite{EL}, we will resurrect this projection as a C*-inner product from a Rieffel equivalence bimodule framework (see equation \\eqref{einnerproduct}). This can certainly be done for consecutive pairs of convergents $\\tfrac{p}{q} < \\theta < \\tfrac{p'}{q'},$ where $p'q - pq' = 1$.\n\n\\medskip\n\n\\begin{rmk}\\label{tracecondition} It is known that there are infinitely many pairs of consecutive rational convergents $\\tfrac{p}{q} < \\theta < \\tfrac{p'}{q'}$ such that $q'(q\\theta-p)$ is bounded away from 0 and 1. For example, by Lemma 3 of Elliott and Evans \\ccite{EE}, for each irrational $\\theta$ one can show that $\\tfrac15 < q'(q\\theta-p) < \\tfrac45$ is satisfied for infinitely many such convergent pairs. Therefore, there are infinitely many pairs satisfying one of the inequalities\n\\[\n\\tfrac15 < q'(q\\theta-p) < \\tfrac12  \\qquad \\text{or} \\qquad \\tfrac12 < q'(q\\theta-p) < \\tfrac45.\n\\]\nThere is no loss of generality in assuming that the irrational $\\theta$ conforms to the latter of these conditions, as we have stipulated in inequality \\eqref{qalpha} below.  One can reduce the latter case to the former case as follows. Let's suppose that $\\tfrac15 < q'(q\\theta-p) < \\tfrac12$ for infinitely many convergent pairs. One easily converts this to the former case by looking at the corresponding AC Powers-Rieffel projection of trace given by the complementary quantity\n\\[\n\\tfrac12 \\ < \\ 1- q'(q\\theta-p) = q(p' - q'\\theta) = q( q'[1-\\theta] - (q'-p')) \\ < \\ \\tfrac45. \n\\]\n\\end{rmk}\n\n\nThe following lemma shows how the morphism $\\zeta$ relates the unbounded traces of $A_\\theta$ and $A_\\tau$ in order to compute the topological invariants of the projection $e$ give by \\eqref{rieffelAC}.\n\n\\begin{lem}\\label{phizetas}\nWith $\\zeta: A_\\tau \\to A_\\theta$ the morphism in \\eqref{zetamorphism}, we have\n\\[\n\\phi_{jk}^\\theta \\zeta\n= \\updelta_2^{q'} \\updelta_2^{j} \\Big[ \\phi_{0k}^\\tau + \\phi_{1k}^\\tau \\Big] \n+ \\updelta_2^{q} \\updelta_2^{k} \\Big[ \\phi_{j0}^\\tau + (-1)^j\\phi_{j1}^\\tau\\Big] \n+ (-1)^{pjk} \\updelta_2^{q'-1} \\updelta_2^{q-1} \\phi_{jk}^\\tau\n\\]\nwhere $\\tau = q'(q\\theta-p)$ and $\\phi_{jk}^\\tau$ are the unbounded $\\Phi$-traces for $A_\\tau$.\n\\end{lem}\n\n\\begin{proof} We have \n\\begin{align*}\n\\phi_{jk}^\\theta \\zeta (U_{\\tau}^m V_{\\tau}^n) \n&= \\phi_{jk}^\\theta (U_\\theta^{q'm} V_\\theta^{qn})\n= e(-\\tfrac{\\theta}2 q'q mn)\\,\\updelta_2^{q'm-j} \\updelta_2^{qn-k}\n\\\\\n&= e(-\\tfrac{1}2 q'[q\\theta - p + p] mn)\\,\\updelta_2^{q'm-j} \\updelta_2^{qn-k}\n\\\\\n&= e(-\\tfrac{\\tau}2 mn) (-1)^{q'pmn} \\,\\updelta_2^{q'm-j} \\updelta_2^{qn-k}\n\\\\\n&= e(-\\tfrac{\\tau}2 mn) (-1)^{jpn} \\,\\updelta_2^{q'm-j} \\updelta_2^{qn-k}\n\\end{align*}\n(since $q'm$ in $(-1)^{q'pmn}$ can be replaced by $j,$ in view of the delta function). Using the identity $\\updelta_2^{am-b} = \\updelta_2^{a} \\updelta_2^{b} + \\updelta_2^{a-1}\\updelta_2^{m-b},$ we have\n\\[\n\\phi_{jk} \\zeta (U_{\\tau}^m V_{\\tau}^n) = \ne(-\\tfrac{\\tau}2 mn) (-1)^{jpn} \n\\Big( \\updelta_2^{q'} \\updelta_2^{j} + \\updelta_2^{q'-1}\\updelta_2^{m-j} \\Big)\n \\Big( \\updelta_2^{q} \\updelta_2^{k} + \\updelta_2^{q-1}\\updelta_2^{n-k} \\Big)\n\\]\n\\[\n= e(-\\tfrac{\\tau}2 mn) (-1)^{jpn}\n\\Big(\n\\updelta_2^{q'} \\updelta_2^{j}\\updelta_2^{q} \\updelta_2^{k} \n+ \\updelta_2^{q'} \\updelta_2^{j} \\updelta_2^{q-1}\\updelta_2^{n-k} +\n\\updelta_2^{q'-1}\\updelta_2^{m-j}\\updelta_2^{q} \\updelta_2^{k} + \\updelta_2^{q'-1}\\updelta_2^{m-j}\\updelta_2^{q-1}\\updelta_2^{n-k} \n\\Big).\n\\]\nThe first term $\\updelta_2^{q'} \\updelta_2^{q} = 0$ vanishes as $q,q'$ are coprime. Also, $\\updelta_2^{q'} \\updelta_2^{q-1} = \\updelta_2^{q'}$ (since if $q'$ is even, $q$ has to be odd, and if $q'$ is odd both vanish), and similarly $\\updelta_2^{q'-1} \\updelta_2^{q} = \\updelta_2^{q}$. Thus we get\n\\begin{align*}\n\\phi_{jk} \\zeta (U_{\\tau}^m V_{\\tau}^n) \n&= e(-\\tfrac{\\tau}2 mn) (-1)^{jpn}\n\\Big(\n \\updelta_2^{q'} \\updelta_2^{j} \\updelta_2^{n-k} +\n\\updelta_2^{q} \\updelta_2^{m-j} \\updelta_2^{k} + \n\\updelta_2^{q'-1} \\updelta_2^{q-1} \\updelta_2^{m-j}\\updelta_2^{n-k} \n\\Big)\n\\\\\n&= e(-\\tfrac{\\tau}2 mn) \n\\Big[\n(-1)^{jpn} \\updelta_2^{q'} \\updelta_2^{j} \\updelta_2^{n-k} +\n(-1)^{jpn}\\updelta_2^{q} \\updelta_2^{m-j} \\updelta_2^{k} + \n(-1)^{jpn} \\updelta_2^{q'-1} \\updelta_2^{q-1} \\updelta_2^{m-j}\\updelta_2^{n-k} \n\\Big]\n\\\\\n&= e(-\\tfrac{\\tau}2 mn) \n\\Big[\n\\updelta_2^{q'} \\updelta_2^{j} \\updelta_2^{n-k} +\n(-1)^{jn}\\updelta_2^{q} \\updelta_2^{m-j} \\updelta_2^{k} + \n(-1)^{pjk} \\updelta_2^{q'-1} \\updelta_2^{q-1} \\updelta_2^{m-j}\\updelta_2^{n-k} \n\\Big]\n\\end{align*}\n(where the middle sign holds since if $q$ is even, $p$ is odd)\n\\[\n= \\updelta_2^{q'} \\updelta_2^{j} e(-\\tfrac{\\tau}2 mn) \\updelta_2^{n-k} +\n\\updelta_2^{q} \\updelta_2^{k} e(-\\tfrac{\\tau}2 mn) (-1)^{jn} \\updelta_2^{m-j}  + \n(-1)^{pjk} \\updelta_2^{q'-1} \\updelta_2^{q-1} e(-\\tfrac{\\tau}2 mn) \\updelta_2^{m-j}\\updelta_2^{n-k} \n\\]\n\\[\n\\ \\ = \\updelta_2^{q'} \\updelta_2^{j} \\Big[ \\phi_{0k}^\\tau + \\phi_{1k}^\\tau\\Big](U_{\\tau}^m V_{\\tau}^n)\n+ \\updelta_2^{q} \\updelta_2^{k} \\Big[ \\phi_{j0}^\\tau + (-1)^j \\phi_{j1}^\\tau\\Big](U_{\\tau}^m V_{\\tau}^n)  + \n(-1)^{pjk} \\updelta_2^{q'-1} \\updelta_2^{q-1} \\phi_{jk}^\\tau(U_{\\tau}^m V_{\\tau}^n)\n\\]\ntherefore we get\n\\[\n\\phi_{jk} \\zeta\n= \\updelta_2^{q'} \\updelta_2^{j} \\Big[ \\phi_{0k}^\\tau + \\phi_{1k}^\\tau \\Big] \n+ \\updelta_2^{q} \\updelta_2^{k} \\Big[ \\phi_{j0}^\\tau + (-1)^j\\phi_{j1}^\\tau\\Big] \n+ (-1)^{pjk} \\updelta_2^{q'-1} \\updelta_2^{q-1} \\phi_{jk}^\\tau\n\\]\nas claimed.\n\\end{proof}\n\n\nIn Appendix A (Section 8) we calculated the unbounded traces of the field $\\mathcal E(t)$ to be\n\\[\n\\phi_{00}(\\mathcal E) = \\phi_{01}(\\mathcal E) = \\phi_{10}(\\mathcal E) = \\phi_{11}(\\mathcal E) =  \\tfrac12.\n\\]\nCombined with Lemma \\ref{phizetas}, we obtain the unbounded traces of our approximately central projection $e = \\zeta \\mathcal E(\\tau)$ to be \n\\[\n\\phi_{jk}^\\theta(e) = \\phi_{jk}^\\tau \\zeta(\\mathcal E(\\tau)) \n= \\updelta_2^{q'} \\updelta_2^{j}  \n+ \\tfrac12 \\updelta_2^{q} \\updelta_2^{k} \\Big[ 1 + (-1)^j  \\Big] \n+ \\tfrac12 (-1)^{pjk} \\updelta_2^{q'-1} \\updelta_2^{q-1} \n\\]\nor\n\\begin{equation}\\label{phie}\n\\phi_{jk}(e) \n= \\updelta_2^{q'} \\updelta_2^{j}  \n+ \\updelta_2^{q} \\updelta_2^{k} \\updelta_2^{j} \n+ \\tfrac12 (-1)^{pjk} \\updelta_2^{q'-1} \\updelta_2^{q-1}.\n\\end{equation}\nWritten out, we have\n\\begin{align}\\label{etraces}\n\\phi_{00}(e) &= \\updelta_2^{q'} + \\updelta_2^{q} + \\tfrac12  \\updelta_2^{q'-1} \\updelta_2^{q-1}, & \\phi_{01}(e) &= \\updelta_2^{q'}  + \\tfrac12 \\updelta_2^{q'-1} \\updelta_2^{q-1},\t\n\\\\\n\\phi_{10}(e) &= \\tfrac12 \\updelta_2^{q'-1} \\updelta_2^{q-1}, \n& \n\\phi_{11}(e) &= \\tfrac12 (-1)^{p} \\updelta_2^{q'-1} \\updelta_2^{q-1}.\t\\notag\n\\end{align}\n\n\n\n\n\\textcolor{blue}{\\Large \\section{The Projection as Rieffel C*-Inner Product}}\n\n\nIn this section our goal is to express the Powers-Rieffel projection $e$ in \\eqref{ezetaE} as a C*-inner product by applying Rieffel's equivalence bimodule theorem \\ccite{MRb}. Doing so will help facilitate the interaction that $e$ has with any projection $P$ so that the topological invariants of the cutdown $\\chi(ePe)$ can be calculated.\n\n\\medskip\n\nFirst, however, we give a quick summary of Rieffel's bimodule background along with needed notation.\n\n\\medskip\n\nLet $G=M\\times \\widehat M$ where $\\widehat M$ is the Pontryagin dual group of characters on the locally compact Abelian group $M,$ and $\\frak h$ the Heisenberg cocycle on $G$ given by \n\\[\n\\frak h((m,s),(m',s')) = \\inner{m}{s'}{}\n\\]\nfor $m,m'\\in M$ and $s,s'\\in \\widehat M$. The Heisenberg projective unitary representation $\\pi: G \\to \\mathcal L(L^2(M))$ is given by phase multiplication and translation\n\\[\n[\\pi_{(m,s)}f](n) = \\langle n,s\\rangle f(n+m)\n\\]\nfor $f\\in L^2(M)$, where $M$ is equipped with its Haar-Plancheral measure (which is unique up to positive scalar multiples). It is projective (with respect to $\\frak h$) in the sense that \n\\begin{equation}\\label{projective}\n\\pi_x \\pi_y = \\frak h(x,y) \\pi_{x+y}, \\qquad \\pi_x^* = \\frak h(x,x)\\pi_{-x}\n\\end{equation}\nfor $x,y\\in G$. We let $S(M)$ denote Schwartz space of $M$.\n\\medskip\n\nIf $D$ is a discrete lattice subgroup of $G$ (i.e. cocompact), it has the associated twisted group C*-algebra $C^*(D,\\frak h)$ of the bounded operators on $L^2(M)$ generated by the unitaries $\\pi_x$ for $x\\in D$. It is the universal C*-algebra generated by unitaries $\\{\\pi_x: x\\in D\\}$ satisfying the projective commutation relations \\eqref{projective}. From the latter relation we have\n\\begin{equation}\\label{pixpiy}\n\\pi_x \\pi_y = \\frak h(x,y)  \\conj{\\frak h(y,x)} \\pi_y \\pi_x\n\\end{equation}\nfor $x,y\\in D$.  Doing the same for the complementary lattice\n\\[\nD^\\perp = \\{y\\in G: \\frak h(x,y)  \\conj{\\frak h(y,x)}  = 1, \\forall x\\in D\\}\n\\]\none obtains the C*-algebra $C^*(D^\\perp,\\conj{\\frak h})$ generated by the unitaries $\\pi_y^*$ for $y\\in D^\\perp$ (which also satisfy the preceding commutation relation with $\\conj{\\frak h}$ in place of $\\frak h$).\n\n\\medskip\n\nRieffel's theorem states that the Schwartz space $S(M)$ can be completed to an equivalence (or imprimitivity) $C^*(D,\\frak h)$-$C^*(D^\\perp,\\conj{\\frak h})$ bimodule, making these algebras strongly Morita equivalent. \n\n\\medskip\n\nOn the C*-algebras $C^*(D,\\frak h)$ and $C^*(D^\\perp,\\conj{\\frak h})$ there are canonical Flip automorphisms defined, respectively, by\n\\[\n\\Phi(\\pi_x) = \\pi_{-x}, \\qquad \\Phi'(\\pi_y) = \\pi_{-y}\n\\]\nfor $x\\in D,\\, y\\in D^\\perp$. These can easily be shown to be multiplicative with respect to the C*-inner products in the sense that \n\\begin{equation}\\label{flipinners}\n\\Phi \\Dinner{f}{g} = \\Dinner{\\tilde f}{\\tilde g}, \\qquad \n\\Phi' \\Dpinner{f}{g} = \\Dpinner{\\tilde f}{\\tilde g}\n\\end{equation}\nwhere $\\tilde f(t) = f(-t)$ and for $f,g \\in S(M)$. It is also easy to see that for the left and right module actions one has\n\\[\n\\widetilde{af}=  \\Phi(a)\\tilde f, \\qquad \\widetilde{hb}=  \\tilde h \\Phi'(b)\n\\]\nfor $a\\in C^*(D,\\frak h)$ and $b\\in C^*(D^\\perp,\\conj{\\frak h})$. The are easy to check by taking, for the first equation, $a = \\pi_x, \\ x\\in D$ (and likewise for the second).\n\n\n\\medskip\n\nWe now apply this construction to the locally compact Abelian group $M=\\Bbb R \\times \\Bbb Z_q \\times \\Bbb Z_{q'}$ and lattice subgroup\n\\[\nD = \\mathbb Z\\varepsilon_1 + \\mathbb Z\\varepsilon_2\n\\]\nof $G=M \\times M$ generated by the basis vectors\n\\begin{equation}\n\\begin{aligned} \n\\varepsilon_1 &= (\\tfrac\\alpha{q}, p, 0 ; \\ 0, 0, 0) \\\\ \n\\varepsilon_2 &= (0, \\ 0, 1 ; \\ 1, 1, 0) \n\\end{aligned}\n\\end{equation}\nwhere $\\alpha = q\\theta - p$. From $\\frak h(\\varepsilon_1,\\varepsilon_2) = e(\\frac\\alpha{q} + \\frac{p}q) = e(\\theta)$, we have associated unitaries generating the irrational rotation algebra:\n\\[\nV = \\pi_{\\varepsilon_1},\\qquad U = \\pi_{\\varepsilon_2}, \\qquad VU = e(\\theta)UV\n\\]\n(in view of \\eqref{pixpiy}) so that the twisted group C*-algebra $C^*(D,\\frak h) \\cong A_\\theta$, generated by $\\pi_{\\varepsilon_1},\\pi_{\\varepsilon_2}$, is just the irrational rotation algebra. The Flip $\\Phi,$ as defined above on the unitaries $\\pi_x$ agrees with that originally defined: $\\Phi(U) = U^{-1}, \\,\\Phi(V) = V^{-1}$.\n\\medskip\n\nRecall that the measure of each element of $\\Bbb Z_q$ is $1\/\\sqrt q$, so that its total measure is $\\sqrt q$. Since a  fundamental domain of the lattice $D$ in $G$ is\n\\[\n[0,\\tfrac\\alpha{q})\\times \\Bbb Z_q \\times \\Bbb Z_{q'} \\times [0,1) \\times \\Bbb Z_q \\times \\Bbb Z_{q'}\n\\]\nwe obtain the covolume of $D$ in $G$ as the product of measures of each component\n\\[\n|G\/D| = \\frac\\alpha{q} \\cdot \\sqrt{q} \\cdot \\sqrt{q'} \\cdot 1 \\cdot \\sqrt{q} \\cdot \\sqrt{q'} = q'\\alpha = q'(q\\theta-p) =: \\tau\n\\]\nwhich will be the trace of the Powers-Rieffel projection $e$.  \n\nA straightforward computation gives the complementary lattice of $D$ as\n\\[\nD^\\perp = \\mathbb Z\\updelta_1 + \\mathbb Z\\updelta_2 + \\mathbb Z\\updelta_3 \n\\]\nwith basis vectors\n\\begin{equation}\n\\begin{aligned} \n\\updelta_1 &= (\\tfrac1{qq'}, p, 0 ;\\ 0, 0, p') \\\\\n\\updelta_2 &= (0, \\ 0, 0 ;\\ \\tfrac1\\alpha, q', 0) \\\\\n\\updelta_3 &= (0, 0, 1 ;\\  0, 0, 0) \\\\\n\\end{aligned}\n\\end{equation}\nas readily checked. (Note that $\\pi_{\\updelta_j}^* = \\pi_{-\\updelta_j}$ since $\\pi_x^* = \\frak h(x,x)\\pi_{-x}$ as in our case $\\frak h(\\updelta_j, \\updelta_j) = 1$.) We have associated unitaries \n\\[\nV_1 = \\pi_{-\\updelta_1}, \\qquad V_2 = \\pi_{-\\updelta_2}, \\qquad \nV_3 = \\pi_{-\\updelta_3}\n\\]\nsatisfying the commutation relations\n\\begin{equation}\\label{TheVs}\nV_1V_2 = e(\\theta') V_2V_1, \\qquad V_3V_1 = e(\\tfrac{p'}{q'})V_1V_3, \n\\qquad V_2V_3=V_3V_2, \\qquad V_3^{q'} = 1.\n\\end{equation}\nThey generate the C*-algebra $C^*(D^\\perp,\\conj{\\frak h})$ isomorphic to a $q'\\times q'$ matrix algebra over some irrational rotation algebra. The Flip $\\Phi'(\\pi_y) = \\pi_{-y}$ on this algebra is easily seen to be given by\n\\[\n\\Phi'(V_1) = V_1^{-1}, \\qquad \\Phi'(V_2) = V_2^{-1}, \\qquad \\Phi'(V_3) = V_3^{-1}.\n\\]\nThe parameter $\\theta'$ in \\eqref{TheVs} is calculated using \\eqref{pixpiy} \n\\[\ne(\\theta') = \\pi_{-\\updelta_1} \\pi_{-\\updelta_2} \\pi_{-\\updelta_1}^*\\pi_{-\\updelta_2}^*\n= \\frak h(\\updelta_1, \\updelta_2)  \\conj{\\frak h(\\updelta_2, \\updelta_1) }\n\\]\ngiving us (modulo the integers)\n\\[\n\\theta' := \\frac1{qq'\\alpha} +\\frac{pq'}q = \\frac1{qq'\\alpha} +\\frac{p'q-1}q \n\\ \\equiv_{\\mathbb Z} \\ \n\\frac1{qq'\\alpha} - \\frac{1}q\n=\\frac{1-q'\\alpha}{qq'\\alpha}\n=\\frac{q\\alpha'}{qq'\\alpha}\n=\\frac{\\alpha'}{q'\\alpha}\n\\]\nsince $p'q - pq' = 1$ and $q'\\alpha + q\\alpha' = 1$, where\n\\[\n\\alpha' = p'-q'\\theta, \\qquad  \\alpha = q\\theta - p.\n\\]\n\n\\bigskip\n\nWe now consider the function (as in \\ccite{EL})  \n\\[\n f(t,r,s) = c \\delta_q^r \\delta_{q'}^s \\sqrt{f_0(t)}, \\qquad c^2 = \\frac{\\sqrt{qq'}}\\alpha\n\\] \nwhere $c$ is a normalizing constant, $f_0$ is continuous and supported on the interval \n$[-\\frac1{2q'},\\frac1{2q'}]$, and $f_1 = 1$ on $[\\frac1{2q'} - \\alpha, \\alpha-\\frac1{2q'}]$, and \n\\[\nf_0(t-\\alpha) = 1 - f_0(t) \\qquad  \\text{for } \\ \\alpha - \\frac1{2q'} \\le t \\le \\frac1{2q'}\n\\]\nas shown in Figure \\ref{fig}. \n\\vskip-10pt\n\n\\begin{figure}[H]\n\\includegraphics[width=2.5in,height=1.5in]\n{f0g0plot.pdf} \n\\caption{\\Small{Graphs of $f_0, g_0$.}}\\label{fig}\n\\end{figure}\n\nAccording to our Standing Condition 1.3 (in the Introduction), we have\n\\begin{equation}\\label{qalpha}\nq\\alpha' < \\frac12 < q'\\alpha = \\tau.