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--- abstract: 'A perturbative QCD treatment of the pion wave function is applied to computing the scattering amplitude for coherent high relative momentum di-jet production from a nucleon.' address: - | School of Physics and Astronomy,\ Tel Aviv University, 69978 Tel Aviv, Israel - | Department of Physics, Box 351560\ University of Washington\ Seattle, WA 98195-1560\ U.S.A. - | Department of Physics, Pennsylvania State University,\ University Park, PA 16802, USA author: - 'L. Frankfurt' - 'G. A. Miller' - 'M. Strikman' title: ' Perturbative Pion Wave function in Coherent Pion-Nucleon Di-Jet Production' --- \#1[[$\backslash$\#1]{}]{} Introduction ============ Consider a process in which a high momentum ($\sim$ 500 GeV/c) pion undergoes a coherent interaction with a nucleus in such a way that the final state consists of two jets (JJ) moving at high transverse relative momentum ($\kappa_\perp>1\sim2 $ GeV/c). In this coherent process, the final nucleus is in its ground state. This process is very rare, but it has remarkable properties[@fms93]. The selection of the final state to be a $q\bar q$ pair plus the nuclear ground state causes the $q\bar q$ component of the pion dominate the reaction process. At very high beam momenta, the pion breaks up into a $q\bar q$ pair well before hitting the nucleus. Since the momentum transfer to the nucleus is very small (almost zero for forward scattering), the only source of high momentum is the gluonic interactions between the quark and the anti-quark. Because $\kappa_\perp$ is large, the quark and anti-quark must be at small separations–the virtual state of the pion is a point-like-configuration[@fmsrev]. But the coherent interactions of a color neutral point-like configuration is suppressed by the cancellation of gluonic emission from the quark and anti-quark[@bb; @fmsrev]. Thus the interaction with the nucleus is very rare, and the pion is most likely to interact with only one nucleon. For this coherent process, the forward scattering amplitude is almost (since the momentum transfer is not exactly zero) proportional to the number of nucleons, $A$ and the cross section varies as $A^2$. This reaction, in which there are no initial or final state interactions, is an example of (color singlet) color transparency [@mueller; @fmsrev]. This is the name given to a high momentum transfer process in which the normal strongly absorbing interactions are absent, and the nucleus is transparent. The term “suppression of a color coherent process" could also be used, because it is the quantum mechanical destructive interference of amplitudes caused by the different color charges of a color singlet that is responsible for the reduced nuclear interaction. The forward angular distribution is difficult to observe, so one integrates the angular distribution, and the $A^2$ variation becomes $\approx A^{4/3}$. But the inclusion of the leading correction to this process, which arises from multiple scatterng of the point-like configuration causes a further increase in the $A$-dependence[@fms93]. Actually at sufficiently small $x_N={2\kappa_t^2\over s}\le {1\over 2m_NR_A}$, the situation changes since the quark-antiquark system scatters off the collective gluon field of the nucleus. Since this field is expected to be shadowed, one expects a gradual disappearance of color transparency for $ x\le 0.01$ - this is the onset of perturbative color opacity [@fms93]. Within the kinematical region of applicability of the QCD factorization theorem, the A dependence of this process is given by the factor: $A^{4/3}\left[G_A(x,Q^2)/G_N(x,Q^2)\right]^2$ [@fms93] Our interest in this curious process has been renewed recently by experimental progress[@danny]. The preliminary result from experiments comparing Pt and C targets is a dependence $\sim A^{1.55\pm0.05}$, qualitatively similar to our 1993 prediction. It is much stronger than the one observed for the soft diffraction of pions off nuclei (for a review and references see [@fmsrev]) , and it is qualitatively different from the behaviour $\sim A^{1/3}$ suggested in [@bb]. Since 1993 many workers have been able to make considerable progress in the theory related to the application of QCD to experimentally relevant observables, and we wish to incorporate that progress and improve our calculation. Our particular aim here is to use perturbative QCD to compute the relevant high-$\kappa_t, q\bar q$ component of wavefunction of the incident pion. We show here that QCD factorization holds for the leading term which dominates at large enough values of $\kappa_\perp$. In the following we discuss the different contributions to the scattering amplitudes as obtained in perturbative QCD. Amplitude for $\pi N\to N JJ$ =============================== Consider the forward ($t=t_{min}\approx 0$) amplitude, ${\cal M}$, for coherent di-jet production on a nucleon $\pi N\to N JJ$[@fms93]: (N)=f,\_,x , \[matel\]where $\widehat{f}$ represents the soft interaction with the target nucleon. The initial $\mid \pi\rangle$ and final $\mid f, \kappa_\perp x \rangle$ states represent the physical states, which generally involve all manner of multi-quark and gluon components. Our notation is that $x$ is the fraction of the total longitudinal momentum of the incident pion, and $1-x$ is the fraction carried by the anti-quark. The transverse momenta are given by $\vec{\kappa}_\perp$ and $-\vec{\kappa}_\perp$. As discussed in the introduction, for large enough values of $\kappa_\perp$, only the $q\bar q$ components of the initial pion and final state wave functions are relevant in Eq. (\[matel\]). This is because we are considering a coherent nuclear process which leads to a final state consisting of a quark and anti-quark moving at high relative transverse momentum. The quark and anti-quark ultimately hadronize at distances far behind the target, and this part of the process is analyzed by the experimentalists using a well-known algorithm[@danny]. We continue by letting the $q\bar q$ part of the Fock space be represented by $\mid \pi\rangle_{q\bar q}$, then \_[q|q]{} =G\_0() V\_[eff]{}\^\_[q|q]{}, \[pieq\] where $G_0(\pi)$ is the non-interacting $q\bar q$ Green’s function evaluated at the pion mass: p\_, yG\_0 ()p’\_,y’= [\^[(2)]{}(p\_-p\_’)(y-y’)m\_\^2- [p\_\^2 +m\_q\^2y(1-y)]{}]{}, where $m_q$ represents the quark mass, $y$ and $y'$ represent the fraction of the longitudinal momentum carried by the quark; and the relative transverse momentum between the quark and anti-quark is $p_\perp$ and $ V_{eff}^\pi$ is the complete effective interaction, which includes the effects of all Fock-space configurations. A similar equation holds for the final state: f,\_,x\_[q|q]{} =\_,x+G\_0(f) V\_[eff]{}\^f f,\_,x \_[q|q]{},\[fstate\] p\_, yG\_0 (f)p’\_,y’= [\^[(2)]{}(p\_-p\_’)(y-y’)m\_f\^2- [p\_\^2 +m\_q\^2y(1-y)]{}]{}, \[gf\] m\_f\^2, in which the first term on the right-hand-side of (\[fstate\]) is the plane-wave part of the wave function. The use of the wave functions (\[pieq\]) and (\[fstate\]) in the equation (\[matel\]) for the scattering amplitude yields $$\begin{aligned} {\cal M}(N)&=&{1\over 2}(T_1+T_2),\nonumber\\ T_1&\equiv& \langle \kappa_\perp,x \mid \widehat{f}\mid \pi\rangle,\quad T_2\equiv _{q\bar q}\langle f,\kappa_\perp,x \mid V_{eff}^f G_0(f)\widehat{f} \mid \pi\rangle_{q\bar q}.\label{tdef}\end{aligned}$$ The term $T_2$ includes the effect of the final state $q\bar q$ interaction; this was not included in our 1993 calculation[@fms93], but its importance was stressed in [@jm] . We shall first evaluate $T_1$, and then turn to $T_2$. Evaluation of $T_1$ =================== The wave function $\mid \pi\rangle_{q\bar q}$ is dominated by components in which the separation between the constituents is of the order of the diameter of the physical pion, but there is a perturbative tail which accounts for short distance part of the pion wave function. This perturbative tail is relevant here because we need to take the overlap with the final state which is constructed from constituents moving at high relative momentum. If we concentrate on those aspects it is reasonable to consider only the one gluon exchange contribution $V^g$ to $V_{eff}^\pi$ and pursue the Brodsky-Lepage analysis [@BL] for the evaluation of this particular component. Their use of the light cone gauge $A^+=0$, simplifies the calculation. We also use their normalization and phase-space conventions. We want to draw attention to the issue of gauge invariance. The pion wave function is not gauge invariant, but the sum of two-gluon exchange diagrams for the pion transition to $q\bar q$ is gauge invariant. This is because only the imaginary part of the scattering amplitude survives in the sum of diagrams, and because two exchanged gluons are vector particles in a color singlet state as a consequence of Bose statistics. So, in this case, conservation of color current has the same form as conservation of electric current in QED. We also note, that in the calculation of hard high-momentum transfer processes, the $q\bar q$ pair in the non-perturbative pion wave function should be considered on energy shell. Corrections to this enter as an additional factor of $1\over \kappa_t^2$ in the amplitude. We define the non-perturbative part of the momentum space wave function as ( l\_,y)l\_, y\_[q|q]{}. We use the one-gluon exchange approximation to the exact wave function of Eq. (\[pieq\]) to obtain an approximate wave funtion, $\chi$, valid for large values of $k_\perp$. $$\begin{aligned} \chi(k_\perp,x)={-4\pi C_F} {1\over m_\pi^2-{ k_\perp^2 +m_q^2\over x(1-x)}} \int_0^1 dy\int {d^2 l_\perp\over(2\pi)^3} V^g(k_\perp,x;l_\perp,y) %%%need to put in wave fucntion \psi(l_\perp,y)\end{aligned}$$ with $$\begin{aligned} % V^g(q_\perp,x;l_\perp,y)=-4\pi C_F\alpha_s %{\bar u (x,q_\perp)\over\sqrt{x}}\gamma_\mu {u(y,l_\perp)\over \sqrt{y}} %{\bar v (x,-q_\perp)\over\sqrt{1-x}}\gamma_\nu % {1\over m_\pi^2-{q_\perp^2-m_q^2\over x} -{l_\perp^2+m_q^2\over 1-y} % -{(q_\perp-l_\perp)^2\over y-x}} +(x\to 1-x, y\to % -{(k_\perp-l_\perp)^2\over y-x}} +(x\to 1-x, y\to %%%\nonumber %\psi(l_\perp,y) V^g(k_\perp,x;l_\perp,y)=-4\pi C_F\alpha_s {\bar u (x,k_\perp)\over\sqrt{x}}\gamma_\mu {u(y,l_\perp)\over \sqrt{y}} {\bar v (x,-k_\perp)\over\sqrt{1-x}}\gamma_\nu {v(1-y,-l_\perp)\over \sqrt{1-y}}d^{\mu\nu} \nonumber\\ \times \left[ {\theta(y-x)\over y-x} {1\over m_\pi^2-{k_\perp^2-m_q^2\over x} -{l_\perp^2+m_q^2\over 1-y}} % LF check 1-y) \right], \end{aligned}$$ and $C_F={n_c^2-1\over 2n_c}={4\over 3}$. The range of integration over $l_\perp$ is restricted by the non-perturbative pion wave function $\psi$. Then we set $l_\perp$ to 0 everywhere in the spinors and energy denominators and evaluate the strong coupling constant at $k_\perp^2$: \_s(k\_\^2)=[4]{} ,where $\beta=11-{2\over 3}n_f$. Then $$\begin{aligned} V^g(k_\perp,x;l_\perp,y)\approx {-4\pi C_F\alpha_s(k_\perp^2)\over x(1-x) y(1-y)}V^{BL}(x,y)\end{aligned}$$ where $V^{BL}(x,y)$ is the Brodsky-Lepage kernal: V\^[BL]{}(x,y)=2,with the operator $\Delta$ defined by ${\Delta \over x-y}\phi(x)={\phi(x)-\phi(y)\over x-y}$. This kernal includes the effects of vertex and quark mass renormalization. The net result for the high $k_\perp$ component of the pion wave function is then (k\_)=[4C\_F \_s(k)k\_\^2]{} \_0\^1 dy V\^[BL]{}(x,y)[(y,k\_\^2)y(1-y)]{} \[pieq1\], where $$\phi(y,q_\perp^2)\equiv \int {d^2 l_\perp\over(2\pi)^3} \theta(q_\perp^2-l_\perp^2)\psi(l_\perp,y).$$ The quark distribution amplitude $\phi$ can be obtained using QCD evolution[@BL]. Furthermore, the analysis of experimental data for virtual Compton scattering and the pion form factor performed in [@tolya; @kroll] shows that this amplitude is not far from the asymptotic one for $k^2_\perp\ge 2-3$ GeV$^2$ (x)=a\_0x(1-x), \[phi\]where $a_0=\sqrt{3}f_\pi$ with $f_\pi\approx 93$ MeV. Equation (\[pieq1\]) represents the high relative momentum part of the pion wave function. Using the asymptotic function (\[phi\]) in Eq. (\[pieq1\]) leads to an expression for $\psi(k_\perp,x)\propto x(1-x)/k^2_\perp$ which is of the factorized form used in Ref. [@fms93]. To compute the amplitude $T_1$, it is necessary to specify the scattering operator $\widehat f.$ For high energy scattering the operator $\widehat {f}$ changes only the transverse momentum and therefore in the coordinate space representation $\widehat f$ depends on $b^2$. The transverse distance operator $\vec b = (\vec{b}_{q}-\vec{b}_{\bar{q}})$ is canonically conjugate to $\vec{\kappa}_\perp$. At sufficiently small values of $b$, the leading twist effect and the dominant term at large $s$ arises from the diagrams when pion fragments into two jets as a result of interactions with the two-gluon component of gluon field of a target, see Figure 1. The perturbative QCD determination of this interaction involves a diagram similar to the gluon fusion contribution to the nucleon sea-quark content observed in deep inelastic scattering. One calculates the box diagram for large values of $\kappa_\perp$ using the wave function of the pion instead of the vertex for $\gamma^*\to q\bar q$. The application of QCD factorization theorem leads [@fmsrev; @bbfs; @frs] to $$%\sigma^{q \bar{q}}_{T} \widehat f(b^2)=i s \frac{\pi^2}{3} b^2 \left[ x_N G_N(x_N, \lambda/b^2) \right] \alpha_{s}(\lambda/b^2), \label{eq:1.27}$$ in which $x_N=2\kappa_\perp^2/s$ where $G_N$ is the gluon distribution function of the nucleon, and $\lambda(x=10^{-3})=9$ according to Frankfurt, Koepf, and Strikman, [@fks]. Accounting for the difference between the pion mass and mass of two jet system requires us to replace the target gluon distribution by the the skewed gluon distribution. The difference between both distributions is calculable in QCD using the QCD evolution equation for the skewed parton distributions[@ffs; @fg]. The most important effect shown in Eq. (\[eq:1.27\]) is the $b^2$ dependence which shows the diminishing strength of the interaction for small values of $b$. In the leading order approximation it is legitimate to rewrite $\sigma$ in the form: (b\^2)=is \_0 [b\^2b\_0\^2 ]{} \[fb2\] in which the logarithmic dependence on $b^2$ is neglected. Our notation is that $ \langle b_0^2 \rangle$ represents the pionic average of the square of the transverse separation, and \_s(\_\^2) \[x\_NG\_N\^[(skewed)]{} (x\_N,\_\^2)\] . The use of Eq. (\[fb2\]) allows a simple evaluation of the scattering amplitude $T_1$ because the $b^2$ operator acts as $-\nabla_{\kappa_\perp}^2$. Using Eqs. (\[pieq\]) and (\[fb2\]) in Eq. (\[tdef\]), leads to the result: T\_1=-4i[\_0b\^2 ]{}[4C\_F\_s(\_\^2) \_\^4]{} (1+[1]{})a\_0x(1-x). \[t1\] This is, except for the small $1\over \ln{\kappa_\perp^2\over \Lambda^2}$ correction ($\kappa_\perp\approx 2$ GeV and $\Lambda\approx 0.2$) arising from taking $-\nabla_{\kappa_\perp}^2$ on $\alpha_s$, is of the same form as the corresponding result of our 1993 paper. The amplitude of Ref. [@bb] varies as a Gaussian in $\kappa_\perp$. The $\kappa_\perp$ dependence: ${d\sigma(\kappa_\perp)\over d\kappa_\perp^2}\propto {1\over \kappa_\perp^8}$ follows from simple reasoning. The probability to find a pion at $b\le {1\over \kappa_\perp}$ is $ \propto b^2$, while the square of the total cross section for small dipole-nucleon interactions is $\propto b^4$. Hence the cross section of productions of jets with sufficiently large values of $\kappa_\perp $ is $\propto {1\over \kappa_\perp^6}$ leading to a differential cross section $\propto {1\over \kappa_\perp{^8}}$. Similar counting can be applied to estimate the $\kappa_\perp$ dependence for diffraction of a nucleon into three jets. Other amplitudes ================ So far we have emphasized that the amplitude we computed in 1993 is calculable using perturbative QCD. However there are four different contributions which occur at the same order of $\alpha_s$. The previous term in which the interaction with the target gluons follows the gluon-exchange represented by the potential $V^g$ in the pion wave function has been denoted by $T_1$. But there is also a term, in which the interaction with the target gluons occurs before the action of $V^g$ is denoted as $T_2$, see Figure 2. However, the two gluons from the nuclear target can also be annihilated by the exchanged gluon (color current of the pion wave function). This amplitude, denoted as $T_3$, is shown in Figure 3. The sum of diagrams when one target gluon is attached before the potential $V^g$ and a second after the potential $V^g$, see E.g. Figure  4, corresponds to an amplitude, $T_4$. We briefly discuss each of the remaining terms $T_2,T_3,T_4$. Their detailed evaluation will appear in a later publication. However we state at the outset that each of these amplitudes is suppressed by color coherent effects, and that each has the same $\kappa_\perp^{-4}$ dependence. The existence of the term $T_2$, which uses an interaction that varies as $b^2$, caused Jennings & Miller[@jm] to worry that the value of ${\cal M}_N$ might be severely reduced due to a nearly complete cancellation. Our preliminary and incomplete estimate obtained by neglecting the term arising from differentiation of the potential, $V^g$ finds instead enhancement. The $T_3$ or meson-color-flow term arises from the attachment of both target gluons to the gluon appearing in $V^g$ as well as the sum of diagrams where one target gluon is attached to potential $V^g$ in the pion wave function and another gluon is attached to a quark. This term is suppressed by color coherent destructive interference caused by the color neutrality of the $q\bar{q} g$ intermediate state. Thus this term has a form which is very similar to that of $T_1$ and $T_2$. The $T_4$ term arises from the sum of diagrams when one target gluon interacts with a quark in the pion wf before exchange by potential $V^g$ and second gluon interacts after that. The sum of these diagrams seems to be O because, for our kinematics, the $q$ and $\bar q$ in the initial and final states are not causally connected in these diagrams. The mathematical origin of this near 0 arises from the sum of diagrams having the form of contour integral: $\int d\nu \frac{1}{(\nu-a-i\epsilon)(\nu-b-i\epsilon)},$ which vanishes because one can integrate using a closed contour in the lower half complex $\nu$-plane. Summary Discussion ================== The purpose of this paper has been to show how to apply leading-order perturbative QCD to computing the scattering amplitude for the process: $\pi N\to JJ$. The high momentum component of the pion wave function, computable in perturbation theory is an essential element of the amplitude. Another essential feature of our result (20) is the $\sim {1\over \kappa_\perp^4}$ dependence of the amplitude manifest in a $\kappa_\perp^{-8}$ behavior of the cross section. This feature needs to be observed experimentally before one can be certain that the experiment [@danny] has verified the prediction of Ref. [@fms93]. Aknowledgements =============== It is a pleasure to dedicate this work to Prof. Kurt Haller, who has been recently interested in color transparency[@kh1], and who has long been interested in the fundamentals of QCD as applied to light-cone physics[@kh2]. Happy birthday, Kurt, and our best wishes for many more to come. This work has been supported in part by the USDOE. L. Frankfurt, G.A. Miller, M.Strikman, Phys. Lett. [**B304**]{} 1, (1993) See the review: L.L. Frankfurt, G.A. Miller and M. Strikman, Ann. Rev. Nucl. Part. Sci. [**44**]{}, 501 (1994) hep-ph/9407274. G.F. Bertsch, S.J. Brodsky, A.S. Goldhaber and J.F. Gunion, Phys. Rev. Lett. [**47**]{}, 297 (1981). A.H. Mueller, in Proceedings of 17’th Rencontre de Moriond, MOriond, 1982, ed. J. Tran Thanh Van (Editions Frontieres, Gif-sur-Yvette, France, 1982) p.13 Fermilab experiment E791, R. Weiss-Babai, talk at Hadron ’97 and D. Ashery, talk at International Workshop of Diffractive physics, Rio de Janeiro, Feb. 1998. B.K. Jennings and G.A. Miller Phys.Rev. [**C50**]{}, 3018 (1994). S.J. Brodsky G.P. Lepage, Phys. Rev. [**D22**]{}, 2157 (1982) B.Blattel, G.Baym, L.L.Frankfurt, and M.Strikman, Phys. Rev. Lett. [**71**]{} 896 (1993). L. Frankfurt, A. Radyushkin, and M. Strikman, Phys. Rev. [**D55**]{} ,98 (1997) L. Frankfurt, W. Koepf, M. Strikman, Phys. Rev. [**D54**]{}, 3194 (1996) L. Frankfurt, A. Freund, V. Guzey, M. Strikman Phys. Lett. [**B418**]{} 345,1998, Erratum-ibid.B429:414,1998. A. Freund, V. Guzey, hep-ph - 9801388; hep-ph - 9806267. A.V. Radyushkin, hep-ph/9707335; A. Szczepaniak, A. Radyushkin and C. Ji, Phys. Rev. [**D57**]{}, 2813 (1998) hep-ph/9708237; I.V. Musatov and A.V. Radyushkin, Phys. Rev. [**D56**]{}, 2713 (1997) hep-ph/9702443. P. Kroll and M. Raulfs, Phys. Lett. [**B387**]{}, 848 (1996) hep-ph/9605264. L. Chen and K. Haller, “Quark confinement and color transparency in a gauge invariant formulation of QCD," hep-th/9803250. K.. Haller, “Gauge Theories In The Light Cone Gauge," Phys. Rev. [**D42**]{}, 2095 (1990). [Figure 1. Contribution to $T_1$. The high momentum component of the pion interacts with the two-gluon field of the target. Only a single diagram of the four that contribute is shown.]{} [Figure 2. Contribution to $T_2$. The high momentum component of the final $q\bar q$ pair interacts with the two-gluon field of the target. Only a single diagram of the four that contribute is shown.]{} [Figure 3. Contribution to $T_3$. The exchanged gluon interacts with the two-gluon field of the target. Only a single diagram of the several that contribute is shown.]{} [Figure 4. Contribution to $T_4$. A gluon interacts with a quark and another with the exchanged gluon. Only a single diagram of the several that contribute is shown.]{}

--- abstract: 'We study direct CP violating asymmetries (CPAs) in the two-body $\Lambda_b$ decays of $\Lambda_b\to pM(V)$ with $M(V)=K^-(K^{*-})$ and $\pi^-(\rho^-)$ based on the generalized factorization method. After simultaneously explaining the observed decay branching ratios of $\Lambda_b\to (p K^-\,,\; p \pi^-)$ with ${\cal R}_{\pi K}\equiv {\cal B}(\Lambda_b\to p \pi^-)/{\cal B}(\Lambda_b\to p K^-)$ being $0.84\pm 0.09$, we find that their corresponding direct CPAs are $(5.8\pm 0.2,\,-3.9\pm 0.2)\%$ in the standard model (SM), in comparison with $(-5^{+26}_{-~5},\, -31^{+43}_{-~1})\%$ and $(-10\pm8\pm4,\, 6\pm7\pm3)\%$ from the perturbative QCD calculation and the CDF experiment, respectively. For $\Lambda_b\to ( p K^{*-},\, p\rho^-)$, the decay branching ratios and CPVs in the SM are predicted to be $(2.5\pm0.5,\,11.4\pm2.1)\times 10^{-6}$ with ${\cal R}_{\rho K^*}=4.6\pm0.5$ and $(19.6\pm1.6,\, -3.7\pm0.3)\%$, respectively. The uncertainties for the CPAs in these decay modes as well as ${\cal R}_{\pi K,\,\rho K^{*-}}$ mainly arise from the quark mixing elements and non-factorizable effects, whereas those from the hadronic matrix elements are either totally eliminated or small. We point out that the large CPA for $\Lambda_b\to p K^{*-}$ is promising to be measured by the CDF and LHCb experiments, which is a clean test of the SM.' author: - 'Y.K. Hsiao$^{1,2}$ and C.Q. Geng$^{1,2,3}$' title: 'Direct CP violation in $\Lambda_b$ decays' --- introduction ============ It is known that one of the main goals in the $B$ meson factories is to confirm the weak CP phase in the Cabbibo-Kobayashi-Maskawa (CKM) paradigm [@CKM] of the Standard Model (SM) through CP violating effects. Needless to say that the origin of CP violation is the most fundamental problem in physics, which may also shed light on the puzzle of the matter-antimatter asymmetry in the Universe. However, the direct CP violating asymmetries (CPAs), ${\cal A}_{CP}$, in $B$ decays have not been clearly understood yet. In particular, the naive result of ${\cal A}_{CP}(\bar B^0\to K^-\pi^+)\simeq {\cal A}_{CP}(B^-\to K^-\pi^0)$ in the SM, cannot be approved by the experiments [@Li_Kpi]. It is known that it is inadequate to calculate the direct CPAs in the two-body mesonic $B$ decays due to the limited knowledges on strong phases [@Hou:2006jy]. Clearly, one should look for CPV effects in other processes, in which the hadronic effects are well understood. Unlike the two-body $B$ meson decays, due to the flavor conservation, there is neither color-suppressed nor annihilation contribution in the two-body baryonic modes of $\Lambda_b\to p K^-$ and $\Lambda_b \to p \pi^-$, providing the controllable nonfactorizable effects and traceable strong phases for the CPAs. In fact, their decay branching ratios have been recently observed, given by [@pdg] $$\begin{aligned} \label{exbr} {\cal B}(\Lambda_b\to p K^-)&=&(4.9\pm 0.9)\times 10^{-6}\,,\nonumber\\ {\cal B}(\Lambda_b\to p \pi^-)&=&(4.1\pm 0.8)\times 10^{-6}\,.\end{aligned}$$ Although the two decays have been extensivily discussed in the leterature [@Lu:2009cm; @Wei:2009np; @Wang:2013upa], the measured values in Eq. (\[exbr\]) cannot be simultaneously explained in the studies. In this paper, we will first examine these two-body baryonic decays based on the configuration of the $\Lambda_b\to p$ transition with a recoiled $K$ or $\pi$, and then calculate ${\cal A}_{CP}(\Lambda_b\to p K^-,p\pi^-)$, which have been measured by the CDF collaboration [@Aaltonen]. We will also extend our study to the corresponding vector modes of $\Lambda_b\to p V$ with $V=K^{*-}(\rho^-)$ as well as other two-body beauty baryons (${\cal B}_b$) decays, such as $\Xi_b$, $\Sigma_b$ and $\Omega_b$. Formalism ========= ![Contributions to $\Lambda_b\to pM(V)$ from (a) color-allowed tree-level and (b) penguin diagrams.[]{data-label="LbtopM"}](LbtopM1.eps "fig:"){width="2.5in"} ![Contributions to $\Lambda_b\to pM(V)$ from (a) color-allowed tree-level and (b) penguin diagrams.[]{data-label="LbtopM"}](LbtopM2.eps "fig:"){width="2.5in"} According to the decaying processes depicted in Fig. \[LbtopM\], in the generalized factorization approach [@ali] the amplitudes of $\Lambda_b\to p M(V)$ with $M(V)=K^-(K^{*-})$ and $\pi^-(\rho^-)$ can be derived as $$\begin{aligned} \label{eq1} {\cal A}(\Lambda_b\to p M)&=&i\frac{G_F}{\sqrt 2}m_b f_M\bigg[ \alpha_{M}\langle p|\bar u b|\Lambda_b\rangle+ \beta_{M}\langle p|\bar u\gamma_5 b|\Lambda_b\rangle\bigg]\,, \nonumber\\ {\cal A}(\Lambda_b\to p V)&=&\frac{G_F}{\sqrt2}m_{V}f_{V} \varepsilon^{\mu*}\alpha_{V}\langle p|\bar u\gamma_\mu(1-\gamma_5) b|\Lambda_b\rangle\;,\end{aligned}$$ where $G_F$ is the Fermi constant and the meson decay constants $f_{M(V)}$ are defined by $\langle M|\bar q_1\gamma_\mu \gamma_5 q_2|0\rangle=-if_M q_\mu~$ and $\langle V|\bar q_1\gamma_\mu q_2|0\rangle=m_{V} f_{V}\varepsilon_\mu^*$ with the four-momentum $q_\mu$ and polarization $\varepsilon_\mu^*$, respectively. The constants $\alpha_{M}$ ($\beta_M$) and $\alpha_{V}$ in Eq. (\[eq1\]) are related to the (pseudo)scalar and vector or axialvector quark currents, given by $$\begin{aligned} \label{eq2} \alpha_{M}(\beta_{M})&=& V_{ub}V_{uq}^*a_1-V_{tb}V_{tq}^*(a_4\pm r_M a_6)\;,\nonumber\\ \alpha_{V}&=& V_{ub}V_{uq}^*a_1-V_{tb}V_{tq}^*a_4\;,\end{aligned}$$ where $r_M\equiv {2 m_M^2}/[m_b (m_q+m_u)]$, $V_{ij}$ are the CKM matrix elements, $q=s$ or $d$, and $a_i\equiv c^{eff}_i+c^{eff}_{i\pm1}/N_c^{(eff)}$ for $i=$odd (even) are composed of the effective Wilson coefficients $c_i^{eff}$ defined in Ref. [@ali]. We note that, as seen from Fig. \[LbtopM\], there is no annihilation diagram at the penguin level for $\Lambda_b\to p M(V)$, unlike the cases in the two-body mesonic $B$ decays. In addition, without the color-suppressed tree-level diagram, the non-factorizable effects in these baryonic decays can be modest. In order to take account of the non-factorizable effects, we use the generalized factorization method by setting the color number as $N_c^{eff}$, which floats from 2 to $\infty$. The matrix elements of the ${\cal B}_b\to {\cal B}$ baryon transition in Eq. (\[eq1\]) have the general forms: $$\begin{aligned} &&\langle {\cal B}|\bar q \gamma_\mu b|{\cal B}_b\rangle= \bar u_{\cal B}[f_1\gamma_\mu+\frac{f_2}{m_{{\cal B}_b}}i\sigma_{\mu\nu}q^\nu+ \frac{f_3}{m_{{\cal B}_b}}q_\mu] u_{{\cal B}_b}\,,\nonumber\\ &&\langle {\cal B}|\bar q \gamma_\mu\gamma_5 b|{\cal B}_b\rangle= \bar u_{\cal B}[g_1\gamma_\mu+\frac{g_2}{m_{{\cal B}_b}}i\sigma_{\mu\nu}q^\nu+ \frac{g_3}{m_{{\cal B}_b}}q_\mu]\gamma_5 u_{{\cal B}_b}\,,\nonumber\\ &&\langle {\cal B}|\bar q b|{\cal B}_b\rangle=f_S \bar u_{\cal B} u_{{\cal B}_b}\,, \langle{\cal B}|\bar q \gamma_5 b|{\cal B}_b\rangle=f_P \bar u_{\cal B}\gamma_5 u_{{\cal B}_b}\,,\end{aligned}$$ where $f_j$ ($g_j$) ($j=1,2,3,S$ and $P$) are the form factors. For the $\Lambda_b\to p$ transition, $f_j$ and $g_j$ from different currents can be related by the $SU(3)$ flavor and $SU(2)$ spin symmetries [@Brodsky1; @Chen:2008sw], giving rise to $f_1=g_1$ and $f_{2,3}=g_{2,3}=0$. These relations are also in accordance with the derivations from the heavy-quark and large-energy symmetries in Ref. [[@CF]]{}. Note that the helicity-flip terms of $f_{2,3}$ and $g_{2,3}$ vanish due to the symmetries. Moreover, as shown in Refs. [@CF; @Wei:2009np; @Gutsche:2013oea], $f_{2,3}$ $(g_{2,3})$ can only be contributed from the loops, resulting in that they are smaller than $f_1(g_1)$ by one order of magnitude, and can be safely ignored. By equation of motion, we get $f_S=[(m_{{\cal B}_b}-m_{\cal B})/(m_b-m_q)] f_1$ and $f_P=[(m_{{\cal B}_b}+m_{\cal B})/(m_b+m_q)] g_1$. In the double-pole momentum dependences, $f_1$ and $g_1$ are in the forms of $$\begin{aligned} f_1(q^2)=\frac{C_F}{(1-q^2/m_{{\cal B}_b}^2)^2}\,,\; g_1(q^2)=\frac{C_F}{(1-q^2/m_{{\cal B}_b}^2)^2}\,,\end{aligned}$$ with $C_F\equiv f_1(0)=g_1(0)$. To calculate the branching ratio of $\Lambda_b\to pM$ or $pV$, we take the averaged decay width $\Gamma\equiv (\Gamma_{M(V)}+\Gamma_{\bar M(\bar V)})/2$ with $\Gamma_{M(V)}$ ($\Gamma_{\bar M(\bar V)}$) for $\Lambda_b\to p M(V)$ ($\bar \Lambda_b\to \bar p\bar M(\bar V)$). From Eq. (\[eq1\]) and Eq. (\[eq2\]), we can derive the ratios $$\begin{aligned} \label{RpiK} {\cal R}_{\pi K} &\equiv &\frac{{\cal B}(\Lambda_b\to p \pi^-)}{{\cal B}(\Lambda_b\to p K^-)} =\frac{f_\pi^2}{f_K^2} \frac{|\alpha_\pi|^2+|\alpha_{\bar \pi}|^2}{|\alpha_K|^2+|\alpha_{\bar K}|^2} \frac{1+\xi^+_{\pi}}{1+\xi^+_{K}}\,, \nonumber\\ {\cal R}_{\rho K^*}&\equiv & \frac{{\cal B}(\Lambda_b\to p \rho^-)}{{\cal B}(\Lambda_b\to p K^{*-})} =\frac{f_\rho^2}{f_{K^*}^2 } \frac{|\alpha_\rho|^2+|\alpha_{\bar\rho}|^2}{|\alpha_{K^*}|^2+|\alpha_{\bar K^*}|^2}\,,\end{aligned}$$ where $\xi^+_M$ ($M=\pi,\,K$) are defined by $$\begin{aligned} \label{Rbeta} \xi^\pm_M\equiv \left(\frac{|\beta_M|^2\pm |\beta_{\bar M}|^2}{|\alpha_M|^2+|\alpha_{\bar M}|^2}\right)R_{\Lambda_b\to p}\,,\end{aligned}$$ with $R_{\Lambda_b\to p}={|\langle p|\bar u\gamma_5 b|\Lambda_b\rangle|^2}/ {|\langle p|\bar u b|\Lambda_b\rangle|^2}$, representing the uncertainty from the hadronization. The direct CP asymmetry is defined by $$\begin{aligned} \label{Acp} {\cal A}_{CP}= \frac{\Gamma_{M(V)}-\Gamma_{\bar M(\bar V)}}{\Gamma_{M(V)}+\Gamma_{\bar M(\bar V)}}\,. %\end{aligned}$$ Explicitly, from Eqs. (\[eq1\]), (\[eq2\]) and (\[Acp\]), we obtain $$\begin{aligned} \label{Acp2} {\cal A}_{CP}(\Lambda_b\to pM)&=& \left(\frac {|\alpha_M|^2-|\alpha_{\bar M}|^2} {|\alpha_M|^2+|\alpha_{\bar M}|^2}+\xi^-_M\right){1\over 1+\xi^+_M}\,,\nonumber\\ {\cal A}_{CP}(\Lambda_b\to pV)&=& \frac{|\alpha_V|^2-|\alpha_{\bar V}|^2}{|\alpha_V|^2+|\alpha_{\bar V}|^2}\,.\end{aligned}$$ It is interesting to point out that for ${\cal R}_{\rho K^*}$ in Eq. (\[RpiK\]), there is no uncertainty from the $\Lambda_b\to p$ transition, while both mesonic and baryonic uncertainties are totally eliminated for ${\cal A}_{CP}(\Lambda_b\to pV)$ in Eq. (\[Acp2\]). Even for ${\cal R}_{\pi K}$ and ${\cal A}_{CP}(\Lambda_b\to pM)$, we will demonstrate later that the hadron uncertainties can be limited. Numerical Results and Discussions ================================== For the numerical analysis, the theoretical inputs of the meson decay constants and the Wolfenstein parameters for the CKM matrix are taken as [@pdg] $$\begin{aligned} % &&(f_\pi,f_K,f_\rho,f_{K^*})=(130.4\pm 0.2,\,156.2\pm 0.7,\,210.6\pm 0.4,\,204.7\pm 6.1)\,\text{MeV}\,,\nonumber\\ &&(\lambda,\,A,\,\rho,\,\eta)= % (0.225,\,0.814,\,0.120\pm 0.022,\,0.362\pm 0.013)\,.\end{aligned}$$ We note that $f_{\rho,K^*}$ are extracted from the $\tau$ decays of $\tau^-\to (\rho^-,K^{*-})\nu_\tau$, and $V_{ub}=A\lambda^3 (\rho-i\eta)$ and $V_{td}=A \lambda^3 (1 - i \eta - \rho)$ are used to provide the weak phase for CP violation, while the strong phases are coming from the effective Wilson coefficients $c^{eff}_i$ ($i=1,2,3, ...,6$). Explicitly, at the $m_b$ scale, one has that [@ali] $$\begin{aligned} c^{eff}_1&=&1.168,\;\; c^{eff}_2=-0.365\,,\nonumber\\ 10^4 \epsilon_1 c^{eff}_3&=&64.7+182.3 \epsilon_1\mp 20.2\eta - 92.6\rho +27.9\epsilon_2\nonumber\\ &&+i(44.2-16.2 \epsilon_1\mp 36.8\eta -108.6\rho + 64.4 \epsilon_2),\,\nonumber\\ 10^4 \epsilon_1 c^{eff}_4&=&-194.1-329.8 \epsilon_1\pm 60.7\eta +277.8\rho -83.7\epsilon_2\nonumber\\ &&+i(-132.6+48.5 \epsilon_1\pm 110.4\eta +325.9\rho -193.3 \epsilon_2),\,\nonumber\\ 10^4 \epsilon_1 c^{eff}_5&=&64.7+89.8 \epsilon_1\mp 20.2\eta - 92.6\rho +27.9\epsilon_2\nonumber\\ &&+i(44.2-16.2 \epsilon_1\mp 36.8\eta -108.6\rho + 64.4 \epsilon_2),\,\nonumber\\ 10^4 \epsilon_1 c^{eff}_6&=&-194.1-466.7 \epsilon_1\pm 60.7\eta +277.8\rho -83.7\epsilon_2\nonumber\\ &&+i(-132.6+48.5 \epsilon_1\pm 110.4\eta +325.9\rho -193.3 \epsilon_2),\,\end{aligned}$$ for the $b\to d$ ($\bar b\to \bar d$) transition, and $$\begin{aligned} &&c^{eff}_1=1.168,\,c^{eff}_2=-0.365\,,\nonumber\\ &&10^4 c^{eff}_3=241.9\pm 3.2\eta + 1.4 \rho + i(31.3\mp 1.4\eta + 3.2\rho),\,\nonumber\\ &&10^4 c^{eff}_4=-508.7 \mp 9.6\eta - 4.2\rho+ i(-93.9 \pm 4.2\eta - 9.6\rho) ,\,\nonumber\\ &&10^4 c^{eff}_5=149.4\pm 3.2\eta + 1.4\rho + i(31.3\mp 1.4\eta + 3.2\rho),\,\nonumber\\ &&10^4 c^{eff}_6=-645.5 \mp 9.6\eta- 4.2\rho + i(-93.9\pm 4.2\eta - 9.6\rho) ,\,\end{aligned}$$ for the $b\to s$ ($\bar b\to \bar s$) transition, where $\epsilon_1=(1-\rho)^2+\eta^2$ and $\epsilon_2=\rho^2+\eta^2$. By adopting $C_F=0.14\pm 0.03$ from the light-cone sum rules in Ref. [@CF], with the central value in agreement with those in Refs. [@Wei:2009np; @Gutsche:2013oea], we find that ${\cal B}(\Lambda_b\to p K^-)=(5.1^{+2.4}_{-2.0})\times 10^{-6}$ and ${\cal B}(\Lambda_b\to p \pi^-)=(4.4^{+2.1}_{-1.7})\times 10^{-6}$, which are consistent with the data in Eq. (\[exbr\]). This is regarded to have a modest nonfactorizable effect, as investigated by the study of $\Lambda_b\to p\pi^-$ in Ref. [@CF]. Nonetheless, since the uncertainties from the predictions exceed those of the data, we fit $C_F$ with the data in Eq. (\[exbr\]), and obtain $$\begin{aligned} \label{CF} C_F=0.136\pm 0.009,\end{aligned}$$ which is able to reconcile the theoretical studies of $C_F$ to the data, and to be used in our study. Theoretical inputs in the SM for $R_{\Lambda_b\to p}$ and $\xi^\pm_M$ in Eq. (\[Rbeta\]) can be evaluated, given by $$\begin{aligned} \label{RKpi} R_{\Lambda_b\to p}&=&1.008\,,\nonumber\\ (\xi^+_{\pi},\,\xi^+_{K})&=&(1.03\pm 0.04\pm 0.00,\,0.11\pm 0.01\pm 0.02)\,,\nonumber\\ (\xi^-_{\pi},\,\xi^-_{K})&=&(-4.0\pm 0.3\pm 0.0,\,-4.0\pm 0.2\pm 0.3)\times 10^{-3}\,,\end{aligned}$$ where the errors for $\xi^\pm_M$ come from the CKM matrix elements and the floating $N^{eff}_c$, respectively. We present the branching ratios and direct CP asymmetries of $\Lambda_b\to p M(V)$ with $M(V)=K^-(K^{*-})$ and $\pi^-(\rho^-)$ in Table \[pre\]. In Refs. [@Aaltonen; @Rosner:2014fda], it is pointed out that the ratio of ${\cal R}_{\pi K}$ observed by CDF [@CDFexbr] or LHCb [@LHCbexbr] has not been realized theoretically, as shown in Table \[BBdata\]. In particular, we note that ${\cal R}_{\pi K}=2.6^{+2.0}_{-0.5}$ in the pQCD prediction [@Lu:2009cm] is about 3 times larger than the data, but better than other QCD calculations, such as ${\cal R}_{\pi K}=10.7$ in Ref. [@Wei:2009np]. However, in Table \[BBdata\] our result of this study shows that ${\cal R}_{\pi K}=0.84\pm 0.09$, which agrees well with the combined experimental value of $0.84\pm 0.22$ by CDF and LHCb. Clearly, our result justifies the theoretical approach based on the factorization in the two-body $\Lambda_b$ decays. We emphasize that the ratio of ${\cal R}_{\rho K^*}$ for the vector meson modes, which is predicted to be around 4.6, is an interesting physical observable as it is free of the hadronic uncertainties from the baryon sectors. A measurement for this ratio will be a firm test of the factorization approach in these baryonic decays. ${\cal R}_{\pi K}$ ${\cal R}_{\rho K^*}$ ------------------- ------------------------ ----------------------- our result $0.84\pm 0.09\pm 0.00$ $4.6\pm 0.5\pm 0.1$ pQCD [@Lu:2009cm] $2.6^{+2.0}_{-0.5}$ — CDF [@CDFexbr]  $0.66\pm 0.14\pm 0.08$ — LHCb [@LHCbexbr] $0.86\pm 0.08\pm 0.05$ — : Ratios of ${\cal R}_{\pi K}$ and ${\cal R}_{\rho K^*}$ from our calculations, the pQCD and experiments, where the errors of our results are from the CKM matrix elements and non-factorizable effects, respectively.[]{data-label="BBdata"} As shown in Table \[pre\], for the first time, the theoretical values of ${\cal B}(\Lambda_b\to p K^-)$ and ${\cal B}(\Lambda_b\to p \pi^-)$ are found to be simultaneously in agreement with the data. Moreover, we demonstrate that the uncertainties from the form factors, the CKM matrix elements and the non-factorizable effects are small and well-controlled. Similarly, for the decays of $\Lambda_b\to (p K^{*-},p \rho^-)$, the predictions of the branching ratios in Table \[pre\] are accessible to the experiments at CDF and LHCb. Note that our results of ${\cal B}(\Lambda_b\to p K^{*-},p\rho^-)$ $\simeq (2.5,\,11.4)\times 10^{-6}$ in Table \[pre\] are larger than those of $(0.3,\,6.1)\times 10^{-6}$ [@Wei:2009np] and $(0.8,\,1.9)\times 10^{-6}$ [@Wang:2012zze] in other theoretical calculations. our result pQCD [@Lu:2009cm] data ---------------------------------------------- ----------------------------- ---------------------- ----------------------------- $10^{6}{\cal B}(\Lambda_b\to p K^-)$ $4.8\pm 0.7\pm 0.1\pm 0.3$ $2.0^{+1.0}_{-1.3}$ $4.9\pm 0.9$ [@pdg] $10^{6}{\cal B}(\Lambda_b\to p \pi^-)$ $4.2\pm 0.6\pm 0.4\pm 0.2$ $5.2^{+2.5}_{-1.9}$ $4.1\pm 0.8$ [@pdg] $10^{6}{\cal B}(\Lambda_b\to p K^{*-})$ $2.5\pm 0.3\pm 0.2\pm 0.3$ — — $10^{6}{\cal B}(\Lambda_b\to p \rho^-)$ $11.4\pm 1.6\pm 1.2\pm 0.6$ — — $10^{2}{\cal A}_{CP}(\Lambda_b\to p K^-)$ $\,\,5.8\pm 0.2\pm 0.1$ $-5^{+26}_{-\;\;5}$ $-10\pm 8\pm 4$ [@Aaltonen] $10^{2}{\cal A}_{CP}(\Lambda_b\to p \pi^-)$ $-3.9\pm 0.2\pm 0.0$ $-31^{+43}_{-\;\;1}$ $6\pm 7\pm 3$ [@Aaltonen] $10^{2}{\cal A}_{CP}(\Lambda_b\to p K^{*-})$ $19.6\pm 1.3\pm 1.0$ — — $10^{2}{\cal A}_{CP}(\Lambda_b\to p \rho^-)$ $-3.7\pm 0.3\pm 0.0$ — — : Decay branching ratios and direct CP asymmetries of $\Lambda_b\to p M(V)$, where the errors for ${\cal B}(\Lambda_b\to pM(V))$ arise from $f_{M(V)}$ and $f_1(g_1)$, the CKM matrix elements and non-factorizable effects, while those for ${\cal A}_{CP}(\Lambda_b\to pM(V))$ are from the CKM matrix elements and non-factorizable effects, respectively. []{data-label="pre"} For CP violation, from Eqs. (\[Acp2\]) and (\[RKpi\]), one can use the reduced forms of ${\cal A}_{CP}(\Lambda_b\to pM)\propto$ $({|\alpha_M|^2-|\alpha_{\bar M}|^2})$$/({|\alpha_M|^2+|\alpha_{\bar M}|^2})$ similar to ${\cal A}_{CP}(\Lambda_b\to pV)$, which indeed present the limited hadron uncertainties, except for the factor of 1/2 for ${\cal A}_{CP}(\Lambda_b\to p \pi^-)$. As shown in Table \[pre\], our predictions of ${\cal A}_{CP}(\Lambda_b\to p \pi,\, pK^-)$ are around $(-3.9,\,5.8)$% with the errors less than $0.2\%$, while the results from the data [@Aaltonen] as well as the pQCD calculations are given to be consistent with zero. For the vector modes, as the uncertainties from the hadronizations have been totally eliminated in Eq. ([\[Acp2\]]{}), we are able to obtain reliable theoretical predictions for ${\cal A}_{CP}$, which should be helpful for experimental searches. In particular, it is worth to note that ${\cal A}_{CP}(\Lambda_b\to p K^{*-})=(19.6\pm 1.6)\%$ is another example of the large and clean CP violating effects without hadronic uncertainties as the process in the baryonic $B$ decays of $B^\pm \to K^{*\pm}\bar{p}p$ [@Geng:2006jt]. Interestingly, one would ask why the CP symmetry in $\Lambda_b\to p K^{*-}$ is larger than those in the other baryonic decay modes. The reason is that the term related to $a_4$ from the penguin diagram in Eq. (\[eq2\]) can be the primary contribution to $\Lambda_b\to p K^{*-}$ in Eq. (\[eq1\]), while allowing the certain contribution to the $a_1$ term from the tree diagram, such that the apparent large interference is able to take place. In contrast, in $\Lambda_b\to p\pi^-(\rho^-)$ and $\Lambda_b\to p K^-$, the $a_1$ and ($a_4+r_M a_6$) terms are dominating the branching ratios, respectively, leaving less rooms for the interferences. Clearly, ${\cal A}_{CP}(\Lambda_b\to p K^{*-})$ as well as the CPAs in other modes should receive more attentions, which have also been emphasized in Ref. [@Gronau:2013mza]. Finally, we remark that our approach can be extended to the two-body decay modes of other beauty baryons (${\cal B}_b$), such as $\Xi_b$, $\Sigma_b$ and $\Omega_b$. Conclusions =========== Based on the generalized factorization method and $SU(3)$ flavor and $SU(2)$ spin symmetries, we have simultaneously explained the recent observed decay branching ratios in $\Lambda_b\to p K^-$ and $\Lambda_b\to p \pi^-$ and obtained the ratio of ${\cal R}_{\pi K}$ being $0.84\pm 0.09$, which agrees well with the combined experimental value of $0.84\pm 0.22$ from CDF and LHCb, demonstrating a reliable theoretical approach to study the two-body $\Lambda_b$ decays. We have also predicted that ${\cal A}_{CP}(\Lambda_b\to p K^-)=(5.8\pm 0.2)\%$ and ${\cal A}_{CP}(\Lambda_b\to p \pi^-)=(-3.9\pm 0.2)\%$ with well-controlled uncertainties, whereas the current data for these CPAs are consistent with zero. We have used this approach to study the corresponding vector modes. Explicitly, we have found that ${\cal B}(\Lambda_b\to p K^{*-},\, p \rho^-)=(2.5\pm0.5,\,11.4\pm2.1)\times 10^{-6}$ with ${\cal R}_{\rho K^*}=4.6\pm0.5$ and ${\cal A}_{CP}(\Lambda_b\to p K^{*-},\,p \rho^-) =(19.6\pm1.6,\, -3.7\pm0.3)\%$. Since our prediction for ${\cal A}_{CP}(\Lambda_b\to p K^{*-})$ is large and free of both mesonic and baryonic uncertainties from the hadron sector, it would be the most promised direct CP asymmetry to be measured by the experiments at CDF and LHCb to test the SM and search for some possible new physics. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== This work was partially supported by National Center for Theoretical Sciences, National Science Council (NSC-101-2112-M-007-006-MY3) and National Tsing Hua University (103N2724E1). [99]{} N. Cabibbo, Phys. Rev. Lett. [**10**]{}, 531 (1963); M. Kobayashi and K. Maskawa, Prog. Theor. Phys. [**49**]{}, 652 (1973). H. n. Li, S. Mishima and A.I. Sanda, Phys. Rev. D [**72**]{}, 114005 (2005); H. n. Li and S. Mishima, Phys. Rev. D [**83**]{}, 034023 (2011). W. S. Hou, H. n. Li, S. Mishima and M. Nagashima, Phys. Rev. Lett.  [**98**]{}, 131801 (2007). K. A. Olive [*et al.*]{} \[Particle Data Group Collaboration\], Chin. Phys. C [**38**]{}, 090001 (2014). C. D. Lu [*et al.*]{}, Phys. Rev. D [**80**]{}, 034011 (2009). S. W. Wang [*et al.*]{}, Chin. Phys. C [**37**]{}, 063103 (2013). Z. T. Wei, H. W. Ke and X. Q. Li, Phys. Rev. D [**80**]{}, 094016 (2009). T. Aaltonen [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett.  [**106**]{}, 181802 (2011); arXiv:1403.5586 \[hep-ex\]. A. Ali, G. Kramer and C. D. Lu, Phys. Rev. D[**58**]{}, 094009 (1998). G. P. Lepage and S. J. Brodsky, Phys. Rev. Lett.  [**43**]{}, 545(1979) \[Erratum-ibid.  [**43**]{}, 1625 (1979)\]. C. H. Chen, H. Y. Cheng, C. Q. Geng and Y. K. Hsiao, Phys. Rev. D [**78**]{}, 054016 (2008). A. Khodjamirian[*et al.*]{}, JHEP [**1109**]{}, 106 (2011); T. Mannel and Y. M. Wang, JHEP [**1112**]{}, 067 (2011). T. Gutsche [*et al.*]{}, Phys. Rev. D [**88**]{}, 114018 (2013). J. L. Rosner, arXiv:1410.6849 \[hep-ex\]. T. Aaltonen [*et al.*]{} \[CDF Collaboration\], Phys. Rev. Lett.  [**103**]{}, 031801 (2009). R. Aaij [*et al.*]{} \[LHCb Collaboration\], JHEP [**1210**]{}, 037 (2012). S. W. Wang [*et al.*]{}, Eur. Phys. J. C [**72**]{}, 1852 (2012). C. Q. Geng, Y. K. Hsiao and J. N. Ng, Phys. Rev. Lett.  [**98**]{}, 011801 (2007). M. Gronau and J. L. Rosner, Phys. Rev. D [**89**]{}, 037501 (2014).

--- author: - | Gonz[á]{}lez-Nuevo J., Cueli M. M., Bonavera L., Lapi A., Migliaccio M.,\ Arg[ü]{}eso F. , Toffolatti L. bibliography: - './XCORR\_ZPH.bib' date: 'Received xxx, xxxx; accepted xxx, xxxx' title: 'Cosmological constraints with the sub-millimetre galaxies Magnification Bias after large scale bias corrections.' --- [The study of the magnification bias produced on high redshift sub-millimetre galaxies by foreground galaxies through the analysis of the cross-correlation function was recently demonstrated as an interesting independent alternative to the weak lensing shear as a cosmological probe.]{} [In the case of the proposed observable, most of the cosmological constraints depends mainly on the largest angular separation measurements. Therefore, we aim at studying and correcting the main large scale biases that affect foreground and background galaxy samples in order to produce a robust estimation of the cross-correlation function. Then we analyse the corrected signal in order to derive updated cosmological constraints.]{} [The large scale bias corrected cross-correlation functions are measured using a background sample of H-ATLAS galaxies with photometric redshifts &gt; 1.2 and two different foreground samples (GAMA galaxies with spectroscopic redshifts or SDSS galaxies with photometric ones, both in the range 0.2 &lt; z &lt; 0.8). They are modelled using the traditional halo model description that depends on both halo occupation distribution and cosmological parameters. These parameters are then estimated by performing a Markov chain Monte Carlo under different scenarios to study the performance of this observable and the way to further improve its results.]{} [After the large scale bias corrections, we get only minor improvements with respect to the Bonavera et al. 2020 results, mainly confirming their conclusions: a lower bound on $\Omega_m > 0.22$ at $95\%$ C.L. and an upper bound $\sigma_8 < 0.97$ at $95\%$ C.L. (results from the $z_{spec}$ sample). Neither the much higher surface density of the foreground photometric sample nor the assumption of gaussian priors for the remaining unconstrained parameters improves significantly the derived constraints. However, by combining both foreground samples into a simplified tomographic analysis, we were able to obtain interesting constraints on the $\Omega_m$-$\sigma_8$ plane: $\Omega_m= 0.42_{- 0.14}^{+ 0.08}$ and $\sigma_8= 0.81_{- 0.09}^{+ 0.09}$ at 68% CL.]{} Introduction ============ The apparent excess number of high redshift sources observed near low redshift mass structures is known as Magnification Bias [see e.g. @SCH92]: the deflections produced by the intervening gravitational field (area stretching and amplification) affecting the light rays coming from distant sources increase, in general, their chances of being included in a flux-limited sample [see for example @ARE11]. An unambiguous manifestation of this bias is the existence of a non negligible cross-correlation function between two source samples with non-overlapping redshift distributions. It has been observed in several contexts: galaxy-quasar cross-correlation function [@SCR05; @MEN10], cross-correlation signal between Herschel sources and Lyman-break galaxies [@HIL13] or the CMB [@BIA15; @BIA16] among others. The cross-correlation signal can be enhanced by optimizing the choice of foreground and background samples. In this paper we use the sub-millimetre galaxies (SMGs) as the background sample because some of their features (steep luminosity function, very faint emission in the optical band and typical redshifts above $ z > 1-1.5 $) make them close to the optimal background sample for lensing studies as confirmed by a long series of publications [see for example @BLA96; @NEG07; @NEG10; @GON12; @BUS12; @BUS13; @FU12; @WAR13; @CAL14; @NAY16; @NEG17; @GON19; @BAK20 among the most important ones]. In early works, the magnification bias produced on SMGs was already observed [@WAN11] and measured with high significance, $> 10\sigma$ [@GON14]. In @GON17 the measurements were further improved, allowing a more detailed study with a Halo model. It was concluded that the lenses are massive galaxies or even galaxy groups/clusters, with minimum mass of $M_{lens}\sim10^{13}M_{\odot}$. Moreover, it was demonstrated that it is possible to split the foreground sample in different redshift bins and to perform a tomographic analysis thanks to the better statistics. Finally, @BON19 use the magnification bias to study the mass properties of a different type of lenses, a sample QSOs at $0.2<z<1.0$. It was possible to estimate the halo mass where the QSOs acting as lenses are located in the sky, $M_{min} = 10^{13.6_{-0.4}^{+0.9}} M_{\odot}$. These mass values indicate that we are observing the lensing effect of a cluster size halo signposted by the QSOs. The interest in magnification bias is driven by the fact that it can be used as an additional cosmological probe to address the estimation of the parameters in the standard cosmological model. In fact, the importance of the magnification bias effect depends on the gravitational deflection caused by low redshift galaxies on light travelling close to such lens, which in turn depends on cosmological distances and galaxy halo properties. Features like the anisotropies in the CMB [e.g., @HIN13; @PLA16_XIII; @PLA18_VI], the big bang nucleosynthesis [e.g. @FIE06] and the SNIa observations of the Universe accelerating expansion [e.g. @BET14] are well handled by the current ‘standard cosmological model’. It is also inclusive of some Large Scale Structure (LSS) significant predictions about galaxies distributions (e.g. [@PEA01]), such as baryon acoustic oscillations (BAOs) (e.g. [@ROS15]). Therefore, measurements based on such observables provides independent and complementary constraints on the cosmological parameters [e.g., @PEA94]. The success of the current model is in the fact that results from different probes are in great accordance. However, with the increase in the quality and quantity of the measurements, some ‘tensions’ and small-scale issues have arisen that might indicate the necessity of modifications of the $\Lambda$CDM model. The main tensions are the value of the Hubble’s constant, $H_0$ [$74.03 \pm 1.42$ km/s/Mpc by @RIE19; @PLA18_VI with $67.4 \pm 0.5$ km/s/Mpc], and the usually degenerate relationship between the $\Omega_m$ and $\sigma_8$ parameters [e.g., @HEY13; @PLA16_XXIV; @HIL17; @PLA18_VI]. In this context, @BON20 (hereafter BON20) test the capability of the Magnification Bias produced on high-z SMGs as an additional independent cosmological probe in the effort to resolve the tensions. With this proof of concept analysis $\Omega_m$ and $H_0$ were not well constrained. However, interesting limits were found: a lower limit of $\Omega_m>0.24$ at 95% CL and an upper limit of $\sigma_8<1.0$ at 95% CL (with a tentative peak around 0.75). Although the derived cosmological constraints from the Magnification Bias are relatively weak, it was confirmed as a new, independent observable making it a valuable new technique. Therefore it is worth making some efforts to improve further such results. In this respect, most of the cosmological analysis that can be performed using the measured cross-correlation function (cosmological parameters, mass function, neutrinos, ...) depends mainly on the observed data at the largest angular scales ($\gtrsim20$ arcmin). On the one hand, this data are the most uncertain ones with large error-bars. Large areas and high source densities are needed in order to derive precise measurements. On the other hand, large scale bias, that can be considered negligible at smaller scales, can affect the data, and, as a consequence, the derived cosmological results. For these reasons the main goal of this work is to deeply study and find the optimal strategy to measure and analyse a precise and unbiased cross-correlation function at cosmological scales. The work is organised as follows. In section \[sec:data\] the background and foreground samples are described and in section \[sec:methodology\] the methodology is presented. The large scale biases and how to correct them are described in \[sec:LS\_bias\]. The derived cosmological constraints and conclusions are discussed in sections \[sec:results\] and \[sec:conclusion\] respectively. In Appendix \[sec:corner\_plots\] we show the posteriors distributions of all the cases analysed and discussed in this work. Data {#sec:data} ==== The different galaxy samples used in this work are described in this section: the background sample, consisting of SMGs sources, and the foreground samples, consisting of two independent samples with spectroscopic and photometric redshifts, respectively. ![Normalised redshift distributions of the three catalogues used in this work: the background sample i.e. H-ATLAS high-z SMGs (red solid line), the GAMA spectroscopic foreground sample (blue solid line) and the SDSS photometric foreground sample (magenta dashed line).[]{data-label="Fig:dNdz_hist"}](PLOTS/zhist){width="\columnwidth"} Background sample ----------------- The Herschel Astrophysical Terahertz Large Area Survey [H-ATLAS; @EAL10] is the largest area extragalactic survey carried out by the Herschel space observatory [@PIL10] covering $\sim 610 deg^2$ with PACS \[43\] and SPIRE \[44\] instruments between 100 and 500 $\mu m$. Details of the H-ATLAS map-making, source extraction and catalogue generation can be found in @IBA10 [@PAS11; @RIG11; @VAL16; @BOU16], and @MAD20. The background sample consists of H-ATLAS sources detected in the three GAMA fields (total area of $\sim 147 deg^2$), the North Galactic Pole (NGP, $\sim 170 deg^2$) and the part of the South Galactic Pole (SGP) that overlaps with the spectroscopic foreground sample ($\sim 60 deg^2$). A photometric redshift selection of 1.2 &lt; z &lt; 4.0 has been applied to ensure no overlap in the redshift distributions of lenses and background sources, and we are thus left with $\sim66000$ ($\sim 24$ per cent of the initial sample and $z_{ph,med} = 2.20$). The redshifts estimation is described in detail in @GON17 [@BON19] and references therein. This is the same background sample used in @GON17, @BON19 and @BON20. Foreground samples ------------------ In this work we use two independent foreground samples. The first one is the same one used by @GON17, BON20 and we name it as the “$z_{spec}$ sample”. It consists of a sample extracted from the GAMA II [@DRI11; @BAL10; @BAL14; @LIS15] spectroscopic survey, with $\sim 150000$ galaxies for 0.2 &lt; $z_{spec}$ &lt; 0.8 ($z_{spec,med} = 0.28$). H-ATLAS and GAMA II surveys were carried out to maximize the common area coverage. Both surveys covered the three equatorial regions at 9, 12 and 14.5 h (referred to as G09, G12 and G15, respectively) and the SGP region was partially observed also by GAMA II. Thus, the resulting common area is of about $\sim 207 deg^2$, surveyed down to a limit of $r \simeq 19.8$ mag. This is the same foreground sample used in @GON17 and BON20. The second foreground sample is selected in the Sixteenth Data Release of the Sloan Digital Sky Survey [SDSS; @BLA17; @AHU19]. It consists of galaxies with photometric redshift between 0.2&lt; $z_{ph}$ &lt; 0.8 and photometric redshift error $z_{err}/(1+z)<1$ (*photoErrorClass*=1). SDSS have completly covered the H-ATLAS equatorial regions and the NGP one (a total area of $\sim 317 deg^2$). The sample, denominated “$z_{ph}$ sample”, comprises $\sim962000$ galaxies in total with median value of $z_{ph,med} = 0.38$. The reason to introduce this second foreground sample is to study the improvements in the final results by increasing the density of potential lenses. The higher uncertainty in the redshift estimation of the foreground photometric redshifts is not very important in the current analysis because we are using a single wide redshift bin. The normalised redshift distributions of the different samples are compared in Figure \[Fig:dNdz\_hist\]. As in @GON17, the random errors in the photometric redshifts are taken into account to estimate the redshift distributions. The main effect is to broadening the distributions beyond the selection limits. Figure \[Fig:dNdz\_hist\] clearly shows the gap in redshift between the background and the foreground sources. The same figure also highlights the different redshift distributions between the two foreground samples. Methodology {#sec:methodology} =========== Tiling area scheme ------------------ The H-ATLAS survey is divided in five different fields: three GAMA fields in the ecliptic (9h, 12h, 15h) and two in the North and South Galactic Poles (NGP, and SGP). The H-ATLAS scanning strategy produced the characteristic diamond repeated shape in most of their fields \[Fig:Tiles\]. Taking into account the available area in each field we have different possibilities to measure the cross-correlation function: - The “All” field area (blue line).- It provide the best statistics, i.e. smaller statistical uncertainties, both at small and large scales. The drawback is that we are limited to 4-5 fields to minimize the cosmic variance. - The “Tile” area (red line).- This is the straightforward shape to be selected taking into account the observational strategy. The area of each tile,  16 sq. deg, should be large enough to avoid a bias in the large scale measurements (normally limited to angular separation below 2 deg). In order to maintain a regular shape for the tiles, a small overlap among such regions is needed, typically lower than 20% of the tiles’ area. The advantage of this area scheme is in the fact that it provides around 24 different tiles, that should help to diminish the cosmic variance. - The “mini-Tile” area (magenta line).- It is built by dividing the tiles in four equal area “mini-Tiles” (each of  2x2 sq. deg). This area scheme typically provides around 96 different tiles. However, the maximum distance allowed by such area scheme is close to the cosmological scales that we want to measure. This was the area scheme used in BON20. Each tiling area scheme has its own strong and weak points and can be affected by different types of large scale biases. Therefore, we perform a detailed analysis in order to compare the measurements from the different tiling area schemes and derive a robust estimation of the cross-correlation function, in particular at the cosmological angular scales. ![Examples of the different area selection to measure the cross-correlation function for the G09 H-ATLAS field. The “All” field area is shown in blue ( 56 sq. deg). The “tile” selection is shown in red ( 4x4 sq. deg.) and the “mini-Tile” one in magenta ( 2x2 sq. deg.) []{data-label="Fig:Tiles"}](PLOTS/tile_schema.png){width="\columnwidth"} Angular cross-correlation function estimation {#sec:cross_corr} --------------------------------------------- As described in detail in [@GON17], BON20, we used a modified version of the [@LAN93] estimator [@HER01]: $$\label{eq:wx} w_x(\theta)=\frac{\rm{D}_1\rm{D}_2-\rm{D}_1\rm{R}_2-\rm{D}_2\rm{R}_1+\rm{R}_1\rm{R}_2}{\rm{R}_1\rm{R}_2}$$ where $\rm{D}_1\rm{D}_2$, $\rm{D}_1\rm{R}_2$, $\rm{D}_2\rm{R}_1$ and $\rm{R}_1\rm{R}_2$ are the normalized data1-data2, data1-random2, data2-random1 and random1-random2 pair counts for a given separation $\theta$. For each selected area, we compute the angular cross-correlation function and the statistical error (averaging between 10 different realizations using different random catalogues each time). Each final measurement corresponds to the mean value of the cross-correlation functions estimated in each individual selected area for a given angular separation bin. The uncertainties correspond to the standard error of the mean, i.e. $\sigma_\mu=\sigma/\sqrt{n}$ with $\sigma$ the standard deviation of the population and $n$ the number of independent areas (each selected region can be assumed as statistically independent due to the small overlap). {width="\columnwidth"} {width="\columnwidth"} Halo Model ---------- As described in detail in the previous related works [@GON17; @BON19], BON20 we adopt the halo model formalism proposed by @COO02 in order to interpret a foreground-background source cross-correlation signal. An halo is defined as spherical regions whose mean over-density with respect to the background at any redshift is given by its virial value, which is estimated following [@WEI03] assuming a flat $\Lambda$CDM model. We used the traditional [@NAV96] density profile with the concentration parameter given in [@BUL01]. The cross-correlation between the foreground and background sources is linked to the low redshift galaxy-mass correlation through the weak gravitational lensing effect. The foreground galaxy sample traces the mass density field that causes the weak lensing, affecting the number counts of the background galaxy sample through Magnification Bias. Following mainly [@COO02] [see @GON17 for details], we compute the correlation between the foreground and background sources adopting the standard Limber [@LIM53] and flat-sky approximations [see e.g. @KIL17 and references therein]. It can be estimated as: $$\begin{split} w_{fb}=2(\beta -1)\int^{z_s}_0 \frac{dz}{\chi^2(z)}\frac{dN_f}{dz}W^{lens}(z) \\ \int_{0}^{\infty}\frac{ldl}{2\pi}P_{gal-dm}(l/\chi^2(z),z)J_0(l\theta) \end{split}$$ where $$W^{lens}(z)=\frac{3}{2}\frac{H_0^2}{c^2}E^2(z)\int_z^{z_s} dz' \frac{\chi(z)\chi(z'-z)}{\chi(z')}\frac{dN_b}{dz'}$$ being $E(z)=\sqrt{\Omega_m(1+z)^3+\Omega_{\Lambda}}$, $dN_b/dz$ and $dN_f/dz$ as the *unit-normalised* background and foreground redshift distribution and $z_s$ the source redshift. $\chi(z)$ is the comoving distance to redshift z. The logarithmic slope of the background sources number counts is assumed $\beta=3$ ($N(S) = N_0 S^{-\beta}$) as in previous works [@LAP11; @LAP12; @CAI13; @BIA15; @BIA16; @GON17; @BON19]. Small variations of its value are almost completely compensated by small changes in the $M_{min}$ parameter. As the Halo Occupation Distribution (HOD) we adopted the three parameters @ZHE05 model: all halos above a minimum mass $M_{min}$ host a galaxy at their centre, while any remaining galaxy is classified as satellite. Satellites are distributed proportionally to the halo mass profile and halos host them when their mass exceeds the $M_1$ mass. Finally, the number of satellites is a power-law function of halo mass with $\alpha$ as the exponent, $N_{sat}(M)=(\frac{M}{M_1}) ^\alpha$. Therefore, $M_\text{min}$, $M_1$ and $\alpha$ are the astrophysical free-parameters of the model. Estimation of parameters {#sec:par_estimate} ------------------------ To estimate the different set of parameters, we performed a Markov chain Monte Carlo (MCMC) using the open source [*emcee*]{} software package [@EMCEE]. It is an MIT licensed pure-Python implementation of [@GOO10] Affine Invariant MCMC Ensemble sampler. For each run, we generated at least 90000 posterior samples to ensure a good statistical sampling after convergence. In the cross-correlation function analysis, we took into account both the astrophysical HOD parameters, and the cosmological parameters. The astrophysical parameters to be estimated are $M_{min}$, $M_1$ and $\alpha$. The cosmological parameters we want to constrain are $\Omega_m$, $\sigma_8$ and $h=H_0/100$. With the current samples, we do not have the statistical power to constrain $\Omega_B$, $\Omega_{\Lambda}$ and $n_s$ in our analysis. As we assume a flat universe, $\Omega_{\Lambda}$ is simply: $\Omega_{\Lambda}=1-\Omega_m$. For the the other two cosmological parameters, we keep them fixed to the *Planck* most recent results, $\Omega_B=0.0486$ and $n_s=0.9667$ (see @PLA18_VI). A traditional Gaussian likelihood function was used during this work. It should be noted that only the cross-correlation data in the weak lensing regime ($\theta \ge 0.2$ arcmin) are being taken into account for the fit since we are in the weak lensing approximation [see @BON19 for a detailed discussion]. In general, we used the same flat priors for all the different analyses. They are based on the ones used in BON20. As for the astrophysical parameters we chose: \[12.0-13.5\] for $\log M_{min}$, \[13.0-15.5\] for $\log M_1$ and \[0.5-1.5\] for $\alpha$. And for the cosmological parameters: \[0.1-0.8\] for $\Omega_m$, \[0.6-1.2\] for $\sigma_8$ and \[0.5-1.0\] for $h$. Large Scale Biases {#sec:LS_bias} ================== The cross-correlation function measurements using the different Tiling area schemes are compared in Figure \[Fig:xcorr\_data\]. The left panel shows the measurements before any correction is applied. While all the different measurements agree almost perfectly within the uncertainties at small scales, there is a widespread variation of estimated values for angular separations above $\sim 10$ arcmin. But the cosmological parameters affect mainly those angular scales (see BON20 appendix figures). Therefore, we need to understand the causes that produce such high variation on our observations at those large angular scales before attempting any robust cosmological analysis. It is well known that the distribution of galaxies in the Universe is not perfectly homogeneous. Therefore, in a field with a limited area, the number of detected galaxies will be somewhat higher or lower than the mean value obtained considering large enough areas. If this variation is not taken into account when building the random catalogues for a particular field, it will affect the DR and RR related terms in equation \[eq:wx\] and the estimated correlation could be stronger or weaker than the intrinsic value (see e.g. @ADE05 for a detailed discussion on this topic). To this respect, there are mainly two different biases that can affect the cross-correlation measurements at large scales: the integral constraint [IC; @ROC99] and the surface density variation [SD; @BLA02]. Integral constraint (IC) ------------------------ When many fields are averaged, the overall effect of the large scale fluctuations tends to make the observed correlation weaker mainly at the largest observed scales. This means that the estimated cross-correlation function is biased low by a constant, the IC: $w_{x\_ideal}(\theta) = w_{x}(\theta) + IC$. Although there are possible theoretical approaches to estimate the IC for a particular scanning strategy (see e.g. @ADE05), it is commonly estimated numerically using the RR counts: $$IC=\frac{\sum_i{\rm{R}_1\rm{R}_2(\theta_i)w_{x\_ideal}(\theta_i)}}{\sum_i{\rm{R}_1\rm{R}_2(\theta_i)}}.$$ As a first approximation of $w_{x\_ideal}(\theta_i)$, we assumed a power-law model, $w_{x\_ideal}(\theta_i)= A \theta^\gamma$. In order to be as much independent as possible of the exact value of the cosmological parameters (that mainly affect the largest angular scales), we estimated the best fit parameters for the power-law using only the observed cross-correlation function below 20 arcmin ($A=10 ^{-1.54}$ arcmin and $\gamma=-0.89$). With the estimated power-law, the derived IC value for the “mini-Tiles” area was $9\times10^{-4}$. We verified that choosing a smaller angular separation upper limit or using different data set did not affect the derived IC value. Moreover, assuming the best fit model of BON20 that can be considered biased low due to the fact that neglected the IC correction, the IC derived was again the same value. Therefore, we can conclude that the “mini-Tiles” estimated cross-correlation functions at the largest scales (&gt;20 arcmin) are biased low but can be safely corrected by adding an $IC = 9\times10^{-4}$. Anyway, as discussed in section \[sec:flat\_priors\], this correction does not introduce any substantial difference with respect to the BON20 results on cosmological parameters. On the other hand, the estimated IC for the the “Tiles” area is $IC = 5\times10^{-4}$, considering both the power-law fit and the BON20 best fit model. As expected, the correction is smaller than in the “mini-Tiles” case taking into account the larger area of the “Tiles”. The IC in the “Tiles” case affects marginally only the measurements above $\sim 40$ arcmin. Considering the large uncertainties at those angular scales, it can be almost considered a negligible correction for the $z_{spec}$ sample measured using the “Tiles” area. However, in the seek of precision, we decided to apply it in any case. On the other hand, the IC results are completely negligible in the case of using the “all” field area scheme, as expected. Surface density variations -------------------------- The results using the “Tiles” area differ for the $z_{spec}$ and the $z_{ph}$ samples. This difference remains after the IC correction because it is the same for both cases. Moreover, the discrepancy is even stronger in the “All” scenario case (since the “All” measurements are almost the same between both samples, we are focusing only in the $z_{ph}$ for simplicity). This is a clear indication that an additional large scale bias is affecting the measurements when larger areas are considered. The fact that the $z_{ph}$ sample is more affected is probably related to the much higher density of sources in this sample. If there is an additional variation of the source density of the foreground or the background sample that is not taken into account when building the random catalogues, it can produce a spurious enhancement of the measured correlation. As explained by @BLA02, the number of close pairs depended on the local surface density while the random pairs are related to the global average surface density. Then, systematic fluctuations produce $DD>RR$ that means a higher correlation (e.g. consider just the simplest estimator of the auto-correlation: $w(\theta)=DD/RR-1$). Therefore, if present, the surface density variation produces the opposite effect with respect to the IC, that is what we are observing with the $z_{ph}$ sample. ### Instrumental Noise variation For the background sample, there is a well known surface density variation related to the instrumental noise due to the scanning strategy (see Figure \[Fig:SD\_maps\], top panel). The overlap between the “Tiles” reduces the instrumental noise that allows fainter SMGs to be detected with respect the rest of the field. For the auto-correlation analysis it was demonstrated that the potential effect can be considered negligible [@AMV19]. Moreover, our results indicate that the relatively low surface density of the $z_{spec}$ sample makes this effect also negligible. In other words, the number of additional pairs due to the fainter background sources in those areas is not relevant enough to affect the measurements for the $z_{spec}$ sample. However, the much higher surface density of the $z_{ph}$ sample could produce a relevant enough enhancement of background-foreground pairs in those regions and, therefore, inducing a large scale surface density variation for the “Tiles” and “All” area schemes (we can consider the “mini-Tile” measurements simply dominated by the IC correction and neglect this other type of large scale bias even for the $z_{ph}$ sample). In order to correct the instrumental noise surface density bias, we adopted the same procedure to generate random catalogues used in @AMV19 for the auto-correlation analysis of the SMGs. First, a flux was chosen randomly from the flux densities of our background sample. Then the simulated galaxy is situated in a random position on the field. At this position the local noise was estimated as the instrumental noise and the confusion noise [see Table 3 of @VAL16 for the GAMA fields]. The estimated local noise is used to introduce a random Gaussian perturbation in the flux density. Finally, the simulated galaxy was kept in the sample if its flux density was greater than four times the local noise, the same detection limit used to produce the official H-ATLAS catalogue. This process was repeated for each random galaxy until the completion of the random catalogue. These newly generated random catalogues correspond only to the background sample, i.e. it was only applied to build the $R_1$ random catalogues (used to estimate the $D_2R_1$ and $R_1R_2$ terms). When the instrumental noise variation is considered, the cross-correlation functions showed a small correction toward lower values at the largest angular scales (not shown individually in Figure \[Fig:xcorr\_data\]). Although this result confirms that this bias is not negligible, it also highlights that it is not enough to explain the stronger correlation observed in the “Tiles” scheme for the $z_{ph}$ sample and the “All” one for both samples. Therefore, we studied additional sources of surface density variations in the foreground samples. ![Top panel: Example of the instrumental noise variation in the G09 field due to the scanning strategy. Bottom panel: Example of the surface density variation for the $z_{ph}$ sample in the G15 filed after been filtered using a gaussian kernel with a standard deviation of 180 arcmin.[]{data-label="Fig:SD_maps"}](PLOTS/ND_map "fig:"){width="\columnwidth"}\ ![Top panel: Example of the instrumental noise variation in the G09 field due to the scanning strategy. Bottom panel: Example of the surface density variation for the $z_{ph}$ sample in the G15 filed after been filtered using a gaussian kernel with a standard deviation of 180 arcmin.[]{data-label="Fig:SD_maps"}](PLOTS/SD_map "fig:"){width="\columnwidth"} ### Surface density variation of the foreground samples. There are different causes of surface density variations in large area galaxy surveys, such as scanning strategy, sensitivity variation with time and foreground contamination. Moreover, the sample selection can amplify or reduce these variations, for example a region where the conditions for spectroscopic observations are different from the mean field ones. The detailed correction of these possible variations is complicated and requires a deep knowledge of the particular details of the instrument and the pipeline used for the production of the catalogue. For the purpose of this work we adopted a simple approach to investigate the existence and correction of surface density variations in the foreground samples. As we can only observe a discrepancy at the largest angular scales, we decided to focus just on this range. First, we created a surface density map by adding $+1$ to the pixel value at the position of each galaxy on the sample. Then we smoothed the map using a Gaussian kernel with a certain standard deviation (see discussion later in this section). Next, we apply the H-ATLAS survey masks (so that we can neglect border effects due to the smoothing step). These surface density maps are then used to generate the Random catalogues, $R_2$, for the foreground samples (used to estimate $D_1R_2$ and $R_1R_2$ terms in equation \[eq:wx\]). The bottom panel in Figure \[Fig:SD\_maps\] shows an example of smoothed surface density map built using the $z_{ph}$ sample, with a standard deviation of 180 arcmin, for the G15 field. As expected, the overall density map at those angular scales is almost homogeneous. However, there are some variations that might be biasing our measurements: the source density in the second “Tile” from the left is higher than the fourth one. However, the exact value to be used as the Gaussian kernel dispersion is an unknown quantity. Using values smaller than 180 arcmin, the resulting density map starts to mimic the two-halo correlation of the foreground data. This means that the obtained $R_2$ contain part of the real auto-correlation and will remove part of this power from the estimated cross-correlation. For this reason and considering that the cross-correlation function decreases steeply for $\theta \sim 100$ arcmin, we can set a Gaussian dispersion of &gt; 150 arcmin as a lower limit. On the other hand, for dispersion values above 180 arcmin, the surface density variation along the area becomes almost negligible in the derived $R_2$. Therefore, we can considered a dispersion of &lt; 200 – 220 arcmin as an upper limit. Overall, we decided to proceed using a dispersion of 180 arcmin as a representative value, but taking into account that it is arbitrarily chosen. At the same time, given the uncertainties of the measurements at the relevant angular scales, small variations around the chosen deviation value became only a second order effect in our large scale measurements. When both surface density variations are taken into account to generate the random catalogues the large scale bias observed in the “Tiles” scheme for the $z_{ph}$ sample or the “All” field area one for both samples disappear. The right panel of Figure \[Fig:xcorr\_data\] shows the estimated cross-correlation functions using different tiling area schemes for the two samples after all the large scale bias corrections. The difference between the mean values at each angular scale is much smaller than the uncertainties. Considering this good agreement, we are confident that the measurements can be considered robust in all the angular scales commonly used for the cosmological analysis. As a final summary, to minimise the number of corrections applied to the data, we recommend to apply just the IC correction to the “mini-Tile” measurements for both samples and to the “Tile” one in the $z_{spec}$ case. In the other cases, the surface density correction is the most relevant one to be considered. Cosmological constraints {#sec:results} ======================== Once the cross-correlation measurements are corrected for the different large scale biases discussed in the previous section, we focus our analysis in their application to the estimate of some relevant parameters as done in BON20: the astrophysical parameters ($M_{min}$, $M_1$ and $\alpha$) and the cosmological ones ($\Omega_m$, $\sigma_8$ and $h$). The higher number of independent smaller sky areas allows to minimise the error contribution given by the cosmic variance resulting in smaller uncertainties. For this reason and considering the almost perfect agreement between the “All” tiling scheme and the “Tiles” ones for both samples, we decided to focus just on the second case in order to simplify the discussion. Therefore, we focus on just four cases (all of them corrected for the relevant large scale biases): “mini-Tiles” and “Tiles” tiling schemes for both samples ($z_{spec}$ and $z_{ph}$). . \[Tab:zspec\] [c c c c c c c c]{} Param & Priors & &\ & $\mathcal{U}$\[a,b\] & $\mu$ & $\sigma$ & peak & $\mu$ & $\sigma$ & peak\ & & $\pm 68 CL$ & $ $ & & $\pm 68 CL$ & &\ \ $\log(M_{min}/M_\odot)$ & \[12.0, 14.0\] & $12.57_{- 0.17}^{+ 0.23}$ & 0.20& 12.61 & $12.61_{- 0.15}^{+ 0.19}$ & 0.18 & 12.56\ \ $\log(M_1/M_\odot)$ & \[12.5, 15.5\] & $14.26_{- 0.38}^{+ 1.24}$ & 0.78& 15.03 & $14.37_{- 0.37}^{+ 1.13}$ & 0.74 & 14.71\ \ $\alpha$ & \[0.5, 1.5\] & – & – & – & – & – & –\ \ $\Omega_m$ & \[0.1, 0.8\] & $ 0.45_{- 0.21}^{+ 0.13}$ & 0.16& 0.38 & $ 0.42_{- 0.24}^{+ 0.14}$ & 0.18 & 0.31\ \ $\sigma_8$ & \[0.6, 1.2\] & $ 0.84_{- 0.18}^{+ 0.11}$ & 0.14& 0.83 & $ 0.82_{- 0.20}^{+ 0.08}$ & 0.14 & 0.75\ \ $h$ & \[0.5, 1.0\] & – & – & – & – & – & –\ \ [c c c c c c c c]{} Param & Priors & &\ & $\mathcal{U}$\[a,b\] & $\mu$ & $\sigma$ & peak & $\mu$ & $\sigma$ & peak\ & & $\pm 68 CL$ & $ $ & & $\pm 68 CL$ & &\ \ $\log(M_{min}/M_\odot)$ & \[12.0, 14.0\] & $12.60_{- 0.13}^{+ 0.20}$ & 0.18& 12.67 & $12.61_{- 0.13}^{+ 0.20}$ & 0.17 & 12.66\ \ $\log(M_1/M_\odot)$ & \[12.5, 15.5\] & $13.81_{- 1.09}^{+ 0.53}$ & 0.76& 13.60 & $13.95_{- 0.95}^{+ 0.74}$ & 0.76 & 13.74\ \ $\alpha$ & \[0.5, 1.5\] & $0.96_{-0.46}^{+0.15}$ & 0.27 & 0.77 & $0.96_{-0.46}^{+0.15}$ & 0.28 & 0.73\ \ $\Omega_m$ & \[0.1, 0.8\] & $ 0.46_{- 0.18}^{+ 0.11}$ & 0.14& 0.38 & $ 0.46_{- 0.19}^{+ 0.12}$ & 0.15 & 0.39\ \ $\sigma_8$ & \[0.6, 1.2\] & $ 0.99_{- 0.11}^{+ 0.12}$ & 0.11& 0.98 & $ 0.98_{- 0.10}^{+ 0.16}$ & 0.12 & 1.00\ \ $h$ & \[0.5, 1.0\] & $ 0.71_{- 0.21}^{+ 0.06}$ & 0.14 & 0.50 & – & – & –\ \ ![Comparison of the derived posterior distributions for the constrained parameters using the four data sets: $\log M_{min}$ (top left panel), $\log M_1$ (top right), $\Omega_m$ (bottom left) and $\sigma_8$ (bottom right). []{data-label="Fig:Comp_1D"}](PLOTS/comp_1D_new){width="\columnwidth"} Flat priors {#sec:flat_priors} ----------- The main results of this work are derived imposing flat priors as described in section \[sec:par\_estimate\]. The full set of posterior distributions can be found in the Appendix \[sec:corner\_plots\]. Figures \[Fig:zspec\_corner\] and \[Fig:zph\_corner\] compare the results derived from the different tiling schemes for the same sample. Moreover, the main statistical quantities that describe the posterior distributions are summarised in Table \[Tab:zspec\] and \[Tab:zph\] for $z_{spec}$ and $z_{ph}$ samples, respectively. The model prediction using the best fit values for both samples using the “mini-Tile” scheme are shown in the right panel of Figure \[Fig:xcorr\_data\] (black solid and green dashed lines, respectively). As in BON20 both $\alpha$ and $h$ are not well constrained. For this reason the comparison will focus on the rest of the parameters (see Figure \[Fig:Comp\_1D\]). All the cases provide similar constraints for $M_{min}$ and $\Omega_m$. On the first one, all of them agree to a mean value of $\log(M_{min}/M_\odot) \simeq 12.6 \pm 0.2$ at 68% CL. This value is very similar with the one found by BON20, $\log(M_{min}/M_\odot)= 12.53^{+0.29}_{-0.16}$. With respect the BON20 results, the introduction of the IC correction did not affect the estimated value of this well constrained parameter. In the case of $\Omega_m$, the new results moved the mean, $\sim0.45$, toward lower, more traditional values. This indicates that the large scale corrections helped to increase slightly the recovered values at the largest angular scales and to reduce their uncertainties. As a consequence, the highest $\Omega_m$ values become less probable based on our current measurements. However, similar lower limits as in BON20 are confirmed, e.g. &gt;0.22 for the $z_{spec}$ cases. On the other hand, the results for $\log M_1$ and $\sigma_8$ are different depending on the sample used. However, the results based on the same sample but using different Tile schemes are consistent between them. For $M_1$, using the $z_{spec}$ sample, we find a preference for $\log(M_1/M_\odot) \geq 13.8$ but only at 68% CL, whereas it shows a clear peak around $\log(M_1/M_\odot) \sim 13.6-13.7$, using the $z_{ph}$ one. In both cases these results are consistent with the BON20 ones. In a similar way, $\sigma_8$ mean estimated value moves from $\sim 0.8$, obtained with the $z_{spec}$ sample, to $\sim 1.0$ using the $z_{ph}$ one. Therefore, with the $z_{spec}$ sample, the same as in BON20, we obtain similar $\sigma_8$ constraints, but not confirmed by the $z_{ph}$ ones. Taking into account that the measurements of the cross-correlation function are almost the same between both samples (see again right panel of Figure \[Fig:xcorr\_data\]), this discrepancy in some of the recovered parameters can only be related to the fact that both samples have different redshift distributions. In fact, @GON17 perform a tomographic analysis of the cross-correlation function using four different redshift bins, bewteen $0.1<z<0.8$, and study the evolution of the same HOD parameters. While the $M_1$ parameter remains almost constant with redhsift, there is a clear evolution of an increasing $M_{min}$ values with redshift. The results of $\alpha$ are inconclusive as it is unconstrained in most of the redshift bins. By using a single wide redshift bin, we are deriving an average of the astrophysical parameters weigthed by the sample redshift distribution. Therefore, by analysing samples with different redshift distributions, it is expected to estimate different astrophysical parameter values, at least for those showing an evolution with redshift as $M_{min}$. Gaussian priors for the unconstrained parameters ------------------------------------------------ As discussed in the previous section, there are two parameters that remain unconstrained with the current data sets: $\alpha$ and $h$. In this section, we study the potential improvements on the results by assuming external constraints on these two parameters. This additional information is introduced in the MCMC as Gaussian priors. For all the analysis in this section we used only the $z_{spec}$ sample with the “mini-Tile” scheme. In the case of $\alpha$ we adopted a normal distribution with mean 1.0 and a dispersion of 0.1 (very similar to the Gaussian priors also used in BON20). The results are summarize in Table \[Tab:Galpha\] and the derived posterior distribution are shown in Figure \[Fig:zspec\_corner\_Galpha\]. In general, adopting a Gaussian prior for the $\alpha$ parameters produces almost no variation with respect to the default case. Only the most related parameters, $\log M_1$ and $\sigma_8$ move slightly toward lower values with a reduction on their dispersion of $\sim9$ and $\sim 21$ %, respectively. For the Hubble constant, we adopted the two popular values given by the local estimation, $74.03\pm 1.42$ km/s/Mpc [@RIE19], and the CMB one, $67.4\pm0.5$ km/s/Mpc [@PLA18_VIII]. The results obtained in these two cases are summarized in Table \[Tab:h\_high\], while the derived posterior distributions are compared in Figure \[Fig:zspec\_corner\_htest\]. The only relevant variation with respect to the default case is that the $\sigma_8$ distribution moves again slightly toward lower values with a reduction on their dispersion of $\sim 29$ %. When comparing between both $h$ priors cases, the results are almost identical. However, as also indicated in BON20, higher values of $h$ seem to perform slightly better: the $\Omega_m$ posterior distribution becomes thinner and moves towards lower, more traditional, values. However, the current uncertainties do not allow us to derive stronger conclusions on this particular topic. Overall, adopting more restrictive priors on the unconstrained parameters does not improve remarkably the results in general. The parameter that seems to benefit more from the reduction of uncertainty in both cases is $\sigma_8$. This is probably due to the fact that it is the parameter that mostly depends on the intermediate angular scales and, therefore, it is the one mostly affected by changes induced both by the smallest scales ($\alpha$’s main influence) and by the largest scales ($h$’s main influence), see appendix in BON20. . \[Tab:Galpha\] [c c c c c]{} Params & Priors & $\mu$ & $\sigma$ & peak\ & & $\pm 68 CL$ & $ $ &\ \ $\log(M_{min}/M_\odot)$ & $\mathcal{U}$\[12.0, 14.0\] & $12.53_{- 0.04}^{+ 0.16}$ & 0.21& 12.59\ \ $\log(M_1/M_\odot)$ & $\mathcal{U}$\[12.5, 15.5\] & $14.31_{- 0.38}^{+ 0.47}$ & 0.71& 14.32\ \ $\alpha$ & $\mathcal{N}$\[1.0, 0.1\] & $0.99_{-0.05}^{+ 0.06}$ & 0.10 & 1.00\ \ $\Omega_m$ & $\mathcal{U}$\[0.1, 0.8\] & $ 0.46_{- 0.16}^{+ 0.01}$ & 0.15& 0.37\ \ $\sigma_8$ & $\mathcal{U}$\[0.6, 1.2\] & $ 0.76_{- 0.16}^{+ 0.01}$ & 0.10& 0.64\ \ $h$ & $\mathcal{U}$\[0.5, 1.0\] & $ 0.75_{- 0.09}^{+ 0.09}$ & 0.14 & 0.66\ \ [c c c c c c c c c]{} Param & &\ & Priors & $\mu$ & $\sigma$ & peak & Priors & $\mu$ & $\sigma$ & peak\ & & $\pm 68 CL$ & $ $ & & & $\pm 68 CL$ & &\ \ $\log(M_{min}/M_\odot)$ & $\mathcal{U}$\[12.0, 14.0\] & $12.54_{- 0.05}^{+ 0.13}$ & 0.19& 12.58 & $\mathcal{U}$\[12.0, 14.0\] & $12.57_{- 0.06}^{+ 0.12}$ & 0.19& 12.61\ \ $\log(M_1/M_\odot)$ & $\mathcal{U}$\[12.5, 15.5\] & $14.29_{- 0.01}^{+ 1.02}$ & 0.76& 14.83 & $\mathcal{U}$\[12.5, 15.5\] & $14.29_{- 0.01}^{+ 1.12}$ & 0.77& 14.93\ \ $\alpha$ & $\mathcal{U}$\[0.5, 1.5\] & – & – & – & $\mathcal{U}$\[0.5, 1.5\] & – & – & –\ \ $\Omega_m$ & $\mathcal{U}$\[0.1, 0.8\] & $ 0.44_{- 0.15}^{+ 0.01}$ & 0.15& 0.35 & $\mathcal{U}$\[0.1, 0.8\] & $ 0.49_{- 0.15}^{+ 0.02}$ & 0.15& 0.41\ \ $\sigma_8$ & $\mathcal{U}$\[0.6, 1.2\] & $ 0.75_{- 0.11}^{+ 0.01}$ & 0.10& 0.69 & $\mathcal{U}$\[0.6, 1.2\] & $ 0.76_{- 0.15}^{+ 0.01}$ & 0.10& 0.68\ \ $h$ & $\mathcal{N}$\[0.74, 0.014\] & $ 0.74_{- 0.01}^{+ 0.01}$ & 0.014 & 0.74 & $\mathcal{N}$\[0.67, 0.005\] & $ 0.67_{- 0.003}^{+ 0.002}$ & 0.005 & 0.67\ \ Combining both data sets {#sec:tomo} ------------------------ {width="\textwidth"} [c c c c]{} Params & $\mu$ & $\sigma$ & peak\ & $\pm 68 CL$ & $ $ &\ \ $\log(M_{min}/M_\odot)$ $z_{spec}$ & $12.48_{- 0.16}^{+ 0.21}$ & 0.18& 12.51\ \ $\log(M_1/M_\odot)$ $z_{spec}$ & $14.37_{- 0.36}^{+ 1.13}$ & 0.74& 15.5\ \ $\alpha$ $z_{spec}$& – & – & –\ \ $\log(M_{min}/M_\odot)$ $z_{ph}$ & $12.60_{- 0.12}^{+ 0.21}$ & 0.19& 12.67\ \ $\log(M_1/M_\odot)$ $z_{ph}$ & $13.69_{- 1.03}^{+ 0.46}$ & 0.71& 13.49\ \ $\alpha$ $z_{ph}$ & $0.97_{- 0.44}^{+ 0.45}$ & 0.27 & 0.88\ \ $\Omega_m$ & $ 0.42_{- 0.14}^{+ 0.08}$ & 0.12& 0.37\ \ $\sigma_8$ & $ 0.81_{- 0.09}^{+ 0.09}$ & 0.09& 0.81\ \ $h$ & $ 0.72_{- 0.22}^{+ 0.09}$ & 0.14 & 0.5\ \ The $z_{ph}$ sample has much better statistics with respect to the $z_{spec}$ one, but we do not see a relevant improvement in the obtained constraints. In addition, even if the measured cross-correlation function is almost the same, each sample provides different results in some of the studied parameters. This is probably linked to the different redshift functions. On this respect, @GON17 tomographic analysis of the cross-correlation function show a strong evolution with redshift at least for the $\log M_{min}$ parameter. As explained before, by using a single wide redshift bin, the derived astrophysical parameters are the average of the evolving values measured by @GON17 weighted by the particular sample redshift distribution. As we saw, the different averaged astrophysical parameter values between the two samples is affecting also the recovered values of some of the cosmological parameters. In particular $\sigma_8$ changes from 0.84 for the $z_{spec}$ sample to 0.99 for the $z_{ph}$ one. A proper tomographic analysis is beyond the scope of this paper, but we can try a simple but interesting analysis: taking into account that the results from both samples are independent and have different redshift distributions, we can try to constraint the cosmological parameters using both samples at the same time. We perform a joint analysis allowing different astrophysical parameters constraints for each sample but keeping the same cosmological parameters. Therefore, we run an additional MCMC analysis but this time with nine parameters to be constrained (three astrophysical ones for each sample and three common cosmological ones). We used for both samples the “mini-Tile” scheme as it needs the simplest large scale bias correction. The results are summarize in Table \[Tab:tomo\] and the derived posterior distributions for the nine parameters are shown in Figure \[Fig:zspec\_corner\_tomo\]. Regarding the astrophysical parameters (see Figure \[Fig:Comb\_astro\]) the main changes of the combined analysis with respect to the individual ones are the following. Imposing a common cosmological parameters values seems not to affect the $\log M_{min}$ constraints for the $z_{ph}$ sample but it produces a shift toward slightly lower mean values for the $z_{spec}$ one (from 12.57 to 12.48). This is probably due to the more peaked redshift distribution of the $z_{ph}$ sample. In the case of the $M_{1}$ parameter, there is only a small reduction in the parameter uncertainty. Finally, there is no improvement in the $\alpha$ parameter, that remains unconstrained. As expected, the most relevant changes are within the cosmological parameters. While for the $\Omega_m$ the estimated posterior distribution is simply less skewed toward high values (from an associated gaussian standard deviation of 0.16 to 0.12), the $\sigma_8$ parameter is the most affected. Its posterior distribution become almost gaussian with a mean value of $\sigma_8=0.81_{- 0.09}^{+ 0.09}$ and a standard deviation of 0.09. However, for the Hubble constant the results give only an upper limit very similar to the one derived from the $z_{ph}$ sample alone ($h<0.8$ at 68% CL). A more detailed discussion on the $\Omega_m$ and $\sigma_8$ results will be presented in the next subsection. Comparison with other results ----------------------------- {width="\columnwidth"} {width="\columnwidth"} The weak gravitational lensing results available in the literature are usually related with a different and complementary observable, the shear. In this section we compare with measurements by cosmic shear of galaxies, focusing on the most constraining, and the CMB lensing by *Planck* [@PLA18_VIII]. In particular the results from the following surveys (with different redshift ranges and affected by different systematic effects) are being taken into account for the comparison: the Canada-France-Hawaii Telescope Lensing Survey presented in CFHTLenS [@JOU17], the Kilo Degree Survey and VIKING based on $450\,$deg$^2$ data [KV450, @HIL20], the first-year lensing data from the Dark Energy Survey (DES, [@TRO18]) and the Subaru Hyper Suprime-Cam first-year data [HSC, @HAM20]. For the comparison, we use publicly released MCMC results. Moreover, the different results we are comparing with have different priors. Since we are not interested in an in-depth comparison, we do not adjust them to our fiducial set-up. In particular, we compare the constraints in the $\Omega_m$ - $\sigma_8$ plane: cosmic shear measures the combination $\sigma_8 \Omega_m^{0.5}$ and CMB lensing the $\sigma_8 \Omega_m^{0.25}$ one. Such combinations highlight degeneracy directions, shown in the marginalised posterior contours ($68\%$ and $95\%\,$C.L.) in Figure \[Fig:External\_OmgM\_sgm8\] for the data-sets described above. To have a direct comparison with literature, the contours of these plots (0.68 and 0.95) are different from those used in the corner plots of this work (0.393 and 0.865, corresponding to the relevant 1-sigma and 2-sigma levels in the 1D histograms in the upper part of the same corner plots). We also show *Planck* CMB temperature and polarisation angular power spectra (dark blue) that, although in certain agreement with the HSC (cyan) and DES (green) constraints, presents the tension issues with the CFHTLenS (red) and KV450 (orange) data. The relevant cosmological constraints derived in this paper are shown in Figure \[Fig:External\_OmgM\_sgm8\] for both samples, $z_{spec}$ and $z_{ph}$, using the “mini-Tiles” scheme. The left panel shows the results from the analysis of each sample individually (grey filled contours for the $z_{ph}$ sample and black dashed curves for the $z_{spec}$ one) while the right panel shows the results from the combination of both samples as described in section \[sec:tomo\]. With respect to the previous BON20 constraints, by analysing each sample individually, the correction of the large scale bias has shifted the constraints on the $\Omega_m$ parameter toward lower values, more in agreement with the rest of the results from other studies. However, even when combining the two data-sets, the Hubble constant remains unconstrained. There is only a mild preference for the lowest values allowed by the flat prior, which is analogous to the one we found from the $z_{ph}$ sample alone. As displayed in Figure \[Fig:External\_OmgM\_sgm8\], it is very relevant to underline that when both samples are analyzed together, the constraints in the $\Omega_m$-$\sigma_8$ plane becomes much more restrictive: $\Omega_m= 0.42_{- 0.14}^{+ 0.08}$ and $\sigma_8= 0.81_{- 0.09}^{+ 0.09}$. It can also be noted, that their almost perpendicular direction with respect to the other lensing results can help to break the typical degeneracy. In any case, the constraints derived in this work confirm the main conclusions from BON20. Finally, we note that the data here discussed cannot be used to place useful constraints on the Hubble constant yet. Conclusions {#sec:conclusion} =========== As discussed in detail in BON20 (see their Figure A.1) the cosmological parameters depend mainly on the largest angular separation measurements. Therefore, the large scale biases can affect the cosmological constraint derived from the analysis of the magnification bias through the cross-correlation function. In this work, we study and correct the main large scale biases that affect our samples in order to product a robust estimation of the cross-correlation function. The result is a remarkable agreement among the different cross-correlation measurements, calculated independently of the used Tiling scheme or foreground samples. Then we analyse these results to estimate cosmological constraints after correcting the different large scale biases. We get minor improvements with respect to the BON20 results, mainly confirming their conclusions: a lower bound on $\Omega_m > 0.22$ at $95\%$ C.L. and an upper bound $\sigma_8 < 0.97$ at $95\%$ C.L. (results from the $z_{spec}$ sample using the “mini-Tile” scheme). Therefore, the large scale biases are a systematic that need to be corrected in order to derive robust and consistent results between different foreground samples or Tiling schemes, but does not help much to improve the precision of the derived constraints. In addition, we compare the estimates derived using two different and independent foreground samples: one consisting of foreground galaxies with spectroscopic redshifts, the $z_{spec}$ sample, and another one with only photometric redshifts, the $z_{ph}$ one. Analysing only one single broad redshift bin, we conclude that the higher errors of the photometric redhsifts do not have a relevant role in our outcomes. The $z_{ph}$ sample here considered has $\sim 6$ times more sources than the $z_{spec}$ one. Its better surface density makes it more sensitive to some large scale biases but helps to reduce the uncertainty in the measured cross-correlation function at intermediate and small angular scales. On the other hand, our current results show that the uncertainty is still dominated by the cosmic variance rather than by the surface density of the specific foreground sample at the largest angular scales. However, the constraints obtained making use of the $z_{ph}$ sample, which provides a more accurate cross-correlation measurements, are generally consistent with those derived using the $z_{spec}$ ones, with similar uncertainties. Moreover, adopting gaussian priors for the unconstrained parameters (i.e. $\alpha$ and the Hubble constant, similarly to BON20) does not improve much the results. Therefore, we are probably reaching the accuracy limit of the cosmological constraints that can be achieved with the analysis of a single redshift bin. Increasing the total area in order to decrease further the cosmic variance is probably an interesting improvement to be considered in the future. Although the measured cross-correlation function is almost the same between both foregrounds samples, we find different constraints for $\log M_1$ and $\sigma_8$ parameters. This is caused by the different redshift distributions between both samples. With a single wide redshift bin, the derived astrophysical parameters, that evolve with time as shown in the tomographic analysis of the cross-correlation function by @GON17, are averaged quantities weighted by the specific redshift distribution of the selected sample. Therefore, taking into account that the measurements of the cross-correlation function from both foreground samples are independent, we make use of the different redshift distributions to perform a simplified tomographic analysis combining both samples into a single MCMC run. We jointly performed the estimation of the cosmological parameters for both samples, but allowed different values of the astrophysical parameters for each sample. In this way, the effect of having different redshift distributions is included in the astrophysical parameters allowing us to determine with higher precision the cosmological parameters. In fact, the improvements on the $\Omega_m$-$\sigma_8$ plane are evident in the right panel of Figure \[Fig:External\_OmgM\_sgm8\]. The cosmological constraints obtained with this independent technique are starting to become competitive with respect to the other lensing results and its particular characteristics make it an interesting possibility in breaking the usual $\Omega_m$-$\sigma_8$ degeneracy. As a general conclusion, we showed that we are probably reaching the limits of the constraints than can be derived using just a single redshift bin, although there are still some ways to improve the results. However, the most promising advances with the study of the SMGs magnification bias are probably going to be obtained by performing a more complex tomographic analysis. JGN, MMC, LB, FA and LT acknowledge the PGC 2018 project PGC2018-101948-B-I00 (MICINN/FEDER). LB and JGN also acknowledge PAPI-19-EMERG-11 (Universidad de Oviedo). MM is supported by the program for young researchers “Rita Levi Montalcini" year 2015. A.L. acknowledges support from PRIN MIUR 2017 prot. 20173ML3WW002, ‘Opening the ALMA window on the cosmic evolution of gas, stars and supermassive black holes’, the MIUR grant ‘Finanziamento annuale individuale attivitá base di ricerca’, and the EU H2020-MSCA-ITN-2019 Project 860744 ‘BiD4BEST: Big Data applications for Black hole Evolution STudies’. We deeply acknowledge the CINECA award under the ISCRA initiative, for the availability of high performance computing resources and support. In particular the projects “SIS19\_lapi”, “SIS20\_lapi” in the framework “Convenzione triennale SISSA-CINECA”.\ The Herschel-ATLAS is a project with Herschel, which is an ESA space observatory with science instruments provided by European-led Principal Investigator consortia and with important participation from NASA. The H-ATLAS web- site is http://www.h-atlas.org. GAMA is a joint European- Australasian project based around a spectroscopic campaign using the Anglo- Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programs including GALEX MIS, VST KIDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA web- site is: http://www.gama-survey.org/.\ Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P. Sloan Foundation, the U.S. Department of Energy Office of Science, and the Participating Institutions. SDSS-IV acknowledges support and resources from the Center for High-Performance Computing at the University of Utah. The SDSS web site is www.sdss.org. SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participation Group, the Carnegie Institution for Science, Carnegie Mellon University, the Chilean Participation Group, the French Participation Group, Harvard-Smithsonian Center for Astrophysics, Instituto de Astrofísica de Canarias, The Johns Hopkins University, Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo, the Korean Participation Group, Lawrence Berkeley National Laboratory, Leibniz Institut für Astrophysik Potsdam (AIP), Max-Planck-Institut für Astronomie (MPIA Heidelberg), Max-Planck-Institut für Astrophysik (MPA Garching), Max-Planck-Institut für Extraterrestrische Physik (MPE), National Astronomical Observatories of China, New Mexico State University, New York University, University of Notre Dame, Observatário Nacional / MCTI, The Ohio State University, Pennsylvania State University, Shanghai Astronomical Observatory, United Kingdom Participation Group, Universidad Nacional Autónoma de México, University of Arizona, University of Colorado Boulder, University of Oxford, University of Portsmouth, University of Utah, University of Virginia, University of Washington, University of Wisconsin, Vanderbilt University, and Yale University.\ In this work, we made extensive use of `GetDist` [@GETDIST], a Python package for analysing and plotting MC samples. In addition, this research has made use of the python packages `ipython` [@ipython], `matplotlib` [@matplotlib] and `Scipy` [@scipy] Posterior distributions of the MCMC results {#sec:corner_plots} =========================================== Posterior distributions for the different analyses discussed during the article. The contours for all these plots are set to 0.393 and 0.865. Notice that the relevant 1-sigma and 2-sigma levels for a 2D histogram of samples is 39.3% and 86.5% not 68% and 95%. Otherwise, there is not a direct comparison with the 1D histograms above the contours. {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"}

--- abstract: 'In this paper, we consider the problem of recovering an unknown sparse matrix $\mathbf X$ from the matrix sketch $\mathbf Y = \mathbf A \mathbf X \mathbf B^T$. The dimension of $\mathbf Y$ is less than that of $\mathbf X$, and $\mathbf A$ and $\mathbf B$ are known matrices. This problem can be solved using standard compressive sensing (CS) theory after converting it to vector form using the Kronecker operation. In this case, the measurement matrix assumes a Kronecker product structure. However, as the matrix dimension increases the associated computational complexity makes its use prohibitive. We extend two algorithms, fast iterative shrinkage threshold algorithm (FISTA) and orthogonal matching pursuit (OMP) to solve this problem in matrix form without employing the Kronecker product. While both FISTA and OMP with matrix inputs are shown to be equivalent in performance to their vector counterparts with the Kronecker product, solving them in matrix form is shown to be computationally more efficient. We show that the computational gain achieved by FISTA with matrix inputs over its vector form is more significant compared to that achieved by OMP.' address: | $^\dagger$Dept. of Electrical Eng. and Comp. Science, Syracuse University, Syracuse NY, USA\ $^*$Dept. of Electrical Engineering, Technion-Israel Institute of Technology, Technion City, Haifa, Israel bibliography: - 'IEEEabrv.bib' - 'bib1.bib' title: Recovery of Sparse Matrices via Matrix Sketching --- Compressive sensing, Sparse matrix recovery, $l_l$ norm minimization, FISTA, OMP Introduction ============ We consider the problem of recovering an unknown matrix $\mathbf X$ from the following observation model $$\begin{aligned} \mathbf Y = \mathbf A \mathbf X \mathbf B^T\label{obs_1}\end{aligned}$$ where $\mathbf X\in \mathbb R^{N\times N}$, $\mathbf A \in \mathbb R^{M\times N}$, $\mathbf B \in \mathbb R^{L\times N}$ and $\mathbf A^T$ denotes the transpose of the matrix $\mathbf A$. This problem has been studied by many researchers in different contexts for arbitrary matrices $\mathbf X$ [@Penrose1; @Dai2; @Peng1]. In many applications dealing with high dimensional data, *sparsity* is one of the low dimensional structures widely observed. Most popular transforms applied to 2-dimensional signals are in the form of (\[obs\_1\]) where compression is obtained by a transformation of rows followed by a transformation of columns of the data matrix [@Caiafa1; @Fang2; @Rivenson1; @Dasarathy1]. With an arbitrarily distributed sparse matrix $\mathbf X$ in which each column/row has only a few non zeros, a natural question to ask is whether it is possible to design sensing matrices in the form of (\[obs\_1\]) so that $\mathbf X$ can be uniquely recovered from $\mathbf Y$ when $M,L < N$. Sparse signal recovery has attracted much attention in the recent literature in the context of *compressive sensing (CS)* [@candes1; @donoho1; @Eldar_B1]. In the standard CS framework, a commonly used mechanism is to stack the high dimensional data into vector form to recover the sparse vector uniquely from an underdetermined linear system [@candes1; @donoho1]. The observation model (\[obs\_1\]) can be equivalently written in vector form using Kronecker products as: $$\begin{aligned} \mathbf y = \mathbf C \mathbf x\label{obs_2}\end{aligned}$$ where $\mathbf y = \mathrm{vec}(\mathbf Y) \in \mathbb R^{ML}$, $\mathbf C = \mathbf B \otimes \mathbf A \in \mathbb R^{ML\times N^2} $, $\mathbf x = \mathrm{vec}(\mathbf X) \in \mathbb R^{N^2}$, $\otimes$ denotes the Kronecker operator and $\mathrm{vec}(\mathbf X)$ is a column vector that vectorizes the matrix $\mathbf X$ (i.e. columns of $\mathbf X$ are stacked one after the other). The sensing matrix in (\[obs\_2\]) has a special structure, i.e., it can be represented as a Kronecker product of two matrices $\mathbf A$ and $\mathbf B$. It has been shown [@Duarte4; @Jokar1; @Duarte5; @Jokar2] that the sparse signal $\mathbf x$ from (\[obs\_2\]) can be recovered by solving the following $l_1$ norm minimization problem $$\begin{aligned} \min ||\mathbf x||_1~ s.t.~ \mathbf C \mathbf x = \mathbf y \label{l_1_norm_min}\end{aligned}$$ under certain conditions on the matrices $\mathbf A$ and $\mathbf B$ where $||\mathbf x||_p$ denotes the $l_p$ norm of $\mathbf x$. In particular, these results imply that the capability of recovering $\mathbf x$ based on (\[obs\_2\]) is ultimately determined by the worst behavior of $\mathbf A$ or $\mathbf B$. Also, this approach is computationally complex especially when the matrix dimension $N$ increases[@Rivenson1; @Dasarathy1]. Several recent papers addressed the problem of recovering a sparse $\mathbf X$ from (\[obs\_1\]) without employing the Kronecker product. In [@Dasarathy1], it was shown that a unique solution for $\mathbf X$ can be found when $\mathbf X$ is distributed sparse under certain conditions on $\mathbf A$ and $\mathbf B$ by solving the following optimization problem: $$\begin{aligned} \min ||\mathbf X||_1 ~ \mathrm{s}. ~\mathrm{t}. ~ \mathbf A \mathbf X \mathbf B^T = \mathbf Y\label{matrix_l1}\end{aligned}$$ where $||\mathbf X||_1$ is the $l_1$ norm of $\mathrm{vec}(\mathbf X)$. The authors derive recovery conditions when the matrices $\mathbf A$ and $\mathbf B$ contain binary elements which are better than that obtained via the Kronecker product approach. In [@Rivenson1], the authors discuss advantages in terms of computational, storage, calibration and implementation while solving (\[matrix\_l1\]) in matrix form compared to that with vector form. However, no specific algorithm was developed to solve for $\mathbf X$. In [@Fang2], a version of orthogonal matching pursuit (OMP) (dubbed 2D OMP) is presented to find a sparse $\mathbf X$ in the matrix form (\[obs\_1\]) when $\mathbf A= \mathbf B$. Our goal in this paper is to develop algorithms to solve for sparse $\mathbf X$ from (\[obs\_1\]) without the employment of Kronecker products. We extend fast iterative shrinkage threshold algorithm (FISTA) [@Beck1; @Yang1] developed for the vector case to sparse matrix recovery with matrix inputs. We further consider a greedy based approach via OMP to find the sparse solution. We show that both algorithms with matrix inputs are equivalent to their vector counterparts obtained via Kronecker products in terms of performance. However, the computational complexity of the matrix approach is shown to be much less, especially with FISTA, compared to solving the problem in vector form. Sparse Matrix Recovery via $\l_1$ Norm Minimization {#matrix_l1norm} =================================================== Vector formulation ------------------ While numerous algorithms have been proposed in the literature to solve (\[l\_1\_norm\_min\]), in this paper we consider FISTA as discussed in [@Beck1; @Yang1]. We consider the noisy observation model so that FISTA with vector inputs as given in Algorithm \[algo\_FISTA\_vec\] [@Yang1], is the solution of $$\begin{aligned} \underset {\mathbf x}{\min} \left\{\frac{1}{2} ||\mathbf y - \mathbf C \mathbf x||_2^2 + \lambda ||\mathbf x||_1\right\}\label{l1_norm}\end{aligned}$$ where $\lambda$ is a regularization parameter. In Algorithm \[algo\_FISTA\_vec\], $L_f=||\mathbf C||_2$ is the Lipschitz constant of $\nabla f(\mathbf x)$ where $||\mathbf C||_2$ denotes the spectral norm of $\mathbf C$, $\nabla$ denotes the gradient operator, and $f(\mathbf x) = \frac{1}{2}||\mathbf y - \mathbf C \mathbf x||_2^2$, and $$\begin{aligned} \mathrm{soft}(\mathbf u, a) = \begin{array}{ccc} \mathrm{sgn}(\mathbf u_i)(|\mathbf u_i| - a)_+ \end{array}\end{aligned}$$ for $i=1,\cdots,N^2$ where $\mathbf u_i$ is the $i$-th element of $\mathbf u$, $x_+$ equals $x$ if $x>0$ and equals $0$ otherwise. **Input:** observation vector $\mathbf y$, measurement matrix $\mathbf C$\ **output:** estimate for signal, $\hat{\mathbf x}$ 1. Initialization: $\mathbf x^{0} =\mathbf 0$, $\mathbf x^{1}=\mathbf 0$, $t_0=1$, $t_1=1$, $k=1$\ Initialize: $\lambda_1$, $\beta\in (0,1)$, $\bar\lambda > 0$ 2. **while** not converged **do** 3. $\mathbf z^{k} = \mathbf x^k + \frac{t_{k-1}-1}{t_k} (\mathbf x^k - \mathbf x^{k-1})$ 4. $ \mathbf u^k = \mathbf z^k - \frac{1}{L_f}\mathbf C^T (\mathbf C \mathbf z^k - \mathbf y) $ 5. $\mathbf x^{k+1} = \mathrm{soft} \left(\mathbf u^k, \frac{\lambda_k}{L_f}\right)$ 6. $t_{k+1} = \frac{1+\sqrt{4t_k^2 + 1}}{2}$ 7. $\lambda_{k+1} = \max(\beta \lambda_k, \bar\lambda)$ 8. $k=k+1$ 9. **end while** $\hat{\mathbf x} = \mathbf x^{k} $ The computational complexity of FISTA is dominated by step 4 in Algorithm \[algo\_FISTA\_vec\]. The matrix-vector multiplications require $\mathcal O(N^4ML +N^4+N^2ML)$ computations. Since $M,L \leq N$, the complexity is in the order of $\mathcal O(N^4ML)$. Thus, FISTA is feasible only when $N, M, L$ are fairly small. Matrix formulation ------------------ With the noisy version of (\[obs\_1\]), we aim to solve the following $l_1$ norm minimization problem: $$\begin{aligned} \underset{\mathbf X}{\min} ~ \left\{\frac{1}{2} ||\mathbf Y - \mathbf A \mathbf X \mathbf B^T||_F^2 + \lambda ||\mathbf X||_1 \right\} \label{FISTA_matrix}\end{aligned}$$ where $\lambda$ is a regularization parameter and $||\mathbf A||_F$ is the Frobenius norm of $\mathbf A$. In generalizing FISTA to solve (\[FISTA\_matrix\]), we follow a similar approach as discussed in [@Toh1]. Consider the more general unconstrained optimization problem: $$\begin{aligned} \underset{\mathbf X\in \mathbb R^{N\times N}}{\min} ~ F(\mathbf X) + \lambda G(\mathbf X)\end{aligned}$$ where $G(\cdot)$ is a proper, convex, lower semicontinuous function, and $F(\cdot)$ is a convex smooth (continuously differentiable) function on an open subset of $\mathbb R^{N\times N}$ containing $\mathrm{dom} G = \{\mathbf X | G(\mathbf X) < \infty\}$. We assume that $\mathrm{dom} G$ is closed and $\nabla F$ is Lipschitz continuous on $\mathrm{dom} G$ with Lipschitz constant $L_f$; i.e. $$\begin{aligned} ||\nabla F(\mathbf X) - \nabla F(\mathbf Z) ||_F \leq L_f ||\mathbf X - \mathbf Z||_F, ~X, Z\in \mathrm{dom}~G. \label{lips_mat}\end{aligned}$$ When $F(\mathbf X) = \frac{1}{2} ||\mathbf Y - \mathbf A \mathbf X \mathbf B^T||_F^2$, it can be shown that $||\nabla F(\mathbf X) - \nabla F(\mathbf Z) ||_F = ||\nabla f(\mathbf x) - \nabla f(\mathbf z) ||_2$ and $||\mathbf X - \mathbf Z||_F = ||\mathbf x - \mathbf z||_2$ where $\mathbf z = \mathrm{vec}(\mathbf Z)$, $f(\mathbf x) = \frac{1}{2}||\mathbf y - \mathbf C \mathbf x||_2^2$ and $\mathbf C=\mathbf B \otimes \mathbf A$ are as defined before. Thus, the Lipschitz constant of $\nabla F(\mathbf X)$ is the same as $\nabla f(\mathbf x)$, and we use the same notation $L_f$ as used in Algorithm \[algo\_FISTA\_vec\]. Consider the following quadratic approximation of $F(\cdot)$ at $\mathbf Z$ for any $\mathbf Z \in \mathrm{dom} G$: $$\begin{aligned} Q_L(\mathbf X, \mathbf Z) &:=& F(\mathbf Z) + \mathrm{tr}(\nabla F(\mathbf Z)^T (\mathbf X - \mathbf Z))\nonumber\\ & +& \frac{L_f}{2} ||\mathbf X - \mathbf Z||_F^2 + \lambda G (\mathbf X)\label{Q_L}\end{aligned}$$ where $\mathrm{tr}(\cdot)$ denotes the trace of a matrix. We can rewrite (\[Q\_L\]) as, $$\begin{aligned} Q_L(\mathbf X, \mathbf Z) = \frac{L_f}{2} ||\mathbf X - \mathbf U(\mathbf Z)||_F^2 + \lambda ||\mathbf X||_1 + \tilde F(\mathbf Z)\label{obj_matrix_Fista_mod}\end{aligned}$$ where $\tilde F(\mathbf Z)$ is a function of only $\mathbf Z$ and $$\begin{aligned} \mathbf U(\mathbf Z) = \mathbf Z - \frac{1}{L_f}\nabla F(\mathbf Z)=\mathbf Z - \frac{1}{L_f} \mathbf A^T (\mathbf A \mathbf X \mathbf B^T - \mathbf Y)\mathbf B.\end{aligned}$$ Thus, $$\begin{aligned} \underset{\mathbf X}{\arg\min}~ Q_L(\mathbf X, \mathbf Z) = \underset{\mathbf X}{\arg\min} \left\{ \frac{L_f}{2} ||\mathbf X - \mathbf U(\mathbf Z)||_F^2 + \lambda ||\mathbf X||_1 \right\}.\end{aligned}$$ Since both terms are element wise separable, we have $$\begin{aligned} \underset{\mathbf X\in \mathrm{dom} G}{\arg\min} ~ Q(\mathbf X, \mathbf Z)= \mathrm{soft}\left(\mathbf U(\mathbf Z), \frac{\lambda}{L_f}\right)\label{P_l_z}\end{aligned}$$ where $\mathrm{soft}\left(\mathbf U(\mathbf Z), \frac{\lambda}{L_f}\right)$ denotes an element wise operation with $$\begin{aligned} \mathrm{soft}\left(\mathbf W, L_0\right)= \begin{array}{ccc} \mathrm{sgn}(\mathbf W_{ij})(|\mathbf W_{ij}| - L_0)_+ \end{array}\end{aligned}$$ for all indices $i,j$ of the $N\times N$ matrix $\mathbf W$. These steps lead to a generalization of FISTA with matrix inputs, as given in Algorithm \[algo\_FISTA\_matrix\]. **Input:** observation matrix $\mathbf Y$, measurement matrices $\mathbf A$ and $\mathbf B$\ **Output:** estimate for sparse signal matrix, $\hat{\mathbf X}$ 1. Initialization: $\mathbf X^{0} =\mathbf 0$, $\mathbf X^{1}=\mathbf 0$, $t_0=1$, $t_1=1$, $k=1$\ Initialize: $\lambda_1$, $\beta\in (0,1)$, $\bar\lambda > 0$ 2. **while** not converged **do** 3. $\mathbf Z^{k} = \mathbf X^k + \frac{t_{k-1}-1}{t_k} (\mathbf X^k - \mathbf X^{k-1})$ 4. $ \mathbf U^k = \mathbf Z^k - \frac{1}{L}\mathbf A^T (\mathbf A \mathbf Z^k\mathbf B^T - \mathbf Y) \mathbf B $ 5. $\mathbf X^{k+1} = \mathrm{soft} \left(\mathbf U^k, \frac{\lambda_k}{L_f}\right)$ 6. $t_{k+1} = \frac{1+\sqrt{4t_k^2 + 1}}{2}$ 7. $\lambda_{k+1} = \max(\beta \lambda_k, \bar\lambda)$ 8. $k=k+1$ 9. **end while** $\hat{\mathbf X} = \mathbf X^{k} $ As in Algorithm \[algo\_FISTA\_vec\], the computational complexity is dominated by step 4. The matrix-matrix multiplication at step 4 in Algorithm \[algo\_FISTA\_matrix\] is performed with $\mathcal O(N^2(N+M+3L) + NML)$ computations. Since $M,L \leq N$, the worst case complexity is $\mathcal O(N^3)$. Recall, that FISTA in vector form has worst case complexity of $\mathcal O(N^4ML)$. Thus, there is a $\mathcal O(NML)$ gain in the matrix version compared to the vector approach. Equivalence of Algorithms \[algo\_FISTA\_vec\] and \[algo\_FISTA\_matrix\] -------------------------------------------------------------------------- It is easy to see that $\mathbf z^k$ and $\mathbf x^{k+1}$ computed in steps 3 and 5 in Algorithm \[algo\_FISTA\_vec\] are the same as $\mathrm{vec}(\mathbf Z^k)$ and $\mathrm{vec } (\mathbf X^{k+1})$, respectively, if $\mathbf u^k = \mathrm{vec}(\mathbf U^k)$ where $\mathbf Z_k$, $\mathbf U_k$ and $\mathbf X^{k+1}$ are as computed at steps 3, 4 and 5 in Algorithm \[algo\_FISTA\_matrix\] . Now, $$\begin{aligned} &~&\mathrm{vec}(\mathbf A^T \mathbf A \mathbf Z^k \mathbf B^T \mathbf B - \mathbf A^T \mathbf Y \mathbf B)\nonumber\\ &=&((\mathbf B^T \mathbf B)\otimes (\mathbf A^T \mathbf A)) \mathrm{vec}(\mathbf Z^k) - (\mathbf B^T\otimes \mathbf A^T )\mathrm{vec}(\mathbf Y)\nonumber\\ &=&(\mathbf B^T \otimes \mathbf A^T)(\mathbf B\otimes \mathbf A) \mathrm{vec}(\mathbf Z^k) - (\mathbf B^T\otimes \mathbf A^T )\mathrm{vec}(\mathbf Y)\nonumber\\ &=&(\mathbf C^T \mathbf C)\mathbf z^k -\mathbf C^T \mathbf y,\end{aligned}$$ where $\mathbf C= \mathbf B \otimes \mathbf A$. Thus, $\mathbf u_k$ computed at step 4 in Algorithm \[algo\_FISTA\_vec\] is the vectorized version of $\mathbf U^k$ of step 4 in Algorithm \[algo\_FISTA\_matrix\]. We conclude that Algorithms \[algo\_FISTA\_vec\] and \[algo\_FISTA\_matrix\] provide the same output, however, Algorithm \[algo\_FISTA\_matrix\] is more efficient. **Input:** observation matrix $\mathbf Y$, measurement matrices $\mathbf A$ and $\mathbf B$\ **Output:** index set $\Lambda$ containing locations of the non zero indices of the matrix $\mathbf X$, estimate for signal matrix $\hat{\mathbf X}$ 1. Initialization: residual $\mathbf R_0 =\mathbf Y$, index set $\Lambda_0=\emptyset$, $t=1$ 2. Find the two indices $\lambda_t=[\lambda_t(1)~ \lambda_t(2)]$ such that $$\begin{aligned} [\lambda_t(1)~\lambda_t(2)]& = &\underset{i,j}{\arg\max}~ |\mathbf b_j^T\mathbf R_{t-1}^T \mathbf a_i|\label{eq_OMP_matx} % &=&\underset{i,j, (i,j)\notin \Lambda_{t-1}}{\arg\max}~ |\mathbf b_j^T\mathbf R_{t-1}' \mathbf a_i | \end{aligned}$$ 3. Augment index set $\Lambda_t = \Lambda_t \cup \{\lambda_t\}$ 4. Find the new signal estimate $$\begin{aligned} % \mathbf x_t = \underset{\mathbf x} {\arg\min}\parallel \mathbf Y - \sum_{m=1}^{t} x_m \mathbf a_{\Lambda_t(m,1)} \mathbf b_{\Lambda_t(m,2)}^T ) \parallel_F\label{eq_frob_norm} \mathbf x_t = \mathbf D_t^{-1} \mathbf d_t\label{x_t} \end{aligned}$$ where $\mathbf D_t$ and $\mathbf d_t$ are as in (\[hat\_t\]) 5. Compute new residual $$\begin{aligned} %\mathbf Q_t &=& \nonumber\\ \mathbf R_t &=& \mathbf Y - \sum_{m=1}^{t} \mathbf x_t(m) \mathbf a_{\Lambda_t(m,1)} \mathbf b_{\Lambda_t(m,2)}^T\label{residual_M} \end{aligned}$$ 6. Increment $t$ and return to step $2$ if $t \leq d $, otherwise stop 7. Estimated support set $\hat\Lambda=\Lambda_d$\ Estimated signal matrix $\hat{\mathbf X}$: $(\Lambda_{d}(m,1), \Lambda_{d}(m,2))$-th component of $\hat{\mathbf X}$ is given by $\mathbf x_{d}(m)$ for $m=1=,\cdots, d$ while rest of the elements are zeros. Sparse Matrix Recovery via OMP ============================== Next, we consider the extension of standard OMP to the matrix form (\[obs\_1\]). We can write the observation $\mathbf Y$ in (\[obs\_1\]) as a summation of $N^2$ matrices as given below: $$\begin{aligned} \mathbf Y =\underset{i,j} {\sum} X_{ij} \mathbf a_i \mathbf b_j^T.\label{obs_sum}\end{aligned}$$ When $\mathbf X$ is sparse with $d$ nonzeros, the summation in (\[obs\_sum\]) has only $d$ terms. Let $\Sigma_d$ denote the support of $\mathbf X$ so that $\mathbf X_{ij}$ is non zero for $i,j=1,\cdots, N$, and let $\bar\Sigma_d$ be its complement. We can write (\[obs\_sum\]) as $ \mathbf Y =\underset{(i,j) \in \Sigma_d} {\sum} X_{ij} \mathbf a_i \mathbf b_j^T. $ Our goal is to recover $\mathbf X_{ij}$ for $(i,j)\in \Sigma_d$ in a greedy manner. The proposed OMP version with matrix inputs is given in Algorithm \[algo\_OMP\_matrix\]. In Algorithm \[algo\_OMP\_matrix\], $\Lambda_t$ contains estimated $(i,j)$ pairs up to $t$-th iteration in which the $m$-th pair is denoted by $(\Lambda_t(m,1), \Lambda_t(m,2))$ for $m=1,\cdots, t$. Once $\Lambda_t$ is updated as in step 3, the signal is estimated solving the following optimization problem: $$\begin{aligned} \mathbf x_t = \underset{\mathbf x} {\arg\min}\parallel \mathbf Y - \sum_{m=1}^{t} x_m \mathbf a_{\Lambda_t(m,1)} \mathbf b_{\Lambda_t(m,2)}^T ) \parallel_F. \label{eq_frob_norm}\end{aligned}$$ The solution of (\[eq\_frob\_norm\]) is given by $$\begin{aligned} \mathbf x_t = \mathbf D_t^{-1} \mathbf d_t\label{hat_t}\end{aligned}$$ where $\mathbf D_t$ is a $t\times t$ matrix in which the $(m,r)$-th element is given by $$\begin{aligned} (\mathbf D_t)_{m,r} = \mathbf b_{\Lambda_t(r,2)}^T \mathbf b_{\Lambda_t(m,2)} \mathbf a_{\Lambda_t(m,1)}^T \mathbf a_{\Lambda_t(r,1)} \end{aligned}$$ for $m,r=1,\cdots, t$ and $$\begin{aligned} \mathbf d_t = [\mathbf b_{\Lambda_t(1,2)}^T \mathbf Y^T \mathbf a_{\Lambda_t(1,1)}~\cdots ~\mathbf b_{\Lambda_t(t,2)}^T \mathbf Y^T \mathbf a_{\Lambda_t(t,1)}]^T\end{aligned}$$ is a $t\times 1$ vector. Then the new approximation at the $t$-th iteration is given by $$\begin{aligned} \mathbf Q_t &=& \sum_{m=1}^{t} \mathbf x_t(m) \mathbf a_{\Lambda_t(m,1)} \mathbf b_{\Lambda_t(m,2)}^T\label{Q_t} \end{aligned}$$ where $\mathbf x_t(m)$ denotes the $m$-th element of $\mathbf x_t$. Algorithm \[algo\_OMP\_matrix\] is a trivial extension of the standard OMP (and was also considered in [@Fang2] for $\mathbf A=\mathbf B$). Computational complexity ------------------------- As shown in [@Fang2] for $\mathbf A= \mathbf B$, it can be easily verified that Algorithm \[algo\_OMP\_matrix\] and the standard OMP [@tropp1] with vector inputs (\[obs\_2\]) provide the same performance at each iteration. However, the computational complexity of Algorithm \[algo\_OMP\_matrix\] is less than that of its vector counterpart. Step 2 in Algorithm \[algo\_OMP\_matrix\] can be implemented as a matrix multiplication $\mathbf A^T \mathbf R_{t-1} \mathbf B$. Thus, the computational complexity of this step is in the order of $\mathcal O(NML+N^2L)$. It is noted that, when implementing the standard OMP as in [@tropp1] with vector form (\[obs\_2\]), the equivalent step is computed with complexity of $\mathcal O(N^2ML)$. With respect to step 4 in Algorithm \[algo\_OMP\_matrix\], the matrix $\mathbf D_t$ requires $\mathcal O(t^2(M+L) )$ computations at the $t$-the iteration. The vector $\mathbf d_t$ requires $\mathcal O(t(ML+M))$ computations. Worst case complexity of the inverse operation is $\mathcal O(t^3)$. Matrix-vector multiplication in (\[x\_t\]) requires $\mathcal O(t^2)$ computations. Thus, at a given iteration, worst case complexity of step 4 in Algorithm \[algo\_OMP\_matrix\] is in the order of $\mathcal O(tML)$. It can be shown that the worst case computational complexity of the equivalent step of standard OMP with Kronecker products to estimate the signal at $t$-th iteration, is in the order of $\mathcal O(t^2 ML)$. Thus, steps 2 and 4 in Algorithm \[algo\_OMP\_matrix\] provide us with a computational gain over the equivalent steps of the standard OMP with Kronecker products. Therefore, we conclude that Algorithm \[algo\_OMP\_matrix\] is an efficient way to find sparse $\mathbf X$ from (\[obs\_1\]) compared to its vector counterpart (\[obs\_2\]) although both provide the same performance. It is further observed that this computational gain is not as significant as with FISTA. Numerical Results ================= In this section, we demonstrate the capability of recovering sparse $\mathbf X$ from observation model (\[obs\_1\]) via different algorithms and provide insights into the computational gains achievable with the matrix version. First, we illustrate the performance of FISTA with different choices for $\mathbf A$ and $\mathbf B$. For numerical results, we assume that $\mathbf X$ is a distributed sparse matrix in which each column has a maximum of $K$ nonzeros and the locations are generated uniformly. The values of nonzero entries are drawn from a uniform distribution in the range $ [-250, ~-200] \cup[200, ~250]$. We consider that the observation matrix $\mathbf Y$ in (\[obs\_1\]) is corrupted by additive noise and the elements of the noise matrix are assumed to be independently and identically distributed Gaussian random variables with mean zero and variance $\sigma_v^2$ ![Normalized reconstruction error vs $M=L$ with FISTA and different projection matrices, $N=40$, $\sigma_v^2=0.01$[]{data-label="fig_1"}](K2_DCT_Matrix_error){width="9.0cm"} In Fig. \[fig\_1\], we plot the normalized reconstruction error $\frac{||\mathbf X - \hat{\mathbf X}||_F}{||\mathbf X||_F}$ vs $M$ when $M=L$ averaging over $500$ trials. We let $N=40$, and $\sigma_v^2=0.01$. In Fig. \[fig\_1\], we illustrate two aspects. First, for given $K$, different types of matrices $\mathbf A$ and $\mathbf B$ are examined. We consider independent random rows of the $N\times N$ DCT matrix, Gaussian, and binary matrices. In the case of a Gaussian, elements are drawn from a normal ensemble and then orthogonalized. By binary matrix, we mean that the elements of the matrix can take values $1/N$ or $0$ with equal probability. Note that, random rows of DCT matrix and Gaussian matrix obey uniform uncertainty principle with good isometry constants in contrast to a binary matrix. When both matrices $\mathbf A$ and $\mathbf B$ are either random rows of DCT matrix or zero mean Gaussian, the recovery of the sparse matrix is guaranteed with less measurements compared to $N$. When $\mathbf A$ and $\mathbf B$ are binary, the recovery is not so good, which is intuitive since binary matrices are not “good” compressive sensing matrices. However, when $\mathbf A$ is binary and $\mathbf B$ is Gaussian, we see an improved performance compared to the case where both are binary. This provides an insight that even when one matrix does not obey uniform uncertainty principle with good isometry constant, still the sparse matrix can be recovered reliably when the other matrix is a “good compressive sensing” matrix. We will further investigate this observation in our future work. Second, with a given type of matrices (in the case of $\mathbf A$, $\mathbf B$ are Gaussian) we vary $K$. It is seen that, recovery capability of FISTA does not degrade significantly as $K$ increases. ![Normalized reconstruction error vs $L$ for given $M$ with FISTA, $\mathbf A$ and $\mathbf B$ contain independent random rows of the $N\times N$ DCT matrix, $N=40$, $\sigma_v^2=0.01$[]{data-label="fig_01"}](DCT_Matrix_error_ML_new){width="9.0cm"} In Fig. \[fig\_01\], we plot the reconstruction error vs $L$ with FISTA keeping $M$ fixed. As a benchmark, the curve corresponding to $M=L$ is also plotted. It can be seen that, when one dimension of the observation matrix is fixed, an improved performance in terms of signal reconstruction error is observed as the other dimension increases. However, when $M$ is very small (or below a certain value), complete recovery is not guaranteed even if $L=N$. This implies that when the dimension of the matrix $\mathbf A$ is fixed, increasing the number of columns of $\mathbf B^T$ does not necessarily guarantee complete recovery when $M$ is very small. $~$ N=20 N=40 N=60 -------- ---------- ----------- ------------ Matrix $0.3863$ $1.3064$ $6.5595$ Vector $1.0987$ $26.4162$ $142.7584$ : Runtime (in $s$) of FISTA with vector and matrix inputs \[table\_comparison\] In Table \[table\_comparison\], we compare the average runtime with MATLAB (in Intel(R) Core(TM) i7-3770 CPU$@$ 3.40GHzz processor with 12 GB RAM) for FISTA for matrix and vector versions as the sparse matrix dimension $N$ varies given that the number of iterations in both Algorithms \[algo\_FISTA\_vec\] and \[algo\_FISTA\_matrix\] is fixed at the same value ($=10000$). We let $K=N/20$ and $M=L=N/2$. Matrices $\mathbf A$ and $\mathbf B$ are assumed to be Gaussian. It reflects the computational efficiency of the matrix approach compared to the vector approach especially as $N$ increases although both algorithms provide the same performance. To illustrate the performance of OMP with matrix inputs, we plot the fraction of the support correctly recovered with Algorithm \[algo\_OMP\_matrix\] for different choices for $\mathbf A$ and $\mathbf B$ with $K=1$ in Fig. \[fig\_3\]. From Fig. \[fig\_3\], it is again observed that, although not with the same scale as with FISTA, the recovery capability can be improved when one projection matrix is binary and the other is Gaussian compared to the case where both $\mathbf A$ and $\mathbf B$ are binary. Another observation is that even with a Gaussian matrix, as $K$ increases the performance of OMP degrades significantly leaving OMP not a better choice when the sparsity level increases. ![Fraction of the support correctly recovered vs $M=L$ with OMP with different projection matrices with no noise[]{data-label="fig_3"}](OMP_2D_spt_A_B){width="9.0cm"} Discussion ========== In this paper, we showed numerically that recovering $\mathbf X$ based on (\[obs\_1\]) in its matrix form is more computationally efficient than solving it after converting to vector form via Kronecker products when $\mathbf X$ is arbitrarily distributed sparse. We developed matrix versions of FISTA to solve $l_1$ norm minimization in (\[matrix\_l1\]) efficiently and OMP to solve for $\mathbf X$ in a greedy manner. It has been shown that a significant computational gain is achieved by FISTA with matrix form compared to its vector counterpart. We further illustrated the recovery capability with different choices for projection operators. The results provide insight into the following. If a linear system of the form (\[obs\_2\]) can be converted into a matrix form as in (\[obs\_1\]), the problem can be solved more efficiently without losing performance with respect to the original vector form. Thus, it is worth investigating such scenarios where the matrix approach can be efficiently used to solve linear systems which are computationally demanding otherwise.

--- abstract: 'We study a distribution of thermal states given by random Hamiltonians with a local structure. We show that the ensemble of thermal states monotonically approaches the unitarily invariant ensemble with decreasing temperature if all particles interact according to a single random interaction and achieves a state $t$-design at temperature $O(1/\log(t))$. For the system where the random interactions are local, we show that the ensemble achieves a state $1$-design. We then provide numerical evidence indicating that the ensemble undergoes a phase transition at finite temperature.' author: - 'Yoshifumi Nakata, and Tobias J. Osborne' title: ' Thermal states of random quantum many-body systems ' --- Introduction ============ In quantum many-body systems, the number of degrees of freedom increases exponentially with the number of particles. This leads to difficulties for analysing their physics. One way to circumvent this difficulty is to assume random interactions, which are understood to be caused by inevitable impurities and disorder present in the physical system, and study typical properties of random many-body Hamiltonians. This idea is developed in random matrix theory and provides a successful description of the complicated physics of heavy atoms, quantum chromodynamics, mesoscopic systems, quantum gravity, and quantum chaotic systems (see e.g. Ref. [@M1990]). A study of random Hamiltonians has recently been extended to quantum spin systems on a lattice [@BS1970; @BS1971; @BF1971; @BF1971-2; @HMH2004; @HMH2005; @KLW2014; @KLW2014-2], where Hamiltonians contain only local interactions and respect the local structure of the system. Such random [*local*]{} Hamiltonians were shown in Refs. [@KLW2014; @KLW2014-2] to have a distribution of eigenvalues different from that of random Hamiltonians without local structure, which we call random [*global*]{} Hamiltonians, implying that random local models are significantly distinct from global ones. The idea of randomisation was applied to a study of the typical properties of quantum states using the unitarily invariant ensemble of states, often called [*random states*]{}. It has been pointed out that random states play a basic role in the foundation of physics, from quantum statistical mechanics [@PSW2006; @GLTZ2006; @R2008; @LPSW2009] to the black hole information paradox [@HP2007; @SS2008; @BF2012; @LSHOH2013]. From the viewpoint of random Hamiltonians, random states are an ensemble of ground states of random global Hamiltonians [@M1990], so that their properties are those typically observed in such [*global*]{} systems at [*zero*]{} temperature. It is then natural to ask whether they are still observed in systems with a [*local*]{} structure at [*finite*]{} temperature. In this Letter, we extend a study of the unitarily invariant ensemble of states (equivalently the ensemble of ground states of random global Hamiltonians) to the ensemble of thermal states of random global/local Hamiltonians. We especially investigate the ensemble of thermal states in comparison with the unitarily invariant ensemble. To this end, we exploit the concept of a [*state $t$-design*]{}, an ensemble of states simulating, up to the order $t$, statistical moments of random states [@RBSC2004; @AE2007], and investigate whether or not a state $t$-design is approximately achievable in random global/local Hamiltonian systems at finite temperature. This provides an insight into the validity of the foundation of physics using random states or a state $t$-design when the system respects a local structure and is at finite temperature. This also has importance in quantum information science since random states have a wide range of applications [@L1997; @EWSLC2003; @RBSC2004; @RRS2005; @S2006; @DCEL2009], and their approximate generation is one of the central issues [@EWSLC2003; @DLT2002; @HL2009; @DJ2011; @HL2009TPE; @BHH2012; @CHMPS2013; @NM2013; @NKM2014; @NM2014]. For an ensemble of thermal states in random [*global*]{} Hamiltonian systems, we show that the ensemble monotonically approaches the unitarily invariant one with decreasing temperature and that a state $t$-design is approximately achieved at $O(1/{\rm log}(t))$ temperature. We then show that, for an ensemble of thermal states in random [*local*]{} Hamiltonian systems, the ensemble is a state $1$-design at any temperature. We numerically study how close the ensemble is to higher designs and show that the ensemble quickly approaches the unitarily invariant one in a high-temperature regime, but converges to a non-uniform distribution at low temperature. We also give numerical evidence that these two regimes of the ensemble are separated by a singular point, indicating a phase transition of the ensemble at finite temperature. Since the singularity is not observed for random [*global*]{} Hamiltonians, this is an intrinsic characteristic of random [*local*]{} Hamiltonians. Random states and State $t$-design ================================== Let $\mathcal{K}$ be a Hilbert space of dimension $D$. Random states $\Upsilon$ are an ensemble of pure states uniformly distributed in Hilbert space with respect to the unitarily invariant measure. Random states play a fundamental role in physics [@PSW2006; @GLTZ2006; @R2008; @LPSW2009; @HP2007; @SS2008; @BF2012; @LSHOH2013], and are important resource in quantum information processing [@L1997; @EWSLC2003; @RBSC2004; @RRS2005; @S2006; @DCEL2009], however, they cannot be efficiently generated. Hence, an ensemble of states, called an [*$\epsilon$-approximate state $t$-design*]{} $\Upsilon_{t}^{(\epsilon)}$ has been studied [@EWSLC2003; @DLT2002; @HL2009; @DJ2011; @HL2009TPE; @BHH2012; @CHMPS2013; @NM2013; @NKM2014; @NM2014]. An $\epsilon$-approximate state $t$-design is defined by $\| \mathbb{E}_{\Psi \in \Upsilon_{t}^{(\epsilon)}}[ \Psi^{\otimes t} ] - \mathbb{E}_{\Psi \in \Upsilon}[ \Psi^{\otimes t} ] \|_1 \leq \epsilon$ [@RBSC2004; @AE2007]. Here, $\Psi={{\left \vert}\Psi \rangle \langle \Psi {\right \vert}}$, $\mathbb{E}$ represents an expectation over an ensemble, i.e. $\mathbb{E}[f(\Psi)]=\int f(\Psi) d\mu(\Psi)$ for the uniform measure $d\mu$, and $\| A \|_1 ={\mathrm{tr}}|A|$ is the trace norm. The $\mathbb{E}_{\Psi \in \Upsilon}[ \Psi^{\otimes t}]$ is calculated to be $\Pi_{\rm sym}^{(t)}/d_{\rm sym}^{(t)}$ using Schur’s lemma [@GR1999], where $\Pi_{\rm sym}^{(t)}$ is a projection operator onto a symmetric subspace of $\mathcal{K}^{\otimes t}$ and $d_{\rm sym}^{(t)} = {\mathrm{tr}}\Pi_{\rm sym}^{(t)} = \binom{D+t-1}{t}$. When $\epsilon=0$, a state $t$-design is called [*exact*]{} and we denote it by $\Upsilon_t$. Since a state $t$-design converges to random states when $t \rightarrow \infty$, the distance between a given ensemble of states and a state $t$-design provides a measure of the uniformity of the ensemble. Random Global and Local Hamiltonians ==================================== We define random Hamiltonians using the Gaussian unitary ensemble GUE$(L)$, which is an ensemble of $L \times L$ Hermitian matrices $\{ H \}$ distributed according to the Gaussian measure $d\mu (H)$ with density proportional to $\exp[-\frac{L}{2} {\mathrm{tr}}H^2]$ [@M1990]. We call the GUE the ensemble of [*random global Hamiltonians*]{} since it has no local structures. An important feature of random global Hamiltonians is that they are invariant under unitary conjugation, i.e. $d\mu (u H u^{\dagger}) = d\mu(H)$ for any $u \in \mathcal{U}(L)$ where $\mathcal{U}(L)$ is the unitary group of degree $L$. Hence, their ground states are random states. We also introduce the ensemble of [*random $k$-local Hamiltonians*]{}: consider a system consisting of $n$ particles, where the dimension of each particle is $d$. We denote by $\mathcal{H}=(\mathbb{C}^d)^{\otimes n}$ the corresponding Hilbert space. A Hamiltonian $H = \sum_{E} h_E$ is called $k$-local if each term $h_E$ acts nontrivially on a set $E$ of at most $k$ particles. An ensemble of $k$-local Hamiltonians $\mathfrak{H}_k$ is called [*random*]{} when each $h_E$ is independently chosen from ${\rm GUE}(d^k)$. Note that $\mathfrak{H}_n$=GUE$(d^n)$ is the ensemble of random global Hamiltonians. Unlike random global Hamiltonians, random $k$-local Hamiltonians for $k\neq n$ do not have global unitary invariance and the ensemble of ground states differs from random states. At finite temperature $T$, a state of a system at thermal equilibrium is given by a thermal state $\rho_H(\beta):=e^{- \beta H}/Z_H(\beta)$, where $\beta =1/T$ is the inverse temperature and $Z_H(\beta)={\mathrm{tr}}e^{- \beta H}$ is the partition function. Although a thermal state is in general not a pure state, we straightforwardly extend the definition of a $t$-design to a mixed state and define a distance between an ensemble of thermal states $\{\rho_H(\beta)\}_{H \in \mathfrak{H}_k}$ and a state $t$-design by $T_t^{(k)}(\beta) := \frac{1}{2} \| \mathbb{E}_{H \in \mathfrak{H}_k} [ \rho_H(\beta)^{\otimes t} ]-\mathbb{E}_{\Psi \in \Upsilon}[ \Psi^{\otimes t} ] \|_1$. If an ensemble of thermal states satisfies $T_t^{(k)}(\beta)=\epsilon/2$ for some $\beta$, average properties of the system can be described by an $\epsilon$-approximate state $t$-design up to order $t$. Random Global Hamiltonian systems ================================= We first study the ensemble of thermal states for random global Hamiltonians $\mathfrak{H}_n$. Since an ensemble of their ground states is unitarily invariant, we investigate how the ensemble converges at finite temperature to the unitarily invariant one with decreasing temperature. When $t=1$, $\mathbb{E}_{H \in \mathfrak{H}_n} [ \rho_H(\beta)^{\otimes t} ]$ reduces to the completely mixed state $I_D/D$, where $D=d^n$ and $I_D$ is the identity matrix in $\mathcal{H}$, since it commutes with all $u \in \mathcal{U}(D)$ due to the unitary invariance of $\mathfrak{H}_n=$GUE($D$). Hence, $T_1^{(n)}(\beta)=0$ for any $\beta$, implying that the ensemble of thermal states is a state $1$-design at any temperature. For $t \neq 1$, we show below that the distance $T_t^{(n)}(\beta)$ for any $t$ monotonically decreases when $\beta$ increases. For simplicity, we denote $\mathbb{E}_{\mathfrak{H}_n}[ (\rho_H (\beta))^{\otimes t}]$ by $X(\beta)$. Due to the unitary invariance of the GUE and the invariance of the partition function under unitary conjugation, $X(\beta)$ commutes with any unitary matrices of the form of $u^{\otimes t}$ ($u \in \mathcal{U}(D)$). From Schur-Weyl duality [@GR1999], $X(\beta) = (\lambda \Pi_{\rm sym}^{(t)}) \oplus A$, where $\lambda = \frac{1}{d_{\rm sym}^{(t)}}{\mathrm{tr}}X(\beta) \Pi_{\rm sym}^{(t)} < 1/d_{\rm sym}^{(t)}$, and $A$ is some operator on the space orthogonal to the symmetric subspace. Hence, we obtain $\Pi_{\rm sym}^{(t)} X(\beta) \Pi_{\rm sym}^{(t)} = \lambda(\beta) \Pi_{\rm sym}^{(t)}$. Recalling that $\mathbb{E}_{\Psi \in \Upsilon^{(t)}}[ \Psi^{\otimes t} ] = \Pi_{\rm sym}^{(t)}/ d_{\rm sym}^{(t)}$, $T_t^{(n)}(\beta)$ is divided into two terms; $T_t^{(n)}(\beta) =|\!| (\mathbb{I}^{(t)} - \Pi_{\rm sym}^{(t)}) X(\beta) |\!|_1/2 + |\!| \Pi_{\rm sym}^{(t)} ( X(\beta) - \Pi_{\rm sym}^{(t)}/ d_{\rm sym}^{(t)} ) |\!|_1/2$, where $\mathbb{I}^{(t)}$ is the identity operator on $\mathcal{H}^{\otimes t}$. Using $\lambda(\beta) \geq 1/d_{\rm sym}^{(t)}$, $T_t^{(n)}(\beta)$ is given by $$T_t^{(n)}(\beta) = 1- {\mathrm{tr}}X(\beta) \Pi_{\rm sym}^{(t)} . \label{Eq:distance}$$ We express the projector $\Pi_{\rm sym}^{(t)}$ as $\frac{1}{t!} \sum_{\sigma \in S_t} V_{\sigma}$, where $S_t$ is the permutation group of order $t$ and $V_{\sigma}$ is a unitary representation of $\sigma$. Using ${\mathrm{tr}}V_{\sigma_c} \rho^{\otimes t} = {\mathrm{tr}}\rho^{|c|}$ for a cyclic element $\sigma_c$ in the permutation group, where $|c|$ is the order of the cycle, $T_t^{(n)}(\beta)$ is rewritten as a function of purities of thermal states as follows; $$T_t^{(n)}(\beta) = 1-\frac{1}{t!} \sum_{\sigma \in S_t} \mathbb{E}_{\mathfrak{H}_n} \prod_{\sigma_c \in \sigma} {\mathrm{tr}}\bigl(\rho_H (\beta) \bigr)^{|c|}, \label{Eq:cycle}$$ where the product is taken over all cycles in $\sigma$. We finally show that $\frac{\partial}{\partial \beta} T_n^{(t)} (\beta) \leq 0$ for any $\beta$, which implies a monotonic decrease of $T_n^{(t)} (\beta)$ with respect to $\beta$. It suffices from Eq.  to show $\frac{\partial}{\partial \beta} {\mathrm{tr}}\bigl(\rho_H (\beta) \bigr)^{m} \geq 0$ for any natural number $m$. This simply holds since $ \frac{\partial}{\partial \beta} {\mathrm{tr}}\bigl(\rho_H (\beta) \bigr)^{m} = m \frac{Z_H(m\beta)}{\bigl( Z_H (\beta) \bigr)^m} [ \langle H \rangle_{\beta} - \langle H \rangle_{m \beta}] $, where $\langle H \rangle_{\beta}:={\mathrm{tr}}[ H \rho_H (\beta)]$ is an internal energy of $H$ at the inverse temperature $\beta$, and the internal energy satisfies $\langle H \rangle_{\beta} \geq \langle H \rangle_{\beta'}$ for $\beta \leq \beta'$. The equality holds if and only if $\beta=0, \infty$. When $\beta=0$, a thermal state $\rho_H(\beta)$ is the completely mixed state $I_D/D$, so that $T_t^{(n)}(0)=1-d_{\rm sym}^{(t)}/D^{t}$, which is approximately given by $1-1/t!$ for a constant $t$. On the other hand, $\lim_{\beta \rightarrow \infty} T_n^{(t)}(\beta)=0$ since ground states of random global Hamiltonians are random states. Hence, $T_t^{(n)}(\beta)$ monotonically decreases from $1-1/t!$ to zero with decreasing temperature. ![ (Color online) Panel (a) shows upper bounds on $T_t^{(n)}(\beta)$ for $D=4$, where $t=11,8,5,2$ from top to bottom. The inset shows the scaling of $T_t^{(n)}(\beta)/t$ with $\beta$ for $t=3,6,9,12$ from top to bottom, which indicates that $T_t^{(n)}(\beta)$ scales as $t e^{-c \beta}$ for large $\beta$. Panel (b) shows a sufficient temperature for thermal states of $\mathfrak{H}_n$ to be an $\epsilon$-approximate $t$-design for $\epsilon=0.5,0.4,0.3,0.2$ from top to bottom. []{data-label="Fig:UBofDistance"}](UpperBound_GUE_d4.eps "fig:"){width="42mm"} ![ (Color online) Panel (a) shows upper bounds on $T_t^{(n)}(\beta)$ for $D=4$, where $t=11,8,5,2$ from top to bottom. The inset shows the scaling of $T_t^{(n)}(\beta)/t$ with $\beta$ for $t=3,6,9,12$ from top to bottom, which indicates that $T_t^{(n)}(\beta)$ scales as $t e^{-c \beta}$ for large $\beta$. Panel (b) shows a sufficient temperature for thermal states of $\mathfrak{H}_n$ to be an $\epsilon$-approximate $t$-design for $\epsilon=0.5,0.4,0.3,0.2$ from top to bottom. []{data-label="Fig:UBofDistance"}](Suff_Temp_for_tDesign.eps "fig:"){width="42mm"} For sufficiently large $t$ and $\beta$, $T_t^{(n)}(\beta)$ can be further calculated from Eq. . Let $H=\sum_{m=0}^{D-1} E_m {{\left \vert}E_m \rangle \langle E_m {\right \vert}}$ be an eigendecomposition of $H$, where the eigenvalues satisfy $E_i \leq E_j$, for $i \leq j$. Using this notation, Eq.  is rewritten as $ T_t^{(n)}(\beta) = 1- {\mathrm{tr}}\mathbb{E}_{\mathfrak{H}_n}[ \sum \prod_{j=1}^t p_{m_j}(\beta) \bigotimes_{j=1}^t {{\left \vert}E_{m_j} \rangle \langle E_{m_j} {\right \vert}} \Pi_{\rm sym}^{(t)} ] $, where the summation runs over all $m_1, \cdots, m_t \in \{ 0,\cdots D -1\}$. Since ${\mathrm{tr}}\bigotimes_{j=1}^t {{\left \vert}E_{m_j} \rangle \langle E_{m_j} {\right \vert}} \Pi_{\rm sym}^{(t)} =1$ when $m_i = m_j$ for all $i,j \in \{1,\cdots, t\}$ and is at most $O(1/t)$ otherwise, $T_t^{(n)}(\beta)$ is simply given by $T_t^{(n)}(\beta) = 1- \mathbb{E}_{\mathfrak{H}_n}[ \sum_{m} (p_{m}(\beta))^t] - O(1/t)$, where we have used that $\mathbb{E}_{\mathfrak{H}_n}[ {{\left \vert}E_{m} \rangle \langle E_{m} {\right \vert}}^{\otimes t}] = \Pi_{\rm sym}^{(t)}/d_{\rm sym}^{(t)}$ for any $m \in \{0,\cdots, D-1\}$. When $\beta$ is sufficiently large such that $t \beta \gg 1/\Delta E$ where $\Delta E=E_1-E_0$, $T_t^{(n)}(\beta)$ is approximately given by $$T_t^{(n)}(\beta) = 1- \mathbb{E}_{\mathfrak{H}_n}[(p_{0}(\beta))^t]. ~\label{Eq:EB}$$ This provides, in general, an upper bound for $T_t^{(n)}(\beta)$, and it becomes exact when $\beta \rightarrow \infty$. Since the joint probability distribution of $\{E_k \}_{k=0}^{D-1}$ for $\mathfrak{H}_n$ is known [@HP2007; @SS2008; @BF2012; @LSHOH2013], an upper bound of $T_t^{(n)}(\beta)$ can be numerically (but exactly) calculated as given in Fig. \[Fig:UBofDistance\]. From Eq. , we also obtain the scaling of a threshold temperature $T_{\epsilon}$, below which the ensemble of thermal states is an $\epsilon$-approximate $t$-design. Since $(p_{0}(\beta))^t \sim 1-t e^{-\Delta E \beta}$ for large $t$ and $\beta$, $T_t^{(n)}(\beta) \sim O( t e^{-c \beta})$, where $c$ is a constant. Thus, we obtain $T_{\epsilon}=O((\log t + \log 1/\epsilon)^{-1})$. The numerics show that this holds even for relatively small $t$ (see the inset of Panel (a) and Panel (b) in Fig. \[Fig:UBofDistance\]). Thus, average properties of random global Hamiltonian systems at temperature $O((\log t + \log 1/\epsilon)^{-1})$ are describable by an $\epsilon$-approximate $t$-design. Since an $\epsilon$-approximate $t$-design has important properties of random states, such as a scrambled feature [@HP2007; @SS2008; @BF2012; @LSHOH2013], even for small $t$, and can be replaced with random states in many quantum informational tasks using them [@L2009], so does the ensemble of thermal states in random global Hamiltonian systems at the corresponding temperature. Random Local Hamiltonian systems ================================ For random local Hamiltonians, the investigation of $T_t^{(k)}$ is not simple since the ensemble of local Hamiltonians $\mathfrak{H}_k$ does not have global unitary invariance. However, the ensemble of thermal states of $\mathfrak{H}_k$ for any $k$ is still an exact state $1$-design at any temperature as shown below. The $\mathfrak{H}_k$ still remains invariant under the conjugation of local unitary operations of the form $\otimes_{l=1}^n u_l$, where $u_l \in \mathcal{U}(d)$. Hence, the $\mathbb{E}_{\mathfrak{H}_k} [ \rho_H (\beta)]$ commutes with all local unitary matrices, implying that it is in the commutant of them; $\mathbb{E}_{\mathfrak{H}_k} [ \rho_H (\beta)] \in ( \otimes_{l=1}^n \mathcal{U}(d) )'$, where $X'$ is the commutant of an algebra $X$. Since the commutant of the tensor products is the same as the tensor product of the commutants of each algebra, $( \otimes_{l=1}^n \mathcal{U}(d) )' = \otimes_{l=1}^n ( \mathcal{U}(d) )'$ [@RD1975], $\mathbb{E}_{\mathfrak{H}_k} [ \rho_H (\beta)]$ is in $\otimes_{l=1}^n (\mathcal{U}(d) )'$. Recalling $(\mathcal{U}(d) )' = \{ I_d \}$ and ${\mathrm{tr}}\mathbb{E}_{\mathfrak{H}_k} [ \rho_H (\beta)]=1$, we obtain $\mathbb{E}_{\mathfrak{H}_k} \rho_H (\beta)= I_D/D$, which implies that $\mathfrak{H}_k$ is an exact $1$-design for any $k$ and for any $\beta$. We numerically study how close the ensemble of thermal states is to higher designs. We particularly consider neighboring interactions on a line of qubits, i.e., $d=2$. The results are given for $n=5$ and $t=2$ in Fig. \[Fig:localNN\]. It is observed that the distance $T_2^{(k)}(\beta)$ quickly decreases with increasing $\beta$ in a small $\beta$ region. However, when $\beta$ is larger than a certain value, $T_2^{(k)}(\beta)$ is almost constant. This limiting values depend on $k$ and are smaller for larger $k$, which is intuitive since the ensemble becomes random states when $k=n$ and $\beta \rightarrow \infty$. It is also observed in Fig. \[Fig:localNN\] that $T_2^{(k)}(\beta)$ monotonically decreases with $\beta$ when $k \neq 2$. In the case of $k=2$, there exists a dip around $\beta =1$, which is also observed for different $n$. ![ (Color online) Panel (a) shows the distance $T_t^{(k)}(\beta)$ for $n=5$, $t=2$, and neighboring interactions. Panel (b) shows its derivative $\partial_{\beta} T_t^{(k)}(\beta)$ in terms of $\beta$. The purple($\bigtriangleup$), orange($\Box$), green($\bigcirc$) represent $k=2,3,4$, respectively, and the blue solid line is for random global Hamiltonians $\mathfrak{H}_n$. The expectation of $\mathbb{E}_{\mathfrak{H}_k}[ (\rho_H (\beta))^{\otimes t}]$ is taken by sampling $2\times 10^4$ Hamiltonians. It is observed that $\beta_{\rm c}^{(2)} \sim 0.8$, $\beta_{\rm c}^{(k)} \sim 0.85$, and $\beta_{\rm c}^{(k)} \sim 1.05$. []{data-label="Fig:localNN"}](DistanceNN.eps "fig:"){width="42mm"} ![ (Color online) Panel (a) shows the distance $T_t^{(k)}(\beta)$ for $n=5$, $t=2$, and neighboring interactions. Panel (b) shows its derivative $\partial_{\beta} T_t^{(k)}(\beta)$ in terms of $\beta$. The purple($\bigtriangleup$), orange($\Box$), green($\bigcirc$) represent $k=2,3,4$, respectively, and the blue solid line is for random global Hamiltonians $\mathfrak{H}_n$. The expectation of $\mathbb{E}_{\mathfrak{H}_k}[ (\rho_H (\beta))^{\otimes t}]$ is taken by sampling $2\times 10^4$ Hamiltonians. It is observed that $\beta_{\rm c}^{(2)} \sim 0.8$, $\beta_{\rm c}^{(k)} \sim 0.85$, and $\beta_{\rm c}^{(k)} \sim 1.05$. []{data-label="Fig:localNN"}](combine.eps "fig:"){width="42mm"} Panel (b) in Fig. \[Fig:localNN\] shows that the two regimes of the ensemble of thermal states, a quickly spreading regime and a converging regime, are likely to be separated by a singular point $\beta_{\rm c}^{(k)}$. This indicates an existence of a phase transition between the two regimes. When $\beta < \beta_{\rm c}^{(k)}$, $\partial_{\beta} T_t^{(k)}(\beta)$ scales quadratically with $\beta$, while it approaches zero exponentially for $\beta > \beta_{\rm c}^{(k)}$ if $k \neq 2$. For $k = 2$, $\partial_{\beta} T_t^{(k)}(\beta)$ approaches a positive value exponentially and then decreases to zero, reflecting the dip of $T_t^{(2)}(\beta)$. Although the kink of $\partial_{\beta} T_t^{(k)}(\beta)$ at $\beta_{\rm c}^{(k)}$ is less prominent for larger $k$, it seems present even for $k=n-1$ but not for $k=n$, where the ensemble converges slowly and smoothly to the unitarily invariant one with decreasing temperature. From these observations, we conjecture that $T_t^{(k)}(\beta)$ ($k\neq n$) has a singular point $\beta_{\rm c}^{(k)}$ in the thermodynamic limit ($n \rightarrow \infty$), leading to a second-order phase transition of the distribution of thermal states. We also numerically checked that most of the features are present for the case of interactions on a complete graph, except that $T_t^{(k)}(\beta)$ and $\partial_{\beta} T_t^{(k)}(\beta)$ do not monotonically decrease in terms of $k$ in a high-temperature region. A possible interpretation of the distinctive temperature $\beta_{\rm c}^{(k)}$, combined with a fact that the expected density of states for the ensemble of random local Hamiltonians is a Gaussian [@HMH2004; @KLW2014], is that the distribution of eigenstates with low energies intrinsically differs from those with intermediate energies. To explain this clearly, let $P(E) \Delta E \propto \exp[-(\frac{E-\bar{E}}{\sigma})^2] \frac{e^{-\beta E}}{Z(\beta)} \Delta E$ be the population of eigenstates between the energy $[E, E+\Delta E]$ for a small $\Delta E$ in a thermal state, where $\bar{E}$ and $\sigma$ is the mean and the standard deviation of the Gaussian density of states, respectively. When $\beta$ is sufficiently small such that the thermal population $e^{-\beta E}/Z(\beta)$ is close to $1/d^n$ for any $E$, the Gaussian term in $P(E)$ is dominant. Hence, the corresponding thermal state is effectively described by a mixture of the eigenstates with eigenenergies in $[\bar{E}- \sigma, \bar{E}+ \sigma]$. On the other hand, the thermal population in $P(E)$ becomes dominant when $\beta$ is large. For instance, the population of eigenstates with energy $E \in [\bar{E}- \sigma, \bar{E}+ \sigma]$ is comparable with that of ground states at $\beta_E = (E-E_0)/\sigma^2$. For $\beta \gg \beta_E$, eigenstates with energy $E$ does not contribute to the thermal state. Due to this trade off between the Gaussian and the thermal factors in $P(E) \Delta E$, $T_t^{(k)}(\beta)$ in a small (large) $\beta$ reflects the properties of eigenstates with intermediate (low) eigenenergies. The $\beta^{(k)}_c$ point is understood as the point where this transition happens. The singularity of $T_t^{(k)}(\beta)$ at $\beta^{(k)}_c$ then indicates that the distribution of intermediate eigenstates for $H \in \mathfrak{H}_k$ is qualitatively different from that of low-energy eigenstates. Since $T_t^{(k)}(\beta)$ rapidly decrease with $\beta$ when $\beta < \beta_{\rm c}^{(k)}$, the distribution of intermediate eigenstates is likely to be similar to the unitarily invariant one. The situation is entirely different for the ensemble of random global Hamiltonians, where no distinctive temperature is observed, for the following two reasons. First, there is no trade off between the density of states and the thermal population in $P(E)$ since the density of states obeys the semi-circle law given by $\sqrt{c - (E-E_{\mu})^2}$ [@M1990]. This is negligible at any temperature compared to the thermal population that exponentially scales with $E$. Second, the ensemble of eigenstates of any eigenenergy is unitarily invariant, which is in sharp contrast to the distribution of eigenstates of random local Hamiltonians. These results show that the distribution of thermal states in random $k$-local Hamiltonian systems has a rich structure and is qualitatively different from that in global Hamiltonian systems, even if $k=O(n)$. This is not only interesting from a theoretical point of view but also physically important since it means that most of the previously known results of random states, equivalently an ensemble of ground states in random global Hamiltonian systems, related to physical situations [@PSW2006; @GLTZ2006; @R2008; @LPSW2009; @HP2007; @SS2008; @BF2012; @LSHOH2013] cannot be directly applied to many-body systems with local interactions. Summary and outlook =================== In this Letter we have investigated a distribution of thermal states in random global/local Hamiltonian systems. For random global Hamiltonians, we analytically showed that the ensemble of thermal states monotonically approaches the unitarily invariant one with decreasing temperature and achieves an $\epsilon$-approximate state $t$-design when the temperature is $O(1/(\log t + \log 1/\epsilon))$. On the other hand, the ensemble of thermal states for random $k$-local Hamiltonians achieves a state $1$-design but not higher designs. We then showed by studying a higher design that the ensemble is divided into two regimes of temperature, a regime where the ensemble quickly spreads toward the uniform one with decreasing temperature and a regime where the ensemble converges to a non-uniform one, which are likely to be separated by a singular point. These studies have revealed the similarities and the differences of random global/local Hamiltonians from the viewpoint of the distribution of thermal states, and have opened a new approach to study random systems by connecting random matrix theory and quantum information science. It is desirable to analytically confirm the features numerically observed in this paper. Proving the phase transition of the ensemble of thermal states is especially important. It is also interesting to derive the probability distribution of ground states in random local Hamiltonian systems, by which an understanding of local Hamiltonian systems will be further deepened. Acknowledgement =============== This work was supported by the ERC grants QFTCMPS, and SIQS, and through the DFG by the cluster of excellence EXC 201 Quantum FQ Engineering and Space-Time Research. Y.N. thanks to R. F. Werner, A. Milsted and C. Bény for helpful discussions. Y. N. acknowledges JSPS Postdoctoral Fellowships for Research Abroad. [99]{} M. L. Metha, *Random Matrices*, (Academic Press, 1990). J. B. French, and S. S. M. Wong, Phys. Lett. B, [**33**]{}, 449 (1970). J. B. French, and S. S. M. Wong, Phys. Lett. B, [**35**]{}, 5 (1971). O. Bohigas, and J. Flores, Phys. Lett. B, [**34**]{}, 261 (1971). O. Bohigas, and J. Flores, Phys. Lett. B, [**35**]{}, 383 (1971). M. Hartmann, G. Mahler, and O. Hess, Lett. Math. Phys, [**68**]{}, 103 (2004). M. Hartmann, G. Mahler, and O. Hess, J. Stat. Phys., [**119**]{}, 1139 (2005). J. P. Keating, N. Linden, and H. J. Wells, preprint, arXiv:1403.1114. J. P. Keating, N. Linden, and H. J. Wells, preprint, arXiv:1403.1121. S. Popescu, A. J. Short and A. Winter, Nature Physics, 2:754-758 (2006). S. Goldstein, J. L. Lebowitz, R. Tumulka and N. Zanghi, Phys. Rev. Lett. [**96**]{}, 050403 (2006). P. Reimann, Phys. Rev. Lett. [**101**]{}, 190403, (2008). N. Linden, S. Popescu, A. J. Short and A. Winter, Phys. Rev. E [**79**]{}, 061103 (2009). P. Hayden and J. Preskill, JHEP 0709:120 (2007). Y. Sekino, and L. Susskind, JHEP10:065 (2008). W. Brown, and O. Fawzi, preprint arXiv:1307.0632 (2012). N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, JHEP 2013:22 (2013). A. Ambainis and J. Emerson, IEEE Conference on Computational Complexity, (2007) 129-140. J. M. Renes, R. Blume-Kohout, A. J. Scott and C. M. Caves, J. Math. Phys., [**45**]{}, 6 (2004). S. Lloyd, Phys. Rev. A [**55**]{}, 1613 (1997). J. Emerson, Y. S. Weinstein, M. Saraceno, S. Lloyd and D. G. Cory. Science, [**302**]{} (2003) 2098-2100. J. Radhakrishnan, M. Rotteler and P. Sen, Algorithmica, [**55**]{}, 490-516 (2009). P. Sen, IEEE conference on Computational Complexity, 274-287 (2006). C. Dankert, R. Cleve, J. Emerson and E. Livine, Phys. Rev. A, [**80**]{}, 012304 (2009). D. P. DiVincenzo, D. W. Leung and B. M. Terhal, IEEE Transactions Information Theory, **48**, 580-599 (2002). A. W. Harrow and R. A. Low, Commun. Math. Phys., [**291**]{}, 257 (2009). I. T. Diniz and D. Jonathan, Commun. Math. Phys., [**304**]{}, 281 (2011). A. W. Harrow and R. A. Low, Proceedings of RANDOM 2009, LNCS, [**5687**]{}, 548 (2009). F. G. S. L. Brandao, A. W. Harrow, M. Horodecki, preprint, arXiv:1208.0692 (2012). P. Ćwikliński, M. Horodecki, M. Mozrzymas, Ł. Pankowski, and M. Studziński, J. Phys. A: Math. Theo., [**46**]{}, 305301 (2013). Y. Nakata and M. Murao, International Journal of Quantum Information [**11**]{} (2013) 1350062. Y. Nakata, M. Koashi, and M. Murao, New J. Phys. [**16**]{} (2014) 053043. Y. Nakata and M. Murao, Eur. Phys. J. Plus (2014) [**129**]{}: 152. R. Goodmamn, and N. R. Wallach, [*Representations and Invariants of the Classical Groups*]{}, Cambridge University Press (1999). R. A. Low, Proc. R. Soc. A 8 vol. 465 no. 2111 3289-3308 (2009). M. A. Rieffel, and A. van Daele, Bull. London Math. Soc. [**7**]{}, 257-260 (1975).