\n\\end{equation}\nIn terms of the function $f_\\tau$ defined in Section 2.2, we could in fact take\n\\begin{equation}\\label{fzero}\nf_0(t) = f_\\tau(q't)\n\\end{equation}\nwhere $\\tau = q'\\alpha$. \n\n\n\\bigskip\n\n\\subsection{Computation of $\\Dpinner{f}{f}$ }\n\nRecall that the $D^\\perp$-inner product of $f$ in our setup is\n\\[\n\\Dpinner{f}{f} =  \n\\rmsumop_{n_1, n_2} \\ \\rmsumop_{n_3=0}^{q'-1} \n\\Dpinner{f}{f}(n_1\\delta_1 + n_2\\delta_2 + n_3\\delta_3) \n\\pi_{n_1\\updelta_1 + n_2\\updelta_2 + n_3\\updelta_3}^*\n\\]\nwhere the coefficients will be worked out soon. First, let's find $\\pi_{n_1\\updelta_1 + n_2\\updelta_2 + n_3\\updelta_3}^*$.\nFrom $\\pi_{u + v} = \\conj{\\frak h(u, v)} \\pi_{u} \\pi_{v}$, we get\n\\begin{align*}\n\\pi_{n_1\\updelta_1 + n_2\\updelta_2 + n_3\\updelta_3} \n&= \\conj{\\frak h(n_1\\updelta_1, n_2\\updelta_2 + n_3\\updelta_3)} \\pi_{n_1\\updelta_1} \\pi_{n_2\\updelta_2 + n_3\\updelta_3}\n\\\\\n&= \\conj{\\frak h(n_1\\updelta_1, n_2\\updelta_2)} \\pi_{\\updelta_1}^{n_1} \n\\conj{\\frak h(n_2\\updelta_2, n_3\\updelta_3)} \\pi_{n_2\\updelta_2} \\pi_{n_3\\updelta_3}\n\\\\\n&= e(-n_1 n_2 \\theta') \\pi_{\\updelta_1}^{n_1} \\pi_{\\updelta_2}^{n_2} \\pi_{\\updelta_3}^{n_3}\n\\end{align*}\nso\n\\[\n\\pi_{n_1\\updelta_1 + n_2\\updelta_2 + n_3\\updelta_3}^*\n= e(n_1 n_2 \\theta') \n\\pi_{\\updelta_3}^{-n_3} \\pi_{\\updelta_2}^{-n_2} \\pi_{\\updelta_1}^{-n_1} \n= e(n_1 n_2 \\theta') V_3^{n_3} V_2^{n_2} V_1^{n_1} \n\\]\nso the inner product becomes\n\\[\n\\Dpinner{f}{f} =  \n\\rmsumop_{n_1, n_2} \\ \\rmsumop_{n_3=0}^{q'-1} \n\\Dpinner{f}{f}(n_1\\delta_1 + n_2\\delta_2 + n_3\\delta_3) \\cdot\ne(n_1 n_2 \\theta') V_3^{n_3} V_2^{n_2} V_1^{n_1} \n\\]\nwhere the coefficients can be worked out as follows:\n\\begin{align*}\n\\Dpinner{f}{f}&(n_1\\delta_1 + n_2\\delta_2 + n_3\\delta_3) \n= \\Dpinner{f}{f}(\\tfrac{n_1}{qq'}, n_1p, n_3 ; \\ \\ \\tfrac{n_2}{\\alpha}, n_2 q', n_1 p')\n\\\\\n&= \\frac1{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(\\tfrac{n_2q'r}q + \\tfrac{n_1p's}{q'}) \n\\rmintop_{\\Bbb R} \\conj{f(t,r,s)} f(t+\\tfrac{n_1}{qq'}, r+n_1p, s+n_3) e(t\\tfrac{n_2}{\\alpha})  dt\n\\\\\n&= \\frac{c^2}{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(\\tfrac{n_2q'r}q + \\tfrac{n_1p's}{q'}) \n\\rmintop_{\\mathbb R} \\delta_q^r \\delta_{q'}^s \\sqrt{f_0(t)}\n\\delta_q^{r+n_1p} \\delta_{q'}^{s+n_3} \\sqrt{f_0(t+\\tfrac{n_1}{qq'})}\n  e(t\\tfrac{n_2}{\\alpha})  dt\n\\\\\n&= \\frac1{\\alpha} \\delta_q^{n_1} \\delta_{q'}^{n_3}\n\\rmintop_{\\mathbb R} \\sqrt{f_0(t) f_0(t+\\tfrac{n_1}{qq'})} e(t\\tfrac{n_2}{\\alpha})  dt.\n\\end{align*}\nFrom \\eqref{fzero} we put $f_0(t) = f_\\tau(x)$ where $x = q't$ to get \n\\[\n\\Dpinner{f}{f}(n_1\\delta_1 + n_2\\delta_2 + n_3\\delta_3)  \n= \\frac1{\\tau} \\delta_q^{n_1} \\delta_{q'}^{n_3}\n\\rmintop_{\\mathbb R} \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{n_1}{q})} e(x\\tfrac{n_2}{\\tau})  dx\n\\]\n(as $\\tau = q'\\alpha$). This gives\n\\[\n\\Dpinner{f}{f} =  \n\\frac1{\\tau} \\rmsumop_{n_1, n_2} \\ \\rmsumop_{n_3=0}^{q'-1} \n \\delta_q^{n_1} \\delta_{q'}^{n_3}\n\\rmintop_{\\mathbb R} \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{n_1}{q})} e(x\\tfrac{n_2}{\\tau})  dx\n \\cdot e(n_1 n_2 \\theta') V_3^{n_3} V_2^{n_2} V_1^{n_1} \n\\]\nsetting $n_1 = qk$, $n_3 = 0$ (and writing $n_2=m$),\n\\[\n= \\frac1{\\tau} \\rmsumop_{k, m} \n\\rmintop_{\\mathbb R} \\sqrt{f_\\tau(x) f_\\tau(x+k)} e(x\\tfrac{m}{\\tau})  dx\n \\cdot e(qk m \\theta')  V_2^{m} V_1^{qk} \n= \\frac1{\\tau} \\rmsumop_{m} \\rmintop_{\\mathbb R} f_\\tau(x)  e(x\\tfrac{m}{\\tau})  dx \\cdot  V_2^{m}\n\\]\nsince the integrand here vanishes for $k\\not=0$. The latter integral can be calculated as follows (cf. Figure \\ref{figfg} with $t=\\tau$)\n\\begin{align*}\n\\rmintop_{\\mathbb R} f_\\tau(x)  e(x\\tfrac{m}{\\tau})  dx \n&= \\rmintop_{-\\tfrac12}^{\\tfrac12-\\tau} f_\\tau(x)  e(x\\tfrac{m}{\\tau})  dx \n+ \\rmintop_{\\tfrac12-\\tau}^{\\tau-\\tfrac12} f_\\tau(x)  e(x\\tfrac{m}{\\tau})  dx \n+ \\rmintop_{\\tau-\\tfrac12}^{\\tfrac12} f_\\tau(x)  e(x\\tfrac{m}{\\tau})  dx \n\\\\\n&= \\rmintop_{-\\tfrac12}^{\\tfrac12-\\tau}  f_\\tau(x)  e(x\\tfrac{m}{\\tau})  dx \n+ \\rmintop_{\\tfrac12-\\tau}^{\\tau-\\tfrac12} e(x\\tfrac{m}{\\tau})  dx \n+ \\rmintop_{-\\tfrac12}^{\\tfrac12-\\tau} f_\\tau(x+\\tau)  e((x+\\tau)\\tfrac{m}{\\tau})  dx \n\\end{align*}\nby making the change of variable $x\\to x+\\tau$ in the third integral. From $f_\\tau(x) + f_\\tau(x+\\tau) = 1$ for $-\\tfrac12 \\le x \\le \\tfrac12-\\tau,$ we get\n\\[\n\\rmintop_{\\mathbb R} f_\\tau(x)  e(x\\tfrac{m}{\\tau})  dx\n= \\rmintop_{-\\tfrac12}^{\\tfrac12-\\tau}e(x\\tfrac{m}{\\tau})  dx \n+ \\rmintop_{\\tfrac12-\\tau}^{\\tau-\\tfrac12}  e(x\\tfrac{m}{\\tau})  dx \n= \\rmintop_{-\\tfrac12}^{\\tau-\\tfrac12}e(x\\tfrac{m}{\\tau})  dx \n= \\tau \\updelta_{m,0}.\n\\]\nTherefore the $D^\\perp$-inner product is $\\Dpinner{f}{f} =  \\rmsumop_{m}  \\updelta_{m,0} \\cdot  V_2^{m} = 1.$  This means that the $D$-inner product\n\\begin{equation}\\label{einnerproduct}\n \\Dinner{f}{f} = e\n\\end{equation}\nis a projection, which we now proceed to calculate and show to be equal to the Powers-Rieffel projection $e$. \n\n\n\\subsection{Computation of $\\Dinner{f}{f}$ }\n\nRecall that the $D$-inner product of Schwartz functions $f, g$ on $M$ is given by\n\\[\n\\inner{f}{g}{D} = |G\/D| \\rmsumop_{w\\in D} \\inner{f}{g}{D}(w) \\pi_w = \n\\tau \\rmsumop_{m,n} \\inner{f}{g}{D}(m\\varepsilon_1+n\\varepsilon_2) \\  U^n V^m\n\\]\nsince $|G\/D| = \\tau$ and \n\\[\n\\pi_w = \\pi_{m\\varepsilon_1+n\\varepsilon_2} = \n\\conj{\\frak h(m\\varepsilon_1, n\\varepsilon_2)} \\pi_{m\\varepsilon_1} \\pi_{n\\varepsilon_2} \n= e(-mn\\theta)V^m U^n = U^n V^m.\n\\]\nThe coefficients are \n\\begin{align*}\n\\Dinner{f}{f}(m\\varepsilon_1+n\\varepsilon_2) \n&= \\Dinner{f}{f}(\\tfrac{m\\alpha}q, mp, n; \\ n, n,0) \n\\\\ \n&= \\rmintop_M f(t,r,s) \\conj{f((t,r,s)+(\\tfrac{m\\alpha}q, mp, n))}\\cdot \n\\conj{\\langle (t,r,s),(n,n,0)\\rangle} dt dr ds\n\\\\\n&= \\rmintop_{\\Bbb R \\times \\Bbb Z_q \\times \\Bbb Z_{q'}} \nf(t,r,s) \\conj{f(t+\\tfrac{m\\alpha}q, r+mp, s+n)} e(-tn) e(-\\tfrac{rn}q) dt \\tfrac1{\\sqrt q}  \\tfrac1{\\sqrt {q'}}\n\\\\\n&= \\frac1{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(-\\tfrac{rn}q) \\rmintop_{\\Bbb R} f(t,r,s) \\conj{f(t+\\tfrac{m\\alpha}q, r+mp, s+n)} e(-tn) dt\n\\\\\n&= \\frac{c^2}{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(-\\tfrac{rn}q) \\rmintop_{\\mathbb R} \\delta_q^r \\delta_{q'}^s  \n\\delta_q^{r+mp} \\delta_{q'}^{s+n} \\sqrt{f_0(t) f_0(t+\\tfrac{m\\alpha}q)} e(-tn) dt\n\\\\\n&= \\frac1{\\alpha} \\delta_q^{m} \\delta_{q'}^{n}\n \\rmintop_{\\mathbb R}  \\sqrt{f_0(t) f_0(t+\\tfrac{m\\alpha}q)} e(-tn) dt\n\\end{align*}\nwhich, again using the change of variable $x=q't$ and $f_0(t) = f_\\tau(x),$ gives\n\\[\n\\Dinner{f}{f}(m\\varepsilon_1+n\\varepsilon_2) \n = \\frac1{\\tau} \\delta_q^{m} \\delta_{q'}^{n}\n \\rmintop_{\\mathbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+m\\tfrac{\\tau}q)} \\ e(-\\tfrac{nx}{q'}) dx.\n\\]\n\n\nThe inner product becomes\n\\begin{align*}\n\\inner{f}{f}{D} &=  \\tau\n\\rmsumop_{m,n} \\inner{f}{f}{D}(m\\varepsilon_1+n\\varepsilon_2) \\  U^n V^m\n= \\rmsumop_{m,n} \\delta_q^{m} \\delta_{q'}^{n}\n  \\rmintop_{\\mathbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+m\\tfrac{\\tau}q)} e(-\\tfrac{nx}{q'}) dx\n\\  U^n V^m\n\\\\\n&=  \\rmsumop_{k,\\ell} \n  \\rmintop_{\\mathbb R}\\sqrt{f_\\tau(x) f_\\tau(x+k\\tau)} e(-\\ell x) dx\n\\  U^{q'\\ell} V^{qk}.\n\\end{align*}\n(by setting $m=qk$ and $n=q'\\ell$). It is easy to see that from condition $\\frac1{2} < \\tau$ (as in \\eqref{qalpha}) the integrand here vanishes for $|k| \\ge 2$, so the sum over $k$ is concentrated at $k=-1,0,1$, hence the inner product can be written  \n\\begin{align*}\n\\inner{f}{f}{D} &=\n \\rmsumop_{\\ell}  \\rmintop_{\\mathbb R} \\sqrt{f_\\tau(x) f_\\tau(x-\\tau)} e(-\\ell x) dx\n\\  U^{q'\\ell} V^{-q}\n+  \\rmsumop_{\\ell} \n  \\rmintop_{\\mathbb R} f_\\tau(x) e(-\\ell x) dx \\  U^{q'\\ell}\n\\\\\n&\\ \\ \\ \\ +  \\rmsumop_{\\ell}  \\rmintop_{\\mathbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+ \\tau)} e(-\\ell x) dx\n\\  U^{q'\\ell} V^{q}\n\\end{align*}\nor \n\\[\n\\inner{f}{f}{D} = \\widetilde G(U^{q'}) V^{-q} +  \\widetilde F(U^{q'}) + V^{q} \\widetilde G(U^{q'})\n\\]\nwhere\n\\[\n\\widetilde F(z) =  \n\\rmsumop_{\\ell}  \\rmintop_{\\mathbb R} f_\\tau(x) e(-\\ell x) dx \\cdot z^\\ell\n=  \\rmsumop_{\\ell}  \\ft f_\\tau(\\ell) \\cdot z^\\ell\n\\]\nand\n\\begin{equation} \\label{Gtilde}\n\\widetilde G(z) =  \\rmsumop_{\\ell}  \\rmintop_{\\mathbb R} \\sqrt{f_\\tau(x) f_\\tau(x - \\tau)} e(-\\ell x) dx \\cdot z^\\ell.\n\\end{equation}\n\n\nIn light of the Poisson Lemma \\ref{poisson}, we see that $ \\widetilde F(z) = F_\\tau(z)$ and $\\widetilde G(z) = G_\\tau(z)$ are the same periodization functions mentioned at the beginning of Section 2.2. Therefore,\n\\[\n\\inner{f}{f}{D} = G_\\tau(U^{q'}) V^{-q} +  F_\\tau(U^{q'}) + V^{q} G_\\tau(U^{q'}) \\ = \\ \n\\zeta \\mathcal E(\\tau) \\ = \\ e\n\\]\nhence the Powers-Rieffel projection $e$ in \\eqref{rieffelAC} is a C*-inner product.\n\n\\medskip\n\n\\subsection{The Morita Isomorphism}\n\n\nNow that the projection $e = \\Dinner{f}{f}$ is an inner product such that $\\Dpinner{f}{f}= 1,$ there is the associated (Morita) isomorphism \n\\begin{equation}\\label{eta}\n\\eta: eA_\\theta e \\longrightarrow C^*(D^\\perp,\\conj{\\frak h}), \n\\qquad\n\\eta(x) = \\Dpinner{f}{xf}, \n\\qquad \\eta^{-1}(y) = \\Dinner{f y}{f}\n\\end{equation}\nwhere we note (and easy to check) that $\\eta$ is a homomorphism with respect to the {\\it opposite} multiplication on $C^*(D^\\perp,\\conj{\\frak h})$. Further, since the canonical normalized traces $\\tau, \\tau'$ of $A_\\theta$ and $C^*(D^\\perp,\\conj{\\frak h})$ (respectively) are related by $\\tau \\Dinner gh = \\tau(e) \\tau' \\Dpinner hg,$ one has\n\\begin{equation}\\label{tracesinnerproducts}\n\\tau(x) = \\tau(e) \\tau'(\\eta(x))\n\\end{equation}\nfor $x\\in eA_\\theta e$. \n\nFurther, this Morita isomorphism intertwines the Flip automorphisms\n\\begin{equation}\\label{etaPhis}\n\\eta \\Phi = \\Phi' \\eta.\n\\end{equation}\nIndeed, from \\eqref{flipinners} for any two Schwartz function $h, k$ we have\n\\[\n\\eta \\Phi \\Dinner{h}{k} = \\eta \\Dinner{\\tilde h}{\\tilde k}\n= \\Dpinner{f}{\\Dinner{\\tilde h}{\\tilde k}f}\n\\]\nwhich, upon applying the Flip $\\Phi'$ (and using $\\widetilde{ah}=  \\Phi(a)\\tilde h$, recalling $\\tilde f = f$ is even), gives \n\\begin{align*}\n\\Phi' \\eta \\Phi \\Dinner{h}{k}\n&= \\Phi' \\Dpinner{f}{\\Dinner{\\tilde h}{\\tilde k}f}\n= \\Dpinner{\\tilde f}{ [ \\Dinner{\\tilde h}{\\tilde k}f ]^{\\sim} }\n\\\\\n&= \\Dpinner{\\tilde f}{ \\Phi(\\Dinner{\\tilde h}{\\tilde k}) \\cdot \\tilde f  }\n= \\Dpinner{f}{ \\Dinner{h}{k} f  }\n\\\\\n&= \\eta( \\Dinner{h}{k}).\n\\end{align*}\n\n\n\\textcolor{blue}{\\Large \\section{Cutdown Approximation}}\n\nIn this section we obtain the cutdown approximations \n\\begin{equation}\\label{cutdownapprox}\n\\eta(eVe) \\approx V_1, \\qquad \\eta(eUe) \\approx V_3\n\\end{equation}\nneeded in the next section. (Recall, $a \\approx b$ means $\\|a - b\\| \\to 0$ as $q,q'\\to\\infty$.)\n\n\\medskip\n\nSince $V_3$ is unitary of order $q'$ and $V_1$ is a unitary with full spectrum, both satisfying $V_3V_1 = e(\\tfrac{p'}{q'})V_1V_3$ (as in \\eqref{TheVs}), they generate a C*-subalgebra isomorphic to the  circle algebra $M_{q'}\\otimes C(\\mathbb T) \\cong M_{q'}(C(\\mathbb T))$ which, in view of \\eqref{cutdownapprox}, approximates the corner algebra $eA_\\theta e$. This makes $e$ a circle algebra projection.\n\n\\bigskip\n\n\nFrom $U = \\pi_{\\varepsilon_2}$, where $\\varepsilon_2 = (0, \\ 0, 1 ; \\ 1, 1, 0)$, and $f(t,r,s) = c \\delta_q^r \\delta_{q'}^s \\sqrt{f_0(t)}$, we get\n\\[\n(Uf)(t,k,\\ell) = (\\pi_{\\varepsilon_2}f)(t,k,\\ell) \n= e(t + \\tfrac{k}q) f(t,k,\\ell+1)\n= c e(t + \\tfrac{k}q) \\delta_q^{k} \\delta_{q'}^{\\ell+1} \\sqrt{f_0(t)}.\n\\]\nWe now calculate the $D^\\perp$-inner product coefficient \n\\begin{align*}\n\\Dpinner{f}{Uf}&(n_1\\delta_1 + n_2\\delta_2 + n_3\\delta_3)\n= \\Dpinner{f}{Uf}(\\tfrac{n_1}{qq'}, n_1p, n_3 ; \\ \\ \\tfrac{n_2}{\\alpha}, n_2 q', n_1 p')\n\\\\\n& = \\frac1{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(\\tfrac{n_2q'r}q + \\tfrac{n_1p's}{q'}) \n\\rmintop_{\\Bbb R} \\conj{f(t,r,s)} (Uf)(t+\\tfrac{n_1}{qq'}, r+n_1p, s+n_3) e(t\\tfrac{n_2}{\\alpha})  dt\n\\\\\n&= \\frac{c}{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(\\tfrac{n_2q'r}q + \\tfrac{n_1p's}{q'}) \n\\rmintop_{\\Bbb R} \\delta_q^r \\delta_{q'}^s \\sqrt{f_0(t)} \n(Uf)(t+\\tfrac{n_1}{qq'}, r+n_1p, s+n_3) \ne(t\\tfrac{n_2}{\\alpha})  dt\n\\\\\n&= \\frac{c}{\\sqrt{qq'}} \n\\rmintop_{\\Bbb R}  \\sqrt{f_0(t)} (Uf)(t+\\tfrac{n_1}{qq'}, n_1p, n_3) \ne(t\\tfrac{n_2}{\\alpha})  dt\n\\\\\n&= \\frac{c^2}{\\sqrt{qq'}} \n\\rmintop_{\\Bbb R}  \\sqrt{f_0(t)}\ne(t + \\tfrac{n_1}{qq'} + \\tfrac{n_1p}q) \\delta_q^{n_1p} \\delta_{q'}^{n_3+1} \\sqrt{f_0(t+\\tfrac{n_1}{qq'})}  e(t\\tfrac{n_2}{\\alpha})  dt\n\\end{align*}\nsince $n_1$ must be divisible by $q$ (if this is nonzero) we can remove $\\tfrac{n_1p}q$ \n\\[\n= \\frac{1}{\\alpha} \\delta_q^{n_1} \\delta_{q'}^{n_3+1} e(\\tfrac{n_1}{qq'}) \n\\rmintop_{\\Bbb R}  \\sqrt{f_0(t) f_0(t+\\tfrac{n_1}{qq'})}  e(t[1+\\tfrac{n_2}{\\alpha}])  dt\n\\]\nwhich, in view of $f_0(t) = f_\\tau(x)$ where $x=q't$, becomes\n\\[\n= \\frac{1}{\\tau} \\delta_q^{n_1} \\delta_{q'}^{n_3+1} e(\\tfrac{n_1}{qq'}) \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{n_1}{q})}  e(\\tfrac{x}{q'}[1+\\tfrac{n_2}{\\alpha}])  dx\n\\]\nas $\\tau = q'\\alpha$ in the first factor. Therefore, the C*-inner product now becomes (as in Section 4.1)\n\\[\n\\Dpinner{f}{Uf} =  \n\\rmsumop_{n_1, n_2} \\rmsumop_{n_3=0}^{q'-1} \n\\frac{1}{\\tau} \\delta_q^{n_1} \\delta_{q'}^{n_3+1} e(\\tfrac{n_1}{qq'}) \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{n_1}{q})}  e(\\tfrac{x}{q'}[1+\\tfrac{n_2}{\\alpha}])  dx\n\\cdot e(n_1 n_2 \\theta') \nV_3^{n_3} V_2^{n_2} V_1^{n_1}  \n\\]\n\\[\n\\ \\ \\ \\ =  \n\\frac{1}{\\tau} V_3^{-1}  \\rmsumop_{n_1, n_2}  \n\\delta_q^{n_1}  e(\\tfrac{n_1}{qq'}) \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{n_1}{q})}  e(\\tfrac{x}{q'}[1+\\tfrac{n_2}{\\alpha}])  dx \\cdot e(n_1 n_2 \\theta') V_2^{n_2} V_1^{n_1}  \n\\]\nnow put $n_1=qk$ (and write $n_2=m$)\n\\[\n=  \n\\frac{1}{\\tau} V_3^{-1}  \\rmsumop_{k, m}  \n  e(\\tfrac{k}{q'}) \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+k)}  e(\\tfrac{x}{q'}[1+\\tfrac{m}{\\alpha}])  dx\n\\cdot e(qk m \\theta') V_2^{m} V_1^{qk}.\n\\]\nSince the product $f_\\tau(x) f_\\tau(x+k)=0$ for $k\\not=0$, we get\n\\[\n\\Dpinner{f}{Uf} = V_3^{-1}  \\cdot \\frac{1}{\\tau} \\rmsumop_{m}  \n\\rmintop_{\\Bbb R}  f_\\tau(x) e(\\tfrac{x}{q'})  e(\\tfrac{mx}{\\tau})  dx\n\\cdot V_2^{m}.\n\\]\nWe now use the Poisson Lemma \\ref{poisson} to evaluate the sum (where $V_2$ is now replaced by the function $e^{2\\pi it},$ for $t\\in[0,1],$ since it is a unitary with full spectrum). Making the substitution $x =\\tau y,$ we get\n\\[\nC(t) := \n\\frac{1}{\\tau} \\rmsumop_{m}  \\rmintop_{\\Bbb R}  f_\\tau(x) e(\\tfrac{x}{q'}) e(\\tfrac{mx}{\\tau}) dx \\cdot e^{2\\pi imt} \n= \\rmsumop_{m}  \\rmintop_{\\Bbb R}  f_\\tau(\\tau y)  e(\\tfrac{\\tau y}{q'}) e(my) dy \\cdot e^{2\\pi imt}\n\\]\nor, letting $h(y) = f_\\tau(\\tau y) e(\\tfrac{\\tau y}{q'}),$ becomes\n\\[\nC(t) =  \\rmsumop_{m}  \\rmintop_{\\Bbb R}  h(y) e(my) dy \\cdot e^{2\\pi imt}\n= \\rmsumop_{m}  \\ft h(-m)\\, e^{2\\pi imt}\n= \\rmsumop_{m}  \\ft h(m)\\, e^{-2\\pi imt}.