--- abstract: 'We propose a novel deep learning model for classifying medical images in the setting where there is a large amount of unlabelled medical data available, but labelled data is in limited supply. We consider the specific case of classifying skin lesions as either malignant or benign. In this setting, the proposed approach – the semi-supervised, denoising adversarial autoencoder – is able to utilise vast amounts of unlabelled data to learn a representation for skin lesions, and small amounts of labelled data to assign class labels based on the learned representation. We analyse the contributions of both the adversarial and denoising components of the model and find that the combination yields superior classification performance in the setting of limited labelled training data.' author: - 'Antonia Creswell[^1] , Alison Pouplin, Anil A Bharath' bibliography: - 'bib.bib' title: 'Denoising Adversarial Autoencoders: Classifying Skin Lesions Using Limited Labelled Training Data' --- Introduction ============ The problem of image classification is one of assigning one or more labels to a given image. Deep learning has been demonstrated to be able to achieve both human and super-human levels of performance [@Esteva2017], on classification tasks. However, achieving competitive levels of performance using deep learning often requires vast numbers of {image, label} pairs, typically in the millions.\ In the medical image setting, it is unlikely that vast amounts of labelled images are available, particularly since medical experts are required to label the data, and this may be very costly and time consuming. Instead, it is often the case that there exists a large corpus of unlabelled data and a smaller dataset of labelled data.\ We propose a model that is able to learn both from labelled data and from unlabelled data by building on previous work involving autoencoders [@bengio2013generalized; @kingma2013auto; @makhzani2015adversarial; @vincent2008extracting; @im2015denoising]. Autoencoders are able to learn data representations from unlabelled data, by jointly learning an encoder and decoder. The encoder maps data samples – in this case images – to a low dimensional encoding space, and the decoder maps the encoding back to image space. An autoencoder is trained to reconstruct its input. There are two key factors that enhance the performance of autoencoders, these are: - Denoising: Before being encoded, an input image is corrupted, and the decoder is trained to recover the clean image. By making the decoding process more challenging, the autoencoder learns more robust representations [@vincent2008extracting; @vincent2010stacked].\ - Regularisation: Rather than allowing encoded data samples to occupy an unconstrained space, the distribution of encoded samples may be shaped to match a desired, *prior* distribution, for example a multivariate standard normal distribution. Regularisation reduces the amount of information that may be held in the encoding, forcing the model to learn an efficient representation for the data. To implement a denoising process, an arbitrary corruption process may be used. For example, white Gaussian noise [@bengio2013generalized] may be added to samples of the training data. Corruption is often trivial to implement. More challenging is the regularisation of the distribution of encoded data samples. There are at least two approaches for shaping the distribution of encoded samples to match a desired distribution. The two key methods for regularising the encoding space are: #### **Variational** Minimising the KL divergence between the distribution of encoded samples and a chosen prior distribution [@kingma2013auto]. For ease of implementation, the prior distribution is often a multivariate standard normal distribution and the encoder is designed to learn parameters of a Gaussian distribution. #### **Adversarial** Rather than using the encoder to parametrise a distribution and calculate the KL divergence, a third, discriminative model is trained to correctly distinguish encoded samples from samples drawn from a chosen prior distribution. The encoder is then updated to encode samples such that the discriminator cannot distinguish encoded data samples from samples drawn from the prior distribution [@makhzani2015adversarial]. We will more formally introduce adversarial training in Section \[sec:AAE\].\ The adversarial [@makhzani2015adversarial] approach allows the encoder to be more expressive than the variational approach [@kingma2013auto], and has achieved superior classification performance in a semi-supervised fashion on several benchmark dataset. While denoising and adversarial training have been used to augment autoencoders in isolation, they have yet to be combined in one model. Here, we propose augmenting an autoencoder with both a denoising criterion and by using adversarial training to shape the distribution of encoded data samples. We augment this model further to make use of labelled data where it is available while still learning from unlabelled data where label information is not available.\ Our contributions are as follows: - We introduce the semi-supervised denoising adversarial autoencoder (ssDAAE) which is able to learn from a combination of labelled and unlabelled data (Section \[sec:SSDAAE\]).\ - We apply our model, the ssDAAE, to the task of classifying skin-lesions as benign or malignant in the setting where the amount of labelled data is limited (Section \[sec:method\]).\ - We compare performance of the ssDAAE with a semi-supervised adversarial autoencoder (ssAAE), a fully supervised AAE (sAAE), a fully supervised DAAE (sDAAE), and a CNN trained with and without corruption. For fair comparison, the CNNs had the same architecture as the encoder of the ssAAE and ssDAAE; that is, the portion of the ssAAE and ssDAAE architecture used to perform classification is the same as the CNN used for standard deep network classification. Additionally, we assessed the effect of additive noise during training of the otherwise standard CNN. Our results show that the ssDAAE consistently out performs the others.\ Although we demonstrate this approach on skin lesions, the semi-supervised approach explored in this paper are not specific to skin lesions, and could potentially be applied to other image datasets where labelled samples are in limited supply, but there is a surplus of unlabelled images that have been captured. Method: Classifying Skin Lesions ================================ In this section, we formulate the ssDAAE. First, we discuss the skin lesion classification problem. Secondly, we describe the Adversarial Autoencoder (AAE) and then we describe how the AAE may be augmented to become the ssDAAE. Finally, we describe how the ssDAAE is trained. Skin lesion classification -------------------------- Skin lesion classification is a non-trivial problem. Even humans have to be specially trained to be able to distinguish benign (not harmful) skin lesions from malignant (harmful) skin lesions. Examples of benign and malignant skin lesions are shown in Figure \[fig:skin\_lesions\]. The high level goal is to train a model to correctly predict whether a skin lesion is benign or malignant. Beyond this, we want to design models, for which we can be confident that we correctly identify a specific proportion of malignant skin lesions as being malignant, while still being able to correctly identify a large number of benign skin lesions as being benign. To this end, in the following sections we describe the model that we propose for skin lesion classification in the setting of limited labelled data. [0.45]{} ![**Examples of Benign and Malignant skin-lesions.** Classifying skin lesions as benign or malignant is non-trivial and requires expert knowledge.[]{data-label="fig:skin_lesions"}](images/benign "fig:"){width="0.9\linewidth"} [0.45]{} ![**Examples of Benign and Malignant skin-lesions.** Classifying skin lesions as benign or malignant is non-trivial and requires expert knowledge.[]{data-label="fig:skin_lesions"}](images/malignant "fig:"){width="0.9\linewidth"} Adversarial Autoencoders {#sec:AAE} ------------------------ An autoencoder consists of two models, and encoder and a decoder, each with their own set of learnable parameters. In our approach, we are using deep convolutional neural networks to embody the encoder and decoder. The encoder, $ E_{\theta_E} :x \rightarrow \hat{z}$ with parameters $\theta_E$, is designed to map an image sample, $x$ to an encoding, $\hat{z}$. The encoding vector, $\hat{z}$, is of much lower dimension than the number of pixels in an image, $x$. The decoder, $D_{\theta_D}: \hat{z} \rightarrow \hat{x}$ is designed to map an encoding $\hat{z}$ back to an image, $\hat{x}$. The parameters, $\theta_E$ and $\theta_D$ of the encoder and decoder respectively are learned such that the difference between the input to the encoder, $x$, and the output of the decoder, $\hat{x}$, are minimised.\ The adversarial autoencoder [@makhzani2015adversarial] incorporates adversarial training [@goodfellow2014generative] to shape the distribution of encoded data samples to match some chosen prior distribution, $p(z)$, such as a multivariate standard normal distribution. Note that we are applying adversarial training to the encoded data samples, rather than the data samples, as more commonly seen in the literature [@goodfellow2014generative; @radford2015unsupervised]. Adversarial training requires the introduction of another model, a discriminator, for which we also use a deep convolutional neural network. The discriminator, $T_{\theta_T}: z \rightarrow (0,1)$ maps encodings (either encoded data samples, $\hat{z}$ or samples drawn from the prior, $z$,) to a probability of whether that sample comes from the chosen prior distribution. The parameters, $\theta_T$ of the discriminator are learned such that high values are assigned to samples that come from the chosen prior distribution and low values are assigned to samples that come from the encoder. To encourage encoded samples to match the chosen prior distribution, the parameters of the encoder, $\theta_E$ are updated such that $T_{\theta_T}(E_{\theta_E}(x))$ are maximised.\ Formally, the following objectives must be optimised during the training of an adversarial autoencoder: $$\label{recCost} \arg \min_{\theta_E, \theta_D} || x - D_{\theta_D}(E_{\theta_E}(x)) ||_2^2$$ [$$\label{disCost_z} \arg \max_{\theta_T} \mathbb{E}_{z \sim p(z)} \log T_{\theta_T}(z) + \mathbb{E}_{\hat{z} \sim p_{g(z)}} \log (1 - T_{\theta_T}(\hat{z}))$$ where $p_g(z)$ is the distributions of encoded data samples, $\hat{z}=E_{\theta_E}(x)$ and $x \sim \mathcal{D}$.]{} $$\label{encCost} \arg \min_{\theta_E} \mathbb{E}_{x \sim \mathcal{D}(x)} \log T_{\theta_T}(E_{\theta_E}(x))$$ where $p(z)$ is some chosen prior distribution, for example a standard normal and $\mathcal{D}(x)$ is the training data distribution.\ Equation (\[recCost\]) is the reconstruction cost that is used to train the encoder and decoder. This cost should be minimised, so that input images may be recovered after encoding and decoding. Equation (\[disCost\_z\]) is the discriminator cost, which, when minimised, means that the discriminator can correctly distinguish between encoded data samples and samples from the chosen prior distribution. Equation (\[encCost\]) is a second regularisation cost used to update the encoder. When minimised – simultaneously with Equation (\[encCost\]) [@goodfellow2014generative] – this regularisation cost encourages the distribution of encoded samples to be similar to the chosen prior distribution.\ An adversarial autoencoder may be trained entirely with unlabelled data and may be evaluated by measuring reconstruction error on a test dataset, or synthesising novel samples by first drawing samples, $z$, from the chosen prior distribution and passing these through the decoder to produce synthetic images, $\hat{x}$. The process of encoding and decoding test samples often reveals whether or not the decoder model has learned a sufficient representation for the data. A further test is to attempt to generate novel samples, by passing random encodings – drawn from the chosen prior distribution – through the decoder. Since a regularised autoencoder is able to generate novel samples, we often refer to it incorporating a [*generative model*]{} of the data.\ In its current form, it is not immediately obvious how an adversarial autoencoder may be used to perform classification. In fact, it is necessary to augment the encoder to predict not only the encoding, but also the label. Semi-Supervised Denoising Adversarial Autoencoder {#sec:SSDAAE} ------------------------------------------------- Before learning to classify skin lesion as benign or malignant, we may first consider learning more about what skin lesion looks like. This could involve learning the colour, general shape and textures of skin lesions. An ssDAAE allows us to do this by incorporating both a generative and classification model in one. The ssDAAE differs from the AAE in two ways.\ Firstly, the AAE is augmented by applying a corruption process. The corruption process, $C : x \rightarrow \tilde{x}$ is a stochastic process in which Gaussian noise, with standard deviation, $\sigma$, is added to a data sample, $x$ to obtain, $\tilde{x}$. This change results in a DAAE.\ Secondly, the encoder of the DAAE is altered to define an ssDAAE by splitting the encoder in to three parts. An initial encoder, $E_{\theta_E}: \tilde{x} \rightarrow \hat{h}$, and two sub-encoders, $E^y_{\theta_y}: \hat{h} \rightarrow \hat{y}$ and $E^z_{\theta_z}: \hat{h} \rightarrow \hat{z}$. The encoder is trained to predict not only an encoding, $\hat{z}$ but also a label vector, $\hat{y} \in (0,1)$. Adversarial training is used (as in an AAE [@makhzani2015adversarial]) to shape both the distribution of encoded samples to match a chosen prior distribution, and the distribution of predicted class labels to match a categorical distribution [@makhzani2015adversarial].\ However, since we are posing skin lesion classification as a binary classification problem, we represent the labels benign and malignant using a single unit and apply a sigmoid function at the end of $E^y_{\theta_y}$. We therefore train a label discriminator $T^y_{\theta_{T_y}}$ to distinguish predicted labels $\hat{y}$ from labels drawn from a binary distribution. This encourages the output of the classifier, $E^y_{\theta_y}$, to be either $0$ or $1$ rather than taking values in between.\ For an input $x$, the output of the decoder, $\hat{x}$ is given by: $$\hat{h}=E_{\theta_E}(C(x))$$ $$\hat{x} = D_{\theta_D}([E^y_{\theta_y}(\hat{h}),E^z_{\theta_z}(\hat{h}) ])$$ where, $[a,b]$ is a concatenation of $a$ and $b$.\ The weights of the encoder, $\theta_E$ and $\theta_z$, are updated via both adversarial training to match the distribution of $\theta_z$’s to a chosen prior and a reconstruction error between $x$ and $\hat{x}$. This forms the generative part of the model, and may be trained on entirely unlabelled data. This property of the model means that we can learn parameters $\theta_E$ and $\theta_z$ using large amounts of unlabelled data to learn more about the structure of skin-lesions. We can also visualise what this model has learned by generating novel images of skin lesions and evaluating them by eye to see whether the model has captured the basic concept of what a skin lesion is.\ Following on from this, we may use the limited labelled training data to “fine tune” the generative model. We may use the labelled data to update the weights $\theta_y$, or additionally to update $\theta_E$ by minimising the classification error between predicted a label $\hat{y}$ and the true label $y$. Experimentally, we found it beneficial to update both $\theta_y$ and $\theta_E$ as this made training more stable.\ For completeness, note that – similar to an adversarial autoencoder (AAE) – the weights of the decoder, $\theta_D$ are learned as part of the minimisation of the reconstruction error between $x$ and $\hat{x}$. In Figure \[fig:ssDAAE\], we present a diagram of our proposed model. ![**ssDAAE model** Image data, $x$, is corrupted before being encoded. The encoding process consists of two sub-mappings of the corrupted image, $\tilde{x}$, yielding an encoding of the image appearance $\hat{z}$ and a label prediction, $\hat{y}$. The decoder uses both of these to reconstruct a version of the uncorrupted image, $\hat{x}$. The blue parts correspond to an AAE model, while the red parts are additions that make this model an ssDAAE.[]{data-label="fig:ssDAAE"}](images/ssDAAE.png){width="0.99\columnwidth"} Training Data {#sec:dataset} ------------- As described above our ssDAAE may be trained using a mixture of both labelled and unlabelled data. The labelled data is obtained from the ISIC archive [@ISIC]. The archive consists of nearly $14$k images, of which over $9$k are benign skin lesions taken from children. The child skin lesion samples contain colour-coded identifier patches - rendering them not suitable for training in their current state. Of the remaining images there are $3419$ examples of benign skin lesions and $1082$ examples of malignant skin lesions.\ To make the $9$k skin lesions taken from children more appropriate for training and classification, we removed the identifier patches as shown in Figure \[fig:preproc\]. This processing step is not considered to be part of the classification framework, rather a means to increase the amount of available training data. These identifier patches are unlikely to be present in real world encounters. The processed child skin lesions are combined with the rest of the benign skin-lesions.\ ![**Pre-processing of child skin lesions** Images have been pre-processed to crop out the area of the skin lesion. First, the skin was detected using a color profile that matches any usual skin color within a certain threshold in order to obtain a binary mask, in which a rectangle was defined such that it possesses the maximal area. The image was then cropped and centered in order to obtain square images of 64x64.[]{data-label="fig:preproc"}](images/child_skin_lesion_segmentation.png){width="0.9\columnwidth"} The ISIC dataset [@ISIC] does not specifically provide distinct labelled and unlabelled datasets. We partition the data into $7$k unlabelled data samples, $5$k labelled data samples to be used for training and the rest, $500$ each for testing and validation. To expand each dataset we performed data augmentation, by flipping the examples of skin-lesions in both the x and y axes and rotating the samples up to $180$ degrees. Experiments and Results {#sec:method} ======================= In this section, we perform exhaustive ablation study to isolate the effectiveness of (a) a model that incorporates denoising, (b) the use of an adversarial autoencoder (generative model) opposed to a CNN (discriminative model) and (c) of utilising additional unlabelled data. From these experiments we will be able to isolate exactly which components of the ssDAAE are necessary to achieve good performance. We start by explaining how the performance of our models is evaluated. Evaluation {#sec:eval} ---------- There is significant label imbalance that can be observed in the ISIC dataset, meaning that the majority of the images ($c.90$%) are benign, choosing a single classification accuracy as a performance metric may have been misleading given that even a system that always outputs the benign class would get, on average, a high score ($c. 0.90$). Instead, we prefer using clinically insightful and interpretable metrics such that the percentage of malignant skin lesions correctly classified as malignant (true positive, or sensitivity) and benign skin lesions correctly classified as benign (true negative, or specificity). Furthermore, in the context of a medical application and because of the label imbalance problem, we are particularly interested in comparing the model performances, in terms of specificity, at high sensitivity values, to avoid miss diagnosing a malignant skin lesion as benign.\ We used similar evaluation metrics to those using in an interrelation skin lesion classification challenge, hosted by the International Skin Imaging Collaboration (ISIC), at the International Symposium on Biomedical Imaging (ISBI). The challenge was composed of three tasks, where the final tasks was skin lesion classification. For the last task, participant’s models were ranked according to the specificity of their model giving a particular sensitivity threshold $\{0.82, 0.89, 0.95, 0.99\}$. We use the same evaluation metrics in our in all our experiments. Training and Architectural Details ---------------------------------- ### Architectural Details {#sub:architectural_details} In this subsection we present the detailed architecture of both the CNN baseline as well as our semi-supervised denoising adversarial autoencoder. #### CNN The baseline CNN model consists of a sequence of 4 convolution layers, a relu non-linearity is applied to the output of each layer before being fed to the next one. The output of the CNN sequence is then flattened and fed to a linear layer containing $1000$ neurons followed by a final linear layer, with one neuron and sigmoid non-linearity that returns values between $(0,1)$, where $0$ is the label for benign and $1$ is the label for malignant. #### Semi-supervised Denoising Adversarial Autoencoder {#semi-supervised-denoising-adversarial-autoencoder} The CNN described above, without the final linear layer, forms the encoder, $E_{\theta_E}$ of our adversarial autoencoder. The output of the 1000 neurons linear layer splits in two: 1. **Latent Encoder, $E^z_{\theta_z}$** : consists of a linear layer with $200$ neurons that is responsible for returning the 200-dimension encoding vector, $\hat{z}$, representing an input image in the learned latent space. 2. **Classifier, $E^y_{\theta_y}$** : consists of a linear layer with a single neuron followed by a sigmoid activation function. The output of this layer is the class prediction, $\hat{y}$, for a given input image. The label output $\hat{y}$ as well as the encoded vector $\hat{z}$ are then fed through three sub-networks (refer to Figure \[fig:ssDAAE\] for a visualisation). - **Latent Space Discriminator, $T_{\theta_T}$** : This model consists of a linear layer with $1000$ neurons and a relu non-linearity, followed by a linear layer with a single neuron and sigmoid non-linearity. - **Binary Label Discriminator, $T^y_{\theta_{T_y}}$** : This model has similar architecture to the the Latent Space Discriminator. - **Decoder, $D_{\theta_D}$** : Finally, this model consists of a linear layer followed by sequence of 4 transposed convolution layers, again a relu non-linearity is applied to the output of each transposed convolution layer before being fed to the next one, finally a sigmoid layer is applied to the last output of the sequence. The input to the decoder is the concatenation of the label and the encoded vector. These models are summarised in the Appendix in Table \[CNN\_architecture\]. ### Preprocessing and Input Corruption All images have been scaled so that all values are between 0 and 1. Furthermore, In order to allow for the partial corruption of the input. A corruption layer has been implemented that is responsible for adding Gaussian noise with mean $0$ and variance $\sigma$ to the input of the system. Various $\sigma$ values between $0$ and $1$ have been attempted. Experimentally the best results were obtained for $\sigma = 0.1$. ### Loss functions and Class Imbalance In this section we describe in detail how we balance the cost functions used to train our networks. For training the baseline CNN model, a single binary cross entropy loss function was used as it is an appropriate loss function for classification tasks. The ssDAAE on the other hand consists of several modules each with their own loss function, these loss functions need to be combined with care. The loss functions include: a classification loss at the output of the classifier, $E^y_{\theta_y}$, a reconstruction loss at the output of the decoder, $D_{\theta_D}$ and both discriminator and regularisation losses at the output of the discriminators, $T_{\theta_T}$ and $T^y_{\theta_{T_y}}$. The latent space discriminator loss is described in Equation (\[disCost\_z\]), this cost may be modified for the label discriminator, by replacing $p(z)$ with a binary distribution and $p_g(z)$ with the output of $E^y_{\theta_y}$. We now describe the encoder loss function, (designed to update the weights, $\{\theta_E, \theta_{E_y}, \theta_{E_z} \}$), which is defined as the weighted combination of the following losses:\ - **Classification Loss $l_{class}$ :** The binary cross-entropy loss between the predicted class and the ground truth label. - **Reconstruction Loss $l_{rec}$ :** The mean squared-error between the decoded image and the input image. - **Latent Regularisation Loss $l_{reg_z}$ :** The binary-cross entropy loss between output of the latent discriminator, $T_{\theta_T}$ and a target label 1. (Where $1$ refers to the discriminator predicting that a samples is from the chosen prior distribution). - **Label Regularisation Loss $l_{reg_y}$ :** The binary-cross entropy loss between output of the binary label discriminator, $T^y_{\theta_{T_y}}$ and a target label $1$. (Where $1$ refers to the discriminator predicting that a sample is from a binary distribution). $l_{encoder} = \beta l_{class} + \eta l_{rec} + \alpha (l_{reg_y} + l_{reg_z})$ where $\alpha$, $\beta$ and $\eta$ are coefficients chosen through experimentation.\ Furthermore, due to the heavy class imbalance in the ISIC dataset (90% of the data is benign), it was also necessary to slightly modify the cross entropy loss function for the classification loss by adding a weight $a$ for label 1 and a different weight $b$ for label 0. which leads to the following expression : $l_{class} = a * (y * log(\hat{y})) + b * ((1-y) log(1-\hat{y}))$ ### Hyper Parameter Choices For both the baseline model and our adversarial auto-encoder model, we used $(a,b) = (9,1)$ for the weighted classification loss. The CNN was trained using an RMSProp Optimizer with a momentum of $0$ and a learning rate of $10^{-4}$. The encoder and decoder of the ssDAAE were trained with the same optimizer with same learning are and momentum as the CNN. We found setting coefficients $(\alpha, \beta, \eta) = (0.1, 1, 0.1)$ to work well. When training the discriminator training, the same optimizer and learning rate were used, but the momentum was set to $0.2$. Ablation Study ============== To appreciate the contributions of our proposed model, we performed ablation studies. We trained $6$ different models listed in Table \[table:ablation\]. Each autoencoding model – consisting of an encoder an decoder – had the same architecture and each CNN had the same architecture as the encoder. The CNN and CNN+noise models act as simple baselines that do not incorporate a generative model, and are trained in a fully supervised way, not making use of any unlabelled data. The sAAE and sDAAE are fully supervised models, that do incorporate a generative model, in the form of an adversarial autoencoder. Finally, the ssAAE and ssDAAE are trained in a semi-supervised fashion to use both labelled and unlabelled data. All models were trained with the same amount of labelled data. The semi-supervised models are trained with the same amount of unlabelled data. To make the comparisons as fair as possible, we used the same hyper parameters [^2] for all models in the study.\ **Model** \(a) **Denoising** \(b) **Autoencoder** \(c) **Unlabelled** ------------- -------------------- ---------------------- --------------------- -- CNN CNN + noise sAAE sDAAE ssAAE ssDAAE : Models used for the ablation study. The semi-supervised DAAE (ssDAAE) has three core components (a) denoising, (b) an adversarial autoencoder and (c) is trained in a semi-supervised fashion, training with additional unlabelled data. The sAAE and sDAAE are fully supervised models.\[table:ablation\] The results of our ablation study are shown in Figure \[fig:ablation\]. At all sensitivity values the ssDAAE outperformed the simple baselines, of the CNN and the CNN with added noise (CNN+noise). At all sensitivity values, the ssDAAE outperformed the ssAAE, suggesting that the corruption process is useful, but perhaps, more so in the semi-supervised model where there are more examples, since the sAAE outperformed the sDAAE only at lower sensitivities $(0.82, 0.89)$.\ Additionally, the CNN outperformed the CNN+noise model at all sensitivity values, further suggesting that many more training examples are needed for denoising to be effective. The fact that the CNN+noise performed less well than a CNN for all sensitivities, in contrast to the sAAE and sDAAE, which do perform well at the lower sensitivities, may be because the CNN+noise network is never exposed to the uncorrupted images, while the autoencoder models are exposed to uncorrupted images when the reconstruction loss is computed.\ It is at the higher sensitivities $(0.89, 0.95, 0.99)$ that we most clearly see that all semi-supervised variants outperformed their supervised variants. The benefits of semi-supervised models over fully supervised suggested by the results, supports our motivation to design models that incorporate unlabelled data with labelled. Further, the additional benefit of incorporating a denoising criterion into semi-supervised models has, as anticipated, also improved performance. Finally, our results suggest that the model that most consistently performs well is our proposed model, the ssDAAE.\ ![**Ablation Study on the ICIS image database.** The results of this study allow us to compare effect of different model variants. The ssDAAE yields the best specificity at high sensitivity levels.[]{data-label="fig:ablation"}](images/results.png){width="0.99\columnwidth"} Effect of different levels of corruption ---------------------------------------- We also explored the effects of using different levels of corruption during training of ssDAAE models. We compared models trained using noise levels, $\sigma=\{0, 0.01, 0.05, 0.1, 0.25, 0.5, 1.0\}$, the results are shown in Figure \[fig:noise\_levels\]. Each model had the same architecture and was trained with the same hyper parameters [^3] to make the comparison as fair as possible. Our results suggest that the optimal corruption level is $\sigma=0.1$ for most sensitivity values. We see that, for all sensitivity values, an ssDAAE trained with a noise level of $\sigma=0.1$ outperformed an ssAAE (a model trained with a noise level of $\sigma=0$). For ssDAAE models trained with noise levels greater than $\sigma=0.25$, inclusive, performance dropped significantly for all sensitivity values, suggesting that too much noise may have an adverse effect on training. ![**Effect of the level of corruption when training an ssDAAE.** We compare models trained with corruption levels, $\sigma=\{0,0.1,0.25,0.5,1\}$. For lesion classification, moderate levels of noise yield the best results.[]{data-label="fig:noise_levels"}](images/noise_levels.png){width="0.99\columnwidth"} Related Work ============ Previous research has been conducted on the ISIC dataset [@ISIC], as part of a challenge hosted by ISBI [@ISBI], however there are two core differences between our work and the approaches currently taken for this and other medical image datasets.\ Firstly, while our work focuses on a single end-to-end classification approach, previous work on skin lesion classification has tended to adopt a three-stage approach [@codella2017skin; @li2017skin; @ramlakhan2011mobile], splitting the task into, (1) a lesion segmentation to extract the relevant parts of the images [@schmid1997colour; @denton1995boundary], followed by (2) a dermoscopic feature classification [@kasmi2016classification; @round2000lesion] that helps to detect clinical patterns, and, finally, (3) a disease classification task aiming to identify “melanoma”, “seborrheic keratosis” and “benign nevi”. The initial preprocessing stages, (1) and (2) require extensive pixel-wise labelling of images – such as by image segmentation – to provide ground truth examples in order to learn to perform these tasks. On the contrary, our approach requires only a small amount of labelled data, where the label is simply “benign” or “malignant”, and makes use of unlabelled data, too.\ The best performances recorded during this challenge have been obtained using fully-supervised deep learning architectures (AlexNet [@ISIC-Berseth]), and transfer learning (VGG-16 nets [@ISIC-Menegola17], ResNet [@ISIC-BiLei]) with networks previously trained on ImageNet. While the success of approaches introduced by Menegola et al. [@ISIC-Menegola17] and Lei et al. [@ISIC-BiLei] highlight the benefits of using additional data to improve performance, the additional data they use is different to the skin lesion data. This because these approaches, [@ISIC-Menegola17], [@ISIC-BiLei] are fully supervised and therefore can only make use of labelled data, the authors were therefore limited to use datasets for which labelled data was available.\ One of our main contributions is in proposing a specific architecture and approach to skin lesion classification which can make use of both labelled and unlabelled medical image data. This method allows classification models to make use of unlabelled data, when the amount of labelled data is in limited supply. This allows us to use additional data that is more similar to the skin lesion data when training our models, unlike Menegola et al. [@ISIC-Menegola17] and Lei et al. [@ISIC-BiLei]. Conclusion ========== Despite the clear success of deep learning techniques in specific image datasets, wide adoption of the many available approaches to training deep networks techniques is highly dependent on the availability of sufficient quantities of $\{label, image\}$ pairs.\ The solution that we propose in this work is a form of semi-supervised learning, in the sense that if ground truth labels are available for only a subset of the data, all the data can still be used to train a deep classification model. Our results show that the additional information that may be learned from the unlabelled data is useful for boosting classification performance.\ Our solution also includes a denoising procedure. While an adversarial autoencoder [@makhzani2015adversarial] is trained to simply to recover its input, our model is trained to recover clean data samples from corrupted ones. This results in our model learning a more robust data representation, which in turn boosts classification performance.\ The approach we suggest is not limited specifically to the form of image data explored in this paper. Currently, we have applied this to dermatological images of skin lesions. Our model is flexible, and may potentially be applied to other datasets, where there is a large amount of image data, but a limited amount of it is labelled. The semi-supervised approach that we have taken in this paper holds significant relevance in developing high specificity classification systems for other medical images. This is because it is often the case that it is very easy to collect many examples of unlabelled images and the availability of experts that provide ground truth labelling is limited. Acknowledgment {#acknowledgment .unnumbered} ============== We like to acknowledge the Engineering and Physical Sciences Research Council for funding through a Doctoral Training studentship as well as Nick Pawlowski and Martin Rajchl for help with providing access to cluster computers. Appendices {#Appendices} ========== Table. \[CNN\_architecture\]) shows details of the architecture of the CNN baseline. **Input** **Real Image** ----------- ----------------------------------------------------------------------------- **conv2D** [\[]{} filterSize : 5, nFilters : 64, stride=2, padding=2[\]]{} **Relu** **conv2D** [\[]{} filterSize : 5, nFilters : 128, stride=2, padding=2[\]]{} **Relu** **conv2D** [\[]{} filterSize : 5, nFilters : 256, stride=2, padding=2[\]]{} **Relu** **conv2D** [\[]{} filterSize : 5, nFilters : 512, stride=2, padding=2[\]]{} **Relu** **Linear** [\[]{}Size : 1000[\]]{} **Linear** [\[]{}Size : 1[\]]{} **Sigmoid** Probability of label = 1 : CNN architecture[]{data-label="CNN_architecture"} The tables bellow show details of the architecture for the ssDAAE. Table \[Encoder\] shows the encoder and Table \[Decoder\] shows the decoder. In addition, the label discriminator is shown in Table \[disY\] and the latent discriminator is shown in Table \[disZ\]. Input ---------------- -- -- Encoder Output : Encoder[]{data-label="Encoder"} **Input** -------------------- -- -- **Decoder Output** : Decoder. Conv2D$^T$ represents transposed 2D convolutions.[]{data-label="Decoder"} **Input** **Label** -------------------------- ------------------------------------ **Linear** [\[]{}Size : 1000[\]]{} **Relu** **Linear** [\[]{}Size : 1[\]]{} **Sigmoid** **Discriminator Output** Label Discriminator Probability : Regularisation of the classifier[]{data-label="disY"} **Input** **Encoded Vector** -------------------------- ---------------------------------------- **Linear** [\[]{}Size : 1000[\]]{} **Relu** **Linear** [\[]{}Size : 1[\]]{} **Sigmoid** **Discriminator Output** Latent Space Discriminator Probability : Regularisation of the encoder[]{data-label="disZ"} [^1]: Corresponding Author: (ac2211@ic.ac.uk), This paper is a preprint of a paper submitted to IET Computer Vision Journal. If accepted, the copy of record will be available at the IET Digital Library [^2]: learning rate, number of training epochs, amount of labelled and unlabelled data, loss function weightings, level of corruption, size of encoding [^3]: learning rate, number of training epochs, amount of labelled and unlabelled data, loss function weightings, size of encoding

--- author: - | Cheng Ka Yue\ chengkayue@gmail.com bibliography: - 'bib.bib' title: Some Infinitary Paradoxes and Undecidable Sentences in Peano Arithmetic --- Introduction ============ In [@Chaitin1995-Berry] there is a conversation between Gregory Chaitin and Kurt Gödel: > \[Chaitin\] said, “Professor Gödel, I’m fascinated by your incompleteness theorem. I have a new proof based on Berry paradox that I’d like to tell you about." Gödel said, “It doesn’t matter which paradox you use." To support this claim, we need to investigate what will happen if we formalize different paradoxes in Peano arithmetic (PA). Most notably, Chaitin proved a version of the First Incompleteness Theorem with a proof resembling the Berry paradox in his [@Chaitin1970-CHACCA], so did George Boolos gave his proof using the same paradox (independently) in [@Boolos1989-Godel]. In this paper[^1], I will present a few infinitary paradoxes and corresponding undecidable sentences. The first three paradoxes are developed, in my master thesis, from a version of the Preface paradox, and the last one is an infinite version of the Surprise Examination paradox from [@Sorensen1993-SORTEU]. We will work in the usual first order Peano arithmetic, though in fact the results hold in any theory that extends PA. The non-logical symbols in the language are the only constant symbol $0$, a unary function symbol $S$ and two binary function symbols $+$ and $\times$. The technique being used to produce undecidable sentences in this paper, involving a general version of the Diagonal Lemma, is mainly from [@Cieslinski2013-CIEGTY]. Preliminaries ============= In this section, I will state a few facts and definitions that are useful in this paper, proofs of those facts and details of the arithmetization of syntax will be skipped. These details can be found in books about Gödel’s Incompleteness Theorems, for examples, [@Smullyan1992-SMUGIT] and [@Smith2007-Intro]. There are formulas in PA that are said to be *provable*. If a formula $\varphi$ is provable in Peano arithmetic, we will denote this fact by $\vdash \varphi$. Then we have some definitions: 1. A formula $\varphi$ is said to be *refutable* if the negation of it, $\neg \varphi$, is provable. 2. A formula is decidable if it is provable or refutable, otherwise it is undecidable. Hence a formula $\varphi$ is undecidable if neither $\varphi$ nor $\neg \varphi$ is provable. 3. Two formulae $\varphi$ and $\psi$ are *provable equivalent* if the formula $\varphi \longleftrightarrow \psi$ is provable. 4. A quantifier is *bounded* in a formula if it is of the form $\exists x (x < t \land \varphi)$ or $\forall x (x < t \rightarrow \varphi)$, where $t$ is a term, and we will write $(\exists x < t)\varphi$ and $(\forall x <t) \varphi$ respectively. 5. A formula is a *$\Delta_0$ formula* if it is provably equivalent to a formula containing only bounded quantifiers. 6. A formula is a *$\Sigma_1$ formula* if it is provably equivalent to a formula of the form $\exists x \varphi$, where $\varphi$ is a $\Delta_0$ formula. We say that a theory is *consistent* if there is no formula $\varphi$ such that both $\varphi$ and $\neg \varphi$ are provable. And we say that a theory is $\omega$-consistent if there is no open formula $\varphi(x)$ such that $\exists x \varphi(x)$ is provable, but for every natural number $n$, $\varphi(n)$ is not provable. In this paper we assume PA is both consistent and $\omega$-consistent.[^2] The following corollary of the assumption of $\omega$-consistency is useful: \[omega-consist-lemma\] Let $\varphi(x)$ be a $\Sigma_1$ formula with a free variable $x$. If $\exists x \varphi(x)$ is provable, then there is a number $n$ such that $\varphi(n)$ is provable. This result simply follows from the definition of $\omega$-consistency and the fact that all $\Delta_0$ formulae are decidable. Another lemma about $\Sigma_1$ formulae is also useful: \[sigma-1-lemma\] If $\varphi$ is a $\Sigma_1$ formula, then for any variable $x$, $\exists x \varphi$ is also a $\Sigma_1$ formula. A proof of this lemma can be found in [@Smullyan1992-SMUGIT]. We can encode each finite sequence of natural numbers into a natural number, call the *code* of the sequence, in a way that we can also decode that number and obtain the original sequence. A number which is the code of a finite sequence is called a *code number*. Then we assign different numbers to the symbols in our object language, hence every expression corresponds to a finite sequence, which can be encoded into a natural number. Such a number is called the *Gödel number* of that expression. Let $\varphi$ be a formula, the Gödel number of $\varphi$ will be denoted by $\ulcorner \varphi \urcorner$. After that, (syntactical) properties and relations of expressions correspond to properties and relations of the Gödel numbers of expressions. Then we can construct the following predicates and functions[^3]: 1. $Code(x)$ is provable if $x$ is a code number. 2. $l(x)=n$ is provable if $x$ is a code number of a sequence with length $n$. 3. $Dec(x,k)=y$ is provable if $x$ is a code number and the $k^{th}$ term of the sequence encoded by $x$ is $y$. 4. $Neg(x)$ is a function such that $Neg(\ulcorner \varphi \urcorner)=\ulcorner \neg \varphi \urcorner$ is provable for any formula $varphi$.[^4] 5. $Subs(x,v,y)$ is a function such that if for any formula $\varphi$, term $t$, variable $v_i$ free in $\varphi$, then $Sub(\ulcorner \varphi \urcorner, \ulcorner v_i \urcorner, \ulcorner t \urcorner)=\ulcorner \varphi(t/v_i) \urcorner$, where $\varphi(t/v_i)$ is the formula obtained from substituting all free occurrence of $v_i$ in $\varphi$ by $t$, is provable. The above relations are $\Delta_0$. We also have an open $\Sigma_1$ formula $Prov(x)$ with one free variable satisfying the following two lemmas: \[Prov-intro\] If $\varphi$ is a provable formula, then $Prov(\ulcorner \varphi \urcorner)$ is provable. \[Prov-elim\] If Peano arithmetic is $\omega$-consistent, and $\varphi$ is a formula such that $Prov(\ulcorner \varphi \urcorner)$ is provable, then $\varphi$ is provable. Since we assume the consistency and $\omega$-consistency of PA, $Prov(\ulcorner \varphi \urcorner)$ is provable if and only if $\varphi$ is provable for any formula $\varphi$. Finally we need two more lemmas. The first one is a generalized version of the usual Diagonal Lemma, the proof of it can be found in [@Boolos1993-BOOTLO-3]: \[Diag-lem\] Let $\varphi(x,y)$ be an open formula with two free variables $x,y$, then there is an open formula $\psi(x)$ with one free variable $x$ such that $\psi(x) \longleftrightarrow \varphi(x, \ulcorner \psi(x) \urcorner)$ is provable. The second one is a consequence of Gödel’s Second Incompleteness Theorem: \[unprov-unprov\] Let $\varphi$ be a sentence, then $\neg Prov(\ulcorner \varphi \urcorner)$ is not provable. The Paradoxes ============= In this section I will present four infinitary paradoxes, the first three of them are from my master thesis, though there are some similar finite version in the literature, I cannot find any name for the infinitary ones. The last one is called the Earliest Class Inspection paradox from [@Sorensen1993-SORTEU], as noted in the introduction. Imagine there are infinitely many people in a room, each of them say one and only one sentence. The following three situations correspond to the first three paradoxes. Paradox 1: Someone is wrong. {#paradox-1-someone-is-wrong. .unnumbered} ---------------------------- If everyone in the room says “Someone is wrong" [^5], then it is impossible for everyone to be right, otherwise none of them is wrong, contradicting their claims. Hence someone must be wrong, but that person also says “Someone is wrong", so “No one is wrong" is true, contradicting him or her being wrong. Paradox 2: Someone else is wrong. {#paradox-2-someone-else-is-wrong. .unnumbered} --------------------------------- If everyone in the room says “Someone else is wrong", then this situation is slightly more complicated. It is consistent that there is exactly one person being wrong, while the rest of them are right. Furthermore, since everyone says the same thing and it is a symmetric situation, it does not matter that which one is wrong. So the truth value assignments of their sentence is arbitrary in this sense, which is similar to the Truth-teller paradox, i.e. “This sentence is true". Paradox 3: Some people are wrong. {#paradox-3-some-people-are-wrong. .unnumbered} --------------------------------- Suppose all people in the room queue up, and the $k^{th}$ person says “There are at least $k$ people wrong". Notice that if the $k^{th}$ person is right, then everyone before this person is also right.[^6] Similarly, if the $k^{th}$ person is wrong, then everyone after this person is also wrong. Using logic we know that either everyone is right or someone is wrong, in both cases the first person is right.[^7] Since the first person is right, someone must be wrong, and there must be someone who is the first person (in the queue) being wrong. Let this person be the $k^{th}$ person. By our observation everyone after her or him is wrong, so there are more than $k$ people wrong, but that means the $k^{th}$ person is right, and we have a contradiction. Paradox 4: The Earliest Class Inspection Paradox {#paradox-4-the-earliest-class-inspection-paradox .unnumbered} ------------------------------------------------ Suppose you are a new teacher, and you are told that there will be a class inspection. There are two conditions on the date of the inspection: first, the sooner the better; second, you do not know and cannot guess the day so that you cannot prepare for it. Therefore, the inspection will be on the first day which you do not believe there will be a class inspection. Now, the next school day is the first available day for class inspection, but then the inspection cannot be on that day since you can reason it. Similarly you can rule out the possibilities of the inspection being on the second day, the third day, the fourth day, and so on. Hence the earliest unexpected class inspection is impossible. Notes on the paradoxes {#notes-on-the-paradoxes .unnumbered} ---------------------- Here are some notes on the paradoxes. The finite version of the first paradox, which is still a paradox, is related to the Liar cycles.[^8] However an existential quantifier is informally used in this paradox, and it can be extended to the infinite case easily, unlike the Liar cycles. The finite version of the second paradox is again similar to the infinite case, that is, we have different consistent truth-value assignments. It is related to a paradox by Jean Buridan, which is “Socrates says that Plato tells a lie, Plato says that Socrates tells a lie" (and they say nothing more), it is also called the No-No paradox in [@Sorensen2004]. In Sorensen’s book, there is a finite version of the second paradox.[^9] The finite version of the third paradox is not necessarily a paradox: if the number of people is even, then the first half people are right and the second half wrong; if the number of people is odd, then the one who is in the exact middle of the queue is in a Liar paradox situation. And the infinite version resembles the Yablo’s paradox[^10] The Earliest Class Inspection Paradox can be regarded as an infinite version of the Surprise Examination Paradox. The Undecidable Sentences ========================= In [@Cieslinski2013-CIEGTY], the authors apply Lemma \[Diag-lem\] to an open formula to obtain undecidable sentences resembling Yablo’s paradox. In the following, we will do the same for the paradoxes in the last section. Formalizing Paradox 1 {#formalizing-paradox-1 .unnumbered} --------------------- Consider the open formula[^11]: $$\exists z \Big(Prov \big(Neg(Subs(y, \ulcorner x \urcorner, \ulcorner z \urcorner))\big) \Big) \land (0 \leq x)$$ By Lemma \[Diag-lem\] there is an open formula $\mathbf{P}(x)$ with one free variable $x$ such that: $$\begin{aligned} & \vdash \mathbf{P}(x) \longleftrightarrow \exists z \Big( Prov\big(Neg(Subs(\ulcorner \mathbf{P}(x) \urcorner, \ulcorner x \urcorner, \ulcorner z \urcorner)) \big) \Big) \land (0 \leq x) \\ \Rightarrow \, & \vdash \mathbf{P}(x) \longleftrightarrow \exists z \big(Prov(\ulcorner \neg \mathbf{P}(z) \urcorner) \big) \land (0 \leq y) \\ \Rightarrow \, & \vdash \mathbf{P}(x) \longleftrightarrow \exists z \big(Prov(\ulcorner \neg \mathbf{P}(z) \urcorner) \big)\end{aligned}$$ Here $\mathbf{P}(k)$ can be read as the sentence that the $k^{th}$ person says, which is, intuitively, “There is someone whose sentence is refutable". Like the undecidable sentence in Gödel original proof, which is a formalization of the Liar paradox, we replace “truth" by “provability", since the latter can be formulated in the object language. We have the following result: For any natural number $k$, $\mathbf{P}(k)$ is undecidable. Let $k$ be a natural number. Suppose $\mathbf{P}(k)$ is provable. Then by the choice of $\mathbf{P}(x)$, $\exists z \big(Prov(\ulcorner \neg \mathbf{P}(z) \urcorner) \big)$ is also provable. By $\omega$-consistency of PA, Lemma \[omega-consist-lemma\] and Lemma \[sigma-1-lemma\], there is a natural number $n$ such that $\neg Prov(\ulcorner \mathbf{P}(n) \urcorner)$ is provable. But by Lemma \[unprov-unprov\] this is impossible. On the other hand, suppose $\mathbf{P}(k)$ is refutable. Then: $$\begin{aligned} \vdash \neg \mathbf{P}(k) \Rightarrow \, & \vdash \neg \exists z \big( Prov(\ulcorner \neg \mathbf{P}(z) \urcorner) \big) & \text{(By the choice of $\mathbf{P}(x)$)} \\ \Rightarrow \, & \vdash \forall z \neg Prov(\ulcorner \neg \mathbf{P}(z) \urcorner)\\ \Rightarrow \, & \vdash \neg Prov(\ulcorner \neg \mathbf{P}(k) \urcorner) & \text{(By universal instantiation)}\end{aligned}$$ While by Lemma \[Prov-intro\], we have $Prov(\ulcorner \neg \mathbf{P}(k) \urcorner)$ provable. Since we assume that PA is consistent, this is impossible. Therefore for any natural number $k$, $\mathbf{P}(k)$ is neither provable nor refutable, hence undecidable. Formalizing Paradox 2 {#formalizing-paradox-2 .unnumbered} --------------------- Consider the following open formula: $$\exists z \Big(z \neq x \land Prov\big( Neg(Subs(y, \ulcorner x \urcorner, \ulcorner z \urcorner)) \big) \Big)$$ By Lemma \[Diag-lem\], there is an open formula $\mathbf{Q}(x)$ with one free variable $x$ such that $$\vdash \mathbf{Q}(x) \longleftrightarrow \exists z \big(z \neq x \land Prov(\ulcorner \neg \mathbf{Q}(z) \urcorner) \big)$$ Similar to the previous formalization, $\mathbf{Q}(k)$ can be read as the sentence that the $k^{th}$ person says, which is, intuitively, “There is someone else whose sentence is refutable". We have the following result: For any natural number $k$, $\mathbf{Q}(k)$ is undecidable. Let $k$ be a natural number. Suppose $\mathbf{Q}(k)$ is refutable. Then: $$\begin{aligned} \vdash \neg \mathbf{Q}(k) \Rightarrow \, & \vdash \neg \exists z \big( z \neq k \land Prov(\ulcorner \neg \mathbf{Q}(z) \urcorner) \big) & \text{(By the choice of $\mathbf{Q}(x)$)} \\ \Rightarrow \, & \vdash \forall z \big( Prov(\ulcorner \neg \mathbf{Q}(z) \urcorner) \rightarrow z=k \big) \\ \Rightarrow \, & \vdash \forall z \big( z \neq k \rightarrow \neg Prov( \ulcorner \neg \mathbf{Q}(z) \urcorner)\big) \\ \Rightarrow \, & \vdash \big( k+1 \neq k \rightarrow \neg Prov(\ulcorner \neg \mathbf{Q}(k+1) \urcorner) \big) & \text{(By Universal Instantiation)}\\ \Rightarrow \, & \vdash \neg Prov(\ulcorner \neg \mathbf{Q}(k+1) \urcorner) & \text{($\vdash k+1 \neq k$)}\end{aligned}$$ But by Lemma \[unprov-unprov\], $\neg Prov(\ulcorner \neg \mathbf{Q}(k+1) \urcorner)$ is not provable. Hence $\mathbf{Q}(k)$ cannot be refutable. On the other hand, suppose $\mathbf{Q}(k)$ is provable, then by the choice of $\mathbf{Q}(x)$, $\exists z \big( z \neq k \land Prov(\ulcorner \neg \mathbf{Q}(z) \urcorner) \big)$ is also provable. By $\omega$-consistency of PA, Lemma \[sigma-1-lemma\] and Lemma \[omega-consist-lemma\], there is a natural number $n$ such that $Prov(\ulcorner \neg \mathbf{Q}(n) \urcorner)$ is provable. By Lemma \[Prov-elim\], $\neg \mathbf{Q}(n)$ is also provable. But this contradicts the first half of this proof, hence $\mathbf{Q}(k)$ cannot be provable. Therefore for any natural number $k$, $\mathbf{Q}(k)$ is undecidable. We have noted the similarity between paradox 2 and the Truth-teller paradox. Nevertheless, the Henkin sentence, the formalized Truth-teller, is provable by Löb’s celebrated theorem, while the formalized version of paradox 2 above is undecidable. Formalizing Paradox 3 {#formalizing-paradox-3 .unnumbered} --------------------- To formalize the third paradox, it is more complicated since we need to refer to a set of numbers (of size $k$) in the object language. So we need a two more definitions: - $HetSeq(x) \longleftrightarrow Code(x) \land \big( \forall y \leq l(x) \big) \big( \forall z \leq l(x) \big) \big(y \neq z \rightarrow Dec(y,x) \neq Dec(z,x) \big)$\ If $HetSeq(x)$ is provable, then $x$ is a code number of a sequence where no two terms are the same. - $Ele(x,y) \longleftrightarrow Code(y) \land \big(\exists u \leq l(y) \big) \big( Dec(u, y)=x \big)$\ If $Ele(x,y)$ is provable, then $x$ represents a number which is a term of the sequence represented by $y$. The idea is to define a kind of sequence, called heterosequence, in which no two terms are the same. Instead of saying there is a set of $k$ natural numbers, we can say there is a heterosequence of length $k$. Note that both $HetSeq(x)$ and $Ele(x,y)$ are $\Delta_0$. Then consider the open formula: $$\exists z \Big[HetSeq(z) \land l(z)=Sx \land (\forall t \leq z) \big[ Ele(t,z) \rightarrow Prov\big( Neg(Subs(y, \ulcorner x \urcorner, \ulcorner t \urcorner))\big) \big] \Big]$$ Again by Lemma \[Diag-lem\] there is an open formula $\mathbf{R}(x)$ such that: $$\vdash \mathbf{R}(x) \longleftrightarrow \exists z \Big(HetSeq(z) \land \big( l(z)=Sx \big) \land (\forall t \leq z) \big( Ele(t,z) \rightarrow Prov(\ulcorner \neg \mathbf{R}(t) \urcorner) \big) \Big)$$ Intuitively, $\mathbf{R}(k)$ is provable if and only if there is a heterosequence of length $k+1$ such that for each element $t$ of that sequence, $\mathbf{R}(t)$ is refutable.[^12] We have the following two lemmas: \[backward-heredity\] If $m, n$ are natural numbers and $m<n$, then $\vdash \mathbf{R}(n) \rightarrow \mathbf{R}(m)$. Let $m,n$ be natural numbers and $m<n$. Suppose $\vdash \mathbf{R}(n)$. Then by the choice of $\mathbf{R}(x)$: $\vdash \exists z \Big(HetSeq(z) \land \big(l(z)= Sn \big) \land (\forall t \leq z) \big( Ele(t,z) \rightarrow Prov(\ulcorner \neg \mathbf{R}(t) \urcorner) \big) \Big)$ By $\omega$-consistency, there is a natural number $N$ such that $\vdash \Big(HetSeq(N) \land \big(l(N)= Sn \big) \land (\forall t \leq N) \big( Ele(t,N) \rightarrow Prov(\ulcorner \neg \mathbf{R}(t) \urcorner) \big) \Big)$ $N$ is the code number of a heterosequence of length $n+1$, then we can take the first $m+1$ terms of the sequence to form a new heterosequence, and let its code number be $M$. By definition, both $l(b)= Sm$ and $Ele(t,M) \rightarrow Ele(t,N)$ are provable, hence $\vdash \Big(HetSeq(M) \land \big(l(b)= Sm \big) \land (\forall t \leq M) \big( Ele(t,M) \rightarrow Prov(\ulcorner \neg \mathbf{R}(t) \urcorner) \big) \Big)$ Therefore $\vdash \mathbf{R}(m)$. By the deduction theorem we get $\vdash \mathbf{R}(n) \rightarrow \mathbf{R}(m)$. \[hereditarily-refutable\] If $m, n$ are natural numbers and $m>n$, then $\vdash \neg \mathbf{R}(n) \rightarrow \neg \mathbf{R}(m)$. By Lemma \[backward-heredity\] we have $\vdash \mathbf{R}(m) \rightarrow \mathbf{R}(n)$, which implies the contrapositive of the formula, therefore $\vdash \neg \mathbf{R}(n) \rightarrow \neg \mathbf{R}(m)$. These two lemmas formalize our previous observations in the situation of paradox 3: “if the $k^{th}$ person is right then everyone before him or her is right" and “if the $k^{th}$ person is wrong then everyone after her or him is wrong". Then we have the following result: For any natural number $n$, $\mathbf{R}(n)$ is undecidable. Let $n$ be a natural number. Suppose $\mathbf{R}(n)$ is refutable, then $\vdash \neg \mathbf{R}(n)$. By Lemma \[hereditarily-refutable\], for every $m>n$, $\mathbf{R}(m)$ is refutable. Therefore the sentences $\neg \mathbf{R}(n), \neg \mathbf{R}(n+1), \ldots, \neg \mathbf{R}(n+n)$ are all provable, by Lemma \[Prov-intro\] the sentences $Prov(\ulcorner \neg \mathbf{R}(n) \urcorner), Prov(\ulcorner \neg \mathbf{R}(n+1) \urcorner), \ldots, Prov(\ulcorner \neg \mathbf{R}(n+n) \urcorner)$ are also provable. Let $c$ be the code number of the sequence $(n,n+1, \ldots, n+n)$. Then the sentences $HetSeq(c)$, $l(c)=Sn$, and $(\forall t \leq c) \big( Ele(t,c) \rightarrow Prov(\ulcorner \neg \mathbf{R}(t) \urcorner) \big)$ are all provable. This implies that $\mathbf{R}(n)$ is provable and we get another contradiction. On the other hand, suppose $\mathbf{R}(n)$ is provable. Then $$\exists z \Big(HetSeq(z) \land \big(l(z)= Sn\big) \land (\forall t \leq z) \big( Ele(t,z) \rightarrow Prov(\ulcorner \neg \mathbf{R}(t) \urcorner) \big) \Big)$$ is also provable. By $\omega$-consistency, there is an number $c$ such that $$\begin{aligned} & \vdash \Big(HetSeq(c) \land l(c)= Sn \land (\forall t \leq c) \big( Ele(t,c) \rightarrow Prov(\ulcorner \neg \mathbf{R}(t) \urcorner) \big) \Big) \\ \Rightarrow \,& \vdash (\forall t \leq c) \big( Ele(t,c) \rightarrow Prov(\ulcorner \neg \mathbf{R}(t) \urcorner))\big) \Big) \quad \qquad \text{(By conjunction elimination)} \\ \Rightarrow \,& \vdash Prov (\ulcorner \neg \mathbf{R}(Dec(1,c) \urcorner) \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \, \text{(By fact 3)}\end{aligned}$$ But it is impossible by the first half of this proof, so $\mathbf{R}(n)$ is not provable. Therefore $\mathbf{R}(n)$ is neither provable nor refutable. Formalizing Paradox 4 {#formalizing-paradox-4 .unnumbered} --------------------- Consider the open formula: $$(\forall z < x) Prov \big(Neg(Subs(y,\ulcorner x \urcorner, \ulcorner z \urcorner))\big) \rightarrow Prov \big(Neg(y)\big)$$ Apply Lemma \[Diag-lem\], we will get an open formula $\mathbf{F}(x)$ with one variable $x$ such that: $$\vdash \mathbf{F}(x) \longleftrightarrow \big[ (\forall z < x) Prov \big(\ulcorner \neg \mathbf{F}(z) \urcorner \big) \rightarrow Prov \big(\ulcorner \neg \mathbf{F}(x) \urcorner) \big) \big]$$ Roughly speaking, $\mathbf{F}(n)$ is related to the proposition “there will be a class inspection at the $(n+1)^{st}$ day"[^13]. And it satisfies the condition that if it is deducible that there is no class inspection at the first $n$ day, then it is deducible that there is no class inspection at the $(n+1)^{st}$ day. Then we have the following result: \[Surprise\] $\exists x \mathbf{F}(x)$ is undecidable. To prove this theorem, we need the following fact: For any open formula $P(x)$ with exactly one free variable $x$, it is provable that $\exists x P(x) \rightarrow \exists x \big( P(x) \land (\forall y<x) \neg P(y) \big)$. The proof of this principle, which can be found in [@Boolos1993-BOOTLO-3], is skipped here. Now we can prove Theorem \[Surprise\]. Suppose $\exists x \mathbf{F}(x)$ is provable, then by the Least Number Principle, $\exists x \big( \mathbf{F}(x) \land (\forall z<x) \neg \mathbf{F}(z) \big)$ is also provable. By the $\omega$-consistency, there is a natural number $n$ such that the formula $(\forall z < n) \neg \mathbf{F}(z) \land \mathbf{F}(n)$ is provable. Then $(\forall z < n) \neg \mathbf{F}(z)$ is provable, by substitution and modus ponens we have $\neg \mathbf{F}(0), \neg \mathbf{F}(1), \ldots, \neg \mathbf{F}(n-1)$ are all provable. Hence by Lemma \[Prov-intro\], $Prov(\ulcorner \neg \mathbf{F}(0) \urcorner), Prov(\ulcorner \neg \mathbf{F}(1) \urcorner), \ldots, Prov(\ulcorner \neg \mathbf{F}(n-1) \urcorner)$ are all provable, so is $(\forall z < n) Prov(\ulcorner \neg \mathbf{F}(z) \urcorner)$. Since $\mathbf{F}(n)$ is also provable, we have: $$\begin{aligned} \vdash \mathbf{F}(n) \Rightarrow \, & \vdash \big[ (\forall z < n) Prov \big(\ulcorner \neg \mathbf{F}(z) \urcorner \big) \rightarrow Prov \big(\ulcorner \neg \mathbf{F}(n) \urcorner) \big) \big] & \text{(By the choie of $\mathbf{F}(x)$)} \\ \Rightarrow \, & \vdash Prov \big(\ulcorner \neg \mathbf{F}(n) \big) & \text{(By modus ponens)} \\ \Rightarrow \, & \vdash \neg \mathbf{F}(n) & \text{(By Lemma \ref{Prov-elim})}\end{aligned}$$ So we get a contradiction, and $\exists x \mathbf{F}(x)$ is not provable. On the other hand, suppose $\exists x \mathbf{F}(x)$ is refutable. Then $\neg \exists x \mathbf{F}(x)$, and equivalently, $\forall x \neg \mathbf{F}(x)$ are provable. For any natural number $n$, we have: $$\begin{aligned} & \vdash \neg \mathbf{F}(n) & \text{(By universal instantiation)} \\ \Rightarrow \, & \vdash (\forall z < n)Prov \big( \ulcorner \neg \mathbf{F}(z) \urcorner \big) \land \neg Prov(\ulcorner \neg \mathbf{F}(n) \urcorner) & \text{(By the choice of $\mathbf{F}(x)$)} \\ \Rightarrow \, & \vdash \neg Prov(\ulcorner \neg \mathbf{F}(n) \urcorner) & \text{(By conjunction elimination)}\end{aligned}$$ But by Lemma \[Prov-intro\], $\neg \mathbf{F}(n)$ is provable implies that $Prov(\ulcorner \neg \mathbf{F}(n) \urcorner)$ is also provable. Again we get a contradiction. Therefore, $\exists x \mathbf{F}(x)$ is undecidable. With some modifications on the definition of $\mathbf{F}(x)$, we can obtain a formalization of the Surprise Examination paradox which is essentially different from the one in [@Fitch1964-FITAGF], since the former is undecidable but the latter is refutable. Summary ======= We have seen four infinitary paradoxes, and four related open formulas, $\mathbf{P}(x)$, $\mathbf{Q}(x)$, $\mathbf{R}(x)$, and $\mathbf{F}(x)$. The first three open formulas lead to infinitely many undecidable sentences, and for the last one we have an undecidable sentence $\exists x \mathbf{F}(x)$. These results partly confirm Gödel’s claim quoted in the first section, and refute a possible counterexample from [@Fitch1964-FITAGF]. Also, it is interesting to investigate whether there is any other paradox like the paradox 2, and to understand the difference between paradox 2 and the Truth-teller. [^1]: This paper was presented in Logic Colloquium 2015, and the results are from my mater’s thesis [@Cheng2015thesis]. [^2]: The latter actually implies the former and can be replaced by weaker a weaker condition called 1-consistency (or $\Sigma_1$ soundness), which also implies consistency. [^3]: Since we usually do not include function symbols other than $S$, $+$ and $\times$, “$f(x)=y$" should be understood as a relation of $x$ and $y$ and “$f(x)$" should be understood as pterms (p for pseudo) in [@Boolos1993-BOOTLO-3]. [^4]: In fact for any expressions, not restricted to formulas, but here we only concern formulas. [^5]: To be more precise, it should be “At least one sentence uttered in this room is false". But for convenience, we simply talk about people being right and wrong instead of the sentences they uttered is true or false. [^6]: Since the $k^{th}$ person is right, there are at least $k$ people wrong. Hence for any $j<k$, it is true that there are at least $j$ people wrong, which means the $j^{th}$ person is right. [^7]: If everyone is right, then of course the first person is right; if someone is wrong, then at least one person is wrong, which is what the first person asserts. [^8]: That is, the people are in a circle, and everyone says the next person is wrong, with an exception that if the number of people is even, then one of them says the next one is right. [^9]: In almost the same form, except that he uses a list of 100 sentences, each of them is the sentence “Some other sentence on this list is false". [^10]: Yablo’s paradox is about an infinite list of sentences where every sentence is “All sentences below are false". If the first sentence is true, then the second one is false, hence some sentence below it is true, contradicting the first one. On the other hand, if the first sentence is false, then some sentence below it is true, and we will get a similar contradiction. [^11]: The subformula $0 \leq x$ is to make sure the formula contains $x$ as a free variable. It is not difficult to prove that $0 \leq x$ is provable. [^12]: The length of the heterosequence is $k+1$, since we count the natural numbers from $0$. [^13]: Note that we count from $0$.

--- author: - 'S. Ole Warnaar' - Wadim Zudilin date: September 2019 title: '$q$-rious and $q$-riouser' --- Dick Askey is known not just for his beautiful mathematics and his many amazing theorems, but also for posing numerous interesting and important open problems. Dick being Dick, these problems are hardly ever isolated, and often intended to demonstrate the unity of analysis, number theory and combinatorics. On this ocassion we wish to take the reader down the rabbit hole created by one such problem, published as *Advanced Problem 6514* by the American Mathematical Monthly in April 1986 [@Askey86]. Dick’s inspiration for the problem was derived from the Macdonald–Morris constant term conjecture for the root system $\mathrm{G}_2$ [@Macdonald82; @Morris82] as well as much earlier work of P. Chebyshev [@Tchebichef82] and E. Landau [@Landau85] on the integrality of factorial ratios. Problem 6514 asks for a proof of the integrality of $$A(m,n)= \frac{(3m+3n)!\,(3n)!\,(2m)!\,(2n)!}{(2m+3n)!\,(m+2n)!\,(m+n)!\,m!\,n!\,n!}$$ for all non-negative integers $m$ and $n$. There are multiple reasons — some of them very deep, see e.g., [@Bober09; @Rodriguez-Villegas05; @Soundararajan19a; @Soundararajan19b] — for wanting to classify integer-valued factorial ratios such as Chebyshev’s $$C(n)=\frac{(30n)!\,n!}{(15n)!\,(10n)!\,(6n)!}\,.$$ Given a particular such ratio, integrality can always be verified by computing the $p$-adic order of the factorials entering the quotient. This is exactly what all eight solvers of Problem 6514 did. Such a verification, however, provides little insight into which ratios are integral and which ones are not, and from the editorial comments to the problem it is clear that Dick would have liked to see other types of solutions too. Indeed, it is remarked that > \[\] the editor \[read: Dick Askey\] feels there is still room for other methods, involving perhaps combinatorial interpretations or manipulation of generating functions. In this particular case, the proposer remarks that $A(m,n)$ should be the constant term of the Laurent polynomial $$\begin{gathered} > \quad\qquad \big((1-x)(1-1/x)(1-y)(1-1/y)(1-y/x)(1-x/y)\big)^m \\[1mm] > \times \big((1-xy)(1-1/xy)(1-y/x^2)(1-x^2/y)(1-y^2/x)(1-x/y^2)\big)^n.\quad\end{gathered}$$ Incidentally, L. Habsieger [@Habsieger86] and D. Zeilberger [@Zeilberger87] both proved the $\mathrm{G}_2$ Macdonald–Morris constant term conjecture shortly after Dick Askey posed his problem. The submission dates of their respective papers (the 12th of May and the 2nd of June 1986) were well within the deadline of the 31st of August for submitting solutions to Problem 6514 to the Monthly. In fact, in the acknowledgement of his paper Zeilberger thanks Dick Askey for “rekindling his interest in the Macdonald conjecture”, so maybe he should belatedly be considered the 9th solver of Askey’s problem. The height of a factorial ratio is the number of factorials in the denominator minus the number of factorials in the numerator, so that the height of $A(m,n)$ is two whereas the height of $C(n)$ is one. A one-parameter family of height-$k$ factorial ratios $$F(n)=\frac{(a_1\,n)!\cdots (a_{\ell}\, n)!} {(b_1\, n)!\cdots (b_{k+\ell}\, n)!}$$ is balanced if $a_1+\dots+a_{\ell}=b_1+\dots+b_{k+\ell}$. All balanced, integral, height-one factorial ratios $F(n)$ were classified in 2009 by J. Bober [@Bober09]. In relation to this classification we should mention F. Rodriguez-Villegas’ observation [@Rodriguez-Villegas05] that if $F(n)$ is a balanced, height-one factorial ratio then the hypergeometric function $\sum_{n{\geqslant}0} F(n) z^n$ is algebraic if and only if $F(n)$ is integral. This observation was key to Bober’s proof, allowing him to use the Beukers–Heckman classification [@BH89] of $_nF_{n-1}$ hypergeometric functions with finite monodromy group. A proof not reliant on the Beukers–Heckman theory was recently found by K. Soundararajan [@Soundararajan19a]. By extending his method he also obtained a partial classification in the height-two case [@Soundararajan19b]. Despite the availability of the number-theoretic, $p$-adic approach to factorial ratios, the question of integrality is very interesting from a purely combinatorial point of view. The simplest example is of course provided by the height-one binomial coefficients $$\frac{(m+n)!}{m!\,n!},$$ whose integrality can be established combinatorially (as well as probabilistically, algebraically, etc.) with little effort. However, to the best of our knowledge, no combinatorial proof is known of the integrality of Chebyshev’s $C(n)$. A related open problem arises from our joint work [@WZ11] from 2011. In [@WZ11] we observed that if each factorial $m!$ in an integral factorial ratio is replaced by a $q$-factorial $$[m]!=[m]_q!=\prod_{i=1}^m\frac{1-q^i}{1-q},$$ then the resulting $q$-factorial ratio is a polynomial with non-negative integer coefficients. The polynomiality and integrality parts are trivial but the positivity — which was referred to in [@WZ11] as ‘$q$-rious positivity’ — is completely open. The only (irreducible) cases that are proven are the three two-parameter families of height one given by $$\frac{[m+n]!}{[m]!\,[n]!},\qquad \frac{[2m]!\,[2n]!}{[m]!\,[n]!\,[m+n]},\qquad \frac{[m]!\,[2n]!}{[2m]!\,[n]!\,[n-m]!}\quad (m{\geqslant}n),$$ where the first family corresponds to the $q$-binomial coefficients and the second family to the $q$-super Catalan numbers. In the $q$-case no arithmetic approach is available, and given the lack of combinatorial methods to deal with integrality, a combinatorial approach to $q$-rious positivity seems hopeless.[^1] Perhaps the most tractable problem is to analytically prove, along the lines of [@WZ11], the positivity of the known two-parameter families of height two, such as $$A_q(m,n)= \frac{[3m+3n]!\,[3n]!\,[2m]!\,[2n]!} {[2m+3n]!\,[m+2n]!\,[m+n]!\,[m]!\,[n]!\,[n]!}\in\mathbb Z[q]$$ and $$C_q(m,n)=\frac{[6m+30n]!\,[n]!} {[3m+15n]!\,[2m+10n]!\,[m]!\,[6n]!}\in\mathbb Z[q].$$ For the first family, which is the $q$-analogue of $A(m,n)$, it is known that [@Cherednik95; @Habsieger86; @Zeilberger87] $$\begin{gathered} A_q(m,n) \\=\operatorname*{CT}\limits_{x,y}\Big[ \big(x,q/x,y,q/y,y/x,qx/y;q\big)_m \big(xy,q/xy,y/x^2,qx^2/y,y^2/x,qx/y^2;q\big)_n\Big],\end{gathered}$$ where $(a_1,\dots,a_k;q)_n:=\prod_{i=1}^k \prod_{j=1}^n (1-a_i q^{j-1})$. This interpretation as a $\mathrm{G}_2$ constant term gives little insight into the positivity of the coefficients. It would appear that the second two-parameter family has not occurred before. For $q=1$ it arose earlier this year in the (partial) classification of height-two factorial ratios by Soundararajan [@Soundararajan19b] mentioned above. It should be noted that if one were to prove the $q$-rious positivity of $C_q(m,n)$ then this immediately would imply the positivity of the $q$-analogue of Chebyshev’s factorial ratio since $$C_q(n)=\frac{[30n]!\,[n]!}{[15n]!\,[10n]!\,[6n]!}=C_q(0,n).$$ [99]{} <span style="font-variant:small-caps;">R. Askey</span>, Advanced Problem 6514, *Amer. Math. Monthly* **93**:4 (1986) 304–305; Solution of Advanced Problem 6514, *Amer. Math. Monthly* **94**:10 (1987) 1012–1014. <span style="font-variant:small-caps;">F. Beukers</span> and <span style="font-variant:small-caps;">G. Heckman</span>, Monodromy for the hypergeometric function $_nF_{n-1}$, *Invent. Math.* **95**:2 (1989) 325–354. <span style="font-variant:small-caps;">J.W. Bober</span>, Factorial ratios, hypergeometric series, and a family of step functions, *J. London Math. Soc.* (2) **79**:2 (2009) 422–444. <span style="font-variant:small-caps;">I. Cherednik</span>, Double affine Hecke algebras and Macdonald’s conjectures, *Ann. of Math.* (2) **141**:1 (1995) 191–216. <span style="font-variant:small-caps;">L. Habsieger</span>, La $q$-conjecture de Macdonald–Morris pour $G_2$, *C. R. Acad. Sci. Paris Sér. I Math.* **303**:6 (1986) 211–213. <span style="font-variant:small-caps;">E. Landau</span>, Sur les conditions de divisibilité d’un produit de factorielles par un autre, in *Collected Works*, Vol. I (Thales-Verlag, Essen, 1985), p. 116. <span style="font-variant:small-caps;">I.G. Macdonald</span>, Some conjectures for root systems, *SIAM J. Math. Anal.* **13**:6 (1982) 988–1007. <span style="font-variant:small-caps;">W.G. Morris</span>, *Constant term identities for finite and affine root systems: conjectures and theorems*, PhD thesis (Univ. Wisconsin-Madison, 1982). <span style="font-variant:small-caps;">F. Rodriguez-Villegas</span>, Integral ratios of factorials and algebraic hypergeometric functions, *Oberwolfach Rep.* **2** (2005) 1814–1816. <span style="font-variant:small-caps;">K. Soundararajan</span>, Integral factorial ratios, *Preprint* [`arXiv:1901.05133 [math.NT]`](http://arxiv.org/abs/1901.05133) (2019). <span style="font-variant:small-caps;">K. Soundararajan</span>, Integral factorial ratios: irreducible examples with height larger than 1, *Preprint* [`arXiv:1906.06413 [math.NT]`](http://arxiv.org/abs/1906.06413) (2019). <span style="font-variant:small-caps;">P.L. Tchebichef</span>, Mémoire sur les nombres premiers, *J. Math. Pures Appl.* **17** (1852) 366–390. <span style="font-variant:small-caps;">S.O. Warnaar</span> and <span style="font-variant:small-caps;">W. Zudilin</span>, A $q$-rious positivity, *Aequat. Math.* **81**:1-2 (2011) 177–183. <span style="font-variant:small-caps;">D. Zeilberger</span>, A proof of the $\mathrm{G}_2$ case of Macdonald’s root system-Dyson conjecture, *SIAM J. Math. Anal.* **18**:3 (1987) 880–883. [^1]: There are of course countless methods to show that the $q$-binomial coefficients have non-negative integer coefficients, but no combinatorial interpretation of the $q$-super Catalan numbers is known. In fact, not even a combinatorial interpretation of the ordinary super Catalan numbers is known.

--- abstract: | The Virtual Observatory is a new technology of the astronomical research allowing the seamless processing and analysis of a heterogeneous data obtained from a number of distributed data archives. It may also provide astronomical community with powerful computational and data processing on-line services replacing the custom scientific code run on user’s computers. Despite its benefits the VO technology has been still little exploited in stellar spectroscopy. As an example of possible evolution in this field we present an experimental web-based service for disentangling of spectra based on code KOREL. This code developed by P. Hadrava enables Fourier disentangling and line-strength photometry, i.e. simultaneous decomposition of spectra of multiple stars and solving for orbital parameters, line-profile variability or other physical parameters of observed objects. We discuss the benefits of the service-oriented approach from the point of view of both developers and users and give examples of possible user-friendly implementation of spectra disentangling methods as a standard tools of Virtual Observatory. author: - Petr Škoda and Petr Hadrava title: Fourier Disentangling Using the Technology of Virtual Observatory --- Introduction ============ The astronomical spectroscopy uses many special techniques to analyse stellar spectra and estimate physical properties of targets studied. Basically they consist in comparison of the observed spectra with theoretical models which, however, may be of very different level of sophistication. For instance, a simple comparison of suitably defined effective centres of spectral lines with their laboratory wavelengths gives Doppler shifts, which in the case of spectroscopic binaries enables one to determine their orbital parameters. Detailed comparison of equivalent widths and shapes of line profiles with synthetic spectra may reveal effective temperatures, gravity acceleration, abundances and other physical parameters of stellar atmospheres. In practice, however, the spectra of components of the binary are blended and the information on orbital and atmospheric parameters are entangled. Several techniques for separation of component spectra from a series of spectra has been proposed which enable also to develop the so called spectra disentangling, i.e. a method of simultaneous separation of the spectra and determination of physical parameters governing their variability. In particular, the method of Fourier disentangling introduced and implemented in program KOREL by @h95 proved to be efficient and viable for a further generalisation. To allow the application of such a powerful method on a number of different objects in a scalable way, we attempted to embed the KOREL in the infrastructure of Virtual Observatory. The Virtual Observatory ======================= Contemporary astronomy faces an enormous amount of data continuously flowing from large telescopes, space missions and supercomputer simulations, that can hardly be analysed (and even previewed) by the traditional scientific methods. Thus the concept of (astronomical) Virtual Observatory (VO) was recently born aiming at federalisation of all astronomical resources (e.g. catalogues, data archives, simulation databases, data processing and analysing tools) using the global infrastructure based on unified data format and set of rigid, yet extensible communication protocols. The development and implementation of these global standards is the role of the International Virtual Observatory Alliance (IVOA). Technically, VO is a collection of inter-operating data archives and software tools which utilise the internet to form a virtual desktop environment in which astronomical research can be conducted in a user friendly manner allowing the astronomer to concentrate on asking the scientific questions instead of spending most of the time with searching in heterogeneous scattered archives, and with homogenisation of data represented by different units in various file formats. Owing to its huge data-mining potential and easy multiwavelength analysis tools, the VO technology allows to tackle problems not feasible by any other means, like the search of very rare astronomical events, candidates of yet unknown classes of objects (e.g. extremely cold brown dwarfs, supermassive stars etc.), statistics of order of tens of millions target or pan-spectral classification as building the spectra energy distributions of radiation from gamma to radio using the archives of all space and ground-based observations together. For the extensive introduction into the VO science see @2006LNEA....2...71S. The Fourier Disentangling ========================= The disentangling of spectra represents nowadays a whole branch of stellar spectroscopy fairly exceeding the scope of our contribution. We thus refer for a detailed explanation of its physical and mathematical principles, astrophysical consequences and for corresponding literature to the review [@h04] or its update [@h09b]. Here we shall only qualitatively characterise the method of Fourier disentangling implemented in code KOREL and we shall list a recent progress. The instantaneous spectra of many variable objects can be in a good approximation expressed as a superposition of their intrinsic (time independent) components convolved with some broadening functions (e.g. Doppler shifted delta-functions) depending on time and some physical parameters of the variability (e.g. the orbital parameters). In the Fourier conjugate space the intrinsic components can thus easily be solved (independently for each Fourier mode) from a more numerous set of observations. Moreover, the values of the free parameters can be fit by the least-square method. To prevent an ill-determination of the problem, a good coverage of the time interval of the characteristic variability is needed. The main task of the development of the method is thus to find a proper theoretical model of the broadening. Already the very simple assumption of line-strength variability with fixed line profiles [@h97] enables many useful applications. To apply the method successfully to real data, the observers should understand the assumptions and properties of the model and to prepare a set of data decisive for the parameters required from the solution. If the solution for the intrinsic component spectra is well over-determined by a great number of observed spectra, their noise can be substantially reduced by the averaging. A recent improvement of the numerical technique [@h09a] enables to retrieve the radial-velocity shifts with an accuracy surpassing the limitations by the step of spectra sampling (this is sometimes called super-resolution). Our recent work [@hss09] opens a disentangling of Cepheid pulsations. The Virtual Observatory Web Services ==================================== As the Fourier disentangling of the large number of spectra may become computation intensive, its full power may be exploited using the modern technology of VO Web Services (WS). The WS is typically complex processing application using the web technology (http protocol and (X)HTML markup) to transfer input data (files, tables, images, spectra etc.) to the main processing back-end (often run in front of queue scheduling and/or parallelising engine on computer clusters or GRIDS) and the results (after intensive number crunching) back to user. All this can be done using only an ordinary web browser (and in principle the science may be done on the fast palmtop or advanced mobile phone). The more detailed analysis about the benefits of GRID technology in stellar spectroscopy is presented by @2009MmSAI..80..484S. This service-oriented approach has many advantages both for the user and developer. Let’s name some of them: - There is the only one, current, well tested version of the code (and documentation), maintained and updated by its author - The user needs not to install anything from the author - The code is optimised for given HW (native compiler), knowing its limits (memory and cache sizes, number of nodes etc.) - The problem is scalable - the more user requirements may be solved by adding more computing nodes and introducing priority queues - The web technology provides the easy way of interaction (forms) and graphics output (in-line images) even produced dynamically (variable refresh rates or event driven - e.g. AJAX) The KOREL Web Service --------------------- The idea of our service is to have an user interface similar to e-shop portal, starting with user registration. Every set of input parameters creates a job, which may be run in parallel with others, the user may stop or remove them, can return to the previous versions etc. Privileged users may even recompile their own version of KOREL code tailored to their needs (e.g. maximum amount and size of spectra). All user communication is encrypted and the user can see only his/her jobs. The service may be accessed from the KOREL portal at Astronomical Institute in Ondřejov[^1]. At the time of preparation of the proceedings the KOREL Web Service requires to upload the files [korel.data]{} and [korel.par]{} in given strict format. Usually, for the preparation of input data the program [PREKOR]{} run at local computer is used, which reads spectra in various formats, rebins them equidistant in radial velocity (logarithmic wavelength) and optionally applies the precisely computed heliocentric correction. In addition to that, its interactive graphics helps to select the proper spectral regions bordered by the clear continuum and allows the removal of bad spectra. In the future, the role of the [PREKOR]{} may be replaced by another set of web services acquiring the spectra directly from VO servers and using proper metadata (e.g. elements of orbits) obtained from proper catalogues published in VO (especially CDS Vizier and Simbad). The interactive capability will be provided by VO spectral tools (e.g. SPLAT or VOSpec). Conclusions =========== The Fourier disentangling is already well-established method of stellar spectra analysis with the wide range of applications. The KOREL web service is probably one of the first attempts to adapt the legacy stellar spectra analysis code for the Virtual Observatory service. The advantages of solution adopted are evident, although some level of user conservatism has to be expected. This work was supported by grants GAČR 202/06/0041, GAČR 202/09/0772 and by projects AV0Z10030501 and LC06014. We thank all the developers of UK’s AstroGrid Virtual Observatory Project for testbeding the new paradigm of scientific research based on joining supercomputing GRID technologies with Virtual Observatory standards. We are greatly indebted to Pavel Škoda and Jan Fuchs for practical implementation of several versions of the KOREL Web Service. Hadrava, P. 1995, , 114, 393 Hadrava, P. 1997, , 122, 581 Hadrava, P. 2004, Publ. Astron. Inst. ASCR, 92, 15 Hadrava, P. 2009, , 494, 399 Hadrava, P. 2009, arXiv:0909.0172 Hadrava, P., Šlechta, M., & Škoda, P. 2009, , in press (arXiv:0909.0610) Solano, E. 2006, Lecture Notes and Essays in Astrophysics, 2, 71 koda, P. 2009, Memorie della Societa Astronomica Italiana, 80, 484 [^1]: http://stelweb.asu.cas.cz/vo-korel

--- abstract: '> Similarity between objects is multi-faceted and it can be easier for human annotators to measure it when the focus is on a specific aspect. We consider the problem of mapping objects into view-specific embeddings where the distance between them is consistent with the similarity comparisons of the form “from the t-th view, object A is more similar to B than to C”. Our framework *jointly* learns view-specific embeddings exploiting correlations between views. Experiments on a number of datasets, including one of multi-view crowdsourced comparison on bird images, show the proposed method achieves lower triplet generalization error when compared to both learning embeddings independently for each view and all views pooled into one view. Our method can also be used to learn multiple measures of similarity over input features taking class labels into account and compares favorably to existing approaches for multi-task metric learning on the ISOLET dataset.' author: - | [**Liwen Zhang**]{}\ University of Chicago\ liwenz@cs.uchicago.edu\ \ UMass Amherst\ smaji@cs.umass.edu\ \ Toyota Technological Institue at Chicago\ tomioka@ttic.edu\ bibliography: - 'reference.bib' title: Jointly Learning Multiple Measures of Similarities from Triplet Comparisons --- Introduction ============ Measure of similarity plays an important role in applications such as content-based recommendation, image search and speech recognition. Therefore a number of techniques to [*learn*]{} a measure of similarity from data have been proposed [@xing2002distance; @DavKulJaiSraDhi07; @WeiBliSau06; @mcfee2011learning]. When the measure of distance is induced by an inner product in a low-dimensional space as is done in many studies, learning a distance metric is equivalent to learning an [*embedding*]{} of objects in a low-dimensional space. This is useful for visualization as well as using the learned representation in a variety of down-stream tasks that require fixed length representations of objects as has been demonstrated by the applications of word embeddings [@mikolov2013efficient] in language. Among various forms of supervision for learning distance metric, similarity comparison of the form ‘object $A$ is more similar to $B$ than to $C$’’, which we call [*triplet comparison*]{}, is extremely useful for obtaining an embedding that reflects a [*perceptual similarity*]{} [@agarwal2007generalized; @tamuz2011adaptively; @van2012stochastic]. Triplet comparisons can be obtained by crowdsourcing, or it may also be derived from class labels if available. The task of judging similarity comparisons, however, can be challenging for human annotators. Consider the problem of comparing three birds as seen in Fig. \[fig:figure1\]. Most annotators will say that the head of bird $A$ is more similar to the head of $B$ while the back of $A$ is more similar to $C$. Such ambiguity leads to noise in annotation and results in poor embeddings. A better approach would be to tell the annotator the desired view or the perspective of the object to use for measuring similarity. Such view-specific comparisons are not only easier for annotators, but they can also enable precise feedback for human “in the loop” tasks, such as, interactive fine-grained recognition [@wah15learning], thereby reducing the human effort. The main drawback of learning view specific embeddings [*independently*]{} is that the number of similarity comparisons scales linearly with the number of views. This is undesirable as even learning a single embedding of $N$ objects may require $O(N^3)$ triplet comparisons [@jamieson2011low] in the worst case. ![**Ambiguity in similarity.** Depending on whether we focus on the back (middle row) or on the head (bottom row), bird $A$ may appear more similar to $B$ or $C$. \[fig:figure1\]](figures/fig1c.pdf){width="0.8\linewidth"} We propose a method for learning embeddings [*jointly*]{} that addresses this drawback. Our method exploits underlying correlations that may exist between the views allowing a better use of the training data. Our method models the correlation between views by assuming that each view is a *low-rank projection* of a common embedding. Our model can be seen as a matrix factorization model in which local metric is defined as ${\boldsymbol{L}}{\boldsymbol{M}}_t{\boldsymbol{L}}^\top$, where ${\boldsymbol{L}}$ is a matrix that parametrizes the common embedding and ${\boldsymbol{M}}_t$ is a positive semidefinite matrix parametrizing the individual view. The model can be efficiently trained by alternately updating the view specific metric and the common embedding. We experiment with a synthetic dataset and two realistic datasets, namely, poses of airplanes, and crowd-sourced similarities collected on different body parts of birds (CUB dataset; Welinder et al., [-@WelinderEtal2010]). On most datasets our joint learning approach obtains lower triplet generalization error compared to the independent learning approach or naively pooling all the views into a single one, especially when the number of training triplets is limited. Furthermore, we apply our joint metric learning approach to the multi-task metric learning setting studied by [@parameswaran2010large] to demonstrate that our method can also take input features and class labels into account. Our method compares favorably to the previous method on ISOLET dataset. Formulation\[sec:formulation\] ============================== In this section, we first review the single view metric learning problem considered in previous work. Then we extend it to the case where there are multiple measures of similarity. Metric learning from triplet comparisons ---------------------------------------- Given a set of triplets $\mathcal{S}=\{(i,j,k)\mid\text{object $i$ is more similar to object $j$ than object $k$}\}$ and possibly input features ${\boldsymbol{x}}_1,\ldots,{\boldsymbol{x}}_N\in \mathbb{R}^H$, we aim to find a positive semidefinite matrix ${\boldsymbol{M}}\in\mathbb{R}^{H\times H}$ such that the pair-wise comparison of the distances induced by the inner product ${\left\langle{\boldsymbol{x}},{\boldsymbol{y}}\right\rangle}_{{\boldsymbol{M}}}={\boldsymbol{x}}^\top{\boldsymbol{M}}{\boldsymbol{y}}$ parametrized by ${\boldsymbol{M}}$ (approximately) agrees with $\mathcal{S}$, [*i*.*e*.]{}, $(i,j,k)\in \mathcal{S}\Rightarrow \|{\boldsymbol{x}}_i- {\boldsymbol{x}}_j\|_{{\boldsymbol{M}}}^2 < \|{\boldsymbol{x}}_i - {\boldsymbol{x}}_k\|_{{\boldsymbol{M}}}^2$. If no input feature is given, we take ${\boldsymbol{x}}_i$ as the $i$th coordinate vector in $\mathbb{R}^{N}$, and learning ${\boldsymbol{M}}$, which would become $N\times N$, would correspond to finding [*embeddings*]{} of the $N$ objects in a Euclidean space with dimension equal to the rank of ${\boldsymbol{M}}$. Mathematically the problem can be expressed as follows: $$\begin{aligned} \label{eq:gnmds-k} \min_{\substack{{\boldsymbol{M}}\in \mathbb{R}^{H\times H},\\ {\boldsymbol{M}}\succeq 0}} \quad & \!\!\!\!\!\sum_{(i,j,k) \in \mathcal{S}}\!\!\!\!\! \ell(\| {\boldsymbol{x}}_i - {\boldsymbol{x}}_j \|_{{\boldsymbol{M}}}^2, \|{\boldsymbol{x}}_i - {\boldsymbol{x}}_k\|_{{\boldsymbol{M}}}^2 ) +\gamma\text{tr}({\boldsymbol{M}}),\end{aligned}$$ where $\|{\boldsymbol{x}}-{\boldsymbol{y}}\|_{{\boldsymbol{M}}}^2=({\boldsymbol{x}}-{\boldsymbol{y}})^\top{\boldsymbol{M}}({\boldsymbol{x}}-{\boldsymbol{y}})$; the loss function can be, for example, logistic [@CoxMilMinPapYia00], or hinge, $\ell(d_{i,j},d_{i,k})=\max(1+d_{i,j}-d_{i,k},0)$ [@agarwal2007generalized; @WeiBliSau06; @CheShaShaBen10]. Other choices of loss functions lead to crowd kernel learning [@tamuz2011adaptively], and $t$-distributed stochastic triplet embedding (t-STE) [@van2012stochastic]. Penalizing the trace of the matrix ${\boldsymbol{M}}$ can be seen as a convex surrogate for penalizing the rank [@agarwal2007generalized; @FazHinBoy01]. $\gamma>0$ is a regularization parameter. After the optimal ${\boldsymbol{M}}$ is obtained, we can find a low-rank factorization of ${\boldsymbol{M}}$ as ${\boldsymbol{M}}={\boldsymbol{L}}{\boldsymbol{L}}^\top$ with ${\boldsymbol{L}}\in\mathbb{R}^{H\times D}$. This is particularly useful when no input feature is provided, because each row of ${\boldsymbol{L}}$, which is $N\times D$ in this case, corresponds to a $D$ dimensional embedding of each object. Jointly learning multiple metrics {#sec:mmte} --------------------------------- Now let’s assume that $T$ sets of triplets $\mathcal{S}_1,\ldots,\mathcal{S}_T$ are available. This can be obtained by asking annotators to focus on a specific aspect when making pair-wise comparisons as in human in the loop tasks [@wah14similarity; @wah15learning]. Alternatively, different measures of similarity can come from considering multiple related metric learning problems as in [@parameswaran2010large; @RaiLiaCar14]. While a simple approach to handle multiple similarities would be to parametrize each aspect or view by a positive semidefinite matrix ${\boldsymbol{M}}_t$, this would not induce any shared structure among the views. Our goal is to learn a global transformation ${\boldsymbol{L}}$ that maps the objects in a common $D$ dimensional space as well as local view-specific metrics ${\boldsymbol{M}}_t$ ($t=1,\ldots,T$). To this end, we formulate the learning problem as follows: $$\begin{aligned} \min_{\substack{ {\boldsymbol{L}}\in\mathbb{R}^{H\times D},\\ {\boldsymbol{M}}_t \in\mathbb{R}^{D\times D},\\ {\boldsymbol{M}}_t \succeq 0\,(t=1,\ldots,T)}} &\!\!\sum_{t=1}^T \sum_{(i,j,k) \in \mathcal{S}_t}\!\!\!\!\!\varphi_{i,j,k}({\boldsymbol{L}},{\boldsymbol{M}}_t)\notag \\[-1.2em] &\quad +\gamma\sum_{t=1}^T {\rm tr}({\boldsymbol{M}}_t) + \beta \|{\boldsymbol{L}}\|_F^2 \, , \label{eq:multi-metric-obj} \end{aligned}$$ where $\varphi_{i,j,k}({\boldsymbol{L}},{\boldsymbol{M}}):=\ell\bigl(\|{\boldsymbol{L}}^\top({\boldsymbol{x}}_i-{\boldsymbol{x}}_j)\|_{{\boldsymbol{M}}}^2,\|{\boldsymbol{L}}^\top({\boldsymbol{x}}_i-{\boldsymbol{x}}_k)\|_{{\boldsymbol{M}}}^2 \bigr)$, and $\ell$ is a loss function as above. We use the hinge loss in the experiments in this paper, but the proposed framework readily generalizes to other loss functions proposed in literature [@tamuz2011adaptively; @van2012stochastic]. Note again that when no input feature is provided, the global transformation matrix ${\boldsymbol{L}}$ becomes an $N\times D$ matrix that consists of $D$ dimensional embedding of the objects. Intuitively the global transformation ${\boldsymbol{L}}$ plays the role of a bottleneck and forces the local metrics to share the common $D$ dimensional subspace because they are restricted in the form ${\boldsymbol{L}}{\boldsymbol{M}}_t{\boldsymbol{L}}^\top$. The proposed model includes various simpler models as special cases. First, if ${\boldsymbol{L}}$ is an $H\times H$ identity matrix, there is no sharing across different views and indeed the objective function will decompose into a sum of view-wise objectives; we call this [*independent learning*]{}. On the other hand, if we constrain all ${\boldsymbol{M}}_t$ to be equal, the same metric will apply to all the views and the learned metric will be essentially the same as learning a single shared metric as in Eq. with $\mathcal{S}=\cup_{t=1}^{T}\mathcal{S}_t$; we call this [*pooled learning*]{}. We employ regularization terms for both the local metric ${\boldsymbol{M}}_t$ and the global transformation matrix ${\boldsymbol{L}}$ in . The trace penalties ${\rm tr}({\boldsymbol{M}}_t)$ are employed to obtain low-rank matrices ${\boldsymbol{M}}_t$ as above. The regularization term on the norm of ${\boldsymbol{L}}$ is necessary to resolve the scale ambiguity. Although the above formulation has two hyperparameters $\beta$ and $\gamma$, we show below in Proposition \[lem:effective\] that the product $\beta\gamma$ is the only hyperparameter that needs to be tuned. To minimize the objective , we update ${\boldsymbol{M}}_t $’s and ${\boldsymbol{L}}$ alternately. Both updates are (sub)gradient descent. The ${\boldsymbol{M}}_t$ update is followed by a projection onto the positive semi-definite (PSD) cone. Note that if we choose a convex loss function, [*e*.*g*.]{}, hinge-loss, then it becomes a convex problem with respect to ${\boldsymbol{M}}_t$’s and ${\boldsymbol{M}}_t$’s can be optimized independently since they appear in disjoint terms. The algorithm is summarized in **Algorithm \[alg:alt-update\]**. ### Effective regularization term {#effective-regularization-term .unnumbered} The sum of the two regularization terms employed in can be reduced into a single effective regularization term with only [*one hyperparameter*]{} $\sqrt{\beta \gamma}$ as we show in the following proposition (we give the proof in the supplementary material). \[lem:effective\] $$\begin{aligned} \!\! \min_{\substack{{\boldsymbol{L}}\in\mathbb{R}^{H\times D},\\ {\boldsymbol{M}}_1,\ldots,{\boldsymbol{M}}_T\in\mathbb{R}^{D\times D}} } \!\!\!\!&\gamma\sum_{t=1}^{T} {\rm tr}({\boldsymbol{M}}_t) + \beta \|{\boldsymbol{L}}\|_F^2 =2\sqrt{ \beta\gamma}{\rm tr}\left(\sum_{t=1}^{T}{\boldsymbol{K}}_t\right)^{\frac{1}{2}},\\ {\rm s.t.}\quad &{\boldsymbol{L}}{\boldsymbol{M}}_t{\boldsymbol{L}}^\top={\boldsymbol{K}}_t\, (\forall t)\end{aligned}$$ where the power $1/2$ in the r.h.s. is the matrix square root. As a corollary, we can always reduce or maintain the regularization terms in without affecting the loss term by the rescaling ${\boldsymbol{M}}_t\leftarrow {\boldsymbol{M}}_t/\alpha^2$ and ${\boldsymbol{L}}\leftarrow\alpha{\boldsymbol{L}}$ with $\alpha = ( \gamma \sum_{t=1}^T \operatorname{tr}({\boldsymbol{M}}_t)/(\beta \|{\boldsymbol{L}}\|_F^2) )^{1/4}$. ### Number of parameters {#number-of-parameters .unnumbered} A simple parameter counting argument tells us that [*independently*]{} learning $T$ views requires to fit $O(DHT)$ parameters, where $H$ is the number of input dimension, which can be as large as $N$, $D$ is the embedding dimension, and $T$ is the number of views. On the other hand, our [*joint learning*]{} model has only $O(HD+D^2T)$ parameters. Thus when $D<H$, our model has much fewer parameters and enables better generalization, especially when the number of triplets is limited. ### Efficiency {#efficiency .unnumbered} Reducing the dimension from $H$ to $D$ by the common transformation ${\boldsymbol{L}}$ is also favorable in terms of computational efficiency. The projection of ${\boldsymbol{M}}_t$ to the cone of $D\times D$ PSD matrices is much more efficient when $D\ll H$ compared to independently learning $T$ views. \[alg:alt-update\] Initialize ${\boldsymbol{L}}$ randomly; initialize ${ {\boldsymbol{M}}_t }$ as identity matrices Learning embeddings from triplet comparisons\[sec:exp\] ======================================================= In this section, we demonstrate the statistical efficiency of our model in both the triplet embedding (no input feature), and multi-task metric learning scenarios (with features). Experimental setup ------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig:birds-views\] (Left) View-specific similarities between poses of planes were obtained by considering subsets of landmarks shown by different colored rectangles and measuring their similarity in configuration up to a scaling and translation. (Right) Perceptual similarities between bird species were collected by showing users either the full image (view $1$), or crops around various parts (view $2, 3, 4, 5, 6$). The average image for each view is also shown.](figures/planes-views2.pdf "fig:"){width=".38\linewidth"} ![\[fig:birds-views\] (Left) View-specific similarities between poses of planes were obtained by considering subsets of landmarks shown by different colored rectangles and measuring their similarity in configuration up to a scaling and translation. (Right) Perceptual similarities between bird species were collected by showing users either the full image (view $1$), or crops around various parts (view $2, 3, 4, 5, 6$). The average image for each view is also shown.](figures/birds-localized.pdf "fig:"){width=".55\linewidth"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- On each dataset, we divided the triplets into training and test and measured the quality of embeddings by the triplet generalization error, i.e., the fraction of test triplets whose relations are incorrectly modelled by the learned embedding. The error was measured for each view and averaged across views. The numbers of training triplets were the same for all the views. The regularization parameter was tuned using a 5-fold cross-validation on the training set with candidate values $\left\{ 10^{-5}, 10^{-4}, \dots , 10^{5}\right\}$. The hinge loss was used as the loss function. We use $m_{\rm max}=20$ as the number of inner iterations in the experiments. In addition, we inspected how the similarity knowledge on existing views could be [*transferred*]{} to a new view where the number of similarity comparisons is small. We did this by conducting an experiment in which we drew a small set of training triplets from one view but used large numbers of training triplets from the other views. We compared our method with the following two baselines. [**Independent**]{}: We conducted triplet embedding on each view treating each of them independently. We parametrized ${\boldsymbol{M}}={\boldsymbol{L}}{\boldsymbol{L}}^\top$ with ${\boldsymbol{L}}\in\mathbb{R}^{N\times D}$ and minimized using the software provided by van der Maaten and Weinberger . [**Pooled**]{}: We learned a single embedding with the training triplets from all the views combined. Synthetic data -------------- #### Description Two synthetic datasets were generated. One consisted of 200 points uniformly sampled from a 10 dimensional unit hypercube, while the other dataset had 200 objects from a mixture of four Gaussian with variance 1 whose centers were randomly chosen in a hypercube with side length 10. Six *views* were generated on each dataset. Each *view* was produced by projecting data points onto a random subspace. The dimensions of the six random subspaces were 2, 3, 4, 5, 6, and 7 respectively. #### Results Embeddings were learned with embedding dimensions $D=5$ and 10. Triplet generalization errors are plotted in Fig. \[fig:syn-and-pose-errors\] (a) and (b) for clustered and uniform data, respectively. Our algorithm achieved lower triplet generalization error than both [**independent**]{} and [**pooled**]{} methods on both datasets. The improvement was particularly large when the number of triplets was limited (less than 10,000 for the clustered case). The simple [**pooled**]{} method was the worst on both datasets. Note that in contrast to the [**pooled**]{} method, the proposed [**joint**]{} method can choose different embedding dimension automatically (due to the trace regularization) for each view while maintaining a shared subspace. Poses of airplanes ------------------ #### Description This dataset was constructed from 200 images of airplanes from the PASCAL VOC dataset [@everingham10pascal] which were annotated with 16 landmarks such as nose tip, wing tips, etc [@boudev10detecting]. We used these landmarks to construct a pose-based similarity. Given two planes and the positions of landmarks in these images, pose similarity was defined as the residual error of alignment between the two sets of landmarks under scaling and translation. We generated 5 views each of which was associated with a subset of these landmarks; see supplementary material for details. Three annotated images from the set are shown in the left panel of Fig. \[fig:birds-views\]. The planes are highly diverse ranging from passenger planes to fighter jets, varying in size and form which results in a slightly different similarity between instances for each view. However, there is a strong correlation between the views because the underlying set of landmarks are shared. #### Results We used $D=3$ and $D=10$ as embedding dimensions. Figure \[fig:pose-results\] shows the triplet generalization errors of the three methods. The proposed joint model performed clearly better than [**independent**]{}. This was not only in average but also uniformly for each view (see supplementary material). The [**pooled**]{} method had a slightly larger error than the proposed joint learning approach but better than the [**independent**]{} approach. CUB-200 birds data ------------------ #### Description We used the dataset [@WelinderEtal2010] consisting of 200 species of birds and use the annotations collected using the setup of Wah et al. . Briefly, similarity triplets among images of each species were collected in a crowd-sourced manner: every time, a user was asked to judge the similarity between an image of a bird from the target specie $z_i$ and nine images of birds of different species $\{ z_k \}_{k\in \mathcal{K}}$ using the interface of Wilber et al. , where $\mathcal{K}$ is the set of all 200 species. For each display, the user partitioned these nine images into two sets, $\mathcal{K}_{sim}$ and $\mathcal{K}_{dissim}$, with $\mathcal{K}_{sim}$ containing birds considered similar to the target and $\mathcal{K}_{dissim}$ having the ones considered dissimilar. Such a partition was broadcast to an equivalent set of triplet constraints on associated species, $\{(i,j,l) \mid j \in \mathcal{K}_{sim}, \, l \in \mathcal{K}_{dissim} \}$. Therefore, for each user response, $\left|\mathcal{K}_{sim}\right|\left|\mathcal{K}_{dissim}\right|$ triplet constraints were obtained. To collect view-specific triplets, we presented 5 different cropped versions ([*e*.*g*.]{} beak, breast, wing) of the bird images as shown in the right panel of Fig. \[fig:birds-views\] and used the same procedure as before to collect triplet comparisons. We obtained about 100,000 triplets from the uncropped original images and about 4,000 to 7,000 triplets from the 5 cropped views. This dataset reflects a more realistic situation where not all triplet relations are available and some of them may be noisy due to the nature of crowd-sourcing. In addition to the triplet generalization error, we evaluated the embeddings in a classification task using a biological taxonomy of the bird species. Note that in Wah et al.  embeddings were used to interactively categorize images; here we simplify this process to enable detailed comparisons. We manually grouped the 200 classes to get 6 super classes so that the number of objects in all classes were balanced. These class labels were not used in the training but allowed us to evaluate the quality of embeddings using the leave-one-out (LOO) classification error. More precisely, at the test stage, we predict the class label of each embedded point according to the labels of its 3-nearest-neighbours (3-NN) in the learned metric. Finally, since more triplets were available from the first (uncropped) view compared to other views, we first sampled equal numbers of triplets in each view up to a total of 18,000 triplets. Afterwards, we added triplets only to the first view. #### Results We used $D=10$ and $D=60$ as embedding dimensions; note that joint learning in 60 dimensions roughly has the same number of parameters as independent learning in 10 dimensions. Figures \[fig:birds-results\] (a) and (b) show the triplet generalization errors and the LOO 3-NN classification errors, respectively. The solid vertical line shows the point (18,000 triplets) that we start to add training triplets only to the first view. Comparing joint learning in 10 dimensions and 60 dimensions, we see that the higher dimension gives the lower error. The error of joint learning was better than independent learning for small number of triplets. Interestingly the error of joint learning in 60 dimensions coincides with that of independent learning in 10 dimensions after seeing 6,000 triplets. This can be explained by the fact that with 6 views, the two models have comparable complexity (see discussion at the end of the previous section) and thus the same asymptotic variance. Our method obtains lower leave-one-out classification errors on all views except for the first view; see supplementary material. #### Learning a new view On the CUB-200 birds dataset, we simulated the situation of learning a new view (or zero-shot learning). We drew a training set that contains 100–1000 triplets from the second view and 3,000 triplets from all other 5 views. We investigated how joint learning helps in estimating a good embedding on a new view with extremely small number of triplets. The triplet generalization errors of both approaches are shown in Fig. \[fig:birds-emb-zeroshot\]. The triplet generalization error of the proposed joint learning was lower than that of the independent learning up to around 700 triplets. The embedding of the second view learned jointly with other views was clearly better than that learned independently and consistent with the quantitative evaluation; see supplementary material. Performance gain and triplet consistency {#sec:relation} ---------------------------------------- In Fig. \[fig:gain\], we relate the performance gain we obtained for the joint/pooled learning approaches compared to the independent learning approach with the underlying between-task similarity. The performance gain was measured by the difference between the area under the triplet generalization errors normalized by that of the independent learning. The between-task similarity was measured by the triplet consistency between two views averaged over all pairs of views. For the CUB-200 dataset in which only a subset of valid triplet constraints are available, we take the independently learned embeddings with the largest number of triplets and use those to compute the triplet consistency. We can see that when the triplet consistency is very high, pooled learning approach is good enough. However, when the triplet consistency is not too high, it may harm to pool the triplets together. The proposed joint learning approach has the most advantage in this intermediate regime. On the other hand, the consistency was close to random (0.5) for the CUB-200 dataset possibly explaining why the performance gain was not as significant as in the other datasets. ![\[fig:gain\] Relating the performance gains of joint and pooled learning with the triplet consistency. ](figures/performance_gain_2.pdf){width="\linewidth"} Incorporating features and class information {#sec:incorp-feature} ============================================ The proposed method can be applied to a more general setting in which each object comes with a feature vector and a loss function not derived from triplet constraints is used. ------ --------- -------------- -------------- -------------- -------------- -------------- Task 378 dim $D=169$ $D=378$ 378 dim $D=169$ $D=378$ 1 4.68 [**3.78**]{} 4.10 4.23 [**3.46**]{} 3.65 2 4.55 3.91 [**3.52**]{} [**3.14**]{} 3.84 3.40 3 6.28 [**5.32**]{} 5.64 3.52 [**3.39**]{} 3.52 4 7.76 [**5.83**]{} [**5.83**]{} 4.23 4.10 [**3.52**]{} 5 6.28 [**5.06**]{} 5.19 4.23 [**3.97**]{} [**3.97**]{} Avg 5.91 [**4.78**]{} 4.86 3.87 3.76 [**3.61**]{} ------ --------- -------------- -------------- -------------- -------------- -------------- As an example, we employ the idea of multi-task large margin nearest neighbor (MT-LMNN) algorithm [@parameswaran2010large] and adapt our model to handle a classification problem. The loss function of MT-LMNN consists of two terms. The first term is a hinge loss for triplet constraints as in but the triplets are derived from class labels. The second term is the sum of squared distances between each object and its “target neighbors” which is also defined based on class labels; see Weinberger et al. for details. The major difference between MT-LMNN and our model is that MT-LMNN parametrizes a local metric as the sum of a global average ${\boldsymbol{M}}_0$ and a view-specific metric ${\boldsymbol{M}}_t$ as ${\boldsymbol{K}}_t={\boldsymbol{M}}_0+{\boldsymbol{M}}_t$; thus the learned metric is generally full rank. On the other hand, our method parametrizes it as a product of global transform and local metric as ${\boldsymbol{K}}_t={\boldsymbol{L}}{\boldsymbol{M}}_t{\boldsymbol{L}}^\top$, which allows the local embedding dimension to be controlled by the trace regularization. We conduct experiments on ISOLET spoken alphabet recognition dataset [@FanCol91] which consists 7797 examples of English alphabets spoken by 150 subjects and each example is described by a 617 dimension feature vector. The task is to recognize the letter of each spoken example as one of the English alphabets. The subjects are grouped into sets of 30 similar speakers leading to 5 tasks. We adapt the experimental setting from the work of MT-LMNN. Data is first projected onto its first 378 leading PCA components that capture 99 % of variance. We train our model in a $H=378$ dimensional space with $D=169$ and $378$, and compare it with a MT-LMNN trained with the code provided by the authors. In the experiment, each task is randomly divided into 60/20/20 subsets for train/validation/test. We tuned the parameters on the validation sets. Test error rates of 3-nearest-neighbor (3-NN) classifiers are reported in Table \[tab:isolet\]. The left panel shows the errors using only the view-specific training data for the classification. The right panel shows those using all the training data with view-specific distance. Results are averaged over 10 runs. Simpler baseline methods, such as, euclidean metric and pooled (single task) learning are not included here because MT-LMNN already performed better than them. We can see that the proposed method performed better than MT-LMNN, while learning in a 378 dimensional space and reducing to a 169 dimensional space led to comparable error rates. A possible explanation for this mild dependence on the choice of embedding dimension $D$ could be given by the fact that both ${\boldsymbol{L}}$ and ${\boldsymbol{M}}_t$ are regularized and the effective embedding dimension is determined by the regularization and not by the choice of $D$; see Prop. \[lem:effective\]. The averaged error rates reported in the original paper using 169 PCA dimensions were 5.19 % for the view-specific case and 4.01 % when all training data were used; our numbers are still better than theirs. Related work\[sec:relatedwork\] =============================== Embedding of objects from triplet or paired distance comparisons goes back to the work of Shepard and Kruskal and studied extensively [@agarwal2007generalized; @tamuz2011adaptively; @mcfee2009partial; @mcfee2011learning; @van2012stochastic] recently. More recently, triplet embedding / metric learning problems that involve multiple measures of similarity have been considered. Parameswaran and Weinberger aimed at jointly solving multiple related metric learning problems by exploiting possible similarities. More specifically, they modeled the inner product in each view by a [*sum*]{} of shared global matrix and a view-specific local matrix. Moreover, Rai, Lian, and Carin proposed a Bayesian approach to multi-task metric learning. Unfortunately, the sum structure in their work typically do not produce a low-rank metric, which makes it unsuitable for learning view-specific embeddings. In contrast, our method models it as a [*product*]{} of them allowing the trace norm regularizer to determine the rank of each local metric. Xie and Xing and Yu, Wang, and Tao studied metric learning problems with multiple input views. This is different from our setting in which the notion of similarity varies from view to view. Amid and Ukkonen considered the task of multi-view triplet embedding in which the view is a latent variable; they proposed a greedy algorithm for finding the view membership of each object as well as its embedding. It could be useful to combine this approach with ours when we do not have enough resource to collect triplets from all possible views. Discussion {#sec:discussion} ========== We have proposed a model for jointly learning multiple measures of similarities from multi-view triplet observations. The proposed model consists of a global transformation, which represents each object as a fixed dimensional vector, and local view-specific metrics. Experiments on both synthetic and real datasets have demonstrate that our proposed joint model outperforms independent and pooled learning approaches in most cases. Additionally, we have shown that the advantage of our joint learning approach becomes the most prominent when the views are similar but not too similar (which can be measured by triplet consistency). Morevoer, we have extended our model to incorporate class labels and feature vectors. The proposed model performed favorably compared to MT-LMNN on ISOLET dataset. Since in many real applications, similarity triplets can be expensive to obtain, jointly learning similarity metrics is preferable as it can recover the underlying structure using relatively small number of triplets. One way to look at the proposed model is to view the shared global transformation as controlling the complexity. However our experiments have shown that generally the higher the dimension, the better the performance (except for the ISOLET dataset tested with view-specific training data). Thus an alternative explanation could be that the regularization on both the global transformation ${\boldsymbol{L}}$ and local metrics ${\boldsymbol{M}}_t$ is implicitly controlling the embedding dimension. Future work includes extension of the current model to other loss functions (e.g., the t-STE loss [@van2012stochastic]) and to the setting in which we do not know which view each triplet came from. [**Supplementary Material**]{} Proof of Proposition 1 ====================== We repeat the statement for convenience. [1]{} $$\begin{aligned} \min_{\substack{{\boldsymbol{L}}\in\mathbb{R}^{H\times D},\\ {\boldsymbol{M}}_1,\ldots,{\boldsymbol{M}}_T\in\mathbb{R}^{D\times D}} } \left(\gamma\sum_{t=1}^{T} {\rm tr}({\boldsymbol{M}}_t) + \beta \|{\boldsymbol{L}}\|_F^2 :\, {\boldsymbol{L}}{\boldsymbol{M}}_t{\boldsymbol{L}}^\top={\boldsymbol{K}}_t\, (\forall t) \right)= 2\sqrt{ \beta\gamma}{\rm tr}\left(\sum\nolimits_{t=1}^{T}{\boldsymbol{K}}_t\right)^{1/2}\end{aligned}$$ Here the power $1/2$ in the right-hand side is the matrix square root. Let’s define $\bar{{\boldsymbol{M}}}=\sum_{t=1}^{T}{\boldsymbol{M}}_t$. For any decomposition ${\boldsymbol{K}}_t={\boldsymbol{L}}{\boldsymbol{M}}_t{\boldsymbol{L}}^\top$, we have $$\begin{aligned} 2\sqrt{\beta\gamma}{\rm tr}\left(\sum_{t=1}^{T}{\boldsymbol{K}}_t\right)^{1/2} &=2\sqrt{\beta\gamma}{\rm tr}\left({\boldsymbol{L}}\bar{{\boldsymbol{M}}}{\boldsymbol{L}}^\top\right)^{1/2}\\ &=2\sqrt{\beta\gamma}\|\bar{{\boldsymbol{M}}}^{1/2}{\boldsymbol{L}}\|_{\ast}\\ &=2\sqrt{\beta\gamma}\sum_{j=1}^{r}\sigma_j(\bar{{\boldsymbol{M}}}^{1/2}{\boldsymbol{L}})\\ &\leq 2\sqrt{\beta\gamma}\sum_{j=1}^{r}\sigma_j(\bar{{\boldsymbol{M}}}^{1/2})\sigma_j({\boldsymbol{L}})\\ &\leq \sum_{j=1}^{r}\left(\gamma\sigma_j^2(\bar{{\boldsymbol{M}}}^{1/2})+\beta\sigma_j^2({\boldsymbol{L}})\right)\\ &=\gamma{\rm tr}(\bar{{\boldsymbol{M}}})+\beta\|{\boldsymbol{L}}\|_F^2,\\ &=\gamma\sum_{t=1}^{T}{\rm tr}({\boldsymbol{M}}_t)+\beta\|{\boldsymbol{L}}\|_F^2\end{aligned}$$ where $\|\cdot\|_\ast$ is the nuclear norm [@FazHinBoy01]; the fourth line follows from Theorem 3.3.14 (a) in Horn & Johnson [@HorJoh91], and the fifth line is due to the arithmetic mean-geometric mean inequality. Let $\bar{{\boldsymbol{K}}}:=\sum_{t=1}^{T}{\boldsymbol{K}}_t$ and $\bar{{\boldsymbol{K}}}={\boldsymbol{U}}{\boldsymbol{\Lambda}}{\boldsymbol{U}}^\top$ be its eigenvalue decomposition. The equality is achieved by choosing $$\begin{aligned} \label{eq:optL} {\boldsymbol{L}}&={\boldsymbol{U}}{\boldsymbol{\Lambda}}^{1/4}(\gamma/\beta)^{1/4} \\ \label{eq:optMt} {\boldsymbol{M}}_t&={\boldsymbol{\Lambda}}^{-1/4}{\boldsymbol{U}}^\top{\boldsymbol{K}}_t{\boldsymbol{U}}{\boldsymbol{\Lambda}}^{-1/4}(\beta/\gamma)^{1/2} \quad (t=1,\ldots,T)\end{aligned}$$ Note that even when $\bar{{\boldsymbol{K}}}$ is singular, ${\boldsymbol{K}}_t$ is spanned by $\bar{{\boldsymbol{K}}}$ and by restricting to the subspace spanned by $\bar{{\boldsymbol{K}}}$, the above discussion is still valid. This lemma can be understood analogously to the identity regarding the nuclear norm[@SreRenJaa05] $$\begin{aligned} \|{\boldsymbol{X}}\|_{\ast} = \min_{{\boldsymbol{U}},{\boldsymbol{V}}}\frac{1}{2}\left(\|{\boldsymbol{U}}\|_F^2+\|{\boldsymbol{V}}\|_F^2\right)\quad \text{subject to} \quad{\boldsymbol{X}}={\boldsymbol{U}}{\boldsymbol{V}}^\top.\end{aligned}$$ Note that the fact that the ratio of the two hyperparameters $\beta/\gamma$ can be absorbed in the scale ambiguity between ${\boldsymbol{L}}$ and ${\boldsymbol{M}}_t$ as in and is special to multiplicative models like our model and the nuclear norm and would not hold for an additive model like MT-LMNN. Additional details and results ============================== Synthetic dataset ----------------- In addition to the results in main paper, we illustrate view-specific triplet generalization error in Figure \[fig:syn-errors\] and leave-one-out classification error for clustered synthetic data in Figure \[fig:syn-loo-errors\]. ![ Triplet generalization errors. The small figures shows errors on individual *views* and the large figures show the average. (Left) Clustered synthetic data. (Right) Uniformly distributed data. []{data-label="fig:syn-errors"}](figures/synthetic/syn-cluster-tripvio-3methods_extended.pdf "fig:"){width="0.45\linewidth"}   ![ Triplet generalization errors. The small figures shows errors on individual *views* and the large figures show the average. (Left) Clustered synthetic data. (Right) Uniformly distributed data. []{data-label="fig:syn-errors"}](figures/synthetic/syn-uniform-tripvio-3methods_extended.pdf "fig:"){width="0.45\linewidth"} ![ Leave-one-out 3-nearest-neighbour classification errors on clustered synthetic data. The small figures shows errors on individual *views* and the large figures show the average.[]{data-label="fig:syn-loo-errors"}](figures/synthetic/syn-cluster-loo-nn-3methods.pdf){width="0.45\linewidth"} Poses of airplanes dataset -------------------------- ### Details of annotations and view generation {#details-of-annotations-and-view-generation .unnumbered} Each of the 200 airplanes were annotated with 16 landmarks namely,\ -------------------- ----------------- -------------------------- --------------------------- 01\. Top\_Rudder 05\. L\_WingTip 09\. Nose\_Bottom 13\. Left\_Engine\_Back 02\. Bot\_Rudder 06\. R\_WingTip 10\. Left\_Wing\_Base 14\. Right\_Engine\_Front 03\. L\_Stabilizer 07\. NoseTip 11\. Right\_Wing\_Base 15\. Right\_Engine\_Back 04\. R\_Stabilizer 08\. Nose\_Top 12\. Left\_Engine\_Front 16\. Bot\_Rudder\_Front -------------------- ----------------- -------------------------- --------------------------- This is also illustrated in the Figure \[f:landmarks\]. The five different views are defined by considering different subsets of landmarks as follows: 1. *all* $ \in \{1,2, \ldots, 20\}$ 2. *back* $ \in \{1,2, 3, 4, 16\}$ 3. *nose* $ \in \{7,8,9\}$ 4. *back+wings* $\in\{1, 2, \ldots, 6, 10, 11, \ldots, 16\}$ 5. *nose+wings* $ \in \{5, 6, \ldots, 15\}$ ![Landmarks illustrated on the several planes[]{data-label="f:landmarks"}](figures/pose/plane1.png "fig:"){width="0.7\linewidth"}\ ![Landmarks illustrated on the several planes[]{data-label="f:landmarks"}](figures/pose/plane2.png "fig:"){width="0.7\linewidth"}\ ![Landmarks illustrated on the several planes[]{data-label="f:landmarks"}](figures/pose/plane3.png "fig:"){width="0.7\linewidth"}\ For triplet $(A, B, C)$ we compute similarity $s_i(A,B)$ and $s_i(A, C)$ by aligning the subset $i$ of landmarks of $B$ and $C$ to $A$ under a translation and scaling that minimizes the sum of squared error after alignment. The similarity is inversely proportional to the residual error. This is also known as “procrustes analysis” commonly used for matching shapes. In addition to the results in main paper, we illustrate view-specific triplet generalization error and leave-one-out 3-nearest-neighbour classification error in Figure \[fig:pose-results-appendix\].   ### Learned embedding {#learned-embedding .unnumbered} Figure \[fig:pose-embd\] shows a 2D projection of the global view of the objects onto their first two principle dimensions. The visualization shows that objects roughly lies on a circle corresponding to the left-right and up-down orientation. ![ The *global view* of embeddings of poses of planes. \[fig:pose-embd\]](figures/pose/pose_hinge_joint_view_global-fromtop.pdf){width="0.8\columnwidth"} CUB-200 birds dataset --------------------- Here, we also include the view-specific generalization errors and leave-one-out classification errors for CUB-200 Birds Dataset. See Figure \[fig:birds-results-appendix\]. Public figures face dataset --------------------------- ### Description {#description-3 .unnumbered} Public Figures Face Database is created by Kumar [*et al*.]{}[@kumar2009attribute]. It consists of 58,797 images of 200 people. Every image is characterized by 75 attributes which are real valued and describe the appearance of the person in the image. We selected 39 of the attributes and categorized them into 5 groups according to the aspects they describe: *hair, age, accessory, shape* and *ethnicity*. We randomly selected ten people and drew 20 images for each of them to create a dataset with 200 images. Similarity between instances for a given group is equal to the dot product between their attribute vectors where the attributes are restricted to those in the group. We describe the details of these attributes below. Each group is considered as a *local view* and *identities* of the people in the images are considered as class labels. ### Attributes {#sec:attributes .unnumbered} Each image in the *Public Figures Face Dataset (Pubfig)* [^1] is characterized by 75 attributes. We used 39 of the attributes in our work and categorized them into 5 groups according to the aspects they describe. Here is a table of the categories and attributes: Category Attributes ----------- ------------------------------------------------------------------------------------------------------------------------------------------ Hair *Black Hair, Blond Hair, Brown Hair, Gray Hair, Bald, Curly Hair, Wavy Hair, Straight Hair, Receding Hairline, Bangs, Sideburns.* Age *Baby,Child,Youth,Middle Aged,Senior.* Accessory *No Eyewear, Eyeglasses, Sunglasses, Wearing Hat, Wearing Lipstick, Heavy Makeup, Wearing Earrings, Wearing Necktie, Wearing Necklace.* Shape *Oval Face, Round Face, Square Face, High Cheekbones, Big Nose, Pointy Nose, Round Jaw, Narrow Eyes, Big Lips, Strong Nose-Mouth Lines.* Ethnicity *Asian, Black, White, Indian.* : List of Pubfig attributes that were used in our work. ### Results {#results-3 .unnumbered} The 200 images are embedded into 5, 10, and 20 dimensional spaces. We draw triplets randomly from the ground truth similarity measure to form training and test sets. Triplet generalization errors and classification errors are shown in Fig. \[fig:pubfig-results\]. In terms of the triplet generalization error, the joint learning reduces the error faster than the independent learning up to around 10,000 triplets where the decrease slows down. Since the error in this regime reduces monotonically with increasing number of dimensions, this can be understood as a [*bias*]{} induced by the joint learning. On the other hand, when we have less than 10,000 triples, the error of the joint learning increases (but not as large as the independent learning) as dimension increases; this can be understood as a [*variance*]{}. When embedding in a 20 dimensional space, the joint learning has lower or comparable error to independent learning even when $10^5$ triplets are available. In terms of the leave-one-out classification error, joint learning continues to be better even when the number of triplets are very large. Learning a new view ------------------- Figure \[fig:birds-emb-zeroshot-appendix\] shows a 2D projection of the embeddings learned by the [**independent**]{} approach and the proposed [**joint**]{} approach in the setting for CUB-200 birds dataset described in the part of [**“learning a new view”**]{} in the main text. Clearly the proposed joint learning approach obtains a better separated clusters compared to the independent approach. ![Learning a new view on CUB-200 birds dataset. Training data contains 100 triplets from the second *local view* and 3,000 triplets from other 5 *views*. Embeddings are learned in a 10 dimensional space and then further embedded in a 2 dimensional plane by using tSNE [@van2008visualizing] for the purpose of visualization. Left: triplet generalization error on the second *local view*. Middle: embedding learned independently. Right: embedding learned jointly. \[fig:birds-emb-zeroshot-appendix\]](figures/birds/birds-zeroshot-tripvio.pdf "fig:"){width="30.00000%"} ![Learning a new view on CUB-200 birds dataset. Training data contains 100 triplets from the second *local view* and 3,000 triplets from other 5 *views*. Embeddings are learned in a 10 dimensional space and then further embedded in a 2 dimensional plane by using tSNE [@van2008visualizing] for the purpose of visualization. Left: triplet generalization error on the second *local view*. Middle: embedding learned independently. Right: embedding learned jointly. \[fig:birds-emb-zeroshot-appendix\]](figures/birds/birds_gnmds_zeroshot_10d_ind_view_2-tSNE.pdf "fig:"){width="30.00000%"} ![Learning a new view on CUB-200 birds dataset. Training data contains 100 triplets from the second *local view* and 3,000 triplets from other 5 *views*. Embeddings are learned in a 10 dimensional space and then further embedded in a 2 dimensional plane by using tSNE [@van2008visualizing] for the purpose of visualization. Left: triplet generalization error on the second *local view*. Middle: embedding learned independently. Right: embedding learned jointly. \[fig:birds-emb-zeroshot-appendix\]](figures/birds/birds_gnmds_zeroshot_10d_joint_view_2-tSNE.pdf "fig:"){width="30.00000%"} Relating the performance gain with the triplet consistency ---------------------------------------------------------- CUB-200 PubFig Synthetic (uniform) Synthetic (clustered) Airplanes ---------------------------------------- --------- -------- --------------------- ----------------------- ----------- Average triplet consistency 0.53 0.59 0.6 0.69 0.85 Performance gain of joint learning(%) -4.6 6.5 26 44 35 Performance gain of pooled learning(%) -8.0 -56 -40 -29 23 : Relating the performance gain of joint and pooled learning with the between-task similarity.[]{data-label="tab:relation"} [^1]: Available at <http://www.cs.columbia.edu/CAVE/databases/pubfig/>

--- abstract: 'We investigate Ramsey properties of a random graph model in which random edges are added to a given dense graph. Specifically, we determine lower and upper bounds on the function $p=p(n)$ that ensures that for any dense graph $G_n$ a.a.s. every 2-colouring of the edges of $G_n\cup G(n,p)$ admits a monochromatic copy of the complete graph $K_r$. These bounds are asymptotically sharp for the cases when $r\geq 5$ is odd and almost sharp when $r \geq 4$ is even. Our proofs utilise recent results on the threshold for asymmetric Ramsey properties in $G(n,p)$ and the method of dependent random choice.' author: - Emil Powierski bibliography: - 'bibthesis.bib' title: Ramsey properties of randomly perturbed dense graphs --- [^1]. Introduction ============ Random graphs and randomly perturbed dense graphs ------------------------------------------------- For $n \in {\mathbb{N}}$ and $0\leq p \leq 1$ we denote by $G(n,p)$ the binomial random graph on $n$ vertices where every edge is present with probability $p$ independently of all other choices. As usual, we say that an event happens *asymptotically almost surely (a.a.s.)* if it holds with probability tending to $1$ as $n \rightarrow \infty$. Given a graph property $\mathcal{P}$, it has been a key question to find a *threshold function*, a function $p^{*} \colon {\mathbb{N}}\to [0,1]$ ensuring that $G(n,p)$ a.a.s. satisfies $\mathcal{P}$ when $p=\omega(p^*)$ and a.a.s. does not satisfy $\mathcal{P}$ when $p=o(p^*).$ Bohman, Frieze and Martin [@bohman2003many] considered a model that combines deterministic graphs and random graphs: In that model of randomly perturbed graphs one starts with an arbitrary dense graph and adds edges in a random manner. More precisely, given $\gamma>0$, we say that a graph $G=(V,E)$ is *$\gamma$-dense* if $|E| \geq \gamma |V|^2.$ Furthermore, we say that $(\gamma,p)$ *ensures* a property $\mathcal{A}$ if $$\tau_p^{\mathcal{A}}= \lim_{n \to \infty} \min_{G_n} \, \Pr(G_n \cup G(n,p(n)) \text{ satisfies } \mathcal{A})=1,$$ where the minimum is taken over all $\gamma$-dense graphs on the same vertex set as $G(n,p(n))$. For a fixed $\gamma>0$, we say that a function $p^*$ is a *threshold* for $\mathcal{A}$ (in the context of randomly perturbed dense graphs) if $\tau_p^{\mathcal{A}}=1$ for $p=\omega(p^*)$ and $\tau_p^{\mathcal{A}}=0$ for $p=o(p^*).$ Throughout, we will assume that $\gamma>0$ is some fixed and small constant. Stricly speaking, working in this model requires to consider sequences of $\gamma$-dense graphs $(G_n)_{n \in {\mathbb{N}}}$. However, for a better presentation, we suppress the sequences and similarly we simply write $p$ for $p(n)$. Recently, several thresholds for this model have been studied in [@bohman2004adding; @krivelevich2006smoothed; @bottcher2017embedding; @krivelevich2017bounded; @balogh2018tilings; @krivelevich2016cycles; @bedenknecht2018powers; @bennett2017adding; @bottcher2018universality; @han2018hamiltonicity; @joos2018spanning; @mcdowell2018hamilton; @dudek2018powers]. Most of the analysis centered around ensuring spanning structures such as trees or (powers of) cycles. Krivelevich, Sudakov and Tetali [@krivelevich2006smoothed] already investigated Ramsey properties of this model (see Section \[subsRP\] below). We continue this line of research (see Section \[ourresults\]). Ramsey properties of random graphs ---------------------------------- For graphs $G, H_1, \dots ,H_k$, we denote by $G \rightarrow (H_1,\dots,H_k)$ the Ramsey-type statement that every colouring of $E(G)$ with colours $\{1 \dots k\}$ yields a monochromatic copy of $H_i$ in colour $i$ for some $i$. In the symmetric case when $H_1=\dots =H_k$, we simply write $G \rightarrow (H)_k$ and if additionally $k=2$, then we write $ G \rightarrow (H)$. Using this notation, Ramsey’s theorem states that for all $k, \ell \in {\mathbb{N}}$ there exists some $n \in {\mathbb{N}}$ such that $K_n \rightarrow \left(K_{\ell}\right)_k$. R[ö]{}dl and Ruci[ń]{}ski established the threshold for the property $G(n,p) \rightarrow~(H)$ which for every fixed graph $H$. For a graph $H=(V,E)$ we define $$d_{2}(H)= \begin{cases} \frac{|E|-1}{|V|-2} & \text{ if } |V|\geq 3 \land |E|>0 \\ \frac{1}{2} & \text{ if } H \cong K_2 \\ 0 & \text{ if } |E|=0 \end{cases}$$ and we let $m_2(H)$ denote the *2-density*, defined by $m_{2}(H)= \max_{J\subseteq H} d_2(J)$. The following is a slightly simplified version of the result mentioned above: \[RRT\] Let $k\geq 2$ be an integer and let $H$ be a graph that is not a forest. Then there exist real constants $c$, $C>0$ such that $$\lim\limits_{n \rightarrow \infty} \Pr(G(n,p)\rightarrow(H)_{k})= \begin{cases} 0 \text{ \, \, if } p=p(n) \leq cn^{-1/m_{2}(H)} \\ 1 \text{ \, \, if } p=p(n) \geq Cn^{-1/m_{2}(H)} \end{cases}$$ Recently, one focus of research in this area is to establish thresholds for asymmetric Ramsey properties. Interestingly enough, some of the recent discoveries will play a key role in the proofs of our results and will be introduced in Section \[proofs\]. Ramsey properties of randomly perturbed dense graphs {#subsRP} ---------------------------------------------------- Concerning the newer model of randomly perturbed graphs, a first reasonable question to address is whether the R[ö]{}dl-Ruci[ń]{}ski threshold can be improved at all. In fact this cannot be achieved for $k\geq 3$ (i.e. more than 2 colours) and small $\gamma $: In case of $\gamma n^2 \leq ex(n,H)$ (for example, when $\gamma<\frac{1}{4}$ and $H$ is a clique) there exists an $H$-free, $\gamma$-dense graph $G_n$. Then we can assign one colour to all edges from $G_n$ without admitting a monochromatic copy of $H$ in that colour. We still have at least two unused colours left to cope with the edges of the random graph, so we will be able to colour the remaining edges without admitting a monochromatic copy of $H$, unless we have $G(n,p) \rightarrow (H)_2$. By Theorem \[RRT\] a threshold for $G(n,p)\rightarrow(H)_k$ is also a threshold for $G(n,p) \rightarrow (H)_2$ and thereby for $G_n \cup G(n,p)\rightarrow (H)_k$ by the above consideration. Hence, we will focus on the case $k=2$ only. We first recall the Ramsey result from [@krivelevich2006smoothed]. \[K3KtThm\] 1. If $p=\omega(n^{-2/(t-1)})$, then for any $0<\gamma <1$, any integer $t \geq 3$ and any $\gamma$-dense $n$-vertex graph $G_n$ we a.a.s have $$G_n \cup G(n,p)\shortrightarrow(K_3,K_t).$$ 2. If $p =o(n^{-2/(t-1)})$, then for any constant $0<\gamma<\frac{1}{4}$ and for every $t \geq 3$ there exists a $\gamma$-dense $n$-vertex graph $G_n$ such that we a.a.s. have $$G_n \cup G(n,p)\nrightarrow(K_3,K_t).$$ Note that in particular this shows that $p(n)=n^{-1}$ is a threshold function for $G_n \cup G(n,p) \rightarrow (K_3)$. Our results {#ourresults} ----------- As mentioned above, our proofs are based on asymmetric Ramsey results and these involve the *asymmetric 2-density* $m_2(G,H)$. For two graphs $G$ and $H$, both having at least one edge, let $$\begin{aligned} d_2(G,H)=\frac{|E(H)|}{|V(H)|-2+1/m_2(G)}\end{aligned}$$ and let $$m_2(G,H)=\max_{J\subseteq H} d_2(G,J)\text{\,.}$$ Then our first result reads as follows. \[mainresult1\] Let $\gamma>0$ be a real constant and let $r\geq 5$ be an integer. 1. \[mainresult1even\] Let $\varepsilon >0$ be a constant and $p=\omega\left(n^{-(1-\varepsilon) /m_{2}\left(K_{\ceil{r/2}},K_{r}\right)}\right)$. For any $\gamma$-dense $n$-vertex graph $G_n$ we a.a.s. have $$G_{n}\cup G(n,p)\shortrightarrow(K_r).$$ 2. \[mainresult1odd\] Let $r$ be odd. Then there exists a real constant $C>0$ such that the following holds for $ p\geq Cn^{-1/m_{2}(K_{\frac{r+1}{2}},K_{r})}$. For any $\gamma$-dense $n$-vertex graph $G_n$ we a.a.s. have $$G_{n}\cup G(n,p)\shortrightarrow(K_r).$$ We complement Theorem \[mainresult1\] by the following lower bound for the threshold. \[mainresult0\] Let $r \geq 5$ be an integer and let $\gamma<\frac{1}{4}$ be a positive constant. Then there is a real constant $c>0$ and an $n$-vertex graph $G_n$ with ${|E(G_{n})| \geq \gamma n^{2}}$ such that for ${p \leq cn^{-1/m_{2}(K_{\ceil{\frac{r}{2}}},K_{r})}}$ we a.a.s. have $$G_{n}\cup G(n,p)\nrightarrow(K_r).$$ Note that Theorem \[mainresult0\] shows that Theorem \[mainresult1\]\[mainresult1odd\] is asymptotically optimal for odd $r\geq 5$ while for even $r\geq 6$ Theorem \[mainresult1\]\[mainresult1even\] leaves a ‘gap’ of an arbitrarily small $\varepsilon$ in the exponent. Independently of our work, Das and Treglown [@DasTreglown] proved a more general result which closes these gaps. Finally, the following theorem covers the remaining case $r=4$. \[K4thm\] 1. \[1statementK4\] Let $\gamma >0$ and $\varepsilon>0$. Then for any graph $G_n$ on $n$ vertices with at least $\gamma n^2$ edges and any $p\geq n^{-\frac{1}{2}+\varepsilon}$ we a.a.s. have $G_n\cup G(n,p)\rightarrow(K_4).$ 2. \[0statementK4\] For $0<\gamma<\frac{1}{4}$ there exists a graph $G_n$ on $n$ vertices with at least $\gamma n^2$ edges such that for $p= o( n^{-\frac{1}{2}})$ we a.a.s. have $G_n\cup G(n,p)\nrightarrow(K_4).$ Again, the result of Das and Treglown [@DasTreglown] improves the upper bound in Theorem \[K4thm\]\[1statementK4\], while Theorem \[K4thm\]\[0statementK4\] improves their lower bound. Thus, both results together establish $n^{-\frac{1}{2}}$ as a threshold for $G_n \cup G(n,p) \rightarrow (K_4)$. Preliminaries & Notation ======================== In this short section we introduce further notations and a few basic results that we will use repeatedly. For a graph $G=(V,E)$ and a subset ${U \subseteq V}$, by $G[U]$ we denote the subgraph of $G$ induced by the vertex set $U$. Furthermore, we will sometimes write $E(U)$ instead of $E(G[U])$. By $\rho(H)$ we denote the *density* of a graph $H$ which is defined as $$\rho(H)=\frac{|E(H)|}{|V(H)|}.$$ The following well-known result establishes the threshold for containing a fixed subgraph. \[[@erdos1960evolution]\] \[subgraphthreshold\] Let $r\geq 3$ be an integer. We have $$\begin{aligned} \lim_{n \to \infty} \Pr (G(n,p) \text{ contains a copy of $K_r$ as a subgraph})\\= \begin{cases} 0 & \text{if } p=o(n^{-\frac{1}{\rho(K_r)}})\\ 1 & \text{if } p=\omega(n^{-\frac{1}{\rho(K_r)}}) \end{cases}\end{aligned}$$ In order to get exponential bounds on the probability of non-existence of a graph we will use the following variant of Janson’s inequality (see [@janson1990poisson], [@janson1988exponential]). \[Janson\] For any $r\geq 3$ there exists some constant $c_r$ such that for any $p<1$ and all $n \in {\mathbb{N}}$ the probability that $G(n,p)$ is $K_r$-free is at most $2^{-c_r n^{r}p^{r \choose 2}}.$ Proofs ====== Proof of Theorem \[mainresult0\] -------------------------------- We start off by showing the optimality of our result for $r\geq 5$ as this proof is easy and still already illustrates the relation between Ramsey properties of the considered model and (asymmetric) Ramsey properties of pure random graphs. The following $0$-statement for an asymmetric Ramsey property is crucial. Note that the result of Marciniszyn, Skokan, Sp[ö]{}hel and Steger [@marciniszyn2009asymmetric] is more general as it addresses an arbitrary number of colours and cliques. \[MSSST\] Let $3 \leq \ell \leq r$ be integers. Then there exists a real constant $ c>0$ such that for $p \leq cn^{-1/m_{2}(K_{\ell}, K_{r})}$ we a.a.s. have ${G(n,p) \rightarrow (K_{\ell},K_r)}.$ Let $G_n$ be the complete bipartite graph $K_{\ceil{\frac{n}{2}}, \floor{\frac{n}{2}}}$. Then, for sufficiently large $n \in {\mathbb{N}}$ we have $$|E(G_n)|\geq \ceil{n/2}\floor{n/2}\geq \frac{(n+1)(n-1)}{4}\geq \gamma n^2$$ as required. We obtain $c=c(\ceil{\frac{r}{2}},r)$ from Theorem \[MSSST\]. We get $$\lim\limits_{n \rightarrow \infty} \Pr(G_{n}\cup G(n,p)\shortrightarrow(K_r)) \leq \lim\limits_{n \rightarrow \infty}\Pr(G(n,p) \rightarrow (K_{\ceil{\frac{r}{2}}},K_r))=0,$$ where the equality follows from Theorem \[MSSST\] and the inequality follows from the following deterministic argument. Suppose that $H_n \nrightarrow (K_{\ceil{\frac{r}{2}}},K_r)$ for an $n$-vertex graph $H_n$ and let $\phi$ be a red-blue-colouring of $E(H_n)$ that does not admit a red copy of $K_{\ceil{\frac{r}{2}}}$ or a blue copy of $K_r$. We extend $\phi$ to a colouring $\Phi$ of $E(G_n \cup H_n)$ by assigning the colour red to all remaining edges. The resulting colouring $\Phi$ does not admit a blue copy of $K_r$, since $\phi$ does not. Futhermore, it does not yield a red copy of $K_r$, because any such copy would need to have at least $ \ceil{\frac{r}{2}}$ vertices in one of the partition classes of the bipartite graph $G_n$, contradicting the choice of $\phi$. Proof of Theorem \[mainresult1\] -------------------------------- The proof of our result for cliques of even size $r\geq 6$ works as follows: We first show that in order to satisfy $G_n \cup G(n,p) \rightarrow (K_r)$ it suffices to have $G(n,p)[U] \rightarrow (K_{\frac{r}{2}},K_r)$ for all ’almost linear’ subsets $U\subseteq V$ (those of size at least $n^{1-\varepsilon}$). In the second step we bound the probability of the above event by means of an asymmetric Ramsey result that was established recently. For odd $r \geq 5$ we can improve the technique by seeking for $G(n,p)[U] \rightarrow (K_{\frac{r+1}{2}},K_r)$ in ’linear subsets’ and $G(n,p)[U] \rightarrow (K_{\frac{r-1}{2}},K_r)$ in the ’almost linear subsets’. Since the latter property generally requires a smaller $p$ to be ensured, the first will be the limiting factor, although we ask for it in slightly bigger subsets. By means of this little trick, the $\varepsilon$ reducing the size of our subsets in the second condition will no longer play a role and we get the asymptotically tight 1-statement. A key element for the first step (of both proofs) is the following lemma given by Fox and Sudakov in their article on the method of dependent random choice [@fox2011dependent]: \[DRCL\] Let $a$, $d$, $m$, $n$ and $r$ be positive integers. Let $G=(V,E)$ be a graph with $|V|=n$ and average degree $d=d(G)=\frac{2 |E|}{|V|}$. If there is a positive integer t such that $$\begin{aligned} \label{DRCI} \frac{d^{t}}{n^{t-1}}-{n \choose r} \left(\frac{m}{n}\right)^{t}\geq a,\end{aligned}$$ then $G$ contains a subset $U\subseteq V$ of at least $a$ vertices such that every $r$ vertices in $U$ have at least $m$ common neighbours. \[RDCCor\] For any $\gamma$, $\varepsilon>0$ and $r \in {\mathbb{N}}$ there is a constant $\alpha>0$ and $n_{0} \in {\mathbb{N}}$ such that the following holds for all integers $n \geq n_{0}$. For every $\gamma$-dense $n$-vertex graph $G=(V,E)$ there is a subset $U \subseteq V$ with $|U| \geq \alpha n$ such that every $r$ vertices in $U$ have at least $n^{1-\varepsilon}$ common neighbours. The following is a straightforward application of the Lemma we just introduced. As usual, for the sake of a less baroque presentation, we do not round our parameters and instead assume they are integers. For $t={\frac{r}{\varepsilon}}$ and $\alpha = \frac{\gamma^{t}}{2}$ let $n_{0} \in {\mathbb{N}}$ be sufficiently large such that $n_{0} \geq \frac{2}{\gamma^{t}}$. For any integer $n\geq n_0$ and a given graph $G$ satisfying the above properties we then get $$\begin{aligned} \frac{d(G)^{t}}{n^{t-1}}-{n \choose r} \left(\frac{n^{1-\varepsilon}}{n}\right)^{t} \geq \frac{(\gamma n)^{t}}{n^{t-1}}-{n \choose r} \left(\frac{n^{1-\varepsilon}}{n}\right)^{t} \geq \gamma^{t} n - n^{r-\varepsilon t} \\ = \gamma^{t}n-1 \geq (\frac{\gamma^{t}}{2}+\frac{\gamma^{t}}{2})n-1 \geq (\alpha+\frac{1}{n_0})n-1 \geq \alpha n\end{aligned}$$ which verifies (\[DRCI\]) for $m=n^{1-\varepsilon}$ and $a=\alpha n$. Hence, the corollary follows from Lemma \[DRCL\]. The following lemma completes step 1. \[mylemma\] Let $\gamma$, $\varepsilon>0$ be real constants and let $r \in {\mathbb{N}}$. Then there exist a real constant $\alpha>0$ and $n_{0} \in {\mathbb{N}}$ such that the following holds for all integers $n \geq n_{0}$ and all $\gamma$-dense $n$-vertex graphs $G=(V,E)$. $$\begin{split} \label{myLemmaImpl} & \text{If we have }G[U] \rightarrow(K_{\ceil{\frac{r}{2}}}, K_{r}) \text{ for all subsets }U \subseteq V\text{ with }|U|\geq \alpha n \text{} \text{ and } \\ & G[U] \rightarrow(K_{\floor{\frac{r}{2}}}, K_{r}) \text{ for all subsets }U \subseteq V \text{ with }|U|\geq n^{1-\varepsilon},\text{ } \text{then }G \rightarrow(K_{r}). \end{split}$$ Let $\alpha=\alpha(\gamma,\varepsilon,r)$ and $n_0=n_0(\gamma,\varepsilon,r)$ be given by Corollary \[RDCCor\]. Let $G=(V,E)$ be a graph satisfying $|V|=n\geq n_0$ and the other assumptions from above. Suppose for contradiction that $\phi$ is a red-blue-colouring of $E(G)$ without a monochromatic copy of $K_{r}$. Without loss of generality we may assume $|\phi^{-1}(\{\text{red}\})|\geq \frac{\gamma}{2} n^{2}$ and let $G_{\text{red}}$ be the subgraph on $V$ that contains only the red edges. By Corollary \[RDCCor\] there exists $U \subseteq V$ with $|U| \geq \alpha n$ such that every $r$ vertices from $U$ have at least $n^{1-\varepsilon}$ common neighbours in $G_{\text{red}}$. It follows from our assumption that we have $G[U]\rightarrow(K_{\ceil{\frac{r}{2}}}, K_{r}).$ Since $G[U]$ does not contain a blue copy of $K_{r}$, there must be a copy of $K_{\ceil{\frac{r}{2}}}$ in $G_{\text{red}}[U]$. Let $X$ be the set of vertices inducing this copy. Owing to $|X|=\ceil{\frac{r}{2}}\leq r$ and the choice of $U \supseteq X$, the common neighbourhood $W$ of $X$ in $G_{\text{red}}$ has size at least $n^{1-\varepsilon}$. Now the second assumption implies $G[W]\rightarrow(K_{\floor{\frac{r}{2}}}, K_{r})$ and by the argument given above we find a copy of $K_{\floor{\frac{r}{2}}}$ in $G_{\text{red}}[W]$. Let $Y$ be the set of vertices forming this copy; then $X \cup Y$ induces a copy of $K_{r}$ in $G_{\text{red}}$, yielding a contradiction. As indicated above, in step 2 we want to bound the probability of the event forming the hypothesis in using asymmetric Ramsey results. Not too long ago, Kohayakawa, Schacht and Sp[ö]{}hel [@kohayakawa2014upper] proved an asymmetric 1-statement for two well-behaved graphs $G$ and $H$. In particular, for cliques their upper bound asymptotically coincides with the lower bound we met in Theorem \[MSSST\]. However, for our purposes their main result does not help much because we need the specific exponential form of the error probability. This is because for our arguments the error probability still needs to converge to 0 when multiplied with the number of subsets of the vertex set, i.e. $2^n$. Instead we want to apply Lemma 23 from [@kohayakawa2014upper] which (in the original form) contains some paper-specific notation that we do not need. Hence, we state a version that is well adapted to our setting and provide a short deduction in the following lines. However, we do not go into any details of [@kohayakawa2014upper]: For our purposes we take $G=K_l$ and $H=K_r$. It is easy to check that $K_r$ is strictly balanced with respect to $d_2(K_l,\cdot)$ (see p.3 from that paper for the definition). Then Lemma 13 from that article assures that $K_l$ and $K_r$ satisfy the hypothesis of Lemma 23. In the lemma you find a graph parameter $x^*(H)>0$ (where, of course, $H$ is a graph). Since we do not intend to use its explicit form anywhere, we refer to [@kohayakawa2014upper] (Definiton 11) in case the reader wishes to have a look at the definiton. \[KSSL\] Let $3 \leq \ell < r$ be integers. Then there exist real constants $C>0$, $ b>0$ and $n_{0} \in {\mathbb{N}}$ such that for any integer $n \geq n_{0}$ and any $p$ satisfying $$n^{-1/x^*(K_r)}\geq p \geq Cn^{-1/m_{2}(K_{\ell}, K_{r})}$$ we have $$\Pr(G(n,p)\rightarrow(K_\ell,K_r)) \geq 1-2^{-bn^{r}p^{r \choose 2}}.$$ Additionally, we need to know that there exists a $p$ that lies between the bounds given in Lemma \[KSSL\]. The following lemma ensures that and it follows from Lemma 13 (in [@kohayakawa2014upper]), once more using that $K_r$ is strictly balanced with respect to $d_2(K_l,\cdot)$. \[x\*Lemma\] For integers $3 \leq \ell < r$ we have $$m_2(K_l,K_r)<x^*(K_r).$$ With this in hand, we will be able to complete step 2. By the union bound, the probability that the hypothesis of fails can now be bounded by a term of the form $$2^n 2^{-bn^{(1-\varepsilon) r} p^{r \choose 2}}$$ (or without the $\varepsilon$ in the ’improved setting’) and it will turn out that this term converges to $0$ for our range of $p$. Note that this is not surprising as $n^r p^{r \choose 2}$ is of the same order as the expected number of copies of $K_r$ which should outdo $n$ (see parts \[calceven\] and \[calcodd\] of Lemma \[Calc\]). We hope that this already outlines the proofs reasonably well. Still, on the next pages we present the proofs for both cases in great detail. In order to make the proofs less technical, we extract some calculations revolving around the asymmetric 2-density into another lemma. \[Calc\] Let $r\geq5$ be an integer and let $C$, $\varepsilon>0$. 1. \[m2clique\] For cliques the asymmetric 2-density has the form $$m_2(K_l,K_r)=\frac{{r \choose 2}}{r-2+2/(l+1)},$$ where $3\leq l\leq r$ is another integer. 2. \[calceven\] If $p \geq C n^{-(1-\varepsilon)/m_2(K_{\ceil{r/2}},K_r)}$ and $\varepsilon< \frac{1}{6}$, then we have $$\begin{aligned} n^{(1-\varepsilon)r}p^{{r \choose 2}}\geq C^{r\choose 2}n^{\frac{5}{4}}.\end{aligned}$$ 3. \[calcodd\] If $r$ is odd, $p \geq Cn^{-1/m_2(K_{\frac{r+1}{2}},K_r)}$ and $\varepsilon<\frac{1}{4r}$, then we have $$\begin{aligned} n^{r}p^{{r \choose 2}} \geq n^{(1-\varepsilon)r}p^{r \choose 2} \geq C^{r \choose 2}n^{\frac{5}{4}}.\end{aligned}$$ 4. \[calctau\] Finally, for odd $r\geq 5$ we have $${m_{2}(K_{\frac{r+1}{2}},K_{r})} >{m_2(K_{\frac{r-1}{2}},K_{r})}.$$ <!-- --> 1. This follows from $$m_2(K_l,K_r)=d_2(K_l,K_r)=\frac{{r \choose 2}}{r-2+1/m_2(K_l)}$$ and $$\begin{aligned} m_2(K_l)= d_2(K_l)=\frac{{l \choose 2}-1}{l-2}= \frac{\frac{(l+1)(l-2)+2}{2}-1}{l-2}= \frac{l+1}{2}.\end{aligned}$$ 2. Using \[m2clique\] we get $$\begin{aligned} n^{(1-\varepsilon)r}p^{{r \choose 2}}&\geq C^{r \choose 2} n^{(1-\varepsilon)(r-(r-2+\frac{2}{\ceil{r/2}+1}))}\\ &\geq C^{r \choose 2} n^{(1-\varepsilon)(2-\frac{2}{3+1})} \geq C^{r \choose 2} n^{\frac{5}{6}\cdot \frac{3}{2}}=C^{r \choose 2} n^{\frac{5}{4}}.\end{aligned}$$ 3. The first inequality in the statement is trivial and the second can be verified by $$\begin{aligned} n^{(1-\varepsilon)r}p^{r \choose 2}\mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily {\ref{m2clique}}}}}{\geq}} C^{r \choose 2} n^{(1-\varepsilon)r-(r-2+\frac{2}{(r+1)/2+1})}\geq C^{r \choose 2} n^{r-\frac{1}{4}-r+2-\frac{1}{2}} = C^{r \choose 2} n^{\frac{5}{4}}.\end{aligned}$$ 4. This can be seen easily after applying \[m2clique\] to both sides. We start with the general case where $r\geq 5$ is an arbitrary integer and $\varepsilon>0$ is given. Throughout this proof let $m_2 \coloneqq m_2(K_{\ceil{\frac{r}{2}}},K_r)$ and . By monotonicity we may assume $\varepsilon \leq \frac{1}{6}$ and $(1-\varepsilon)x^*>m_2$. We start by applying our preparatory lemmas: - Lemma \[KSSL\] yields constants $C=C(\ceil{\frac{r}{2}},r)$, $b=b(\ceil{\frac{r}{2}},r)$ and $n_1=n_1(\ceil{\frac{r}{2}},r)$ such that $$\begin{aligned} \label{Proofr} \Pr (G(\tilde{n},\tilde{p}) \nrightarrow(K_{\ceil{r/2}}, K_{r}))\leq 2^{-b\tilde{n}^{r} \tilde{p}^{ {r \choose 2}}}\end{aligned}$$ for all integers $\tilde{n} \geq n_1$ and any $\tilde{p}$ with $$\tilde{n}^{-1/x^*} \geq \tilde{p} \geq C\tilde{n}^{-1/m_{2}}.$$ - Lemma \[mylemma\] yields $\alpha =\alpha(\gamma,\varepsilon,r)$ and $n_2 =n_2(\gamma,\varepsilon,r)$ such that holds for graphs $G$ with $|V(G)| \geq n_2$. Let $n_0 \in {\mathbb{N}}$ satisfy $$n_0\geq \max (n_1^{1/(1-\varepsilon)},n_2, \alpha^{-1/\varepsilon})$$ and additionally $$n^{-1/x^*} \geq C n^{-(1-\varepsilon) /m_{2}}$$ for all integers $n \geq n_0;$ the latter is possible due to the assumption we made in the beginning of the proof. Now let an integer $n\geq n_0$ be given. By monotonicity we may assume $$p= C n^{-(1-\varepsilon) /m_{2}}.$$ Then for any $U \subseteq \{1,...,n\}$ with $|U| \geq n^{1-\varepsilon}$ we may apply (\[Proofr\]) to $G(n,p)[U]$ (which we may identify with the random graph $G(|U|,p)$), since we have $$|U|\geq n^{1- \varepsilon}\geq n_0^{1- \varepsilon} \geq n_1$$ and $$\begin{aligned} |U|^{-1/x^*} \geq n^{-1/x^*} \geq p= C n^{-(1-\varepsilon) /m_{2}} \geq C|U|^{-1/m_{2}}.\end{aligned}$$ yields $$\begin{aligned} \label{forfinalBlockeven} \Pr (G(n,p)[U] \nrightarrow(K_{\ceil{r/2}}, K_{r}))\leq 2^{-bn^{(1-\varepsilon)r} p^{ {r \choose 2}}}.\end{aligned}$$ Owing to $\alpha n = \alpha n^{\varepsilon}n^{1-\varepsilon}\geq \alpha \alpha^{(-1/\varepsilon)\varepsilon}n^{1-\varepsilon}= n^{1-\varepsilon}$, the following statement implies : $$\begin{aligned} & \text{If we have }G(n,p)[U] \rightarrow(K_{\ceil{r/2}}, K_{r})\text{ for all subsets }U \subseteq V \text{ with } |U|\geq n^{1-\varepsilon}, \\ & \text{then }G(n,p) \rightarrow(K_{r})\text{.}\end{aligned}$$ Therefore, we have $$\begin{aligned} & \Pr (G_{n}\cup G(n,p) \nrightarrow (K_{r})) \\ \leq \,\,& \Pr( \exists \, U \subseteq V: |U|\geq n^{1-\varepsilon} \land G(n,p)[U]\nrightarrow (K_{\ceil{r/2}},K_r))\\ \leq \,\,& \sum_{U \subseteq V: |U|\geq n^{1-\varepsilon}}\Pr(G(n,p)[U]\nrightarrow (K_{\ceil{r/2}},K_r))\\ \mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily {(\ref{forfinalBlockeven})}}}}{\leq}} \,\,& \sum_{U \subseteq V: |U|\geq n^{1-\varepsilon}} 2^{-b\left(n^{1-\varepsilon}\right)^{r} p^{ {r \choose 2}}} \leq 2^{n-b C^{{r \choose 2}}n^{\frac{5}{4}}},\\\end{aligned}$$ where the last inequality uses Lemma \[Calc\]\[calceven\]. This proves the theorem since the last term converges to $0$ when $n$ goes to infinity. Now as to the more specific case where $r\geq 5$ is odd: Throughout the proof let $m_2^+ \coloneqq m_2(K_{\frac{r+1}{2}},K_r)$, $m_2^- \coloneqq m_2(K_{\frac{r-1}{2}},K_r)$ and $x^* \coloneqq x^*(K_r).$ By Lemma \[Calc\]\[calctau\] there exists a $\tau>0$ such that $m_2^+= m_2^-\cdot\frac{1}{1- \tau}$ and we let $\varepsilon= \min(\tau,\frac{1}{4r})>0$. Again, we start by applying our preparatory lemmas: - Lemma \[KSSL\] yields constants $C_1=C_1(\frac{r+1}{2},r)$, $b_1=b_1(\frac{r+1}{2},r)$ and $n_1=n_1(\frac{r+1}{2},r)$ such that $$\begin{aligned} \label{Proofr+1} \Pr (G(\tilde{n},p) \nrightarrow(K_{\frac{r+1}{2}}, K_{r}))\leq 2^{-b_1\tilde{n}^{r} p^{ {r \choose 2}}}\end{aligned}$$ for all integers $\tilde{n} \geq n_1$ and any $p$ with $$\tilde{n}^{-1/x^*} \geq p \geq C_1\tilde{n}^{-1/m_{2}^+}.$$ - Lemma \[KSSL\] yields constants $C_2=C_2(\frac{r-1}{2},r)$, $b_2=b_2(\frac{r-1}{2},r)$ and $n_2=n_2(\frac{r-1}{2},r)$ such that $$\begin{aligned} \label{Proofr-1} \Pr (G(\tilde{n},p) \nrightarrow(K_{\frac{r-1}{2}}, K_{r}))\leq 2^{-b_2\tilde{n}^{r} p^{ {r \choose 2}}}\end{aligned}$$ for all integers $\tilde{n} \geq n_2$ and any $p$ satisfying $$\tilde{n}^{-1/x^*} \geq p \geq C_2\tilde{n}^{-1/m_{2}^-}.$$ - Lemma \[mylemma\] yields $\alpha =\alpha(\gamma,\varepsilon,r)$ and $n_3 =n_3(\gamma,\varepsilon,r)$ such that holds for graphs $G$ with $|V(G)|\geq n_3$. We may assume $\alpha \leq 1$. Now let $$C=\max(C_1,C_2)\cdot\alpha^{-1/m_2^+}\geq \max(C_1,C_2)$$ and let $n_0$ satisfy $n_0\geq \max(\frac{n_1}{\alpha},n_2^{1/(1-\varepsilon)},n_3)$ and additionally $$n^{-1/x^*}\geq Cn^{-1/m_{2}^+}$$ for all integers $n \geq n_0$, which is possible by Lemma \[x\*Lemma\]. Now let an integer $n \geq n_0$ be given. By monotonicity we may assume $$p= Cn^{-1/m_{2}^+}.$$ - For any $U \subseteq \{1,...,n\}$ with $|U|\geq \alpha n$ we may apply to $G(n,p)[U]$ (which we may identify with $G(|U|,p)$), as we have $|U|\geq \alpha n \geq n_1$ and $$\begin{aligned} |U|^{-1/x^*} &\geq n^{-1/x^*} \geq p= Cn^{-1/m_{2}^+}\\ & \geq C_1(\alpha n)^{-1/m_{2}^+}\geq C_1|U|^{-1/m_{2}^+}.\end{aligned}$$ (\[Proofr+1\]) shows $$\begin{aligned} \label{forfinalblock1} \Pr (G(n,p)[U] \nrightarrow(K_{\frac{r+1}{2}}, K_{r}))\leq 2^{-b_1(\alpha n)^{r} p^{ {r \choose 2}}}.\end{aligned}$$ - Analogously, if $r\geq 7$, for any $U \subseteq \{1,...,n\}$ with $|U|\geq n^{1-\varepsilon}$ we may apply to $G(n,p)[U]$ since we have $|U| \geq n^{1-\varepsilon}\geq n_2$ and $$\begin{aligned} |U|^{-1/x^*}\geq n^{-1/x^*} \geq p = Cn^{-1/m_{2}^+} = Cn^{-(1-\tau)/m_{2}^-} &\geq C_2n^{-(1-\varepsilon)/m_{2}^-}\\ &\geq C_2|U|^{-1/m_{2}^-},\end{aligned}$$ where we used the definitions of $\tau$ and $\varepsilon$. We get $$\begin{aligned} \label{forfinalblock2} \Pr (G(n,p)[U] \nrightarrow(K_{\frac{r-1}{2}}, K_{r}))\leq 2^{-b n^{(1-\varepsilon)r} p^{ {r \choose 2}}},\end{aligned}$$ where $b=\min(b_2, c_5)$ with $c_5$ being the constant yielded by Theorem \[Janson\]. This follows from (\[Proofr-1\]) for $r\geq 7$ and from Theorem \[Janson\] for $r=5$. Note that in the latter case $\frac{r-1}{2}=2$, and thus $G \shortrightarrow (K_{\frac{r-1}{2}},K_r)$ is just containing a copy of $K_r$. Therefore, the asymmetric Ramsey result is not applicable in that case and we have to go for Janson’s. Using (\[myLemmaImpl\]) we get $$\begin{aligned} & \Pr (G_{n}\cup G(n,p) \nrightarrow (K_{r})) \\ & \leq \Pr ((\exists \, U \subseteq V :|U| \geq \alpha n \land G(n,p)[U] \nrightarrow(K_{\frac{r+1}{2}}, K_{r})) \lor \\ & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(\exists \, U \subseteq V :|U| \geq n^{1-\varepsilon} \land G(n,p)[U] \nrightarrow(K_{\frac{r-1}{2}}, K_{r}) )) \\ & \leq \sum_{U \subseteq V: |U|\geq \alpha n} \Pr (G_(n,p)[U] \nrightarrow(K_{\frac{r+1}{2}}, K_{r}))+\\ & \,\,\,\,\,\,\, \sum_{U \subseteq V: |U|\geq n^{1-\varepsilon}} \Pr (G(n,p)[U] \nrightarrow(K_{\frac{r-1}{2}}, K_{r})) \\ & \mathrel{\overset{\makebox[0pt]{\mbox{\normalfont\tiny\sffamily {\eqref{forfinalblock1},\eqref{forfinalblock2}}}}}{\leq}} \sum_{U \subseteq V: |U|\geq \alpha n} 2^{-b_1(\alpha n)^r p^{{ r \choose 2}}}+\sum_{U \subseteq V: |U|\geq n^{1-\varepsilon}} 2^{-b\left(n^{1-\varepsilon}\right)^{r} p^{ {r \choose 2}}}\\ & \leq 2^{n-b_1 C^{{r \choose 2}}\alpha^{r}n^{\frac{5}{4}}}+ 2^{n-bC^{r \choose 2} n^{\frac{5}{4}}}, \end{aligned}$$ where the last inequality follows from Lemma \[Calc\]\[calcodd\], which is applicable due to $\varepsilon \leq \frac{1}{4r}$. This proves the theorem because the last term goes to $0$ as $n$ approaches infinity. Brief discussion of $\boldsymbol{r\leq 4}$ and the proof of Theorem \[K4thm\] ----------------------------------------------------------------------------- Considerations are of a different nature for the clique sizes ${r=3}$ and $r=4$. In these cases ${G\shortrightarrow (K_{\ceil{\frac{r}{2}}},K_r})$ comes down to containing a copy of $K_r$. Proceeding as in Theorem \[mainresult0\], we obtain a lower bound of $p=n^{-\frac{1}{\rho(K_r)}}$ (see Theorem \[subgraphthreshold\]). We also obtain an upper bound based on Lemma \[mylemma\]; however, we need $p$ to be of order $$n^{-\frac{1}{\rho(K_r)}+\frac{1}{|E(K_r)|}}$$ as we can only apply this technique if the expected number of copies of $K_r$ is linear. This leaves us with a significant gap between the bounds. For $r=3$ the threshold coincides with the lower bound (see Theorem \[K3KtThm\]), whereas for $r=4$ we will improve the lower bound so that it matches the order of the upper bound. Let us start with a brief proof of the upper bound; the method is the same as in the proof of Theorem \[mainresult1\] where we went into great detail. Following the lines of the proof of Theorem \[mainresult1\]\[mainresult1even\] and applying Theorem \[Janson\] yields $$\begin{aligned} \Pr(G_n \cup G(n,p)\nrightarrow (K_4)) &\leq \Pr(\exists U \subseteq V : |U| \geq n^{1- \varepsilon} \land G(n,p)[U] \text{ is } K_4 \text{-free})\\ &\leq 2^n 2^{-c_r n^{(1-\varepsilon)4}n^{(-\frac{1}{2}+\varepsilon)6}}= 2^{n-c n^{1+2 \varepsilon}}, \end{aligned}$$ for a sufficiently large $n$ and an appropriate constant $c>0$. Clearly, the last term converges to $0$ as desired. We now turn our attention to the lower bound and start by introducing our key lemma which we will prove later. We write $G\rightarrow (H)^v$ if every 2-colouring of $V(G)$ admits a monochromatic copy of $H$. \[basicLemmaK\_4\] Let $G=(V_1 \mathop{\dot{\cup}} V_2,E)$ be a graph such that $G[V_1]\nrightarrow (K_3)$, ${G[V_2]\nrightarrow (K_4)^v}$ and $G[V_2]$ does not contain any subgraph $H$ with $\rho(H)\geq \frac{1}{2}$ and $|V(H)|\leq 8$. Then we have $G\nrightarrow (K_4,K_4)$. Let us now first show that Lemma \[basicLemmaK\_4\] implies Theorem \[K4thm\]\[0statementK4\]. We need the following special case of a result on the threshold for vertex-Ramsey properties by [Ł]{}uczak, Ruci[ń]{}ski and Voigt [@luczak1992ramsey]. \[vertexRamseyK4\] There exists a real constant $c>0$ such that for $p\leq cn^{-\frac{1}{2}}$ we have $$\lim\limits_{n \rightarrow \infty} \Pr(G(n,p)\shortrightarrow(K_4)^v)=0.$$ As in the proof of Theorem \[mainresult0\] we choose $G_n=K_{\ceil{\frac{n}{2}},\floor{\frac{n}{2}}}$ and we denote the partition classes by $V_1$ and $V_2$. We show that $H_n=G_n \cup G(n,p)$ a.a.s. satisfies the hypothesis of Lemma \[basicLemmaK\_4\]. By Theorem \[RRT\] we a.a.s. have ${G(n,p)\nrightarrow (K_3)}$ and thus $H_n[V_1] \nrightarrow (K_3).$ Theorem \[vertexRamseyK4\] ensures ${G(n,p)\nrightarrow (K_4)^v}$ and thus $H_n[V_2]\nrightarrow (K_4)^v$. Finally, by Theorem \[subgraphthreshold\] $G(n,p)$ (and thus $H_n[V_2]$) a.a.s. does not contain any of the finitely many graphs $H$ with $\rho(H)\geq 2$ and $|V(H)|\leq 8$ as a subgraph. Hence the theorem follows from Lemma \[basicLemmaK\_4\]. It remains to prove Lemma \[basicLemmaK\_4\]. We let $${\mathcal{C}_G=\lbrace M \subseteq V^{(4)}\, |\, G[M] \text{ is a copy of }K_{4}\rbrace }$$ be the family of vertex subsets inducing a copy of $K_4$ in a graph $G$. We will define a red-blue-colouring $\phi$ of $E(G)$ in three steps. By assumption, there is a colouring $\phi_1$ of $E(V_1)$ which yields no monochromatic copies of $K_3$. There also exists a vertex colouring of $V_2$ that does not admit monochromatic copies of $K_4$. Let $U \subseteq V_1$ be one of its two colour classes, then $G[U]$ is $K_4$-free and for any $L \in \mathcal{C}_{G[V_2]}$ we have $U \cap L\neq \varnothing$. For any such $L$ we choose an arbitrary vertex $a_L \in U \cap L$ and an arbitrary edge $e_L \in E(L)$ incident with $a_L$. Let $$E_s=\bigcup\limits_{L \,\in \, \mathcal{C}_{G[V_2]}} e_L.$$ Now we define a colouring $\phi_2$ of $E(V_2)$ as follows. For $e=vw \in E(V_2)$ we let $$\begin{aligned} \phi_2(e)= \begin{cases} \text{red, if } v,w \in U \lor e \in E_s \\ \text{blue, otherwise} \end{cases}\end{aligned}$$ Finally, for an edge $e=av$ with $a \in V_1$, $v \in V_2$ define $$\begin{aligned} \phi_3(e)= \begin{cases} \text{blue, if } v \in U\\ \text{red, otherwise} \end{cases}\end{aligned}$$ We claim that $\phi=\phi_1 \cup \phi_2 \cup \phi_3$ does not yield any monochromatic copy of $K_4$. Suppose first that $\phi$ admits a blue copy $L$ of $K_4$. Then, owing to our choice of $\phi_1$, we have $|L\cap V_1| \leq 2$. On the other hand, $|L\cap V_2| \leq 3$, since $L \subseteq V_2$ would imply that $e_L \in E(L)$ is a red edge. It follows from the two above observations that there are two distinct vertices $v,w \in L \cap V_2$ and $a \in L \cap V_1$. Since $av$ and $aw$ are blue, it follows from the defintion of $\phi_3$ that $v,w \in U$. However, this means that $vw$ is red in $\phi_2$, yielding a contradiction. Let us now suppose that $L \in \mathcal{C}_G$ is a red copy. Again, we have $|L\cap V_2| \geq 2$ due to our choice of $\phi_1$. In case of $|L\cap V_2| \leq 3$ we again find two distinct vertices $v,w \in L\cap V_2$ and $a \in L \cap V_1$. Since $vw$ is red, by the defintion of $\phi_2$, at least one of the vertices $v$ and $w$ is in $U$, and thus either $va$ or $wa$ is blue. It remains to check the case $|L\cap V_1| =4$. Recall that there are no copies of $K_4$ in $G[U]$. Therefore, there exists a vertex $z \in L\setminus U$. Hence, all edges in $E(L)$ which are incident with $z$ were chosen to be $e_{L'}$ for some $L' \in \mathcal{C}_{G[V_2]}$. This yields two further sets $L_1, L_2 \in \mathcal{C}_{G[V_2]}$ with $|L \cap L_i| \geq 2$ for $i=1,2$ and $L \neq L_1 \neq L_2 \neq L$. We will show that for $M= L \cup L_1 \cup L_2$ we have $$\begin{aligned} \label{EM2M} |E(M)|\geq 2|M|,\end{aligned}$$ contradicting the last assumption of the lemma because of $|M| \leq 8$. Let $$\begin{aligned} H=G[L], \, H_1=G[L] \cup G[L_1], \, H_2=H_1 \cup G[L_2], \, s_1=|L_1 \setminus L|, \, s_2=|L_2 \setminus (L \cup L_1)|\end{aligned}$$ and similarly $$t_1=|E(H_1) \setminus E(H)|, \, t_2=|E(H_2) \setminus E(H_1)|.$$ Let us prove that $$\begin{aligned} \label{tisi} t_i\geq 2 s_i+1 \text{ for } i \in \{1,2\}.\end{aligned}$$ This inequality can be obtained immediately for any $i$ with $s_i \in \{1,2\}$ by counting the edges incident to the vertices counted by $s_i$. We have $0 \leq s_1,s_2\leq 2$ and $s_1 \neq 0$, hence for the proof of it is left to check $i=2$ in case of $s_2=0$. It is easy to confirm that $\mathcal{C}_{H_1}=\{L,L_1\}$, thus $L_2$ induces an edge $e \notin E(H_1)$, i.e. $t_2\geq 1=2s_2+1$. This completes the proof of which yields due to $$|E(M)|=6+t_1+t_2\geq 6+ 2 s_1+1 + 2s_2 +1=2(4+s_1+s_2)=2|M|.$$ Acknowledgements {#acknowledgements .unnumbered} ================ I am very grateful to my supervisor Mathias Schacht for his time and encouragement. I would like to thank Shagnik Das and Andrew Treglown for sharing a simplification that strengthened part \[0statementK4\] of Theorem \[K4thm\]. [^1]: A large part of this work forms the Bachelor thesis of the author which was supervised by Mathias Schacht (schacht@math.uni-hamburg.de) and handed in at the Department of Mathematics at the University of Hamburg in March 2018. While we prepared this manuscript, Das and Treglown [@DasTreglown] independently obtained similar and more general results (see Section \[ourresults\])