\n\\]\nWe will show that $C(t)$ converges uniformly in $t$ to 1 for large $q,q',$ where we can take $0\\le t \\le1$. By Lemma \\ref{poisson} we have\n\\[\nC(t) = \\rmsumop_{n}  h(n - t)\n= \\rmsumop_{n}  f_\\tau(\\tau n - \\tau t) e(\\tfrac{\\tau n - \\tau t}{q'}).\n\\]\nSince $f_\\tau$ is supported on $[-\\tfrac12,\\tfrac12],$ the only $n$'s that contribute to this sum are those satisfying $\\tau |n - t| < \\tfrac12$. Since $\\tau > \\tfrac12,$ we have $\\tfrac12 |n-t| \\le \\tau |n - t| < \\tfrac12$ which gives $-1 < n-t < 1$. Adding this inequality to $0 \\le t \\le 1$ gives $-1 < n < 2$ so that $n = 0,1$. Using the fact that $f_\\tau$ is even, we obtain\n\\[\nC(t) \\ = \\ f_\\tau(\\tau t) e(\\tfrac{ - \\tau t}{q'}) + f_\\tau(\\tau - \\tau t) e(\\tfrac{\\tau  - \\tau t}{q'})\n\\ \\approx \\ f_\\tau(\\tau t) + f_\\tau(\\tau - \\tau t) \\ = \\ 1\n\\]\nuniformly in $t$ for large $q'$. The last equality here can be seen by looking at the cases $0 \\le \\tau t \\le \\tfrac12$ and $\\tfrac12 < \\tau t \\le \\tau$ separately. The former case follows from our observation in \\eqref{ftx1}, and in the latter case we have $ 0 \\le \\tau - \\tau t < \\tau - \\tfrac12$ where $f_\\tau(\\tau - \\tau t) = 1$ and $f_\\tau(\\tau t) = 0$.\n\nThis gives us the cutdown approximation\n\\[\n\\eta(eUe) =  \\Dpinner{f}{Uf} \\approx V_3^{-1}.\n\\]\n\nWe next calculate $\\Dpinner{f}{Vf}$ where $V = \\pi_{\\varepsilon_1} \n= \\pi_{(\\frac\\alpha{q}, p, 0 ; \\ 0, 0, 0)}$. We have\n\\[\n(Vf)(t,r,s) = f(t+\\tfrac\\alpha{q}, r+p, s) \n= c \\delta_q^{r+p} \\delta_{q'}^{s} \\sqrt{f_0(t+\\tfrac\\alpha{q})}.\n\\]\nThe C*-inner product coefficients are\n\\begin{align*}\n\\Dpinner{f}{Vf}&(n_1\\delta_1 + n_2\\delta_2 + n_3\\delta_3)\n\\\\\n&= \\frac1{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(\\tfrac{n_2q'r}q + \\tfrac{n_1p's}{q'}) \n\\rmintop_{\\Bbb R} \\conj{f(t,r,s)} Vf(t+\\tfrac{n_1}{qq'}, r+n_1p, s+n_3) e(t\\tfrac{n_2}{\\alpha})  dt\n\\\\\n&= \\frac{c}{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(\\tfrac{n_2q'r}q + \\tfrac{n_1p's}{q'}) \n\\rmintop_{\\Bbb R} \\delta_q^r \\delta_{q'}^s \\sqrt{f_0(t)} \\cdot Vf(t+\\tfrac{n_1}{qq'}, r+n_1p, s+n_3) e(\\tfrac{n_2 t}{\\alpha})  dt\n\\intertext{in which both $r, s$ have to be 0}\n&= \\frac{c}{\\sqrt{qq'}} \\rmintop_{\\Bbb R} \\sqrt{f_0(t)} Vf(t+\\tfrac{n_1}{qq'}, n_1p, n_3) e(\\tfrac{n_2 t}{\\alpha})  dt\n\\\\\n&= \\frac1{\\alpha} \\delta_q^{n_1+1} \\delta_{q'}^{n_3} \\rmintop_{\\Bbb R}\n\\sqrt{f_0(t) f_0(t+\\tfrac{n_1}{qq'}+\\tfrac\\alpha{q})} \\ e(\\tfrac{n_2 t}{\\alpha})  dt\n\\\\\n&= \\frac1{\\tau} \\delta_q^{n_1+1} \\delta_{q'}^{n_3} \n\\rmintop_{\\Bbb R} \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{n_1}{q}+\\tfrac{\\tau}{q})} \\ e(\\tfrac{n_2 x}{\\tau})  dx\n\\end{align*}\n(again using the substitution $x=q't$ and $f_0(t) = f_\\tau(x)$), therefore\n\\[\n\\Dpinner{f}{Vf} =  \\frac1{\\tau} \n\\rmsumop_{n_1, n_2} \\rmsumop_{n_3=0}^{q'-1} \n\\delta_q^{n_1+1} \\delta_{q'}^{n_3} \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{n_1+\\tau}{q})} \\ e(\\tfrac{n_2 x}{\\tau})  dx\n\\cdot e(n_1 n_2 \\theta')\\ V_3^{n_3} V_2^{n_2} V_1^{n_1}  \n\\]\n\\[\n=  \\frac1{\\tau} \n\\rmsumop_{n_1, n_2} \\delta_q^{n_1+1}  \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{n_1+\\tau}{q})} \\ e(\\tfrac{n_2 x}{\\tau})  dx \n  \\cdot   V_1^{n_1}  V_2^{n_2}\n\\]\nhere we put $n_1 = -1 + qn$ (and $n_2=m$)\n\\[\n=  \nV_1^{-1} \\frac1{\\tau}  \\rmsumop_{n, m} \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x- \\alpha' + n )} \\ e(\\tfrac{mx}{\\tau})  dx \n\\cdot  V_1^{nq}  V_2^{m}\n\\]\nsince $\\tfrac{1-\\tau}{q} = \\alpha'$. It is easy to see that the integrand here vanishes for $n\\not=0,1$. Thus, \n\\[\n\\Dpinner{f}{Vf} = \nV_1^{-1} \\frac1{\\tau}  \\rmsumop_{m} \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x- \\alpha')} \\ e(\\tfrac{mx}{\\tau})  dx \n\\cdot  V_2^{m}\n\\]\n\\[ \\qquad \n+ V_1^{-1} V_1^{q} \\frac1{\\tau}  \\rmsumop_{m} \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x- \\alpha' + 1 )} \\ e(\\tfrac{mx}{\\tau})  dx \n\\cdot  V_2^{m}\n\\]\nmaking the substitution $y = x\/\\tau,$\n\\[\n\\Dpinner{f}{Vf} = \nV_1^{-1}  \\rmsumop_{m} \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(\\tau y) f_\\tau(\\tau y - \\alpha')} \\ e(my)  dy \n\\cdot  V_2^{m}\n\\]\n\\[\\ \\  \\qquad \\qquad \\qquad\n+ V_1^{-1} V_1^{q}  \\rmsumop_{m} \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(\\tau y) f_\\tau(\\tau y - \\alpha' + 1 )} \\ e(my)  dy \n\\cdot  V_2^{m}\n\\]\n\\[\\ \\  \\qquad \\qquad \n= V_1^{-1}  \\rmsumop_{m} \\ft h_0(-m) \\cdot  V_2^{m} + V_1^{-1} V_1^{q}  \\rmsumop_{m} \\ft h_1(-m)  \\cdot  V_2^{m}\n\\]\nor\n\\begin{equation}\\label{innerfVf}\n\\Dpinner{f}{Vf} \n= V_1^{-1}  \\rmsumop_{m} \\ft h_0(m) \\cdot  V_2^{-m} + V_1^{-1} V_1^{q}  \\rmsumop_{m} \\ft h_1(m)  \\cdot  V_2^{-m}\n\\end{equation}\nwhere we have written\n\\[\nh_0(y) = \\sqrt{f_\\tau(\\tau y) f_\\tau(\\tau y - \\alpha')}, \\qquad\nh_1(y) = \\sqrt{f_\\tau(\\tau y) f_\\tau(\\tau y - \\alpha' + 1)}.\n\\]\nBy Lemma \\ref{poisson}, the first sum is (as done earlier)\n\\[\n\\rmsumop_{m} \\ft h_0(m) \\ e^{-2\\pi i m t} = \\rmsumop_{m} h_0(m-t) = h_0(-t) + h_0(1 - t)\n\\]\nsince $h_0(m-t)=0$ for $m\\not=0,1$ and $0\\le t \\le1$. (Note $f_\\tau(\\tau(m-t)) = 0$ for $m\\not=0,1$ since $\\tfrac12|m-t| \\le \\tau|m-t| < \\tfrac12$ gives $-1 < m - t < 1,$ and adding $0\\le t \\le1$ gives $-1 < m  < 2$ hence $m=0,1$.)  Since $\\alpha' \\to 0$ for large $q,q',$ it follows that $h_0(-t) \\to f_\\tau(\\tau t)$ and $h_0(1 - t) \\to f_\\tau(\\tau - \\tau t)$ (both uniformly) hence their sum\n\\[\nh_0(-t) + h_0(1 - t) \\ \\to \\ f_\\tau(\\tau t) + f_\\tau(\\tau - \\tau t) = 1 \n\\]\nby equation \\eqref{ftx1} for $f_\\tau$. This shows the the first term in \\eqref{innerfVf} for $\\Dpinner{f}{Vf}$ converges to $V_1^{-1}$ in norm. It remains now to check that the second term converges to 0 for large $q,q'$. For the second term we likewise have only the $m=0,1$ terms\n\\[\n\\rmsumop_{m} \\ft h_1(m) \\ e^{-2\\pi i m t} = \\rmsumop_{m} h_1(m-t) = h_1(-t) + h_1(1 - t).\n\\]\nThe fact that this converges uniformly to 0 follows from $f_\\tau(s) f_\\tau(s + 1 - \\alpha') \\to 0$ uniformly in $s$. Since $f_\\tau$ is supported on $[-\\tfrac12,\\tfrac12],$ this product is 0 unless $-\\tfrac12 \\le s \\le -\\tfrac12 + \\alpha',$ and on this interval (which shrinks to $-\\tfrac12$ as $\\alpha'\\to0$) one has $f_\\tau(s) \\to f_\\tau(-\\tfrac12) = 0$. Therefore, we have obtained the norm approximation\n\\[\n\\eta(eVe) = \\Dpinner{f}{Vf}  \\approx V_1^{-1} \n\\]\nfor large $q, q'$.\n\n\\textcolor{blue}{\\Large \\section{K-theory of Powers-Rieffel Projections}}\n\n\nIn this section we prove Theorem \\ref{MaintheoremA}.  Theorem \\ref{Kmatrixthm} is then proved from it and Lemma \\ref{phiV1V3} (which is proved in Section 7).  All norm approximations ``$\\approx$\" here are understood to hold for large enough integer parameters $q,q'$. \n\n\\medskip\n\nWe will denote by $\\Xi$ the linear *-anti-automorphism of the continuous field of rotation algebras $\\{A_t\\}$ defined by\n\\[\n\\Xi(U_t^{m} V_t^{n} ) = U_t^{n} V_t^{m}\n\\]\non the canonical unitary basis. It is a vector space linear transformation satisfying\n\\[\n\\Xi(ab) = \\Xi(b)\\Xi(a), \\qquad \\Xi(a^*) = \\Xi(a)^*\n\\]\nfor $a,b \\in A_t$. It follows that we also have $\\Xi(V_t^{r} U_t^{s} ) = V_t^{s} U_t^{r} $. (We simply write $\\Xi$ instead of $\\Xi_t$ since it will present no confusion.) Further, $\\Xi$ commutes with the Flip \n\\[\n\\Xi \\Phi = \\Phi \\Xi\n\\]\nso it leaves the Flip orbifild invariant.\n\n\nLet $P(\\theta)$ denote any continuous field of Flip-invariant smooth projections.  For example, $P$ could be any of the Powers-Rieffel projection fields forming the basis for $K_0(A_\\theta^\\Phi) = \\mathbb Z^6$ given in \\eqref{basis}. For convenience we write $P$ as a  continuous section\n\\[\nP(t) = \\rmsumop_{m,n} c_{m, n}(t) U_t^mV_t^{n}\n\\]\nof the continuous field of rotation C*-algebras $\\{A_t\\},$ where $c_{m,n}(t)$ are rapidly decreasing coefficients; and from its Flip-invariance one has $c_{-m,-n} = c_{m,n}$.\n\nFor large $q$, the cut down $e P(\\theta) e$ is close to the Flip-invariant projection\n\\[\n\\chi(e P(\\theta) e) \\ \\approx \\ e P(\\theta) e.\n\\]\nwhere $\\chi$ is the characteristic function of the interval $[\\tfrac12,\\infty]$.\n\\medskip\n\nLet $A_{p'\/q'}$ denote the rational rotation algebra generated by the unitaries $U' = U_{p'\/q'}$ and $V' = V_{p'\/q'}$ satisfying\n\\[\nV' U' = e(\\tfrac{p'}{q'}) U' V'.\n\\]\nLet $\\pi$ denote the canonical representation \n\\[\n\\pi: A_{p'\/q'} \\to C^*(V_1,V_3), \\qquad  \n\\qquad \\pi(U') = V_1,\\qquad \\pi(V') = V_3\n\\]\nwhich exists since $V_3V_1 = e(\\tfrac{p'}{q'})V_1V_3$ from \\eqref{TheVs}. \n\nWe will use the well-known result of Elliott (\\ccite{GE1984}) that all normalized traces on a rational rotation algebra agree on projections. In particular, $\\tau' \\pi$ and the canonical trace $\\tau_{p'\/q'}$ of $A_{p'\/q'}$ are equal on projections. The canonical trace of $\\chi(e P(\\theta) e)$ is therefore obtainable from the approximations $\\eta(eUe) \\approx V_3^{-1}, \\ \\eta(eVe) \\approx V_1^{-1}$, as follows. Since $P(t)$ is Flip-invariant we can write it as $P(t) = \\rmsumop_{m,n} c_{m, n}(t) U_t^{-m}V_t^{-n},$ hence for sufficiently large $q,q',$ we have \n\\begin{align*}\n\\eta(\\chi(e P(\\theta) e)) \n&\\approx \\eta(eP(\\theta) e)\n =  \\eta \\rmsumop_{m,n} c_{m,n}(\\theta) eU_\\theta^{-m}V_\\theta^{-n}e \n \\approx  \\eta \\rmsumop_{m,n} c_{m,n}(\\theta) (eU_\\theta e)^{-m} (eV_\\theta e)^{-n} \n\\\\\n&= \\rmsumop_{m,n} c_{m,n}(\\theta)  \\eta(eV_\\theta e)^{-n} \\eta (eU_\\theta e)^{-m}\n\\qquad \\text{(opposite multiplication)}\n\\\\\n&\\approx  \\rmsumop_{m,n} c_{m,n}(\\theta) V_1^{n} V_3^{m}  \n\\\\\n&= \\pi\\ \\rmsumop_{m,n} c_{m,n}(\\theta) {U'}^n {V'}^m  \n\\\\\n&\\approx \\pi \\ \\rmsumop_{m,n} c_{m,n}(\\tfrac{p'}{q'}) {U'}^n {V'}^m\n\\\\\n&\\approx \\pi \\Xi \\ \\ \\rmsumop_{m,n} c_{m,n}(\\tfrac{p'}{q'}) {U'}^m {V'}^n\n\\\\\n&= \\pi \\Xi P(\\tfrac{p'}{q'}).\n\\end{align*}\nThis shows that the projections $\\chi(e P(\\theta) e)$ and  $\\eta^{-1}\\pi \\Xi P(\\tfrac{p'}{q'})$, being close, are therefore unitarily equivalent in the Flip orbifold $A_\\theta^\\Phi,$ and in particular they have the same canonical and unbounded trace invariants. Thus  \n\\[\n\\tau'(\\eta(\\chi(e P(\\theta) e))) = \\tau'\\pi \\Xi P(\\tfrac{p'}{q'}) = \\tau_{p'\/q'}(\\Xi P(\\tfrac{p'}{q'})) \n= \\tau_{p'\/q'}(P(\\tfrac{p'}{q'})).\n\\]\nwhere the last equality holds since $\\tau_{p'\/q'} \\Xi$ is a normalized trace on the rational rotation algebra $A_{p'\/q'}$ so it agrees with $\\tau_{p'\/q'}$ on the projections.\n\nHence from \\eqref{tracesinnerproducts}, we get\n\\begin{equation}\\label{canontraceP}\n\\tau(\\chi(e P(\\theta) e)) = \\tau(e) \\tau'(\\eta(\\chi(e P(\\theta) e))) = q'(q\\theta-p) \\tau_{p'\/q'}(P(\\tfrac{p'}{q'})).\n\\end{equation}\n\n\\medskip\n\nWe now compute the unbounded traces $\\phi_{jk}$ of the cutdown projection $\\chi(e P(\\theta)e)$ (which requires more work). We have\n\\[\n\\phi_{jk}(\\chi(e P(\\theta) e)) = \\phi_{jk}( \\eta^{-1} \\pi \\Xi P(\\tfrac{p'}{q'}))\n= (\\phi_{jk} \\eta^{-1}) \\pi \\Xi P(\\tfrac{p'}{q'}).\n\\]\nHere, it is easily checked that $\\phi_{jk} \\eta^{-1}$ is a $\\Phi'$-trace when restricted to the C*-algebra generated by $V_1,V_3$ since $\\phi_{jk}$ are $\\Phi$-traces and using the intertwining relation \\eqref{etaPhis}. In Section 9 (see equations \\eqref{Phiprimetraces}) we showed that the vector space of $\\Phi'$-traces on $C^*(V_1,V_3)$ is 2-dimensional with basis the $\\Phi'$-traces \n\\[\n\\psi_{1} ( V_3^{n} V_1^m ) = e(\\tfrac{p'mn}{2q'})  \\updelta_2^{m},\n\\qquad\n\\psi_{2} ( V_3^{n} V_1^m ) = e(\\tfrac{p'mn}{2q'}) (-1)^{p'n}\\ \\updelta_2^{m-q'}.\n\\]\nTherefore, on $C^*(V_1,V_3)$ (the range of $\\pi$) there are scalars $a_{jk}^-, a_{jk}^+$ such that\n\\[\n\\phi_{jk} \\eta^{-1} = a_{jk}^- \\psi_1 + a_{jk}^+ \\psi_2.\n\\]\nWe then have\n\\[\n\\phi_{jk}(\\chi(e P(\\theta) e)) \n= a_{jk}^- \\psi_1 \\pi \\Xi P(\\tfrac{p'}{q'}) + a_{jk}^+ \\psi_2 \\pi \\Xi P(\\tfrac{p'}{q'}).\n\\]\nThe maps $\\psi_j \\pi \\Xi$ ($j=1,2$) are readily found on the basis elements as follows\n\\begin{align*}\n\\psi_1 \\pi \\Xi ({U'}^m {V'}^n) &= \\psi_1 \\pi({U'}^n {V'}^m)  = \\psi_1(V_1^n V_3^m)\n= e(-\\tfrac{p'mn}{q'}) \\psi_1(V_3^m V_1^n)\n\\\\\n&= e(-\\tfrac{2p'mn}{2q'})  e(\\tfrac{p'mn}{2q'})  \\updelta_2^{n} \n= e(-\\tfrac{p'mn}{2q'})  \\updelta_2^{n}\n\\ =  (\\phi_{00} + \\phi_{10})({U'}^m {V'}^n)\n\\end{align*}\nhence \n\\[\n\\psi_1 \\pi \\Xi = \\phi_{00}^{} + \\phi_{10}^{}\n\\]\non $A_{p'\/q'}$, where here $\\phi_{00} \\equiv \\phi_{00}^{p'\/q'}, \\phi_{10} \\equiv \\phi_{10}^{p'\/q'}$ -- two of the four basic unbounded traces on the rotation algebra, namely\n$\\phi_{ij}({U'}^m{V'}^n) = e(-\\tfrac{p'mn}{2q'})\\,\\updelta_2^{m-i} \\updelta_2^{n-j}$.\nSimilarly,\n\\begin{align*}\n\\psi_2 \\pi \\Xi ({U'}^m {V'}^n)  &= \\psi_2(V_1^n V_3^m)\n= e(-\\tfrac{p'mn}{q'}) \\psi_2(V_3^mV_1^n)\n= e(-\\tfrac{p'mn}{2q'}) (-1)^{p'm}\\ \\updelta_2^{n-q'}\n\\\\\n&= (\\phi_{0,q'} + (-1)^{p'} \\phi_{1,q'}) ({U'}^m {V'}^n)\n\\end{align*}\nsince $(-1)^{p'm} = \\updelta_2^{m} + (-1)^{p'} \\updelta_2^{m-1}$ -- where, of course, $\\phi_{0,s}$ is $\\phi_{00}$ or $\\phi_{01}$ depending on parity of $s$. Therefore, \n\\[\n\\psi_2 \\pi \\Xi =  \\phi_{0,q'} + (-1)^{p'} \\phi_{1,q'}.\n\\]\nWe have therefore obtained\n\\begin{equation}\\label{phicutdowns}\n\\phi_{jk}(\\chi(e P(\\theta) e)) = a_{jk}^- C_0(P) + a_{jk}^+ C_1(P)\n\\end{equation}\nwhere\n\\begin{equation}\\label{CP}\nC_0(P) = \\phi_{00}(P) + \\phi_{10}(P), \\qquad\nC_1(P) = \\phi_{0,q'}(P) + (-1)^{p'} \\phi_{1,q'}(P)\n\\end{equation}\nwhere $P = P(\\tfrac{p'}{q'})$ on the right sides. The invariants $\\phi_{jk}P(\\tfrac{p'}{q'})$  depend only on the field $P(\\theta)$ and do not depend specifically on $\\theta, p',q'$ - for instance, this can be seen from unbounded trace values of the basis fields listed in \\eqref{Kbasis}. Note however, how $C_1$ depends on the parities of $p'$ and $q'$ in \\eqref{CP}.\n\\medskip\n\nIt now remains to find the coefficients $a_{jk}^-, a_{jk}^+$ which depend only on the AC projection $e$. First, evaluate the equation\n\\[\n\\phi_{jk} \\eta^{-1} = a_{jk}^- \\psi_1 + a_{jk}^+ \\psi_2\n\\]\nat the identity $\\eta(e) = 1$ to get\n\\begin{equation}\\label{stare}\n\\phi_{jk} (e) = a_{jk}^-   + a_{jk}^+  \\updelta_2^{q'}.\t\t\n\\end{equation}\nNext, evaluate it at $V_3,$ where $\\psi_{1} ( V_3 ) =  1, \\ \\psi_{2} ( V_3 ) =  (-1)^{p'} \\updelta_2^{q'} = -\\updelta_2^{q'}$ (since $p',q'$ are coprime), to get\n\\begin{equation}\\label{star3}\n\\phi_{jk} \\eta^{-1}(V_3) = a_{jk}^-  - a_{jk}^+  \\updelta_2^{q'}.\t\n\\end{equation}\nNow evaluate it at $V_1$ (noting $\\psi_{1} (V_1) = 0,\\, \\psi_{2} (V_1) =  \\updelta_2^{q'-1}$)\n\\begin{equation}\\label{star1}\n\\phi_{jk} \\eta^{-1}(V_1) = a_{jk}^+  \\updelta_2^{q'-1}.\t\n\\end{equation}\n\nWe now consider the two parity cases for $q'$.