--- abstract: 'We show that Sarnak’s conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.' author: - Jon Chaika - Alex Eskin title: Mobius disjointness for interval exchange transformations on three intervals --- Introduction ============ Let $\mu: {{{\mathbb}N}}\to \{-1, 0, 1\}$ denote the Möbius function, namely, $\mu(n) = 0$ if $n$ is not square-free, $\mu(n) = 1$ if $n$ is square-free and has an even number of prime factors, and $\mu(n) = -1$ if $n$ is square-free and has an odd number of prime factors. Let $X$ be a topological space, and let $T: X \to X$ be an invertible map. We think of the map $T$ as a dynamical system. Peter Sarnak made the following far-reaching conjecture: \[conj:Sarnak\] Suppose the topological entropy of $T$ is $0$. Then, for any $x \in X$, and any (continuous) function $f: X \to {{{\mathbb}R}}$, $$\label{eq:Mobius:disjointness} \lim_{N \to \infty} \frac{1}{N} \sum_{n=0}^{N-1} f(T^n x) \mu(n) = 0.$$ \[def:IET\] An interval exchange transformation (IET) is given by a vector $\vec{\ell} = (\ell_1, \dots, \ell_d) \in {{{\mathbb}R}}^d_+$ and a permutation $\pi$ on $\{ 1, \dots, d \}$. From $\vec{\ell}$ we obtain $d$-subintervals of $[0, \sum_{i=1}^d \ell_i)$ as follows: $$I_1 = [0,\ell_1), \quad I_2 = [\ell_1, \ell_1 + \ell_2), \dots, I_d = [\sum_{i=1}^{d-1} \ell_d, \sum_{i=1}^d \ell_d).$$ Now we obtain a $d$-Interval Exchange Transformation $T = T_{\pi, \vec{\ell}} :[0, \sum_{i=1}^d \ell_i) \to [0, \sum_{i=1}^d \ell_i)$ which exchanges the intervals according to $\pi$. More precisely, if $x \in I_j$, then $$T(x) = x - \sum_{k < j} \ell_k + \sum_{\pi(k) < \pi(j)} \ell_k.$$ It is well known that the topological entropy of any interval exchange transformation is $0$. Thus, if Conjecture \[conj:Sarnak\] is true, then (\[eq:Mobius:disjointness\]) should hold for any interval exchange transformation. In this paper, we consider only the case $d=3$. Extending our results e.g. to $d=4$ will require fundamental new ideas. \[lemma:3iet:to:rotation\] If $T$ is a $3$-IET with permutation $\begin{pmatrix}1&2&3\\ 3 & 2 & 1\end{pmatrix}$, then $T$ is also the induced map of a rotation on an interval. Let $\hat{R}: [0, \ell_1 + 2\ell_2 + \ell_3) \to [0,\ell_1 + 2 \ell_2 + \ell_3)$ be given by $$\hat{R}(x) = \begin{cases} x + \ell_2 + \ell_3 & \text{if $x \le \ell_1 + \ell_2$} \\ x + \ell_2 + \ell_3 - (\ell_1 + 2 \ell_2 + \ell_3) & \text{otherwise.} \end{cases}$$ Then $\hat{R}$ is a $2$-IET (hence a rotation), and the induced map of $\hat{R}$ on the interval $[0,\ell_1+\ell_2+\ell_3)$ is $T$. [[**The maps $T$ and $R$, the number $\alpha$ and the interval $J$.** ]{}]{} Let $R: [0,1) \to [0,1)$ denote rotation by $\alpha$ (i.e. $R(x) = x+\alpha$ mod $1$). Let $J = [0,z)$ be a subset of $[0,1]$. In the rest of the paper, we assume that the $3$-IET $T$ is the induced map of $R$ to $J$ and that $x \notin J$ implies $Rx \in J$. [[**The numbers $a_k$, $p_k$ and $q_k$.** ]{}]{} Let $a_0, a_1, \dots, $ denote the continued fraction expansion of $\alpha$. Let $p_k/q_k$ denote the continued fraction convergents of $\alpha$. Then, $$q_{k+1} = a_{k+1} q_k + q_{k-1}.$$ [[**Connection to tori and tori with marked points:** ]{}]{} Let ${{\mathcal M}}_1$ denote the space of flat tori of area $1$. The space ${{\mathcal M}}_1$ admits a transitive action by the Lie group $SL(2,{{{\mathbb}R}})$. Let $\hat{Y} \in {{\mathcal M}}_1$ denote the square torus. Then, the stabilizer of $\hat{Y}$ is $SL(2,{{{{\mathbb}Z}}})$, and thus ${{\mathcal M}}_1$ can we identified with $SL(2,{{{\mathbb}R}})/SL(2,{{{{\mathbb}Z}}})$. Under this identification, a torus with a fundamental domain the parallelogram whose vertices are the points $0$, $v_1$, $v_2$ and $v_1 + v_2$ corresponds to the coset $M SL(2,{{{{\mathbb}Z}}})$ where $M \in SL(2,{{{\mathbb}R}})$ is the matrix whose columns are $v_1$ and $v_2$. The $SL(2,{{{\mathbb}R}})$ action on ${{\mathcal M}}_1$ coincides with the left multiplication action on $SL(2,{{{\mathbb}R}})/SL(2,{{{{\mathbb}Z}}})$. Let ${{\mathcal M}}_{1,2}$ denote the space of tori with two marked points. This space also admits an action by $SL(2,{{{\mathbb}R}})$. If $g \in SL(2,{{{\mathbb}R}})$ and $X \in {{\mathcal M}}_{1,2}$ is the torus with fundamental domain the parallelogram with vertices $0$, $v_1$, $v_2$ and $v_1 + v_2$ and with the marked points $p_1$, $p_2$, then $g X$ is the torus with fundamental domain the parallelogram with vertices $0$, $g v_1$, $g v_2$ and $g(v_1 + v_2)$, and with the marked points $g p_1$ and $g p_2$. Recall that $R:[0,1] \to [0,1]$ denotes the rotation by $\alpha$. Let $\hat{X}=\begin{pmatrix} 1&-\alpha\\0&1 \end{pmatrix} \hat{Y} \in {{\mathcal M}}_1$. Observe that the first return of the vertical flow on $\hat{X}$ to the horizontal side coincides with $R$. If $T$ is a 3-IET given by the induced map of $R$ to an interval $J=[0,z)$ then $T$ is also the first return of the vertical flow on $\hat{X}$ to a horizontal segment of length $|J|$. Let $X$ denote the torus $\hat{X}$ with two marked points, one at each endpoint of the horizontal segment of length $J$. Let $g_{t} = \begin{pmatrix} e^{t} & 0 \\ 0 & e^{-t} \end{pmatrix} \in SL(2,{{{\mathbb}R}})$. We refer to the action of the $1$-parameter subgroup $g_t$ as the geodesic flow on ${{\mathcal M}}_1$ (or ${{\mathcal M}}_{1,2}$). The action of $g_t$ on both ${{\mathcal M}}_1$ and ${{\mathcal M}}_{1,2}$ is ergodic. [[**Renormalization.** ]{}]{} We will need to put a diophantine condition on the IET $T$. In terms of $X \in {{\mathcal M}}_{1,2}$, we want the geodesic ray $\{ g_t X {\;\: : \;\:}t > 0 \}$ to spend significant time in compact subsets of ${{\mathcal M}}_{1,2}$. Directly in terms of the IET data, our conditions are the following: ASSUMPTIONS: There exist constants $C_1,C_2,C_3,C_4 >1$ such that the following holds: Suppose $\ell \in {{{\mathbb}N}}$ and $0<\eta$ are small enough. Then there exists $\hat{c}_\eta>0$ and $c_\ell$ (depending on $\eta$ and $\ell$ respectively) so that for every $0<c<c_\ell$ there exists a constant $k_c \in {{{\mathbb}N}}$ and infinite sequences $L_i,k_i$ so that: 1. $q_{k_i-1}<c{L_i}<q_{k_i}$. 2. $a_{k_i}<C_1$, 3. $a_{k_i+1}<C_2$ and 4. $a_{k_i+2}<C_3$. 5. The shortest vertical trajectory on the torus from one marked point to a ${\| q_k\alpha\|}$ neighborhood of the other has length at least $\frac{q_k}{C_4}$. \[Separated\] 6. There exists $u_i$ so that either $$\lambda(\psi_{L_i}^{-1}(u_i))>\hat{c}_{\eta} \text{ and } \lambda(\psi_{L_i}^{-1}((-\infty,u_i)))<\eta \hat{c}_\eta$$ or $$\lambda(\psi_{L_i}^{-1}(u_i))>\hat{c}_\eta \text{ and } \lambda(\psi_{L_i}^{-1}((u_i,\infty)))<\eta \hat{c}_\eta$$ where $\psi_r(x) = \sum_{\ell=0}^{r -1} \chi_J(R^\ell x)$ and our 3-IET is the first return map of $R$ to $J$. 7. $\underset{i \to \infty}{\lim}\, \int_0^1d(R^{L_i}x,x)d\lambda=0$. 8. We have $L_i>{q_{k_{i}+\ell}}$. 9. We have $\max \{j:\psi_{L_i}^{-1}(j)\neq \emptyset\}- \min \{j:\psi_{L_i}^{-1}(j)\neq \emptyset\}<k_c.$ 10. There exists $v_i$ so that $q_{v_i}\leq L_i<q_{v_i+1}$ and either $L_i=q_{v_i}$ or $a_{v_i+1}>4$ and $L_i=pq_{v_i}$ where $p\leq \lfloor \frac{a_{v_i+1}}4\rfloor$. Our main result is the following: \[theorem:3iet:mobius:disjoint\] Suppose $T$ is a $3$-IET satisfying the assumptions (A0)-(A9). Then, Möbius disjointness, i.e. (\[eq:Mobius:disjointness\]) holds. Assumptions (A0)-(A9) are reasonable in view of the following: \[prop:good:assump\] Let $X \in \mathcal{M}_{1,2}$. Let $\nu_T$ be the measure on $\mathcal{M}_{1,2}$ given by $\int f d\nu_T=\frac 1 T\int_0^T f(g_tX)dt$ for all $f \in \mathcal{C}_c(\mathcal{M}_{1,2})$. If there exists a weak-\* limit of $\nu_T$ that is not the zero measure then the corresponding 3-IET satisfies assumptions (A0)-(A9). Proposition \[prop:good:assump\] is proved in §\[sec:renorm\]. From the Birkhoff ergodic theorem and Proposition \[prop:good:assump\] it is clear that almost all $3$-IET’s satisfy the assumptions (A0)-(A9). Thus, an immediate corollary of Theorem \[theorem:3iet:mobius:disjoint\] is the following: \[cor:almost:all:mobius\] For almost all 3-IET’s, Möbius disjointness (i.e. (\[eq:Mobius:disjointness\]) holds. [[**Disjointness.** ]{}]{} As in e.g. [@Borgain:Sarnak:Ziegler] we derive the Möbius disjointness result from a result about joinings of powers of $T$. In fact, we prove the following: \[theorem:disjoint:powers\] If $T$ is a 3-IET that satisfies assumptions (A0)-(A9) then there exists $\kappa>1$ so that for all $n>0$, $B_n=\{m<n:T^m \text{ is not disjoint from }T^n\}$ has the property that $m_1<m_2 \in B_n$ then $\frac{m_2}{m_1}>\kappa$. See Appendix \[sec:appendix:A\] for a proof that Theorem \[theorem:disjoint:powers\] implies Theorem \[theorem:3iet:mobius:disjoint\]. This is a straightforward modification of a note of Harper [@Harper:note]. It is included for completeness. In Appendix \[sec:appendix:B\], we prove that for almost every 3-IET, $T$, $T^n$ is disjoint from $T^m$ for all $0<n<m$. This gives an alternative (and much easier) proof of Corollary \[cor:almost:all:mobius\]. However, the proof in Appendix \[sec:appendix:B\] does not give a useful diophantine condition under which Möbius disjointness holds. [[**Related work:** ]{}]{} Möbius disjointness has been shown for a variety of systems see for example [@ELR], [@GT] and [@W] among others. Most closely related to this work is [@D], where Vinogradov’s circle method is used to prove that every rotation (2-IET) is disjoint from Möbius; [@Bou] which shows a set of 3-IETs satisfying a certain measure 0 condition are disjoint from Möbius and [@Borgain:Sarnak:Ziegler] where a slightly stronger version of our criterion is introduced to show that the time 1 map of horocycle flows are disjoint from Möbius. This last paper motivated our approach. [[**Further questions and conjectures.** ]{}]{} What is the Hausdorff codimension of the set of $X\in \mathcal{M}_{1,2}$ so that any weak-\* limit point of $\nu_T$ is the zero measure? For almost every IET that is not of rotation type and $n<m\in \mathbb{N}$ we have $T^n$ is disjoint from $T^m$. In fact if $U_T$ is the unitary operator associated to (composition with) $T$ on $L^2$ function of inttegral zero then there is a sequence $k_1,...$ so that $U_{T^{nk_i}}$ converges to the 0 operator in the weak operator topology and $U_{T^{mk_i}}$ converges to the identity operator in the strong operator topology. [[**Outline:** ]{}]{} The Section 2 establishes an abstract disjointness criterion, Proposition \[prop:fact:qfact\]. Sections 4 uses this to prove Theorem \[theorem:disjoint:powers\]. Section 3 recalls standard facts about rotations used in Section 4. Section 5 proves Proposition \[prop:good:assump\]. Appendix A proves that Theorem \[theorem:disjoint:powers\] implies Theorem \[theorem:3iet:mobius:disjoint\]. Appendix B proves that almost every 3-IET has the property that all of its distinct positive powers are disjoint. [[**Acknowledgments:** ]{}]{} J. C. was supported in part by NSF grants DMS-135500 and DMS-1452762 and the Sloan foundation. A. E. is supported in part by NSF grant DMS 1201422 and the Simons Foundation. The authors thank Adam Harper for graciously letting us modify his note and a helpful correspondence. A. E. thanks Princeton University and the Institute for Advanced Study for support during part of this work. The authors would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the program Dynamics of Group Actions and Number Theory where work on this paper was undertaken. This work was supported by EPSRC grant no EP/K032208/1. Disjointness criterion {#sec:criterion} ====================== Let $(X,d)$ be a metric space. We set $X_1 = X_2 = X$, and write the product $X {\times}X$ as $X_1 {\times}X_2$. Let $\lambda$ be a measure on $X$, and let $T_1: X_1 {\times}X_1$ and $T_2: X_2 \to X_2$ be $\lambda$-preserving maps. Let $\sigma$ be a joining of $(X_1,T_1, \lambda)$ and $(X_2, T_2,\lambda)$, i.e. $\sigma$ is an ergodic $T_1 {\times}T_2$-invariant measure on $X_1 {\times}X_2$ which projects to $\lambda$ in either factor. Our basic strategy is due to Ratner [@Ratner:Joinings]. In fact, we use the following proposition: \[prop:Ratner:disjointness\] Suppose $S: X \to X$ is a $\lambda$-preserving map which commutes with $T_1$ and $T_2$. Suppose $d_1 \ge 0$, $d_2 > 0$, and for every $\delta > 0$ and any compact set $K \subset X_1 {\times}X_2$ with $\sigma(K) > 1-\delta$ and for every $\delta > \epsilon > 0$ there exist points $(x,y) \in X_1 {\times}X_2$, $(x', y') \in X_1 {\times}X_2$ and $r \in {{{\mathbb}N}}$ so that the following conditions hold: - $(T_1{\times}T_2)^r(x',y') \in K$. - $(T_1{\times}T_2)^r(x,y) \in K$. - $d(T_1^{r} x', S^{d_1} T_1^{r} x) + d(T_2^{r} y', S^{-d_2} T_2^{r} y) < \epsilon$. Then, $\sigma$ is $S^{d_1} {\times}S^{-d_2}$-invariant. Suppose $\sigma$ is not $S^{d_1} {\times}S^{-d_2}$-invariant. Then $(S^{d_1} {\times}S^{-d_2})\sigma$ is an ergodic $T_1 {\times}T_2$-invariant measure which is distinct from $\sigma$. Thus, $(S^{d_1} {\times}S^{-d_2})\sigma$ and $\sigma$ are mutually singular. It follows that for any $\delta > 0$ there exists a compact set $K$ with $\sigma(K) > 1-\delta$ such that $(S^{d_1} {\times}S^{-d_2}) K \cap K = \emptyset$. Then there exists $\epsilon$ such that $$d((S^{d_1} {\times}S^{-d_2}) K, K ) > \epsilon.$$ This is not consistent with conditions (a)-(c). \[prop:fact:qfact\] Suppose $S$ is continuous except for finitely many points, and suppose $\lambda$ gives zero measure to the points of discontinuity of $S$. Assume 1. There exists a sequence of measurable partitions of $X_1$, $U^{(i)}_{-k},...,U^{(i)}_k$ and a sequence of numbers $r_i$ so that $$\underset{i \to \infty}{\lim}\, \int_{U_j^{(i)}}d(T_1^{r_i}(x),S^jx)\, d\lambda(x)=0.$$ 2. There exists $\ell \in \{1,2,3\}$, a sequence of measurable sets $A_i$, and functions $F_i$ preserving the measure $\lambda$, so that 1. $\underset{i \to \infty}{\lim}\int_{A_i}d(F_iy,y)\,d\lambda(y)=0$. 2. $\underset{i \to \infty}{\lim}\int_{A_i}d(S^{-\ell}(T_2^{r_i}y),T_2^{r_i}F_iy)\, d\lambda(y)=0$.\ This implies 3. $\underset{i \to \infty}{\lim}\int_{F_i(A_i)}d(F_i^{-1}y,y)\,d\lambda(y)=0$. 4. $\underset{i \to \infty}{\lim}\int_{F_i(A_i)}d(S^{\ell}(T_2^{r_i}y),T_2^{r_i}F_i^{-1}y)\, d\lambda(y)=0$. 3. There exists an absolute constant $\delta_0 > 0$ such that the following holds: for any $0 < \delta < \delta_0$, either there exists $a \in {{{{\mathbb}Z}}}$ so that for infinitely many $i$, $$\label{eq:a:is:good} \sigma(\{(x,y):x\in U_{a}^{(i)} \text{ and } y \in A_i\})> 27 \delta + 14 \lambda( \bigcup_{\ell < a} U_\ell^{(i)})$$ or there exists $a' \in {{{{\mathbb}Z}}}$ so that for infinitely many $i$, $$\label{eq:a:prime:is:good} \sigma(\{(x,y):x\in U_{a'}^{(i)} \text{ and } y \in F_i(A_i)\})> 27 \delta + 14 \lambda(\bigcup_{\ell>a'}U_\ell^{(i)}).$$ Under the assumptions (1)-(3) with (\[eq:a:is:good\]) there exists $d\geq 0$ so that $\sigma$ is $S^d\times S^{-\ell}$ invariant. Also, under the assumptions (1)-(3) with (\[eq:a:prime:is:good\]) there exists $d\leq 0$ so that $\sigma$ is $S^d\times S^{\ell}$ invariant. \[lemma:find:friend\] Suppose $\epsilon' > 0$ and $\delta > 0$. Then, for any compact set $K \subset X_1 {\times}X_2$ with $\sigma(K) > 1-\delta$ and all $i \in {{{\mathbb}N}}$ sufficiently large, there exists a compact set $K_i' \subset X_1 {\times}X_2$ with $\sigma(K_i') > 1- 7 \delta$ such that for all $(x,y) \in K_i'$ with $y \in A_i$, there exists $x' \in X_1$ with $(x', F_i y) \in K$ and $d(x', x ) < \epsilon'$. Similarly, for all $(x,y) \in K_i'$ with $y \in F(A_i)$, there exists $x' \in X_1$ with $(x',F_i^{-1} y) \in K$ and $d(x',x) < \epsilon'$. Define a probability measure $\tilde{\sigma}$ on $X_1 {\times}X_2$ by $$\tilde{\sigma}(E) = \frac{\sigma(E \cap K)}{\sigma(K)}.$$ For $y \in X_2$, let $\tilde{\sigma}_y$ be the conditional measure of $\tilde{\sigma}$ along $X_1 {\times}\{ y\}$. Let $B(x,\epsilon')$ denote the open ball of radius $\epsilon'$. For $\tilde{\sigma}$- almost all $(x,y) \in X_1 {\times}X_2$, $\tilde{\sigma}_y(B(x,\epsilon'/2)) > 0$. Therefore, there exists $\rho(\epsilon',\delta) > 0$ and a set $K_1 \subset K$ with $\tilde{\sigma}(K_1) > 1-\delta$ such that for all $(x,y) \in K_1$, $$\tilde{\sigma}_y(B(x,\epsilon'/2)) > \rho(\epsilon',\delta).$$ Let $\pi_2: X_1 {\times}X_2 \to X_2$ denote projection to the second factor. Since the function $y \to \tilde{\sigma}_y$ is measurable, by Lusin’s theorem there exists a compact set $K_2 \subset X_2$ with $\pi_2^*(\tilde{\sigma})(K_2) > 1 - \delta$ on which it is uniformly continuous relative to the Kantorovich-Rubinstein metric, $$d(\mu,\nu) = \sup_{f} \left|\int_{X_1} f \, d\nu -\int_{X_1} f \, d\nu \right|,$$ where the sup is taken over all $1$-Lipshitz functions $f: X_1 \to {{{\mathbb}R}}$ with $\sup |f(x)| \le 1$. Then, there exists $\delta' > 0$ such that for all $y,y' \in K_2$ with $d(y',y) < \delta'$ and for all $x \in X_1$ such that $(x,y) \in K_1$ $$\tilde{\sigma}_{y'}(B(x,\epsilon')) > \rho(\epsilon',\delta)/2.$$ Then, for all $y,y' \in K_2$, with $d(y,y') < \delta$ and all $x$ with $(x,y) \in K_1$, $$\label{eq:sigma:y:prime:B:x:epsilon} \sigma_{y'}(B(x,\epsilon') \cap K) > 0.$$ We now estimate $\lambda(K_2)$. For $E \subset X_2$, $$\pi_2^*\tilde{\sigma}(E) = \tilde{\sigma}(\pi_2^{-1}(E)) = \frac{\sigma(\pi_2^{-1}(E) \cap K)}{\sigma(K)} \le \frac{\sigma(\pi_2^{-1}(E))}{1-\delta} = \frac{\lambda(E)}{1-\delta}.$$ Therefore, $$\lambda(K_2) \ge (1-\delta) \pi_2^*\tilde{\sigma}(K_2) \ge (1-\delta)^2 \ge 1 - 2 \delta.$$ By condition (2a) in Proposition \[prop:fact:qfact\], for $i$ sufficiently large, there exists a compact set $K_{3,i} \subset X_1 {\times}X_2$ with $\sigma(K_{3,i}) > 1-\delta$ such that for $(x,y) \in K_{3,i}$, $y \in A_i$ and $i$ sufficiently large, $$\label{eq:T2Liy:close:to:y} d(F_i y, y) < \delta'.$$ Now let $$K'_i = (id \times F_i)^{-1}\left(K_1 \cap (X_1 {\times}K_2) \right) \cap K_{3,i} \cap (X_1 {\times}K_2).$$ Then, $$\sigma(K'_i)> 1 - 7 \delta.$$ Suppose $(x,y) \in K'_i$, with $y \in A_i$. For large enough $i$, (\[eq:T2Liy:close:to:y\]) holds, and also $(x,y) \in K_1 {\times}K_2$ and $F_i y \in K_2$. Thus (\[eq:sigma:y:prime:B:x:epsilon\]) holds (with $y' = F_i y$). This implies the first statement of the lemma. The proof of the second statement is identical. We establish the (\[eq:a:is:good\]) case. The (\[eq:a:prime:is:good\]) case is analogous. The basic strategy is to choose $(x,y) \in U_a^{(i)} {\times}A_i$, and apply Proposition \[prop:Ratner:disjointness\] with $r=r_i$ to the points $(x,y)$ and $(x',F y)$, where $x'$ is as in Lemma \[lemma:find:friend\]. We now give the details. Suppose $\delta > 0$ and $0 < \epsilon < \delta$ are arbitrary. Let $\Delta$ denote the union of the points of discontinuity of $S^j$, $1 \le j \le k$. There exists $c_1(\epsilon) > 0$ such that if we let $$K_0 = \{ x \in X_1 {\;\: : \;\:}d(x,\Delta) > c_1(\epsilon) \}$$ then $\lambda(K_0) > 1-\epsilon > 1 - \delta$. Let $$K_{00} = \{ x \in X_1 {\;\: : \;\:}d(x,\Delta) > c_1(\epsilon)/2 \}.$$ Since $K_{00}$ is compact and $S$ is continuous on $K_{00}$, there exists $\epsilon' > 0$ such that if $x_1, x_2 \in K_{00}$, with $d(x_1,x_2) < \epsilon'$ then for all $1 \le j \le k$, $d(S^j x_1, S^j x_2) < \epsilon/6$. Without loss of generality, we may assume that $\epsilon' < c_1(\epsilon)/2$. Then, we have, for all $1 \le j \le k$, $$\label{eq:S:uniformly:cts} d(S^{j} x, S^{j} x') < \epsilon/6 \qquad\text{if $x \in K_0$ and $d(x',x) < \epsilon'$}.$$ Let $a$ be as in Proposition \[prop:fact:qfact\] (3). Write $$\gamma = \lambda(\bigcup_{\ell < a} U_\ell^{(i)})$$ We may assume that $i$ is large enough so that there exists a compact set $$K_{1b} \subset X_1 \setminus \bigcup_{\ell < a} U_\ell^{(i)}$$ with $\lambda(K_{1b}) > 1- 2\gamma$. In view of assumption (1) of Proposition \[prop:fact:qfact\], there exists a compact set $K_{1a} \subset X_1$ with $\lambda(K_{1a}) > 1-\delta$ such that $$d(T_1^{r_i} x, S^j x) < \epsilon/6 \qquad\text{ for $x \in U^{(i)}_j$.}$$ In view of assumption (2b) of Proposition \[prop:fact:qfact\], there exists a compact set $K_{2b} \subset X_2$ with $\lambda(K_{2b}) > 1-\delta$ such that for $y \in K_{2b} \cap A_i$ and $i \in {{{\mathbb}N}}$ sufficiently large, $$|S(T_2^{r_i}y)-T_2^{r_i}F_iy| < \frac{\epsilon}{2}.$$ As in the proof of Proposition \[prop:Ratner:disjointness\], let $K$ be a compact set so that $\sigma(K)>1-\delta$ and $(T_1^{d}\times T_2^{-\ell})K$ are compact and disjoint for all $0\leq d\leq k$ and $0\leq \ell\leq 3$. Formally, $K$ may depend on $d$, but without loss of generality we may assume that the same $K$ works for all $0 \le d \le k$. Let $$K''_i = ((K_{1a} \cap K_{1b}) {\times}X)\cap (T_1{\times}T_2)^{-r_i} K$$ Note that $\sigma(K''_i) > 1 - 3 \delta - 2\gamma$. Let $K'_i$ be as in Lemma \[lemma:find:friend\] for $K''_i$ instead of $K$. We have $$\sigma(K'_i) > 1 - 21 \delta - 14 \gamma.$$ Let $$\Omega= (T_1{\times}T_2)^{-r_i}(K \cap (K_0 {\times}K_0)) \cap K'_i \cap (X_1 {\times}K_{2b}) \cap (K_0 {\times}K_0).$$ Then, $$\sigma(\Omega) \ge 1 - 3\delta - (21 \delta + 14 \gamma) - \delta - 2\delta = 1 - 27 \delta - 14 \gamma.$$ Let $$G_i = \{(x,y):x\in U_{a}^{(i)} \text{ and } y \in A_i\}.$$ By (\[eq:a:is:good\]), $\sigma(\Omega \cap G_i) > 0$. Now let $(x,y)$ be any point in $\Omega \cap G_i$. Then, by Lemma \[lemma:find:friend\], there exists $x' \in K_{1a} \cap K_{1b}$ with $(x',F_i y) \in (T_1{\times}T_2)^{-r_i} K$. Because $$(x',y') = (x', F y) \in ((K_{1a} \cap K_{1b}) {\times}X)\cap (T_1{\times}T_2)^{-r_i} K$$ letting $r=r_i$ conditions (a) and (b) of Proposition \[prop:Ratner:disjointness\] hold. Also, since $x' \in K_{1b}$, $x' \not\in \bigcup_{\ell < a} U_a^{(i)}$, and thus we may assume $x' \in U_b^{(i)}$ for some $b \ge a$. Then, since $x' \in K_{1a}$, $$d(T_1^{r_i} x', S^{b} x') < \epsilon/6.$$ Also, in view of Lemma \[lemma:find:friend\], $$d(x', x) < \epsilon',$$ and since $x \in K_{1a}$, $$d(T_1^{r_i} x, S^a x) < \frac \epsilon 6.$$ We have $x \in K_0$ and $T_1^{r_i} x \in K_0$. Therefore, by (\[eq:S:uniformly:cts\]), $$\begin{gathered} d(T_1^{r_i} x', S^{b-a} T_1^{r_i} x) \le d(T_1^{r_i} x', S^{b} x') + d(S^{b} x', S^{b} x) + d(S^{b} x, S^{b-a} T_1^{r_i} x) \\ < \frac{\epsilon}{3} + \frac{\epsilon}{6} + \frac{\epsilon}{6} = \frac{\epsilon}{2}.\end{gathered}$$ Similarly, $$d(T_2^{r_i} y', S^{-r} T_2^{r_i} y) = d(T_2^{r_i} F_iy, S^{-r} T_2^{r_i} y) < \frac{\epsilon}{2}.$$ Therefore, assumption (c) in Proposition \[prop:Ratner:disjointness\] also holds with $d_1=b-a$ and $d_2=\ell$, and Proposition \[prop:Ratner:disjointness\] can be applied. Let $\lambda$ denote Lebesgue measure on $[0,1]$. Let $S: [0,1] \to [0,1]$ be a 3-IET, $T_1=S^n$ and $T_2=S^m$. Let $\sigma$ be an ergodic joining of $T_1$ and $T_2$. \[cor:trivial:joining\] If $S$ is weakly mixing and the conditions of Proposition \[prop:fact:qfact\] are satisfied then $\sigma = \lambda {\times}\lambda$. Note that by [@BN] all the 3-IETs we consider are weakly mixing. This corollary uses the following standard result: (See for example [@Rudolph:book Lemma 6.14].) \[lemma:rudolph\] If $(X,B,\mu,T)$ is ergodic and $\sigma$ is a joining of $(X,M_1,\mu,S_1)$ and $(Y,M_2,\nu_2,S_2)$ that is $T \times id$ invariant then $\sigma =\nu_1\times \nu_2$. We include a proof because the statement in [@Rudolph:book] is slightly more specific. Given $A \subset Y$ with positive $\nu_2$ measure let $\sigma_A(B)=\sigma (B \times A)$. This is a measure on $X$. Because $\sigma$ has marginals $\mu,\nu_2$ this measure is absolutely continuous with respect to $\mu$. So it has a Radon-Nikodym derivative $f_A$. By our assumption this is a $T$ invariant function and so it is constant. This implies any two rectangles with the same dimensions have the same measure and thus $\sigma$ is the product measure. We show only the (\[eq:a:is:good\]) case, since the (\[eq:a:prime:is:good\]) case is similar. Because $\sigma$ is $S^{d_1} \times S^{-r}$ invariant it is $S^{md_1}\times S^{-m}$ invariant. Using the fact that $\sigma$ is a joining of $S^n$ and $S^m$, this implies that $\sigma$ is $S^{md_1+nr}\times id$ invariant. Since $S$ is weak mixing and thus totally ergodic, $S^{md_1+nr}$ is ergodic and so by Lemma \[lemma:rudolph\], $\sigma=\lambda {\times}\lambda$. In §\[sec:facts:rotation\]-§\[sec:work\] we will show that Proposition \[prop:fact:qfact\] can be applied to prove Theorem \[theorem:disjoint:powers\]. Facts about rotations {#sec:facts:rotation} ===================== Let $\|q_k\alpha\|=dist(q_k\alpha,\mathbb{Z})$. Let $\lambda$ be Lebesgue measure. $d(x,y)=\min\{|x-y|,1-|x-y|\}$ \[lemma:q:k:plus:1:ak:qk\] $q_{k+1}=a_{k+1}q_k+q_{k-1}$ \[lemma:good:bound\] $\frac 1 {q_{k+1}+q_k}<\|q_{k}\alpha\|<\frac 1 {a_{k+1}q_k}$ See [@Khinchin:book], 4 lines before equation 34. \[lemma:orbit:dense\] $\{R^ix\}_{i=0}^{q_k-1}$ is $2\|q_{k-1}\alpha\|$ dense for all $x$. Because $R$ is an isometry, it suffices to prove this for $x=0$. We approximate $\{R^n(0)\}_{n=0}^{q_k-1}$ by $\{n\frac{p_k}{q_k} \text{ mod 1}\}_{n=0}^{q_k-1}$, a set that is $\frac 1 {q_k}$ dense. Now $R^\ell(0)$ is within $\|q_k\alpha\|=q_k|\alpha-\frac{p_k}{q_k}|$ of $\ell \frac{p_k}{q_k}$ mod 1 for $0\leq \ell\leq q_k$. Since $\|q_k\alpha\|<\|q_{k-1}\alpha|$ and by Lemma \[lemma:good:bound\] we have $\frac{1}{q_k}<2\|q_{k-1}\alpha\|$. This establishes the lemma. \[lemma:orbit:sep\] $\{R^ix\}_{i=0}^{q_k-1}$ is $\|q_{k-1}\alpha\|$ separated for all $x$. \[lemma:ret:time\] If $x$ is in an interval $I$ of size $\|q_k\alpha\|$ then the return time of $x$ to $I$ is either $q_{k+1}$ or $q_{k+1}+q_k$. If $k$ is even and $I = [-\|q_k \alpha \|,0)$ then the return time of $q_{k+1} + q_k$ takes place on $[-\|q_{k+1} \alpha\|,0)$. If $k$ is odd and $I=[0,\|q_k\alpha\|)$ then the return time of $q_{k+1}+q_k$ takes place on $[\|q_k\alpha\|-\|q_{k+1}\alpha\|,\|q_k\alpha\|)$. First, we assume $k+1$ is odd. If $x\in I$ then $R^{q_{k+1}}x=x-\|q_{k+1}\alpha\|$. So if $x$ is not in the leftmost $\|q_{k+1}\|$ of $I$ then $R^{q_{k+1}}x\in I$. Otherwise, $R^{q_k}x=x+\|q_\alpha\|$ is on the right of $I$ and within $\|q_{k+1}\alpha|$ of $I$. So $R^{q_{k+1}+q_k}x\in I$. The case of $k+1$ even is similar. \[lemma:DK\](Denjoy-Koksma) If $f$ is bounded variation then $|\sum_{i=0}^{q_k-1}f(R^ix)-q_k\int f \, d\lambda|\leq var(f)$. Note if $f$ is the characteristic function of an interval $var(f)=2$. This is the only case we use in the sequel and we present the proof of this case below. A similar argument will be used to prove the more general Lemma \[lemma:happy:times\]. Following the paragraph in the introduction ‘Connection to tori and tori with marked points’ we want to understand the intersections of a (half open) vertical line segment of length $q_k$ to a horizontal line segment of length $z$ on $\hat{X}$, see Figure \[fig:original\]. (Indeed, $R^{q_k}$ is given by a vertical trajectory of length $q_k$ and $\sum_{j=0}^{q_k-1}\chi_J(R^jx)$ is given by the intersection of the corresponding vertical trajectory of length $q_k$ with a horizontal trajectory of length $z$.) This is equivalent to understanding the intersections of a vertical segment of length $1$ to a horizontal line segment of length $q_kz$ on $g_{\log(q_k)}\hat{X}$. Call these segments $\gamma_1$ and $\gamma_2$ respectively, see Figure \[fig:renorm1\]. We close up these two curves as pictured in Figure \[fig:renorm2\] using the following observations:\ 1. Any vertical trajectory of length $q_k$ on $\hat{X}$ has that its endpoints differ by a horizontal vector of length at most $\|q_k\alpha\|<\frac 1{a_{k+1}q_k}.$ This implies we can close $\gamma_1$ up by a horizontal segment, $\zeta_1$, of length less than $\frac 1 {a_{k+1}}\leq1$. Call the resulting closed curve $\hat{\gamma}_1$. 2. We may close up $\gamma_2$ by a vertical segment $\zeta_2'$ of length at most $1$, union a horizontal segment $\zeta_2$ of length at most $\frac 1 2 $ which is either contained in $\gamma_2$ or disjoint from it. Call the resulting closed curve $\hat{\gamma}_2$. Any vertical segment of length 1 on $g_{\log(q_k)}\hat{X}$ is a translate of $\gamma_1$ and so we may close it up so that it is a translate of $\hat{\gamma}_1$. The intersection of any translate of $\hat{\gamma}_1$ with $\hat{\gamma}_2$ is constant (it is a topological invariant of these curves). So now we study the intersection of translates of $\hat{\gamma}_1$ and $\zeta_2\cup \zeta_2'$. $\gamma_1$ can intersect $\zeta_2$ either $0$ and $1$ times. $\gamma_1$ does not intersect $\zeta_2'$ and $\zeta_1$ does not intersect $\zeta_2$. Once again $\zeta_1$ intersects $\zeta_2'$ at most once. To summarize the intersections with $\gamma_2$ of any two translates of $\gamma_1$ differ by at most $2$. ![The torus $\hat{X}$. A vertical segment of length $q_k$ intersects a horizontal slit of length $z$.[]{data-label="fig:original"}](original){width="30.00000%"} ![The torus $g_{\log(q_k)} \hat{X}$: A vertical segment $\gamma_1$ of length $1$ (drawn in red) intersects a horizontal slit $\gamma_2$ of length $q^k z$ (drawn in blue).[]{data-label="fig:renorm1"}](renorm1){width="30.00000%"} ![Closing the curves. We complete the vertical segment $\gamma_1$ to a closed curve $\hat{\gamma_1}$ by adding a horizontal segment $\zeta_1$ (drawn in green). Simularly, we close up the horizontal slit $\gamma_2$ to obtain a closed curve $\hat{\gamma}_2$ by adding in a horizontal segment $\zeta_2$ and a vertical segment $\zeta_2'$ (drawn in purple).[]{data-label="fig:renorm2"}](renorm2){width="30.00000%"} \[lemma:happy:times\] For all $k\in \mathbb{N}$ with $a_{k+1}>4$, and $i \in \mathbb{N}$ with $i \leq \lfloor \frac {a_{k+1}}{4}\rfloor$ we have that there exists $j$ with $\lambda(\psi_{iq_k}^{-1}(j))>\frac 1 {12}$ and either $j-\min\{\ell: \psi_{jq_k}^{-1}(\ell) \neq \emptyset\}\leq 1$ or $\max\{\ell:\psi_{jq_k}^{-1}(\ell)\neq \emptyset\}-j\leq 1$. Moreover $\psi_{iq_k}$ is at most $i+2$ valued. This is similar to the proof of Lemma \[lemma:DK\], but the vertical segment on $\hat{X}$ has length $iq_k$. We once again work on $g_{\log(q_k)}\hat{X}$, where the vertical segment $\gamma_1$ has length $i$, and the slit $\gamma_2$ has length $q_k z$ (See Figure \[fig:renorm2\]). Thus, we need to estimate the number of intersections between $\gamma_1$ and $\gamma_2$. As in the proof of Lemma \[lemma:DK\], we make the following observations (see Figure \[fig:renorm2\]): 1. Any vertical trajectory of length $iq_k$ on $\hat{X}$ has that its endpoints differ by a horizontal vector of length at most $i\|q_k\alpha\|<\frac i{a_{k+1}q_k}.$ This implies we can close $\gamma_1$ up by a horizontal segment, $\zeta_1$, of length less than $\frac i {a_{k+1}}<\frac 1 4 $. Call the resulting closed curve $\hat{\gamma}_1$. 2. We may close up $\gamma_2$ by a vertical segment of length at most $1$, $\zeta_2'$, union a horizontal segment of length at most $\frac 1 2$, $\zeta_2$, which is either contained in $\gamma_2$ or disjoint from it. Call the resulting closed curve $\hat{\gamma}_2$. Any vertical segment of length $i$ on $g_{\log(q_k)}\hat{X}$ is a translate of $\gamma_1$ and so we may close it up so that it is a translate of $\hat{\gamma}_1$. As in the proof of Lemma \[lemma:DK\], the intersection of any translate of $\hat{\gamma}_1$ with $\hat{\gamma}_2$ is constant (it is a topological invariant of these curves). So now we study the intersection of translates of $\hat{\gamma}_1$ and $\zeta_2\cup \zeta_2'$. $\gamma_1$ can intersect $\zeta_2$ between $0$ and $i$ times. $\gamma_1$ does not intersect $\zeta_2'$ and $\zeta_1$ does not intersect $\zeta_2$. Also $\zeta_1$ intersects $\zeta_2'$ at most once. To summarize the intersections with $\gamma_2$ of any two translates of $\gamma_1$ differ by at most $i+1$. Observe that on every horizontal line, a segment of at least $\frac{1}{4}$ has that all the corresponding translates of $\hat{\gamma}_1$ for this line segment intersect $\zeta_2$ or all of them do not. (Indeed there is a segment of size at least $\frac 1 2$ so that a vertical segment of length 1 from any point on this segment misses $\zeta_2$ and $\{j\|q_k\alpha\|:0\leq j\leq i\}$ is contained in an interval of length at most $\frac 1 4 $. That is, there is a subinterval of size $\frac 1 4$ so that for each $x$ in this subinterval we have that $j\|q_k\alpha\|+x$ is in the subinterval of size $\frac 1 4 $ for all $0\leq j\leq i$.) So on a subset of this set of measure at least $\frac 1 8$ the translates of $\hat{\gamma}_2$ either all intersect $\zeta_2'$ or all miss $\zeta_2'$. This set satisfies the lemma and it is either within one of the maximal or within $1$ of the minimal. \[lem:hit interval\] If $\hat{I}$ is an interval of size at least $\gamma \|q_k\alpha\|$ and $q_L>12 \gamma^{-1}\|q_k\alpha\|$ then for all $x$ we have $\frac 1 {q_L}\sum_{j=0}^{q_L-1}\chi_{\hat{I}}(R^j x)\in [\frac 1 2 \lambda(\hat{I}),2 \lambda(\hat{I})]$. Also for all $\gamma>0$ there exists $u$ so that if $\hat{I}$ is an interval of size at least $\gamma \|q_k\alpha\|$ then $\frac 1 {q_L}\sum_{j=0}^{q_L-1}\chi_{\hat{I}}(R^j x)\in [\frac 1 2 \lambda(\hat{I}),2 \lambda(\hat{I})]$ for all $L>k+u$ and if $t>q_{k+u}$ we have $\frac 1 t \sum_{j=0}^{t-1}\chi_{\hat{I}}(R^jx)\geq \frac 1 4 \lambda(\hat{I})$. This follows by Lemma \[lemma:DK\]. Indeed $\sum_{j=0}^{q_L-1}\chi_{\hat{I}}(R^j x)\in [q_L\lambda(\hat{I})-2,q_L\lambda(\hat{I})+2]$ so if the lemma follows if $q_L\lambda(\hat{I})>4$. Since $\|q_k\alpha\|<\frac 1 {3q_{k+1}}$ this is the case. To see the second claim first notice that $q_{k+2}=q_{k+1}+q_k>2q_k$. So if $2^b>12\gamma^{-1}$ then $q_{L}>12\gamma^{-1}\|q_k\alpha\|$ whenever $L\geq k+2b$, and the second claim follows from the first with $u=2b$. To see the last claim, notice $\sum_{j=0}^{t-1}\chi_{\hat{I}}(R^jx) \geq \sum_{i=0}^{\lfloor \frac t {q_{k+u}}\rfloor-1} \sum_{j=0}^{q_{k+u}-1}\chi_{\hat{I}}(R^{j}R^{iq_{k+u}}x)$ and apply the previous sentence to obtain that this is at least $\lfloor \frac t {q_{k+u}}\rfloor q_{k+u}\frac 1 2 \lambda(\hat{I})$. Since $t<2 \lfloor \frac t {q_{k+u}}\rfloor q_{k+u}$ we have the final claim. \[lem:sum bound\] $\underset{k \to \infty}{\lim}\frac{\sum_{i=1}^ka_i}{q_k}=0.$ By Lemma \[lemma:q:k:plus:1:ak:qk\] we have that $q_\ell>a_{\ell}q_{\ell-1}$ and $q_{\ell+2}>2q_\ell.$ So by induction we have that $q^k>2^{\frac{j}2}\prod_{i=1}^{k}a_i$ where $j=|\{i<k:(a_i,a_{i-1})=(1,1)\}|$. Applying the Criterion {#sec:work} ====================== In this section we show that (A0)-(A9) imply the assumptions of Proposition \[prop:fact:qfact\]. Proposition \[prop:technical:cond3\] connects between rotations and powers of 3-IETs. Lemma \[lemma:first:sum\] is an intermediate step in showing that (A0)-(A9) imply the assumptions of Proposition \[prop:technical:cond3\] and Section \[sec:subsec:proof:of:prop:technical\] completes the argument. In view of Proposition \[prop:fact:qfact\] and Corollary \[cor:trivial:joining\], to prove Theorem \[theorem:disjoint:powers\] it is enough to prove the following: \[prop:satisfy:2a:2b:2c\] Suppose assumptions (A0)-(A9) are satisfied. Then, there exists a constant $\epsilon_* > 0$ (depending only on the constants in (A0)-(A9)) such that the following holds: Suppose $n \in {{{\mathbb}N}}$, $m'<m<n$, and $$\label{eq:sublacunary} \left|\frac{m'}{m} - 1\right| \le \epsilon_*.$$ Then the assumptions of Proposition \[prop:fact:qfact\] can be satisfied for $X_1 = X_2 = J$, $T_1 = S^n$ and $T_2$ either $S^m$ or $S^{m'}$. (Note that we view $T_1$ and $T_2$ as maps from $J$ to $J$). [[**Notation.** ]{}]{} Before starting the proof of Proposition \[prop:satisfy:2a:2b:2c\] we introduce some notation. Let $$\psi_M(x) = \sum_{\ell=0}^{M -1} \chi_J(R^\ell x).$$ Then, for any $x \in J$ so that $R^Mx\in J$, $$\label{eq:time:change:general} S^{\psi_M(x)} x = R^M x.$$ [[**Picking parameters.** ]{}]{} Let $$\label{eq:def:c} c = \frac{m - m'}{n}.$$ Since we have $0 < m' < m \le n$, note that $$\frac{m}{m'} = 1 + \frac{c n}{m'} \ge 1 + c.$$ Thus, in order to prove Theorem \[theorem:disjoint:powers\], we may assume that $c$ is small. Let $k$, $L$ be such that assumptions (A0)-(A9) are satisfied. Let $r_k= \lfloor \frac{\lambda(J)L}n \rfloor$. Let $$w_k = \lfloor \frac{r_k}{\lambda(J)}\rfloor.$$ Then, $$(m-m') w_k = \frac{(m-m')}{n } L + O(\frac{m-m'}{2 \lambda(J)}) = c L + O(n).$$ Thus, in view of (A0) and (\[eq:def:c\]), we have $$\label{eq:m:mprime:wk:range} (m-m') w_k \in [q_{k-1},q_k).$$ The proof of Proposition \[prop:satisfy:2a:2b:2c\] relies on the following technical result: \[prop:technical:cond3\] There exists $c_0, \tilde{c}>0$, $\hat{C} > 0$ and $u \in {{{\mathbb}N}}$ depending only on the constants $C_1,...,C_4$ of the assumptions (A0)-(A4) so that if $c < c_0$ (where $c$ is as in (\[eq:def:c\])), then there exists $\hat{m}\in \{m,m'\}$ and $d \in \{-3,-2,-1,1,2,3\}$ so that, after passing to a subsequence, for all large enough $k$, 1. there exists $\tilde{A} \subset J$ with $\lambda(\tilde{A})>\tilde{c}$ so that for all $y \in \tilde{A}$ we have $$\label{eq:Rqk:S:hat:m:rk} R^{q_k}S^{\hat{m}r_k}y = S^d S^{\hat{m}r_k} R^{q_k}y.$$ 2. If $N>\frac{q_{k+u}}{2\hat{m}}$ then for any $y \in [0,1]$ we have $$\frac{1}{N} \sum_{i=0}^{N-1}\chi_{\tilde{A}}(S^{i\hat{m}}y) > \frac{\lambda(\tilde{A})}{\hat{C}} \text{ and } \frac{1}{N} \sum_{i=0}^{N-1}\chi_{R^{q_k}\tilde{A}}(S^{i\hat{m}}y) > \frac{\lambda(\tilde{A})}{\hat{C}}.$$ Proof of Proposition \[prop:satisfy:2a:2b:2c\] assuming Proposition \[prop:technical:cond3\] -------------------------------------------------------------------------------------------- We now prove Proposition \[prop:satisfy:2a:2b:2c\] assuming Proposition \[prop:technical:cond3\]. We assume that the constant $c$ in Proposition \[prop:technical:cond3\] is small enough, and $d=1$. The case of $d\in\{3,2,-1,-2,-3\}$ is similar. In view of assumption (A8) there exists an interval $K_1(L)\subset \mathbb{N}$ of size at most $k_c$ so that for any $x \in J$, $\psi_L(x) \in K_1(L)$. Since $R(J^c) \subset J$, there exists an interval $K_2(L)$ of size $(k_c+1)$ such that for all $x \in [0,1]$, $\psi_L(x) \in K_2(L)$. We have $$\int_0^1 \psi_L(x) \, d\lambda(x) = \lambda(J) L.$$ Therefore, $\lfloor \lambda(J) L \rfloor \in K_2(L)$. Since $R(J^c) \subset J$, it follows that for all $x \in J$, $$\lfloor \lambda(J) L \rfloor = \psi_{L'(x)}(x) + \delta(x) \qquad\text{ where $0 \le \delta(x) \le 1$, and $|L'(x) - L| < (k_c+1)$. }$$ Write $$n r_k = \lfloor \lambda(J) L \rfloor + \epsilon_k, \qquad \text{ where $|\epsilon_k| < n$.}$$ Now, for $x \in J$, by (\[eq:time:change:general\]), $$T_1^{r_k} x = S^{n r_k} x = S^{\epsilon_k} S^{\lfloor \lambda(J) L \rfloor} x = S^{\epsilon_k} R^{L'(x)} x = S^{\epsilon_k} R^{L'(x) - L} R^L x$$ Thus, by (A6) condition (1) of Proposition \[prop:fact:qfact\] follows, (with the size of the partition dependent on $n$). Let $F_k$ be the first return map of $R^{q_k}$ to $J$. (Essentially we want $F_k$ to be $R^{q_k}$, but we want $F_k$ to be a map from $J$ to $J$). Since $R^{q_k}$ tends to the identity map as $k \to \infty$, condition (2a) of Proposition \[prop:fact:qfact\] follows. For $x \in \hat{A}$, and since we are assuming $d=1$, (\[eq:Rqk:S:hat:m:rk\]) becomes $$R^{q_k} T_2^{r_k} x = S T_2^{r_k} R^{q_k} x.$$ Since $R^{q_k}$ tends to the identity as $k \to \infty$, there exists a subset $E \subset J$ of almost full measure such that for $x \in E$, $R^{q_k} x = F_k x$. Then, for $x \in E \cap \hat{A}$, $$R^{q_k} T_2^{r_k} x = S T_2^{r_k} F_k x.$$ Condition (2b) of Proposition \[prop:fact:qfact\] follows. We now begin the proof of Condition (3) of Proposition \[prop:fact:qfact\]. In (A0)-(A9) we choose $\eta < (96\cdot 25)^{-1}$. Let $\rho = \hat{c}_\eta$, and choose $\delta_0 < \frac{1}{12} \eta \rho$. Then, by (A5), we can either choose $a$ such that for $i$ sufficiently large, $$\label{eq:choice:of:a} \frac{1}{96\hat{C}} \lambda(U_a^{(i)}) > 14 \delta_0 + 27 \lambda(\bigcup_{\ell < a} U_\ell^{(i)}) \text{ and } \lambda(U_a^{(i)}) > \rho,$$ or choose $a'$ so that for $i$ sufficiently large, $$\label{eq:choice:of:a:prime} \frac{1}{96\hat{C}} \lambda(U_{a'}^{(i)}) > 14 \delta_0 + 27 \lambda(\bigcup_{\ell > a'} U_\ell^{(i)}) \text{ and } \lambda(U_{a'}^{(i)}) > \rho,$$ We need a lemma to obtain Condition (3) of Proposition \[prop:fact:qfact\] from Proposition \[prop:technical:cond3\]: \[lemma:strectch:in\] For every $\rho>0$ there exists $b \in {{{\mathbb}N}}$ so that if for some $s \in {{{{\mathbb}Z}}}$, $\lambda(\psi_L^{-1}(s))>\rho$ where $L\geq q_\ell$ so that either $L=q_\ell$ or $a_{\ell+1}>4$ and $L=iq_\ell$ for $i\leq \lfloor \frac{a_{\ell+1}}4 \rfloor$ then there exists a measurable set $V$ with the following properties: - We have $$\lambda\left(\bigcup_{j=0}^{q_{\ell-b}-1}R^j(V)\right)>\frac{1}{2} \lambda(\psi_L^{-1}(s)).$$ - We have $R^j(V) \subset \psi_L^{-1}(s)$ for all $0\leq j<q_{\ell-b}$. - The sets $R^j(V)$, $0\leq j<q_{\ell-b}$ are pairwise disjoint. We claim that there exists $b' \in {{{\mathbb}N}}$ so that $$\label{eq:claim:stretch:in} \lambda(\{x: R^jx\in \psi^{-1}_L(s) \text{ for all }0\leq j<5q_{j-b'}\})> \frac 4 5\lambda(\psi_L^{-1}(s)).$$ To prove (\[eq:claim:stretch:in\]), note that for any $x \in [0,1]$ there exist at most $2$ different $0\leq j<q_\ell-1$ so that $\psi_L(R^{j+1}x)\neq \psi_L(R^jx)$. (Indeed the set where $\psi_L(x) \neq \psi_L(Rx)$ is two intervals of length $i\|q_\ell \alpha\|<\frac i {a_{\ell+1}q_\ell }\leq \frac 1 {4q_\ell}<\|q_{\ell-1}\|$. Any orbit of length $q_\ell-1$ can hit each of these intervals at most once.) Thus for any $r \in {{{\mathbb}N}}$, $$\label{eq:tmp:claim:stretch:in} \lambda(\{x:\psi_L(x)\neq \psi_L(R^jx) \text{ for some }0<j<r\})\leq \frac{3r}{q_\ell-1}.$$ (Indeed each orbit of length $q_\ell-1$ can have at most 3 consecutive stretches in this set. These stretches have length at most $r$.) Now (\[eq:tmp:claim:stretch:in\]) implies (\[eq:claim:stretch:in\]). Now we build $V$. For each $x \in G= \{x: R^jx\in \psi^{-1}_L(s) \text{ for all }0\leq j<5q_{\ell-b'}\}$ let $m_x=\max\{j:R^jx\notin \psi_L^{-1}(s)\}+1$ and $M_x= \min\{j:R^j\notin \psi_L^{-1}(s)\}-1$. Let $$V=\bigcup_{x\in G}\bigcup_{j=0}^{\lfloor \frac{M_x-m_x}{q_{\ell-b}}\rfloor-1} R^{m_x+jq_{\ell-b}}x.$$ We assume that (\[eq:choice:of:a\]) holds. (The proof in the case (\[eq:choice:of:a:prime\]) holds is virtually identical). Let $a$ and $\rho$ be as in (\[eq:choice:of:a\]). We then apply Lemma \[lemma:strectch:in\] with this $\rho$ and $s = a$. Then, let $V$, $\ell$ and $b$ be as in Lemma \[lemma:strectch:in\]. We assume $\epsilon_*$ (and thus $c$) is small enough so that (A7) implies that $$\label{eq:L:prime:big} q_{\ell - b} >q_{k+u+4}.$$ Let $\sigma$ be any joining of $S^n \times S^{\hat{m}}$ and for each $x$ let $\Sigma_x$ denote the points $y$ so that $(x,y)$ is $\sigma$-generic. Let $$E_1 = \bigcup_{i=0}^{n-1} R^i V \cap J.$$ We are assuming that $x \in V \subset J$, and also we are assuming that $Ry \in J$ whenever $y \not\in J$. Then, for at least half of $0 \le i < n$ we have $R^i x \in J$, it follows that $$\lambda(E_1) > \frac{n}{2} \lambda(V) > \frac{n}{4 q_{\ell-b}} \lambda(\psi_L^{-1}(a)).$$ We can choose $N \in {{{\mathbb}N}}$ so that $N \ge \frac{q_{\ell-b}}{12n}$ and also $$\label{eq:phi:L:Snj:x:a} \psi_L(S^{nj} x ) = a \quad\text{ for all $x \in E_1$ and all $0 \le j < 2N$.}$$ Let $$E_1' = \bigcup_{j=0}^{N-1} S^{nj} E_1.$$ Then, in view of (\[eq:phi:L:Snj:x:a\]), $$\label{eq:E1:subset:phi:L:inverse:a} S^{nj} E_1' \subset \psi_L^{-1}(a) \qquad\text{for all $0 \le j < N$.}$$ Note that $R^ix$, $R^jx$ are in distinct $S^n$ orbits if $|i-j|<n$ and $R^ix,R^jx \in J$. This means that the above union is disjoint, and thus $$\lambda(E_1') = N \lambda(E_1) > \frac{N n}{4q_{\ell-b}} \lambda(\psi_L^{-1}(a)) > \frac{1}{48} \lambda(\psi_L^{-1}(a)).$$ Since $\sigma$ is a self joining of $\lambda$, we can find $E \subset E_1 {\times}[0,1]$ so that $$\label{eq:measure:sigma:E} \sigma(E) > \frac{1}{2} \lambda(E_1) > \frac{1}{96} \lambda(\psi_L^{-1}(a)).$$ We have, by (\[eq:L:prime:big\]), $$\label{eq:N:big} N \ge \frac{q_{\ell-b}}{12 n} > \frac{q_{k+u+4}}{12 n} > \frac{q_{k+u}}{\hat{m}}.$$ where $\hat{m} \in \{m,m'\}$ and $u$ is as in Proposition \[prop:technical:cond3\]. Let $\chi_{\tilde{A}}$ denote the characteristic function of $\tilde{A}$, we have, in view of (\[eq:N:big\]) and conclusion (2) of Proposition \[prop:technical:cond3\], for any $y \in [0,1)$, $$\label{eq:most:in:hatA} \frac{1}{N} \sum_{j = 0}^{N - 1} \chi_{\tilde{A}}(S^{\hat{m} j} y) \ge \frac{\lambda(\tilde{A})}{\hat{C}}.$$ Then, since $\sigma$ is $S^{n} {\times}S^{\hat{m}}$ invariant, $$\begin{aligned} \sigma(\psi_L^{-1}(a) \times \tilde{A}) & = \frac{1}{N} \sum_{j=0}^{N-1} \sigma(S^{-n j} \psi_L^{-1}(a) \times S^{-\hat{m} j } \tilde{A}) & \\ & \geq \frac{1}{N} \sum_{j=0}^{N-1} \sigma((S^{-n j} \psi_L^{-1}(a) \times S^{-\hat{m} j } \tilde{A}) \cap E) & \\ & = \int_{E} \left(\frac{1}{N} \sum_{j=0}^{N-1} \chi_{\tilde{A}}(S^{\hat{m}j} y)\right) \, d\sigma(x,y) & \text{by (\ref{eq:E1:subset:phi:L:inverse:a})} \\ & \ge \frac{\lambda(\tilde{A}) \sigma(E)}{\hat{C}} & \text{by (\ref{eq:most:in:hatA})} \\ & > \frac{1}{96 \hat{C}} \lambda(\psi_L^{-1}(a)) \, \lambda(\tilde{A}), &\text{by (\ref{eq:measure:sigma:E}).} \end{aligned}$$ Condition (3) of Proposition \[prop:fact:qfact\] now follows immediately from (\[eq:choice:of:a\]). The main lemma -------------- The next lemma about rotations is the key step in the proof of Proposition \[prop:technical:cond3\]. \[lemma:first:sum\] Assume (A0)-(A4) are satisfied and also that $(m-m')w_k \in [q_{k-1},q_k)$. Let $C_1, \dots, C_4$ be as in assumptions (A0)-(A4). Then there exist $c_2 > 0$ and $C' > 0$ depending only on $C_1, \dots, C_4$ such that for all $k \in {{{\mathbb}N}}$ there exists an interval $I \subset [0,1)$ and a set of natural numbers $E=\{e,...,e+c_2q_k\}$ so that 1. $|I| \ge C'{\| q_{k}\alpha\|}$ 2. For all $x \in \bigcup_{i \in E}R^iI$. $$\label{eq:E:property} \sum_{\ell=0}^{mw_k-1}(\chi_J(R^\ell x)-\chi_J(R^{\ell+q_k}x))-\sum_{\ell=0}^{m'w_k-1}(\chi_J(R^{\ell}x)-\chi_J(R^{\ell+q_k}x))\in \{-1,1\}.$$ In the rest of this subsection, we will prove Lemma \[lemma:first:sum\]. We will derive Proposition \[prop:technical:cond3\] from Lemma \[lemma:first:sum\] in §\[sec:subsec:proof:of:prop:technical\]. The proof of this lemma is complicated and so we provide a brief sketch: We use (\[eq:difference:non:zero\]) to have a criterion for $\sum_{\ell=0}^{mw_k-1}(\chi_J(R^\ell x)-\chi_J(R^{\ell+q_k}x))-\sum_{\ell=0}^{m'w_k-1}(\chi_J(R^{\ell}x)-\chi_J(R^{\ell+q_k}x))\in \{-1,1\}.$ Claims \[claim:hit:both\], \[claim:choose:E0\] and \[claim:E0:works\] use this criterion to prove the claim. Claim \[claim:hit:both\] and subsequent comments identify $I$. Claim 4.6 identifies $E$. Claim 4.7 is used to show the critierion given by (\[eq:difference:non:zero\]) holds for these $I$ and $E$. Let $$c_2 \leq \frac{1}{C_4} \qquad\text{ so that $c_2 < 1$.}$$ Recall that $J=[0,z]$. Assume $k$ is odd. (This is an assumption of convenience of exposition. If $k$ is odd then $R^{q_k} {0} =-\|q_k\alpha\|$, if $k$ is even it is $\|q_k\alpha\|$. Thus if $k$ is even all sets $[-\|q_k\alpha\|,0)$ should be $[0,\|q_k\alpha\|)$ and $[z-\|q_k\alpha\|,z)$ should be $[z,z+\|q_k\alpha\|)$). Observe $$\begin{gathered} \label{eq:sum:rearrangement} \sum_{\ell=0}^{mw_k-1}\chi_J(R^\ell x)-\chi_J(R^{\ell+q_k}x)-(\sum_{\ell=0}^{m'w_k-1}\chi_J(R^{\ell}x)-\chi_J(R^{\ell+q_k}x))=\\ \sum_{\ell=0}^{(m-m')w_k-1}\chi_J(R^\ell R^{m'w_k}x)-\sum_{\ell=0}^{(m-m')w_k-1}\chi_J(R^{\ell+q_k}R^{m'w_k}x)) \equiv F(R^{m' w_k} x), \end{gathered}$$ where $$F(y) = \sum_{\ell=0}^{(m-m')w_k-1}\chi_J(R^\ell y)- \sum_{\ell=0}^{(m-m')w_k-1}\chi_J(R^{\ell+q_k}y)).$$ Recall that $J=[0,z)$. We have $$\label{eq:F:alternative} F(y) = \sum_{\ell=0}^{(m-m')w_k-1}\chi_{[-\|q_k \alpha\|,0)}(R^\ell y) - \sum_{\ell=0}^{(m-m')w_k-1}\chi_{[z-\|q_k \alpha\|,z)}(R^\ell y).$$ Note that, since $(m-m')w_k<q_k$, by Lemma \[lemma:orbit:sep\], each of the sums in (\[eq:F:alternative\]) is at most $1$. Thus, $F(y) \in \{-1, 0, 1\}$, and $$\begin{gathered} \label{eq:difference:non:zero} F(y) \in \{ 1, -1 \} \\ \text{if and only if $\{R^\ell y \}_{\ell=0}^{(m-m')w_k}$ hits $[-{\| q_k\alpha\|},0) \cup [z-{\| q_k\alpha\|},z)$ exactly once. }\end{gathered}$$ Consider $[-{\| q_k\alpha\|},0)$. By Lemma \[lemma:ret:time\], the function that assigns to a point in $[-\|q_k\alpha\|,0)$ its first return time takes two values, $q_{k+1}$ and $q_{k+1}+q_k$. The return time of $q_{k+1}+q_k$ occurs on $[-{\| q_{k+1}\alpha\|},0)$. \[claim:hit:both\] - For every $x \in [-\|q_k\alpha\|,0)$ there exists $j \in \{1,...,q_{k+1}+q_k\}$ so that $$R^j x \in [z-{\| q_k\alpha\|},z).$$ - We have $$|\{0\leq j \leq q_{k+1}+q_k:\exists x \in[-\|q_k\alpha\|,0) \text{ with }R^j x \in [z-\|q_k\alpha\|,z)\}|\leq 4.$$ - There exists $j \in \{1,...,q_{k+1}+q_k\}$ such that $$\lambda(R^j([-\|{q_{k+1}\alpha}\|,0))\cap [z-{\| q_k\alpha\|},z)) > \frac{1}{4}\|q_{k+1} \alpha\|$$ Since $R$ is minimal, if $J'$ is an interval then for every $x$, $R^ix\in J'$ for some $0\leq i< \underset{x \in J'}{\max}\min\{j>0:R^jx \in J'\}$. For any interval of size $\|q_k\alpha\|$ this is $q_{k+1}+q_k$, see Lemma \[lemma:ret:time\]. This proves (a). By Lemma \[lemma:orbit:sep\], for any $\ell$ there exists at most two $j$ in $\{\ell,...,\ell+q_{k+1}-1\}$ so that there exists $x \in[-\|q_k\alpha\|,0) \text{ with }R^j x \in [z-\|q_k\alpha\|,z)$. Since $q_{k+1}+q_k<2q_{k+1}$ this implies (b). The statement (c) follows from (a) and (b). Indeed, let $\Delta$ denote the set of $j$ in part (b). Then, in view of (b), $|\Delta| \le 4$. For each $j \in \Delta$, let $I_j$ denote the set of $x \in [-\|q_{k+1} \alpha\|, 0)$ such that $R^j x \in [z-{\| q_k\alpha\|},z)$. Then, by (a), $[-\|q_{k+1} \alpha\|,0) = \bigcup_{j \in \Delta} I_j$. Thus, there exists $j \in \Delta$ such that $\lambda(I_j) \ge \frac 1 4 \|q_{k+1} \alpha\|$. We now continue the proof of Lemma \[lemma:first:sum\]. We choose $j$, $0 \le j \le q_{k+1} + q_k$ so that $$\lambda(R^j([-\|q_{k}\alpha\|,0))\cap [z-\|q_k\alpha\|,z)) \text{ is maximal.}$$ Let $$I=R^{-j}([z-\|q_k\alpha\|,z)) \cap [-\|q_{k+1}\alpha\|,0)).$$ Note that by Claim \[claim:hit:both\](c), $\lambda(I)>\frac{1}{4} \|q_{k+1}\alpha\|$. \[claim:choose:E0\] Either $$\label{eq:j:small} (j - c_2 q_k, j+q_k) \subset \{1, q_{k+1}+q_k - 1\}$$ or $$\label{eq:j:big} (j - q_k, j+ c_2 q_k) \subset \{1, q_{k+1}+q_k - 1\}$$ Note that by (A4), for $0<i<q_k/C_4$, $$\label{eq:first:comment} \text{ if $x \in [-\|q_k\alpha\|,0)$ then $R^i x \not\in [z-\|q_k\alpha\|,z)$.}$$ Therefore, $j > q_k/C_4 \ge c_2 q_k$. Similarly, for $0<i<q_k/C_4$, $$\label{eq:first:comment:prime} \text{if $x \in [z-\|q_k\alpha\|,z)$ then $R^i x \not\in [-\|q_k\alpha\|,0)$.}$$ Therefore, $j < q_{k+1} + q_k - \frac{q_k}{C_4} \le q_{k+1} + q_k - c_2 q_k$. Now if $j < q_{k+1}$ then (\[eq:j:small\]) holds, and if $j > q_k$ then (\[eq:j:big\]) holds. This completes the proof of Claim \[claim:choose:E0\]. \[claim:E0:works\] Suppose (\[eq:j:small\]) holds and $\ell \in (j - c_2 q_k, j+q_k)$ or (\[eq:j:big\]) holds and $\ell \in (j-q_k, j+c_2 q_k)$. Also assume that $\ell \ne j$. Then, $$R^\ell I \cap (([-\|q_k\alpha\|,0) \cup [z-\|q_k\alpha\|,z)) = \emptyset.$$ Recall that $I \subset [-\|q_{k+1} \alpha \|,0)$ and thus by Lemma \[lemma:ret:time\], $$R^\ell I \cap [-\|q_k\alpha\|,0) = \emptyset \qquad \text{ for $1 \le \ell \le q_{k+1} + q_k - 1$.}$$ Also, by Lemma \[lemma:ret:time\], the return time of any point in the interval $[z-\|q_k\alpha\|,z)$ to itself is at least $q_{k+1} > q_k$. Thus, for $\ell$ such that $|\ell - j| < q_k$, $$R^\ell I \cap [z-\|q_k\alpha\|,z) = \emptyset.$$ Claim \[claim:E0:works\] follows. We now continue the proof of Lemma \[lemma:first:sum\]. Let $r = \min(c_2 q_k, (m-m') w_k)$. Recall that $(m-m')w_k < q_k$. If (\[eq:j:small\]) holds, let $$E = (j-r, j-r+c_2 q_k), \qquad \text{ so that $E+[0,(m-m')w_k) \subset (j-c_2 q_k, j+q_k)$.}$$ If (\[eq:j:big\]) holds, let $$\begin{gathered} E = (j-(m-m')w_k, j-(m-m')w_k + c_2 q_k) \\ \text{ so that $E+[0,(m-m')w_k) \subset (j- q_k, j+c_2 q_k)$.}\end{gathered}$$ Then, for all $i \in E$, $$i \le j \le i+(m-m')w_k.$$ Hence, by Claim \[claim:E0:works\], for all $x \in I$ and for all $i \in E$, $$\label{eq:tmp:Rellix} |\{R^{i+\ell} x\}_{\ell=1}^{(m-m')w_k}\cap ([-\|q_k\alpha\|,0)\cup [z-\|q_k\alpha\|,z))|=1.$$ (the only contribution is from the case where $i + \ell = j$). Therefore, in view of (\[eq:sum:rearrangement\]) and (\[eq:difference:non:zero\]), for $x \in R^{-m' w_k} I$ and $\ell \in E$, (\[eq:E:property\]) holds. From the definition, $|E| \ge c_2 q_k$. We now estimate $\lambda(I)$. By Lemma \[lemma:q:k:plus:1:ak:qk\], $$q_{k+2} = a_{k+2} q_{k+1} + q_{k} < (a_{k+2}+1) q_{k+1}.$$ $$q_{k+1} = a_{k+1} q_k + q_{k-1} < (a_{k+1} + 1) q_{k}$$ By Lemma \[lemma:good:bound\], $$\begin{gathered} \lambda(I) \ge \frac{1}{4} \|q_{k+1} \alpha\| \ge \frac{1}{4} \frac{1}{q_{k+2} + q_{k+1}} \ge \frac{1}{4} \frac{1}{(a_{k+2} + 2)q_{k+1}} \ge \\ \ge \frac{1}{4} \frac{1}{(a_{k+2} + 2)(a_{k+1}+1) q_k} \ge \frac{1}{4} \frac{a_{k+1}}{(a_{k+2} + 2)(a_{k+1}+1)}\|q_k \alpha\|.\end{gathered}$$ Thus, by (A3), $$\lambda(I) \ge \frac{1}{8(C_3+2)} \|q_k \alpha \|.$$ This completes the proof of Lemma \[lemma:first:sum\]. Proof of Proposition \[prop:technical:cond3\] from Lemma \[lemma:first:sum\] {#sec:subsec:proof:of:prop:technical} ---------------------------------------------------------------------------- Recall $w_k =\lfloor \frac{r_k}{\lambda(J)} \rfloor$ as above. (Corollary to Lemma \[lemma:first:sum\]) Given $w_k$ so that $q_k <(m-m')w_k<q_{k+1}$ as before there exists $\hat{m}\in \{m,m'\}$ and a set $A_k$ with $\lambda(A_k) \geq \tilde{c}$ (depending on our non-divergence condition, that is $C_1,...,C_4,C'$) so that for all $x \in A_k$ we have $$\label{eq:cor:first:sum} \sum_{i=0}^{\hat{m}w_k-1}\chi_J(R^ix)-\sum_{i=0}^{\hat{m}w_k-1}\chi_J(R^{i+q_k}x)=d\in \{-3,-2,-1,1,2,3\}.$$ Lemma \[lemma:first:sum\] establishes that there exists $\bar{c} > 0$ and $\bar{c}_1 > 0$, and for an infinite sequence of $k \in {{{\mathbb}N}}$ there exists an interval $I' \subset [0,1]$ with $\lambda(I')>\bar{c}\|q_k \alpha\|$ so that for any $x\in I'$ there exists $H_x \subset \{0,1, \dots ,q_k-1\}$ with $|H_x|>\bar{c}_1 q_k$ so that for any $x \in I'$ and any $\ell \in H_x$, and any $w_k$ with $(m-m')w_k \in [q_{k-1},q_k)$ we have $$\begin{gathered} \label{eq:the:main:sum} \sum_{i=0}^{mw_k-1}(\chi_J(R^i R^\ell x)-\chi_J(R^{i+q_k}R^\ell x)) \\ - \sum_{i=0}^{m'w_k-1}(\chi_J(R^i R^\ell x)-\chi_J(R^{i+q_k}R^\ell x)) \in \{-1,1\}.\end{gathered}$$ Also note that by Lemma \[lemma:DK\], $\psi_{q_k}$ takes at most 5 values (which are also consecutive) and so for any $s \in {{{\mathbb}N}}$ and any $x$ we have $$\label{eq:rearrange:psi} \psi_s(x)-\psi_s(R^{q_k}x)=\psi_{q_k}(x)-\psi_{q_k}(R^s x)\in \{-4,\dots, 4\}.$$ Note that the left-hand-side of (\[eq:the:main:sum\]) is $$(\psi_{mw_k}(R^\ell x) - \psi_{m w_k}(R^{q_k} R^\ell x)) - (\psi_{m'w_k}(R^\ell x) - \psi_{m' w_k}(R^{q_k} R^\ell x)) \equiv S_1(R^\ell x) - S_2(R^\ell x).$$ By (\[eq:the:main:sum\]), for all $x \in I'$ and for all $\ell \in H_x$, $S_1(R^\ell x) - S_2(R^\ell x) \in \{-1,1\}$, and by (\[eq:rearrange:psi\]), we have $|S_1(R^\ell x)| \le 4$, and $|S_2(R^\ell x)| \le 4$. It follows that for all $x \in I'$ and all $\ell \in H_x$, $$S_i(R^\ell x) \in \{-3,-2,-1,1,2,3\} \qquad\text{ for some $i \in \{1,2\}$.}$$ Thus, there exists $\hat{m}\in \{m,m'\}$ and $d \in \{-3,-2,-1,1,2,3\}$ and a set $A_k$ with $$\lambda(A_k) \ge \frac{1}{12} |H_x| \lambda(I') = \frac{\bar{c}\bar{c}_1 q_k \|q_k\alpha\|}{12},$$ so that for for $x \in A_k$ (\[eq:cor:first:sum\]) holds. We frequently use the following trivial result in this section. \[lem:size bound\] If $\lambda(B)\geq \gamma$ and $B$ is the union of at most $\ell$ intervals then there exists $B'\subset B$ with $\lambda(B')\geq\frac 1 2 \lambda(B)$ and $B'$ is the union of intervals of size at least $\frac{\gamma}{2\ell}$. There exists $A_k'\subset A_k$ with $\lambda(A_k')>\frac{1}{2} \lambda(A_k)$ and so that $A_k'$ is made of at most $4q_k+1$ intervals with length at least $\frac {\tilde{c}}2 \frac 1 {4q_k+1}$. Recall that $A_k$ is a level set of $$\sum_{i=0}^{\hat{m}w_i-1}\chi_J(R^ix)-\sum_{i=0}^{\hat{m}w_i-1}\chi_J(R^iR^{q_k}x)=\sum_{i=0}^{q_k-1}\chi_J(R^ix)-\sum_{i=\hat{m}w_i}^{\hat{m}w_i +q_k -1}\chi_J(R^ix),$$ a function which has at most $4q_k$ discontinuities. The lemma follows from Lemma \[lem:size bound\] since this implies that any level set is made of at most $4q_k+1$ intervals. In the previous results we have proved properties of a level set of $\sum_{i=0}^{\hat{m}w_k-1}\chi_J(R^ix)-\sum_{i=0}^{\hat{m}w_k-1}\chi_J(R^{i+q_k}x)$. In this lemma we relate that to proving nice properties about a set, $G_\ell=\{x:S^{\hat{m}n}x=R^\ell x\}$ for some $\ell$, to obtain the set $\tilde{A}$ in Proposition \[prop:technical:cond3\]. \[lemma:Ak:twoprime\] For all large enough $k$ there exists $\hat{m} \in \{m,m'\}, \, d\in \{-3,-2,-1,1,2,3\}$ and a set $\tilde{A}_k$ with $\lambda(\tilde{A}_k)> \frac{\tilde{c}}4$ and which is the union of at most $8q_k$ intervals of size at least $\frac {\tilde{c}}2 \frac 1 {8\cdot 4q_k}$ so that for $x \in \tilde{A}_k$, $$\label{eq:Ak:twoprime} R^{q_k}S^{\hat{m}r_k}(x)=S^d S^{\hat{m}r_k}(R^{q_k}x).$$ By Lemma \[lemma:DK\], for all $h,j \in {{{\mathbb}N}}$, and any $x \in [0,1]$, $$\label{eq:iterated:djk} \left|-h q_j \lambda(J)+ \sum_{i=0}^{h q_j-1} \chi_J(R^i x)\right| \le 2h.$$ Let $0 < N < q_b$ be a positive integer, and write $$N = \sum_{i=0}^{b-1} h_i q_i, \qquad\text{ where $h_i \in {{{{\mathbb}Z}}}$, $0 \le h_i \le a_{i+1}$ and $h_{b-1}<\frac{N}{q_{b-1}}$.}$$ Let $D_0 = 0$, and for $0 < j \le b$, let $$D_j = \sum_{i=0}^{j} h_i q_i, \qquad\text{where $q_0 = 1$.}$$ Note that $D_b = N$. Then, by (\[eq:iterated:djk\]), for $x \in [0,1]$, $$\begin{aligned} \label{eq:bounding} \left| - N \lambda(J)+\sum_{i=0}^{N-1}\chi_J(R^i x)\right| & = \notag \left| \sum_{j=0}^{b-1} \left( -(D_{j+1}-D_j) \lambda(J) + \sum_{i = D_j}^{D_{j+1}-1} \chi_J(R^i x) \right) \right| \notag \\ & = \left| \sum_{j=0}^{b-1} \left( -h_j q_j \lambda(J) + \sum_{i = 0}^{h_j q_j-1} \chi_J(R^i R^{D_j} x) \right) \right| \notag \\ & \le 2 \sum_{j=0}^{b-2} a_{j+1}+\frac{N}{q_{b-1}} \qquad\text{ by (\ref{eq:iterated:djk})} \notag \\ &=o(N) + \frac{N}{q_{b-1}}=o(N). \end{aligned}$$ For $x \in J$ define $N(x)$ so that $$\hat{m} r_k = \sum_{i=0}^{N(x)} \chi_J(R^i x).$$ We now apply (\[eq:bounding\]) with $N(x)$ instead of $N$. We obtain that for each $x \in J$ there exists $N(x) \in {{{\mathbb}N}}$ so that $$\label{eq:sum:is:hatm:rk} \hat{m}r_k = \sum_{i=0}^{N(x)} \chi_J(R^i x), \qquad\text{ and $|N(x) - \hat{m} w_k| \le 4\sum_{i=0}^{b-2} a_i+\frac{\hat{m}w_k}{q_{b-1}}$.}$$ Since $$\hat{m}w_k<q_{k+1}\frac{\max\{m,m'\}}{m-m'}<(C_2+1)q_{k}\frac{\max\{m,m'\}}{m-m'}$$ we have that there exists $D$ so that for all $k$, $\hat{m}w_k<Dq_k$. Also by Lemma \[lem:sum bound\] $$\sum_{j=0}^{b-2} a_{j+1} +\frac{N(x)}{q_{b-1}} = o(q_{b-1})+o(N(x))=o(N(x))$$ and so $N(x)=\hat{m}w_k+o(\hat{m}w_k)$. Therefore, for all large enough $k$ we have $|N(x) - \hat{m} w_k|<\frac{\tilde{c}}{32}q_k$. Observe that if $$\label{eq:Nx:different:from:mwk} \sum_{i=0}^{N(x)}\chi_J(R^ix)-\sum_{i=0}^{N(x)}\chi_J(R^iR^{q_k}x)\neq \sum_{i=0}^{\hat{m} w_k}\chi_J(R^ix)-\sum_{i=0}^{\hat{m}w_k} \chi_J(R^iR^{q_k}x),$$ then $R^jx $ is in one of two intervals of size at most $\|q_k\alpha\|$ for some $$j \in [\min(N(x), \hat{m} w_k), \max(N(x), \hat{m} w_k)].$$ It follows that there exists $\tilde{A}_k \subset A_k'$ with $\lambda(\tilde{A}_k) > \frac{\tilde{c}}4$ which is a union of intervals each of size at least $\frac {\tilde{c}}2 \frac 1 {8\cdot 4q_k}$, such that for $x \in \tilde{A}_k$ (\[eq:Nx:different:from:mwk\]) does not hold. Indeed, we are removing at most $2\frac{\tilde{c}}{32}q_k$ intervals of size $\|q_k\alpha\|$ so we obtain a set of measure at least $\lambda(A_k')-\frac{\tilde{c}}{16}$ that is a union of at most $4q_k+1+4\frac{\tilde{c}}{32}q_k$ intervals and can invoke Lemma \[lem:size bound\]. Suppose $x \in \tilde{A}_k \subset A_k$. Then, in view of (\[eq:cor:first:sum\]), $$\sum_{i=0}^{N(x)}\chi_J(R^ix)-\sum_{i=0}^{N(x)}\chi_J(R^iR^{q_k}x) = -d,$$ where $d \in \{-3,-2,-1,1,2,3\}$. In view of (\[eq:sum:is:hatm:rk\]), this can be rewritten as $$\label{eq:sum:is:d:plus:hatm:rk} d+\hat{m} r_k= \sum_{i=0}^{N(x)}\chi_J(R^iR^{q_k}x).$$ Now, in view of (\[eq:time:change:general\]), (\[eq:sum:is:hatm:rk\]) and (\[eq:sum:is:d:plus:hatm:rk\]), $$S^{\hat{m} r_k} x = R^{N(x)} x \qquad\text{and}\qquad S^{d+\hat{m} r_k} R^{q_k} x = R^{N(x)} R^{q_k} x.$$ The equation (\[eq:Ak:twoprime\]) follows. To complete the proof we need to show $\{S^{i\hat{m}}x\}$ hits $\tilde{A}_k$ frequently enough. Lemma \[lem:hit interval\] lets us that show $R$ orbits hit $\tilde{A}_k$ frequently enough. The key observation we use is that if we define $j_i$ by $S^{\hat{m}i}x=R^{j_i}x$ then $j_{i+1}-j_i\leq 2 \hat{m}$. We call a set with this property *$2\hat{m}$ dense*. This motivates us to build an auxiliary set, $\hat{A}_k$, so that the hits of an $R$ orbit to $\hat{A}_k$ give a lower bound for the hits of an $S^{\hat{m}}$ orbit to $\tilde{A}_k$. We assume $k$ is large enough so that Lemma \[lemma:Ak:twoprime\] holds, $q_k>16 \hat{m}$ and $4\hat{m}\|q_k\alpha\|<\frac{\tilde{c}}{16}$. Consider $\tilde{A}_k$ and remove from it all $x$ so that $$\sum_{i=0}^{\hat{m}w_i}\chi_J(R^i(x))-\sum_{i=0}^{\hat{m}w_i}\chi_J(R^{i+q_k}x)\neq \sum_{i=0}^{\hat{m}w_i}\chi_J(R^iR^j(x))-\sum_{i=0}^{\hat{m}w_i}\chi_J(R^{i+q_k}R^jx)$$ for some $j<2\hat{m}$. This means that if $x$ remains in this set then $$\label{eq:stretch} \exists\ j\leq 0\leq k \text{ so that } k-j\geq 2\hat{m} \text{ and }R^\ell x\in \tilde{A}_k \text{ for all } \ell \in \{j,\dots, k\}.$$ The set remaining, $A'''_k$ has measure at least $\lambda(\tilde{A}_k)-4\hat{m}\|q_k\alpha\|$ and is made up of at most $8\hat{m}+8q_k$ intervals. (Indeed, we are removing at most $2\hat{m}$ pre-images of 2 intervals of size $\|q_k\alpha\|$.) Let $\hat{A}_k$ be a subset of $A'''_k$ of measure at least $\frac 1 2 \lambda(A'''_k)$ made of intervals of length at least $\frac{\tilde{c}}{32 \cdot 64(q_{k})}$. (Indeed, we invoke Lemma \[lem:size bound\] using that $A'''_k$ is a set of measure at least $\frac{\tilde{c}}8$ which is made up of at most $16q_k$ intervals.) By the estimate of the size of intervals in $\hat{A}_k$, Lemma \[lem:hit interval\] implies that if $\frac{q_{k+u}}{q_k}$ is large enough we have for $t > q_{k+u}$, $$\label{eq:sep:hit:bound} \frac{1}{t}\sum_{i=0}^{t} \chi_{\hat{A}_k}(R^ix)>\frac 1 {4} \lambda(\hat{A}_k).$$ Because $R^ix \in \hat{A}_k$, the equation (\[eq:stretch\]) implies there exists $j\leq i\leq k$ with $k-j\geq 2\hat{m}$ so that $R^\ell x\in \tilde{A}_k$ for all $\ell \in \{j,\dots,k\}$. Then, we have $$\frac 1 {2\hat{m}}\sum_{i=0}^t \chi_{\hat{A}_k}(R^ix)\leq |\{i \in \mathcal{C}:R^ix\in \tilde{A}_k\}|,$$ where $\mathcal{C}$ is any $2\hat{m}$ dense subset of $\{0,...,t+2\hat{m}\}$. Observing that $\{j\in [0,k+2m]: \exists i \text{ with }S^{\hat{m}i}x=R^jx\}$ is $2\hat{m}$ dense this implies that $$\frac 1 {2\hat{m}}\sum_{i=0}^t \chi_{\hat{A}_k}(R^ix)\leq \sum_{i=0}^{N_t(x)} \chi_{\tilde{A}_k}(S^{i\hat{m}}x)$$ where $N_t(x)=\min\{j:S^{\hat{m}j}x=R^\ell x \text{ with }\ell\geq t\}$. We obtain the proposition with $\hat{C}=16$. Indeed, $\lambda(\hat{A})\geq \frac 1 4 \lambda(\tilde{A})$ and so by (\[eq:sep:hit:bound\]) we have that for $t>q_{k+u}$, we have that $$\label{eq:new:sep:hit:bound} \frac{1}{t}\sum_{i=0}^t \chi_{\hat{A}_k}(R^ix)>\frac{\lambda(\tilde{A})}{16}.$$ Lastly, $G_x:=\{j:\exists i \text{ with }S^{\hat{m}i}x=R^jx\}$ is at least $\hat{m}$ separated (that is if $j \in G_x$ and $|i-j|<\hat{m}$ then $i \notin G_x$) and so $N_t(x)\leq \frac{t}{\hat{m}}$, letting us obtain, using (\[eq:new:sep:hit:bound\]), $$\frac{1}{2\hat{m} \cdot 16}\lambda(\tilde{A})\leq \frac{1}{2\hat{m}t}\sum_{i=0}^t \chi_{\hat{A}_k}(R^ix)\leq \frac 1 {t}\sum_{i=0}^{N_t(x)} \chi_{\tilde{A}_k}(S^{i\hat{m}}x)\leq \frac {1} {\hat{m}N_t(x)}\sum_{i=0}^{N_t(x)} \chi_{\tilde{A}_k}(S^{i\hat{m}}x).$$ Multiplying the sequence of inequalities by $\hat{m}$ completes the estimate. Renormalization {#sec:renorm} =============== Recall that $X$ is a torus with two marked points related to a 3-IET, $T$ and $\hat{X}$ is the torus obtained by forgetting the two marked points. [[**Divergence in the space of tori, $\mathcal{M}_1$:** ]{}]{} By Mahler’s compactness criterion the divergence of $g_t\hat{X}$ is controlled by the shortest (non-homotopicaly trivial) simple closed curve on $g_t\hat{X}$. This sequence is given by curves $\gamma_k$ with vertical holonomy $q_k$ and horizontal holonomy $\pm |q_k\alpha-p_k|=\pm {\| q_k\alpha\|}$. Coarsely, this curve is contracted from $t=0$ to $t= \log(q_k\sqrt{a_{k+1}})$ and then expanded. Additionally, there is a fixed compact set $\hat{K}$ so that $g_{\log(q_k)}\hat{X} \in \hat{K}$ for all $k$ (and in particular $|\gamma_k|$ is proportional to 1 at $g_{\log(q_k)}$). \[lem:rot divergence\] For any $\hat{K} \subset \mathcal{M}_1$ there exists $\ell_{\hat{K}}$ so that for all $k$ we have $|\{t\in [\log(q_k),\log(q_{k+1})):g_t\hat{X} \in \hat{K}\}|<\ell_{\hat{K}}$. For any $\hat{K}$ there exists $\delta$ so that if $ g_s\hat{X} \in \hat{K}$ then the shortest simple closed curve on $g_s\hat{X}$ is at least $ \delta$. As in the previous paragraph, consider the curve $\gamma_k$ on $\hat{X}$, with vertical holonomy $q_k$ and horizontal holonomy $\pm |q_k \alpha - p_k|$. On $g_s \hat{X}$ the curve $g_s \gamma_k$ has vertical holonomy $e^{-s} q_k$ and horizontal holonomy $\pm e^s |q_k \alpha - p_k|$. If $s\in [\log(q_k),\log(q_{k+1})]$ then, since we are assuming that the length of $g_s \gamma_k$ is at least $\delta$, we must have $e^s |q_k\alpha-p_k|\geq \frac \delta 2 $ or $e^{-s}q_k\geq \frac \delta 2 $. By Lemma \[lemma:good:bound\] the first condition can only hold if $e^s>\frac{\delta}2 {a_{k+1}q_k}$. Noticing that $a_{k+1}q_k>\frac 1 2 q_{k+1}$ this implies $s>\log(q_{k+1}) +2\log(2)+\log(\delta)$. The second condition can only hold if $s<2q_k \frac1{\delta}$. The lemma follows with $\ell_{\hat{K}}=-2\log(\delta)-3\log(2).$ We now assume the assumption of Proposition \[prop:good:assump\]. This means there exists a compact set $\mathcal{K}\subset \mathcal{M}_{1,2}$ so that $\underset{T \to \infty}{\limsup} \, \frac 1 T |\{0<t<T:g_tX \in \mathcal{K}\}|=c>0$. Let $D_1,...$ be a sequence chosen so that $$\frac 1 {D_i} |\{0<t<D_i:g_tX\in \mathcal{K}\}|>\frac {99c}{100}$$ and $$\underset{\zeta>D_1}{\sup}\, |\{0<t<\zeta: g_tX\in \mathcal{K}\}|-c<\frac c {100}.$$ The next lemma is used to obtain (A7). \[lem:cont in cpct\] For all $r>0$ for all $i$ large enough we have $$|\{t<D_i:g_tX \in \mathcal{K}, |\{s\in [t,t+r]:g_sX \in \mathcal{K}\}|>\frac c {99} r\}|>\frac {8c} {9}.$$ This is a standard application of the Vitali covering lemma. Indeed let $$B=\{t<D_i:g_tX \in \mathcal{K}, \lambda(\{s\in [t,t+r]:g_sX \in \mathcal{K}\})>\frac c {99} r\}$$ and so for each $t \in B$ we have $\lambda(\{s\in [t,t+r]:g_sX \in \mathcal{K}\})<\frac c {99}r.$ By applying the Vitali covering lemma to the intervals $[t,t+r]$ where $t\in B$, we may take a disjoint subcollection of these intervals $I_1,\dots I_\ell$ so that $$\label{eq:vitali}\lambda(\cup_{i=1}^\ell \{s\in I_i:g_s{X}\in \mathcal{K})>\frac 1 3 \lambda( \cup_{t \in B}\{s\in[t,t+r]:g_sX\in \mathcal{K})\geq \frac{\lambda(B)}3.$$ Indeed let $U_1=\{[t,t+r]:t\in B\}$ and choose $I_1$ to be an interval $[t,t+r]$ in this set so that $\lambda(\{s\in [t,t+r]:g_sX\in \mathcal{K}\}$ is maximal. Let $U_2=\{[t,t+r]:t \in B \text{ and }[t,t+r]\cap I_1=\emptyset$ and let $I_2$ be an interval $[t,t+r]$ in this set so that $\lambda(\{s\in [t,t+r]:g_sX\in \mathcal{K}\}$ is maximal. Also observe $$\lambda(\{s: s\in [\tau,\tau+r] \text{ with }\tau \in B, \, [\tau,\tau+r]\cap I_1\neq \emptyset \text{ and } g_sX\in \mathcal{K}\})\leq 2\lambda(\{s\in I_1:g_sX\in \mathcal{K}\}).$$ Repeating this procedure we obtain our intervals $I_1,\dots, I_\ell$. Having established (\[eq:vitali\]) we see at most $\frac c {99}r$ of the points in each interval are in $\mathcal{K}$ and the measure of the union of these intervals is at most $D_i$. This is a contradiction unless $\lambda(B)\leq \frac c {33}D_i+r$. By the same proof we obtain: \[lem:vit\]For all $\epsilon,\gamma>0$ there exists $\delta>0$ so that if $|A\cap[0,R]|>\gamma R$ then $$|\{t\in [0,R]\cap A:\lambda(\{t+s\in A\}_{t\in [0,T]})>\delta \gamma\}|>(1-\epsilon)\gamma R-2T.$$ Let $f(t)=\max\{j: q_j\leq e^{t}\}$. The next lemma is used to obtain (A8). \[lem:in cpct\] For all $r,\epsilon>0$ there exists $\hat{K}\subset \mathcal{M}_{1}$ so that for all $i$ large enough we have $$|\{t<D_i:g_tX\in \mathcal{K}, g_\ell \hat{X} \in \hat{K} \text{ for all }\ell\in [\log(q_{f(t+r)}),\log(q_{f(t+r)+1})]\}|\geq (1-\epsilon) \frac {99c} {100} D_i.$$ We use the following straightforward consequence of Lemma \[lem:vit\].\ **Sublemma:** If $h:[0,\infty)\to \{0,1\}$ has $ \frac 1 R \int_0^R h(t)\geq \frac{99c}{100}$ then for all $\epsilon'>0,r, \ell$ there exists $L$ so that $$\begin{gathered} |\{t<R:h(t)=1 \text { and there exists }0<\rho\leq r \text{ so that }\\ h(t+s)=0 \text{ for all but a set of measure $\ell$ of } s \in [t+\rho,t+\rho+L] \} |<\epsilon' cR+2L.\end{gathered}$$ Apply Lemma \[lem:vit\] with $\epsilon=\epsilon'$ and $\gamma =\frac {99c}{100}$ to obtain $\delta$. Choose $L$ so that $\delta L>\ell+r$. Let $\hat{C}$ be the compact set in $\mathcal{M}$ given by projecting $\mathcal{K}$ to $\mathcal{M}$ by forgetting the marked points. Let $\ell=\ell_{\hat{C}}$ as in Lemma \[lem:rot divergence\] and $\delta$ be the shortest simple closed curve on any surface in $\hat{C}$. Obtain $L$ from the sublemma with $r=r$, $\epsilon'=\frac{\epsilon}2$, $\ell=\ell$. Let $\hat{\mathcal{K}}$ denote the set of all tori whose shortest simple closed curve is at least $\delta e^{-L}$. The lemma holds for this $\hat{\mathcal{K}}$. Indeed, if $g_s\hat{X} \notin \hat{\mathcal{K}}$ then by examining the size of the shortest simple closed curve we see $g_{s+\tau}X\notin \mathcal{K}$ for all $-L<\tau<L$. That is, considering $A=\{s:g_sX \in \mathcal{K}\}$ and $\rho=\log(q_{f(t+r)})-t$ we are asking that $|\{s\in [t+\rho,t+\rho+L]:s\in A\}|<\ell$. So by the sublemma the set of such $t$ has small density and so we have the lemma. We now begin the derivation of (A5), (A6) and (A8). \[lemma:the:time\] For all $t$ there exists $-2\leq s\leq 2$ so that either - there exists $k$ with $a_{k+1}>4$ and $i\leq \lfloor \frac{a_{k+1}}4 \rfloor$ so that $e^{t+s}=iq_k$ - or there exists $k$ so that $e^{t+s}=q_k$. Let $j=f(t)$. If $a_{j+1}\leq4$ then since $e^2>5$ we may choose $s$ so that $e^{t+s}=q_j$ (and so $k=j$). If $a_{j+1}>4$ and $i>\frac {q_{j+1}}2$ choose $s$ so that $e^{t+s}=q_{j+1}$ (and so $k=j+1$). Otherwise choose $s$ so that $e^{s+t}=iq_j$ with $i\leq \lfloor \frac{a_{j+1}}4\rfloor$ (and so $k=j$). The following is a corollary of Lemma \[lemma:the:time\] and Lemma \[lemma:happy:times\]: \[cor:happy times\] For all $\eta>0$ there exists $\rho>0$ so that for all $t$, there exists $-2\leq s\leq 2$ with $\lambda(\psi_{e^{t+s}}(j))> \rho$ for some $j$ so that $$\begin{cases} \text{either } j-\min\{\ell: \psi_{e^{s+t}}^{-1}(\ell) \neq \emptyset\}\leq 2 \text{ and } \lambda(\psi_{e^{s+t}}^{-1}((0,j))<\rho \eta\\ \text{or } \max\{\ell:\psi_{e^{s+t}}^{-1}(\ell)\neq \emptyset\}-j\leq 2 \text{ and } \lambda(\psi_{e^{s+t}}^{-1}(j,\infty))<\rho\eta. \end{cases}$$ For the proof we use the following trivial result:\ **Sublemma:** For all $\eta>0$ there exists $\rho>0$ so that if $x_0,x_1,x_2 >0$ and $\sum x_i>\frac 1 {12}$ then there exists $i$ so that $x_i>\rho$ and $\sum_{j=0}^{i-1}x_j<\eta \rho$. Also there exists $\ell$ so that $x_\ell>\rho$ and $\sum_{i=\ell+1}^2 x_i<\eta \rho$. Applying Lemma \[lemma:the:time\] we obtain $e^{t+s}$. If $q_k=e^{t+s}$ then by Lemma \[lemma:DK\] we have $\psi_{q_k}$ takes at most 5 values that are consecutive. Letting $x_i$ be the measure of the $i^{th}$ level set and appying the sublemma implies the corollary. Otherwise by Lemma \[lemma:happy:times\] and the sublemma imply the corollary. The next lemma is used to obtain (A0)-(A4). Its proof is similar to Lemma \[lem:in cpct\] and is omitted. \[lem:A0 to A4\] Given any $\epsilon>0$ there exists $M$ so that $\lambda(E_2)=\lambda(\{t<D_i:g_tX\in \mathcal{K} \text{ and for all } s\in [-3,3] \text{ we have }a_{f(t+s)},a_{f(t+s)+1},a_{f(t+s)+2}<M \})>D_i(1-\epsilon)\frac{99c}{100}$. By choosing $\epsilon=\frac 1 9$ in Lemmas \[lem:in cpct\], \[lem:cont in cpct\] and \[lem:A0 to A4\], for each $r$, we may choose a sequence of $t$ going to infinity which is simultaneously in the three sets whose measure is bounded from below in these Lemmas. For each $t$ there exists $s$ as in Lemma \[lemma:the:time\] and this choice verifies (A6) and (A9). Consider $L=e^{s+t}$ and $c=e^{-r}$. Since $|s|<3$, by Lemma \[lem:A0 to A4\] assumptions (A0-4) hold for this $L$ and $c$. Indeed, $C_1,C_2,C_3=M$ and $C_4$ is $e^{-3}$ times the minimum of the shortest distance between the marked points taken over surfaces in $\mathcal{K}$. By Lemma \[lem:cont in cpct\] (and the fact that the projection of $\mathcal{K}$ to $\mathcal{M}_1$ is compact) (A7) holds. Indeed if $\hat{C}$ is the projection of $\mathcal{K}$ to $\mathcal{M}$ and $\frac{c}{99}r>\ell_{\hat{C}}(u+2)$ (where $\ell_{\hat{C}}$ is as in Lemma \[lem:rot divergence\]) then $L>q_{k+u}$. Moreover, by Corollary \[cor:happy times\] for each $\eta>0$ there exists $\hat{c}_\eta$ so that (A5) holds. We now just need to show (A8) holds. If $e^{s+t}\in \{q_{f(t)},q_{f(t)+1}\}$ this is by Lemma \[lemma:DK\]. Otherwise note $\psi_{nq_i}$ is at most $5+2n$ valued by Lemma \[lemma:happy:times\]. By Lemma \[lem:in cpct\] there exist $N_{r,\epsilon}$ so that $a_{f(t)+1}<N_{r,\epsilon}$ and thus $e^{s+t}=\ell q_j$ for some $\ell<N_{r,\epsilon}$. We obtain (A8) with $k_{e^{-r}}=5+2N_{r,\epsilon}$. The Sarnak conjecture and joinings of powers {#sec:appendix:A} ============================================ The following result is a trivial modification of a note [@Harper:note] of Harper, which is included for completeness. What is below is a lightly edited version of his note. See that note for connections with the work of other authors. Let $(X, T)$ be a topological dynamical system. Assume that there exists $C>1$ so that for every $n$, the set $B_n=\{m<n:T^m \text{ is not disjoint from }T^n\}$ has the property that if $m>m'\in B_n$ then $\frac m {m'}>C$ then $T$ is disjoint from Möbius. Indeed for any continuous compactly supported function with integral 0, $F$, we have $\sum_{n=1}^M \mu(n)F(T^nx)=o(M)$. Let $\mu$ denote the Möbius function. To prove Theorem 1 we shall require a lemma concerning the additive function $$\omega_\tau (n) := \underset{ p|n, p\leq e^{ \frac 1 \tau}}{\sum}1.$$ Define $\mu_\tau := \underset{ p\leq e ^{\frac 1 \tau}}{\sum}\frac 1 p$ and let $N$ be any natural number. Then we have the following variance estimate: $$\underset{ n\leq N}{\sum} (\omega_\tau (n) - \mu_\tau )^ 2 \leq N\mu_\tau + O(e^{ \frac 1 \tau }).$$ Lemma 1 is a special case of the Turán-Kubilius inequality, but since the proof is just a short calculation we shall give it in full. Expanding the sum in the statement we obtain $$\underset{ p,q\leq e ^{\frac 1 \tau}} {\sum} \underset{n\leq N}{\sum} 1_{p,q|n} - 2\mu_\tau \underset{ p\leq e^{ \frac 1 \tau}}{\sum} \lfloor \frac N p \rfloor + N\mu_\tau^ 2$$ and on removing the square brackets, and paying attention to the diagonal contribution in the double sum, we see that is at most $$\underset{ p,q\leq e^ {\frac 1 \tau}}{\sum} [N/pq] - N\mu_\tau^2 + N\mu_\tau + 2\mu_\tau \pi(e ^{\frac 1 \tau}),$$ Completion of proof ------------------- Let $F(n)=F(T^nx)$ and in view of Lemma 1 and the Cauchy-Schwarz inequality, we have that $$\begin{gathered} \label{eq:key estimate}|\sum_{n=1}^N \mu(n)F(n)|=\\ |\frac 1 {\mu_\tau} \sum_{n\leq N}\mu(n)F(n) \sum_{p |n, p\leq e^{\frac 1 \tau}}1 +\sum_{n=1}^N\mu(n)F(n) \frac{\mu_\tau-\omega_\tau(n)}{\mu_\tau}|\\\leq \frac 1 {\mu_\tau}|\sum_{n=1}^N \mu(n)F(n)\sum_{p|n,p\leq e^{\frac 1 \tau}}1|+\sqrt{\frac{N(N\mu_\tau+O(e^{\frac 1 \tau}))}{\mu_\tau^2}}.\end{gathered}$$ Observe that for each $\tau$ there exists $N_0$ so that for all $N>N_0$ we have $\sqrt{\frac{N(N\mu_\tau+O(e^{\frac 1 \tau}))}{\mu_\tau^2}}<\sqrt{N}\sqrt{\frac{2N}{\mu_\tau}}=N\sqrt{\frac{2}{\mu_\tau}}.$ So now we control $$|\sum_{n\leq N} \mu(n)F(n) \sum_{p| n, p<e^{\frac 1 \tau}} 1|\leq |\sum_{n\leq N}\sum_{p|n,p<e^{\frac 1 \tau}}\mu(p)\mu(\frac n p)F(n)|+\sum_{n\leq N}2|\{p<e^{\frac 1 {\tau}}:p^2|n\}|\cdot \|F\|_{\sup} .$$ Because $\sum_{n\leq N}2|\{p<e^{\frac 1 {\tau}}:p^2|n\}|$ is $O(N)$ we focus on the other term, $$\label{eq:appendix bound}| \sum_{n\leq N}\sum_{p|n,p<e^{\frac 1 \tau}}\mu(p)\mu(\frac n p)F(n)|\leq \sum_{j\leq \log_2(N)} \sum_{2^j\leq k<2^{j+1}}| \mu(k)\sum_{p\leq \min\{e^{\frac 1 \tau},\frac Nk\}} \mu(p)F(pk)|.$$ We apply Cauchy-Schwartz to $ \sum_{2^j\leq k<2^{j+1}}| \mu(k)\sum_{p\leq \min\{e^{\frac 1 \tau},\frac Nk\}} \mu(p)F(pk)|$ and bound (\[eq:appendix bound\]) by $$\begin{gathered} \sum_{j\leq\log_2(N)}\sqrt{2^j\sum_{2^j\leq m<2^{j+1}}|\sum_{p\leq \min\{e^{\frac 1 \tau},\frac N m\}}\mu(p)F(pm)|^2}\leq \\ \sum_{j\leq \log_2(N)} \sqrt{2^j \sum_{p_1,p_2\leq \min\{e^{\frac 1 \tau}, \frac N {2^j}\}} |\sum_{m=2^j}^{\min\{2^{j+1}, \frac N{p_1},\frac N {p_2}\}}F(p_1m)\overline{F(p_2m)} |}.\end{gathered}$$ The contribution of the diagonal terms ($p_1=p_2$) is at most $2^j \sqrt{\pi (\min \{e^{\frac 1 \tau}, \frac N {2^j})\}} \|F\|_{\sup} $ where $\pi(n)$ is the number of primes less than or equal to $n$. The contribution of the $p_1\neq p_2$ where $p_1$ and $p_2$ are not disjoint is at most $$2^j\sqrt{C \log (\min\{e^{\frac 1 \tau},\frac N {2^j}\})\pi (\min\{e^{\frac 1 \tau},\frac {N}{2^j})\}}\|F\|_{\sup}.$$ Summing over $j$ these terms give a contribution that is $O(N)$. Indeed, we estimate by $\sum_{j\leq \log_2(N)} 2^j\sqrt{C\log(\pi(\frac{N}{2^j}))\pi(\frac{N}{2^j})}\leq \sum_{j=1}^k C 2^j O((k-j)+1)2^{\frac 1 2 (k-j)}$ for $k=\lceil \log_2(N)\rceil$. This is clearly $O(N)$. For $\tau$ fixed we choose $M_0$ large enough so that for any $M>M_0$, $p_1,p_2<e^{\frac 1 \tau}$ with $T^{p_1}$ disjoint from $T^{p_2}$, and $L\leq M$ we have $$|\sum_{n\leq L}F(p_1n)\overline{F(p_2n)}|<\tau M.$$ The contribution of the $p_1,p_2$ where $T^{p_1}$ and $T^{p_2}$ are disjoint and $2^j>M_0$ is at most $$\sqrt{2^j(\tau 2^j) \pi(\min \{e^{\frac 1 \tau},\frac N {2^j}\})^2}.$$ For fixed $\tau$, summing over $j$, this is also $O(N)$. Indeed we focus on $$\sum_{j:\frac{N}{2^j}<e^{\frac 1 \tau}}\sqrt{2^j(\tau 2^j) \pi(\min \{e^{\frac 1 \tau},\frac N {2^j}\})^2}$$ and observe that this is bounded by $O(N\tau \log(\frac 1 {\tau}))$. If $N$ is large enough the terms when $2^j<M_0$ are also $O(N)$. Since $\mu_n \to \infty$ plugging this into the last line of (\[eq:key estimate\]) and possibly choosing an even larger $N$ so that $\sqrt{\frac{N(N\mu_\tau+O(e^{\frac 1 \tau}))}{\mu_\tau^2}}<N\sqrt{\frac{2}{\mu_\tau}}$ completes the proof. Disjointness of powers for generic $3$-IET’s {#sec:appendix:B} ============================================ \[thm:disjoint powers\]For almost every $3$-IET, $T$ we have that $T^n$ is disjoint from $T^m$ for all $0<n<m$. We prove this by the following straightforward disjointness criterion: \[prop:criterion\] Let $T$ be an ergodic 3-IET, $R$ be an irrational rotation and $0<n<m$ be natural numbers. Assume there exists $c>0$, $r\in \mathbb{N}$, a sequence $k_1,...$ sets $F_i$, $G_i$ so that for all $i$ 1. $\underset{i \to \infty}{\lim}\, \underset{x \in F_i}{\max} |T^{nk_i}x-x|=0$ 2. $\underset{i \to \infty}{\lim}\, \underset{x \in G_i}{\max} |T^{mk_i}x-R^{-1}x|=0$ 3. $1-\lambda(F_i)<\lambda(G_i)-c.$ Then $T^n$ and $T^m$ are disjoint. Let $\sigma$ be an ergodic joining of $T^n \times T^m$ that is a probability measure. Because is $T$ is ergodic it suffices to show that $\sigma$ is $id \times T^{-1}$ invariant. By the fact that ergodic probability measures are mutually singular or the same it suffices to show that $(id \times R^{-1})_*\sigma$ is not singular with respect to $\sigma$. By our assumptions, for any $i$ we have $\sigma(F_i \times G_i)\geq c $. Similarly to Section 2, $\sigma$ is not singular with respect to $(id \times R^{-1})_* \sigma$. For any $\alpha$ let ${\langle\langle q_j\alpha \rangle\rangle}=(-1)^j\|q_j\alpha\|$, the signed distance of $R^{q_j}x$ from $x$. If $x \in [0,1)$ there exists $b_1,...$ so that $b_i\leq a_i$, if $b_i=a_i$ then $b_{i+1}=0$ and $x=\sum_{i=1}^{\infty}b_i{\langle\langle q_{i-1}\alpha \rangle\rangle}$. Notice that for any fixed $\alpha$ the set of $x$ with (an allowable) Ostrowski expansion $b_1,...,b_k$ is an interval of size at least ${\| q_{k+1}\alpha\|}$. \[lem:all ostrowski\]Given a 3-IET consider it as rotation by $\alpha$ induced on an interval $[0,x)$. Let $[a_1,\dots]$ be the continued fraction of $\alpha$ and $(b_1,...)$ be the $\alpha$-Ostrowski expansion of $x$. For $\lambda^2$ almost every $(\alpha,x)$ we have that for any ordered k-tuple of pairs $(c_1,d_1),...,(c_k,d_k)$ of natural numbers so that $d_i\leq c_i-1$ we have that there are infinitely many $i$ with $((a_i,b_i),...,(a_{i+k-1},b_{i+k-1}))=((c_1,d_1),...,(c_k,d_k))$. For almost every $\alpha$ any $(k+1)$-tuple of natural numbers occurs infinitely often in its continued fraction expansion by the ergodicity of the Gauss map with respect to a fully supported finite invariant measure and the fact that having a fixed initial $(k+1)$-tuple $(c_1,\dots,c_{k+1})$ is a set of positive measure. For any $\alpha$ with this property, the set of $x$ so that the pair $(\alpha,x) $ satisfies the proposition is a set of full measure because the complement has no Lebesgue density points. Indeed let $\alpha$ have $a_{j+i}=c_i$ for $i\leq k+1$ and $y\in [0,1)$, then an interval of size at least $\|q_{j+k+1}\alpha\|$ in $B(y,\|q_j\alpha|)$ have that the $j+1$ through $j+k$ terms of their Ostrowski expansion are $d_1,\dots d_{k-1}$. Since $\frac{\|q_{j+k+1}\alpha\|}{\|q_j\alpha\|}>3^{-(j+1)}c_1\cdots c_{k+1}$ we have the claim. By Lemma \[lem:all ostrowski\] it suffices to show that any 3-IET given by inducing rotation by $\alpha$ on $[0,x)$ where the sequence $(a_1,b_1),...$ contains all $k$-tuples $(c_1,d_1),...,(c_k,d_k)$ with the condition that $c_i-1\geq d_i$ infinitely often satisfies the assumptions of Proposition \[prop:criterion\] for some $c>0$. Let $(10m,0),(10m,0),(4m,1),(10m,0)$ be the pairs of $($continued fraction expansion, Ostrowski expansion$)$. Let $\ell$ be an index so that $(a_{\ell+1},b_{\ell+1}),(a_{\ell+2},b_{\ell+2}),(a_{\ell+3}, b_{\ell+3}),(a_{\ell+4},b_{\ell+4})=(10m,0),(10m,0),(4m,1),(10m,0)$ and $\ell+2$ is even (this can be done because the 8-tuple $$(10m,0),(10m,0),(4m,1),(10m,0),(10m,0),(4m,1),(10m,0),(10m,0)$$ occurs infinitely often). Let $x_{\ell-i}=\sum_{i=0}^{\ell-1}b_i{\langle\langle q_i\alpha \rangle\rangle}$. Note there exists $r<q_\ell$ so that $x_{\ell-1}=R^{r}0.$ **Sublemma:** There exists $j$ so that $\lambda(\{x:\sum_{i=0}^{q_{\ell+2}-1}\chi_{[0,x_{\ell-1})}(R^ix)\neq j\})<\frac 1 {100m^3}.$ Let $\phi_\ell(x)=\sum_{i=0}^{q_{\ell+2}-1}\chi_{[0,x_{\ell-1})}(R^ix)$ If $\phi_\ell(x)\neq \phi_\ell (Rx)$ then $\chi_{[0,q_{\ell-1}(x))}(x)-\chi_{[0,x_{\ell-1})}(R^{q_k}x)\neq 0$. Since $\ell$ is even this means that $x \in [-\|q_{\ell+2}\alpha\|,0)\cup [x_{\ell-1}-\|q_{\ell+2}\alpha\|,x_{\ell-1}\|)$. Observe that $R^{r}([-\|q_{\ell+2}\alpha\|,0))=[x_{\ell-1}-\|q_{\ell+2}\alpha\|,x_{\ell-1})$. It follows that $\phi_\ell$ has two level sets, one on $\cup_{i=1}^{r} R^i([-\|q_{\ell+2}\alpha\|,0))$ and the other on its complement. $r\lambda([-\|q_{\ell+2}\alpha\|,0))\leq q_\ell \|q_{\ell+2}\|<\frac 1 {100m^3}$. Now consider $J=[0,x_{\ell-1}) \cup (J\setminus [0,x_{\ell-1})$. By the sublemma there exists $j$ so that $\lambda(\{x:\sum_{i=0}^{mq_{\ell+2}-1}\chi_{ [0,x_{\ell-1})}(R^ix)=mj\})>1-\frac m {100m^3}$ and $\lambda(\{x:\sum_{i=0}^{nq_{\ell+2}-1}\chi_{ [0,x_{\ell-1})}(R^ix)=nj\})>1-\frac n {100m^3}>1-\frac m {100m^3}$. Now since $b_{\ell+3}=0$ $$\begin{gathered} {\| q_{\ell+2}\alpha\|}-\frac 1 {10m}{\| q_{\ell+2}\alpha\|} <{\| q_{\ell+2}\alpha\|}-{\| q_{\ell+4}\alpha\|}\leq|(J\setminus [0,x_{\ell-1})|<\\ {\| q_{\ell+2}\alpha\|}+{\| q_{\ell+3}\alpha\|}<{\| q_{\ell+2}\alpha\|}+\frac 1 {10m}{\| q_{\ell+2}\alpha\|}\end{gathered}$$ and $R^i(J \setminus [0,x_{\ell-1})$ are disjoint for all $j<q_{\ell+3}$ we have $$\lambda(\{x:\sum_{i=0}^{mq_{\ell+2}-1}\chi_J(R^ix)=mj+1\})>mq_{\ell+2}{\| q_{\ell+2}\alpha\|}-\frac 1 {100m^2}-mq_{\ell+2}\frac 1 {10m}{\| q_{\ell+2}\alpha\|}$$ and $$\lambda(\{x:\sum_{i=0}^{nq_{\ell+2}-1}\chi_J(R^ix)=nj\})\geq 1-nq_{\ell+2}{\| q_{\ell+2}\alpha\|}-\frac 1 {100m^2}-nq_{\ell+2}\frac 1 {10m}{\| q_{\ell+2}\alpha\|}.$$ We have verified the assumptions of Proposition \[prop:criterion\] with $c=(m-n-\frac{m+n}{10m})\|q_{\ell+2}\alpha\|q_{\ell+2}-\frac{2}{100m^2}$. Since $q_{\ell+2}\|q_{\ell+2}\alpha\|>\frac{1}{10m+3}$ this is positive. [xxxxxxxx]{} M. Boshernitzan, A. Nogueira. Generalized eigenfunctions of interval exchange maps. [*Erg. Th. Dynam. Sys.*]{} [**24**]{} (2004), no. 3, 697–705. J. Bourgain. [*On the correlation of the Moebius function with rank-one systems.*]{} J. Anal. Math. 120 (2013), 105–130. J. Bourgain, P. Sarnak, T. Ziegler. [*Disjointness of Moebius from horocycle flows.*]{} From Fourier analysis and number theory to Radon transforms and geometry, pp. 67–83, Dev. Math., 28, Springer, New York, 2013. H. Davenport, [*On some infinite series involving arithmetical functions 2.*]{} Quart. J. of Math. 8 313–320. E. H. El Abdalaoui, M. Lemańczyk, T. de la Rue. [*On spectral disjointness of powers for rank-one transformations and Möbius orthogonality*]{} J. Funct. Anal. 266 (2014), no. 1, 284–317. B. Green, T. Tao. [*The Möbius function is strongly orthogonal to nilsequences.*]{} Ann. of Math. (2) 175 (2012), no. 2, 541–566. I. Katai, [*A remark on a theorem of H. Daboussi.*]{} Acta Math. Hungar. 47 (1986), no. 1-2, 223–225. A. Khinchin. [*Continued fractions.*]{} With a preface by B. V. Gnedenko. Translated from the third (1961) Russian edition. Reprint of the 1964 translation. Dover Publications, Inc., Mineola, NY, 1997. xii+95 pp. A. Harper. A different proof of a finite version of VinogradovÕs bilinear sum inequality. <https://www.dpmms.cam.ac.uk/~ajh228/FiniteBilinearNotes.pdf> M. Ratner. Horocycle flows, joinings and rigidity of products. [*Ann. of Math. (2)*]{} [**118**]{} (1983), no. 2, 277–313. D. Rudolph. [*Fundamentals of measurable dynamics.*]{} Ergodic theory on Lebesgue spaces. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1990. Z. Wang. [*Möbius disjointness for analytic skew products*]{}. Preprint http://arxiv.org/pdf/1509.03183.pdf.