\n\n{\\bf CASE: $q'$ is even.} Then $p'$ is odd and equations \\eqref{stare} and \\eqref{star3} become\n\\[\n\\phi_{jk} (e) = a_{jk}^-   + a_{jk}^+, \\qquad\n\\phi_{jk} \\eta^{-1}(V_3) = a_{jk}^-  - a_{jk}^+\n\\]\nwhich give\n\\[\na_{jk}^- = \\tfrac12 [ \\phi_{jk} (e) + \\phi_{jk} \\eta^{-1}(V_3) ], \\qquad\na_{jk}^+ = \\tfrac12 [ \\phi_{jk} (e) - \\phi_{jk} \\eta^{-1}(V_3) ]\t\\qquad (q' \\text{ even}).\n\\]\nFrom \\eqref{phie} we have $\\phi_{jk} (e) = \\delta_2^j$ ($q'$ even, $q$ odd), and from Lemma \\ref{phiV1V3}, and its consequent equation \\eqref{phiV3remark} in this case, we have\n\\[\n\\phi_{jk} \\eta^{-1}(V_3) = \\phi_{jk} \\inner{fV_3}{f}{D} \\ = \\  (-1)^{pk} \\updelta_2^{j-1} . \n\\]\nWe then get\n\\[\na_{jk}^- = \\tfrac12 [ \\delta_2^j + (-1)^{pk} \\updelta_2^{j-1}  ], \\qquad\na_{jk}^+ = \\tfrac12 [ \\delta_2^j - (-1)^{pk} \\updelta_2^{j-1} ]\t      \\qquad (q' \\text{ even})\n\\]\nwhich simplify to\n\\begin{equation}\\label{aqprimeeven}\na_{jk}^- = \\tfrac12 (-1)^{pjk}, \\qquad\na_{jk}^+ = \\tfrac12  (-1)^{j+pjk}  \t\t\\qquad (q' \\text{ even})\n\\end{equation}\nThis, together with \\eqref{CP}, give us the results indicated in Theorem \\ref{MaintheoremA} for the case where $q'$ is even.\n\\medskip\n\n{\\bf CASE: $q'$ is odd.} In this case, equation \\eqref{star1} gives \n\\[\na_{jk}^+ = \\phi_{jk} \\eta^{-1}(V_1)\n\\]\nand \\eqref{stare} gives \n\\begin{equation}\\label{aqprimeoddminus}\na_{jk}^- = \\phi_{jk} (e)  \n= \\updelta_2^{q} \\updelta_2^{j} \\updelta_2^{k}  + \\tfrac12 (-1)^{pjk}\\updelta_2^{q-1}\n\\end{equation}\nfrom \\eqref{phie} (since $q'$ is odd). By Lemma \\ref{phiV1V3}, and consequent equation \\eqref{phiV1remark}, we have\n\\begin{equation}\\label{aqprimeoddplus}\na_{jk}^+ = \\phi_{jk} \\eta^{-1}(V_1) = \\phi_{jk} \\inner{fV_1}{f}{D}  = \\tfrac12 (-1)^{p'j}\\, [ \\updelta_2^{k-1} + (-1)^{pj} \\updelta_2^{q-k-1}  ]\n\\end{equation}\nwhich are the values given in Theorem \\ref{MaintheoremA} in the case that $q'$ is odd. This completes the proof of Theorem \\ref{MaintheoremA} (the canonical traces having already been obtained above).\n\n\\bigskip\n\nWe now proceed to prove Theorem \\ref{Kmatrixthm} by calculating the K-matrix of the  projection $e$, which we do for three parity cases. As stated in the Introduction, for simplicity we let $\\chi_i := \\chi(e P_i(\\theta) e)$ denote the cutdown projection of the $i$-th basis generator $P_i$ by $e$. The trace vector of $e$ consists of the traces of these cutdowns\n\\[\n\\vec \\tau(e) =  \\begin{bmatrix} \n\\tau \\chi_1 & \\tau \\chi_2 & \\tau \\chi_3 & \\tau \\chi_4 & \\tau \\chi_5& \\tau \\chi_6\n\\end{bmatrix}.\n\\]\nIn view of \\eqref{canontraceP}, they are\n\\[\n\\tau\\chi_i = q'(q\\theta-p) \\tau_{p'\/q'}(P_i(\\tfrac{p'}{q'})).\n\\]\nInserting the traces of the six fields $P_i$ as indicated in \\eqref{Kbasis}, we get the trace vector\n\\[\n\\vec \\tau(e_q) = \n\\begin{bmatrix} \nq'(q\\theta-p) & \\tau \\chi_2 & p'(q\\theta-p) & p'(q\\theta-p) & p'(q\\theta-p) & p'(q\\theta-p) \n\\end{bmatrix}  \n\\]\nwhere\n\\[\n\\tau \\chi_2 =  \\begin{cases} \n2p'(q\\theta-p) &\\text{for } 0 < \\theta < \\tfrac12\n\\\\\n2(q' - p') (q\\theta-p)  &\\text{for } \\tfrac12 < \\theta < 1\n\\end{cases}.\n\\]\nThis gives the canonical traces side of the $K$-theory of the AC projection $e$ as stated in Theorem \\ref{Kmatrixthm}.\n\n\n\\medskip\n\nIn view of equation \\eqref{phicutdowns}, it is convenient to write the full K-matrix $K(e) = [\\phi_{jk}(\\chi_i)]_{jk, i}$ of $e$ (relative to the ordered $K_0$ basis \\eqref{basis}) as a matrix product\n\\[\nK(e) = \n\\begin{bmatrix} \n\\phi_{00}(\\chi_1) & \\phi_{00}(\\chi_2) & \\phi_{00}(\\chi_3) & \\phi_{00}(\\chi_4) & \n\\phi_{00}(\\chi_5) & \\phi_{00}(\\chi_6)\n\\\\\n\\phi_{01}(\\chi_1) & \\phi_{01}(\\chi_2) & \\phi_{01}(\\chi_3) & \\phi_{01}(\\chi_4) & \n\\phi_{01}(\\chi_5) & \\phi_{01}(\\chi_6)\n\\\\\n\\phi_{10}(\\chi_1) & \\phi_{10}(\\chi_2) & \\phi_{10}(\\chi_3) & \\phi_{10}(\\chi_4) & \n\\phi_{10}(\\chi_5) & \\phi_{10}(\\chi_6)\n\\\\\n\\phi_{11}(\\chi_1) & \\phi_{11}(\\chi_2) & \\phi_{11}(\\chi_3) & \\phi_{11}(\\chi_4) & \n\\phi_{11}(\\chi_5) & \\phi_{11}(\\chi_6)\n\\end{bmatrix}\n\\ = \\ AC\n\\]\nwhere\n\\[\nA = \\begin{bmatrix} \na_{00}^- & a_{00}^+  \\\\ \na_{01}^- & a_{01}^+  \\\\ \na_{10}^- & a_{10}^+  \\\\ \na_{11}^- & a_{11}^+ \n \\end{bmatrix}, \t\t\\qquad\nC = \\begin{bmatrix} \nC_0(P_1) & C_0(P_2) & C_0(P_3) & C_0(P_4) & C_0(P_5) & C_0(P_6) \n\\\\ \nC_1(P_1) & C_1(P_2) & C_1(P_3) & C_1(P_4) & C_1(P_5) & C_1(P_6) \n\\end{bmatrix}.\n\\]\nOf course, $A$ depends only the AC projection $e$, and $C$ is a matrix of topological invariants of the ordered basis (given in \\eqref{basis}).\n \n\\medskip\n\nIt is more convenient to consider the following three cases separately: \n\n(i) $q'$ even,\n\n(ii) $q'$ odd and $q$ even, and \n\n(iii) $q'$ and $q$ both odd.\n\n\\medskip\n\n\\noindent{\\bf Case (i): $q'$ even.} From \\eqref{aqprimeeven} we have $a_{jk}^- = \\tfrac12 (-1)^{pjk},\\ a_{jk}^+ = \\tfrac12  (-1)^{j+pjk},$ so\n\\[\nA = \\frac12 \\begin{bmatrix} \n1 & 1  \\\\ \n1 & 1  \\\\ \n1 & -1  \\\\ \n(-1)^{p} & (-1)^{p+1} \n \\end{bmatrix}\n \\]\n Equations \\eqref{CP} in the even $q'$ case (so $p'$ is odd) become\n\\[\nC_0(P) = \\phi_{00}(P) + \\phi_{10}(P), \\qquad\nC_1(P) = \\phi_{00}(P) - \\phi_{10}(P)\n\\]\nwhich, in view of the unbounded traces in \\eqref{Kbasis}, give \n\\[\nC = \\begin{bmatrix} \n1 & 0 & 1 & 0 & 1 & 0 \n\\\\ \n1 & 0 & 0 & 1 & 0 & 1\n\\end{bmatrix}\n\\]\n\nTherefore we obtain the K-matrix in the even $q'$ case to be\n\\[\nK(e) = \n\\frac12 \\begin{bmatrix} \n1 & 1  \\\\ \n1 & 1  \\\\ \n1 & -1  \\\\ \n(-1)^{p} & (-1)^{p+1} \n \\end{bmatrix}\n\\begin{bmatrix} \n1 & 0 & 1 & 0 & 1 & 0 \n\\\\ \n1 & 0 & 0 & 1 & 0 & 1\n\\end{bmatrix}\n\\]\nor\n\\[\n\\qquad \\qquad \\qquad K(e) = \n\\frac12 \n\\begin{bmatrix} \n2 & 0 & 1 & 1 & 1 & 1 \n\\\\ \n2 & 0 & 1 & 1 & 1 & 1 \n\\\\ \n0 & 0 & 1 & -1 & 1 & -1\n\\\\ \n0 & 0 & (-1)^p & (-1)^{p+1} & (-1)^p & (-1)^{p+1} \n\\end{bmatrix}_{q' \\text{ even}}\t\n\\]\n(where we have subscripted the matrix with the parity case to which it applies).\n\n\\medskip\n\n\\noindent{\\bf Case (ii): $q$ even.} In this case $p$ and $q'$ are odd and equations \\eqref{aqprimeoddminus} and \\eqref{aqprimeoddplus} become \n\\[\na_{jk}^- = \\updelta_2^{j} \\updelta_2^{k}, \\qquad a_{jk}^+ = \\updelta_2^j \\updelta_2^{k-1} \n\\] \nsince\n\\[\na_{jk}^+ = \\tfrac12 (-1)^{p'j}\\, [ \\updelta_2^{k-1} + (-1)^{pj} \\updelta_2^{q-k-1}  ]\n= \\tfrac12 (-1)^{p'j}\\, [ 1 + (-1)^{j}   ] \\updelta_2^{k-1} \n=  (-1)^{p'j}\\, \\updelta_2^j \\updelta_2^{k-1} \n=  \\updelta_2^j \\updelta_2^{k-1}.\n\\]\nThis gives\n\\[\nA = \\begin{bmatrix} \n1 & 0  \\\\ \n0 & 1  \\\\ \n0 & 0  \\\\ \n0 & 0 \n \\end{bmatrix}\n \\]\nand \\eqref{CP} becomes\n\\[\nC_0(P) = \\phi_{00}(P) + \\phi_{10}(P), \\qquad\nC_1(P) = \\phi_{01}(P) + (-1)^{p'} \\phi_{11}(P)\n\\]\nwhich lead to\n\\[\nC = \\begin{bmatrix} \n1 & 0 & 1 & 0 & 1 & 0 \n\\\\ \n0 & 0 & \\updelta_2^{p'} & \\updelta_2^{p'-1} & -\\updelta_2^{p'} & -\\updelta_2^{p'-1}\n\\end{bmatrix}\n\\]\nwhere we made use of $\\tfrac12(1+(-1)^{p'}) = \\updelta_2^{p'}$ and \n$\\tfrac12(1- (-1)^{p'}) = \\updelta_2^{p'-1}$. Therefore,\n\\[\nK(e) = \n\\begin{bmatrix} \n1 & 0  \\\\ \n0 & 1  \\\\ \n0 & 0  \\\\ \n0 & 0 \n \\end{bmatrix}\n \\begin{bmatrix} \n1 & 0 & 1 & 0 & 1 & 0 \n\\\\ \n0 & 0 & \\updelta_2^{p'} & \\updelta_2^{p'-1} & -\\updelta_2^{p'} & -\\updelta_2^{p'-1}\n\\end{bmatrix}\n\\]\nor\n\\[\nK(e) = \\begin{bmatrix} \n1 & 0 & 1 & 0 & 1 & 0 \n\\\\ \n0 & 0 & \\updelta_2^{p'} & \\updelta_2^{p'-1} & -\\updelta_2^{p'} & -\\updelta_2^{p'-1}\n\\\\ \n0 & 0 & 0 & 0 & 0 & 0\n\\\\ \n0 & 0 & 0 & 0 & 0 & 0\n\\end{bmatrix}_{q \\text{ even}}\n\\]\n\\medskip\n\n\\noindent{\\bf Case (iii): $q',q$ both odd.} Here, equations \\eqref{aqprimeoddminus} and \\eqref{aqprimeoddplus} give\n\\[\na_{jk}^-   \n=  \\tfrac12 (-1)^{pjk}, \\qquad \na_{jk}^+ =  \\tfrac12 (-1)^{j + pjk}\n\\]\nsince\n\\[\na_{jk}^+ \n= \\tfrac12 (-1)^{p'j}\\, [ \\updelta_2^{k-1} + (-1)^{pj} \\updelta_2^{k}  ]\n= \\tfrac12 (-1)^{p'j} \\, (-1)^{pj(k+1)} \n= \\tfrac12 (-1)^{j(p'+p + pk)}\n= \\tfrac12 (-1)^{j(1 + pk)}\n\\]\nwhere the last equality holds because one of $p,p'$ will be even and the other odd (from $qp' - q'p = 1,$ where $q,q'$ are both odd). Therefore,\n\\[\nA = \\frac12\n\\begin{bmatrix} \n1 & 1  \\\\ \n1 & 1  \\\\ \n1 & -1\\\\ \n(-1)^p & (-1)^{p-1}\n \\end{bmatrix}.\n \\]\n The matrix $C$ is the same as in the previous case (ii) (where $q'$ is odd), thus\n\\[\nK(e) = \n\\frac12\n\\begin{bmatrix} \n1 & 1  \\\\ \n1 & 1  \\\\ \n1 & -1\\\\ \n(-1)^p & -(-1)^{p}\n \\end{bmatrix}\n \\begin{bmatrix} \n1 & 0 & 1 & 0 & 1 & 0 \n\\\\ \n0 & 0 & \\updelta_2^{p'} & \\updelta_2^{p'-1} & -\\updelta_2^{p'} & -\\updelta_2^{p'-1}\n\\end{bmatrix}\n\\]\n\\[\n= \\frac12 \\begin{bmatrix} \n1 & 0 & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1} & 1-\\updelta_2^{p'} & -\\updelta_2^{p'-1} \n\\\\ \n1 & 0 & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1} & 1-\\updelta_2^{p'} & -\\updelta_2^{p'-1} \n\\\\ \n1 & 0 & \\updelta_2^{p'-1} &  -\\updelta_2^{p'-1} & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1}\n\\\\ \n(-1)^p & 0 & (-1)^p\\updelta_2^{p'-1} &  -(-1)^{p}\\updelta_2^{p'-1} & (-1)^p(1+\\updelta_2^{p'}) & (-1)^p\\updelta_2^{p'-1}\n\\end{bmatrix}_{q',q \\text{ both odd}}\n\\] \n\\medskip\n\nSince $q',q$ are both odd, one of $p$ or $p'$ will be even which would simplify the matrix a bit.\n(E.g., $(-1)^{p}\\updelta_2^{p'-1} = \\updelta_2^{p'-1}$ since if $p'$ is odd $p$ must be even. Also, $1 - \\updelta_2^{p'} = \\updelta_2^{p'-1}$.) With this in mind, the preceding K-matrix becomes\n\\[\nK(e) =  \\frac12 \\begin{bmatrix} \n1 & 0 & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1} & \\updelta_2^{p'-1} & -\\updelta_2^{p'-1} \n\\\\ \n1 & 0 & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1} & \\updelta_2^{p'-1}& -\\updelta_2^{p'-1} \n\\\\ \n1 & 0 & \\updelta_2^{p'-1} &  -\\updelta_2^{p'-1} & 1+\\updelta_2^{p'} & \\updelta_2^{p'-1}\n\\\\ \n(-1)^p & 0 & \\updelta_2^{p'-1} &  -\\updelta_2^{p'-1} & (-1)^p(1+\\updelta_2^{p'}) & \\updelta_2^{p'-1}\n\\end{bmatrix}_{q',q \\text{ both odd}}\n\\]\nThese all give us the matrices in Theorem \\ref{Kmatrixthm} and therefore complete its proof.\n\n\n\n\\textcolor{blue}{\\Large \\section{Unbounded Traces of C*-Inner Products}}\n\nIn this section we calculate the unbounded traces of the inner products $\\inner{fV_1}{f}{D}$ and $\\inner{fV_3}{f}{D}$ given by the following lemma. These quantities are needed for the calculations in Section 6 in computing the coefficients $a^-, a^+$.\n\n\\bigskip\n\n\\begin{lem}\\label{phiV1V3} For $ij = 00, 01, 10, 11,$ we have\n\\[\n\\phi_{ij} \\inner{fV_1}{f}{D}  \n=  \\frac12 \\updelta_2^{j-1} \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{p'}  \\right] \n+ \\frac12 \\updelta_2^{q-j-1} \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{pq' + p'}  \\right]\n\\]\nand\n\\[\n\\phi_{ij} \\inner{fV_3}{f}{D} \\ = \\ \\frac12 (\\updelta_2^{i-1}  + \\updelta_2^{q'-1-i}) (\\updelta_2^{j} + (-1)^{pi} \\updelta_2^{q-j}).\n\\]\n\\end{lem}\n\n\n\\begin{rmk}\nIn Section 6 the computation for $\\phi_{ij} \\inner{fV_1}{f}{D}$ is needed for the case that $q'$ is odd (see equations \\eqref{aqprimeoddminus} and \\eqref{aqprimeoddplus}), so in this case it simplifies to\n\\begin{equation}\\label{phiV1remark}\n\\phi_{ij} \\inner{fV_1}{f}{D}  = \\frac12 (-1)^{p'i} \\left[ \\updelta_2^{j-1} + (-1)^{pi} \\updelta_2^{q-j-1}  \\right]\t.\t\\qquad (q' \\text{ odd})\n\\end{equation}\nWe have also used the computation for $\\phi_{ij} \\inner{fV_3}{f}{D}$ for when $q'$ is even (so $q,p'$ are odd), which simplifies it to\n\\begin{equation}\\label{phiV3remark}\n\\phi_{ij} \\inner{fV_3}{f}{D} \t\n \\ = \\ \\updelta_2^{i-1} (-1)^{pj} \t\t\\qquad (q' \\text{ even})\n\\end{equation}\n\\end{rmk}\n\n\\medskip\n\n\\subsection{Computation of $\\phi_{ij} \\inner{fV_1}{f}{D}$}\n\nWe will first need to compute $\\eta^{-1}(V_1) = \\inner{fV_1}{f}{D}$ where (as in Section 4) \n\\[\nV_1 = \\pi_{-\\updelta_1}, \\qquad \\updelta_1 = (\\tfrac1{qq'}, p, 0 ;\\ 0, 0, p').\n\\]\nRecalling that $f(t,r,s) = c \\delta_q^r \\delta_{q'}^s \\sqrt{f_0(t)}$, where $c^2 = \\frac{\\sqrt{qq'}}\\alpha,$ we have\n\\begin{align*}\nfV_1(t,r,s) &= f\\pi_{(\\tfrac{-1}{qq'}, -p, 0 ;\\ 0, 0, -p')} (t,r,s) =  e(\\tfrac{-p's}{q'})  f(t - \\tfrac{1}{qq'}, r-p, s) \n= c e(\\tfrac{-p's}{q'})   \\delta_q^{r-p} \\delta_{q'}^{s} \\sqrt{f_0(t - \\tfrac{1}{qq'})}.\n\\end{align*}\nFrom the delta factor $\\delta_{q'}^{s}$ the exponential appearing here can be replaced by 1, thus\n\\[\nfV_1(t,r,s) = c \\delta_q^{r-p} \\delta_{q'}^{s} \\sqrt{f_0(t - \\tfrac{1}{qq'})}.\n\\]\nWe therefore get the $D$-inner product coefficients\n\\begin{align*}\n\\Dinner{fV_1}{f}&(m\\varepsilon_1+n\\varepsilon_2) \n= \\frac1{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(-\\tfrac{rn}q) \\rmintop_{\\Bbb R} fV_1(t,r,s) \\conj{f(t+\\tfrac{m\\alpha}q, r+mp, s+n)} e(-tn) dt\n\\\\\n&= \\frac{c^2}{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(-\\tfrac{rn}q) \\delta_q^{r-p} \\delta_{q'}^{s} \\delta_q^{r+mp} \\delta_{q'}^{s+n}\n\\rmintop_{\\Bbb R}   \\sqrt{f_0(t - \\tfrac{1}{qq'}) f_0(t+\\tfrac{m\\alpha}q)} e(-tn) dt\n\\intertext{which, in view of the first two delta functions, we set $r=p$ and $s=0$ }\n&= \\frac1{\\alpha} e(-\\tfrac{pn}q) \\delta_q^{m+1} \\delta_{q'}^{n}  \n\\rmintop_{\\Bbb R} \\sqrt{f_0(t - \\tfrac{1}{qq'}) f_0(t+\\tfrac{m\\alpha}q)} e(-tn) dt.\n\\intertext{Using the equality $f_0(t) = f_\\tau(q't)$, by \\eqref{fzero}, and making the change of variable $x=q't$, this becomes (using $q'\\alpha = \\tau$)}\n&= \\frac1{\\tau } e(-\\tfrac{pn}q) \\delta_q^{m+1} \\delta_{q'}^{n}  \n\\rmintop_{\\Bbb R} \\sqrt{f_\\tau(x - \\tfrac{1}{q}) f_\\tau(x+\\tfrac{m\\tau}q)}  e(-\\tfrac{nx}{q'})dx.\n\\end{align*}\nThis gives the $D$-inner product (noting that $|G\/D|=\\tau=q'\\alpha$)\n\\begin{align*}\n\\inner{fV_1}{f}{D} &=  \\tau\n\\rmsumop_{m,n} \\inner{fV_1}{f}{D}(m\\varepsilon_1+n\\varepsilon_2) \\  U^n V^m\n\\\\\n&= \\rmsumop_{m,n} e(-\\tfrac{pn}q) \\delta_q^{m+1} \\delta_{q'}^{n}  \n\\rmintop_{\\Bbb R} \\sqrt{f_\\tau(x - \\tfrac{1}{q}) f_\\tau(x+\\tfrac{m\\tau}q)}  e(-\\tfrac{nx}{q'})dx\n\\cdot U^n V^m.\n\\intertext{Now we set $m=qk-1$ and $n=q'\\ell,$}\n&= \\rmsumop_{k,\\ell} e(-\\tfrac{pq'\\ell}q)  \n\\rmintop_{\\Bbb R} \\sqrt{f_\\tau(x - \\tfrac{1}{q}) f_\\tau(x+ k\\tau - \\tfrac{\\tau}q)} \ne(-\\ell x) dx \\cdot  U^{q'\\ell} V^{qk-1}.\n\\end{align*}\nMaking the translation $x \\to x+\\tfrac{1}{q}$ in the integral gives\n\\[\n= \\rmsumop_{k,\\ell} e(-\\tfrac{pq'\\ell}q)  \n\\rmintop_{\\Bbb R} \\sqrt{f_\\tau(x) f_\\tau(x + k\\tau + \\tfrac{1-\\tau}q)} \ne(-\\ell [x+\\tfrac{1}{q}]) dx \\cdot  U^{q'\\ell} V^{qk-1}\n\\]\n\\[\n= \\rmsumop_{k,\\ell} e(-\\tfrac{pq'\\ell}q)  e(- \\tfrac{\\ell}{q}) \n\\rmintop_{\\Bbb R} \\sqrt{f_\\tau(x) f_\\tau(x + k\\tau + \\tfrac{1-\\tau}q)} \ne(-\\ell x) dx \\cdot  U^{q'\\ell} V^{qk-1}.\n\\]\nFrom $p'q-pq'=1$ we have $e(-\\tfrac{pq'\\ell}q)  e(- \\tfrac{\\ell}{q}) = e(-\\tfrac{(pq'+1)\\ell}q) = e(-\\tfrac{p'q\\ell}q) = e(-p'\\ell) = 1$, and from $1 = q'\\alpha + q\\alpha' = \\tau + q\\alpha'$ we have $\\tfrac{1-\\tau}q = \\alpha',$ hence\n\\[\n= \\rmsumop_{k,\\ell} \n\\rmintop_{\\Bbb R} \\sqrt{f_\\tau(x) f_\\tau(x + k\\tau + \\alpha')} \ne(-\\ell x) dx \\cdot  U^{q'\\ell} V^{qk-1}.