--- abstract: 'We show that under gravity the effective masses for neutrino and antineutrino are different which opens a possible window of neutrino-antineutrino oscillation even if the rest masses of the corresponding eigenstates are same. This is due to CPT violation and possible to demonstrate if the neutrino mass eigenstates are expressed as a combination of neutrino and antineutrino eigenstates, as of the neutral kaon system, with the plausible breaking of lepton number conservation. In early universe, in presence of various lepton number violating processes, this oscillation might lead to neutrino-antineutrino asymmetry which resulted baryogenesis from the B-L symmetry by electro-weak sphaleron processes. On the other hand, for Majorana neutrinos, this oscillation is expected to affect the inner edge of neutrino dominated accretion disks around a compact object by influencing the neutrino sphere which controls the accretion dynamics, and then the related type-II supernova evolution and the r-process nucleosynthesis.' address: 'Department of Physics,Indian Institute of Science, Bangalore-560012, India ' author: - BANIBRATA MUKHOPADHYAY title: ' POSSIBLE NEUTRINO-ANTINEUTRINO OSCILLATION UNDER GRAVITY AND ITS CONSEQUENCES ' --- Introduction {#intro} ============ The neutrino oscillation, in the flat space, is due to difference in rest masses between two mass eigenstates. However, in late eighties, it was first pointed out [@gas] that presence of gravitational field affects different neutrino flavors differently which violates equivalence principle and thus governs oscillation, even if neutrinos are massless or of degenerate mass. The neutrino oscillation with LSND data [@mansar] indeed can be explained by degenerate or massless neutrinos with flavor non-diagonal gravitational coupling. It was further argued [@ab] that the flavor oscillation is possible in weak gravitational field with the probability phase proportional to the gravitomagnetic field. The oscillation was also shown to be feasible when the maximum velocities of different neutrino differ each other, even if they are massless [@cg]. All the above results are for flavor oscillation or/and without considering rigorous general relativistic effects. However, properties of neutrino in curved spacetime have already been discussed [@schw; @pal; @mukh] in literature. Here we address the [*neutrino$-$antineutrino oscillation*]{}, which violates lepton number conservation, focusing on the nature of space-time curvature and its special effect. While the neutrino$-$antineutrino oscillation under gravity is an interesting issue on its own right, the present result is able to address two long-standing mysteries in astrophysics and cosmology: (1) Source of abnormally large neutron abundance to support the r-process nucleosynthesis in astrophysical site. (2) Possible origin of baryogenesis. Oscillation probability {#prob} ======================== Let us recall the fermion Lagrangian density in curved spacetime [@schw; @mukh] $$\begin{aligned} {\cal L}=\sqrt{-g}\,\overline{\psi}\left[(i\gamma^a\partial_a-m)+\gamma^a\gamma^5 B_a\right]\psi ={\cal L}_f+{\cal L}_I, ~~~ \label{lagf}\end{aligned}$$ where $$\begin{aligned} B^d=\epsilon^{abcd} e_{b\lambda}\left(\partial_a e^\lambda_c+\Gamma^\lambda_{\alpha\mu} e^\alpha_c e^\mu_a\right), \,\,\,\, e^\alpha_a\, e^\beta_b\,\eta^{ab}=g^{\alpha\beta}, \label{bd}\end{aligned}$$ the choice of unit is $c=\ch=k_B=1$. ${\cal L}_I$ may be a CPT violating interaction and thus the corresponding dispersion energy [@mukh] for neutrino and antineutrino in standard model $$\begin{aligned} %\nonumber E_{\nu} = \sqrt{({\vec p} - {\vec B})^2 + m^2} + B_0, ~~~~ E_{\overline{\nu}} = \sqrt{({\vec p} + {\vec B})^2 + m^2} - B_0. \label{edis}\end{aligned}$$ Eq. (\[edis\]) tells us that under gravity neutrino energy is split up from antineutrino energy. The CPT status of ${\cal L}_I$ has been discussed in detail in our previous works [@mukh]. Now motivated by the neutral kaon system, we consider two distinct orthonormal eigenstates $|E_\nu>$ and $|E_{\overline \nu}>$ for a neutrino and an antineutrino type respectively. Further we introduce a set of neutrino mass eigenstates at $t=0$ as [@baren] $$\begin{aligned} |m_1>=cos\theta\, |E_\nu>+sin\theta\, |E_{\overline \nu}>,\hskip0.5cm |m_2>=-sin\theta\, |E_\nu>+cos\theta\, |E_{\overline \nu}>. \label{fl2}\end{aligned}$$ Therefore, in presence of gravity, the oscillation probability for $|m_1(t)>$ at $t=0$ to $|m_2(t)>$ at a later time $t=t_f$ can be found as $$\begin{aligned} %\hskip-1.5cm %\nonumber P_{12} %&=&\left|\left[-sin\theta\, <E_\nu|+cos\theta\, <E_{\overline \nu}|~\right]\left[e^{-iE_\nu t_1} %cos\theta\,|E_\nu>+e^{-iE_{\overline \nu} t_1}sin\theta\,|E_{\overline \nu}>\right]\right|^2\\ =sin^22\theta\,sin^2\delta,\,\,\,\,\,\, \delta=\frac{(E_\nu-E_{\overline \nu})t_f}{2}= \left[(B_0-|\vec{B}|)+\frac{\Delta m^2}{2|{\vec p}|}\right]\,t_f, \label{pab}\end{aligned}$$ where we consider ultra-relativistic neutrinos. Normally, the rest mass difference of particle and antiparticle is zero and thus possible $\delta\neq 0$ is mostly due to $B_a\neq 0$ i.e. due to gravitational coupling. Therefore, the neutrino-antineutrino oscillation may be possible in presence of gravity provided there is a lepton number violating process. If neutrinos exhibit Majorana mass, then lepton number violation is automatically taken care. Hence, the CPT violating nature of background curvature coupling generates effective mass difference, while lepton number violating process leads to oscillation between neutrino and antineutrino. The oscillation probability is maximum at $\theta=\pi/4$ and is zero at $\theta=0,\pi/2$. From eqn. (\[pab\]), the oscillation length, $L_{osc}$, by appropriately setting dimensions, is obtained as $$\begin{aligned} %L_{osc}\sim t_1=\frac{\pi}{B_0} %\label{ol} L_{osc}= c\,t_f=\frac{\pi\,\ch\,c}{\tilde{B}}\sim\frac{6.3\times 10^{-19}GeV}{\tilde{B}}{\rm km}, \label{old1}\end{aligned}$$ where $\tilde{B}=B_0-|\vec{B}|$ is expressed in GeV unit and the neutrino is considered to be moving in the speed of light. Consequence and Discussion {#dicu} ========================== One of the situations where the gravity induced neutrino-antineutrino oscillation may occur is the GUT era of anisotropic phase of early universe when $\tilde{B}\sim 10^5$ GeV [@mukh]. From eqn. (\[old1\]), this leads to $L_{osc}\sim 10^{-24}$km which is $10^{14}$ orders of magnitude larger than the Planck length. This has an important implication as the size of universe at the GUT era is within $\sim 10^{26}$ times of the Planck. Therefore, the oscillation may lead to leptogenesis and then to baryogenesis by electro-weak sphaleron processes due to $B-L$ conservation, what we see today. Another plausible region for an oscillation of this kind to occur is the inner accretion disk of the neutrino dominated accretion flow (NDAF) [@ndaf] around a rotating compact object which can be extended upto several thousand Schwarzschild radius. From eqns. (\[bd\]) and (\[old1\]) we can obtain $$\begin{aligned} B^0 =-\frac{4a\sqrt{M}z}{\bar{\rho}^2\sqrt{2r^3}},\,\,\,\,\, L_{osc}\sim \frac{1.8\,x^{7/2}\,M_s}{a\,H}{\rm km}=\frac{1.2\,x^{7/2}}{a\,H}M, \label{b0z}\end{aligned}$$ for the Kerr geometry, where $\bar{\rho}^2=2r^2+a^2-x^2-y^2-z^2$. The detailed calculation and discussion are presented elsewhere [@mukh2]. Here we choose the mass of the compact object $M=M_s\,M_\odot$, radius and height of the disk orbit where oscillation takes place respectively $r\sim\bar{\rho}= x\,M$ and $z=H\,M$, and we assure $\tilde{B}\sim B_0$. Any oscillation at the inner edge of NDAF is expected to be influenced by gravity what affects the accretion dynamics and outflow. From eqn. (\[b0z\]), $L_{osc}$ varies from a few factors to several hundreds of Schwarzschild radii at $x\le 10$ for a fast spinning compact object. Supernova is thought to be the astrophysical site of the r-process nucleosynthesis. During supernova, neutron capture processes for radioactive elements take place in presence of abnormally large neutron flux. However, how does the large neutron flux arise is still an open question. There are two related reactions: $$\,\,n+\nu_e\rightarrow p+e^-,\,\,\,\, \,\,\,\,\,p+\bar{\nu}_e\rightarrow n+e^+. \label{nuc}$$ If $\bar{\nu}_e$ is over abundant than $\nu_e$, then, from eqn. (\[nuc\]), neutron production is expected to be more than proton production into the system. Therefore, the possible conversion of $\nu_e$ to $\bar{\nu}_e$ due to the gravity induced oscillation explains the overabundance of neutron. [00]{} M. Gasperini, [*Phys. Rev.*]{} [**D 38**]{}, 2635 (1988). R. B. Mann, & U. Sarkar, [*Phys. Rev. Lett.*]{} [**76**]{}, 865 (1996). D. V. Ahluwalia, & C. Burgard, [*Gen. Rel. Grav.*]{} [**28**]{}, 1161 (1996); [*Phys. Rev.*]{} [**D57**]{}, 4724 (1998). S. R. Coleman, & S. L. Glashow, [*Phys. Lett.*]{} [**B405**]{}, 249 (1999). J. Schwinger, [*Particles, Sources and Fields*]{}, (Addison-Wesley, 1970). J. F. Nieves, & P. B. Pal, [*Phys.Rev.* ]{} [**D58**]{}, 096005 (1998). P. Singh, & B. Mukhopadhyay, [*Mod. Phys. Lett.*]{} [**A18**]{}, 779 (2003); B. Mukhopadhyay, [*Mod. Phys. Lett.*]{} [**A20**]{}, 2145 (2005); U. Debnath, B. Mukhopadhyay, & N. Dadhich, [*Mod. Phys. Lett.*]{} [**A21**]{}, 399 (2006). G. Barenboim, J. F. Beacom, L. Borissov, & B. Kayser, [*Phys. Lett.*]{} [**B537**]{}, 227 (2002). R. Narayan, T. Piran, & P. Kumar, [*Astrophys. J.*]{} [**557**]{}, 949 (2001); K. Kohri, R. Narayan, & T. Piran, [*Astrophys. J.*]{} [**629**]{}, 341 (2005). B. Mukhopadhyay, [*Class. Quantum Grav.*]{} [**24**]{}, 1433 (2007).

--- abstract: 'The parametrized post Newtonian formalism for 5-dimensional metric theories with a compact extra dimension is developed. The relation of the 5-dimensional and 4-dimensional formulations is then analyzed, in order to compare the higher dimensional theories of gravity with experiments. It turns out that the value of post Newtonian parameter $\gamma$ in the reduced 5-dimensional Kaluza-Klein theory is two times smaller than that in 4-dimensional general relativity. The departure is due to the existence of an extra dimension in the Kaluza-Klein theory. Thus the confrontation between the reduced 4-dimensional formalism and Solar system experiments raises a severe challenge to the classical Kaluza-Klein theory.' author: - 'Peng Xu[^1] and Yongge Ma[^2]' title: 'Five-dimensional PPN formalism and experimental test of Kaluza-Klein theory' --- As a candidate of fundamental theory, Kaluza-Klein (KK) theory unifies gravity with electromagnetic field (or Yang-Mills field) by certain higher dimensional general relativity (GR) [@kk] [@bla]. Since the original 5-dimensional (5D) KK theory was proposed by Kaluza [@kk2] and Klein [@kk1], considerable works have been done along this line [@duff] [@kk3] [@wesson]. The fantastic idea that our spacetime has extra dimension, promotes various higher dimensional theories, including the well-known string theory [@str]. Besides the potential function to unify the fundamental interactions, higher dimensional gravity theories are also shown to be effective in accounting for the dark constituent of the universe (see e.g. [@qq]). Given the fascinating virtues of extra dimensions, it becomes very desirable to confront higher dimensional theories of gravity with experiments. Works on this subject can be traced back to 1980’s [@pe] [@Div], while no agreement has been obtained in the literature. Different classes of solutions to higher dimensional GR are designed to represent Solar system (for soliton-like solution see [@we] [@we2] [@liu2000], for Schwarzschild-like solution see [@pe2] [@rah]). However, whether the available experimental data permit higher dimensional theories gets quite different answers in different approaches. These ambiguities are caused by the freedom in choosing higher dimensional solutions which are supposed to represent the Solar system in 4 dimensions. On the other hand, in 4-dimensional (4D) case, a general framework, called Canonical Parameterized Post-Newtonian (PPN) Formalism, was established by Nordtvedt, Will et al. [@nor] [@will1] [@will2] in 1970s as a basic tool to connect gravitational theories with the Solar system experiments. In PPN formalism, the perturbative metric of a gravitational theory, which is generated by the matter distribution of the Solar system, is expanded by orders in terms of linear combinations of post Newtonian potentials. The differences among various metric theories are represented by the coefficients (the PPN parameters) of these post Newtonian potentials. Because of its high accuracy and well-defined procedure, PPN formalism has attained great achievements in testing 4D metric theories by Solar system experiments [@wilbook] [@willliving]. Thus, some crucial issues arise naturally. Is there a higher-dimensional PPN formalism? If there is, what is the relation between the higher dimensional formalism and the 4D one? More crucially, can one test higher dimensional theories by the accurate Solar system experiments without the ambiguities motioned above? The purpose of this letter is to address these issues first in terms of 5D gravity theories with a compact extra dimension. A 5D PPN formalism will be developed. Its relation with the 4D formalism will be set up. As one will see without any ambiguities, the concrete analysis reveals a severe contradiction between KK theory and the Solar system experiments. The $5$D gravitational theories which we consider are defined on some 5-manifold with topology $\mathbf{M}^{4}\times\mathbf{S}^{1}$, where $\mathbf{S}^{1}$ is a compact extra dimension of radii $R$. Both gravity and matter fields are assumed to be distributed over the 5-manifold. Similar to 4D PPN formalism, the post Newtonian coordinates system is chosen as certain asymptotic (in 4D sense) flat system $\{t,x^{m}\},m=1,2,3,5$, where $x^{5}$ is the coordinate of extra space. Since the compactification radii $R$ is sufficiently small, a killing vector field $\xi^\mu$ arises naturally along the extra dimension in the low energy regime [@bla]. It is convenient to take an adapted coordinate system such that its fifth coordinate basis vector $(\frac{\partial}{\partial x^{5}})^{\mu}$ coincides with $\xi^\mu$. The 5-metric reads $\widetilde{g}_{\mu\nu}=\widetilde{\eta}_{\mu\nu}+\widetilde{h}_{\mu\nu }$ with signature (-,+,+,+,+), where $\widetilde{h}_{\mu\nu}$ is the perturbative metric generated by the matter distribution, e.g., the Solar system. The gauge is chosen so that the spatial part of $\widetilde{h}_{\mu \nu}$ is diagonal. As in Canonical PPN Formalism, we will expand $\widetilde{h}_{\mu\nu}$ by orders in terms of linear combinations of our generalized post Newtonian potentials which are functionals of matter variables. We assume that the matter composing the Solar system can be idealized as a perfect fluid. The matter variables which we considered for the 5D perfect fluid in Solar system include: 5D rest mass density $\widetilde{\rho}$, 5D pressure $\widetilde{p}$ for the matter flow, the ratio $\widetilde{\Pi}$ of 5D specific energy (including compressional energy, radiation, thermal energy, etc.) density to 5D rest mass density, and the coordinate velocity $\widetilde{v}^{m}$ of material particles or matter flow in post Newtonian frame. The first three 5D matter variables give the corresponding effective 4D matter variables as$$\int\sqrt{\widetilde{g}_{55}}\widetilde{\rho}dx^{5}=\rho,\text{\ }\int \sqrt{\widetilde{g}_{55}}\widetilde{p}dx^{5}=p\text{,\ }\int \sqrt{\widetilde{g}_{55}}\widetilde{\rho}\widetilde{\Pi}dx^{5}=\rho \Pi.\label{55}%$$ The general 5D post Newtonian potentials which we used for KK-like theories are $\widetilde{U},\widetilde{\Phi}_{1},\widetilde{\Phi}_{2},\widetilde{\Phi }_{3},\widetilde{\Phi}_{4},$ and $\widetilde{V}_{m}$, which satisfy respectively the 5D Poisson equations with respect to the flat spatial background as: $$\begin{aligned} \nabla^{2}\widetilde{U} =-\frac{16}{3}\pi\widetilde{G}\widetilde{\rho }, \ \ \nabla^{2}\widetilde{\Phi}_{1} =-\frac{16}{3}\pi\widetilde{G} \widetilde{\rho}v^{2}, \nonumber \\ \nabla^{2}\widetilde{\Phi}_{2} =-\frac{16}{3}\pi\widetilde{G} \widetilde{\rho}\widetilde{U},\ \ \nabla^{2}\widetilde{\Phi}_{3} =-\frac{16}{3}\pi\widetilde{G} \widetilde{\rho}\widetilde{\Pi}, \nonumber \\ \nabla^{2}\widetilde{\Phi}_{4} =-\frac{16}{3}\pi\widetilde{G}\widetilde {p}, \ \ \nabla^{2}\widetilde{V}_{m} =-\frac{16}{3}\pi\widetilde{G}\widetilde{\rho }\widetilde{v}_{m},\nonumber\end{aligned}$$ where $\widetilde{G}$ denotes the 5D gravitational constant and we use the unit where the velocity of light $c=1$. Note that one may add more potentials in this framework in order to consider more complicated 5D theories. Note also that the upper bound of the compactification radii $R$ is constrained by the tests of gravitational inverse-square law to be about $10^{-4}% %TCIMACRO{\unit{m}}% %BeginExpansion \operatorname{m}% %EndExpansion $ [@liu], which is sufficiently small compared with the characteristic length $10^{12}% %TCIMACRO{\unit{m}}% %BeginExpansion \operatorname{m}% %EndExpansion $ of Solar system. With this condition we can estimate the order relations of matter variables and potentials. Since $|\widetilde{v}|\ll1$, we denote its order of smallness as $\widetilde{v}\sim\mathcal{O}(1)$. Note that in the adapted coordinate system the 5-metric components take the form [@kk2] [@ma]:$$\widetilde{g}_{\mu\nu}=\left( \begin{array} [c]{cc}% g_{\alpha\beta}+\phi B_{\alpha}B_{\beta} & \phi B_{\alpha}\\ \phi B_{\beta} & \phi \end{array} \right) ,$$ where $\alpha,\beta=0,1,2,3$. Thus, the “effective” 4-spacetime can be understood as $(M^{4},g_{\alpha\beta})$ with the local coordinate system $\{x^{\alpha}\}$ [@ma] [@yang]. Denote the 5-velocity of a test particle as $\widetilde{U}^{\mu}$, then the 4-velocity of the particle in $M^{4}$ is defined as [@ma] $$U^{\alpha}=\frac{\widetilde{U}^{\alpha}}{\sqrt{-\widetilde{U}^{\alpha }\widetilde{U}_{\alpha}}},\label{u4}%$$ where $\widetilde{U}^{\alpha}\widetilde{U}_{\alpha}\equiv g_{\alpha\beta }\widetilde{U}^{\alpha}\widetilde{U}^{\beta}$. From Eq.(\[u4\]) one can estimate the order relation between the coordinate velocities in five and four dimensions as $\widetilde{v}^{i}=v^{i}+\mathcal{O}(3)$. From Virial’s theorem we have $\widetilde{v}^{2}\sim\widetilde{U}\sim\mathcal{O}(2).$ Since the scale of the extra dimension is very small, one can approximate the solution of 5D Poisson equations by that of the corresponding 4D equations. Hence the Newtonian gravitational potentials in five and four dimensions are of the same order, i.e., $\widetilde{U}\sim U$. The order relations between the matter variables in five and four dimensions can be estimated from Eq.(\[55\]) as $\widetilde{\Pi}\sim\Pi$ and $\frac{\widetilde{p}}% {\widetilde{\rho}}\sim\frac{p}{\rho}$. Therefore, in the light of the order relations in 4D PPN theory [@wilbook], we obtain $\widetilde{U}% \sim\widetilde{\Pi}\sim U\sim\Pi\sim\mathcal{O}(2)$ and $\frac{\widetilde{p}% }{\widetilde{\rho}}\sim\frac{p}{\rho}\sim O(2).$ Moreover, the $5$D continuous equation of perfect fluid ensures $\frac{|\partial/\partial t|}{|\partial /\partial x|}\sim\mathcal{O}(1).$ With all these instruments we can parametrize any 5D metric theories. Just as in canonical 4D PPN framework [@wilbook], to get nontrivial results, we should expand the components of a metric in terms of the linear combinations of our generalized post Newtonian potentials to the following orders: $\widetilde{g}_{00}\sim\mathcal{O}% (4),\widetilde{g}_{0m}\sim\mathcal{O}(3),\widetilde{g}_{mn}\sim\mathcal{O}(2)$. The concrete relations between the 5D post Newtonian potentials and the 4D ones can be worked out by means of the Green function. Let $|\overrightarrow {x}-\overrightarrow{x}^{\prime}|$ be the spatial distance between the source and field points in the post Newtonian coordinate system measured by the 4D flat spatial metric, and $|\vec{x}-\vec{x}^{\prime}|$ be its 3D projection. When $|\overrightarrow{x}-\overrightarrow{x}^{\prime}|\gg R$, the Green function $\widetilde{\mathbf{G}}(\overrightarrow{x},\overrightarrow{x}% ^{\prime})$ of the 5D Poisson equation can be approximated as [@FL]$$\widetilde{\mathbf{G}}(\overrightarrow{x},\overrightarrow{x}^{\prime}% )=\frac{G}{|\vec{x}-\vec{x}^{\prime}|}+\frac{2G}{|\vec{x}-\vec{x}^{\prime}|}e^{-\frac{|\vec{x}-\vec{x}^{\prime}|}{R}},$$ where $G$ is the 4D gravitational constant. Thus we have$$\begin{aligned} \widetilde{U}(\overrightarrow{x}) = \int\widetilde{\mathbf{G}}% (\overrightarrow{x},\overrightarrow{x}^{\prime})\widetilde{\rho}% (\overrightarrow{x}^{\prime})dx^{3\prime}dx^{5\prime}=U(\overrightarrow{x})-\gamma\Phi_{2}(\overrightarrow{x}) +\mathcal{O}(6),\end{aligned}$$ where we used in general $\widetilde{g}_{55} = 1+2\gamma\widetilde {U}+\mathcal{O}(4)$. By similar ways, we obtain the following relations: $$\begin{aligned} \widetilde{\Phi}_{1} & =\Phi_{1}+\mathcal{O}(6)=\int\frac{G\rho(\vec {x}^{\prime})v^{2}(\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}%&| d^{3}x^{\prime}+\mathcal{O}(6),\label{re22}\\ \widetilde{\Phi}_{2} & =\Phi_{2}+\mathcal{O}(6)=\int\frac{G\rho(\vec {x}^{\prime})U(\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}d^{3}x^{\prime }+\mathcal{O}(6),\label{re33}\\ \widetilde{\Phi}_{3} & =\Phi_{3}+\mathcal{O}(6)=\int\frac{G\rho(\vec {x}^{\prime})\Pi(\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}d^{3}x^{\prime }+\mathcal{O}(6),\\ \widetilde{\Phi}_{4} & =\Phi_{4}+\mathcal{O}(6)=\int\frac{Gp(\vec{x}^{\prime })}{|\vec{x}-\vec{x}^{\prime}|}d^{3}x^{\prime}+\mathcal{O}(6),\\ \widetilde{V}_{i} & =V_{i}+\mathcal{O}(5)=\int\frac{G\rho(\vec{x}^{\prime })v_{i}(\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}d^{3}x^{\prime }+\mathcal{O}(5),\\ \widetilde{V}_{5} & =\int\frac{G\rho(\vec{x}^{\prime})\widetilde{v}_{5}% (\vec{x}^{\prime})}{|\vec{x}-\vec{x}^{\prime}|}d^{3}x^{\prime}+\mathcal{O}(5).\end{aligned}$$ The procedure of parametrizing 5D theories is similar to that of 4D ones [@wilbook]. Here we just outline the main steps and key points. The field equation of KK theory with matter fields reads$$\widetilde{R}_{\mu\nu}-\frac{1}{2}\widetilde{g}_{\mu\nu}\widetilde{R}% =8\pi\widetilde{G}\widetilde{T}_{\mu\nu},\label{kkequ}%$$ where $\widetilde{T}_{\mu\nu}=(\widetilde{\rho}+\widetilde{\rho}\widetilde {\Pi}+\widetilde{p})\widetilde{U}_{\mu}\widetilde{U}_{\nu}+\widetilde {p}\widetilde{g}_{\mu\nu}$ is the energy-momentum tensor of the 5D perfect fluid. Eq.(\[kkequ\]) is equivalent to$$\widetilde{R}_{\mu\nu}=8\pi\widetilde{G}(\widetilde{T}_{\mu\nu}-\frac{1}% {3}\widetilde{g}_{\mu\nu}\widetilde{T}),\label{kkeq2}$$ where $\widetilde{T}\equiv\widetilde{g}^{\mu\nu}\widetilde{T}_{\mu\nu}.$ The Ricci tensor can be expanded in terms of the perturbative metric to the necessary order around the flat background as$$\begin{aligned} \widetilde{R}_{00} & =-\frac{1}{2}\nabla^{2}\widetilde{h}_{00}-\frac{1}% {2}\sum_{m}\partial_{0}\partial_{0}\widetilde{h}_{mm}+\partial_{m}\partial _{0}\widetilde{h}_{m0}\nonumber\\ & +\frac{1}{2}\partial_{m}\widetilde{h}_{00}(\partial _{n}\widetilde{h}_{mn}-\frac{1}{2}\sum_{n}\partial_{m}\widetilde{h}% _{nn}) -\frac{1}{4}\nabla\widetilde{h}_{00}\cdot\nabla\widetilde{h}_{00}\nonumber\\ & +\frac {1}{2}\widetilde{h}_{mn}\partial_{m}\partial_{n}\widetilde{h}_{00}+\frac{1}% {2}(\partial_{m}\widetilde{h}_{n0}\partial_{m}\widetilde{h}_{n0}-\partial _{m}\widetilde{h}_{n0}\partial_{n}\widetilde{h}_{m0}),\label{R00}%\end{aligned}$$$$\begin{aligned} \widetilde{R}_{mn} & = -\frac{1}{2}(\nabla^{2}\widetilde{h}_{mn}-\partial_{m}\partial _{n}\widetilde{h}_{00}+\sum_{l}\partial_{m}\partial_{n}\widetilde{h}% _{ll}-\partial_{l}\partial_{n}\widetilde{h}_{ml}\nonumber\\ & -\partial_{l}\partial _{m}\widetilde{h}_{nl}),\label{rij}%\end{aligned}$$$$\begin{aligned} \widetilde{R}_{0m} = -\frac{1}{2}(\nabla^{2}\widetilde{h}_{0m}-\partial_{n}\partial _{m}\widetilde{h}_{0n}+\sum_{n}\partial_{0}\partial_{m}\widetilde{h}% _{nn}-\partial_{n}\partial_{0}\widetilde{h}_{mn}).\label{r0i}%\end{aligned}$$ The components of the perturbative metric can be solved order by order. - $\widetilde{h}_{00}$ to $\mathcal{O}(2)$: To the required order, $$\widetilde{R}_{00}\approx-\frac{1}{2}\nabla^{2}\widetilde{h}_{00},\text{ \ \ }\widetilde{T}_{00}=-\widetilde{T}=\widetilde{\rho},\text{ \ \ }% \widetilde{g}_{00}=-1.$$ Thus we have $$\nabla^{2}\widetilde{h}_{00}=-\frac{32}{3}\pi\widetilde{G}\widetilde{\rho},\text{ \ \ \ \ } \widetilde{h}_{00} =2\widetilde{U}.$$ Which justifies that $\widetilde{U}$ is the 5D Newtonian potential. Note that the $\mathcal{O}(2)$ term of $\widetilde{h}_{00}$ should be same for any 5D metric theories in order to have the same 5D Newtonian limitation. - $\widetilde{h}_{mn}$ to $\mathcal{O}(2)$: Here we impose the gauge condition$$\frac{1}{2}\partial_{m}\widetilde{h}_{\mu}^{\mu}-\partial_{\mu}\widetilde {h}_{m}^{\mu}=0.$$ From Eq.(\[rij\]) we have$$\widetilde{R}_{mn}=-\frac{1}{2}\nabla^{2}\widetilde{h}_{mn},$$ and then $$-\frac{1}{2}\nabla^{2}\widetilde{h}_{mn}=\frac{8\pi}{3}\widetilde {G}\widetilde{\rho}\delta_{mn}.\nonumber$$ Hence we gets$$\widetilde{h}_{mn}=\widetilde{U}\delta_{mn}=U\delta_{mn}.\label{hmn}%$$ - $\widetilde{h}_{0m}$ to $\mathcal{O}(3)$: By imposing the gauge condition$$\frac{1}{2}\partial_{0}\widetilde{h}_{\mu}^{\mu}-\partial_{\mu}\widetilde {h}_{0}^{\mu}=\frac{1}{2}\widetilde{h}_{00,0},\label{gauge}%$$ from Eq.(\[r0i\]) we get $$\widetilde{R}_{0m}=-\frac{1}{2}\nabla^{2}\widetilde{h}_{0m},$$ and thus$$-\frac{1}{2}\nabla^{2}\widetilde{h}_{0m}+\widetilde{U}_{,0m}=-8\pi \widetilde{G}\widetilde{\rho}\widetilde{v}^{m}.$$ Hence we obtain$$\widetilde{h}_{0i}=-\frac{5}{2}V_{i}-\frac{1}{2}W_{i},\text{ \ }% h_{05}=3\widetilde{V}_{5},\label{h0m}%$$ where $W_{i}\equiv\int\frac{G\rho(\vec{x}^{\prime})[\vec{v}^{\prime}\cdot (\vec{x}-\vec{x}^{\prime})](x_{i}-x_{i}^{\prime})}{|\vec{x}-\vec{x}^{\prime }|^{3}}d^{3}x^{\prime}$ is another 4D post Newtonian potential [@wilbook]. - $\widetilde{h}_{00}$ to $\mathcal{O}(4)$: To evaluate this part we use all the lower-order solutions of $h_{\mu\nu}$. From Eqs.(\[R00\]), (\[hmn\]) and (\[h0m\]) we get$$\widetilde{R}_{00}=-\frac{1}{2}\nabla^{2}\widetilde{h}_{00}-\nabla ^{2}\widetilde{U}^{2}+3\nabla^{2}\widetilde{\Phi}_{2}.$$ Thus from Eq.(\[kkeq2\]) we have $$\nabla^{2}\widetilde{h}_{00}=2\nabla^{2}\widetilde{U}-2\nabla^{2}\widetilde {U}^{2}+3\nabla^{2}\widetilde{\Phi}_{1}+2\nabla^{2}\widetilde{\Phi}% _{2}+2\nabla^{2}\widetilde{\Phi}_{3}+4\nabla^{2}\widetilde{\Phi}_{4},\nonumber$$ and hence$$\widetilde{h}_{00}=2\widetilde{U}-2\widetilde{U}^{2}+3\widetilde{\Phi}_{1}+\widetilde{\Phi}_{2}+2\widetilde{\Phi}_{3} +4\widetilde{\Phi}_{4}.$$ Now we are facing the problem how to relate the parametrized KK theory to the experiments. For most gravitational experiments in Solar system, we may consider only the free test particles without electric charge. From the viewpoint of the KK theory, this implies that the test particles do not mover along the extra dimension, i.e., $\widetilde{U}^{\mu}\xi_{\mu}=0$. In this case, it is easy to show that the 5D geodesic equations for both massive and massless test particles are reduced to the 4D geodesic equations in the effective 4D spacetime. Thus the reduced 4D theory behaves just like a metric theory for these particular test particles or photons. Along the reduction procedure previously discussed, we can reduce the parametrized 5-metric to the effective 4-metric $g_{\alpha\beta}$ as$$\begin{aligned} g_{00} & =-1+2U-2U^{2}+3\Phi_{1}+\Phi_{2}+2\Phi_{3}+4\Phi_{4}\label{kk00},\\ g_{0i} & =-\frac{5}{2}V_{i}-\frac{1}{2}W_{i}\label{kk0i},\\ g_{ij} & =(1+U)\delta_{ij}.\label{kkij}%\end{aligned}$$ According to the general form of the post Newtonian metric [@wilbook] $$\begin{aligned} g_{00}&=&-1+2U-2\beta U^2-2\xi\Phi_{w}\nonumber\\ &&+(2\gamma+2+\alpha_{3}+\zeta_{1}-2\xi)\Phi_{1}\nonumber\\ && +2(3\gamma-2\beta+1+\zeta_{2}+\xi)\Phi_{2}+2(1+\zeta_{3})\Phi_{3}\nonumber\\ &&+2(3\gamma+3\zeta_{4}-2\xi)\Phi_{4}-(\zeta_{1}-2\xi)\mathcal{A}\nonumber\\ &&-(\alpha_{1}-\alpha_{2}-\alpha_{3})w^{2}U-\alpha_{2}w^{i}w^{j}U_{ij} \nonumber\\ &&+(2\alpha_{3}-\alpha_{1})w^{i}V_{i}+\mathcal{O}(6),\\ g_{0i}&=&-\frac{1}{2}(4\gamma+3+\alpha_{1}-\alpha_{2}+\zeta_{1}-2\xi)V_{i}\nonumber\\ &&-\frac{1}{2}(1+\alpha_{2}-\zeta_{1} +2\xi)W_{i}-\frac{1}{2}(\alpha_1-2\alpha_2)w^{i}U\nonumber\\ &&-\alpha_2 w^j U_{ij}+\mathcal{O}(5),\\ g_{ij}&=&(1+2\gamma U)\delta_{ij}+\mathcal{O}(4),\end{aligned}$$ the relevant post Newtonian parameters for the reduced KK theory and 4D GR are compared in Table 1. ------------ --------------- --------- ------- ------------------------------------ -------------------------------------------- **Theory** $\gamma$ $\beta$ $\xi$ ($\alpha_1$,$\alpha_2$,$\alpha_3$) ($\zeta_1$,$\zeta_2$,$\zeta_3$, $\zeta_4$) **GR** 1 0 0 (0,0,0) (0,0,0,0) **KK** $\frac{1}{2}$ 0 0 (0,0,0) (0,0,0,$\frac{1}{6}$) ------------ --------------- --------- ------- ------------------------------------ -------------------------------------------- : The PPN parameters of GR and KK Therefore, given the same (reduced) 4D matter distribution such as a 4D perfect fluid, the detail comparison between the above two theories leads to significant conclusions. Firstly, the metric component $g_{00}$ in the post Newtonian coordinates system in KK theory is smaller than that in 4D GR. But the departure appear only in $\mathcal{O}(4)$ terms, and hence the reduced 5D KK theory has the right Newtonian limitation. Secondly, the metric component $g_{0i}$ in KK theory are $\frac{7}{5}$ times smaller than those in 4D GR in an $\mathcal{O}(3)$ term. This departure may in principle be detected by the current precise gravitational experiments in Solar system. At last, the metric components $g_{ij}$ together with $g_{00}$ and $g_{0i}$ in KK theory determine the post Newtonian parameter $\gamma = \frac{1}{2}$, which is obviously different from $\gamma = 1$ in 4D GR. It is obvious that the above departure is due to the existence of an extra dimension in KK theory. The disaster of KK theory is that the value of the parameter $\gamma$ has been accurately measured by Solar system experiments. In time delay experiment one obtains $\gamma -1= (2.1\pm2.3)\times10^{-5}$ [@r3] [@willliving], and in light deflection experiment one gets $\gamma -1=(-1.7\pm4.5)\times10^{-4}$ [@r1] [@r2] [@willliving]. Hence there is a severe contradiction between 5D KK theory and the Solar system experiments. Our PPN formalism and related discussion can be generalized straightforwardly to higher dimensional KK theories with compact extra dimensions. Therefore, although the original idea of Kaluza and Klein is rather beautiful, the classical KK theories can not survive the experiments. Acknowledgements {#acknowledgements .unnumbered} ================ This work is a part of project 10675019 supported by NSFC. [99]{} T. Appelquist, A. Chodos, and P. G. O. Freund, ed., *Modern Kaluza-Klein Theories*, (Addison-Wesley, Reading, MA, 1986). M. Blagojevic, *Gravitation and Gauge Symmetries*, (IOP Publishing, Bristol, 2002). T. Kaluza, Sitz. Preuss. Akad. Wiss. **33**, 966 (1921). O. Klein, Z. Phys. **37,** 895 (1926). M.J. Duff, B.E.W. Nilsson, and C.N. Pope, Phys. Rep. [**1**30]{}, 1 (1986). J. M. Overduin and P. S. Wesson, Phys. Pep. **283,** 305 (1997). P.S. Wesson, *Space-Time-Matter: Modern Kaluza-Klein Theory*, (World Scientific, Singapore, 1999). J. Polchinski, *String Theory I, II*, (Cambridge University Press, 1998). L. Qiang, Y. Ma, M. Han and D. Yu, Phys. Rev. D **71,** 061501(R) (2005). D.J. Gross, M.J. Perry, Nucl. Phys. B **226**, 29 (1983). D. Davidson, D. Owen, Phys. Lett. B **155**, 247 (1985). D. Kalligas, P. S. Wesson and C. W. F. Everitt, Astrophys. J. **439**, 548 (1995). P. H. Lim, J. M. Overduin, and P. S. Wesson, J. Math. Phys. **36**, 6907 (1995). H. Liu, and J. M. Overduin, Astrophys. J. **538**, 386 (2000). R. Mayers and M. Perry, Annal. Phys. **172**, 304 (1986) F. Rahaman, S. Ray, M. Kalam and M. Sarker, arXiv: 0707.0951v2. K.J. Nordtvedt, Astrophys.J. **161**, 1059 (1970). K.S. Throne and C.M. Will, Astrophys. J. **163**, 595 (1971). C.M. Will, Astrophys. J. **163**, 611 (1971); **169**, 125 (1971). C.M. Will, *Theory and experiment in gravitational physics*, Revised edition (Cambridge University Press, 1993). C.M. Will, *The Confrontation between General Relativity and Experiment*, Living Rev. Relativity **9** (2006), 3; http://www.livingreviews.org/lrr-2006-3. C. D. Hoyle et al, Phys. Rev. D **70**, 042004 (2004). Y. Ma and J. Wu, Int. J. Mod. Phys. A **19**, 5043 (2004). X. Yang, Y. Ma, J. Shao, and W. Zhou, Phys. Rev. D **68**, 024006 (2003). E.G. Floratos and G.K. Leontaris, Phys. Lett. B **465**, 95 (1999). B. Bertotti, L. Iess, and P. Tortora, Nature **425**, 374 (2003). S.S. Shapiro, J.L. Davis, D.E. Lebach, and J.S. Gregory, Phys. Rev. Lett. **92**, 121101 (2004). R.N. Treuhaft, and S.T. Lowe, Astron. J. **102**, 1879 (1991). [^1]: Email: moooonbird@gmail.com [^2]: Email: mayg@bnu.edu.cn

--- abstract: 'By precisely monitoring the “ticks" of Nature’s most precise clocks (millisecond pulsars), scientists are trying to detect the “ripples in spacetime" (gravitational waves) produced by the inspirals of supermassive black holes in the centers of distant merging galaxies. Here we describe a relatively simple demonstration that uses two metronomes and a microphone to illustrate several techniques used by pulsar astronomers to search for gravitational waves. An adapted version of this demonstration could be used as an instructional laboratory investigation at the undergraduate level.' author: - 'Michael T. Lam' - 'Joseph D. Romano' - 'Joey S. Key' - Marc Normandin - 'Jeffrey S. Hazboun' title: 'An acoustical analogue of a galactic-scale gravitational-wave detector' --- Introduction {#s:introduction} ============ A [*pulsar timing array*]{} is a galactic-scale gravitational-wave detector, which can be used to search for gravitational waves from the inspiral of supermassive black-hole binaries (of order $10^9$ solar masses) in the centers of distant galaxies[@PTACQG; @Detweiler1979; @hd1983]. The array consists of a set of galactic millisecond [*pulsars*]{}—rapidly-rotating neutron stars, which have masses of order the mass of the Sun and magnetic fields of order a billion times stronger than that of the Earth[@handbook]. Millisecond pulsars rotate nearly a thousand times each second (faster than a kitchen blender), emitting a narrow beam of radio waves along the magnetic axes that sweep across the sky similar to a revolving beacon on top of a lighthouse. If this radio beam crosses our line of sight to the pulsar, a radio telescope on Earth will observe pulses of radiation, which arrive with a regularity that rivals (or even exceeds) that of the best atomic clocks[@Hobbs+2012]. By precisely monitoring the pulse arrival times, radio astronomers can determine what the rotation period of the pulsar is, how the rotation is slowing down, whether the pulsar is orbiting a companion star, as well as how the interstellar medium affects the propagation of the pulses[@handbook]. The difference between the [*measured*]{} times of arrival and the [*expected*]{} times of arrival (taking all of these effects into account) are called [*timing residuals*]{}. If the pulsar timing model is good, the residuals should be randomly scattered around zero with a root-mean-square (rms) amplitude determined by measurement noise in the radio receiver and statistical fluctuations in the pulses themselves. The residuals for an individual pulsar may be correlated in time[@cd1985; @IPTADR1noise; @NG9EN; @CordesCQG] (so-called red noise), but residuals associated with different Earth-pulsar baselines should not be correlated with one another in the absence of any common external influence. Deviations from this expected behavior could be due to either an incomplete timing model (e.g., not realizing that the pulsar is in a binary) or the presence of gravitational waves[@ERA]. A gravitational wave passing between the Earth and a pulsar will stretch and squeeze space transverse to its motion, slightly advancing or retarding the arrival times of the individual pulses[@ew1975]. Unlike the measurement noise or intrinsic pulsar timing noise discussed above, the modulation of pulse arrival times induced by a gravitational wave will be [*correlated*]{} across different pulsars in the array, due to its common influence in the vicinity of the Earth. Moreover, this correlation will have a very specific dependence on the angle between a pair of Earth-pulsar baselines, the so-called [*Hellings and Downs curve*]{}[@hd1983] shown in Figure \[f:HDcurve\]. ![Expected correlation coefficient between the timing residuals for a pair of Earth-pulsar baselines separated by angle $\zeta$.[]{data-label="f:HDcurve"}](HDcurve_flat){width="60.00000%"} The detection of this expected correlation in the timing residuals from an array of pulsars would be evidence for the presence of gravitational waves, similar to the recent detections by the advanced LIGO and Virgo detectors[@GW150914; @LIGOPapers; @GW170817]. Metronomes and microphones {#s:metronomes_microphones} -------------------------- In order to illustrate how gravitational-wave astronomers are using correlation methods to search for gravitational waves, we have developed a demonstration using metronomes and a microphone, which serves as an [*acoustical analogue*]{} of a pulsar timing array. In this demonstration, radio pulses from an array of Galactic pulsars are represented by ticks of an array of metronomes (only two metronomes are needed for this demonstration); radio receivers on Earth are represented by a single microphone; and the passage of a gravitational wave is represented by the motion of the microphone around its nominal position. The analogy is not perfect as the motion of the microphone does not represent a wave of any kind, and the correlations that it induces have a different angular dependence than that induced by a real gravitational wave[@jr2015]. But what is important is that there [*are*]{} correlations, as the microphone motion modulates the arrival times of the metronome pulses by changing the distance between the metronomes and the microphone. And although the angular dependence of the correlations for the microphone motion is different than that for gravitational waves, it is, nonetheless, a specific function of the angle between a pair of microphone-metronome baselines, which can be calculated theoretically and also verified experimentally by doing the demonstration. In the following sections, we will describe the metronome-microphone demonstration in detail. In Section \[s:hardware\_software\], we describe the specific hardware (i.e., metronomes and microphone) and software routines that we use to do the analysis. In Section \[s:techniques\], we list the techniques used in real pulsar timing analyses that are illustrated by the demonstration. They can be thought of as the *learning outcomes* for the demonstration. In Sections \[s:analysis1\] and \[s:analysis2\], we discuss the two main parts of the demonstration (the single-metronome and double-metronome analyses), listing the steps needed to perform the analysis and the function of the graphical user interface (GUI) buttons used to execute each step. In Section \[s:discussion\], we conclude with a discussion of some caveats and possible improvements to the demonstration, and how it might be adapted for use in the collection of high school and undergraduate laboratory[@Rubbo+2007; @Newburgh2008; @fsa2015; @Burko2017] and classroom[@Farr+2012; @Kassner2015; @Mathur+2017; @Kaur+2017; @Hilborn2018] investigations centered around understanding gravitational physics. \[Sample data files and analysis routines are available for download from URL <http://github.com/josephromano/pta-demo>.\] Required hardware and software {#s:hardware_software} ============================== The metronome-microphone pulsar-timing-array demonstration requires two metronomes. Our preferred choice is Seiko model SQ50-V quartz metronomes (Figure \[f:metronome-microphone\]), as this model has adjustable beats-per-minute (bpm) up to 208 bpm, adjustable volume, and two different tempo sounds—mode $a$ and mode $b$, with mode $b$ having a slightly higher pitch. Having two modes is helpful in distinguishing the pulses from the individual metronomes when both metronomes are on simultaneously, since the pulse shapes (profiles) are different. ![Two Seiko metronomes and one Logitech USB noise-canceling microphone used for the demonstration.[]{data-label="f:metronome-microphone"}](metronome "fig:"){width=".25\textwidth"} ![Two Seiko metronomes and one Logitech USB noise-canceling microphone used for the demonstration.[]{data-label="f:metronome-microphone"}](metronome "fig:"){width=".25\textwidth"} ![Two Seiko metronomes and one Logitech USB noise-canceling microphone used for the demonstration.[]{data-label="f:metronome-microphone"}](microphone "fig:"){width=".3\textwidth"} One also needs some type of microphone, either an external USB microphone or an internal microphone, connected to a laptop that is set up to run the relevant data analysis routines (described below). We have found that the internal microphone on a MacBook Pro works best since it has ambient noise reduction, although it is somewhat inconvenient to physically move the laptop to simulate the passage of a gravitational wave. (We move the microphone is a small circle of radius $\sim\!10~{\rm cm}$ at constant speed, for reasons we will describe below.) We have also used a Logitech USB Desktop noise-canceling microphone (Figure \[f:metronome-microphone\]), which is a little easier to maneuver. In addition, one needs an open space covering an area of about $10~{\rm ft}\times 5~{\rm ft}$ for the placement of the two metronomes and microphone. A schematic diagram of the setup is shown in Figure \[f:setup\]. A photograph of an actual real-world setup used to take the data is shown in Figure \[f:actualsetup\]. ![Schematic diagram showing the location of the microphone and metronomes for the different analyses. The stationary microphone is located at the origin; the moving microphone undergoes uniform circular motion, indicated by the counter-clockwise circle. Metronome 2 is placed at angular location $45^\circ$ in this figure, but will be placed at the other angular locations for different parts of the demonstration.[]{data-label="f:setup"}](setup-circular){width="\textwidth"} ![Photograph of an actual setup for data taking. Metronome 1 is shown at angular location $0^\circ$; metronome 2 at angular location $135^\circ$. The separation between the microphone (located at the origin) and the metronomes at the edge of a semi-circle is approximately 5 feet.[]{data-label="f:actualsetup"}](actualsetup){width="\textwidth"} We have written Python-based routines to do the relevant data analysis calculations. These includes routines for: (i) audio recording and play back of metronome pulses, (ii) pulse data folding, (iii) pulse profile calculation, (iv) matched-filter estimation of pulse arrival times, (v) timing residual calculation, (vi) linear detrending of timing residuals, (vii) fitting of sinusoids to the timing residuals, and (viii) correlation coefficient calculation. Two Python-based GUIs exist for performing the two main data taking and data analysis tasks: [PTAdemo1GUI.py]{} (for analyzing the single-metronome data) and [PTAdemo2GUI.py]{} (for analyzing the double-metronome data). The two GUIs and the data analysis techniques are described in more detail in the following sections. Pulsar timing techniques illustrated by the demonstration {#s:techniques} ========================================================= The metronome-microphone demonstration is useful as an educational tool since it illustrates several techniques used in real pulsar timing analyses. It does this in the simplified context of metronome pulses recorded by a microphone, which students or the general public can more easily identify with. To set the stage for the analyses that will be described in Secs. \[s:analysis1\] and \[s:analysis2\], we describe below the key techniques illustrated by the demonstration. Folding data {#s:folding} ------------ Folding data is a technique that can be used to find both the pulse period and pulse shape (profile) in noisy time-series data [@handbook]. The time-series (of total duration $T_{\rm tot}$) is first split into smaller segments, each of duration $T$, which are then averaged together. If $T$ matches the true pulse period $T_{\rm p}$, then the pulse contributions in each segment combine coherently when the segments are summed, while the noise contributions combine incoherently (positive and negative values tending to cancel one another). If $T$ does not match the true pulse period, the pulse contributions will effectively cancel out when averaged against the noise. So the basic procedure is to systematically try different values of $T$ until a pulse profile “sticks out of the data", which will occur when $T$ equals $T_{\rm p}$. The signal-to-noise ratio of the recovered pulse profile grows like $\sqrt{N}$, where $N$ is the number of segments or, equivalently, the number of individual pulses combined[@cs2010]. Figure \[f:folding\] illustrates what happens when you fold data with an incorrect and the correct pulse period. ![Illustration of folding data with both an incorrect and the correct choice for the pulse period $T_{\rm p}$. Top panel: Noisy time-series data with several injected pulses having $T_{\rm p}=2~{\rm s}$. Left column: Segments of the original time-series data each of duration $1.6$ s (in blue), and the average of these segments (in red). Right column: Same as for the left column but for segments of duration 2 s. Note that when the data are folded with the correct pulse period $T_{\rm p}=2~{\rm s}$, the signal components combine coherently and the pulse profile is easily visible in the average of the segments (bottom right plot).[]{data-label="f:folding"}](folding_flat){width="\textwidth"} Matched-filter determination of pulse arrival times {#s:matchedfilter} --------------------------------------------------- The measured times of arrival (TOAs) are determined by correlating a time-shifted version of the pulse profile with the time-series data[@Taylor1992; @NG9WN]. Mathematically, one calculates the correlation function C() dty(t)p(t-), \[e:correlation-time\] where $y(t)$ is the timeseries and $p(t-\tau)$ is the pulse profile shifted forward in time by $\tau$. (${\cal N}$ is a normalization constant, defined below.) In the frequency domain, we have C() =[N]{} df y(f) p\^\*(f)e\^[i2f]{}, \^[-1]{}, \[e:correlation-freq\] where $\tilde y(f)$ and $\tilde p(f)$ are the Fourier transforms[@FourierFootnote] of $y(t)$ and $p(t)$. The correlation function $C(\tau)$ has local maxima at the arrival times of the pulses (Figure \[f:matchedfilterdemo\]). ![Illustration on simulated data showing that how the correlation function $C(\tau)$ has local maxima at the arrival times of the pulses. An animation showing the calculation of $C(\tau)$ as a function of the timeshift of the pulse profile is available at <http://github.com/josephromano/pta-demo/tree/master/code/>, with filename [matchedfilterdemo.avi]{}.[]{data-label="f:matchedfilterdemo"}](matchedfilterdemo_flat){width="60.00000%"} In what follows, we will denote these measured arrival times by $\tau^{\rm meas}[i]$, where $i=1,2,\cdots$. The normalization constant ${\cal N}$ is included so that the values of the correlation function at the measured TOAs are estimates of the amplitudes of the pulses. Calculating timing residuals based on a timing model {#s:timingresiduals} ---------------------------------------------------- Calculating the timing residuals is a simple matter of subtracting the expected TOAs from the measured TOAs of the pulses: = \^[meas]{}\[i\] - \^[exp]{}\[i\], \[e:timingresiduals\] where $i=1,2,\cdots$ labels the individual pulses. As mentioned in Sec. \[s:introduction\], for real pulsar timing analyses the expected TOAs are determined by a rather sophisticated timing model, which takes into account the rotation period of the pulsar, its spin-down rate, its location in the sky, etc. But for the metronome-microphone demonstration, the timing model is exceedingly simple: \^[exp]{}\[i\] = \^[meas]{}\[i\_0\] + (i-i\_0)T\_[p]{}, \[e:timingmodel\] which is just the measured TOA of the pulse having the largest correlation with the pulse profile, indexed by $i_0$, plus integer multiples of the pulse period $T_{\rm p}$ of the metronome. Improving the timing model by removing a linear trend in the residuals {#s:detrend} ---------------------------------------------------------------------- A linear trend in the timing residuals is an indication that the pulse period (determined by folding the data in Step 1 above) is not quite right. This is because $\delta\tau[i]$ involves a term $-(i-i_0)T_{\rm p}$, and if there is an error $\epsilon$ in $T_{\rm p}$, this term grows linearly with the pulse number, $i$. By fitting a line to the timing residuals and removing this trend, we obtain an improved estimate of the pulse period, which we can be used for subsequent timing model calculations. Figure \[f:residuals\_trend\_detrended\] show timing residuals for metronome pulses before and after removing a linear trend. Calculating correlation coefficients between pairs of timing residuals {#s:corrcoeff} ---------------------------------------------------------------------- The correlation coefficient between a pair of timing residuals is simply the time-averaged product of the timing residuals for a pair of microphone-metronome baselines. More generally, we can define the [*binned correlation function*]{} &C\_[12]{}\[k\] \_[i,j]{} \_[1]{}\[i\]\_[2]{}\[j\],\ &i, j \^[meas]{}\_[1]{}\[i\] - \^[meas]{}\_[2]{}\[j\] t, for two sets of timing-residuals $\delta\tau_1$, $\delta\tau_2$, where $\Delta t$ is the chosen bin size. (The condition on the indices $i$ and $j$ is simply that the difference in the measured times of arrival for the two timing residuals must lie in the $k$th lag bin.) But for the purposes of the demonstration: (i) we are interested in only the zero-lag result (so $k=0$), and (ii) to simplify the calculation, we can fit smooth curves $x_1(t)$, $x_2(t)$ to the two sets of timing residuals, for which C\_[12]{}\[0\] x\_1 x\_2 \_0\^[T\_[tot]{}]{} dt x\_1(t)x\_2(t). Normalizing by the rms values $\sqrt{\langle x_1^2\rangle}$ and $\sqrt{\langle x_2^2\rangle}$, we get . \_[12]{}/, \[e:corrcoeff\] for the correlation coefficient, which takes values between $-1$ and 1. Note that for timing residuals induced by uniform circular microphone motion (see Secs. \[s:expectedcorrelation\] and \[s:UCM\]), the best-fit smooth functions to the timing residuals will be sinusoids. Microphone-motion-induced timing residuals {#s:microphonemotion} ------------------------------------------ Similar to calculating the response of an Earth-pulsar baseline to a passing gravitational wave, one can calculate the response of a metronome timing residual to the motion of the microphone: \_I\^[mic]{}(t) = - u\_Ir(t) \[e:microphonemotion\] where $L_I(t)$ is the distance between metronome $I=1,2$ and the location of the microphone $\vec r(t)$ at time $t$, $L_I$ is the nominal distance between the metronome and the microphone pointing in direction $\hat u_I$, and $c_{\rm s}$ is the speed of sound in air. This response is just the change in the metronome pulse propagation time due to the motion of the microphone relative to metronome $I$. The last (approximate) equality above is valid if we ignore correction terms of order $A/L\sim 0.1$, where $A\sim 10~{\rm cm}$ is the amplitude of the microphone motion and $L\sim 1~{\rm m}$ is the distance from the metronome to the microphone when it is at the origin. Such an approximation amounts to ignoring the curvature of the pulse wavefronts. Expected correlations in the timing residuals induced by uniform circular motion {#s:expectedcorrelation} -------------------------------------------------------------------------------- For the microphone undergoing uniform circular motion with amplitude $A$ and frequency $f_0$, r(t) = A, \[e:UCM\] it follows immediately that \_I\^[mic]{}(t) -(2f\_0 t+\_0-\_I), \[e:microphonemotion-approx\] where $\theta_I$ is the angle that the location of metronome $I$ makes with respect to the $x$-axis, and where (as before) we have ignored the higher-order correction terms to the residual. Using the trigonometric identity AB =, it is fairly easy to show that the time-averaged correlation coefficient of the microphone-induced timing residuals for metronomes 1 and 2 is \_[12]{}, \[e:rho12\] where $\zeta\equiv \theta_1-\theta_2$ is the angle between the two microphone-metronome baselines. This equality is correct to order $(A/L)^2$. This dependence of the correlation coefficient on the angle between the two metronomes is what we confirm experimentally with the double-metronome analysis described in Sec. \[s:analysis2\]. Since the timing residuals for the two metronomes typically are evaluated at different times, we actually correlate the best-fit sinusoids to the timing residuals, as described in Sec. \[s:corrcoeff\]. Justification of the choice of uniform circular motion for the microphone {#s:UCM} ------------------------------------------------------------------------- In principle, we could move the microphone in any way whatsoever, and we would still see correlations in the timing residuals associated with the two metronomes. But the form of the expected correlation will be more complicated than the simple $\rho_{12} \simeq\cos\zeta$ dependence that we found above. For example, if instead of uniform circular motion we let the microphone swing back-and-forth sinusoidally in a plane (i.e., along some line in the $xy$-plane), then the correlation coefficient will also depend on the angle that this plane makes with the $x$-axis. Said another way, uniform circular motion has the nice property that the $x$ and $y$ components of its motion are statistically equivalent, but uncorrelated with one another. It turns out that this is also the assumption that goes into the calculation of the Hellings and Downs correlation curve (Figure \[f:HDcurve\]) for the real pulsar-timing gravitational-wave case—i.e., the Hellings and Downs curve is derived under the assumption that the pulsar timing array is responding to a stochastic background of gravitational waves that is both [*isotropic*]{} (no preferred direction) and [*unpolarized*]{} (statistically equivalent and uncorrelated linear polarization components)[@hd1983]. From a different perspective, the effect of uniform circular motion on the timing residuals is exactly what one sees in real pulsar-timing timing residuals if the yearly orbital motion of the Earth around the Sun is not taken into account in the timing models for the pulsars. Single-metronome analysis {#s:analysis1} ========================= The purpose of the single-metronome analysis is to calculate the pulse period and pulse profile for each metronome separately in the absence of microphone motion. The pulse periods and pulse profiles calculated here can be considered as [*reference*]{} periods and profiles, to be used as inputs for the double-metronome analysis, where both metronomes are running simultaneously and the microphone is moving when it is recording pulses. A screenshot of the GUI for this analysis, [PTAdemo1GUI.py]{}, is shown in Figure \[f:PTAdemo1GUI\]. The GUI has space for plots of: (i) pulses from the individual metronomes, (ii) pulse profiles (obtained by folding the pulse data), and (iii) timing residuals. The GUI also has several text entry fields and buttons, whose functions are described below: ![Python GUI for the single-metronome analysis. Plots, text entry fields, and buttons are described in the main text.[]{data-label="f:PTAdemo1GUI"}](PTAdemo1GUI.png){width="\textwidth"} i[**Record pulses**]{}: Records audio pulse data from a metronome, and save the corresponding timeseries to an ascii [.txt]{} file with file prefix specified by the [PULSE DATA FILENAMES]{} text entry boxes (default [m208a]{} or [m184b]{}). The [bpm]{} text entry boxes have the beats-per-minute settings for the two metronomes. The pulse recording routine is hard-coded to record 8 seconds of data. i[**Playback pulses**]{}: Plays back and plots the audio pulse data saved in the ascii files, again defaulted to [m208a.txt]{} or [m184b.txt]{}. i[**Calculate profile**]{}: Either (i) calculates the pulse period $T_{\rm p}$ and pulse profile directly by folding the metronome data, and then saves the profile to the file [m208a\_profile.txt]{} or [m184b\_profile.txt]{}, or (ii) reads in the pulse profile data that has already been saved in these files. Method (i) is used only the first time the analysis is run. If the pulse profiles are read-in from the files [m208a\_profile.txt]{} and [m184b\_profile.txt]{}, the text entry boxes for the pulse periods need to be entered by hand. For both cases, the pulse profile is plotted from 0 to $T_{\rm p}$. i[**Calculate residuals**]{}: Calculates the timing residuals by subtracting the expected TOAs from the measured TOAs of the pulses as described in Secs. \[s:timingresiduals\] and \[s:matchedfilter\]. i[**Detrend residuals**]{}: Improves the estimate of the pulse period for a metronome by removing a linear trend from the timing residuals as described in Sec. \[s:detrend\]. Detrending may change the estimate of the pulse period by 1-2 microseconds. The updated period is displayed in the [Pulse period]{} text entry box. Steps for doing the analysis ---------------------------- iRecord pulses from each metronome separately, keeping the microphone stationary. The microphone should be located at the origin of coordinates and the two metronomes should be at angular location $0^\circ$. The file prefixes and bpms in the text entry boxes should be chosen to match the physical settings of the metronome. iAfter recording the pulse data for each metronome, you can play it back, calculate the corresponding pulse profile and period, calculate the residuals, and detrend the residuals, by simply pressing the relevant GUI buttons. This analysis produces two pulse profile data files (e.g., [m208a\_profile.txt]{} and [m184b\_profile.txt]{}) and the associated pulse periods for the two metronomes, which are needed for the double-metronome analysis described in the next section. Double-metronome analysis {#s:analysis2} ========================= The purpose of the double-metronome analysis is to experimentally verify the expected $\rho_{12} \simeq \cos\zeta$ correlation coefficient for the timing residuals for the two metronomes when the microphone is undergoing uniform circular motion. This is analogous to real pulsar timing analyses looking for evidence of the Hellings and Downs correlation curve when correlating timing residuals for pairs of Earth-pulsar baselines. A screenshot of the GUI for this analysis, [PTAdemo2GUI.py]{}, is shown in Figure \[f:PTAdemo2GUI\]. The GUI has space for plots of: (i) pulses from the two metronomes running simultaneously, (ii) reference pulse profiles for the two metronomes, which were calculated using [PTAdemo1GUI.py]{}, and (iii) timing residuals for the two metronomes. The GUI also has several text entry fields and buttons, whose functions are described below: ![Python GUI for the double-metronome analysis. Plots, text entry fields, and buttons are described in the main text.[]{data-label="f:PTAdemo2GUI"}](PTAdemo2GUI.png){width="\textwidth"} i[**Record pulses**]{}: Records audio pulse data from the two metronomes running simultaneously, and then saves the corresponding timeseries to an ascii [.txt]{} file with file prefix specified by the [Data file]{} text entry box under the [FILENAMES]{} label (default [m208a184b0]{}). i[**Playback pulses**]{}: Plays back and plots the audio pulse data saved in [m208a184b0.txt]{}. i[**Load pulse profiles**]{}: Reads-in and plots the reference pulse profiles for the two metronomes, which were calculated by [PTAdemo1GUI.py]{} and were saved in ascii [.txt]{} files specified by the [Profile 1,2]{} text entry boxes under the [FILENAMES]{} label (default [m208a\_profile]{} and [m184b\_profile]{}). The text entry boxes for the pulse periods should be filled with in with the values calculated by [PTAdemo1GUI.py]{}. i[**Calculate residuals**]{}: Calculates the timing residuals as described previously for [PTAdemo1GUI.py]{}. i[**Fit sinusoids & remove offsets**]{}: Simultaneously removes constant offsets and calculates best-fit sinusoids to the timing residuals for the two metronomes, using initial estimates for the amplitude, frequency, and phase of the sinusoids, and the constant offset given in the text entry boxes labeled [INITIAL ESTIMATES (1,2)]{} (defaults $2\times 10^{-4}$, $0.4~{\rm Hz}$, 0 radians, and 0 sec, respectively). The constant offset arises from the arbitrariness of setting the timing residual of the pulse with the highest correlation to zero. The best-fit parameter values calculated by [Fit sinusoids & remove offsets]{} are written to the text entry boxes labeled [BEST-FIT VALUES (1,2)]{}. i[**Calculate corr coeff**]{}: Calculates the time-averaged correlation coefficient between the best-fit sinusoids for the two sets of timing residuals, as described in Sec. \[s:corrcoeff\]. Theoretically, the value of the correlation coefficient should equally $\cos\zeta$, where $\zeta$ is the separation angle between the line-of-sights to the two metronomes, as described in Sec \[s:expectedcorrelation\]. Steps for doing the analysis ---------------------------- iStart by placing both metronomes at the same angular location $0^\circ$ and at the same distance $L\sim 1~{\rm m}$ from the origin. With both metronomes running simultaneously, record the audio data while moving the microphone in uniform circular motion about the origin: Typically, it is best to have $A\approx 10~{\rm cm}$ (=0.1 m) and period of oscillation $T_0\equiv 1/f_0\approx 2~{\rm s}$. This leads to microphone-induced timing residuals of order $A/c_{\rm s}\approx 3\times 10^{-4}~{\rm s}$, where $c_{\rm s}=340~{\rm m/s}$ is the speed of sound in air. This timing precision turns out to be more than an order-of-magnitude larger than the precision to which we can estimate the TOAs of the metronome pulses, meaning that we can easily observe the effect of microphone motion in the timing-residual data. iAfter recording the double-metronome data, you can play it back, load the pulse profiles, and calculate the residuals for each metronome. The timing residuals induced by the microphone motion should be sinusoidal and have large signal-to-noise ratio. You should then fit sinusoids to the residuals for each metronome, adjusting the [INITIAL GUESS]{} amplitudes, frequencies, and phases if necessary. (The initial guesses just have to be close, not exact.) Finally, you should calculate the correlation coefficient, which should have a value very close to 1 for this case, since the two microphones are at the same angular location. iRepeat the above two steps but with metronome 2 at different angular locations ($45^\circ$, $90^\circ$, $135^\circ$, $180^\circ$) with respect to metronome 1 (which should always remain at $0^\circ$). The motion of the microphone should be as similar as possible to that for Step 1. Change the name of the file prefix in the [Data file]{} text entry box to [m208a184bXX]{}, where [XX]{} is [45]{}, [90]{}, [135]{}, [180]{}, to reflect the change in the angular location of metronome 2. You should find that the correlation coefficient is approximately equal to $\cos\zeta$, where $\zeta=45^\circ$, $90^\circ$, $135^\circ$, $180^\circ$ is the angular separation of the two metronomes. This analysis produces data files ([m208a184bXX.txt]{}, where [XX]{} is [0]{}, [45]{}, [90]{}, [135]{}, [180]{}), containing the double-metronome pulse timeseries. Discussion {#s:discussion} ========== We have described a demonstration using two metronomes and a microphone that serves as an acoustical analogue of a Galactic-scale gravitational-wave detector, i.e., a pulsar timing array. This demonstration also serves as an educational tool, illustrating several techniques used in real pulsar timing analyses, but in the simplified context of metronome pulses recorded by a microphone. From our experience, we have found that the demonstration is best suited for undergraduates or senior-level high-school students who already have some familiarity with basic physics and astronomy. For less mathematically-inclined audiences, the mathematical discussion of the underlying data analysis techniques needs to be reduced accordingly. But the main idea that a common disturbance (in this case, the microphone motion) can induce correlations in the pulse arrival times, and a graphical display showing the timing residuals from the two metronomes being shifted by an amount equal to their angular separation is accessible to nearly all audiences. Some caveats {#s:caveats} ------------ The tricky technical aspect of the double-metronome analysis is to properly extract the pulse arrival times when the two metronomes are running simultaneously, producing pulses that can significantly overlap with one another. The fact that the pulse profiles $p_I(t)$ for the two metronomes ($I=1,2$) are different for different tempo modes $a$ and $b$ is crucial for distinguishing the pulses from the two metronomes. Still, the correlation functions $C_I(\tau)$ have several local maxima, and we need to find the largest local maxima in the vicinity of the expected pulse arrival times to determine the measured TOAs $\tau_{I}^{\rm meas}[i]$. If the search window is not properly centered on the expected arrival time or if it includes a local maximum of the correlation function that doesn’t correspond to the true arrival time of the pulse, then the returned measured TOA will deviate from its true value, thus causing errors in the corresponding timing residual and the subsequent fit to the residuals. To help mitigate such problems, the routine that calculates the measured TOAs currently uses an adaptive width for the search window, which increases in size if it originally does not include a peak in the correlation function (this is usually a sign that the window was not large enough to include the true pulse arrival time). Even with this adaptive-search-window technique, we sometimes do not get good agreement between the measured and theoretical correlation coefficients for intermediate separation angles between the two metronomes, i.e., $\zeta$ close to 90 degrees. A possible alternative reason for this might be reflections of the sound waves off of the table top or parts of the laptop, when using the laptop’s internal microphone to do the recordings. Recovery of the expected correlation is usually better if we use a USB microphone, which does not have many intervening parts to interfere with the sound waves. Possible use as an instructional laboratory investigation {#s:lab} --------------------------------------------------------- Although we have not tried to use this demonstration in its full form as an instructional laboratory, we suspect that some variant of this demonstration might be useful for an undergraduate physics or astronomy lab. We have used the single-metronome demonstration at public outreach events and for a high-school Advanced Placement Physics demonstration with good success in getting students to understand the fundamentals of pulsar timing based on questions asked throughout. In a lab for more advanced students, the usefulness of the full demonstration comes in the form of learning the data analysis techniques of folding, matched-filtering, cross-correlation, etc., which are very general and have widespread applicability in many branches of science. Students who are comfortable with computer programming could be asked to code up their own data folding and matched-filtering routines, etc., and apply them to the metronome pulse data. Or the students could take the routines that already exist, but create their own customized GUIs to perform custom analyses on other recorded sound files. Of course, one could simply try to use the existing demonstration (more or less as is) as a lab, but we suspect that it would be best if it were done as a “communal effort", at least as far as the metronome data taking is concerned. In other words, the two metronomes and single microphone would be shared amongst all lab groups, but each group would be responsible for performing one of the single-metronome or double-metronome analyses (e.g., a double-metronome analysis for a specific angular separation). Otherwise, there would be too much noise contamination from $\sim\!10$ pairs of metronomes running simultaneously! Enhancements under development {#s:enhancements} ------------------------------ To make it easier for people who are not computer savvy to perform the demonstration, we are currently developing a web-based interface for running the data analysis part of the demonstration. This will eliminate the need for the demonstrator to have a working installation of all the requisite Python routines and Python packages on his/her own computer, and should simplify the operation of the GUIs. Although for this scheme the data analysis will be done remotely on the web server, the data taking will still be done locally using the two (physical) metronomes and e.g., a smartphone for recording the pulses. The sound files recorded by the smartphone would then need to be uploaded to the web server for the subsequent single-metronome and double-metronome analyses. Moving further in this direction, we also have an implementation of the metronome-microphone demonstration that exclusively uses readily-available smartphones to drive the demonstration[@kv2013; @mg2014; @Osorio+2018]. We have written a smartphone app, called [*TableTopPTA*]{}, which allows a smartphone to operate as either a metronome or a microphone, as well as to perform all of the data analysis calculations needed for the single-metronome and double-metronome analyses. The demonstration can then be done using just three smartphones, two of which run in metronome mode; the other running in microphone mode and performing the subsequent data analysis calculations. The app is written in Javascript and runs on Android smartphones; we have not yet written an iPhone version of the app. The current code is available for public download from URL <https://github.com/marcnormandin/tabletop_pta>. Although some of the analysis routines for the smartphone app are not currently as up-to-date as those for the Python implementation of the demonstration, we have decided to make the code publicly available in case people want to experiment with what we currently have and possibly improve things in the process. We would like to acknowledge support from NSF Physics Frontier Center award number 1430284. JDR and MN would also like to acknowledge support from NSF grant PHY-1505861. [5]{} Detweiler, S. 1979, “Pulsar Timing Measurement and the Search for Gravitational Waves”, , 234, 1100 Hellings, R. W., & Downs, G. S. 1983, “Upper Limits on the Isotropic Gravitational Radiation Background from Pulsar Timing Analysis”, [Astrophys. J. L.]{}, 265, L39 Bizouard, M. A., Jenet, F., Price, R., & Will, C. M. 2013, “Pulsar Timing Arrays”, Classical and Quantum Gravity, 30, 220301 Lorimer, D. R., & Kramer, M. 2012, [*Handbook of Pulsar Astronomy*]{}, by D. R. Lorimer , M. Kramer, Cambridge, UK: Cambridge University Press, 2012 Hobbs, G., Coles, W., Manchester, R. N., et al. 2012, “Development of a Pulsar-based Time-scale”, [Mon. Not. Royal Ast. Soc.]{}, 427, 2780 Cordes, J. M., & Downs, G. S. 1985, “JPL Pulsar Timing Observations. III. Pulsar Rotation Fluctuations”, [Astrophys. J. S.]{}, 59, 343 Lam, M. T., Cordes, J. M., Chatterjee, S., et al. 2017, “The NANOGrav Nine-Year Data: Excess Noise in Millisecond Pulsar Arrival Times”, , 834, 35 Lentati, L., Shannon, R. M., Coles, W. A., et al. 2016, “From Spin Noise to Systematics: Stochastic Processes in the First International Pulsar Timing Array Data Release”, [Mon. Not. Royal Ast. Soc.]{}, 458, 2161 Cordes, J. M. 2013, “Limits to PTA Sensitivity: Spin Stability and Arrival Time Precision of Millisecond Pulsars”, Classical and Quantum Gravity, 30, 224002 Condon, J. J., & Ransom, S. M. 2016, “Essential Radio Astronomy”, Princeton University Press Estabrook, F. B., & Wahlquist, H. D. 1975, “Response of Doppler Spacecraft Tracking to Gravitational Radiation”, General Relativity and Gravitation, 6, 439 Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2016, “Observation of Gravitational Waves from a Binary Black Hole Merger”, Physical Review Letters, 116, 061102 <https://www.ligo.caltech.edu/page/detection-companion-papers> Abbott, B. P., Abbott, R., Abbott, T. D., et al. 2017, “Observation of Gravitational Waves from a Binary Neutron Star Inspiral”, Physical Review Letters, 119, 161101 Jenet, F. A., & Romano, J. D. 2015, “Understanding the Gravitational-Wave Hellings and Downs Curve for Pulsar Timing Arrays in Terms of Sound and Electromagnetic Waves”, [AJP]{}, 83, 635 Rubbo, L. J., Larson, S. L., Larson, M. B., Ingram, D. R. 2007, “Hands-on Gravitational Wave Astronomy: Extracting Astrophysical Information from Simulated Signals”, [AJP]{}, 75, 597 Newburgh, R. 2008, “A Demonstration of Einstein’s Equivalence of Gravity and Acceleration”, European Journal of Physics, 29, 2 Ford, J., Stang, J., & Anderson, C. 2015, “Simulating Gravity: Dark Matter and Gravitational Lensing in the Classroom”, The Physics Teacher, 53, 557 Burko, L. M. 2017, “Gravitational Wave Detection in the Introductory Lab”, The Physics Teacher, 55, 288 Farr, B., Schelbert, G., & Trouille, L. 2012, “Gravitational Wave Science in the High School Classroom”, [AJP]{}, 80, 898 Kassner, K. 2015, ”Classroom Reconstruction of the Schwarzschild Metric”, European Journal of Physics, 36, 6 Mathur, H., Brown, K., & Lowenstein, A. 2017, “An Analysis of the LIGO Discovery Based on Introductory Physics”, [AJP]{}, 85, 676 Kaur, T., Blair, D., Moschilla, J., Stannard, W., & Zadnik, M. 2017, “Teaching Einsteinian Physics at Schools: Part 1, Models and Analogies for Relativity”, Physics Education, 52, 6 Hilborn, R. C. 2018, “Gravitational Waves from Orbiting Binaries Without General Relativity”, [AJP]{}, 86, 186 Cordes, J. M., & Shannon, R. M. 2010, “A Measurement Model for Precision Pulsar Timing”, arXiv:1010.3785 Taylor, J. H. 1992, “Pulsar Timing and Relativistic Gravity”, Royal Soc. of London Phil. Trans. Series A, 341, 117 Lam, M. T., Cordes, J. M., Chatterjee, S., et al. 2016, “The NANOGrav Nine-Year Data Set: Noise Budget for Pulsar Arrival Times on Intraday Timescales”, , 819, 155 The Fourier transform $\tilde y(f)$ of the time-series $y(t)$ is defined by $\tilde y(f) = \int dt\> y(t)e^{-i2\pi f t}$ or, equivalently, $y(t) = \int df\> \tilde y(f)e^{i2\pi f t}$. Kuhn, J., Vogt, P. 2013, “Applications and Examples of Experiments with Mobile Phones and Smartphones in Physics Lessons”, Frontiers in Sensors, 1, 4 Martinez, L. & Garaizar, P. 2014, “Learning Physics Down a Slide: A Set of Experiments to Measure Reality Through Smartphone Sensors”, International Journal of Interactive Mobile Technologies, 8, 3 Osorio, M., Pereyra, C. J., Gau, D. L., & Laguarda, A. 2018, “Measuring and characterizing beat phenomena with a smartphone”, European Journal of Physics, 39, 2

--- abstract: 'In this work we investigate the orbital distribution of interstellar objects (ISOs), observable by the future wide-field National Science Foundation Vera C. Rubin Observatory (VRO). We generate synthetic population of ISOs and simulate their ephemerides over a period of 10 years, in order to select those which may be observed by the VRO, based on the nominal characteristics of this survey. We find that the population of the observable ISOs should be significantly biased in favor of retrograde objects. The intensity of this bias is correlated with the slope of the size-frequency distribution (SFD) of the population, as well as with the perihelion distances. Steeper SFD slopes lead to an increased fraction of the retrograde orbits, and also of the median orbital inclination. On the other hand, larger perihelion distances result in more symmetric distribution of orbital inclinations. We believe that this is a result of Holetschek’s effects, which is already suggested to cause observational bias in orbital distribution of long period comets. The most important implication of our findings is that an excess of retrograde orbits depends on the sizes and the perihelion distances. Therefore, the prograde/retrograde orbits ratio and the median inclination of the discovered population could, in turn, be used to estimate the SFD of the underlying true population of ISOs.' author: - | Dušan Marčeta,[^1] Bojan Novaković\ Department of Astronomy, Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000 Belgrade, Serbia\ bibliography: - 'references.bib' date: 'Accepted XXX. Received YYY; in original form ZZZ' title: Retrograde orbits excess among observable interstellar objects --- \[firstpage\] Planetary systems – comets: general – minor planets, asteroids: general Introduction {#sec:intro} ============ The existence of galactic population of the objects ejected from the planetary systems has been long hypothesized [e.g. @Sekanina]. The expelling of a large number of planetesimals during the early stages of the Solar System is predicted by its evolution models [e.g. @Charnoz2003; @Bottke2005; @2011Natur.475..206W], and is reasonable to assume that this process is also at work in other planetary systems throughout the Galaxy. Some authors claim that ejections in the early phase is not sufficient to match their estimated number density, and proposed other ejection mechanisms, including ejection of the planetesimals during the late phases of the stellar evolution process [@Veras2014; @Stone2015]. The discovery of 1I/(2017 U1) ’Oumuamua, the first macroscopic interstellar object (ISO) by Pan-STARRS survey [@MPC-oumuamua], not only confirmed their existence, but also indicated that the population of these objects is relatively numerous. In turn, as discussed by @2018ApJ...855L..10D, this enabled setting better constraints on their number density and size-frequency distribution (SFD). This is further supported by more recent discovery of the object 2I/(2019 Q4) Borisov [@MPC-borisov], which is also confirmed to have the interstellar origin . One can say that ’Oumuamua was the exact opposite of what we expected from an interstellar object. This is primarily related to its extremely elongated shape and asteroidal nature. The estimates of its aspect ratio go from 3.5:1 [@2018ApJ...852L...2B] to 10:1 [@2017Natur.552..378M]. Although, there are small objects with comparable aspects ratios in the Solar System, such as asteroid (1865) Cerberus, whose aspect ratio is estimated to 4.5:1 , they are generally rare. Therefore, highly elongated shape of the very first known interstellar object ’Oumuamua, was highly unexpected. On the other hand, although models of planetary systems evolution predict that the large number of planetesimals should escape their mother systems, it is expected that large majority of these object should originate from the outer parts of the systems, far beyond the snow-line [@2018ApJ...852L..15C]. Hence, it was reasonable to expect that ISOs show cometary activity close to the perihelion. Although coma around ’Oumuamua was not detected directly, astrometric measurements showed deviation from a purely gravity driven trajectory, which may be explained with an additional force induced by cometary activity [@2018Natur.559..223M]. However, @2018ApJ...867L..17R argues that this amount of activity should have led to significant evolution of the object’s rotational state, and probably to its disruption, but no significant evolution of the light curve was observed during this period. Unlike the latter study, @2019ApJ...876L..26S suggest that out-gassing activity that followed the sub-solar point of an elongated body could produce the observed non-gravitational acceleration, without causing extreme spin up. This, and many other questions about the ’Oumuamua, are still open [@2019NatAs...3..594O]. The lack of observed typical cometary activity was not only surprising because of the disagreement with an expected nature of a vast majority of ISOs, but also because the probability of their discovery should be significantly biased in favor of cometary-like objects, due to increased brightness caused by the sublimation of volatile materials. Still, the second ISO (2I/Borisov) shows cometary activity, suggesting that we should expect a large variety of characteristics among ISOs, which will hopefully be discovered in the near future, especially after the start of the National Science Foundation Vera C. Rubin Observatory’s (VRO) Legacy Survey of Space and Time (LSST)[^2]. Recent studies about ISOs number density and number of objects expected to be detected by the current and future surveys give large variety of results. A comprehensive analysis of ISOs number density by @2009ApJ...704..733M indicated that the probability for the VRO to detect an ISO during its operating period is very small, on the order of 0.001-1 %. This result is based on a consideration of the expected ISOs number density, which included the number density of stars, the amount of solids available to form planetesimals, the frequency of planets and planetesimals formation, the efficiency of planetesimals ejection, and the possible size distribution of these small bodies. However, the analysis was limited only to the ISOs orbiting beyond the orbit of Jupiter, and did not take into account a possibility that ISOs become active when approach closer to the Sun, that may significantly increase their brightness, and therefore chances to be detected. @2016ApJ...825...51C extended this analysis by taking into account gravitational focusing by the Sun (which increases the number of ISOs per unit volume closer to the Sun), the effect of different observing angles (photometric phase functions), comet brightening, and more precise definition of the observing constraints (such as solar elongation and air mass). These improvements allowed consideration of the detection of closer ISOs, leading to an estimation of 0.001 to 10 expected detections of ISOs by the VRO during the 10 years of its nominal operating period. Such a small number of expected detections is mainly a consequence of the estimated number density of ISOs. However, @2017AJ....153..133E determined the upper limit for the ISOs number density to be several orders of magnitudes larger than previously estimated. Their analysis is based on a modeling of ISOs population around the Sun, that naturally includes the effect of gravitational focusing. The authors exposed this population to detectability simulation based on the performances of three surveys (Pan-STARRS1, Mt. Lemmon Survey, and Catalina Sky Survey), and considered the different effects, including cometary activity, photometric phase functions, observing constellations, and various SFD functions. In addition, @2017AJ....153..133E based their findings on the fact that no single ISO was discovered at that time. Therefore, the recent discoveries of ’Oumuamua and Borisov, suggest that it may not be a surprise if the VRO detects even larger number of ISOs, than expected in the most optimistic predictions [see @2019Sci...366..558G]. While a nominal number of the detectable ISOs is definitely an important parameter to know, the observational selection effects may play important role in analyzing and modeling the underlying populations [@2002aste.book...71J]. Still, many aspects of the observational selection effects on ISOs population have received very little attention in the literature so far. The goal of the work presented in this paper is twofold: i) to determine the orbit and size-frequency distribution of the ISOs observable by the VRO, and ii) to analyze how these distributions depend on the same properties of the underlying true population. The population of interstellar objects {#sec:population} ====================================== In order to perform the analysis, it is necessary to define some input parameters, make some assumptions and adopt some methodologies. Below we outline our approach. Number density and size distribution of ISOs {#ss:num-density} -------------------------------------------- A total number of objects which can be detected by an observation program primarily depends on how many of them are in the observable volume of the space, and how large (bright) they are. Hence, the two most important parameters that determine the detection probability of ISOs are their number density and SFD. However, due to the lack of observational data, the estimations of these parameters are based primarily on theoretical assumptions and, consequently, are very uncertain. There is a large dispersion of the assumptions for the ISOs number density, and for objects larger than 1 km in diameter, it ranges from $10^{-9}$ au$^{-3}$ [@2009ApJ...704..733M] to $10^{-2}$ au$^{-3}$ [@2017AJ....153..133E]. This is a consequence of limited knowledge about how efficiently planetary systems populate the interstellar space with asteroids and comets. A number of ejections in the early phases depends on various characteristics of the systems, such are their orbit architectures, or masses of the planets. In addition, as mentioned before, it is possible that other mechanisms characteristic for the late phases of planetary systems evolution also contribute to this process. A similar situation is also with the SFD of ISOs. For instance, it is unknown if it represents their initial population, as they were expelled from their mother planetary systems, or it is significantly altered during their interstellar phase. These concerns naturally arise from the attempts to explain the lack of cometary activity and extremely elongated shape of ’Oumuamua. As an example, @2019MNRAS.484L..75V suggest that the elongated shape is a consequence of isotropic erosion. If true, this should also shrinks sizes of ISOs, significantly altering their SFD, to such an extent that it may be even responsible for lack of the observable objects. The main goal of this work is not to estimate the exact number of objects which will be detected and eventually discovered by the current and future survey programs, but to analyze the orbital and size distributions of the detectable objects. To this purpose, we assumed that a cumulative size-frequency distribution of ISOs is given in the standard, single slope power-law form $ N(>D)\propto D^{-\gamma}$, where $D$ and $\gamma$ are objects’ diameters and SFD slope, respectively. The analyzes were then performed assuming the range of SFD slopes $\gamma$ between $1.4$ and $4$, with a discrete step of $0.1$. A population of main-belt asteroids, larger than about 2 km in diameter, has a cumulative size-frequency distribution characterized by a slope of $\gamma = 2.4$ . For Jupiter Family Comets (JFC), @2013Icar..226.1138F found a shallow $\gamma$ slope of $1.9$ in the size range $2.8 - 18$ km, while for Long Period Comets (LPCs), @2019Icar..333..252B found a slope of $\gamma = 3.6$ for objects larger than $1$ km. Therefore, although the interval of slopes analyzed here is selected somewhat arbitrary, it covers the values available in the literature for possibly representative populations, and even extends for about 0.5 on both sides with respect to the interval of quoted slopes.[^3] The range of analyzed slopes allows to highlight any possible connection between the orbit distribution and the SFD. Furthermore, we generated the population with a number density of $10^{-4}$ au$^{-3}$ objects larger than 1 km in diameter, which is between the extremes of the previous assumptions [@2009ApJ...704..733M; @2017AJ....153..133E]. For the SFD slope of 2.5, which is expected for the so-called self-similar collisional cascade [@1969JGR....74.2531D], this number density corresponds to 10 ISOs per au$^3$ larger than 10 meters in diameter. While the number density is the crucial parameter for the estimation of the absolute number of objects which could be observed, it is not expected that this parameter influence the orbital and size distributions of the observable objects, because it equally impacts all objects from the population, regardless of their sizes and orbits. Having this in mind, we chose the number density that is within the bounds of the previous estimates, can be treated with available computing resources, and provides a sufficient sample for statistical analysis. Orbital elements of ISOs {#ss:orb_ele} ------------------------ It seems appropriate to suppose that the population of ISOs throughout the Galaxy, far from any massive body, is homogeneous, and that their velocity vectors are isotropic. Also, it is reasonable to assume that the distribution of their speeds mimic that of the nearby stars. However, in the area close to the Sun, these distributions will be altered due to the effect of gravitational focusing. In order to generate the steady-state population of ISOs in the vicinity of the Sun, we applied a modified method of @2011PASP..123..423G. In particular, for the purpose of the analyses performed in this work, we use a concept of three spheres: observable, model and initialization [see also @2017AJ....153..133E]. The idea behind this concept is the following. We are interested in the ISOs potentially observable by the VRO. Therefore, we define the radius of the [*observable sphere*]{} in such a way that at least brightest objects from the population should be visible at the edge of this sphere. However, in order to determine the population of the ISOs situated inside the observable sphere, we need to model the population in a larger volume of space, from which objects can enter the observable sphere during the 10 yr operational period of the VRO. This sphere that should feed the observable sphere is called [*model sphere*]{}. The radius of the model sphere should be large enough to include all the objects that can reach the observable sphere. Therefore, it is defined based on the distance that the fastest objects will cross in 10 yr. The distribution of the objects inside the model sphere is not uniform, due to the gravitational focusing in vicinity of the Sun. For this reason, we need to define an additional volume of space that in turn feeds the model sphere, but which is far enough from the Sun that the gravitational focusing may be neglected. This is what we called [*initialization sphere.*]{} A detailed technical description of our methodology is given below. 1. We used simulation time of 10 years, that is nominal operating period for the VRO. This means that our population should have unchanging characteristics, at least, within this period. 2. We set limiting apparent visual magnitude of $m = 24.5$, which is the nominal limiting magnitude of the VRO [@2016IAUS..318..282J]. This practically means that the brightest object from the population can reach this limiting magnitude at the edge of the observable space, under ideal observing conditions. 3. We worked with objects between $10$ m and $10$ km in diameter. Based on the predictions of ISOs number density and SFD, there should be comparatively few ISOs larger than $10$ km, and it is therefore unlikely they will penetrate the inner Solar System. For the most optimistic assumptions of the number density [e.g. @2017AJ....153..133E] and moderate SFD slopes, we can expect on the order of $10^{-1}$ ISOs larger than $10$ km, inside the observable volume of space at any instance. On the other hand, an object of $10$ m in size has to pass within $0.1$ au from the Earth, while being in the opposition, in order to reach the limiting apparent magnitude. For this reason, it is highly unlikely that objects smaller than $10$ m will be detected, due to their faintness, but also extremely large apparent velocities if they appear close enough to the Earth. 4. We assumed that the distribution of speeds of these objects relative to the Sun, when they are at infinity (hyperbolic excess velocity, $v_{\infty}$), mimics the distribution of speeds of the nearby stars. We adopted the normal distribution with mean value of $v_0 = 25$ km/s, and standard deviation of $\sigma=5$ km/s [e.g. @1998MNRAS.298..387D]. This means that 99.75$\%$ of the objects have speeds between $10$ and $40$ km/s. 5. The relation between diameter and absolute magnitude is calculated according to the following relation [@1997Icar..126..450H]: $$H=15.618-2.5\log\left(p_v\right)-5\log\left(D\right), \label{eq:H(D)}$$ where diameter ($D$) is given in kilometers and $p_v$ is geometric albedo, for which we adopted a value of 0.04 for the whole population, since it is the estimated value for ’Oumuamua [@2017Natur.552..378M]. The apparent magnitude of an ISO is calculated from the equation: $$m = H+5log(r_g r_h)+\Phi \left(\alpha \right), \label{eq:m(H)}$$ where $m$ is apparent visual magnitude, $H$ is absolute magnitude, $r_g$ and $r_h$ are geocentric and heliocentric distances, respectively, and $\Phi \left(\alpha \right)$ is the integral phase function of the phase angle ($\alpha$) [@1989aste.conf..524B]. We adopted simplified linear darkening function $\Phi \left(\alpha \right)=\beta \alpha$ with a slope of $\beta = 0.04$ degree$^{-1}$, which neglects brightening due to opposition surge [@2017ApJ...850L..36J]. According to Eq. \[eq:H(D)\], the brightest object from the generated population, with a diameter of 10 km, has absolute magnitude of $H = 14.1$. From Eq. \[eq:m(H)\], we obtained the limiting heliocentric distance of $11.5$ au, at which the brightest object can reach the limiting apparent magnitude, when it is at the opposition ($\Phi = 0$; $r_g=r_h-1$ au). Based on this, we adopted the value of $12$ au as the radius of the sphere that represents the observable volume of space. 6. We calculated the heliocentric distance from which the fastest objects from the population ($v_{\infty} = 40$ km/s), assuming they are on direct paths toward the Sun, can reach the observable sphere during the 10 years of the simulation interval. This heliocentric distance is found to be at $97$ au, which we adopted for the radius of the model sphere. 7. As mentioned before, far enough from the Sun, it is reasonable to expect that population of ISOs is homogeneous and that their velocity vectors are isotropic. However, in vicinity of the Sun this assumption will be disturbed as a result of gravitational focusing. A parameter that determines the intensity of the gravitational focusing is defined as $F=1+ {{v_{esc}}^2 / {v_{\infty}}^2}$ [see e.g. @2017ApJ...850L..36J], where in our case $v_{esc}$ is the escape velocity from the Sun, at a given heliocentric distance. For the average hyperbolic excess velocity of the generated population ($v_{\infty} = 25$ km/s), a value of this parameter at the edge of the model sphere ($r_m = 97$ au, $v_{esc} = 4.28$ km/s) is only $F \approx 1.03$. Hence, we assumed that outside the model sphere population of ISOs is unaffected by the gravity of the Sun, and thus it is homogeneous and isotropic. To obtain the population inside the model sphere, where the gravitational focusing cannot be neglected, this space has to be populated only with the objects initially located outside it. The simulation of the process of populating has to last long enough, that the population inside the model sphere can reach approximately a constant number. To achieve this, all objects at the edge of the model sphere, should have enough time to cross this sphere, regardless of their initial velocity vectors. We called this time the [*initialization time*]{}. On the other hand, the size of the initialization sphere, that populates the model sphere, should coincide with the distance from which the fastest objects can reach the model sphere during the initialization time. To estimate the initialization time, we determined the longest time taken by an object from the population to cross the entire model sphere. This calculation is performed over the whole range of values of $v_{\infty}$ and $q$. For $v_{\infty}$ this range goes from $10$ to $40$ km/s, while for the perihelion distances we considered the range of distances between the radius of the Sun ($0.005$ au) and $97$ au, which is the radius of the model sphere. These ranges of $v_{\infty}$ and $q$ were then sampled on equidistant grid points, and for any possible combination of these two parameters we calculated the semi-major axis ($a$) and eccentricity ($e$), according to the equations [@kemble_2006]: $$a = - {{\mu} \over {v_{\infty}^2}}, \label{eq:a(v)}$$ $$e=1-{{q} \over {a}} \label{eq:e(q)}$$ where $\mu$ is the gravitational parameter of the Sun. Having $a$ and $e$, a critical hyperbolic anomaly ($H_{cr}$) that corresponds to the edge of the model sphere, can be obtained from the equation for a hyperbolic orbit: $$r=a \left(1 - e\cosh{H} \right), \label{eq:r(H)}$$ where $H$ is hyperbolic anomaly. Finally, we calculated the time needed for an object to move along the hyperbolic trajectory from $-H_{cr}$ to $H_{cr}$ according to the hyperbolic Kepler equation $$M=e\sinh{H}-H, \label{eq:H}$$ where M is mean anomaly. The obtained times taken by objects from our generated population to cross the model sphere are shown in Fig. \[fig:time\_spent\], as a function of perihelion distance, eccentricity and hyperbolic excess velocity. ![Surface shows the time that an interstellar object spends inside the model sphere, of $97$ au radius, depending on $v_{\infty}$ and perihelion distance. The color scale represents orbital eccentricity.[]{data-label="fig:time_spent"}](01.pdf){width="\columnwidth"} We found that the longest time to cross the model sphere of 80 years needs an object with $v_{\infty} = 10$ km/s, $q = 21.45$ au and $e = 3.43$. Therefore, all objects inside $3\sigma$ limits of $v_{\infty}$, at the edge of the model sphere, have enough time to leave it in 80 years. 8. To calculate the radius of the initialization sphere, we calculated the distance from which the fastest object from the population ($v_{\infty} = 40$ km/s), on its direct path to the Sun, can reach the model sphere in 80 years, and we obtained the value of $800$ au. This means the model sphere in the time of 80 years will be almost entirely populated with the objects which were initially outside this sphere, but inside the sphere of $800$ au in radius[^4]. The main characteristics of the three spheres described above are summarized in Table \[tab:spheres\]. [| l | l |]{} observable & largest object visible\ sphere - 12 au & at the edge under ideal conditions\ model & fastest object from the population\ sphere - 97 au & can reach observable sphere in\ & 10 years simulation time\ \ & all objects from the population leave this\ & sphere in initialization time of 80 years\ initialization & fastest object at the edge of this sphere\ sphere - 800 au & can reach the model sphere in initialization\ & time of 80 years\ \ & Inside this sphere (and outside the model sphere) is\ & assumed that the gravitational focusing is negligible,\ & and that distribution of ISOs is homogeneous\ & and isotropic\ 9. Assuming number density of 10 objects per au$^3$ for object larger than 10 m (described above), we generated $\approx21$ billion objects randomly distributed inside the initialization sphere, with isotropicaly distributed velocity vectors whose intensities follow the already mentioned normal distribution ($v_0=25$ km/s , $\sigma=5$ km/s). The Cartesian state vectors of these objects were then converted to the Keplerian orbital elements [@Bate_1971], and their positions after 80 years were determined by solving hyperbolic Kepler equation (Eq. \[eq:H\]). Finally, objects located inside the model sphere ($\approx43$ million) were selected for further analysis. Fig. \[fig:number-density\] shows the variation of the number density of the selected population, compared to the value in the initialization sphere where we assumed that the gravitational focusing is negligible, versus the distance from the Sun. It can be seen that due to the gravitational focusing at $1$ au from the Sun the number density should be twice higher than the assumed value in the initialization sphere. On the other hand, one can notice that at the edge of the model sphere, the number density is already very close to the value in the initialisation sphere, which means that the assumption about homogeneous and isotropic population outside this sphere is valid. ![Variation of the number density of ISOs with heliocentric distance. The values on y-axis are normalized to the assumed value in the initialisation sphere, outside of the model sphere, which is unaffected by the gravitational focusing.[]{data-label="fig:number-density"}](02.pdf){width="\columnwidth"} ![Number of objects that enter and leave the model sphere as a function of time. We stress that these numbers are roughly equal after the initialization time of 80 years.[]{data-label="fig:in-out"}](03.pdf){width="\columnwidth"} Another important aspect to consider is how the number of objects that enter and exist from the model sphere evolves with time. As can be seen in Fig. \[fig:in-out\], an initial flux towards the model sphere, of about 1.7 millions of objects per year, remains constant till about 85 years since the beginning of the simulation.[^5] A number of objects exiting from the model sphere is initially zero, because the sphere is originally empty, and increases till it reaches the inward flux, that happens after 80 years. Therefore, after the initialization time interval of 80 years, the inward and outward flux are in balance, meaning that the population of objects inside the model sphere is in the steady-state. ![Time variations in the number density (upper panel) and in the ratio of the number of objects (bottom panel) inside the model and observable spheres.[]{data-label="fig:model-obsrvations"}](04.pdf){width="\columnwidth"} Moreover, as Fig. \[fig:model-obsrvations\] shows, the ratio between the number of objects inside the model ($r=97$ au) and observational ($r=12$ au) spheres, is stable around the end of the initialization interval. At this point, the value of the ratio is about $500$, that is somewhat below the ratio of the volumes of the two spheres, due to the increased number density of objects closer to the Sun, as a result of the gravitational focusing. While the model sphere starts to fill immediately after the simulation begins, the observable sphere remains completely empty until the first objects manage to reach it, after more than 10 years. Since this point, as the consequence of the gravitational focusing, the observable sphere is filled faster than the model sphere, due to the larger number density of objects just outside the observable sphere than just outside the model sphere. Because of this, the observable sphere reaches an equilibrium number of objects a little earlier, about 30 years after the start of the simulation, leading to a rise in the number density ratio, until a stable value is reached. Finally, in order to estimate a role of planetary perturbations, we also randomly selected a smaller sample of $1$ million objects, and propagated their trajectories using Bulirsch-Stoer algorithm as implemented in a public domain *Mercury* software package [@1999MNRAS.304..793C]. The orbits of these objects were followed for $80$ yr, within the dynamical model that includes gravitational effects of the Sun and eight major planets. As expected, no statistically significant differences were noticed between this sample and the overall population which is propagated by means of hyperbolic Kepler equation. ### Orbital distributions The distributions of the orbital elements of the resulting population of all objects within the model sphere is shown in Fig. \[fig:ISO orbital elements\]. The sinusoidal distribution of orbital inclinations is a consequence of the fact that the orbital normal vectors are randomly distributed over the sphere, which result in larger number of highly inclined orbits. Beside this, as expected, longitudes of nodes and arguments of perihelions are uniformly distributed. ![Distribution of orbital elements for the generated population of ISOs inside the model sphere, taken at the end of the initialization time.[]{data-label="fig:ISO orbital elements"}](05.pdf){width="\columnwidth"} Longitudes of nodes, arguments of perihelions and inclinations depend only to the initial positions and orientations of the velocity vectors, and are therefore expected to be mutually independent. On the other hand, perihelion distance and eccentricity are related through the equation [@kemble_2006]: $$q=a+\sqrt{a^2+B_{dist}^2}, \label{eq:q(a)}$$ where $a$ is semi-major axis, and $B_{dist}$ is distance, measured in the b-plane[^6], between the trajectory defined by the initial velocity vector ($\vec{v}_{\infty}$) and the Sun. Taking into account Eq. \[eq:e(q)\], a relation between perihelion distance and eccentricity can be written in the form: $$q=B_{dist} \sqrt{{e-1} \over {e+1}}. \label{eq:q(e)}$$ [We noticed that distribution of orbital eccentricities, for sample of orbits inside certain range of perihelion distances, excellently follows Gamma distribution of the form]{} $$f \left( e \right)=\frac{\left(e-\mu \right)^{\alpha-1}}{\beta^{\alpha} \Gamma \left(\alpha \right)} \exp{\left(-\frac{e-\mu}{\beta}\right)}, \label{eq:gama_pdf}$$ where $e$ is orbital eccentricity, and $\alpha$, $\beta$ and $\mu$ are shape, scale and location parameters of the Gamma distribution, respectively. The approximations for different ranges of perihelion distances are shown in Fig. \[fig:gamma\_2D\]. ![Orbital eccentricities of ISOs approximated with Gamma functions, for different ranges of perihelion distances. The two panels show normalized histograms for 6 samples of the generated population of ISOs. All histograms excellently follow Gamma distributions (shown as black lines). The samples are created according to perihelion distance, since it is directly related to eccentricity for a given $v_{\infty}$. The assumed normal distribution of $v_{\infty}$ ($v_0=25$ km/s, $\sigma=5$ km/s) results in Gamma-like distribution of eccentricities, with the parameters depending on the perihelion distances.[]{data-label="fig:gamma_2D"}](06.pdf){width="\columnwidth"} Based on our numerical experiments (see Fig. \[fig:gamma\_parameters\]), we found that the parameters of these Gamma distributions are in simple relations with the perihelion distance, as given by the formulas: $$\begin{aligned} \alpha &= 11.042-4.966 / q \\ \beta &= 0.087q-0.022 \\ \gamma &= -0.203q+0.377 \label{eq:gama_parameters} \end{aligned}$$ The Eqs. \[eq:gama\_pdf\] and \[eq:gama\_parameters\] define the analytical expression for the bi-variate distribution of perihelion distance and eccentricity of our generated population. The obtained distribution is presented in Fig. \[fig:bi-variate distribution\]. ![Dependence of the parameters of the fitted Gamma distributions on the perihelion distance. The dots present parameters obtained for consecutive intervals of perihelion distance of $1$ au, while the solid lines are their corresponding fits. The parameters are fitted by appropriate rational functions (for shape parameter - $\alpha$) and linear functions (for scale and location parameters - $\beta$ and $\mu$, respectively).[]{data-label="fig:gamma_parameters"}](07.pdf){width="\columnwidth"} ![Bi-variate distribution of perihelion distances and eccentricities. The surface is obtained by generating Gamma distribution defined by Eq. \[eq:gama\_pdf\], using distribution’s parameters defined by Eq. \[eq:gama\_parameters\].[]{data-label="fig:bi-variate distribution"}](08.pdf){width="\columnwidth"} The analytic expressions for orbital distributions allow direct sampling of orbits from these distributions, without further need for the previously described complex algorithm. Analytical models of the populations are very useful in estimating the observational constraints and selection effects of the populations. Similar expressions are already determined for some populations in the Solar System such as the model of Centaur objects [@1997Icar..127..494J] incorporated in the comprehensive model of the Solar System (S3M)[^7] by @2011PASP..123..423G. Analysis and Discussion ======================= Orbit and size distribution of observable objects ------------------------------------------------- To analyze the objects from our ISOs population, which could be detected by the VRO, we conducted the simulation in which we calculated geocentric coordinates, solar elongation and apparent brightness of the objects for every hour inside the simulation period of ten years. In order to identify potentially observable objects, we adopted detectability conditions (summarized in Table \[tab:observational constraints\]) based on the nominal characteristics of the so-called Wide, Fast, Deep (WFD) observational proposal of the LSST [@2017arXiv170804058L; @2016IAUS..318..282J]. Finally, we identified all the objects that satisfy the detectability conditions in at least one simulation time step. Apparent visual magnitude $m<24.5$ ----------------------------- --------------------------------------------------------------- Declination $-65^{\circ} < \delta <5^{\circ}$ Elongation $>60^{\circ}$ Galactic coordinates limits $\left| b\right|=\left(1-l/90^{\circ}\right)\times10^{\circ}$ $0^{\circ}<l<90^{\circ}$ $270^{\circ}<l<360^{\circ}$ where $b$ and $l$ are galactic latitude and longitude, respectively : Detection constraints based on the nominal characteristics of the LSST observational program. An object is considered as observable if in at least one time step within the 10 years of the simulation it satisfies the constraints given in this table. The limitations guarantee that the object is bright enough while located in the appropriate part of the sky to be observed by LSST. We emphasize that a part of the sky around the Galactic plane is excluded.[]{data-label="tab:observational constraints"} Clearly, the restrictions given in Table \[tab:observational constraints\] are far from being sufficient to make any object detected, and especially identified as unknown. A probability that the object will be detected depends on many other factors, such as seeing conditions, effects of the Moon, detection and trailing losses, observing cadence, etc. Moreover, a chance that the detected object will be identified as interesting for follow-up, that would lead to its orbit determination and classification as interstellar, depends on complex set of parameters included in the Minor Planet Center’s so-called digest score [@2019PASP..131f4501K]. However, the constraints given in Table \[tab:observational constraints\], are for sure the necessary conditions that any object potentially detectable by LSST must satisfy. To take into account only objects that may satisfy these conditions, from the previously described global ISOs population in the model sphere ($\approx43$ million), we selected only those ($\approx380 000$) which were initially inside the observable sphere, or appear inside this sphere during the simulation time of 10 years, based on the hyperbolic Kepler equation. This does not guarantee that these objects will reach the defined limiting apparent magnitude, and/or to appear in the appropriate part of sky to be observed. It only implies that no other objects from our synthetic population can be observed during the VRO operational period. In Fig. \[fig:detectable\_objects\], orbital elements of potentially observable objects are shown. ![Distribution of orbital elements of potentially detectable objects. The graphs present normalized histograms of orbital elements for objects which are located inside the observable sphere ($r=12$ au) at the beginning of the simulation, or appear inside it during the simulation period.[]{data-label="fig:detectable_objects"}](09.pdf){width="\columnwidth"} In the bottom panel of this figure one can see that the distribution of the times of perihelion passages is uniform over the simulation period, which, combined with steadiness of distributions of other orbital elements, means that the generated population is time independent during this period. Taking into account the size range of the generated population, the distribution of orbital elements (primarily the perihelion distance), and the assumed albedo of $0.04$, these objects are expected to be very faint, that is the main difficulty for their detection. Fig. \[fig:apparent magnitudes\] shows the distribution of the apparent magnitudes for different SFD slopes, at an arbitrary epoch. ![Graph shows the distribution of the apparent magnitudes of the whole synthetic population, at an arbitrary epoch, for different SFD slopes. The mode of the distribution is highlighted to emphasize the apparent faintness of ISOs.[]{data-label="fig:apparent magnitudes"}](10.pdf){width="\columnwidth"} It is obvious that regardless of the SFD slope, the mode of this distribution, is at any epoch far beyond capabilities of the current and planned wide-field surveys. Only objects from the far tail of the distribution may be detected. This is further illustrated in Fig. \[fig:detectable\_frequency\], where one can see the frequency of the objects, among the whole population, which satisfy the conditions given in Table \[tab:observational constraints\], and may possibly be detected. ![Dependence of the fraction of detectable objects on the population’s SFD slope.[]{data-label="fig:detectable_frequency"}](11.pdf){width="\columnwidth"} The results show that for steeper SFD slopes, at best, only one in several thousands objects should be expected to fulfill the minimum detectability criteria. Analyzing the orbital elements of the detectable objects (those which happen to satisfy constraints given in Table \[tab:observational constraints\] during the simulation interval) we noticed that the most prominent feature is asymmetric distribution of their orbital inclinations[^8], reflected through a larger number of retrograde ($i>90^{\circ}$) than direct orbits ($i<90^{\circ}$), as shown in Fig. \[fig:inclinatios\]. ![Distributions of orbital inclinations. The upper panel shows the distribution of orbital inclinations of the observable objects depending on the SFD slope of the underlying true population. The lower panel shows the inclination distributions (normalized histograms and their interpolated curves) for the three selected values of SFD slope ($1.5$, $2.5$ and $4$), which are also highlighted in the upper panel. It can be seen that, as the SFD slope increases, the maximum of the distribution shifts toward larger inclinations.[]{data-label="fig:inclinatios"}](12.pdf){width="\columnwidth"} In order to identify which factors affect this asymmetry, we examined how the distribution of orbital inclinations depends on possibly relevant parameters. More in particular, to explore if the ratio of the numbers of retrograde and direct objects (R/D ratio) is size dependant, we analyzed a set of detectable objects for 27 different SFD slopes (from $1.4$ to $4$). This set of objects was divided in subsets based on diameters, with a step of $200$ m, and the R/D ratio for each of these subsets was calculated. The obtained results are shown in Fig. \[fig:D-depandance\]. In this figure, the data for all the SFD slopes are shown together. This is because we would like to highlight how the R/D ratio changes for different sizes of objects, and plotting all the objects together provides a better statistical sample. ![R/D ratio for detectable objects of different sizes, binned by $200$ m.[]{data-label="fig:D-depandance"}](13.pdf){width="\columnwidth"} It is noticeable that, for objects of several kilometers in diameter, there is almost no difference, while there is a large asymmetry for sub-kilometer objects, with the latter group making 3 out of 5 to be retrograde. The consequence of this phenomenon is that the R/D ratio, as well as the median inclination, is correlated with the SFD slope of the true population, because steeper SFDs have more smaller objects that, in turn, results in a higher R/D ratio. To clearly illustrate this fact, in Fig. \[fig: correlation\] we plot the median inclination and the R/D ratio as a function of the SFD slope. The results shown in this figure are basically the same ones as those shown in Fig. \[fig:D-depandance\], but this time separated based on the slopes, instead of the diameters. As can be seen in Fig. \[fig: correlation\], both parameters depend on the SFD slope. This fact could allow preliminary estimation of the SFD slope of the true population, based on the orbital inclinations of known population, once a sufficient number of objects have been discovered. ![Dependence of R/D ratio (upper panel) and median inclination (lower panel) of detectable objects on the SFD slope of the underlying true population.[]{data-label="fig: correlation"}](14.pdf){width="\columnwidth"} Beside the SFD slope, we also examined the dependence of the asymmetry on other parameters, and found a potentially interesting dependence of the R/D ratio on the perihelion distance. Similar to the analysis shown in Fig. \[fig:D-depandance\], from the set of all detectable objects we took the subsets of objects inside perihelion distance limits, with a step of $0.5$ au, and calculated R/D ratios of these subsets. The obtained results for three different slopes of the SFD are shown in Fig. \[fig:perihelion\]. It should be noticed that the objects with smaller perihelion distances are the most strongly influenced, with maximum around 1-2 au. In addition, this dependence is more pronounced for steeper SFD slopes. ![R/D ratio for detectable objects for different perihelion distances, binned by $0.5$ au. We plot only data for bins containing at least 100 objects.[]{data-label="fig:perihelion"}](15.pdf){width="\columnwidth"} The role of Holetschek’s effect ------------------------------- The question is what causes that the majority of the detectable ISOs have retrograde orbits, and through which mechanism(s) the R/D asymmetry is related to the SFD and perihelion distances? To answer these questions we turn our attention to long period comets, that could be affected by the same orbital biases as interstellar objects. Still, we need to keep in mind a fact that the LPCs are periodic (returning objects), while ISOs are not. Moreover, in this work we are considering only asteroid-like ISOs, so brightness increase due to the activity is not taken into account here. Nevertheless, some results about the LPCs seem to be useful to explain some of our findings. An excess of the retrograde orbits has been noticed among the LPCs [see e.g. @1967AJ.....72.1002E; @1981MNRAS.197..265F; @1991Natur.352..506M; @2016AJ....152..103S], although more recent results suggest that it may not be so pronounced [@2019AJ....157..181V]. There is still no consensus if this phenomenon is a result of observational selection effect, or there is a real asymmetry in the population, as a consequence of some dynamical mechanism. For instance, on one side, argued that direct comets are exposed to stronger action of planetary perturbations, leading to their faster dynamical evolution, and consequently elimination from the Solar System. This claim seems however to be disputed by findings of @1981MNRAS.197..265F, who found the same asymmetry among the orbits of young comets, that have not had time to evolve due to the planetary perturbations, indicating that the pattern cannot be explained by an aging effect. Also, @1991Natur.352..506M suggested dynamical explanation for the excess of observed comets in retrograde orbits. These authors proposed it could be due to enhanced volatility of retrograde comets, as a result of more energetic collisions with direct meteoroids, comparing to direct comets. The latter explanation however can not be applied to our results for ISOs, because it involves cometary activity. An alternative explanation for a possible excess of retrograde objects among known LPCs is Holetschek’s effect. According to this effect, objects which reach perihelion on the side of the Sun opposite to the position of the Earth are less likely to be discovered [@Holetschek; @1967AJ.....72..716E; @1983MNRAS.204...23H; @2002MNRAS.335..641H]. This is because in this configuration objects are both, too close to the Sun (small elongation), and further away from the Earth (fainter). Therefore, a probability for an object to be discovered depends on the difference ($\Delta \lambda$) between the heliocentric longitude of the Earth and that of the object at the time of the perihelion passage of this object. This is illustrated in Fig. \[fig:longitudes\]. ![Illustration of the orbital and position constellation related to Holetschek’s effect. Quantity $\Delta \lambda$ is the difference between the heliocentric ecliptic longitudes of an interstellar object and the Earth, at the epoch of the object’s perihelion passage.[]{data-label="fig:longitudes"}](16.pdf){width="\columnwidth"} This effect is more important for direct than for retrograde orbits. The qualitative explanation is as follows: after perihelion, the Earth and the retrograde object are, on the average, moving towards each other, reducing quickly $\Delta \lambda$ angle. Therefore, although object’s heliocentric distance is somewhat increasing, its geocentric distance is dropping comparatively quickly, which in turn allow some of the retrograde objects to be discovered after their perihelion passage. On contrary, the objects on direct orbits are, after perihelion, moving typically away from the Earth, leaving almost no chance to be discovered. This means that retrograde objects which are not in observable position when they are at perihelion, have much better chance to take a more observable position, before moving too far from the Sun. Having in mind that our population of ISOs is generated as symmetric, there is no doubt that the asymmetry of orbital inclinations is due to a selection effect. We examined the distribution of the angle $\Delta \lambda$ for the observable ISOs and found clear distinction between direct and retrograde orbits, as shown in Fig. \[fig:Holetschek 1\]. For direct orbits there is a strong concentration around $\Delta \lambda=0$, while for the retrograde objects this distribution is almost uniform. This is a strong indication that Holetschek’s effect is responsible for the R/D asymmetry in our data. ![Holetschek’s effect for direct and retrograde objects. This figure shows normalized histograms of $\Delta \lambda$ of the detectable objects, separately for those on direct ($i<90^{\circ}$) and retrograde ($i>90^{\circ}$) orbits. The vertical plane denotes objects which pass through their perihelions while they are on the same heliocentric ecliptic longitude as the Earth.[]{data-label="fig:Holetschek 1"}](17.pdf){width="\columnwidth"} @1975BAICz..26...92K found Holetschek’s effect to be strongly dependent on the perihelion distance [see also @Holetschek]. Briefly, the author found that between $q=0.5$ and $q=2$ au, the distribution of orbits is strongly biased, and the effect reaches its maximum for perihelion distances around $1$ au. On the other hand, for $q\leq0.5$ Holetschek’s effect should be negligible, while beyond $q\approx2$ au it almost disappears. Therefore, if the observed asymmetry of the R/D ratio is a consequence of Holetschek’s effect, our data should exhibit similar patterns. In this respect, we note that the results shown in Fig. \[fig:perihelion\] already point out in this direction. Still, to further clarify this we analyzed the distribution of $\Delta \lambda$ of the detectable objects, for three different ranges of perihelion distances. Our data shown in Fig. \[fig:Holetschek 2\] are exhibiting a very similar pattern as the one found by @1975BAICz..26...92K. For $q\in[0,2]$ au, there is a peak in the distribution of detectable objects in terms of $\Delta \lambda$. Some deviation from random distribution is also visible for $q\in[2,4]$ au, while for $q\in[4,6]$ au the distribution is uniform. The fact that Holetschek’s effect is more important for direct than for retrograde orbits, and that it mainly affects the orbits with $q\in[0.5,2.5]$ au, fully explains the results presented in Fig. \[fig:perihelion\]. The excess of retrograde objects is the most pronounced for orbits with [$q\approx 1.5$ au]{}, the ones strongly affected by observational bias caused by the aforementioned effect. Taken together, these results clearly indicate that Holetschek’s effect is responsible for the excess of the retrograde orbits among the interstellar objects observable by the VRO. Having saying that, we do not exclude that other observational selection effect also contribute to the excess of retrograde orbits, but to a somewhat lesser extent [see e.g. @1975BAICz..26...92K; @2002MNRAS.335..641H for a review on other observational biases]. ![Holetschek’s effect for different ranges of perihelion distances. The normalized histograms of $\Delta \lambda$ of the detectable objects, for three ranges of perihelion distances. The vertical plane denotes objects which pass through their perihelions while they are on the same ecliptic longitude as the Earth. Data shown in the plot include both, direct and retrograde orbits.[]{data-label="fig:Holetschek 2"}](18.pdf){width="\columnwidth"} In addition, as noted by @1983MNRAS.204...23H, the effect is stronger for smaller than for larger objects, because the larger objects are on average brighter, and visible at larger heliocentric and geocentric distances. For this reason, the observational window of larger objects is longer, and therefore they are less sensitive to the effect. This practically means that Holetschek’s effect is size dependent. Our results shown in Fig. \[fig:D-depandance\] fully support this fact. This is also the reason for the dependence of the R/D ratio on the SFD slope. A stepper size-distribution implies more small objects, which makes the considered population more affected by Holetschek’s effect. This concept explains the dependence of the R/D ratio and the median inclination on the SFD slope shown in Fig. \[fig: correlation\]. On some limitations of our approach ----------------------------------- We would like here to discuss some limitations of the obtained results and prospects for future work. As already mentioned in Section \[ss:num-density\], a single power-law approximation of the size-frequency distribution used here, may not actually be the best option. For some populations of small objects in the Solar System, as for instance in the Kuiper belt, it is well known that the slope becomes significantly shallower at smaller sizes [e.g. @2019Sci...363..955S]. This would imply that there should be comparatively less smaller objects than in our simulation. As our analysis show that the R/D ratio is less significant for larger objects, the observed excess of retrograde orbits would be somewhat less pronounced in population described at smaller sizes with the shallower SFD slope. Though we think our overall conclusions would be still valid, the results would be definitely different, and this deserve to be studied in future work. Another limitation of the results presented here stems from the fact that a probability to detect an ISO depends on several factors that we did not considered here. These factors include seeing conditions, effects of the Moon, detection and trailing losses, observing cadence, digest score etc. For instance, @2017AJ....153..133E have also noticed an excess of retrograde objects in their data, but attributed this to the digest score flag that may favor the retrograde orbits. On the other hand, we found here that the excess of retrograde orbits exists even without the efficiency of the Moving Object Processing System taken into account. Therefore, it would be important to investigate what is the R/D ratio when all the factors are taken into account simultaneously. Finally, we considered here only asteroid-like interstellar objects, and neglected any cometary activity. To calculate apparent magnitudes we assumed simplified linear darkening function of the phase angle, and neglected any possible brightening in comet-like objects. Recently, @2019AJ....157..162H found that comet C/2010 U3 (Boattini) was active at a new record heliocentric distance of $25.8$ au. The second most distant activity is observed in comet C/2017 K2, found to be active at $23.75$ au [@2017ApJ...849L...8M]. CO-driven comets activity at large heliocentric distances has been also predicted by some models. There are several possible mechanisms for activity at these large distances. Sublimation rate is a nonlinear function of temperature, and can occur at low rates at large distances. The sublimation temperatures of the most abundant ices that can drive activity, $CO$, $CO_2$, and $H_{2}O$, are $25$ K, $80$ K, and $160$ K, respectively. The distance at which surface-ice sublimation becomes effective at driving comet activity is when the gas flow lifts sufficient dust from the surface to be detected from Earth. For water this is within the distance of Jupiter; for $CO_2$, this is at the distance between Saturn and Uranus; and for $CO$, it is at distances within the Kuiper Belt [@2009Icar..201..719M see also @2019AJ....157...65J], or even up to heliocentric distances of $85$ au [@2020AA....in..press]. Having in mind that our estimated observable sphere has radius of $12$ au, a cometary-like activity might occur outside this sphere, and a comet-like ISO may be potentially observable at larger heliocentric distances. This would affect our results to some extent, because a larger observable sphere should be used, that would also imply larger model and initialization spheres. Still, for vast majority of comets, brightening should occur only at smaller heliocentric distances [@2009Icar..201..719M]. The total brightness of a comet is the sum of the brightness of the comet nucleus and the brightness of the coma. found that beyond $\sim5$ au contribution of the coma brightness tends to zero, and beyond this distance the comet brightness depends on the heliocentric distance in a way very similar to asteroid-like objects. Therefore, the size of the observable sphere that we used here would be reasonably appropriate also for comet-like ISOs. However, due to the increased brightness within $\sim5$ au, cometary-like objects should be on average discovered at somewhat larger heliocentric distances. As a result, Holetschek’s effect would be less important for these objects, and this would affect to some degree the estimated R/D ratio of observable ISOs. Finally, the results regarding the orbital distribution of observable objects should not be significantly affected. Anyway, some caution is need here, and we underline that our results are strictly speaking valid only for asteroid-like ISOs. Let us also note here that it is suggested by @2012MNRAS.423.1674F that Holetschek’s effect should be less relevant for modern sky-surveys. The reason why we see the consequences of this effect in our simulations, might be because we work with asteroid-like objects, or due to the fact that ISOs are passing only once through the observable sphere. However, as the LSST survey is planned to operate for elongations larger than $60\circ$ we believe the observed excess of retrograde objects is still mainly due to Holetschek’s effect. Summary and Conclusions ======================= We study the distribution of the orbital elements of the generated synthetic population of the interstellar objects, specifically focusing on those observable by the VRO, based on the nominal characteristics of this survey. While a several other authors performed a similar investigation, they focused on the expected number of detectable ISOs, rather than on their orbital characteristics. Our main conclusions can be summarized in the following: - The gravitational focusing should not affect significantly the orbital distribution of the observable ISOs. At 1 au distance from the Sun only twice as many objects as in the interstellar space should be expected, but outside of the model sphere, the number density is very close to the assumed value in the initialisation sphere, meaning that the assumption about homogeneous and isotropic population outside the model sphere is valid. - The perturbation by the planets do not produce any significant effect on the orbital distribution of ISOs. - [We found that the distribution of orbital eccentricities can be very well approximated with Gamma distributions, whose parameters are functions of perihelion distance. These analytical expressions for orbital distributions allow simple direct sampling of objects without the need for applying complex technique described above.]{} - Among the potentially observable ISOs, there is an asymmetry in the distribution of orbital inclinations, with an overabundance of retrograde objects. - The excess is the result of Holetschek’s effect which is already suggested to be responsible for the oversupply of retrograde objects among the observed long-periodic comets. Holetschek’s effect depends on the objects’ sizes and their perihelion distances. - The excess of retrograde objects depends on objects’ sizes and perihelion distances. This should allow estimation of the SFD of the underlying true population based on R/D ratio and median inclination of the discovered population. Acknowledgements {#acknowledgements .unnumbered} ================ We sincerely thank the reviewer for constructive criticisms and valuable comments, which were of great help in revising the manuscript. The authors acknowledge financial support from the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project ON176011 “Dynamics and kinematics of celestial bodies and systems”. \[lastpage\] [^1]: E-mail: dmarceta@matf.bg.ac.rs (DM) [^2]: Formerly known as the Large Synoptic Survey Telescope (LSST) [^3]: We note that, generally, it seems that population of the small Solar System objects has a shallower SFD at small than at larger sizes [@2009Icar..202..104G; @2014Icar..231..168B; @2019Sci...363..955S], and therefore should be represented as a broken power-law. In this work we did not consider these findings, but it would be worth to model also populations with broken power-law slopes in the future work. [^4]: We note that it would be possible to use also an initialization sphere that extends beyond the adopted limit, that would also require to use longer initialization time and significantly larger number of objects. However, this would notably increases computational cost, with no obvious benefit for our work. [^5]: This drop in a number of objects entering the model sphere is simply a consequence of limited size of the initialization sphere. For our purpose here, the stable flux towards the model sphere is lasting long enough, but in principle it could be extended to any desirable time, by changing appropriately a size of the initialization sphere. [^6]: The b-plane is defined to contain the focus of an idealized two-body trajectory that is assumed to be a hyperbola, and is perpendicular to the incoming asymptote of the hyperbola. [^7]: This model is used to evaluate the performance of Pan-STARRS survey in discovering objects from various populations of the Solar System, including also the interstellar comets. [^8]: We recall here that @2017AJ....153..133E have also noticed similar fact, but attributed this to the digest score flag that may favor the retrograde orbits. However, although the digest score may play a role in the distribution of detected ISOs, there must be other reason as well, because in our study we did not consider efficiency of the Moving Object Processing System.

--- abstract: 'In parametric sequence alignment, optimal alignments of two sequences are computed as a function of the penalties for mismatches and spaces, producing many different optimal alignments. Here we give a $3/(2^{7/3}\pi^{2/3})n^{2/3} +O(n^{1/3} \log n)$ lower bound on the maximum number of distinct optimal alignment summaries of length $n$ binary sequences. This shows that the upper bound given by Gusfield et. al. is tight over all alphabets, thereby disproving the “$\sqrt{n}$ conjecture". Thus the maximum number of distinct optimal alignment summaries (i.e. vertices of the alignment polytope) over all pairs of length $n$ sequences is $\Theta(n^{2/3})$.' address: 'Department of Mathematics, University of California, Berkeley 94720' author: - Cynthia Vinzant title: Lower Bounds for Optimal Alignments of Binary Sequences --- sequence alignment ,parametric analysis ,computational biology Introduction and Notation ========================= Finding optimal alignments of DNA or amino acid sequences is often used in biology to measure sequence similarity (homology) and determine evolutionary history. For a review of many problems relating to sequence alignment, see [@G; @book] and [@ASCB]. Here we deal with the question of how many different alignment summaries can be considered optimal for a given pair of sequences (though many different alignments may correspond to the same alignment summary). Given sequences $S$, $T$, an *alignment* $\Gamma$ is a pair $(S', T')$ formed by inserting spaces, “$-$", into $S$ and $T$. In each position, there is a *match*, in which $S'$ and $T'$ have the same characters, a *mismatch*, in which they have different characters, or a space in one of the sequences. Then for any alignment, we have an *alignment summary* $(w, x,y)$, where $w$ is the number of matches, $x$ is the number of mismatches, and $y$ is the number of spaces in one of the sequences. Notice that $n = w+x+y$, where $n$ is the length of both sequences. Given a pair of sequences, the convex hull of all such points $(w,x,y)$ is called their *alignment polytope*. We can score alignments by weighting each component. Since we have $w+x+y=n$, we can normalize so that the weight of $w$ is 1, the weight of $x$ is $-\alpha$ and the weight of $y$ is $-\beta$. Then $$score_{(\alpha, \beta)}(w,x,y) = w - \alpha x - \beta y.$$ A sequence is *optimal* if it maximizes this score. For biological relevance, we will only consider non-negative $\alpha$ and $\beta$, which penalizes mismatches and spaces. It is also possible to weight other parameters, such as *gaps* (consecutive spaces) or mismatches between certain subsets of characters. Here we will consider only the two parameter model described above. \[ex1\] For the sequences 111000 and 010110, we have an alignment $$\begin{matrix} - & 1 & - & 1&1& 0 &0 &0\\ 0 & 1 & 0 & 1&1&1&-&- \end{matrix} \qquad$$ which has 3 matches, 1 mismatch, and 2 spaces. So for a given $\alpha$ and $\beta$ the score of the this alignment would be $3 - \alpha - 2\beta$. Any value of $\alpha$ and $\beta$ will give an optimal alignment. Given $\alpha$ and $\beta$, we can use the Needleman-Wunsch algorithm to effectively compute optimal alignments [@NW] (for a review, see [@ASCB Ch. 2, 7]). Unfortunately, different choices for $\alpha, \beta$ give different optimal alignments, leaving the problem of which weights to use. To resolve this, Waterman, Eggert, and Lander proposed *parametric alignment*, in which the weights $\alpha$, $\beta$ are viewed as parameters rather than constants [@beginnings]. Since alignments are discrete, this creates a partition of the $(\alpha, \beta)$ plane into *optimality regions*, so that for each region $R$, there is an alignment that is optimal for all the points on its interior and $R$ is maximal with this property [@G]. Each optimality region is a convex cone in the plane [@G], [@ASCB Ch. 8]. Notice that because our scoring function is linear, the vertices of the alignment polytope are our optimal alignment summaries. Also, if we let $P_{xy}$ be the convex hull of all $(x,y)$ occurring in alignment summaries, then $$score_{(\alpha, \beta)} = w - \alpha x - \beta y = n - (\alpha +1)x - (\beta +1)y,$$ since $n= w+x+y$. Thus the vertices of $P_{xy}$ will be those that minimize $(x,y)\cdot (\alpha+1, \beta+1)$ for some $(\alpha, \beta$), thus maximizing $score_{(\alpha, \beta)}$ and corresponding to optimal alignments [@ASCB]. From this we can see that the the decomposition of the $(\alpha, \beta)$ plane into optimality regions can be obtained by shifting the normal fan of $P_{xy}$ by $(-1, -1)$ [@ASCB Ch. 8]. The goal of parametric alignment is to find all these optimality regions with their corresponding optimal alignments. The Needleman-Wunsch algorithm is also an effective method of computing the alignment polytope of sequences (and thus optimal alignments and the decomposition of the $(\alpha, \beta)$ plane) [@ASCB]. Gusfield et. al. showed that for two sequences of length $n$, the number of optimality regions of the $(\alpha, \beta)$ plane (equivalently the number of vertices in their alignment polytope) is $O(n^{2/3})$[@G]. Indeed for larger dimensional models (say with $d$ free parameters), this bound was extended to $O(n^{d-(1/3)})$ by Fernández-Baca et. al. [@Baca2] and improved to $O(n^{d(d-1)/(d+1)})$ by Pachter and Sturmfels [@alg; @bounds]. For $d=2$, Fernández-Baca et. al. refined this bound to $3(n/2\pi)^{2/3} + O(n^{1/3}\log (n))$ and showed it to be tight over an infinite alphabet [@Baca]. They also provide a lower bound of $\Omega(\sqrt{n})$ over a binary alphabet. Using randomly-generated sequences, Fernández-Baca et. al. observed that the average number of optimality regions closely approximates $\sqrt{n}$. This led them to conjecture that, over a finite alphabet, the expected number of optimality regions is $\Theta(\sqrt{n})$[@Baca]. The question remained of whether or not the upper bound of Gusfield et. al. was tight over a finite alphabet. For a discussion, see [@ASCB Ch. 8], which conjectures that the maximum number of optimality regions induced by any pair of length-$n$ binary strings is $\Theta(\sqrt{n})$ [@ASCB]. Here we construct a counterexample to this conjecture, which together with the above upper bounds shows it instead to be $\Theta(n^{2/3})$. Our main theorem is that Gusfield’s bound is tight for binary strings. The maximum number of optimality regions induced by binary strings of length $n$ is $\Theta(n^{2/3})$. Ideally, sequences would have few optimal alignments, making the “best" one more apparent. While this result may not tell us about the expected number of optimal alignments (or be biologically relevant), it does provide a worst case scenario for sequence alignment and show that the bound from [@G] cannot be improved. Luckily, the bound is still sublinear. Indeed parametric sequence alignment can be practical and has been achieved for whole genomes [@fly]. This paper is mainly motivated by [@Baca], [@G], and [@ASCB]. We largely follow their notation and presentation. Decomposing the $(\alpha, \beta)$ plane ======================================= Alignment Graphs ---------------- We can represent every alignment of two length-$n$ sequences as a path through their *alignment graph*. The graph can be thought of as an $(n+1) \times (n+1)$ grid, with rows and columns numbered consecutively from top to bottom (left to right), from 0 to $n$ [@Baca]. An *alignment path* is a path on these vertices, starting at $(0,0)$, ending at $(n,n)$, and only moving down, right or diagonally down and to the right. Each path corresponds to a unique alignment. In this path, a move down (or left) corresponds to a space in the first (or second) sequence, and a diagonal move corresponds to a match or mismatch (depending on the characters). See Figure \[fig: example\] for the alignment graph of our above example alignment. (45, 28) (4,4)(2,0)[7]{}[(0,1)[12]{}]{} (4,4)(0,2)[7]{}[(1,0)[12]{}]{} (2.5,4.5)(0,2)[3]{}[1]{} (2.5, 10.5)(0,4)[2]{}[0]{} (2.5, 12.5)[1]{} (4.5, 16.5)(2,0)[3]{}[1]{} (10.5, 16.5)(2,0)[3]{}[0]{} (4,14)[(1, -1)[2]{}]{} (6,10)[(1,-1)[6]{}]{} (4,16)[(0, -1)[2]{}]{} (6,12)[(0,-1)[2]{}]{} (12,4)[(1,0)[4]{}]{} (4,10)(.4,0)[15]{}[(0,-1)[6]{}]{} (4, 14)(.4,0)[15]{}[(0,-1)[2]{}]{} (10, 16)(.4,0)[15]{}[(0,-1)[2]{}]{} (10, 12)(.4,0)[15]{}[(0,-1)[2]{}]{} (31,0)[(0,1)[22]{}]{} (25,6)[(1,0)[18]{}]{} (30.5,2)(0,4)[5]{}[(1,0)[1]{}]{} (27,5.5)(4,0)[4]{}[(0,1)[1]{}]{} (27,2) (25,0)[(1,1) [18]{}]{} (26, 0)[(1,2)[11]{}]{} (32,19)[$\Gamma_1$]{} (35,14)[$\Gamma_2$]{} (38, 9)[$\Gamma_3$]{} (29,20)[$\beta$]{} (42,4)[$\alpha$]{} \[fig: example\] Optimality regions ------------------ Gusfield et. al. observed that the boundaries between optimality regions in the $(\alpha, \beta)$ plane must be lines passing through the point $(-1,-1)$. \[Gusfield et. al., [@G]\] All optimality regions on the $(\alpha, \beta)$ plane are semi-infinite cones, and are delimited by lines of the form $\beta = c + (c+1)\alpha$ for some constant $c$. In general, a boundary between two optimality regions consists of the $(\alpha, \beta)$ for which the optimal sequences from each region have equal, optimal scores. Since $$score_{(-1, -1)}(w,x,y) = w +x +y \equiv n,$$ for every $w, x, y$, each such line (specifically these boundary lines) must pass through the point $(-1,-1)$. They also note that all of these boundary lines must intersect the non-negative $\beta$-axis because none of them cross the positive $\alpha$-axis [@G]. This comes from observing that in any alignment, we can change a mismatch to a space (in each sequence) without affecting the number of matches. Thus all along the line $\beta =0$, the optimal alignment will have the maximum number of matches possible, without regard to spaces (since those are not penalized). So no boundary line can separate the nonnegative $\alpha$-axis into distinct optimality regions. Since all boundary lines must pass through the point $(-1,-1)$ and cannot intersect the positive $\alpha$-axis, we indeed have that \[Gusfield et. al., [@G]\] Each of the optimality regions must have nontrivial intersection with the non-negative $\beta$-axis. That is, for any path $\Gamma$ that is optimized by some $(\alpha, \beta)$, there must be some $\beta '$ so that $\Gamma $ is optimized by $(0, \beta')$. This allows us to restrict our attention to optimality regions on the $\beta$-axis. Then boundary regions are just points, $(0,\beta)$, for which consecutive optimal alignments have optimal $score_{(0,\beta)}$. Note that alignments with summaries $(w_1, x_1, y_1)$ and $(w_2, x_2, y_2)$ will have equal $score_{(0,\beta)}$ when $$w_1 - \beta y_1 = w_2 - \beta y_2,$$ meaning that $$\beta = \frac{\Delta w}{\Delta y} := \frac{w_2-w_1}{y_2-y_1}.$$ In order to find different optimality regions, we will find distinct $\frac{\Delta w}{\Delta y}$ forming boundary points on the $\beta$-axis. The Lower Bound =============== For each $2 \leq r$, define $F_r$ as $F_r:= \{\frac{a}{b}\leq 1 \; : \; \frac{a}{b}$ is reduced and $a+b = r\}$. Since $a/b$ is reduced and $a+b=r$, $a$ and $b$ must be relatively prime to $r$. Then each number relatively prime to $r$ will show up exactly once (in either the numerator or the denominator), so $|F_r| = \phi(r)/2$ for $r >2$ where $\phi$ is the Euler totient function, and $|F_2| = |\{1/1\}| =1$. Let $$\mathcal{F}_q = \bigcup_{r=2}^q F_r,$$ giving us $|\mathcal{F}_q| = \frac{1}{2}\sum_{r=3}^{q}\phi(r) +1$.\ \ Fixing $q$, let $a_1/b_1 < a_2/b_2 < \ldots < a_m/b_m=1$ be the elements of $\mathcal{F}_q$. We’re going to construct two sequences of length $n = 4 \sum_kb_k $, $S = s_1s_2 \hdots s_n$ and $T=t_1t_2\hdots t_n$. Since $b_k < a_k+ b_k$, this gives us $$n = 4\sum_{k=1}^m b_k < 4 \sum_{k=1}^m (a_k + b_k) = 4\sum_{r=2}^s r |F_r| = 2 \sum_{r=2}^s r \phi(r).$$ The Sequences ------------- Let’s construct the first sequence, $S$. To start, let the first $b_1+a_1$ elements of $S$ be 0, followed by $b_1-a_1$ 1’s. Then repeat for $k>1$ (i.e. next place $b_2+a_2$ 0’s followed by $b_2-a_2$ 1’s). Notice that for each $a_k/b_k \in \mathcal{F}_q$, we use $(b_k+a_k)+ (b_k-a_k)=2b_k$ places. To get the second half of the sequence, take the reverse complement of the first half (reflecting it and switching all the 1’s and 0’s). So $$S = 0^{b_1+a_1}1^{b_1 - a_1} 0^{b_2+a_2}\hdots 0^{b_m+a_m}1^{b_m - a_m} \;\; 0^{b_m-a_m}1^{b_m + a_m}\hdots 0^{b_1-a_1}1^{b_1 + a_1}.$$ More formally, define $$i(r) = \sum_{k=1}^r 2b_k \;\;\;\;\; \text{ and } \;\;\;\;\; j(r) = \sum_{k=r}^m 2b_k.$$ (So $n = 2i(m) = 2j(1)$). Then $$s_{i(r-1) +k} = \left\{ \begin{array}{rl} 0 & \text{for } 1 \leq k \leq b_r +a_r\\ 1 & \text{for } b_r+a_r +1\leq k \leq 2b_r \end{array} \right.$$ and $$s_{\frac{n}{2}+j(r+1)+k} = \left\{ \begin{array}{rl} 0 & \text{for } 1 \leq k\leq b_r -a_r\\ 1 & \text{for } b_r-a_r +1\leq k \leq 2b_r. \end{array} \right.$$ The second sequence, $T$, will just be $n/2$ 1’s followed by $n/2$ 0’s, that is, $$t_k = \left\{ \begin{array}{rl} 1 & \text{for } 1 \leq k\leq n/2\\ 0 & \text{for } n/2 +1 \leq k \leq n. \end{array} \right.$$ \[ex: main\]For $q = 4$, $\mathcal{F}_4 = \{1/3, 1/2, 1/1 \}$. Then $n = 4(3+2+1)= 24$. Our sequences are $$S = 000011000100\; 110111001111$$ $$T = 111111111111\; 000000000000$$ The Alignment Paths ------------------- We are going to construct $m+1$ alignment paths, $\Gamma_{m+1}, \Gamma_m, \hdots, \Gamma_1$. Let $\Gamma_{m+1} $ be the path along the main diagonal (corresponding to the alignment with no spaces). To get $\Gamma_r$, align the first $j(r) = \sum_{k=r}^m 2b_k$ 0’s of $S$ with spaces and align its remaining elements without spaces, ending by aligning the last $j(r)$ 0’s of $T$ with spaces. Note that because there are $n/2$ 1’s in both $S$ and $T$, we’ll have enough room to do this. In fact, in the last alignment, $\Gamma_1$, all the 1’s of $S$ will be matched with all the 1’s of $T$. See Figure \[fig: gammas\] for the graphs of the optimal alignments of our example. (56,51) (5,0)(2,0)[25]{}[(0,1)[48]{}]{} (5,0)(0,2)[25]{}[(1,0)[48]{}]{} (3.5,.5)(0,2)[4]{}[1]{} (3.5,8.5)(0,2)[2]{}[0]{} (3.5,12.5)(0,2)[3]{}[1]{} (3.5, 18.5)[0]{} (3.5,20.5)(0,2)[2]{}[1]{} (3.5, 24.5)(0,2)[2]{}[0]{} (3.5,28.5)[1]{} (3.5,30.5)(0,2)[3]{}[0]{} (3.5,36.5)(0,2)[2]{}[1]{} (3.5,40.5)(0,2)[4]{}[0]{} (5.7,48.5)(2,0)[12]{}[1]{} (29.7,48.5)(2,0)[12]{}[0]{} (5,48)[(1, -1)[48]{}]{} (5,44)[(1,-1)[44]{}]{} (5,40)[(1,-1)[4]{}]{} (9,32)[(1,-1)[32]{}]{} (9,30)[(1,-1)[2]{}]{} (11, 24)[(1,-1)[4]{}]{} (15, 18)[(1,-1)[6]{}]{} (21, 8)[(1,-1)[8]{}]{} (5,48)[(0, -1)[8]{}]{} (9,36)[(0,-1)[6]{}]{} (11,28)[(0,-1)[4]{}]{} (15, 20)[(0,-1)[2]{}]{} (21, 12)[(0,-1)[4]{}]{} (29,0)[(1,0)[24]{}]{} (29,48)(.4,0)[60]{}[(0,-1)[8]{}]{} (29,36)(.4,0)[60]{}[(0,-1)[6]{}]{} (29,28)(.4,0)[60]{}[(0,-1)[4]{}]{} (29,20)(.4,0)[60]{}[(0,-1)[2]{}]{} (29,12)(.4,0)[60]{}[(0,-1)[4]{}]{} (5,40)(.4,0)[60]{}[(0,-1)[4]{}]{} (5,30)(.4,0)[60]{}[(0,-1)[2]{}]{} (5,24)(.4,0)[60]{}[(0,-1)[4]{}]{} (5,18)(.4,0)[60]{}[(0,-1)[6]{}]{} (5,8)(.4,0)[60]{}[(0,-1)[8]{}]{} (29,0) (29,12) (29,20) (29,24) \[fig: gammas\] (20, 32) (4,0)[(0,1)[30]{}]{} (0,4)[(1,0)[18]{}]{} (3.5,8)[(1,0)[1]{}]{} (3.5,10)[(1,0)[1]{}]{} (3.5, 16)[(1,0)[1]{}]{} (3.5, 28)[(1,0)[1]{}]{} (16, 3.5)[(0,1)[1]{}]{} (0,7.5)[1/3]{} (0,9.5)[1/2]{} (2,15.5)[1]{} (2,29)[$\beta$]{} (17, 2)[$\alpha$]{} (4,8)[(3,4)[14]{}]{} (4,10)[(2,3)[12]{}]{} (4,16)[(1,2)[6]{}]{} (5,24)[$\Gamma_4$]{} (8,20)[$\Gamma_3$]{} (13.5,16)[$\Gamma_2$]{} ( 13,10)[$\Gamma_1$]{} (13,17)[(-2,1)[2.5]{}]{} \[fig: ab decomp\] Alignment Scores ---------------- Let $w_r^1$ denote the number of matching 1’s in $\Gamma_r$ and similarly $w_r^0$ denote the number of matching 0’s in $\Gamma_r$, with $w_r$ being the total number of matches. Note that $$w_r^1 - w_{r+1}^1 = b_r +a_r \;\; \text{ and }\;\;w_r^0 - w_{r+1}^0 = - (b_r-a_r).$$ Since $w_r = w_r^1+w_r^0$, we have that $$w_r - w_{r+1} = (b_r+a_r) - (b_r-a_r) = 2a_r.$$ Let $y_r$ denote the number of spaces in $\Gamma_r$ (which equals $j(r)$). Then $$y_r - y_{r+1} = j(r) - j(r+1) = 2b_r.$$ Putting these together, we get that for every $r$, $$\frac{\Delta w_r}{\Delta y_r} := \frac{w_r - w_{r+1}}{y_r - y_{r+1}} = \frac{a_r}{b_r}.\label{a/b achieved}$$ Optimality ---------- We need to show that each of these paths is optimal for distinct optimality regions, which will be accomplished by the next two lemmata. \[optimal\] Let $\Gamma$ be any alignment of $S$ and $T$. Then for any $\beta \geq 0$, there is some $\Gamma_r$ so that $score_{(0,\beta)}(\Gamma_r) \geq score_{(0,\beta)}(\Gamma)$. Say that $\Gamma$ has alignment path $\sigma$ and alignment summary $(w,x,y)$. Let the coordinates of the alignment graph be $(t,s)$, with $(0,0)$ starting in the upper left corner. Say that $(n/2, n/2+k)$ is the first time $\sigma$ meets the vertical line $t=n/2$. Because of the symmetry of our sequences, we can take $k$ to be nonnegative (meaning that $\sigma$ hits the line $t=n/2$ below or at $s=n/2$). If $\sigma$ has $k<0$, we can rotate our picture $180^o$ to get another alignment path with the same summary and $k \geq 0$. So suppose $k \geq 0$ and take $r$ so that $j(r+1) < k \leq j(r)$.\ \ *(Case 1:* $k-j(r+1) \leq b_r - a_r$*)*. Since there are only $w_{r+1}^1$ 1’s above $s=n/2+k$, we have $w^1\leq w^1_{r+1}$. Similarly, there are at most $w_{r+1}^0$ 0’s below $s=n/2+k$, so $w^0 \leq w_{r+1}^0$. Furthermore, by going through the point $(n/2, n/2+k)$, $\sigma$ must have at least $k$ spaces, so $y \geq k \geq j(r+1) = y_{r+1}$. Putting these together gives that for any $\beta \geq 0$, $$score_{(0,\beta)}(\Gamma_{r+1}) - score_{(0,\beta)}(\Gamma) = (w_{r+1} - w) - \beta (y_{r+1} - y) \geq 0 .$$ Intuitively, $\Gamma$ can have at most as many matches and must have at least as many spaces as $\Gamma_{r+1}$, and thus cannot have a higher score.\ \ *(Case 2: $k-j(r+1) > b_r - a_r$ and $\beta \leq 1$)* There are $w_r^0$ 0’s in $S$ below $s= n/2+k$, so we have $w^0 \leq w^0_r$. In addition to the $w_{r+1}^1$ 1’s in $S$ above $s=n/2 + j(r+1)$, there are another $k-j(r)+(b_r+a_r)$ 1’s in $S$ between $s=n/2+j(r+1)$ and $s=n/2+k$. So $$w^1 \leq w_{r+1}^1 +k-j(r)+(b_r +a_r) = w_r^1 + k -j(r),$$ since $w_{r+1}^1 + (b_r +a_r)= w_r^1$. Thus $$w = w^0 +w^1 \leq w^0_r + w^1_r +k - j(r) = w_r +k -j(r).\label{w_r}$$ As is case 1, we have that $y\geq k$, so $$\begin{aligned} score_{(0,\beta)}(\Gamma_r) - score_{(0,\beta)}(\Gamma) &= (w_r - w) - \beta(y_r - y) \nonumber\\ &\geq (j(r) - k) - \beta(j(r) - k) \tag{by \eqref{w_r}} \nonumber \\ & \geq 0 \tag{as $\beta \leq 1$} \nonumber \end{aligned}$$\ *(Case 3: $k-j(r+1) > b_r - a_r$ and $\beta > 1$)* We’ll show that $score_{(0,\beta)}(\Gamma_{m+1}) \geq score_{(0,\beta)}(\Gamma)$. Remember that $\Gamma_{m+1}$ is the alignment with no spaces ($y_{m+1} = 0$), corresponding to the main diagonal of the alignment graph. Note for any $r$, $$\label{w_m+1} w_r = w_{m+1}+\sum_{k=r}^m 2a_k,$$ so using equation from case 2, we get $$w_{m+1} - w \geq j(r)-k - \sum_{k=r}^m 2a_k.$$ As in previous cases, $y \geq k$. Then, $$\begin{aligned} score_{(0,\beta)}(\Gamma_{m+1}) - score_{(0,\beta)}(\Gamma) & = (w_{m+1} - w) - \beta(y_{m+1} -y) \nonumber\\ & \geq j(r)-k - \sum_{k=r}^m 2a_k + \beta k \tag{by \eqref{w_m+1}} \nonumber\\ &\geq j(r) - \sum_{k=r}^m 2a_k \tag{as $\beta >1$} \nonumber \\ & = \sum_{k=r}^m 2b_k - \sum_{k=r}^m 2a_k \nonumber \\ & \geq 0. \nonumber\end{aligned}$$ Lemma \[optimal\] tells us that any optimality region has one of the $\Gamma_r$ as an optimal alignment. Now we need to check that each $score_{(0,\beta)}(\Gamma_r)$ is optimized by a different region. To see this, we use equation and following lemma. \[Fernández-Baca, et. al., [@Baca]\] Let $\Gamma_1, \Gamma_2, \hdots, \Gamma_q$ be paths in the alignment graph. Assume $score(\Gamma_i)=w_i - \beta y_i$, where $y_1 > y_2 > \hdots > y_q$. Let $\beta_0 = 0, \beta_q = \infty$, and for $r = 1, \hdots, q-1$, $\beta_r = (w_r - w_{r+1})/(y_r - y_{r+1})$. Suppose $\beta_0 < \beta_1 < \hdots < \beta_q$. Then for $\beta \in (\beta_{r-1}, \beta_r)$ and $p \neq r$, $score_{(0,\beta)}(\Gamma_r) > score_{(0,\beta)}(\Gamma_p)$. So each of the $\Gamma_r$ do indeed represent each of the different optimality regions on the $\beta$-axis, and thus in the $(\alpha, \beta)$ plane. The Actual Lower Bound ---------------------- The maximum number of optimality regions induced by any pair of length-$n$ sequences is $\Omega(n^{2/3})$. Above we have constructed sequences of length $n \leq 2 \sum_{r=2}^q r\phi(r)$ that gave $m = \frac{1}{2}\sum_{r=2}^q \phi(r)$ optimality regions. From analytic number theory, as calculated in [@Baca], $$m= \frac{1}{2}\sum_{r=3}^q \phi(r) +1 = \frac{3}{2 \pi^2}q^2 +O(q \log q),$$ and $$n \leq 2\sum_{r=2}^q r\phi(r) = \frac{4}{\pi^2}q^3 + O(q^2\log q).$$ Then $q \geq (\frac{\pi^2 n}{4})^{1/3} +O(\log n)$, meaning $$\begin{aligned} m= \frac{1}{2} \sum_{r=3}^q \phi(r) +1 & \geq \frac{3}{2\pi^2}\left((\frac{\pi^2 n}{4})^{1/3}\right)^2 +O(n^{1/3} \log n) \nonumber\\ & = \frac{3}{2^{7/3}\pi^{2/3}}n^{2/3} +O(n^{1/3} \log n). \nonumber\end{aligned}$$ With the upper bounds from [@G] and [@Baca], this gives The maximum number of optimality regions over all pairs of length-$n$ sequences is $\Theta(n^{2/3})$, and more specifically is between $\frac{3}{2^{7/3}\pi^{2/3}}n^{2/3} +O(n^{1/3} \log n)$ and $\frac{3}{(2\pi)^{2/3}}n^{2/3} +O(n^{1/3} \log n)$. It’s unclear whether the current bounds on optimality regions for scoring with $d>2$ parameters, $O(n^{d(d-1)/(d+1)})$, are also tight or whether better upper bounds exist. Another interesting open question (perhaps with more practical relevance) is the order of the expected number of optimality regions, rather than the maximum. Acknowledgements ================ Thanks to Lior Pachter for his advice and suggestion of this problem. This paper came out of his class at U.C. Berkeley, “Discrete Mathematics for the Life Sciences", in the spring of 2008. Thanks also to Bernd Sturmfels and Peter Huggins for their useful suggestions. [00]{} C. Dewey, P. Huggins, K. Woods, B. Sturmfels, L. Pachter, Parametric alignment of Drosophila genomes. *PLoS Computational Biology*, 2(6):e73 (2006). D. Fernández-Baca, T. Seppäläinen, G. Slutzki, Bounds for parametric sequence alignment. *Discrete Applied Math* 118 (2002), 181-198. D. Fernández-Baca, T. Seppäläinen, G. Slutzki, Parametric multiple sequence alignment and phylogeny construction. *Journal of Discrete Algorithms.* 2(2)(2004), 271-287. D. Gusfield, K. Balasubramanian, D. Naor, Parametric Optimization of Sequence Alignment. *Algorithmica* 12 (1994), 312-326. D. Gusfield, Algorithms on Strings, Trees, and Sequences: Computer Science and Computational Biology, Cambridge University Press, Cambridge, New York, Melbourne, 1997. S. Needleman, C. Wunsch, A general method applicable to the search for similarities in the amino acid sequence of two proteins. *Journal of Molecular Biology*, 48 (1970) 443-445, 1970. L. Pachter, B. Sturmfels, Parametric inference for biological sequence alignment. *Proc. of the National Academy of Sciences, USA*, 101(46) (2004) 16138-43, 2004. L. Pachter, B. Sturmfels, editors. *Algebraic Statistics for Computational Biology.* Cambridge University Press, 2005. M.S. Waterman, M. Eggert, E.S. Lander, Parametric sequence comparisons. *Proc. of the National Academy of Sciences, USA*, 89 (1992) 6090-6093.