\n\\]\nSince the function $f_\\tau$ is supported on the interval $-\\tfrac12 < x < \\tfrac12,$ we must also have $-\\tfrac12 < x + k\\tau + \\alpha' < \\tfrac12$ (otherwise the integrand vanishes). From these inequalities, we have $k\\tau < k\\tau + \\alpha' < \\tfrac12 - x < 1$ which implies $k < \\frac1\\tau < 2$ since $\\tau > \\tfrac12$ (in view of Standing Condition 1.3). Further, these inequalities also imply $k\\tau + \\alpha' > -\\tfrac12 - x > -1,$ so $ k\\tau > -1 - \\alpha'$. But as $\\alpha' < \\tfrac1{2q}$ (since from $\\tau > \\tfrac12$ and $1 = \\tau + q\\alpha'$ one has $q\\alpha' < \\tfrac12$), we have $k\\tau  > -1 - \\tfrac1{2q} \\ge -\\tfrac3{2}$ for $q \\ge1$ and hence $k > -\\tfrac3{2\\tau} > -3$ (from $\\tau > \\tfrac12$). This shows that the preceding sum runs only over $k = -2, -1, 0, 1$:\n\\[\n\\inner{fV_1}{f}{D} = \\rmsumop_{k=-2}^1 \\rmsumop_{\\ell} \n\\rmintop_{\\Bbb R} H_k(x) e(-\\ell x) dx \\cdot  U^{q'\\ell} V^{qk-1}\n= \\rmsumop_{k=-2}^1 \\rmsumop_{\\ell} \\ft H_k(\\ell) \\cdot  U^{q'\\ell} V^{qk-1}.\n\\]\nwhere we have written $H_k(x) = \\sqrt{f_\\tau(x) f_\\tau(x + k\\tau + \\alpha')}$ for simplicity and used its Fourier transform. \nApplying the unbounded trace $\\phi_{ij}(U^mV^n)\\ =\\ e(-\\tfrac{\\theta}2 mn)\\,\\updelta_2^{m-i} \\updelta_2^{n-j}$ we get\n\\begin{align*}\n\\phi_{ij} \\inner{fV_1}{f}{D} \n&= \\rmsumop_{k=-2}^1 \\rmsumop_{\\ell} \\ft H_k(\\ell)\\,  \\phi_{ij}(U^{q'\\ell} V^{qk-1})\n= \\rmsumop_{k=-2}^1 \\updelta_2^{qk-1-j} \n\\rmsumop_{\\ell} \\ft H_k(\\ell) \\cdot  e(-\\tfrac{\\theta}2 q'\\ell(qk-1))\\,\\updelta_2^{q'\\ell - i} \n\\\\\n&  = \\  \\updelta_2^{j-1} (\\Omega_0 + \\Omega_{-2}) + \\updelta_2^{q-1-j} (\\Omega_{-1} + \\Omega_1)\n\\end{align*}\nwhere\n\\[\n\\Omega_k = \\rmsumop_{\\ell} \\ft H_k(\\ell) \\cdot  e(-\\tfrac{\\theta}2 q'\\ell(qk-1))\\,\\updelta_2^{q'\\ell - i}.\n\\]\nWriting $\\Omega_k$ with respect to even and odd $\\ell = 2n, 2n+1,$ gives\n\\begin{align*}\n\\Omega_k &= \\updelta_2^{i}  \\rmsumop_{n} \\ft H_k(2n) \\cdot  e(-q' \\theta(qk-1)n)\n + \\updelta_2^{q' - i} e(-\\tfrac{\\theta}2 q'(qk-1)) \n\\rmsumop_{n} \\ft H_k(2n+1) \\cdot  e(-q' \\theta(qk-1)n) \n\\\\\n& = \\updelta_2^{i}  \\rmsumop_{n} \\ft H_k(2n) \\cdot  e(nx')\n+ \\updelta_2^{q' - i} e(\\tfrac{1}2 x' ) \\rmsumop_{n} \\ft H_k(2n+1) \\cdot  e(nx') \n\\intertext{where $x' = -q' \\theta(qk-1)$. By Lemma \\ref{poissonparity} this becomes}\n& = \\frac12 \\updelta_2^{i} \\rmsumop_n  H_k(\\tfrac{x'}2+n) + H_k(\\tfrac{x'}2+ \\tfrac12+n)\n + \\ \\frac12 \\updelta_2^{q' - i} \\rmsumop_n H(\\tfrac{x'}2+n) - H(\\tfrac{x'}2+ \\tfrac12+n)\n\\end{align*}\nso that \n\\[\n\\Omega_k = \\frac12 (\\updelta_2^{i} + \\updelta_2^{q' - i}) \\ \\rmsumop_n  H_k(\\tfrac{x'}2+n) \n+  \\frac12 (\\updelta_2^{i} - \\updelta_2^{q' - i}) \\rmsumop_n  H_k(\\tfrac{x'}2+ \\tfrac12+n).\n\\]\nWe work out the two sums in $\\Omega_k$ by working out the sum\n\\[\n\\rmsumop_n  H_k(\\tfrac{x'}2+\\tfrac{\\epsilon}2 +n) \n= \\rmsumop_n  \\sqrt{f_\\tau(\\tfrac{x'}2+\\tfrac{\\epsilon}2+n) \nf_\\tau(\\tfrac{x'}2 + \\tfrac{\\epsilon}2 + n + k\\tau + \\alpha')} \n\\]\nwhere $\\epsilon = 0,1$. (This Gossamer of a monster is mostly fur!) It is convenient to write $x'$ (using $q \\alpha' = 1 - q' \\alpha = 1-\\tau$) as follows\n\\[\nx' \\ = \\ -q' \\theta (qk-1) \\ = \\ -kpq' + p' - k\\tau -  \\alpha' \n\\]\nwhich is easily checked.\n\n\\bigskip\n\nThe function values $f_\\tau(\\tfrac{x'}2+\\tfrac{\\epsilon}2+n)$ and \n$f_\\tau(\\tfrac{x'}2 + \\tfrac{\\epsilon}2 + n + k\\tau + \\alpha')$ are nonzero when both arguments lie in $(-\\tfrac12,\\tfrac12),$ i.e. when the inequalities \n\\[\n-1 < x' + \\epsilon + 2n < 1, \\qquad \n-1< x' + \\epsilon + 2n + 2k\\tau  + 2\\alpha' < 1\n\\]\nhold. Using the above form for $x'$, these inequalities become\n\\[\n-1 <  -kpq' + p' - k\\tau -  \\alpha' + \\epsilon + 2n < 1,\n\\]\n\\[\n-1< -kpq' + p' - k\\tau -  \\alpha'  + \\epsilon + 2n + 2k\\tau + 2\\alpha' < 1.\n\\]\nAdding these inequalities gives $-2 < 2[ -kpq' + p'  + \\epsilon + 2n ] < 2,$ or\n\\[\n-1 <  -kpq' + p'  + \\epsilon + 2n < 1.\n\\]\nSince middle number is an integer we get $-kpq' + p'  + \\epsilon + 2n = 0$. For such integer $n$ to exist, the integer $-kpq' + p' + \\epsilon$ must be even and the following sum involves only the term with $n = \\tfrac12(kpq' - p'  - \\epsilon),$ thus we have\n\\begin{align*}\n\\rmsumop_n  H_k(\\tfrac{x'}2+\\tfrac{\\epsilon}2 +n)\n& = \\delta_2^{-kpq' + p' + \\epsilon} \nH_k(\\tfrac{x'}2+\\tfrac{\\epsilon}2 + \\tfrac12 kpq' - \\tfrac12p'  - \\tfrac12\\epsilon)\n\\\\\n&= \\delta_2^{kpq' + p' + \\epsilon} \nH_k(\\tfrac{-kpq' + p' - k\\tau -  \\alpha'}2 + \\tfrac12 kpq' - \\tfrac12p')\n\\\\\n&= \\delta_2^{kpq' + p' + \\epsilon} f_\\tau(\\tfrac{k\\tau + \\alpha'}2)\n\\end{align*}\nusing fact that $f_\\tau$ is even in the last equality. This yields\n\\begin{align*}\n\\Omega_k &= \\frac12 (\\updelta_2^{i} + \\updelta_2^{q' - i}) \\ \\delta_2^{kpq' + p'} f_\\tau(\\tfrac{k\\tau + \\alpha'}2) \n+  \\frac12 (\\updelta_2^{i} - \\updelta_2^{q' - i}) \\delta_2^{kpq' + p' + 1} f_\\tau(\\tfrac{k\\tau + \\alpha'}2)\n\\\\\n& = \\frac12 \\left[ (\\updelta_2^{i} + \\updelta_2^{q' - i}) \\ \\delta_2^{kpq' + p'}  \n+  (\\updelta_2^{i} - \\updelta_2^{q' - i}) \\delta_2^{kpq' + p' + 1} \n\\right] f_\\tau(\\tfrac{k\\tau + \\alpha'}2)\n\\\\\n& = \\frac12 \\left[ \\updelta_2^{i}\\delta_2^{kpq' + p'} + \\updelta_2^{q' - i} \\delta_2^{kpq' + p'}  \n+  \\updelta_2^{i} \\delta_2^{kpq' + p' + 1} - \\updelta_2^{q' - i} \\delta_2^{kpq' + p' + 1} \n\\right] f_\\tau(\\tfrac{k\\tau + \\alpha'}2)\n\\\\\n& = \\frac12 \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} ( \\delta_2^{kpq' + p'}  \n -  \\delta_2^{kpq' + p' + 1} ) \\right] f_\\tau(\\tfrac{k\\tau + \\alpha'}2)\n\\end{align*}\ntherefore\n\\[\n\\Omega_k = \\frac12 \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{kpq' + p'}  \\right] f_\\tau(\\tfrac{k\\tau + \\alpha'}2).\n\\]\nSetting $k=0$ and $k=-2$ gives\n\\[\n\\Omega_0 = \\frac12 \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{p'}  \\right] f_\\tau(\\tfrac{\\alpha'}2), \\qquad\n\\Omega_{-2} = \\frac12 \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{p'}  \\right] f_\\tau(\\tfrac{\\alpha'}2 - \\tau)\n\\]\nwhich sum to\n\\[\n\\Omega_{0} + \\Omega_{-2} = \\frac12 \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{p'}  \\right]\n\\]\nsince $f_\\tau(\\tfrac{\\alpha'}2) + f_\\tau(\\tfrac{\\alpha'}2 - \\tau) = 1$ (which follows from \\eqref{ftx1} by taking $x = \\tfrac{\\alpha'}2$ and $t = \\tau$ there). Similarly, for $k = -1, 1$ we have\n\\[\n\\Omega_{-1} = \\frac12 \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{pq' + p'}  \\right] f_\\tau(\\tfrac{-\\tau + \\alpha'}2), \\qquad\n\\Omega_1 = \\frac12 \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{pq' + p'}  \\right] f_\\tau(\\tfrac{\\tau + \\alpha'}2)\n\\]\nwhich sum to\n\\[\n\\Omega_{1} + \\Omega_{-1} = \\frac12 \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{pq' + p'}  \\right] \n\\]\nsince $f_\\tau(\\tfrac{-\\tau + \\alpha'}2) + f_\\tau(\\tfrac{\\tau + \\alpha'}2) = 1$ (which follows from \\eqref{ftxt} by taking $x = \\tfrac{-\\tau + \\alpha'}2 \\in (-\\tfrac12,0)$ and $t = \\tau$). We have therefore obtained\n\\begin{align*}\n\\phi_{ij} \\inner{fV_1}{f}{D} \n&= \\updelta_2^{j-1} (\\Omega_0 + \\Omega_{-2}) + \\updelta_2^{q-j-1} (\\Omega_{-1} + \\Omega_1) \n\\\\\n&=  \\frac12 \\updelta_2^{j-1} \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{p'}  \\right] +\n \\frac12 \\updelta_2^{q-j-1} \\left[ \\updelta_2^{i}+ \\updelta_2^{q' - i} (-1)^{pq' + p'}  \\right]\n\\end{align*}\nwhich establishes the equation for $\\phi_{ij} \\inner{fV_1}{f}{D}$ in the statement of Lemma \\ref{phiV1V3}.\n\nNow if $q'$ is odd, as in fact is needed for the computation in Section 6, this result simplifies to\n\\begin{align*}\n\\phi_{ij} \\inner{fV_1}{f}{D}  &= \\frac12  \\updelta_2^{q-j-1}  \\left[ \\updelta_2^{i}+ \\updelta_2^{i-1} (-1)^{p + p'}  \\right]\n+ \\frac12 \\updelta_2^{j-1}  \\left[ \\updelta_2^{i}+ \\updelta_2^{i-1} (-1)^{p'}  \\right]\n\\\\\n&= \\frac12 \\left[ \\updelta_2^{q-j-1}   (-1)^{pi + p'i}  + \\updelta_2^{j-1}  (-1)^{p'i}  \\right]\n\\\\\n&= \\frac12 (-1)^{p'i}  \\left[ \\updelta_2^{q-j-1}   (-1)^{pi}  + \\updelta_2^{j-1} \\right]\n\\qquad\t(q'\\text{ odd})\n\\end{align*}\nas noted in the remark following Lemma \\ref{phiV1V3}.\n\n\\subsection{Computation of $\\phi_{ij} \\inner{fV_3}{f}{D}$} \n\nHere we compute the unbounded traces $\\phi_{ij}$ of $\\eta^{-1}(V_3) = \\Dinner{fV_3}{f}$. We have, using $V_3 = \\pi_{-\\updelta_3}$ and $\\updelta_3 = (0, 0, 1 ;\\  0, 0, 0)$, \n\\[\nfV_3(t,r,s) = \\pi_{-\\updelta_3}(f)(t,r,s)\n= \\pi_{(0, 0, -1 ;\\  0, 0, 0)}(f)(t,r,s) \n= f(t,r,s-1) \n\\]\nso\n\\begin{align*}\n\\Dinner{fV_3}{f}(m\\varepsilon_1+n\\varepsilon_2) \n&= \\frac1{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(-\\tfrac{rn}q) \\rmintop_{\\Bbb R} f(t,r,s-1) \\conj{f(t+\\tfrac{m\\alpha}q, r+mp, s+n)} e(-tn) dt\n\\\\\n&= \\frac{c^2}{\\sqrt{qq'}} \\rmsumop_{r=0}^{q-1} \\rmsumop_{s=0}^{q'-1} \ne(-\\tfrac{rn}q) \\rmintop_{\\Bbb R} \\delta_q^r \\delta_{q'}^{s-1}  \n\\delta_q^{r+mp} \\delta_{q'}^{s+n} \\sqrt{f_0(t) f_0(t+\\tfrac{m\\alpha}q)} e(-tn) dt\n\\\\\n&= \\frac1{\\alpha} \\delta_q^{m} \\delta_{q'}^{n+1}\n \\rmintop_{\\Bbb R}  \\sqrt{f_0(t) f_0(t+\\tfrac{m\\alpha}q)} e(-tn) dt\n\\intertext{in view of \\eqref{fzero}, and using a change of variable $x=q't$, this becomes}\n&= \\frac1{\\tau} \\delta_q^{m} \\delta_{q'}^{n+1}\n \\rmintop_{\\Bbb R}   \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{m\\tau}q)} \\ e(-\\tfrac{nx}{q'}) dx.\n\\end{align*}\n\nThis gives inner product (noting that $|G\/D|=\\tau=q'\\alpha$)\n\\begin{align*}\n\\inner{fV_3}{f}{D} &=  \\tau\n\\rmsumop_{m,n} \\inner{fV_3}{f}{D}(m\\varepsilon_1+n\\varepsilon_2) \\  U^n V^m\n\\\\\n&= \\rmsumop_{m,n} \\delta_q^{m} \\delta_{q'}^{n+1}\n \\rmintop_{\\Bbb R}   \\sqrt{f_\\tau(x) f_\\tau(x+\\tfrac{m\\tau}q)} e(-\\tfrac{nx}{q'}) dx\n\\  U^n V^m\n\\intertext{setting $m=qk$ and $n=q'\\ell - 1$}\n&=  \\rmsumop_{k,\\ell} \n\\rmintop_{\\Bbb R}  \\sqrt{f_\\tau(x) f_\\tau(x+k\\tau)} e(-\\ell x) e(\\tfrac{x}{q'})  dx\n\\  U^{q'\\ell - 1} V^{qk}\n\\intertext{and noting that the integrand vanishes for $|k|\\ge2,$}\n&=   \\rmsumop_{k=-1}^1 \\rmsumop_{\\ell} \n\\rmintop_{\\Bbb R}  L_k(x) e(-\\ell x) dx\n\\  U^{q'\\ell - 1} V^{qk}\n=   \\rmsumop_{k=-1}^1 \\rmsumop_{\\ell} \\ft L_k(\\ell) \\  U^{q'\\ell - 1} V^{qk}\n\\end{align*}\nwhere we put $L_k(x) = e(\\tfrac{x}{q'}) \\sqrt{f_\\tau(x) f_\\tau(x+k\\tau)}$.  Applying $\\phi_{ij}(U^mV^n)\\ =\\ e(-\\tfrac{\\theta}2 mn)\\,\\updelta_2^{m-i} \\updelta_2^{n-j}$ one gets\n\\[\n\\phi_{ij} \\inner{fV_3}{f}{D} \n= \\rmsumop_{k=-1}^1  \\updelta_2^{qk-j} \\rmsumop_{\\ell} \\ft L_k(\\ell) \\cdot\ne(-\\tfrac{\\theta}2 (q'\\ell - 1)qk)\\, \\updelta_2^{q'\\ell -1-i} \n= \\rmsumop_{k=-1}^1  \\updelta_2^{qk-j} e(\\tfrac{\\theta}2 qk) \\ \\Lambda_k\n\\]\nwhere\n\\[\n\\Lambda_k = \\rmsumop_{\\ell} \\ft L_k(\\ell) \\cdot e(-\\tfrac{\\theta}2 qq' k\\ell)\\, \\updelta_2^{q'\\ell -1-i}.\n\\]\nThus,\n\\[\n\\phi_{ij} \\inner{fV_3}{f}{D} \n= \\updelta_2^{j} \\ \\Lambda_0 \n+ \\updelta_2^{q-j} \\left[ e(\\tfrac{\\theta}2 q) \\ \\Lambda_1\n+ e(-\\tfrac{\\theta}2 q) \\ \\Lambda_{-1} \\right].\n\\]\nWe have\n\\[\n\\Lambda_k = \n\\updelta_2^{i -1}  \\rmsumop_{n} \\ft L_k(2n) \\cdot e(-\\theta qq' k n)\n+ \\updelta_2^{q' -1-i} e(-\\tfrac{\\theta}2 qq' k) \n\\rmsumop_{n} \\ft L_k(2n+1) \\cdot e(-\\theta qq' k n) \n\\]\nwhich by the Poisson formulas in Lemma \\ref{poissonparity} becomes\n\\[\n\\Lambda_k  = \\frac12 \\updelta_2^{i -1} (A_k + B_k) \n+  \\frac12  \\updelta_2^{q' -1-i} (A_k - B_k)\n\\]\nwhere\n\\[\nA_k = \\rmsumop_n L_k(\\tfrac{-\\theta qq' k}2+n), \\qquad\nB_k = \\rmsumop_n L_k(\\tfrac{-\\theta qq' k}2+ \\tfrac12+n).\n\\]\nFirst, consider $k=0$:\n\\[\nA_0 = \\rmsumop_n L_0(n) = \\rmsumop_n e(\\tfrac{n}{q'}) f_\\tau(n) = 1\n\\]\nsince $f_\\tau(n) = 1$ for $n=0$ and 0 otherwise, and\n\\[\nB_0 = \\rmsumop_n L_0(\\tfrac12+n) \n= \\rmsumop_n e(\\tfrac{\\tfrac12+n}{q'}) f_\\tau(\\tfrac12+n)  = 0\n\\]\nsince $f_\\tau(\\tfrac12+n) = 0$ for all integers $n$. This gives\n\\[\n\\Lambda_0 = \\frac12 (\\updelta_2^{i -1} + \\updelta_2^{q' -1-i}).\n\\]\nTo compute $\\Lambda_1$ and $\\Lambda_{-1},$ it will suffice to find\n\\[\nA_1 = \\rmsumop_n L_1(\\tfrac{-\\theta qq' }2+n), \\qquad\nB_1 = \\rmsumop_n L_1(\\tfrac{-\\theta qq' }2+ \\tfrac12+n).\n\\]\nFirst, it is easy to see that $L_{-1}(x) = \\conj{L_1(-x)}$ (since $f_\\tau$ is even), and consequently $A_{-1} = \\conj{A_1}$ and $B_{-1} = \\conj{B_1},$ hence $\\Lambda_{-1} = \\conj{ \\Lambda_1}$. For example, to see $B_{-1} = \\conj{B_1},$ we have \n\\[\nB_{-1} = \\rmsumop_n L_{-1}(\\tfrac{\\theta qq'}2+ \\tfrac12+n)\n= \\rmsumop_n \\conj{ L_1(\\tfrac{-\\theta qq'}2 - \\tfrac12 - n) }\n= \\rmsumop_n \\conj{ L_1(\\tfrac{-\\theta qq'}2 + \\tfrac12 + n) } = \\conj{B_1}\n\\]\nusing the substitution $n \\to -1-n$. We now show that\n\\[\nA_1 = \\frac12 \\updelta_2^{q'p} e(\\tfrac{-\\tau }{2q'}), \\qquad B_1 = \\frac12 \\updelta_2^{q'p-1} e(\\tfrac{-\\tau}{2q'}). \n\\]\nFor the first, write\n\\[\nA_1 = \\rmsumop_n L_1(\\tfrac{-\\tau}2 - \\tfrac{q'p}2 + n).\n\\]\nLetting $\\epsilon = 0$ when $q'p$ is even, and $\\epsilon = 1$ when $q'p$ is odd, the preceding sum becomes, after appropriate translation of the index $n,$\n\\[\nA_1 = \\rmsumop_n L_1(\\tfrac{-\\tau}2 + \\tfrac{\\epsilon}2 + n)\n= \\rmsumop_n e\\left(\\tfrac{\\tfrac{-\\tau}2 + \\tfrac{\\epsilon}2 + n}{q'}\\right) \n\\sqrt{f_\\tau(\\tfrac{-\\tau}2 + \\tfrac{\\epsilon}2 + n) f_\\tau(\\tfrac{\\tau}2 + \\tfrac{\\epsilon}2 + n)}.\n\\]\nThe function values under the square-root here is nonzero when both of its arguments lie in the open interval $(-\\tfrac12, \\tfrac12),$ so their sum $\\epsilon+2n$ (which is an integer) is in the open interval $(-1, 1),$ so $\\epsilon+2n = 0$. This means that if $q'p$ is odd ($\\epsilon=1$) then no such $n$ exists and hence $A_1 = 0$. And if  $q'p$ is even, so $\\epsilon=0,$ then $n = 0$:\n\\[\nA_1 =  e(\\tfrac{-\\tau}{2q'}) \\sqrt{f_\\tau(\\tfrac{-\\tau}2) f_\\tau(\\tfrac{\\tau}2)} \n= \\frac12 e(\\tfrac{-\\tau}{2q'})\n\\]\nsince $f_\\tau(\\tfrac{\\tau}2 ) = \\tfrac12$ from \\eqref{ftt2}. We may then write \n\\[\nA_1 =  \\frac12 \\updelta_2^{q'p} e(\\tfrac{-\\tau }{2q'})\n\\]\nin either parity case.\n\nLikewise (with $\\epsilon$ as before), we have\n\\begin{align*}\nB_1\n&= \\rmsumop_n L_1( \\tfrac{-\\tau}2 - \\tfrac{q'p}2 + \\tfrac12+n )\n= \\rmsumop_n L_1( \\tfrac{-\\tau}2 + \\tfrac{\\epsilon}2 + \\tfrac12+n )\n\\\\\n&= \\rmsumop_n e\\left(\\tfrac{ \\tfrac{-\\tau}2 + \\tfrac{\\epsilon}2 + \\tfrac12+n}{q'}\\right) \n\\sqrt{f_\\tau( \\tfrac{-\\tau}2 + \\tfrac{\\epsilon}2 + \\tfrac12+n) \nf_\\tau( \\tfrac{\\tau}2 + \\tfrac{\\epsilon}2 + \\tfrac12+n )}.\n\\end{align*}\nBy the same argument as before, the function values under the square-root are nonzero when their arguments are in $(-\\tfrac12, \\tfrac12),$ so their sum $1+\\epsilon+2n = 0$. Therefore, $B_1 = 0$ when $q'p$ is even ($\\epsilon=0$). And when $q'p$ is odd we must have $n = -1$ and $\\epsilon=1$ which gives $B_1 = e(\\tfrac{-\\tau}{2q'})  \\sqrt{f_\\tau( \\tfrac{-\\tau}2) f_\\tau( \\tfrac{\\tau}2) } = \\frac12 e(\\tfrac{-\\tau}{2q'}),$\nso that in either parity case we have\n\\[\nB_1 = \\frac12 \\updelta_2^{q'p-1} e(\\tfrac{-\\tau}{2q'}).\n\\]\nWe thus get\n\\begin{align*}\n\\Lambda_1  &= \\frac12 \\updelta_2^{i -1} (A_1 + B_1) +  \\frac12  \\updelta_2^{q' -1-i} (A_1 - B_1)\n\\\\\n& = \\frac14 \\updelta_2^{i -1} \\left[  \\updelta_2^{q'p}  +  \\updelta_2^{q'p-1} \\right] \ne(\\tfrac{-\\tau }{2q'})\n+  \\frac14  \\updelta_2^{q' -1-i} \\left[  \\updelta_2^{q'p} -  \\updelta_2^{q'p-1}  \\right] e(\\tfrac{-\\tau }{2q'})\n\\\\\n&  = \\frac14 \\updelta_2^{i -1} e(\\tfrac{-\\tau }{2q'})\n+  \\frac14  \\updelta_2^{q' -1-i} (-1)^{q'p} e(\\tfrac{-\\tau }{2q'})\n\\end{align*}\nand since $e(\\tfrac{\\theta}2 q) e(\\tfrac{-\\tau }{2q'}) = (-1)^p$ we have\n\\[\ne(\\tfrac{\\theta}2 q) \\Lambda_1\n= \\frac14 \\updelta_2^{i -1} (-1)^p\n+  \\frac14  \\updelta_2^{q' -1-i} (-1)^{q'p} (-1)^p\n= \\frac14 \\updelta_2^{i -1} (-1)^p\n+  \\frac14  \\updelta_2^{q' -1-i} (-1)^{pi}\n\\]\n(where the last term holds in view of the $\\updelta_2^{q' -1-i}$ factor)\nwhich is real, and as seen above $\\Lambda_{-1} = \\conj{\\Lambda_1},$ we get\n\\begin{align*}\n\\phi_{ij} \\inner{fV_3}{f}{D} \n&= \\updelta_2^{j} \\ \\Lambda_0 \n+ \\updelta_2^{q-j} \\left[ e(\\tfrac{\\theta}2 q) \\ \\Lambda_1\n+ e(-\\tfrac{\\theta}2 q) \\ \\Lambda_{-1} \\right]\n\\\\\n&= \\frac12 \\updelta_2^{j}  (\\updelta_2^{i -1} + \\updelta_2^{q' -1-i}) \n+ \\frac12 \\updelta_2^{q-j} \\cdot \\left[ \\updelta_2^{i -1} (-1)^p\n+   \\updelta_2^{q' -1-i} (-1)^{pi} \\right]\n\\\\\n&= \\frac12 \\updelta_2^{j}  (\\updelta_2^{i -1} + \\updelta_2^{q' -1-i}) \n+ \\frac12 \\updelta_2^{q-j} \\cdot \\left[ \\updelta_2^{i -1} \n+   \\updelta_2^{q' -1-i} \\right] (-1)^{pi} \n\\\\\n&= \\frac12 (\\updelta_2^{i-1}  + \\updelta_2^{q'-1-i}) (\\updelta_2^{j} + (-1)^{pi} \\updelta_2^{q-j})\n\\end{align*}\nwhich is the expression in the statement of Lemma \\ref{phiV1V3}, the proof of which is now complete. \n\n\\textcolor{blue}{\\Large \\section{Appendix A: Unbounded Traces of $\\mathcal E(t)$}}\n\nIn this section we show that the Connes-Chern character of the continuous field $\\mathcal E(t)$ is\n\\[\n\\bold T(\\mathcal E(t)) = (t; \\tfrac12, \\tfrac12, \\tfrac12, \\tfrac12)\n\\]\nfor $\\tfrac12 \\le t < 1$.\n\nRecall that the continuous field $\\mathcal E: [\\frac12,1) \\to \\{A_t\\}$ of Flip-invariant Powers-Rieffel projections is given by\n\\begin{equation}\n\\mathcal E(t) = G_t(U_t) V_t^{-1} + F_t(U_t) + V_t G_t(U_t) \n\\end{equation}\nwhere $F_t, G_t$ are smooth functions, as in Section 2.2. Fix $j,k$ and write $\\phi := \\phi_{jk}^t,$ which is defined on $A_t^\\infty$ by\n\\begin{equation}\n\\phi_{jk}^t(U_t^mV_t^n)\\ =\\ e(-\\tfrac{t}2 mn)\\,\\updelta_2^{m-j} \\updelta_2^{n-k}.\n\\end{equation}\nBy the $\\Phi$-trace property of $\\phi,$ we have $\\phi(G_t(U_t) V_t^{-1}) = \\phi( \\Phi(V_t^{-1}) G_t(U_t)) = \\phi(V_t G_t(U_t))$ so that\n\\[\n\\phi(\\mathcal E(t)) = \\phi(F_t(U_t)) + 2 \\phi(G_t(U_t)V_t^{-1} ).\n\\]\nExpressing $F_t$ in terms of its Fourier transform\n\\[\nF_t(U_t) = \\rmsumop_{n\\in \\mathbb Z} \\ft F_t(n) U_t^n \n\\]\nwhere $\\ft F_t(n) = \\ft f_t(n)$ (see proof of Lemma \\ref{poisson}), we obtain\n\\[\n\\phi(F_t(U_t)) = \\rmsumop_{n\\in \\mathbb Z} \\ft F_t(n) \\phi(U_t^n) \n= \\rmsumop_{n\\in \\mathbb Z} \\ft F_t(n) \\delta_2^{n-j}  \\delta_2^{0-k}\n= \\delta_2^{k} \\rmsumop_{n\\in \\mathbb Z} \\ft F_t(n) \\delta_2^{n-j}.  \n\\]\nExpanding this series into its even and odd indices, one has\n\\[\n\\rmsumop_{n\\in \\mathbb Z} \\ft F_t(n) \\delta_2^{n-j}  \n\\ =\\ \\delta_2^{j}\\rmsumop_{n\\in \\mathbb Z} \\ft F_t(2n)   +\n\\delta_2^{j-1} \\rmsumop_{n\\in \\mathbb Z} \\ft F_t(2n+1) \n\\ =\\ \\delta_2^{j}\\rmsumop_{n\\in \\mathbb Z} \\ft f_t(2n)   +\n\\delta_2^{j-1} \\rmsumop_{n\\in \\mathbb Z} \\ft f_t(2n+1) \n\\]\nwhich, in view of the Poisson Lemma \\ref{poissonparity}, become\n\\[\n= \\delta_2^{j} \\left(\n\\frac12 \\rmsumop_{n=-\\infty}^\\infty f_t(n) + f_t(\\tfrac12+n)\n\\right)\n+ \\delta_2^{j-1} \\left(\n\\frac12  \\rmsumop_{n=-\\infty}^\\infty f_t(n) - f_t(\\tfrac12+n).\n\\right)\n\\]\nThe sums here are (see Figure \\ref{figfg})\n\\[\n\\rmsumop_n f_t(n) = f_t(0) = 1, \\qquad \\rmsumop_n f_t(\\tfrac12+n) = 0\n\\]\nso that\n\\[\n\\phi(F_t(U_t)) = \\delta_2^{k} \\frac12 (\\delta_2^{j} + \\delta_2^{j-1}) = \\frac12\\delta_2^{k}.\n\\]\nWe similarly compute $\\phi(G_t(U_t)V_t^{-1} )$ using the Fourier series for $G_t$\n\\[\nG_t(U_t) = \\rmsumop_{n\\in \\mathbb Z} \\ft G_t(n) U_t^n. \n\\]\nWe have \n\\begin{align*}\n\\phi(G_t(U_t)V_t^{-1}) &= \\rmsumop_{n\\in \\mathbb Z} \\ft G_t(n) \\phi(U_t^n V_t^{-1})\n= \\delta_2^{-1-k} \\rmsumop_{n\\in \\mathbb Z} \\ft G_t(n) e(\\tfrac{t}2n) \\delta_2^{n-j} \n\\\\\n&= \\delta_2^{k-1} \\delta_2^{j} \n\\rmsumop_{n\\in \\mathbb Z} \\ft G_t(2n) e(tn) \n+\n\\delta_2^{k-1} \\delta_2^{j-1} e(\\tfrac{t}2)\\rmsumop_{n\\in \\mathbb Z} \\ft G_t(2n+1) e(tn) \n\\end{align*}\nwhich again by Lemma  \\ref{poissonparity} (with $H=g_t$ and using $\\ft G_t = \\ft g_t$) is \n\\[\n= \\frac12 \\delta_2^{k-1} \\delta_2^{j} \\left( \\rmsumop_n g_t(\\tfrac{t}2+n) + g_t(\\tfrac{t}2+ \\tfrac12+n)\n\\right)\n+ \\frac12 \\delta_2^{k-1} \\delta_2^{j-1} \\left( \\rmsumop_n g_t(\\tfrac{t}2+n) - g_t(\\tfrac{t}2+ \\tfrac12+n) \\right).\n\\]\nThe individual sums here are (note $f_t(\\tfrac{t}2) = \\tfrac12 = g_t(\\tfrac{t}2)$ from Section 2.2) \n\\[\n\\rmsumop_n g_t(\\tfrac{t}2+n) = g_t(\\tfrac{t}2) = \\frac12, \\qquad\n\\rmsumop_n  g_t(\\tfrac{t}2+ \\tfrac12+n) = 0  \n\\]\nso\n\\[\n\\phi(G_t(U_t)V_t^{-1}) = \\frac14 \\delta_2^{k-1}(\\delta_2^{j} + \\delta_2^{j-1}) \n= \\frac14 \\delta_2^{k-1}.\n\\]\nTherefore we obtain the desired unbounded traces\n\\[\n\\phi_{jk}^t (\\mathcal E(t)) = \\frac12\\delta_2^{k} + 2 \\frac14 \\delta_2^{k-1} = \\frac12\n\\]\nwhich gives the Connes-Chern character of $\\mathcal E(t)$ as\n\\[\n\\bold T(\\mathcal E(t)) = (t; \\tfrac12, \\tfrac12, \\tfrac12, \\tfrac12)\n\\]\nfor $\\tfrac12 \\le t < 1$ (the parameter range over which the field $\\mathcal E$ is defined).\n\\medskip\n\nWe note that the unbounded traces of the projection depend on the following values of the underlying function: $f_t(0) = 1,\\ f_t(\\tfrac12) = 0, \\ f_t(\\tfrac{t}2)=\\tfrac12, \\ f_t(\\tfrac{t}2+\\tfrac12) = 0$ (where the last of these holds since $f_t$ is compactly supported on $[-\\tfrac12,\\tfrac12],$ where it is an even function). So any homotopic deformation of $f_t$ which preserves these boundary conditions (and of course maintaining the equations between $F,G$ that make $\\mathcal E$ is a projection) would still give us the same unbounded traces.\n\n\n\\textcolor{blue}{\\Large \\section{Appendix B: Flip-Traces on $M_{q'}(C(\\mathbb T))$}}\n\n\nIn this section we show that the two functionals\n\\begin{align}\\label{Phiprimetraces}\n\\psi_{1} ( V_3^{n} V_1^m ) &= e(\\tfrac{p'mn}{2q'})  \\updelta_2^{m} \n\\\\\n\\psi_{2} ( V_3^{n} V_1^m ) &= e(\\tfrac{p'mn}{2q'}) (-1)^{p'n}\\ \\updelta_2^{m-q'}\t\\notag\n\\end{align}\nform a basis for all $\\Phi'$-traces on the circle algebra $B = C^*(V_1,V_3) \\cong M_{q'}\\otimes C(\\mathbb T)$ generated by unitaries $V_3, V_1$ satisfying\n\\[\nV_3 V_1 = e(\\tfrac{p'}{q'}) V_1 V_3, \\qquad V_3^{q'} = I\n\\]\nas in \\eqref{TheVs}, with $V_1$ of full spectrum, where $\\Phi'$ is the Flip automorphism of $B$ defined by\n\\[\n\\Phi'(V_3) = V_3^{-1}, \\qquad \\Phi'(V_1) = V_1^{-1}.\n\\]\n\n\\medskip\n\n\\begin{proof} We begin with the canonical epimorphism of the rational rotation algebra onto $B,$\n\\[\n\\pi: A_{p'\/q'} \\to C^*(V_1,V_3), \\qquad \\pi(U') = V_1,\\qquad \\pi(V') = V_3\n\\]\nwhere $U' = U_{p'\/q'}, \\ V' = V_{p'\/q'}$ satisfy $V'U' = e(\\tfrac{p'}{q'}) U'V'$. This surjection intertwines the two Flips\n\\[\n\\pi \\Phi = \\Phi'\\pi \n\\]\nwhich is easy to verify, where $\\Phi(U')={U'}^{-1}, \\Phi(V') = {V'}^{-1}$ is the Flip on $A_{p'\/q'}$.\n\n\\medskip\n\nFix a $\\Phi'$-trace $\\psi$ on $B$. Since $\\psi \\pi$ is a $\\Phi$-trace on $A_{p'\/q'},$ it is a linear combination of the four basic unbounded $\\Phi$-traces defined on the basic unitaries ${U'\\,}^m{V'\\,}^n$ by \n\\begin{equation}\n\\phi_{ij}({U'\\,}^m{V'\\,}^n)\\ =\\ e(-\\tfrac{p'mn}{2q'})\\,\\updelta_2^{m-i} \\updelta_2^{n-j}\n\\end{equation}\n($ij = 00, 01,10,11$). Thus\n\\[\n\\psi \\pi = a \\phi_{00} + b\\phi_{01} + c\\phi_{10} + d\\phi_{11}\n\\]\nfor some constants $a,b,c,d$. Evaluating this gives on the basic unitaries,\n\\[\n\\psi( V_1^{m} V_3^n ) = \\psi \\pi ({U'}^m{V'}^n) \n= a \\phi_{00}({U'}^m{V'}^n) + b\\phi_{01}({U'}^m{V'}^n) + c\\phi_{10}({U'}^m{V'}^n) \n+ d\\phi_{11}({U'}^m{V'}^n) \n\\]\n\\[\n= e(-\\tfrac{p'mn}{2q'})\n\\Big( a\\updelta_2^{m} \\updelta_2^{n} + b\\updelta_2^{m} \\updelta_2^{n-1}\n+ c\\updelta_2^{m-1} \\updelta_2^{n} + d\\updelta_2^{m-1} \\updelta_2^{n-1}\n\\Big)\n\\]\nfrom $e(\\tfrac{p'mn}{q'}) V_1^{m} V_3^n = V_3^n V_1^{m},$ this gives\n\\begin{equation}\\label{psiabcd}\n\\psi( V_3^{n} V_1^m ) = e(\\tfrac{p'mn}{2q'}) \n\\Big( a\\updelta_2^{m} \\updelta_2^{n} + b\\updelta_2^{m} \\updelta_2^{n-1}\n+ c\\updelta_2^{m-1} \\updelta_2^{n} + d\\updelta_2^{m-1} \\updelta_2^{n-1}\n\\Big).\n\\end{equation}\nSince this expression should be invariant under the translation $n \\to n+q'$ (as $V_3$ has order $q'$), we get\n\\begin{align*}\na\\updelta_2^{m} \\updelta_2^{n} &+ b\\updelta_2^{m} \\updelta_2^{n-1}\n+ c\\updelta_2^{m-1} \\updelta_2^{n} + d\\updelta_2^{m-1} \\updelta_2^{n-1}\n\\\\\n&=\n(-1)^{p'm} a\\updelta_2^{m} \\updelta_2^{n+q'} + (-1)^{p'm} b\\updelta_2^{m} \\updelta_2^{n+q'-1}\n+ (-1)^{p'm} c\\updelta_2^{m-1} \\updelta_2^{n+q'} + (-1)^{p'm} d\\updelta_2^{m-1} \\updelta_2^{n+q'-1}\n\\end{align*}\nor\n\\begin{align*}\na\\updelta_2^{m} \\updelta_2^{n} + b\\updelta_2^{m} \\updelta_2^{n-1}\n&+ c\\updelta_2^{m-1} \\updelta_2^{n} + d\\updelta_2^{m-1} \\updelta_2^{n-1}\n\\\\\n& =\na\\updelta_2^{m} \\updelta_2^{n+q'} + b\\updelta_2^{m} \\updelta_2^{n+q'-1}\n+ (-1)^{p'} c\\updelta_2^{m-1} \\updelta_2^{n+q'} + (-1)^{p'} d\\updelta_2^{m-1} \\updelta_2^{n+q'-1}\n\\end{align*}\nwhich must be satisfied for all four parities of $m,n$. Setting $m=n=0,$ it gives\n\\[\na =  a\\updelta_2^{q'} + b\\updelta_2^{q'-1} \n\\]\nand for $m=0, n=1$:\n\\[\nb =  a \\updelta_2^{q'-1} + b \\updelta_2^{q'}\n\\]\nfor $m=1, n=0$:\n\\[\nc = (-1)^{p'} c \\updelta_2^{q'} + (-1)^{p'} d \\updelta_2^{q'-1}\n\\] \nand for $m=n=1$:\n\\[\nd =  (-1)^{p'} c \\updelta_2^{q'-1} + (-1)^{p'} d \\updelta_2^{q'}\n\\] \n\n\nWe consider separately two parity cases for $q'$. \n\nIf $q'$ is even (so $p'$ is odd), these become $a =  a, \\ b =  b, \\ c = - c, \\ d = - d,$\nso $c=d=0,$ and equation \\eqref{psiabcd} becomes\n\\[\n\\ \\ \\qquad \\psi( V_3^{n} V_1^m ) = e(\\tfrac{p'mn}{2q'}) \n\\Big( a \\updelta_2^{n} + b \\updelta_2^{n-1} \\Big) \\updelta_2^{m},\\qquad (q' \\ \\text{even})\n\\]\nwhere $a,b$ are arbitrary scalars. Taking $a=b=1$ and $a=1,b=-1$, gives us the two basic unbounded traces\n\\[\n\\ \\ \\qquad \n\\psi_{1} ( V_3^{n} V_1^m ) = e(\\tfrac{p'mn}{2q'})  \\updelta_2^{m}, \\qquad \n\\psi_{2} ( V_3^{n} V_1^m ) = e(\\tfrac{p'mn}{2q'}) (-1)^{n} \\updelta_2^{m} \n\\qquad (q' \\ \\text{even}).\n\\]\nThese are easily verified to be well-defined under $n\\to n+q',$ where $q'$ is even here, and that both are $\\Phi'$-traces. \n\n\\bigskip\n\nIf $q'$ is odd, we get $a = b, \\  d =  (-1)^{p'} c,$ from which we can take $a$ and $c$ to be independent parameters, and equation \\eqref{psiabcd} in this case becomes\n\\begin{align*}\n\\psi( V_3^{n} V_1^m ) &= e(\\tfrac{p'mn}{2q'}) \n\\Big( a\\updelta_2^{m} \\updelta_2^{n} + a\\updelta_2^{m} \\updelta_2^{n-1}\n+ c\\updelta_2^{m-1} \\updelta_2^{n} + c (-1)^{p'} \\updelta_2^{m-1} \\updelta_2^{n-1}\n\\Big)\n\\\\\n& = a e(\\tfrac{p'mn}{2q'}) ( \\updelta_2^{n} + \\updelta_2^{n-1})  \\updelta_2^{m}\n+ c e(\\tfrac{p'mn}{2q'}) ( \\updelta_2^{n} +  (-1)^{p'} \\updelta_2^{n-1})  \\updelta_2^{m-1} \n\\\\\n& = a e(\\tfrac{p'mn}{2q'})  \\updelta_2^{m}\n+ c e(\\tfrac{p'mn}{2q'})  (-1)^{p'n}  \\updelta_2^{m-1} \n\\end{align*}\ngiving us the two basic unbounded traces in the odd $q'$ case\n\\[\n\\psi_1( V_3^{n} V_1^m ) = e(\\tfrac{p'mn}{2q'})  \\updelta_2^{m}, \t\\qquad\n\\psi_2( V_3^{n} V_1^m ) = e(\\tfrac{p'mn}{2q'}) (-1)^{p'n} \\updelta_2^{m-1}.\n\\]\nCombining the two parity cases for $q'$, we can write the two basic $\\Phi'$-traces as in \\eqref{Phiprimetraces} above. (Note that $\\psi_2$ here agrees with the odd $q'$ case, and when $q'$ is even, $p'$ has to be odd so $(-1)^{p'n} = (-1)^{n}$ as in $\\psi_2$ in the even case.) \n\\end{proof}\n\nNotice that by contrast with the unbounded $\\Phi$-trace functionals $\\phi_{jk}$ for the smooth rotation algebra, the $\\Phi'$-traces $\\psi_1,\\psi_2$ are continuous linear functionals on the circle algebra $M_{q'}(C(\\mathbb T))$. Thus, the ``unbounded traces\" here turn out to be bounded.\n\n\\medskip\n\n\\subsection*{Acknowledgement}\nThis paper and \\ccite{WaltersModular} were written at about the time the author retires. He is therefore most grateful to his home institution of 26 years, the University of Northern British Columbia, for many years of research and so much other support. The author expresses his nontrivial gratitude to the many referees who made helpful review reports over the years (including critical ones). Thank you.\n\n\n\\bigskip\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAvalanche processes are present in a vast number of out-of-equilibrium physical phenomena \\citep{Sethna2001,Salje2014a,Bak1996}. These processes are characterized by intense bursts of activity preceded by periods of silence. Some properties that characterize this kind of processes can be described in terms of probability density functions (PDFs) that exhibit lack of finite moments due to their power-law shape. Consequently, fitting PDFs to different avalanche properties such like sizes, energies, or amplitudes is a task that requires a rigorous treatment \\citep{Clauset2009,Deluca2013}. \nOne of the most important features of this kind of functions is their invariance under any scale transformation. This property of scale invariance can be written as $f \\left( \\lambda x \\right) = \\lambda^{-\\gamma}f\\left( x \\right)$ for $x \\in \\left(0,+\\infty \\right)$. The only solution for all $\\lambda$ of this functional equation \\citep{Christensen_Moloney} is a power-law $f(x)=k x^{-\\gamma}$, where the exponent $\\gamma$ can take any real value and $k$ is a constant.\n\nSome experimental works confirm the presence of scale invariance in data by assuming power-law \nbehavior for which\ndata scarcely covers few orders of magnitude \\citep{Friedman2012,Papanikolaou2011,Uhl2015}. The broader the distribution range the more reliable the property of scale invariance in experimental data.\nAmplitude distributions have been studied in different experimental works based on the amplitude of acoustic emission (AE) avalanches \\citep{Vives1994,Petri1994,Vives1995,Carrillo1998,Weiss2007}. However, experimental fitted distributions expand at most two orders of magnitude in voltages \\citep{Vives1994,Weiss2000,Weiss2001,Weiss2007} due to the limitations in the observation windows. Typically the existence of noise and\/or under-counting effects affects the smallest observable values, whereas saturation and\/or lack of statistics due to under-sampling limits the largest observable values. In most cases, these experimental limitations are not sharp due to electronic uncertainties. Recent studies regarding the AE in compression experiments of porous materials \\citep{Baro2013,Nataf2014,Navas-Portella2016}, wood \\citep{Makinen2015}, ethanol-dampened charcoal \\citep{Ribeiro2015}, confined-granular\nmatter under continuous shear \\citep{Lherminier2015}, etc. have focused the attention in the energy \ndistribution of avalanches due to the similarities with the Gutenberg-Richter law for earthquakes \\citep{Serra_Corral}. \n\nIn this work, we provide a procedure to broaden the range of validity of power-law like behavior of the distributions corresponding to avalanche amplitudes and energies. From a set of $n_{\\rm cat}$ catalogs of events whose measured properties span different observation windows, data analysis is performed by assembling them in order to obtain global exponents that characterize the distribution of these avalanche properties. Through this procedure, the fitted global distribution spans a broader range than the one from the fit of every individual catalog.\n\nThis manuscript is organized as follows: In Section \\ref{sec:methodology} an overview of the fitting procedure is shown. In Section \\ref{sec:vycor} we present the experimental methodology in the recording of AE during displacement-driven compression of porous glasses \\citep{Navas-Portella2016}. In Sections \\ref{seq:amplitudes}   and \\ref{seq:energy} avalanche amplitudes and energies are studied respectively by applying the methodology exposed in Sec. \\ref{sec:methodology}. Finally, a brief summary of the results will be presented in Sec. \\ref{Conclusions}.\n\\section{General Methodology}\n\\label{sec:methodology}\nBy considering $n_{\\rm cat}$ catalogs of $N_{i}$ ($i=1,...,n_{\\rm cat}$) events each, corresponding to different experiments (or different observation windows) and characterized by a set of variables (amplitude, energy, duration, etc.), one wants to fit a general power-law type PDF with a global exponent for all the catalogs. Note that, in  the $i$-th catalog, the variable $\\mathcal{X}$ can acquire values in a range typically spanning several orders of magnitude. \nThe first step consist in fitting a power-law PDF in a range $\\left[ a_{i},b_{i} \\right]$ for each catalog via maximum likelihood estimation (MLE) and goodness-of-fit testing \\citep{Deluca2013}. Details of this fitting-procedure are explained in Appendix \\ref{sec:AP1}. By this method we correct for problems close to the limits of the observation windows although discarding some experimental data. In this situation, one may be able to state that, for the $i$-th catalog, the variable $\\mathcal{X}$ follows a power-law PDF $f^{(i)}_{\\mathcal{X}}(x; \\hat{\\gamma}_{i},a_{i},b_{i})$ in a certain range $\\left[a_{i},b_{i}\\right]$ with exponent $\\hat{\\gamma}_{i}$ and a number $\\hat{n}_{i}$ of data entering into the fit $\\left(\\hat{n}_{i} \\leq N_{i}\\right)$. Under these conditions, the next null hypothesis $\\rm H_{0}$ is formulated: the variable $\\mathcal{X}$ is power-law distributed with a global exponent $\\Gamma$ for all the catalogs. \n\nThe log-likelihood function of this global distribution can be written as:\n\\begin{equation}\n\\log \\mathcal{L}= \\sum_{i=1}^{n_{\\rm cat}} \\sum_{j=1}^{\\hat{n}_{i}} \\log f^{(i)}_{\\mathcal{X}} \\left( x_{ij}; \\Gamma, a_{i},b_{i} \\right)\n\\label{eq:general}\n\\end{equation}\n where $x_{ij}$ corresponds to the values of the variable $\\mathcal{X}$ in the $i$-th catalog, $\\hat{n}_{i}$ is the number of data  between $a_{i}$ and $b_{i}$ in the $i$-th catalog and $\\Gamma$ is the global exponent. Since the particular ranges $\\left[ a_{i},b_{i} \\right]$ and the number of data $\\hat{n}_{i}$ are known, one has to find the value of the exponent $\\hat{\\Gamma}$ that maximizes the log-likelihood expression in Eq. (\\ref{eq:general}).\n\nIntuitively, one could be tempted to think that the null hypothesis will not be rejected if the values of the particular exponents $\\gamma_{i}$ do not differ too much. Nevertheless, a more rigorous treatment is required. Statistical procedures, such as a permutational test \\citep{Deluca2016}, could be used in order to check whether the exponents are the same or not. However, since we propose a global distribution characterized by a global exponent $\\hat{\\Gamma}$, a goodness-of-fit test for this global distribution is performed in order to determine whether the null hypothesis can be rejected or not \\citep{Deluca2013,Clauset2009}. If the goodness-of-fit test yields a high enough $p$-value, one is able to state that the variable $\\mathcal{X}$ is power-law distributed with exponent $\\hat{\\Gamma}$ along all the different catalogs or experiments, with ranges $\\left[ a_{i},b_{i} \\right]$ each.  Details of the goodness-of-fit test are exposed in Appendix \\ref{sec:AP4}. In this way, if these intervals span different orders of magnitude,  one can increase the power-law range in several decades. An alternative procedure, where the ranges $\\left[ a_{i},b_{i} \\right]$ are optimized directly from Eq. (\\ref{eq:general}), is disregarded for being enormously computer-time consuming.\n \nThis methodology is applied to avalanche amplitudes and energies on AE data in failure-under-compression experiments of nanoporous silica glasses. Since the experimental set-up records discrete values for the amplitude (in  dB) and almost continuous values of the energy (in aJ), particular expressions for the log-likelihood Eq. (\\ref{eq:general}) as well as the different ways of implementing the goodness-of-fit test will be explained in Sections \\ref{seq:amplitudes} and \\ref{seq:energy}.\n\\section{Failure under compression of porous glasses}\n\\label{sec:vycor}\nUni-axial compression experiments of porous glass Vycor (a nanoporous silica glass with $40\\%$ porosity) are performed in a conventional test machine ZMART.PRO (Zwick\/Roell). Cylindrical samples with no lateral confinement are placed between two plates that approach each other at a certain constant rate $\\dot{z}$. We refer to such a framework as displacement-driven-compression. With the aim of having the same conditions for all the experiments, samples have the same diameters $\\Phi = 4.45$mm and heights $H=8$mm, and the compression rate is fixed at $\\dot{z}=0.005$mm\/min. Before compression, samples were cleaned with a $30\\%$ solution of $\\rm H_{2}O_{2}$, during 24 h and dried at 130$^{\\circ}$C.  Simultaneous to the compression, recording of an AE signal is performed by using a piezoelectric transducer embedded in one of the compression plates. The electric signal $U(t)$ is pre-amplified, band filtered (between 20 kHz and 2 MHz), and analysed by means of a PCI-2 acquisition system from Euro Physical Acoustics (Mistras Group) with an AD card working at 40 Megasamples per second with 18 bits precision \\citep{PCI2}. This should be kept in mind when considering some of the measures as continuous (energy or voltage). Recording of data stops when a big failure event occurs and the sample gets destroyed.\n\nWe prescribe that an AE avalanche event (often called AE hit in specialized AE literature) starts at the time $t_{j}$ when the signal $U(t)$ crosses a fixed detection threshold and finishes at time $t_{j}+\\tau_{j}$ when the signal remains below threshold from $t_{j}+\\tau_{j}$ to at least $t_{j}+\\tau_{j}+200\\mu$s. The amplitude $A$ recorded in  dB follows the expression $A=\\left[20 \\log_{10}\\left( \\vert V \\vert \/ V_{0} \\right) \\right]$, where $V$ is the peak voltage achieved by the AE signal during the event, $V_{0}= 1  \\mu$V is a reference voltage, and the brackets round the value to its nearest integer in  dB. Such a procedure is extensively used in electronic systems.\nNote that in our terminology $A$ will be called amplitude in  dB,\nwhereas the peak voltage $V$ will be refered to simply as amplitude,\nin agreement with previous literature.\nFrom the values of $A$ one can obtain the values $y$ of the discretized peak-voltage:\n\\begin{equation}\ny=g(A)=V_{0}10^{A\/20}.\n\\label{eq:canvia}\n\\end{equation}\nAs the values of $A$ are integer, \nthe values of $y$ will no longer be integer but they will collapse into a set of values $\\lbrace y_{1},y_{2},...,y_{j},...,y_{k}  \\rbrace$ measured in $\\mu V$.\nThe energy $E_{j}$ of each avalanche or event is determined as $E_{j}=\\frac{1}{R}\\int_{t_{j}}^{t_{j}+\\tau} U^{2}(t) dt$ where $R$ is a reference resistance of $10$ k$\\Omega$. At the end of one experiment, one has a catalog or collection of events each of them characterized by a time of occurrence $t$, amplitude in  dB $A$, energy $E$, and duration $\\tau$.\n\n\nIn order to obtain catalogs that span different observation windows, displacement-driven compression experiments with Vycor cylinders have been performed for different values of the pre-amplification and the detection threshold. In this case, $n_{\\rm cat}=4$ experiments have been performed with the following pre-amplification values: 60  dB, 40 dB, 20 dB and 0 dB, and the respective values of the detection threshold 23 dB, 43 dB, 63 dB and 83 dB referring to the signal $U(t)$, not the preamplified signal (in such a way that after preamplification the threshold always moves to $83$ dB). This value of the threshold is as low as possible in order to avoid parasitic noise.\n\nSignal pre-amplification is necessary if one wants to record small AE events. Some values of the pre-amplified signals are so large that can not be detected correctly by the acquisition system. This fact leads to a saturation in the amplitude and, consequently, an underestimated energy of the AE event. This effect can be immediately observed in the distributions of the amplitude in  dB, Fig. \\ref{fig:fig1}, where there is an excess of AE events in the last bin of amplitudes for the experiments at $60$ dB and $40$ dB. Note that the thresholding we perform turns out to be of the same kind as that in Refs.\\citep{Deluca2015,Font-Clos2015}.\n\\section{Amplitudes}\n\\label{seq:amplitudes}\nIn Fig. \\ref{fig:fig1} we show the probability mass functions of the amplitude in  dB for the complete datasets of all the experiments performed at different values of the pre-amplification. \n\\begin{figure}\n\\includegraphics[scale=0.75]{amplitude.eps}\n\\caption{\\label{fig:fig1} Estimated probability mass functions of the amplitude in  dB for the complete datasets of the different experiments performed at different pre-amplifications (PRE). Error bars are estimated as the standard deviation for each bin \\citep{Deluca2013}. }\n\\end{figure}\n\\subsection{Particular fits}\n\nWe consider that, for each experiment, the random variable $\\mathcal{V}$\ncorresponding to amplitude \n(i.e., the peak voltage, whose values are denoted by $V$)\nfollows a truncated continuous power-law distribution,\n\\begin{equation}\nf_{\\mathcal{V}}(V) dV=\\frac{1-\\alpha}{V_{\\rm max}^{1-\\alpha}-V_{\\rm min}^{1-\\alpha}}V^{-\\alpha} dV.\n\\label{eq:ampdis}\n\\end{equation}\nHowever, the true value of $\\mathcal{V}$ is not accessible from the experiments, \nand what we have instead is its discretized counterpart $\\mathcal{Y}$\n(the discretized peak voltage, whose values are denoted by $y$),\nwhich is  concentrated in $k$ discrete values (but not equispaced).\nIn fact, the values $V$ that the variable $\\mathcal{V}$ \ncan take are the real values of the voltages read by the AD card, but they are transformed into  dB, losing precision. \n\nUnder the assumption of a power-law distributed $\\mathcal{V}$, \nwe are able to state that the variable $\\mathcal{Y}$ has probability mass function\n$$\nf_{\\mathcal{Y}}\\left( y\\right) = P\\left( \\mathcal{Y}=  y  \\right) =\nP \\left[ g\\left(A-\\Delta\\right) \\leq \\mathcal{V} < g\\left(A+\\Delta \\right) \\right]\n$$\n\\begin{equation}\n= \\frac{ g^{1-\\alpha}\\left( A + \\Delta \\right) - g^{1-\\alpha}\\left( A - \\Delta \\right) }{V_{\\rm max}^{1-\\alpha}-V_{\\rm min}^{1-\\alpha}}\n=\\frac{2 \\sinh [2.30(1-\\alpha) \\Delta \/ 20]}\n{V_{\\rm max}^{1-\\alpha}-V_{\\rm min}^{1-\\alpha}} \\, \\frac 1 {y^{\\alpha-1\n}},\n\\label{eq:conc}\n\\end{equation}\nwhere $y=g\\left( A \\right)$, \n$\\Delta =0.5$  dB is a vicinity around the values of $A$, \n$2.30 \\simeq \\log 10$,\nand \n$P$ refers to a probability. \nNote that $f_{\\mathcal{Y}}\\left( y\\right)$ is a power law\nbut with exponent $\\alpha-1$.\n\nThe log-likelihood function for a particular experiment can be written as:\n\\begin{equation}\n\\log \\mathcal{L} = \\sum_{\\rm l= A_{min}}^{\\rm A_{max}} \\omega_{l} \\log  f_{\\mathcal{Y}}\\left( g(l) \\right)  = \n  \\sum_{\\rm l= A_{min}}^{\\rm A_{max}} \\omega_{l} \\log P \\left( g\\left(l-\\Delta\\right) \\leq \\mathcal{V} < g\\left(l+\\Delta \\right)  \\right) \n\\label{eq:mleamp}\n\\end{equation}\nwhere $\\rm A_{min}$ and $\\rm A_{max}$ are the values of the amplitude in  dB corresponding to the cutoffs imposed on the sample for the analysis (see Appendix \\ref{sec:AP1} for further details). The frequency $\\omega_{l}$ is the number of events with discretized peak voltage $y_{l}=g(l)$. \nThe next step consists in finding the value of $\\alpha$ that maximizes Eq. (\\ref{eq:mleamp}) using a numerical method. The values of the fitted exponent $\\alpha$ for different values of $A_{\\rm min}$ and $A_{\\rm max}$ are shown in Appendix \\ref{sec:AP3} by using MLE exponent maps \\citep{Baro2012}. Once the exponent is found, one has to determine whether the fit is appropriate to data or not. All the details concerning the fitting procedure and the statistical test are exposed in Appendix \\ref{sec:AP1}.\n\n\nIn Table \\ref{tab:1} we present the fitting values of the particular fits: exponents $\\hat{\\alpha}$, ranges $\\left[ A_{\\rm min}, A_{\\rm max}  \\right]$, number of events $\\hat{n}$ included in the fit and an estimated $p$-value. Numbers in parenthesis in the columns specifying the ranges $\\left[ A_{\\rm min}, A_{\\rm max} \\right]$ correspond to the total range of the sample. Each experiment detects avalanches within $2.8$ decades in amplitude but all the experiments together would yield a total range of $5.8$ decades. This range is broader than other ranges of AE amplitudes \\citep{Petri1994,Vives1995,Carrillo1998,Weiss2000,Koslowski2004,Richeton2005,Weiss2007}.\nIt must be mentioned that performing these particular fits by simply assuming that the discrete variable $\\mathcal{Y}$ directly follows a truncated continuous power-law leads to the rejection of this hypothesis in the goodness-of-fit test. \n\nAccording to the range of detection for the experiment performed at 0  dB, one could be able to observe events up to $139$  dB. Nevertheless, the maximum in this sample corresponds to $123$  dB. Under the hypothesis that a power-law distribution with the same exponent  ($\\hat{\\alpha}=1.61 \\pm 0.04$) can be extended for larger values of the amplitude, the probability $P \\left( 123 \\rm{ dB} < A < 140 \\rm{ dB} \\right)$ turns out to be  $P=0.042$. For $N_i=548$ trials, the probability of having no events in this range can be estimated by $\\left( 1-0.042 \\right)^{548}= 6.2 \\times 10^{-11}$. Based on these simple calculations, one could justify the existence of a corner value due to the finite size of the sample properties \\citep{Serra_Corral}. \nHowever, this corner value would be only visible for the experiment at zero amplification;\nthe same calculation for the experiment at 20 dB gives a probability of having no events\nabove the maximum observed of 0.14, \nwhich is not an extremal value at all.\n\\begin{table}[htbp]\n\\begin{tabular}{| l | r | rr | rr | rr | r |c| }\n\\hline\n\\multicolumn{1}{| l |}{PRE in dB} & \\multicolumn{1}{| c |}{$\\hat{\\alpha}$} & \\multicolumn{2}{| c |}{$A_{\\rm min}$ in dB}  & \\multicolumn{2}{| c |}{$A_{\\rm max}$ in dB} & \\multicolumn{2}{| c |}{$\\hat{n}(N)$} & \\multicolumn{1}{| c |}{$p$-value}  \\\\ \\hline\n60 & $1.743\\pm 0.007$  & $32$ & $(23)$ & $78$ & $(79)$ & 21414 &(28614) & $0.92$  \\\\ \\hline\n40 & $1.75 \\pm 0.01$ & $46$ & $(43)$ & $72$ & $(99)$ & 9146 &(11717) & $0.20$  \\\\ \\hline\n20 & $1.67\\pm 0.04$ & $64$ & $(63)$ & $114$ & $(115)$ & 353 &(376) & $0.50$ \\\\ \\hline\n0 & $1.61\\pm 0.04$ & $84$ & $(83)$ & $122$ & $(123)$ & 528 &(548) & $0.64$ \\\\ \\hline\n\\textbf{Global} & 1.740 $\\pm$ 0.006 & $32$ & $(23)$ & $ 122$ & $(123)$ & 31441 & (41255) & 0.36 \\\\ \\hline\n\\end{tabular}\n\\caption{Fitted parameters for Eq. (\\ref{eq:ampdis}) for each particular experiment and for the global fit. $\\hat{\\alpha}$ corresponds to the fitted exponent in the range $\\left[ A_{\\rm min},A_{\\rm max} \\right]$ for which the goodness-of-fit test exceeds the significance level $p_{c}=0.2$. The error of the exponent is computed as the standard deviation of the MLE \\citep{Deluca2013}. Numbers in parentheses correspond to the maximum and minimum value of the amplitude in  dB for each sample. $\\hat{n}$ is the number of data entering into the fit and $N$ is the total number of events in the dataset. }\n\\label{tab:1}\n\\end{table}\n\n\n\\subsection{Global Fit}\n\\label{seq:globalfita}\nOnce the particular fits have been performed, the ranges for which the power-law hypothesis cannot be rejected are known for each experiment $\\left[ A_{\\rm min_{i}},A_{\\rm max_{i}}\\right]$ (see Table \\ref{tab:1}). For each catalog $i$, we have $\\hat{n}_{i}$ events that follow the distribution in Eq. (\\ref{eq:ampdis}) and we assume that there exists a global exponent $\\hat{\\alpha}_{g}$ that characterizes a global distribution that includes the power-law regimes for all the experiments. \n\nUnder these assumptions, for the particular case of amplitudes in  dB, the general log-likelihood function in Eq. (\\ref{eq:general}) reads:\n\\begin{equation}\n\\begin{split}\n\\log \\mathcal{L} =& \\sum_{i=1}^{n_{\\rm cat}} \\sum_{l=A_{\\rm min_{i}}}^{\\rm A_{max_{i}}}  \\omega_{il} \\log  f^{(i)}_{\\mathcal{Y}}\\left( g(l) \\right) = \\\\\n&  \\sum_{i=1}^{n_{\\rm cat}} \\sum_{l=A_{\\rm min_{i}}}^{\\rm A_{max_{i}}}  \\omega_{il} \\log  P\\left( g\\left(l-\\Delta\\right) \\leq \\mathcal{V}_{i} < g\\left(l+\\Delta \\right) \\right)  \n\\end{split}\n\\label{eq:mleglobalAmplitude}\n\\end{equation}\nwhere $\\omega_{il}$ is the number of events with amplitude in  dB $l$ in the $i$-th experiment from a set of $n_{cat}$ catalogs $\\left( \\sum_{l=A_{\\rm min_{i}}}^{A_{\\rm max_{i}}}  \\omega_{il} = \\hat{n}_{i}   \\right)$. \n\nAfter maximization of the log-likelihood, \nnext step consists in determining whether the null hypothesis of considering a global exponent $\\hat{\\alpha}_{g}$ is compatible with the values of the particular fits shown in Table \\ref{tab:1}. This procedure is explained in more detail in Appendix \\ref{sec:AP4}.\n\\begin{figure}\n\\includegraphics[scale=0.75]{distributiobonaA.eps}\n\\caption{\\label{fig:globalhista} Aggregated amplitude probability density \nof the global distribution with exponent $\\alpha_{g}=1.740\\pm 0.006$, $p$-value$=0.36$, and number of fitted data $\\mathcal{N}=31441$. Error bars are estimated as the standard deviation for each bin \\citep{Deluca2013}. \nBlack solid line shows the fit of a truncated power-law with exponent $\\hat{\\alpha}_{g}=1.740$\nand ranging almost $5$ decades, from $g(32 -\\Delta)$ to $g(122 +\\Delta)$. }\n\\end{figure}\nThe global fit yields a global exponent $\\hat{\\alpha}_{g}=1.740 \\pm 0.006$ with a $p$-value$=0.36$ for $\\mathcal{N}=\\sum_{i=1}^{n_{\\rm cat}}\\hat{n}_{i}=31441$ events. Note that the value of the global exponent is in agreement with the weighted harmonic mean \n$$\n\\hat{\\alpha}_{g}= 1.740 \\simeq 1+\\frac{\\mathcal{N}}{\\sum_{i=1}^{n_{\\rm{cat}}} \\frac{\\hat{n}_{i}}{\\hat{\\alpha}_{i}-1}}= 1.741,\n$$ \nsee Appendix \\ref{sec:AP5} for a justification of this result. This procedure has been tested over simulated power-law data with the same parameters as in Table \\ref{tab:1}, yielding  acceptable $p$-values.\n\n\nFigure \\ref{fig:globalhista} shows the global PDF for the amplitudes and the global fit.  \nObserve how the global exponent is valid along $4.5$ orders of magnitude, giving an unprecedented broad fitting-range in amplitudes.\nThe procedure to construct this aggregated histogram is explained in Appendix \\ref{sec:AP2}.\nAs the estimation of the probability density is done using bins \\citep{Deluca2013}, \nnote that\none can safely replace the unknown values of the random variable $\\mathcal{V}$ \nby the known discretized values of $\\mathcal{Y}$.\nThe only requirement is that the width of the bins is not smaller than the discretization of \n$\\mathcal{Y}$.\n\\section{Energies}\n\\label{seq:energy}\n\\subsection{Particular fits}\nFigure \\ref{fig:rawe} (a) shows the energy distributions for the complete dataset of all the experiments performed at different values of the pre-amplification. \nContrarily to the case of amplitudes, continuous values of the energy are collected \n(see Fig. \\ref{fig:rawe}). Due to the problems of saturation for large amplitudes and the presence of noise for small amplitudes, the energy corresponding to these events is not well estimated. In the following analysis we only consider events whose amplitude lies in  $\\left[ V_{\\rm min_{i}}, V_{\\rm max_{i}} \\right]$, where the ranges are those that have been found in the particulars fits for amplitude PDF in Table \\ref{tab:1} (see Fig. \\ref{fig:rawe} (b)). We propose that the energy follows a truncated continuous power-law PDF:\n\\begin{equation}\nf_{\\mathcal{E}}(E) dE=\\frac{1-\\epsilon}{E_{\\rm max}^{1-\\epsilon}-E_{\\rm min}^{1-\\epsilon}}E^{-\\epsilon} dE.\n\\label{eq:energy}\n\\end{equation}\nBy fixing the values of the range $\\left[ E_{\\rm min}, E_{\\rm max}  \\right]$ we find the value of $\\epsilon$ that maximizes the next log-likelihood function:\n\\begin{equation}\n\\log \\mathcal{L} = n \\log \\left( \\frac{1-\\epsilon}{E_{\\rm max}^{1-\\epsilon}-E_{\\rm min}^{1-\\epsilon}}   \\right) - \\epsilon \\sum_{j=1}^{n} \\log E_{j},\n\\end{equation}\nwhere $E_{j}$ are the particular values of the energy and $n$ is the number of data in $\\left[ E_{\\rm min}, E_{\\rm max} \\right]$. Note that $E_{\\rm min}$ and $E_{\\rm max}$ do not have a direct correspondence with $V_{\\rm min}$ and $V_{\\rm max}$. Explicit details of this particular fit are exposed in Appendix \\ref{sec:AP1}. The values of the fitted exponent $\\hat{\\epsilon}$ for different values of $E_{\\rm min}$ and $E_{\\rm max}$ are shown in Appendix \\ref{sec:AP3} by using MLE exponent maps \\citep{Baro2012}. In Table \\ref{tab:tab2} we present fitting parameters when the minimum significance level is set as $p_{c}=0.20$. \nThe values of the exponents are in rough agreement with the one reported in Ref. \\citep{Baro2013}, \nin particular the value for 60 dB.\n\\begin{figure}\n\\includegraphics[scale=0.65]{distrise.eps}\n\\caption{\\label{fig:rawe} Estimated energy PDFs for the different experiments performed at different pre-amplifications (PRE). (a) Complete datasets. (b) Events with amplitude in the power-law range  $V \\in \\left[ V_{\\rm min_{i}}, V_{\\rm max_{i}} \\right]$. Vertical bars of the same color as the PDFs correspond to the power-law ranges exposed in Table \\ref{tab:tab2}. Error bars are estimated as the standard deviation for each bin \\citep{Deluca2013}.}\n\\end{figure}\n\\begin{table}[htbp]\n\\begin{tabular}{| l | r | rr | rr | rr | r |c| }\n\\hline\n\\multicolumn{1}{| l |}{PRE in dB} & \\multicolumn{1}{| c |}{$\\hat{\\epsilon}$} & \\multicolumn{2}{| c |}{$E_{\\rm min}$ in aJ}  & \\multicolumn{2}{| c |}{$E_{\\rm max}$ in aJ} & \\multicolumn{2}{| c |}{$\\hat{n}(N)$} & \\multicolumn{1}{| c |}{$p$-value}  \\\\ \\hline\n60 & $1.360\\pm 0.004$  & $4.642$ & $(1.001)$ & $10^{5}$ & $(3.005 \\times 10^{5})$ & 16342 &(21414) & $0.43$  \\\\ \\hline\n40 & $1.32 \\pm 0.02$ & $146.780$ & $(3.133)$ & $6.812 \\times 10^{3}$ & $(9.378 \\times 10^{4})$ & 4814 &(9146) & $0.32$  \\\\ \\hline\n20 & $1.29\\pm 0.02$ & $4641.589$ & $(270.446)$ & $2.15 \\times 10^{9}$ & $(1.588 \\times 10^{9})$ & 284 &(353) & $0.33$ \\\\ \\hline\n0 & $1.27 \\pm 0.02$ & $4.642 \\times 10^{5}$ & $(1.545 \\times 10^{4})$ & $10^{10}$ & $(9.156 \\times 10^{9})$ & 396 &(528) & $0.55$ \\\\ \\hline\n\\textbf{Global} & 1.352 $\\pm$ 0.004 & $4.642$ & $(1.001)$ & $ 10^{10}$ & $(9.156 \\times 10^{9})$ & 21836 & (31441) & 0 \\\\ \\hline\n\\end{tabular}\n\\caption{Fitted parameters for Eq. (\\ref{eq:energy}) for each experiment. $\\hat{\\epsilon}$ corresponds to the fitted exponent in the range $\\left[ E_{\\rm min},E_{\\rm max} \\right]$ for which the goodness-of-fit test exceeds the significance level $p_{c}=0.2$. $\\hat{n}$ is the number of data entering into the fit and $N$ is the total number of events in the dataset. Numbers in parentheses refer to \nvalues of the energy in the range\n$\\left[ V_{\\rm min_{i}}, V_{\\rm max_{i}} \\right]$. \nError bars of the exponent correspond to the standard deviation of the MLE.}\n\\label{tab:tab2}\n\\end{table}\n\\subsection{Global fit}\nIn order to write the log-likelihood function of the global fit, one has to consider that each experiment contributes with $\\hat{n}_{i}$ data which are distributed according to Eq. (\\ref{eq:energy}) in the range $\\left[ E_{\\rm min_{i}},E_{\\rm max_{i}} \\right]$ with a global exponent $\\epsilon_{g}$:\n\\begin{equation}\n\\log \\mathcal{L} = \\mathcal{N} \\log \\left(  1-\\epsilon _{g} \\right) - \\epsilon_{g} \\sum_{i=1}^{n_{\\rm cat}} \\sum_{j=1}^{\\hat{n}_{i}} \\log E_{ij} - \\sum_{i=1}^{n_{\\rm cat}} \\hat{n}_{i} \\log \\left( E_{\\rm max_{i}}^{1-\\epsilon_{g}} - E_{\\rm min_{i}}^{1-\\epsilon_{g}}   \\right)\n\\end{equation}\nwhere  $\\hat{n}_{i}$ is the number of data in the $i$-th catalog, $\\mathcal{N}=\\sum_{i}^{n_{\\rm cat}} \\hat{n}_{i}$ and $E_{ij}$ are the values of the energy in each power-law regime $i$. The values of the ranges $\\left[ E_{\\rm min_{i}},E_{\\rm max_{i}} \\right]$ are taken from the particular fits in Table \\ref{tab:tab2}. Details of the goodness-of-fit test for this global fit are explained in Appendix \\ref{sec:AP4}.\nBy considering the particular ranges shown in Table \\ref{tab:tab2}, the global fit of the energy exhibits an exponent $\\hat{\\epsilon}_{g}=1.352\\pm 0.004$ ($\\mathcal{N}=\\sum_{i=1}^{n_{\\rm cat}} \\hat{n}_{i}=21836$) along more than nine decades. As it happens for the case of the global amplitude distribution, the value of the global exponent is in agreement with the weighted harmonic mean $$\\hat{\\epsilon}_{g}= 1.352 \\simeq 1+\\frac{\\mathcal{N}}{\\sum_{i=1}^{n_{\\rm{cat}}} \\frac{\\hat{n}_{i}}{\\hat{\\epsilon}_{i}-1}}=1.347.$$ \nAs we have mentioned, this result is justified in Appendix \\ref{sec:AP5}. \n\nNevertheless, this global fit does not fulfil the goodness-of-fit test and the null hypothesis $\\rm H_{0}$ that all the catalogs share a common exponent $\\hat{\\epsilon}_{g}$ is rejected.\nIn Fig. \\ref{fig:globale} we show the aggregated empirical probability density for the energy of the AE events. This histogram has been constructed following the procedure explained in Appendix \\ref{sec:AP2}. Simulated data with the same parameters as in Table \\ref{tab:tab2} also yield the same rejection of the null hypothesis. \nWe have performed the same analysis without the restriction of just considering events whose amplitude $V \\in \\left[ V_{\\rm min_{i}}, V_{\\rm max_{i}} \\right]$ as well as sparing some catalogs. In all the cases, the rejection of the null-hypothesis occurs.\n\nThis result could be explained by a biased measurement of the energy caused by the interplay between the measured event duration and the detection threshold. The higher the threshold, the shorter the duration and the lower the energy. This fact would not be significant for the case of the amplitudes, since these are independent of the duration, but it should be for the energy since it corresponds to the integrated (squared) AE signal along the registered duration.\n\n\\begin{figure}\n\\includegraphics[scale=0.75]{dis.eps}\n\\caption{\\label{fig:globale} Aggregated empirical energy PDF for all the experiments.  \nError bars are estimated as the standard deviation for each bin \\citep{Deluca2013}. Black solid line corresponds to the fit of the truncated power-law with exponent $\\hat{\\epsilon}_{g}=1.352$ in the range  $\\left[ 4.642 \\mbox{ aJ} , 10^{10} \\mbox{ aJ} \\right]$. }\n\\end{figure}\n\n\\section{Conclusions}\n\\label{Conclusions}\nIn this work we have presented a methodology to estimate a global exponent for the PDF of certain avalanche observables by using different catalogs of events. This methodology has been applied to amplitudes and energies in AE avalanches recorded during different compression experiments of porous glasses. For the case of the amplitude PDF, a global exponent has been found spanning $4.5$ orders of magnitude. To our knowledge, this is the broadest fitting range that has been found for the amplitude distribution of AE events. For the case of the energies, we graphically obtain an apparent power-law spanning $9.5$ decades. However, precise statistical analysis shows that the hypothesis of the existence of a global exponent does not hold. Experimental limitations due to the set of thresholds and definitions of AE avalanches could justify the rejection of the null hypothesis. We expect this methodology to be useful to broaden the range of power-law fits in different distributions that appear in experimental works in condensed matter physics and in other complex systems, for instance, earthquakes.\n\\begin{acknowledgments}\nThe research leading to these results has received founding from ``La Caixa\" Foundation.\nWe also acknowledge financial support \nthrough the ``Mar\\'{\\i}a de Maeztu'' Programme for Units of Excellence in R\\&D (MDM-2014-0445), as well as from projects\nFIS2015-71851-P, MAT2013-40590-P, and the\nProyecto Redes de Excelencia 2015 MAT 2015-69-777-REDT, \nall of them from the Spanish Ministry of Economy and Competitiveness,\nand from 2014SGR-1307 from AGAUR.   \n\\end{acknowledgments}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}