--- abstract: 'Recently, more and more works have proposed to drive evolutionary algorithms using machine learning models. Usually, the performance of such model based evolutionary algorithms is highly dependent on the training qualities of the adopted models. Since it usually requires a certain amount of data (i.e. the candidate solutions generated by the algorithms) for model training, the performance deteriorates rapidly with the increase of the problem scales, due to the curse of dimensionality. To address this issue, we propose a multi-objective evolutionary algorithm driven by the generative adversarial networks (GANs). At each generation of the proposed algorithm, the parent solutions are first classified into *real* and *fake* samples to train the GANs; then the offspring solutions are sampled by the trained GANs. Thanks to the powerful generative ability of the GANs, our proposed algorithm is capable of generating promising offspring solutions in high-dimensional decision space with limited training data. The proposed algorithm is tested on 10 benchmark problems with up to 200 decision variables. Experimental results on these test problems demonstrate the effectiveness of the proposed algorithm.' author: - | Cheng He, Shihua Huang, Ran Cheng *IEEE Member*,\ Kay Chen Tan *IEEE Fellow*, and Yaochu Jin *IEEE Fellow* [^1] [^2] [^3] [^4] bibliography: - 'Reference.bib' title: 'Evolutionary Multi-Objective Optimization Driven by Generative Adversarial Networks [(GANs)]{}' --- [He : Evolutionary Multi-Objective Optimization Driven by Generative Adversarial Networks]{} Multi-objective optimization, evolutionary algorithm, machine learning, deep learning, generative adversarial networks Introduction {#sec:introduction} ============ Multi-objective optimization problems (MOPs) refer to the optimization problems with multiple conflicting objectives [@app-network], e.g., structure learning for deep neural networks [@liu2018structure], energy efficiency in building design [@app-build], and cognitive space communication [@ferreira2017multi]. The mathematical formulation of the MOPs is presented as follows [@deb2014multi]: $$\begin{aligned} \label{eq:MOP} \text{Minimize}& F(\mathbf{\mathbf{x}})=\!(f_1(\mathbf{x}),f_2(\mathbf{x}),\dots,f_M(\mathbf{x}))&\\ \text{subject to}&\mathbf{x}\in X, \nonumber\end{aligned}$$ where $X$ is the search space of decision variables, $M$ is the number of objectives, and $\mathbf{x}$$=$$(x_1,\dots,x_D)$ is the decision vector with $D$ denoting the number of decision variables [@tian2017effectiveness]. Different from the single-objective optimization problems with single global optima, there exist multiple optima that trade off between different conflicting objectives in an MOP [@PD]. In multi-objective optimization, the Pareto dominance relationship is usually adopted to distinguish the qualities of two different solutions [@ENS]. A solution $\mathbf{x}_A$ is said to Pareto dominate anther solution $\mathbf{x}_B$ ($\mathbf{x}_A\prec \mathbf{x}_B$) *iff* $$\left\{ \begin{array}{lr} \forall i\in 1,2,\dots,M, f_i(\mathbf{x}_A) \leq f_i(\mathbf{x}_B)\\ \exists j\in 1,2,\dots,M, f_j(\mathbf{x}_A) < f_j(\mathbf{x}_B). \end{array} \right.$$ The collection of all the Pareto optimal solutions in the decision space is called the Pareto optimal set (PS), and the projection of the PS in the objective space is called the Pareto optimal front (PF). The goal of multi-objective optimization is to obtain a set of solutions for approximating the PF in terms of both convergence and diversity, where each solution should be close to the PF and the entire set should be evenly spread over the PF. To solve MOPs, a variety of multi-objective evolutionary algorithms (MOEAs) have been proposed, which can be roughly classified into three categories [@RVEA]: the dominance-based algorithms (e.g. the elitist non-dominated sorting genetic algorithm (NSGA-II) [@NSGA-II] and the improved strength Pareto EA (SPEA2) [@SPEA2]); the decomposition-based MOEAs (e.g., the MOEA/D [@MOEAD] and MOEA/D using differential evolution (MOEA/D-DE) [@MOEADDE]); and the performance indicator-based algorithms (e.g., the $\mathcal{S}$-metric selection evolutionary multi-objective optimization algorithm (SMS-EMOA) [@SMSEMOA] and the indicator based EA (IBEA) [@IBEA]). There are also some MOEAs not falling into the three categories, such as the third generation differential algorithm (GDE3) [@GDE3], the memetic Pareto achieved evolution strategy (M-PAES) [@knowles2000m], and the two-archive based MOEA (Two-Arc)  [@praditwong2006new], etc. ![The general framework of MOEAs.[]{data-label="fig:EA"}](EAframework.eps){width="0.75\linewidth"} In spite of the various technical details adopted in different MOEAs, most of them share a common framework as displayed in Fig. \[fig:EA\]. Each generation in the main loop of the MOEAs consists of three operations: offspring reproduction, fitness assignment, and environmental selection [@eiben2015evolutionary]. To be specific, the algorithms start from the population initialization; then the offspring reproduction operation will generate offspring solutions; afterwards, the generated offspring solutions are evaluated using the real objective functions; finally, the environmental selection will select some high-quality candidate solutions to survive as the population of the next generation. In conventional MOEAs, since the reproduction operations are usually based on stochastic mechanisms (e.g. crossover or mutation), the algorithms are unable to explicitly learn from the environments (i.e. the fitness landscapes). For instance, conventional EAs use the mating selection strategy to select some promising parent solutions based on their fitness values, and then randomly crossover two of them to generate offspring solutions. For conventional crossover operators such as SBX [@PM], the offspring solutions will distribute around the vertices of a hyper-rectangle in parallel with the axes of decision variables, and its longest diagonal is the line segment of the two chosen parent solutions. If the PS of an MOP is not parallel with any axis of decision variable, especially when the PS has a 45$^\circ$ angle to all of the axes (e.g. IMF1 to IMF3 problems in [@IM-MOEA]), there is only a little chance that the offspring solutions will fall around the PS, resulting in the inefficiency of conventional crossover in offspring generation. An example of the SBX based offspring generation in a 2-D decision space is given in \[fig:rotate\], where the generated offspring solutions $\mathbf{s}_1, \mathbf{s}_2$ are far from their parents $\mathbf{p}_1,\mathbf{p}_2$ and the PS. ![An example of the genetic operator (SBX [@PM]) based offspring generation in a 2-D decision space, where $\mathbf{p}_1$ and $\mathbf{p}_2$ denote the parent solutions, and $\mathbf{s}_1$ and $\mathbf{s}_2$ denote the offspring solutions.[]{data-label="fig:rotate"}](rotate.eps){width="0.7\linewidth"} To address the above issue, a number of recent works have been dedicated to designing EAs with learning ability, known as the model based evolutionary algorithms (MBEAs) [@MBEA; @zhang2011evolutionary]. The basic idea of MBEAs is to replace the heuristic operations or the objective functions with computationally efficient machine learning models, where the candidate solutions sampled from the population are used as training data. Generally, the models are used for the following three main purposes when adopted in MOEAs. First, the models are used to approximate the real objective functions of the MOP during the fitness assignment process. MBEAs of this type are also known as the surrogate-assisted EAs [@Jin2000On], which use computationally cheap machine learning models to approximate the computationally expensive objective functions [@jin2009systems]. They aim to solve computationally expensive MOPs using a few real objective function evaluations as possible [@jin2011review; @SA2017]. A number of surrogate-assisted MOEAs were proposed in the past decades, e.g., the $S$-metric selection-based EA (SMS-EGO) [@SMS-EGO], the Pareto rank learning based MOEA [@seah2012pareto], and the MOEA/D with Gaussian process (GP) [@GP] (MOEA/D-EGO) [@MOEADEGO]. Second, the models are used to predict the dominance relationship [@ParetoSVM] or the ranking of candidate solutions [@lu2012classification; @bhatt2015novel] during the reproduction or environmental selection process. For example, in the classification based pre-selection MOEA (CPS-MOEA) [@CPSMOEA], a k-nearest neighbor (KNN) [@KNN] model is adopted to classify the candidate solutions into *positive* and *negative* classes. Then the *positive* candidate solutions are selected to survival [@zhang2018preselection]. Similarly, the classification based surrogate-assisted EA (CSEA) used a feedforward neural network [@svozil1997introduction] to predict the dominance classes of the candidate solutions in evolutionary multi-objective optimization [@CSEA]. Third, the models are used to generate promising candidate solutions during the offspring reproduction process. The MBEAs of this type mainly include the multi-objective estimation of distribution algorithms (MEDAs) [@EDA] as well as the inverse modeling based algorithms [@giagkiozis2014pareto]. [The MEDAs estimate the distribution of promising candidate solutions by training and sampling models in the decision space [@karshenas2013multiobjective]. Instead of generating offspring solutions via crossover or mutation from the parent solutions, the MEDAs explore the decision space of potential solutions by building and sampling explicit probabilistic models of the promising candidate solutions [@eda-initial; @sun2018improved].]{} Typical algorithms include the Bayesian multi-objective optimization algorithm (BMOA) [@BMOA], the naive mixture-based multi-objective iterated density estimation EA (MIDEA) [@MIDEA], the multi-objective Bayesian optimization algorithm (mBOA) [@mBOA], and the regularity model based MEDA (RM-MEDA) [@RM-MEDA], etc. For example, in the covariance matrix adaptation based MOEA/D (MOEA/D-CMA) [@MOEADCMA], the covariance matrix adaptation model [@loshchilov2013cma] is adopted for offspring reproduction. As for the inverse modeling based algorithms, they sample points in the objective space and then build inverse models to map them back to the decision space, e.g., the Pareto front estimation method [@giagkiozis2014pareto], the Pareto-adaptive $\epsilon$-dominance-based algorithm ($pa\lambda$-MyDE) [@hernandez2007pareto], the reference indicator-based MOEA (RIB-EMOA) [@martinez2014using], and the MOEA using GP based inverse modeling (IM-MOEA) [@IM-MOEA]. Despite that existing MBEAs have shown promising performance on a number of MOPs, their performance deteriorates rapidly as the number of decision variables increases. There are mainly two difficulties when applying existing MBEAs to multi-objective optimization. First, the requirement of training data for building and updating the machine learning models increases exponentially as the number of decision variables becomes larger, i.e., the MBEAs severely suffer from the curse of dimensionality  [@cd; @wang2015memetic]. Second, since there are multiple objectives involved in MOPs, it is computationally expensive to employ multiple models for sampling different objectives. The generative adversarial networks (GANs) are generative models which have been successfully applied in many areas, e.g., image generation [@GAN], unsupervised representation learning [@radford2015unsupervised], and image super-resolution [@ledig2017photo]. They are capable of learning the regression distribution over the given/target data in an adversarial manner. [Meanwhile, the candidate solutions can be seen as samples by the distribution of the PS in evolutionary multi-objective optimization. Under mild conditions, a PS is an ($M$$-$1)-dimensional manifold, given that $M$ is the number of the objectives [@IM-MOEA]. Hence, there are two main motivations of using GANs for reproduction in evolutionary multi-objective optimization. First, it is intuitive to sample candidate solutions using GANs for the estimation of the distribution of the solution set in multi-objective optimization. Second, it is a natural character of GANs that the samples can be divided into fake and real ones, which is somehow consistent with the nature that the candidate solutions can be divided in multi-objective optimization (i.e. dominated and non-dominated solutions).]{} Furthermore, it is naturally suitable to drive evolutionary multi-objective optimization using GANs due to the following reasons. First, the pairwise generator and discriminator in GANs are capable of distinguishing and sampling promising candidate solutions, which is particularly useful in multi-objective optimization in terms of the Pareto dominance relationship. Second, thanks to the adversarial learning mechanism, the GANs are able to learn high-dimensional distributions efficiently with limited training data. By taking such advantages of GANs, we propose a GAN-based MOEA, termed GMOEA. To the best of our knowledge, it is the first time that the GANs are used for driving evolutionary multi-objective optimization. The main new contributions of this work can be summarized as follows: 1. In contrast to conventional MBEAs which are merely dependent on given data (i.e. the candidate solutions), the GANs are able to reuse the data generated by themselves. To take such an advantage, in GMOEA, we propose a classification strategy to classify the candidate solutions into *real* and *fake* samples which are reused as training data. This is particularly meaningful for data enhancement in high-dimensional decision space. 2. We sample a multivariate normal Gaussian distribution as the input of the GANs in the proposed GMOEA. Specifically, the distribution is learned from the promising candidate solutions which approximate the non-dominated front obtained at each generation. The rest of this paper is organized as follows. In Section \[sec:related\], we briefly review the background of the GANs and other related works. The details of the proposed GMOEA are presented in Section \[sec:algorithm\]. Experimental settings and comparisons of GMOEA with the state-of-the-art MOEAs on the benchmark problems are presented in Section \[sec:result\]. Finally, conclusions are drawn in Section \[sec:conclusion\]. Background {#sec:related} ========== Generative Adversarial Networks ------------------------------- The generative adversarial networks have achieved considerable success as a framework of generative models [@GAN]. In general, the GANs produce a model distribution $P_{\hat{\mathbf{x}}}$ (i.e. the distribution of the fake/generated data) that mimics a target distribution $P_{\mathbf{x}}$ (i.e. the distribution of the real/given data). A pair of GANs consist of a generator and a discriminator, where the generator maps Gaussian noise $\mathbf{z}$ ($\mathbf{z}\in P_{\mathbf{z}}$) to a model distribution $G(\mathbf{z})$ and the discriminator outputs probability $D(\mathbf{x})$ with $\mathbf{x} \in P_{\mathbf{x}}$ $\bigwedge$ $\mathbf{x} \notin P_{\hat{\mathbf{x}}}$. Generally speaking, the discriminator seeks to maximize probability $D(\mathbf{x})$ ($\mathbf{x}\in P_{\mathbf{x}}$) and minimize probability $D(G(\mathbf{z}))$, while the generator aims to generate more realistic samples to maximize probability $D(G(\mathbf{z}))$, trying to cheat the discriminator. To be more specific, those two networks are trained in an adversarial manner using the min-max value function $V$: $$\begin{aligned} \label{eq:GAN} &\min\limits_{G} \max\limits_{D}V(D, G) =\\ &\mathbb{E}_{\mathbf{x}\in{P_{\mathbf{x}}}}[logD(\mathbf{x})] + \mathbb{E}_{\mathbf{z} \in {P_{\mathbf{z}}}}[log(1-D(G(\mathbf{z})))].\end{aligned}$$ Algorithm \[al:miniBP\] presents the detailed procedures of the training process. First, a number of $m$ samples are sampled from a Gaussian distribution and the given data (target distribution), respectively. Second, the discriminator is updated using the gradient descending method according to: $$\label{eq:dis2} \bigtriangledown \theta_d \frac{1}{m} \sum_{i=1}^m[logD(\mathbf{x}_i) + log\left(1-D(G(\mathbf{z}_i))\right)].$$ Sequentially, the generator is updated using the gradient descending method according to: $$\label{eq:gen2} \bigtriangledown \theta_g \frac{1}{m} \sum_{i=1}^m[log\left(1-D(G(\mathbf{z}_i))\right),$$ where $\mathbf{z}_i$ is a vector randomly sampled from a Gaussian distribution. The above procedures are repeated for a number of iterations [@kingma2014adam].   \ $P_{\mathbf{x}}$ (given data), $P_{\mathbf{z}}$ (Gaussian noise), $m$ (batch size). /\*\*\*\*\* Update the discriminator \*\*\*\*/ Randomly sample $m$ samples $\{\mathbf{z}_1,\dots,\mathbf{z}_m\}$ from $P_{\mathbf{z}}$ Randomly sample $m$ samples $\{ \mathbf{x}_1,...,\mathbf{x}_m \}$ from $P_{\mathbf{x}}$ Update the discriminator according to (\[eq:dis2\]) /\*\*\*\*\*\* Update the generator \*\*\*\*\*\*/ Sample $m$ samples $\{\mathbf{z}_1,\dots,\mathbf{z}_m\}$ from $P_{\mathbf{z}}$ Update the generator according to (\[eq:gen2\]) Improved Strength Pareto Based Selection {#sec:spea2} ---------------------------------------- The improved strength Pareto based EA (SPEA2) [@SPEA2] is improved from its original version (SPEA) [@SPEA] by incorporating a tailored fitness assignment strategy, a density estimation technique, and an enhanced truncation method. In the tailored fitness assignment strategy, the dominance relationship between the pairwise candidate solutions are first detected, and then a strength value is assigned to each candidate solution. This value indicates the number of candidate solutions it dominates: $$\label{eq:dominance} Str(\mathbf{x}_i)=|\{j| j\in P\wedge \mathbf{x}_i \prec \mathbf{x}_j \}|,$$ where $P$ is the population and $\mathbf{x}_i,\mathbf{x}_j$ are the candidate solutions in it. Besides, the raw fitness can be obtained as: $$\label{eq:rf} Raw(\mathbf{x}_i)=\sum^N_{j\in P \wedge \mathbf{x}_j \prec \mathbf{x}_i}{Str(\mathbf{x}_j)}.$$ Moreover, the additional density information, termed $Den$, is used to discriminate the candidate solutions having identical raw fitness values. The density of a candidate solution is defined as: $$\label{eq:df} Den(\mathbf{x}_i)=\frac{1}{\sigma^k_i+2},$$ where $k$ is the square root of the population size, and $\sigma^k_i$ denotes the $k$th nearest Euclidean distance from $\mathbf{x}_j$ to the candidate solutions in the population. Finally, the fitness can be calculated as $$\label{eq:fit} Fit(\mathbf{x}_i)= Raw(\mathbf{x}_i)+Den(\mathbf{x}_i).$$ [The environmental selection of SPEA2 aims to select $N$ solutions from population $P$. It first selects all the candidate solutions with $Fit$$<$$1$ into set $A$. If the size of $A$ is smaller than $N$, $N$ solutions with the best $Fit$ are selected from $P$; otherwise, a truncation procedure is invoked to iteratively remove candidate solutions from $A$ until its size equals to $N$, where the candidate solution with the minimum Euclidean distance to the solutions in $A$ is removed each time.]{} Since the density information is well used, the environmental selection in SPEA2 maintains a set of diverse candidate solutions. In this work, we adopt it for solution classification and environmental selection in our proposed GMOEA, where the details will be presented in Section \[sec:algorithm\].B. The Proposed Algorithm {#sec:algorithm} ====================== The main scheme of the proposed GMOEA is presented in Algorithm \[al:framework\]. First, a population $P$ of size $N$ and a pair of GANs are randomly initialized, respectively. Then the candidate solutions in $P$ are classified into two different datasets with equal size (labeled as *fake* and *real*) and used to train the GANs. Next, a set $Q$ of $N$ offspring solutions is generated by the proposed hybrid reproduction strategy. Afterwards, $N$ candidate solutions are selected from the combination of $P$ and $Q$ by environmental selection. Finally, the solution classification, model training, offspring reproduction, and environmental selection are repeated until the termination criterion is satisfied. We will not enter the details of the environmental selection as it is similar to the solution classification, except that the environmental selection takes [$N$ solutions from the combination of $P$ and $Q$ as input (instead of selecting half of the solutions from $P$ only)]{} and only outputs the *real* solutions.   \ $N$ (population size), $m$ (batch size for training the GAN) $P\leftarrow$ Initialize a population of size $N$ $GAN\leftarrow$ Initialize the GANs $\mathbf{X}\leftarrow \text{Solution Classification}$ [/\*Half of the solutions in $P$ are classified as *fake* samples\*/]{} $\mathbf{net} \leftarrow \text{Model Training}$ [/\*Use $\mathbf{X}$ to train the model\*/]{} $Q\leftarrow \text{Offspring Reproduction}$ [/\*Generate $N$ offspring solutions by the proposed reproduction method\*/]{} $P\leftarrow \text{Environmental Selection}$ [/\*Select $N$ solutions from the combination of $P$ and $Q$\*/]{} $P_0$ Solution Classification ----------------------- Solution classification is used to divide the population into two different datasets (*real* and *fake*) for training the GANs. The *real* solutions are those better-converged and evenly distributed candidate solutions; by contrast, the *fake* ones are those of relatively poor qualities. We use the environmental selection strategy as introduced in Section \[sec:spea2\] to select half of the candidate solutions in the current population as *real* samples and the rest as *fake* ones. [ $N$ (number of fake sample), $P$ (population).]{} $Fit\leftarrow$ Calculate the fitness values of candidate solutions in $P$ according to (\[eq:fit\]) $A\leftarrow \arg\limits_{\mathbf{x}_i\in P}{Fit(\mathbf{x}_i)<1}$ $\leftarrow$ Select $N$ candidate solutions with the minimal $Fit$ Delete $\arg\min\limits_{\mathbf{x}_j\in A}{\min{dis(\mathbf{x}_j, A\backslash \mathbf{x}_j)}}$ in $A$ $A\leftarrow$ *real* $P\backslash A\leftarrow$ *fake* The pseudo codes of the solution classification are presented in Algorithm \[al:select\]. Generally, the purpose of solution classification is to select a set of high-quality candidate solutions in terms of convergence and diversity. The first term is intuitive, which aims to enhance the selection pressure for pushing the population towards the PF. The second term aims to satisfy the identity independent distribution assumption for better generalization of the GANs [@DL]. Model Training -------------- The structures of the generator and discriminator adopted in this work are feedforward neural networks [@ANN] with two hidden layers and one hidden layer (the number of neurons in each layer is $D$), respectively. [Here we adopt this simple structure for the following two reasons. First, if a more powerful model is adopted, the amount of training data should be increased, resulting in the rapid increase in the computational cost in terms of both CPU time and the number of function evaluations. For the MOPs we try to solve in this work, the current model is good enough. Second, in the area of evolutionary optimization, similar simple networks have also been validated in other recent works, e.g.  [@CSEA; @zhang2018computational]. We surmise that it is due to fact that the problem scales in evolutionary optimization are much smaller than those in other applications such as image processing, such that a simple network will work properly.]{} The general scheme of the GANs is given in Fig. \[fig:GANs\], where the distributions of the *real* and *fake* datasets are denoted as $P_{\mathbf{r}}$ and $P_{\mathbf{f}}$, respectively. The activation functions of the output layers in these two networks are sigmoid functions to ensure that the output values vary in $[0, 1]$. Here, we propose a novel training method to take advantage of the labeled samples. ![The general scheme of model training in the proposed GMOEA.[]{data-label="fig:GANs"}](GANs.eps){width="\linewidth"} First, the mean vector and covariance matrix of the *real* samples are calculated by: $$\label{eq:mean} \begin{aligned} \boldsymbol{\mu}&= \frac{\sum_{i=1}^{\lfloor N/2\rfloor}{\mathbf{r}_i}}{\lfloor N/2\rfloor},\\ \boldsymbol{\Sigma} &= \frac{\sum_{i=1}^{\lfloor N/2\rfloor}{(\mathbf{r}_i-\boldsymbol{\mu})(\mathbf{r}_i}-\boldsymbol{\mu})^\text{T}}{\lfloor N/2\rfloor-1},\\ \end{aligned}$$ where $\mathbf{r}_i$ is the $i$th member of the *real* dataset and $N$ is the population size. Then the GANs are trained for several iterations. At each iteration, the discriminator is updated using three different types of training data, i.e., the *real* samples, the *fake* samples, and the samples generated by the generator. The loss function for training the discriminator is given as follows: $$\label{eq:d1} \begin{split} &\max\limits_{D} V(D) = \mathbb{E}_{\mathbf{r}\in{P_{\mathbf{r}}}}[log(D(\mathbf{r}))] + \\ &\mathbb{E}_{\mathbf{f} \in {P_{\mathbf{f}}}}[log(1-D(\mathbf{f}))]+ \mathbb{E}_{\mathbf{z} \in {P_{\mathbf{z}}}}[log(1-D(G(\mathbf{z})))], \end{split}$$ where $D(\mathbf{r})$, $D(\mathbf{r})$ and $D(G(\mathbf{z}))$ denote the outputs of the discriminator with the *real* sample, the *fake* sample and the sample generated by the generator being the inputs, respectively. The input of the generator is vector $\mathbf{z}$ sampled from a multivariate normal distribution. Finally, the generator is updated according to (\[eq:gen2\]) using the samples generated by itself. [Note that we have greedily used the imbalanced training set, aiming to enhance the convergence of GMOEA by pushing the target distribution away from the $\textit{fake}$ distribution. In other words, we prefer a model with higher accuracy in distinguishing the fake samples, such that there is a clear margin between the target distribution and the fake one.]{} [ $\mathbf{X}$ (given data), $m$ (number of samplings).]{} $\boldsymbol{\mu}\leftarrow mean{(P_{\mathbf{r}})}$ /\*Mean vector of the data\*/ $\boldsymbol {\Sigma}\leftarrow cov(P_{\mathbf{r}})$ /\*Covariance matrix of the data\*/ Randomly sample $m$ samples $\{ \mathbf{x}_1,...,\mathbf{x}_m \}$ from $\mathbf{X}$ $\{\mathbf{z}_1,\dots,\mathbf{z}_m\}\leftarrow multivariate\_normal(m, \boldsymbol{\mu}, \boldsymbol {\Sigma})$ Update the discriminator according to (\[eq:d1\]) $\{\mathbf{z}_1,\dots,\mathbf{z}_m\}\leftarrow multivariate\_normal(m, \boldsymbol{\mu}, \boldsymbol {\Sigma})$ Update the generator according to (\[eq:gen2\]) The detailed procedure of the model training in GMOEA is given in Algorithm \[al:GAN\]. Here, we use the multivariate normal Gaussian distribution [@balakrishnan2014continuous], which is specified by its mean vector and covariance matrix, to generate training data. The mean vector represents the location where samples are most likely to be generated, and the covariance indicates the level to which two variables are correlated. This modification is inspired by the generative method in variational auto-encoder (VAE) [@VAE], which aims to generate data that approximates the given distribution. More importantly, this modification will potentially reduce the amount of data required for training the generator, since the distributions of $P_{\mathbf{z}}$ and $G(\mathbf{z})$ are similar. Offspring Reproduction ---------------------- In this work, we adopt a hybrid reproduction strategy for offspring generation in GMOEA, which aims at balancing the exploitation and exploration of the proposed algorithm. The general idea of the proposed reproduction strategy is simple and efficient. At each generation, $N$ offspring solutions will be generated either by the GAN model or the genetic operators (i.e. crossover and mutation) with equal probability. [Since there is a risk of mode collapse in training a GAN model [@arjovsky2017wasserstein], the trained model may generate some poor solutions. To remedy this issue, we propose to mix the candidate solutions generated by both the GAN model and genetic operators as training data.]{} To generate a candidate solution using the GANs, we first calculate the mean vector $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$ of the *real* samples according to (\[eq:mean\]). Then, a $D$-dimensional vector $\mathbf {x}$ is sampled with each element being independently sampled from a continuous uniform distribution $U(0,1)$. Afterwards, a $D$-dimensional vector $\mathbf{y}$ satisfying the multivariate normal distribution is generated according to the following probability density function: $$\label{eq:PDF} \mathbf{y} =\frac{\exp{\left(-{\frac {1}{2}}(\mathbf {x} - \boldsymbol{\mu})^{\mathrm {T} }{\boldsymbol{\Sigma}}^{-1}(\mathbf {x} - \boldsymbol{\mu}) \right)}}{\sqrt{(2\pi )^{D}|\boldsymbol{\Sigma}|}},$$ where $D$ denotes the dimensionality of the decision space. Finally, the output of the generator, $G(\mathbf{y})$, is restricted according to the lower and upper boundaries (i.e., $\mathbf{l}$ and $\mathbf{u}$) of the decision space as follows: $$\mathbf{x'}=G(\mathbf{y})(\mathbf{u}-\mathbf{l})+\mathbf{l},$$ where $\mathbf{x'}$ is the candidate solution generated by the GANs. Experimental Study {#sec:result} ================== To empirically examine the performance of the proposed GMOEA, we mainly conduct two different experiments to examine the properties of our proposed GMOEA. Among these experiments, six representative MOEAs are compared, namely, NSGA-II [@NSGA-II], MOEA/D-DE [@MOEADDE], MOEA/D-CMA [@MOEADCMA], IM-MOEA[@IM-MOEA], GDE3 [@GDE3], and SPEA2 [@SPEA2]. NSGA-II and SPEA2 are selected as they both adopt crossover and mutation operators for offspring generation. MOEA/D-DE and GDE3 are selected as they both adopt the differential evolution operator. MOEA/D-CMA is chosen as it is a representative MBEA, which uses the covariance matrix adaptation evolution strategy for multi-objective optimization. Besides, IM-MOEA is selected as it is an MBEA using the inverse models to generate offspring solutions for multi-objective optimization. The two experiments are summarized as follows: - The effectiveness of our proposed training method is examined according to the qualities of the offspring solutions generated by the GANs which are trained by different methods. - The general performance of our proposed GMOEA is compared with the six algorithms on ten IMF problems with up to 200 decision variables. In the remainder of this section, we first present a brief introduction to the experimental settings of all the compared algorithms. Then the test problems and performance indicators are described. Afterwards, each algorithm is run for 20 times on each test problem independently. Then the Wilcoxon rank sum test  [@haynes2013wilcoxon] is used to compare the results obtained by the proposed GMOEA and the compared algorithms at a significance level of 0.05. Symbols ‘$+$’, ‘$-$’, and ‘$\approx$’ indicate the compared algorithm is significantly better than, significantly worse than, and statistically tied by GMOEA, respectively. Experimental settings --------------------- For fair comparisons, we adopt the recommended parameter settings for the compared algorithms that have achieved the best performance as reported in the literature. The six compared algorithms are implemented in PlatEMO using Matlab [@PlatEMO], and our proposed GMOEA is implemented in Pytorch using Python 3.6. All the algorithms are run on a PC with Intel Core i9 3.3 GHz processor, 32 GB of RAM, and 1070Ti GPU. *1) Reproduction Operators.* In this work, the simulated binary crossover (SBX) [@review-book] and the polynomial mutation (PM) [@PM] are adopted for offspring generation in NSGA-II and SPEA2. The distribution index of crossover is set to $n_c$$=$20 and the distribution index of mutation is set to $n_m$$=$20, as recommended in [@review-book]. The crossover probability $p_c$ is set to 1.0 and the mutation probability $p_m$ is set to $1/D$, where $D$ is the number of decision variables. In MOEA/D-DE, MOEA/D-CMA, and GDE3, the differential evolution (DE) operator [@DE] and PM are used for offspring generation. Meanwhile, the control parameters are set to $CR$$=$1, $F$$=$0.5, $p_m$$=$$1/D$, and $\eta$$=$20 as recommended in [@MOEADDE]. *2) Population Size.* The population size is set to 100 for test instances with two objectives and 105 for test instances with three objectives. *(3) Specific Parameter Settings in Each Algorithm.* In MOEA/D-DE, the neighborhood size is set to 20, the probability of choosing parents locally is set to 0.9, and the maximum number of candidate solutions replaced by each offspring solution is set to 2. In MOEA/D-CMA, the number of groups is set to 5. As for IM-MOEA, the number of reference vectors is set to 10 and the size of random groups is set to 3. [In our proposed GMOEA, the training parameter settings of the GANs are fixed, where the batch size is set to 32, the learning rates for our discriminator and generator are 0.0001 and 0.0004 respectively, the total number of iterations is set to 625 (i.e. $200\times 100/32$), and the Adam optimizer [@Adam] with $\beta_{1}$$=$0.5, $\beta_{2}$$=$0.999 is used to train our GAN. Note that the specified model in GMOEA is suitable for the benchmark investigated in this work, and its structure can be revised accordingly to fit different problems.]{} *(4) Termination Condition.* The total number of FEs is adopted as the termination condition for all the test instances. The number of FEs is set to 5000 for test problems with 30 decision variables, 10000 for problems with 50 decision variables, 15000 for problems with 100 decision variables, and 30000 for problems with 200 decision variables. Test Problems and Performance Indicators ---------------------------------------- In this work, we adapt ten problems selected from [@IM-MOEA], termed IMF1 to IMF10. Among these test problems, the number of objectives is three in IMF4, IMF8 and two in the rest ones. We adopt two different performance indicators to assess the qualities of the obtained results. The first one is IGD [@IGD], which can assess both the convergence and distribution of the obtained solution set. Suppose that $P^*$ is a set of relatively evenly distributed reference points [@RPgeneration] in the PF and $\Omega$ is the set of the obtained non-dominated solutions. The IGD can be mathematically defined as follows. $$\label{eq:IGD} \text{IGD}(P^*,\Omega)=\frac{\sum_{\mathbf{x}\in P^*}dis(\mathbf{x},\Omega)}{|P^*|},$$ where $dis(\mathbf{x},\Omega)$ is the minimum Euclidean distance between $\mathbf{x}$ and points in $\Omega$, and $|P^*|$ denotes the number of elements in $P^*$. The set of reference points required for calculating IGD values are relatively evenly selected from the PF of each test problem, and a set size closest to 10000 is used in this paper. The second performance indicator is the hypervolume (HV) indicator [@HV]. Generally, hypervolume is favored because it captures in a single scalar both the closeness of the solutions to the optimal set and the spread of the solutions across objective space. Given a solution set $\Omega$, the HV value of $\Omega$ is defined as the area covered by $\Omega$ with respect to a set of predefined reference points $P^*$ in the objective space: $$\label{eq:HV} \text{HV}(\Omega,P^*)=\lambda (H(\Omega,P^*)),$$ where $$H(\Omega,P^*)=\{z\in Z|\exists x\in P,\exists r\in P^*:f(x)\leq z \leq r\} \nonumber,$$ and $\lambda$ is the Lebesgue measure with $$\lambda (H(\Omega,P^*))=\int_{\mathbb{P^*}^n}1_{H(\Omega,P^*)}(z)dz\nonumber,$$ where $1_{H(\Omega,P^*)}$ is the characteristic function of $H(\Omega,P^*)$. Note that, a smaller value of IGD will indicate better performance of the algorithm; in contrast, a greater value of HV will indicate better performance of the algorithm. Effectiveness of the Model Training Method ------------------------------------------ To verify the effectiveness of our proposed model training method in GMOEA, we compare the offspring solutions generated by our modified GANs [(where the data augmentation via multivariate Gaussian model is adopted)]{} and the original GANs during the optimization of IMF4 and IMF7. We select IMF4 since its PS is complicated, and this problem is difficult for existing MOEAs to maintain diversity. IMF7 with 200 decision variables is tested to examine the effectiveness of our proposed training method in solving MOPs with high-dimensional decision variables. The numbers of FEs for these two problems are set to 5000 and 30000, respectively. Besides, each test instance is tested for 10 independent runs to obtain the statistic results. In each independent run, we sample the offspring solutions every 10 iterations for IMF4 and every 50 iterations for IMF7. {width="\linewidth"} Fig. \[fig:compare2\] presents the offspring solutions obtained on tri-objective IMF4. It can be observed that the original GANs tend to generate offspring solutions in a smaller region of the objective space (e.g., near the top center in Fig. \[fig:compare2\]). By contrast, our modified GANs have generated a set of widely spread offspring solutions with better convergence in most iterations. Fig. \[fig:compare3\] presents the offspring solutions obtained on IMF7 with 200 decision variables. It can be observed that our modified GANs have generated a set of better-converged and spreading offspring solutions; by contrast, the original GANs have generated offspring solutions mostly in the left corner. {width="0.95\linewidth"} It can be concluded from the three comparisons that our proposed training method is effective in diversity maintenance and convergence enhancement, even on MOPs with complicated PSs and up to 200 decision variables. ![The trajectories of generator and discriminator’s training losses of the original GAN (with multivariate Gaussian model disabled) and our modified GAN during the evolution, respectively.[]{data-label="fig:traject"}](trajectory_Ori.eps "fig:"){width="0.85\linewidth"} ![The trajectories of generator and discriminator’s training losses of the original GAN (with multivariate Gaussian model disabled) and our modified GAN during the evolution, respectively.[]{data-label="fig:traject"}](trajectory_Ours.eps "fig:"){width="0.85\linewidth"} [ Furthermore, we display the trajectories of generator and discriminator’s training losses during the evolution in Fig. \[fig:traject\], where GMOEA is adopted to optimize IMF1 with 30 decision variables. In this figure, the horizontal denotes the epoch number from the first generation to the last generation of the evolution, where each epoch is averaged over 20 independent runs. It can be observed that the training loss of each discriminator rises while the training loss of each generator drops; nevertheless, the generator in our modified GAN trends to have a lower and more stable training loss than that of the original GAN. It can be attributed to the fact that the generator in our modified GAN generates more realistic samples that the discriminator cannot distinguish, and thus the generator is powerful in generating promising samples. This is consistent with the design principle of offspring generators (i.e. generating promising candidate solutions) in EAs. ]{} General Performance ------------------- The statistical results of the IGD and HV values achieved by the seven compared MOEAs on IMF1 to IMF10 are summarized in Table \[tab:IMMOEA-IGD\] and Table \[tab:IMMOEA-HV\], respectively. Our proposed GMOEA has performed the best on these ten problems, followed by IM-MOEA, NSGA-II, and MOEA/D-CMA. It can be concluded from these two tables that GMOEA shows an overall better performance compared with the model-free MOEAs, i.e., NSGA-II, MOEA/D-DE, GDE3, and SPEA2, on IMF problems. Meanwhile, GMOEA has shown a competitive performance in compared with MOEA/D-CMA and IM-MOEA on these IMF problems. ’$+$’, ’$-$’ and ’$\approx$’ indicate that the result is significantly better, significantly worse and statistically similar to that obtained by GMOEA, respectively. \[tab:IMMOEA-IGD\] ’$+$’, ’$-$’ and ’$\approx$’ indicate that the result is significantly better, significantly worse and statistically similar to that obtained by GMOEA, respectively. \[tab:IMMOEA-HV\] The final non-dominated solutions achieved by the compared algorithms on bi-objective IMF3 and tri-objective IMF8 with 200 decision variables in the runs associated with the median IGD value are plotted in Fig. \[fig:IMF3\] and Fig. \[fig:IMF8\], respectively. It can be observed that GMOEA has achieved the best results on these problems, where the obtained non-dominated solutions are best converged. {width="\linewidth"} {width="\linewidth"} The convergence profiles of the seven compared algorithms on nine IMF problems with 200 decision variables are given in Fig \[fig:IMF\_Convergence\]. It can be observed that GMOEA converges faster than the other six compared algorithms on most problems. The results have demonstrated the superiority of our proposed GMOEA over the six compared algorithms on MOPs with up to 200 decision variables in terms of convergence speed. ![The statistics of the runtime results achieved by the original IBEA and GMOEA.[]{data-label="fig:runtime"}](runtime.eps "fig:"){width="0.8\linewidth"}\ Conclusion {#sec:conclusion} ========== In this work, we have proposed an MOEA driven by the GANs, termed GMOEA, for solving MOPs with up to 200 decision variables. Due to the learning and generative abilities of the GANs, GMOEA is effective in solving these problems. The GANs in GMOEA are adopted for generating promising offspring solutions under the framework of MBEAs. In GMOEA, we first classify candidate solutions in the current population into two different datasets, where some high-quality candidate solutions are labeled as *real* samples and the rest ones are labeled as *fake* samples. Since the GANs mimic the distribution of target data, the distribution of *real* samples should consider two issues. The first issue is the diversity of training data, which ensures that the data could represent the general distribution of the expected solutions. The second issue is the convergence of training data, which ensures that the generated samples could satisfy the target of minimizing all the objectives. A novel training method is proposed in GMOEA to take full advantage of the two datasets. During the training, both the *real* and *fake* datasets, as well as the data generated by the generator, are used to train the discriminator. It is highlighted that the proposed training method is demonstrated to be powerful and effective. Only a relatively small amount of training data is used for training the GANs (a total number of 100 samples for an MOP with 2 objectives and 105 samples for MOPs with 3 objectives). Besides, we also introduce an offspring reproduction strategy to further improve the performance of our proposed GMOEA. By hybridizing the classic stochastic reproduction and generating sampling based reproduction, the exploitation and exploration can be balanced. To assess the performance of our proposed GMOEA, a number of empirical comparisons have been conducted on a set of MOPs with up to 200 decision variables. The general performance of our proposed GMOEA is compared with six representative MOEAs, namely, NSGA-II, MOEA/D-DE, MOEA/D-CMA, IM-MOEA, GDE3, and SPEA2. The statistical results demonstrate the superiority of GMOEA in solving MOPs with relatively high-dimensional decision variables. This work demonstrates that the MOEA driven by the GAN is promising in solving MOPs. Therefore, it deserves further efforts to introduce more efficient generative models. Besides, the extension of our proposed GMOEA to MOPs with more than three objectives (many-objective optimization problems) is highly desirable. Moreover, its applications to real-world optimization problems are also meaningful. [^1]: C. He, S. Huang, and R. Cheng are with the Shenzhen Key Laboratory of Computational Intelligence, University Key Laboratory of Evolving Intelligent Systems of Guangdong Province, Department of Computer Science and Engineering, Southern University of Science and Technology, Shenzhen 518055, China. E-mail: chenghehust@gmail.com, shihuahuang95@gmail.com, ranchengcn@gmail.com. (*Corresponding author: Ran Cheng*) [^2]: K. C. Tan is with the Department of Computer Science, City University of Hong Kong, Hong Kong. E-mail: kaytan@cityu.edu.hk. [^3]: Y. Jin is with the Department of Computer Science, University of Surrey, Guildford, Surrey, GU2 7XH, United Kingdom. Email: yaochu.jin@surrey.ac.uk. [^4]: This work was supported in part by the National Key R$\&$D Program of China grant (No. 2017YFC0804002), in part by the National Natural Science Foundation of China (No. 61903178 and 61906081) , in part by the Program for Guangdong Introducing Innovative and Entrepreneurial Teams grant (No. 2017ZT07X386), in part by the Shenzhen Peacock Plan grant (No. KQTD2016112514355531), in part by the Science and Technology Innovation Committee Foundation of Shenzhen grant (No. ZDSYS201703031748284), and in part by the Program for University Key Laboratory of Guangdong Province grant (No. 2017KSYS008).

--- abstract: 'We present an accurate and efficient method to calculate the gravitational potential of an isolated system in three-dimensional Cartesian and cylindrical coordinates subject to vacuum (open) boundary conditions. Our method consists of two parts: an interior solver and a boundary solver. The interior solver adopts an eigenfunction expansion method together with a tridiagonal matrix solver to solve the Poisson equation subject to the zero boundary condition. The boundary solver employs James’s method to calculate the boundary potential due to the screening charges required to keep the zero boundary condition for the interior solver. A full computation of gravitational potential requires running the interior solver twice and the boundary solver once. We develop a method to compute the discrete Green’s function in cylindrical coordinates, which is an integral part of the James algorithm to maintain second-order accuracy. We implement our method in the [Athena++]{} magnetohydrodynamics code, and perform various tests to check that our solver is second-order accurate and exhibits good parallel performance.' author: - Sanghyuk Moon - 'Woong-Tae Kim' - 'Eve C. Ostriker' bibliography: - 'mybib.bib' title: 'A FAST POISSON SOLVER OF SECOND-ORDER ACCURACY FOR ISOLATED SYSTEMS IN THREE-DIMENSIONAL CARTESIAN AND CYLINDRICAL COORDINATES' --- INTRODUCTION ============ There are a number of astronomical systems, such as galactic and protostellar disks, where self-gravity and rotation play an essential role in dynamical evolution. For instance, starburst activity occurring in massive circumnuclear disks can not only inflate the natal disks to form thick tori surrounding active galactic nuclei (AGN) [@wada02; @wada09] but also drive large-scale galactic winds and outflows [@strick00; @strick04a; @strick04b; @sch18a; @sch18b]. Accretion disks around AGN may be gravitationally unstable at some radii to form stars [@goodman03; @goodman04; @levin07; @nayakshin07; @jiang11]. Self-gravity is also important in formation of large-scale spiral structure [@goldreich65; @baba13; @onghia13] and giant molecular clouds [@kim03; @dobbs08; @tasker09] on larger scales in galactic disks. In addition, recent observations of young stellar objects indicate that at least in the early stage of evolution, protostellar disks are massive enough to be self-gravitating [@kratter16; @tobin16]. Gravitational instability of such disks may form trailing spirals that can redistribute the mass and angular momentum and induce heat via shocks [@mejia05; @evans15] and may be responsible for the formation of giant planets [@boss07; @zhu12]. To follow evolution of self-gravitating disks, one needs to solve the Poisson equation $$\label{eq:Poisson} \nabla^2\Phi = 4\pi G \rho,$$ in cylindrical coordinates $(R,\phi,z)$ subject to a proper boundary condition. In Equation , $\Phi$, $\rho$, and $G$ refer to the gravitational potential, mass density, and gravitational constant, respectively. For an isolated system, $\Phi$ has to satisfy vacuum (or “open”) boundary conditions (i.e., $\Phi$ vanishes at infinite distances), for which the formal solution of Equation is given by $$\label{eq:intPoisson} \Phi({\bf x}) = \iiint {\cal G}_\infty({\bf x,x'}) \rho({\bf x'}) \,d^3x',$$ where ${\cal G}_\infty({\bf x,x'}) \equiv - G/ |{\bf x-x'}|$ is the gravitational potential per unit mass due to a point source situated at ${\bf x'}$. Hereafter, we call ${\cal G}_\infty({\bf x,x'})$ the continuous Green’s function (CGF) to distinguish it from the discrete Green’s function (DGF) based on the *discrete* Laplace operator (e.g., @burk97) discussed in Section \[s:dgf\]. In simulating dynamics of geometrically thin disks, it has been customary to assume that the disk density along the vertical direction follows a simple function such as Dirac’s delta function for a razor-thin disk and a Gaussian function for a slightly extended disk (e.g., @kal71 [@miller76; @li09; @wang15]). In this case, the integral along the $z$-direction in Equation can be performed analytically, and finding $\Phi(R, \phi)$ at $z=0$ reduces to numerical evaluation of the remaining two-dimensional (2D) integral in the $R$–$\phi$ plane. For example, @miller76 solved the gravitational potential of an infinitesimally-thin disk by using a fast Fourier transform (FFT) technique along the azimuthal direction, while directly summing the individual contributions from concentric rings. He introduced a constant softening factor in order to avoid singularity at $\bf x=x'$ of the CGF. @li09 applied this method to develop an efficient gravity solver for disks with finite thickness on a 2D *uniform* polar grid. They improved the parallel efficiency of their method by cutting off the high azimuthal Fourier modes based on the energy criterion. When the grid spacing is *logarithmic* in the radial direction, a suitable change of variables recasts the integral in Equation to a 2D convolution [@kal71; @bt], for which the standard FFT convolution method works efficiently [@hoc88]. For example, @bm08 applied this technique to a razor-thin disk by taking a softening factor proportional to $R$ to avoid divergence of the CGF. @so08 extended this method to a slightly vertically-extended disk, in which finite disk thickness naturally provides the required softening. Noting that softening reduces the accuracy of a gravity solver, @wang15 avoided singularity by using the force kernels integrated over cells, and achieved a second-order accuracy for self-gravity of a razor-thin disk. Although the methods described above are useful and efficient, they are all limited to 2D polar geometry in the $R$–$\phi$ plane. To our knowledge, there is no efficient method available for fully three-dimensional (3D) cylindrical systems with the vacuum (open) boundary conditions. This is presumably because the Green’s function integral takes a convolution form only along the azimuthal and vertical directions: there is no variable transformation that can cast the integral to a full 3D convolution. One may still attempt to perform the radial integral in Equation by direct summation, while applying the FFT convolution along the azimuthal and vertical directions. But, the associated computational cost is of order ${\cal O}(N^4+N^3\log N)$, with $N$ being the typical number of cells in one spatial dimension [@pfen93; @sell97], making the method computationally prohibitive. In many cases, it is computationally more efficient to solve Equation directly, rather than evaluating the integral in Equation . For example, @gupta97 discretized Equation using a fourth-order scheme in 2D Cartesian coordinates and employed a V-cycle multigrid method to solve the resulting linear system. @lai07 adopted another fourth-order formula to discretize Equation in cylindrical coordinates and solved the resulting linear system using FFT combined with a varient of the Bi-Conjugate Gradient iterative method. Perhaps, the most efficient and robust method to solve a discretized Poisson equation may be a full multigrid algorithm [e.g., @mat03], which can in principle be implemented in either Cartesian or cylindrical coordinates. However, all the methods mentioned above in turn require provision of appropriate potentials at the domain boundaries in advance. Since Equation naturally satisfies vacuum boundary conditions, it is reasonable to use it to find the desired boundary potentials for Equation . Still, the computational cost of ${\cal O}(N^4+N^3\log N)$ would be inevitable if radial summation is employed for the boundary potentials. One way to reduce the computational cost is to expand the Green’s function in eigenfunction series and truncate it at some point. For instance, the so-called “multipole expansion method” [@bb75; @zeus; @boley08; @katz16] in spherical polar coordinates costs ${\cal O}(l_{\rm max}m_{\rm max}N^3)$ operations for the boundary potential calculations, where $l_{\rm max}$ and $m_{\rm max}$ refer to the maximum meridional and azimuthal mode numbers, respectively. Although this method appears feasible for small $l_{\rm max}$ and $m_{\rm max}$, the computational cost would increase to ${\cal O}(l_{\rm max}m_{\rm max}N^4)$ for a flattened mass distribution with surfaces lying close to domain boundaries. An additional $N$ factor in the computational cost arises from the fact that the interior and exterior multipole moments for such a flattened mass distribution are different at most boundary points (see, e.g., @cohl99). @cohl99 derived an alternative expansion of the Green’s function in cylindrical coordinates, which they termed the compact cylindrical Green’s function (CCGF). Their CCGF method can perfectly resolve a highly flattened mass distribution, effectively using $l_{\rm max}=\infty$. Coupled with FFT, the CCGF method requires ${\cal O}(m_{\rm max}N^3 + N^3\log N)$ operations for the boundary potential calculations, and has been applied to 3D simulations of self-gravitating disks [e.g., @from05; @mel08; @mar12; @motl17]. When the mass distribution is highly non-axisymmetric and/or dominated by small-scale modes, however, the simulation outcomes depend rather sensitively on the choice of $m_{\rm max}$ [@from04b; @from05]. In such situations, an accurate force evaluation requires $m_{\rm max}$ comparable to $N$ [@from05], and the associated computational cost of ${\cal O}(m_{\rm max}N^3)$ as well as the memory requirement to store four-dimensional arrays would be overwhelming for high-resolution simulations. A very powerful method to deal with vacuum boundary conditions is the four-step algorithm developed by @james77 [see also @mag07 for more compact description]. In the first step, a preliminary solution $\Psi$ is computed based on the interior density distribution and subject to the $\Psi=0$ boundary condition. In the second step, $\Psi$ is used to compute a screening surface charge, $\sigma = \Delta^2 \Psi/(4 \pi G)$, and in the third step a boundary potential $\Theta$ generated by the surface charge and consistent with vacuum boundary conditions is computed via convolution with the DGF. In the fourth step, the final interior solution $\Phi$ is computed, making use of $\Theta$ to enforce a new (nonzero) Dirichlet boundary condition. The key physical principle underlying James’s method is best understood via an electrostatic analogy. Consider the potential $\Phi$ produced by an isolated, charged box with a metal surface. If the box is then grounded, charges would flow in and be distributed over the surface to enforce zero potential. The potential $\Psi$ of the grounded box is then the sum $\Psi = \Phi + \Theta$ of the potential produced by the original internal charge distribution plus the potential generated by the surface charges. On the surface of the box, the desired potential produced by the original interior charge distribution is then given by $\Phi^{\rm B}=-\Theta$, since by definition $\Psi=0$ on the surface. While direct calculation of $\Phi$ in the whole interior via a volume integration with a Green’s function that enforces vacuum BCs would be computationally very expensive (and would also require storing large arrays), calculation of $\Theta$ on the surface requires only integration over the surface. Once the surface potential $\Phi^{\rm B}$ is known, it can be used to compute the interior $\Phi$ via an efficient numerical method. The DGF and its convolution presented in @james77 are valid for Cartesian coordinates, but it is straightforward to extend James’s method to cylindrical coordinates. @snytnikov11 was the first to adapt James’s method for 3D cylindrical problems, but he used the CGF in place of DGF. Since the difference between CGF and DGF is quite large at small $\bf |x - x'|$ (see Appendix \[s:calc\_dgf\]), this forced him to make a computational domain larger than the original volume in order to improve accuracy. We note, however, that the domain cannot be extended arbitrarily across the inner radial boundary $R_{\rm min}$ in cylindrical coordinates, especially when $R_{\rm min}$ is small. The James algorithm with CGF loses accuracy when the ratio of the outer to inner radial boundary is large, which is frequently encountered in various astronomical applications. In such situations, it is desirable to use the DGF rather than the CGF for accurate potential computations. In this paper, we develop an efficient, accurate, and scalable algorithm to calculate the gravitational potential of an isolated system in cylindrical coordinates. Our algorithm is fully 3D, and considers both uniform and logarithmic cylindrical grids. For completeness, and to connect with James’s original method, we also present the method for Cartesian coordinates. Our method utilizes the James algorithm with the DGF for the boundary potential from screening charges. For the interior solver, for cylindrical grids we employ a hybrid method that incorporates the eigenfunction expansion in azimuthal and vertical directions and a tridiagonal solver in the radial direction. For Cartesian grids our interior solver employs standard Fourier methods. We implement our algorithm in the [Athena++]{} code framework (e.g., @white16), and parallelize it on a distributed memory platform using the message passing interface (MPI). Using various test problems, we confirm that our method is efficient and second-order accurate. The remainder of the paper is organized as follows. In Section \[s:equation\], we introduce our 3D computational domain in Cartesian and cylindrical coordinates and discretize the Poisson equation. In Section \[s:interior\_solver\], we describe the hybrid method we adopt to solve the discrete Poisson equation in the interior of the computational domain for a given Dirichlet boundary condition. In Section \[s:bc\], we introduce the James algorithm in Cartesian coordinates and its extension to cylindrical coordinates. In Section \[s:tests\], we present the results of our Poisson solver on various test problems to demonstrate its accuracy and efficiency. In Section \[s:discussion\], we summarize and discuss the present work. DISCRETE POISSON EQUATION {#s:equation} ========================= A standard way to solve Equation is a finite-difference method in which the differential operator $\nabla^2$ is replaced by the *difference* operator $\Delta^2$, yielding the *discrete* Poisson equation $\Delta^2\Phi = 4\pi G \rho$. The definition of $\Delta^2$ depends on the coordinates and the desired level of approximation. This section defines our computational domains and the finite-difference representations of $\Delta^2$, which are second-order accurate, in uniform Cartesian, uniform cylindrical, and logarithmic cylindrical coordinate systems. Uniform Cartesian Grid {#s:uniform_cartesian_grid} ---------------------- In Cartesian coordinates, we discretize the computational domain $[x_{\rm min},x_{\rm max}]\times[y_{\rm min},y_{\rm max}]\times[z_{\rm min},z_{\rm max}]$ with size $L_x\times L_y\times L_z$ uniformly into $N_x\times N_y\times N_z$ cells. We define the face-centered coordinates as $x_{i+1/2} = x_{\rm min} + i \delta x$ where $\delta x \equiv L_x/N_x$ and similarly for $y_{j+1/2}$ and $z_{k+1/2}$. We also define the cell-centered coordinates as $x_i = (x_{i-1/2} + x_{i+1/2})/2$ with index $i$ running from $1$ to $N_x$, and similarly for $y_j$ with $j=1,2,\cdots, N_y$ and $z_k$ with $k=1,2,\cdots,N_z$. We denote the cells inside the nominal index range given above as “active cells", because these are the places where other fluid variables are updated by a hydrodynamics solver. To the active cells, we add one extra layer of “ghost cells" to the boundaries of the computational domain, with their cell-centered coordinates are denoted, for example, by $i=0$ and $i=N_x+1$ in the $x$-direction. The boundary conditions for the Poisson equation and other equations of hydrodynamics are provided using these ghost cells. We similarly define the ghost cells in the other coordinate systems described below. The second-order accurate, finite-difference approximation to Equation can be written as $$\label{eq:fd_uniform_cartesian} \left(\Delta_x^2 + \Delta_y^2 + \Delta_z^2\right)\Phi_{i,j,k} = 4\pi G \rho_{i,j,k},$$ where $\Phi_{i,j,k}$ and $\rho_{i,j,k}$ are the cell-centered potential-density pair and the difference operators $\Delta_x^2$, $\Delta_y^2$, and $\Delta_z^2$ are defined by $$\begin{aligned} \Delta_x^2\Phi_{i,j,k} &= \frac{\Phi_{i-1,j,k}-2\Phi_{i,j,k}+\Phi_{i+1,j,k}}{(\delta x)^2}, \\ \Delta_y^2\Phi_{i,j,k} &= \frac{\Phi_{i,j-1,k}-2\Phi_{i,j,k}+\Phi_{i,j+1,k}}{(\delta y)^2}, \\ \Delta_z^2\Phi_{i,j,k} &= \frac{\Phi_{i,j,k-1}-2\Phi_{i,j,k}+\Phi_{i,j,k+1}}{(\delta z)^2}. \label{eq:delz2}\end{aligned}$$ Uniform Cylindrical Grid ------------------------ In uniform cylindrical coordinates, we discretize the computational domain $[R_{\rm min},R_{\rm max}]\times[\phi_{\rm min},\phi_{\rm max}]\times[z_{\rm min},z_{\rm max}]$ with size $L_R\times L_\phi\times L_z$ uniformly into $N_R\times N_\phi\times N_z$ cells. We require that $L_\phi=\phi_{\rm max}-\phi_{\rm min}$ should be an integer fraction of 2$\pi$ to impose periodic boundary condition along the azimuthal direction. We define the face-centered radial and azimuthal coordinates as $R_{i+1/2} = R_{\rm min} + i \delta R$ and $\phi_{j+1/2} = \phi_{\rm min} + j \delta \phi$, where $\delta R \equiv L_R/N_R$ and $\delta \phi \equiv L_\phi/N_\phi$. Unlike in Cartesian coordinates, the definition of the radial cell-center is ambiguous in cylindrical coordinates because the geometric center does not coincide with the volumetric center. When the radial grid is uniform, finite difference of quantities defined at the geometric centers can retain second-order accuracy. We thus define the cell-centered coordinates as $R_{i} = (R_{i-1/2} + R_{i+1/2})/2$ with $i=1,2,\cdots,N_R$ and $\phi_{j} = (\phi_{j-1/2} + \phi_{j+1/2})/2$ with $j=1,2,\cdots,N_\phi$. Discretization in the vertical direction is the same as in the uniform Cartesian coordinates. The second-order finite-difference approximation to Equation can be written as $$\label{eq:fd_uniform_cylindrical} \left( \Delta_R^2 + \Delta_\phi^2 + \Delta_z^2 \right)\Phi_{i,j,k} = 4\pi G \rho_{i,j,k},$$ where the difference operators $\Delta_R^2$ and $\Delta_\phi^2$ are defined by $$\begin{aligned} \Delta_R^2\Phi_{i,j,k} =& \frac{\Phi_{i-1,j,k}-2\Phi_{i,j,k}+\Phi_{i+1,j,k}}{(\delta R)^2} + \frac{\Phi_{i+1,j,k}-\Phi_{i-1,j,k}}{2R_i\delta R},\label{eq:radial_operator_uniform} \\ \Delta_\phi^2\Phi_{i,j,k} =& \frac{\Phi_{i,j-1,k}-2\Phi_{i,j,k}+\Phi_{i,j+1,k}}{R_i^2(\delta\phi)^2},\end{aligned}$$ while $\Delta_z^2$ is defined through Equation . Logarithmic Cylindrical Grid ---------------------------- In logarithmic cylindrical coordinates, we discretize a cylindrical computational domain in the same way as in the uniform cylindrical grid, but with logarithmic radial spacing. We define the face-centered radial coordinates as $R_{i+1/2} = f^i R_{\rm min}$, with a common multiplication factor $f\equiv(R_{\rm max} / R_{\rm min})^{1/N_R} >1$. Since the radial zone width, given by $R_{i+1/2}-R_{i-1/2} = (f-1)f^{i-1}R_{\rm min}$ shrinks toward small radii, a logarithmic cylindrical grid is advantageous in resolving the central regions of a disk with high accuracy. We also define the cell-centered radial coordinates using the volumetric centers as $$R_i \equiv \frac{\int_{R_{i-1/2}}^{R_{i+1/2}} R^2 dR} {\int_{R_{i-1/2}}^{R_{i+1/2}} R dR} = %\frac{2}{3} \frac{R_{i+1/2}^3 - R_{i-1/2}^3}{R_{i+1/2}^2 - R_{i-1/2}^2} = \frac{2(f^2+f+1)}{3(f+1)} f^{i-1}R_{\rm min},$$ for $i=1,2,\cdots,N_R$. Note that the radial cell spacing $\delta R_i \equiv R_{i+1} - R_i = (f-1)R_i$ increases with $R_i$. The second-order finite-difference approximation to Equation takes the same form as Equation , but with the radial difference operator defined as $$\label{eq:radial_operator} \Delta_R^2\Phi_{i,j,k} = \frac{\Phi_{i-1,j,k}-2\Phi_{i,j,k}+\Phi_{i+1,j,k}}{(R_i\ln f)^2}.$$ Appendix \[s:fd\] shows that Equation makes the finite difference approximation second-order accurate. Calculation of Interior Potential for Dirichlet Boundary Conditions {#s:interior_solver} =================================================================== In this section, we provide the general method that we use to obtain the interior potential within the original domain, given Dirichlet boundaries for the potential on the surface. We describe our methods for both Cartesian grids (Section \[sec:cart\_int\]) and cylindrical grids (Section \[s:interior\_solver\_cylindrical\]). We note that in principle, alternative fast and efficient solvers (such as multigrid) could be employed for computing the interior potential given Dirichlet boundary conditions for the potential. The interior solver is employed in three different instances in our method. The first instance is our use of the interior solver to obtain the numerical DGF, as described in Appendix \[s:calc\_dgf\]; this is done once at the beginning of any simulation. The other two instances are in the first and fourth step of the James’s method; each time the Poisson solution is required, two calls to the interior solver are made. In the first step of James’s method, the interior potential from the original density distribution is computed subject to the zero boundary condition. In the fourth step of James’s method, the interior potential must be computed subject to a known boundary potential $\Phi^{\rm B}$; a method to obtain $\Phi^{\rm B}$ will be presented in Section \[s:bc\]. Formally, we define $\Phi^{\rm B}$ as having nonzero value only in a single layer of ghost zones immediately outside the active domain. Then we can write the desired potential as $\Phi = \widetilde{\Phi} + \Phi^{\rm B}$ where the required boundary condition is $\widetilde{\Phi}=0$. From the definition of $\Phi^{\rm B}$, $\Delta^2 \Phi^{\rm B}$ will be nonzero only in the single layer of active zones adjoining the domain boundaries. We can thus define a modified density distribution $\rho \rightarrow \rho - \Delta^2 \Phi^{\rm B}/(4\pi G)$ which is the same as the original density distribution everywhere except in the layer just inside the domain boundaries. We then employ this modified density distribution following the procedure of Section \[sec:cart\_int\] or Section \[s:interior\_solver\_cylindrical\] to compute $\widetilde{\Phi}$. Within the interior, where $\Phi^{\rm B}=0$, this solution is then the desired final solution $\Phi$. Note that in Sections \[sec:cart\_int\] and \[s:interior\_solver\_cylindrical\] below, $\rho_{i,j,k}$ is any arbitrary density distribution on the grid, and in fact represents a different quantity for each of the three instances where we solve for the interior potential. Cartesian Grid Solution with Zero Boundary Value {#sec:cart_int} ------------------------------------------------ It is conventional to utilize the eigenfunctions of a differential operator in solving an elliptic partial differential equation. The same technique can be applied to the discretized Poisson equation, if the eigenfunctions of the corresponding discrete Laplace operator can be found. Let ${\cal X}^l_i$, ${\cal Y}^m_j$, and ${\cal Z}^n_k$ be the eigenfunctions of the discrete Laplace operators $\Delta_x^2$, $\Delta_y^2$, and $\Delta_z^2$ satisfying $\Delta_x^2{\cal X}^l_i = \lambda_x^l{\cal X}^l_i$, $\Delta_y^2{\cal Y}^m_j = \lambda_y^m{\cal Y}^m_j$, and $\Delta_z^2{\cal Z}^n_k = \lambda_z^n{\cal Z}^n_k$, with respective eigenvalues $\lambda^l_x$, $\lambda_y^m$, and $\lambda_z^n$. It is straightforward to show that $${\cal X}^l_i = \sin\left(\frac{\pi li}{N_x+1}\right),\label{eq:car_eigen_x}$$ $${\cal Y}^m_j = \sin\left(\frac{\pi mj}{N_y+1}\right),\label{eq:car_eigen_y}$$ $${\cal Z}^n_k = \sin\left(\frac{\pi nk}{N_z+1}\right),\label{eq:car_eigen_z}$$ are the desired eigenfunctions satisfying the zero boundary condition at the ghost cells. The corresponding eigenvalues are $$\lambda_x^l = -k_l^2\left[ \sin\left( \frac{\pi l}{2(N_x+1)} \right) \bigg/\left( \frac{\pi l}{2N_x} \right) \right]^2,$$ $$\lambda_y^m = -k_m^2\left[ \sin\left( \frac{\pi m}{2(N_y+1)} \right) \bigg/\left( \frac{\pi m}{2N_y} \right) \right]^2,$$ $$\lambda_z^n = -k_n^2\left[ \sin\left( \frac{\pi n}{2(N_z+1)} \right) \bigg/\left( \frac{\pi n}{2N_z} \right) \right]^2,$$ where $k_l \equiv \pi l / L_x$, $k_m \equiv \pi m / L_y$, and $k_n \equiv \pi n / L_z$. In the limit of $l/N_x, m/N_y, n/N_z \ll 1$, the discrete eigenvalues reduce to the counterpart of the continuous Laplace operator ($-k^2$). The discrete analog of the Sturm-Liouville theory (e.g., @hil68 [chap. 1.10–1.16]; see also, @atk64) guarantees that the eigenfunctions given in Equation – satisfy discrete orthogonality relations, for example, $$\label{eq:orthogorel} \frac{2}{N_x+1} \sum_{i=1}^{N_x}{\cal X}_i^l{\cal X}_i^{l'} = \delta_{ll'}\quad\text{and}\quad\frac{2}{N_x+1} \sum_{l=1}^{N_x} {\cal X}_i^l{\cal X}_{i'}^l = \delta_{ii'},$$ where the symbol $\delta$ denotes the Kronecker delta. These orthogonality relations allow us to expand $\widetilde{\Phi}_{i,j,k}$ and $\rho_{i,j,k}$ as $$\begin{aligned} \widetilde{\Phi}_{i,j,k} = \frac{8}{(N_x+1)(N_y+1)(N_z+1)}\sum_{l=1}^{N_x}\sum_{m=1}^{N_y}\sum_{n=1}^{N_z}\widetilde{\Phi}^{lmn}{\cal X}_i^l{\cal Y}_j^m{\cal Z}_k^n,\label{eq:car_Phi_forward}\\ \rho_{i,j,k} = \frac{8}{(N_x+1)(N_y+1)(N_z+1)}\sum_{l=1}^{N_x}\sum_{m=1}^{N_y}\sum_{n=1}^{N_z}\rho^{lmn}{\cal X}_i^l{\cal Y}_j^m{\cal Z}_k^n,\label{eq:car_rho_forward}\end{aligned}$$ where the expansion coefficients $\widetilde{\Phi}^{lmn}$ and $\rho^{lmn}$ are given by $$\begin{aligned} \widetilde{\Phi}^{lmn} &= \sum_{i=1}^{N_x}\sum_{j=1}^{N_y}\sum_{k=1}^{N_z}\widetilde{\Phi}_{i,j,k}{\cal X}_i^l{\cal Y}_j^m{\cal Z}_k^n,\label{eq:car_Phi_backward}\\ \rho^{lmn} &= \sum_{i=1}^{N_x}\sum_{j=1}^{N_y}\sum_{k=1}^{N_z}\rho_{i,j,k}{\cal X}_i^l{\cal Y}_j^m{\cal Z}_k^n.\label{eq:car_rho_backward}\end{aligned}$$ Plugging Equations – in Equation , we obtain a simple algebraic relation $$\label{eq:car_Poisson_transform} \widetilde{\Phi}^{lmn} = \frac{4\pi G \rho^{lmn}}{\lambda_x^l + \lambda_y^m + \lambda_z^n}.$$ Therefore, the Poisson equation in Cartesian coordinates can be solved by the following three steps: 1. Perform a forward transform $\rho_{i,j,k}\to\rho^{lmn}$ using Equation : ${\cal O}(N_xN_yN_z\log_2[N_xN_yN_z])$. 2. Convert $\rho^{lmn} \to \widetilde{\Phi}^{lmn}$ using the kernel in Equation : ${\cal O}(N_xN_yN_z)$. 3. Perform a backward transform $\widetilde{\Phi}^{lmn}\to\widetilde{\Phi}_{i,j,k}$ using Equation : ${\cal O}(N_xN_yN_z\log_2[N_xN_yN_z])$. In practice, the discrete transforms in Equations – can be performed efficiently with an FFT algorithm. The public [FFTW]{} library[^1] performs sine transforms of various kinds, among which we use [FFTW\_R0DFT00]{} consistent with the zero boundary condition. To perform 3D FFTs in parallel, we decompose the computational domain into 2D pencils along, for example, the $y$-axis and execute 1D sine transforms locally in each pencils (e.g., @li10). We then transpose the pencils to the $z$- and $x$-axes sequentially, each time by executing corresponding sine transforms, which completes a 3D FFT. The parallel transpose among different pencil decompositions are done with the [remap\_3d]{} function in Steve Plimpton’s parallel FFT package[^2]. We note that, since mass density and the gravitational potential in general are distributed as blocks rather than pencils in real applications, we have to transpose between block and pencil decompositions at the input and output stage of the Poisson solver. Plimpton’s remap routine provides this functionality as well. Cylindrical Grid Solution with Zero Boundary Value {#s:interior_solver_cylindrical} -------------------------------------------------- A natural boundary condition in the azimuthal direction is that both mass density and gravitational potential are periodic, with period $L_\phi$. This holds true even when the problem under study has $P$-fold symmetry in the $\phi$-direction, with a domain size $L_\phi = 2\pi / P$. The algorithm presented below is applicable for such systems as long as the $P$-fold symmetry is considered in the boundary condition for the DGF (Equation ). The eigenfunction ${\cal P}^m_j$ for the discrete Laplace operator $\Delta_\phi^2$ and the corresponding eigenvalue $\lambda_\phi^m$ are given by $$\label{eq:cyl_eigenfunction} {\cal P}^m_j = \exp\left[ \frac{2\pi\sqrt{-1}mj}{N_\phi} \right],$$ $$\lambda^m_\phi = -\frac{m^2}{R_i^2}\left[ \sin\left( \frac{\pi m}{N_\phi} \right) \bigg/\left( \frac{\pi m}{N_\phi} \right) \right]^2.$$ Note that $\lambda^m_\phi \rightarrow -m^2/R^2$ for $m/N_\phi \ll 1$. The eigenfunction ${\cal P}^m_j$ satisfies the discrete orthogonality relation $$\frac{1}{N_\phi} \sum_{j=1}^{N_\phi}({\cal P}_j^m)^* {\cal P}^{m'}_j = \delta_{mm'}\quad\text{and}\quad\frac{1}{N_\phi} \sum_{m=1}^{N_\phi}({\cal P}_j^m)^* {\cal P}^{m'}_j = \delta_{jj'}.$$ We expand $\widetilde{\Phi}_{i,j,k}$ and $\rho_{i,j,k}$ only along the azimuthal and vertical directions as $$\begin{aligned} \widetilde{\Phi}_{i,j,k} = \frac{2}{N_\phi (N_z+1)}\sum_{m=1}^{N_\phi}\sum_{n=1}^{N_z}\widetilde{\Phi}^{mn}_{i}{\cal P}_j^m{\cal Z}_k^n,\label{eq:cyl_Phi_forward}\\ \rho_{i,j,k} = \frac{2}{N_\phi (N_z+1)}\sum_{m=1}^{N_\phi}\sum_{n=1}^{N_z}\rho^{mn}_{i}{\cal P}_j^m{\cal Z}_k^n,\label{eq:cyl_rho_forward}\end{aligned}$$ where the expansion coefficients $\widetilde{\Phi}^{mn}_{i}$ and $\rho^{mn}_{i}$ satisfy the inverse transforms $$\begin{aligned} \widetilde{\Phi}^{mn}_{i} &= \sum_{j=1}^{N_\phi}\sum_{k=1}^{N_z}\widetilde{\Phi}_{i,j,k} ({\cal P}^m_j)^* {\cal Z}^n_k,\label{eq:cyl_Phi_backward}\\ \rho^{mn}_{i} &= \sum_{j=1}^{N_\phi}\sum_{k=1}^{N_z}\rho_{i,j,k} ({\cal P}^m_j)^* {\cal Z}^n_k.\label{eq:cyl_rho_backward}\end{aligned}$$ One cannot analytically expand $\widetilde{\Phi}_{i,j,k}$ and $\rho_{i,j,k}$ along the radial direction because radial eigenfunction ${\cal R}^l_i$, defined through $\Delta_R^2{\cal R}^l_i = \lambda^l_R {\cal R}^l_i$, has no closed-form expression and is not compatible with FFT.[^3] Plugging Equation – into Equation yields $$\label{eq:tridiagonal} \left( \Delta_R^2 + \lambda^m_\phi + \lambda^n_z \right) \widetilde{\Phi}^{mn}_i = 4\pi G \rho^{mn}_i,$$ which, using Equations and , can be written as $$\label{eq:tridiagonal_uni} \left[ \frac{1}{(\delta R)^2} - \frac{1}{2R_i \delta R} \right] \widetilde{\Phi}_{i-1}^{mn} + \left[ \lambda_\phi^m + \lambda^n_z - \frac{2}{(\delta R)^2} \right] \widetilde{\Phi}_i^{mn} + \left[ \frac{1}{(\delta R)^2} + \frac{1}{2R_i \delta R} \right] \widetilde{\Phi}_{i+1}^{mn} = 4\pi G \rho^{mn}_i$$ in uniform cylindrical coordinates, and $$\label{eq:tridiagonal_log} \frac{1}{(R_i\ln f)^2} \widetilde{\Phi}_{i-1}^{mn} + \left[ \lambda_\phi^m + \lambda^n_z - \frac{2}{(R_i\ln f)^2} \right] \widetilde{\Phi}_i^{mn} + \frac{1}{(R_i\ln f)^2} \widetilde{\Phi}_{i+1}^{mn} = 4\pi G \rho^{mn}_i$$ in logarithmic cylindrical coordinates. Note that Equations and are tridiagonal matrix equations subject to the zero boundary conditions of $\widetilde{\Phi}_{0}^{mn} = \widetilde{\Phi}_{N_R+1}^{mn} = 0$, which can easily be solved via the Thomas algorithm involving back substitutions (e.g., @nr). Therefore, the Poisson equation in cylindrical coordinates can be solved by the following three steps: 1. Perform a forward transform $\rho_{i,j,k}\to\rho^{mn}_i$ using Equation : ${\cal O}(N_RN_\phi N_z\log_2[N_\phi N_z])$. 2. Solve Equation or for $\widetilde{\Phi}^{mn}_i$ : ${\cal O}(N_RN_\phi N_z)$. 3. Perform a backward transform $\widetilde{\Phi}^{mn}_i\to\widetilde{\Phi}_{i,j,k}$ using Equation : ${\cal O}(N_RN_\phi N_z\log_2[N_\phi N_z])$. In actual computation, the discrete transforms in Equations – can be carried out efficiently with an FFT algorithm. For transforms involving ${\cal P}^m_j$, we use the real-to-complex transform in [FFTW]{}, which halves the size of the output by utilizing the Hermitian symmetry. For transforms involving ${\cal Z}^n_k$, we use the sine transform as in Cartesian coordinates. For parallel computations, we employ the pencil decomposition technique along with the Steve Plimpton’s parallel transpose routines, similarly to the Cartesian solver. CALCULATION OF THE BOUNDARY POTENTIAL {#s:bc} ===================================== Overview of the James Algorithm {#s:James_overview} ------------------------------- We adopt the James algorithm to calculate the boundary potential $\Phi^{\rm B}$ which is second-order accurate. As explained in Introduction, James’s method first solves for the preliminary potential $\Psi=\Phi+\Theta$ with zero boundary condition, where $\Phi({\bf x}) = -\int G\rho({\bf x'})/|{\bf x-x'}|\,d^3x'$ is the gravitational potential generated from the original density distribution (i.e., the desired solution) and $\Theta({\bf x}) = -\oint G\sigma({\bf x'})/|{\bf x-x'}|\,d^2x'$ is that from the screening charges. Since $\Psi^{\rm B}=0$ by definition, one only needs to compute $\Theta^{\rm B}=-\oint G\sigma({\bf x'})/|{\bf x}^{\rm B} - {\bf x'}|\,d^2x'$ to obtain $\Phi^{\rm B} = -\Theta^{\rm B}$. Once the preliminary gravitational potential $\Psi$ with the zero boundary condition is obtained (using the method of Section \[s:interior\_solver\] and the original density distribution), the screening charges are found by applying the discrete Laplace operators at the ghost cells. It should be noted that at the ghost cells, the discrete Laplace operator calls for the value of $\Psi$ *outside* the ghost cells, which is set to zero in @james77. Figure \[fig:james\] depicts this situation, where ${\mathscr R}$ and $\partial{\mathscr R}$ represent the active and the ghost cells defined in Section \[s:equation\]. Encompassing this, one can imagine the infinite domain ${\mathscr R}_\infty$ where $\Psi=\rho=0$ everywhere exterior to $\partial{\mathscr R}$. It is evident that the $\Psi$ satisfies the discrete Poisson equation at every cell in ${\mathscr R}_\infty$ if one adds $\sigma=\Delta^2\Psi/(4\pi G)$ at $\partial{\mathscr R}$. Since $\Delta^2\Psi = 4\pi G (\rho+\sigma)$ in $\mathscr R_\infty$, one can solve $\Delta^2\Theta = 4\pi G\sigma$ to obtain $\Phi = \Psi - \Theta$ that satisfies the original Poisson equation $\Delta^2\Phi = 4\pi G\rho$ subject to the vacuum boundary condition (i.e., $\Phi$ is the desired solution for an isolated mass distribution). Note that one needs $\Theta$ only at the domain boundary, which can be efficiently calculated using the Green’s function for the discrete Laplace operator subject to vacuum boundary condition (i.e., the DGF). Then, $\Phi^{\rm B} = -\Theta^{\rm B}$ gives a new boundary condition for the final step of the James’s method. To summarize, the James algorithm consists of the following four steps: (1) Solve the Poisson equation with the zero boundary condition to obtain $\Psi$; (2) Evaluate the screening charge $\sigma$ by applying the discrete Laplace operator to the ghost cells ($\sigma = \Delta^2\Psi/4\pi G$); (3) Use the DGF to calculate the gravitational potential $\Theta^{\rm B}$ at the domain boundary due to $\sigma$; (4) Solve the Poisson equation with the Dirichlet boundary condition $\Phi^{\rm B} = - \Theta^{\rm B}$. In section \[s:interior\_solver\] we have presented the method we adopt for the interior Poisson solver, which is employed in Steps (1) and (4) and also is used to pre-compute the DGF (see Appendix \[s:calc\_dgf\]), which enters in Step 3. In what follows, we describe Steps (2) and (3) in more detail. Computation of the Screening Charges {#s:James_boundary_charges} ------------------------------------ Once the preliminary gravitational potential $\Psi$ with the zero boundary condition is obtained (using the method of Section \[s:interior\_solver\] and the original density distribution), one can readily apply the discrete Laplace operators at the ghost cells in each boundary to calculate the screening charges. ### Cartesian Grid A Cartesian grid has six boundary surfaces consisting of the loci of ghost zones immediately outside the problem domain: bottom (bot; $k=0$), top (top; $k=N_z+1$), south (sth; $j=0$), north (nth; $j=N_y+1$), west (wst; $i=0$), and east (est; $i=N_x+1$). With $\Psi=0$ in both the first and second layer of ghost zones outside the domain, the screening charges on these boundary surfaces are given by $$\begin{aligned} \sigma_{i,j}({\rm bot}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=1}, & \sigma_{i,j}({\rm top}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=N_z},\nonumber\\ \sigma_{j,k}({\rm wst}) &= \frac{1}{4\pi G (\delta x)^2} \Psi|_{i=1}, & \sigma_{j,k}({\rm est}) &= \frac{1}{4\pi G (\delta x)^2} \Psi|_{i=N_x},\nonumber\\ \sigma_{i,k}({\rm sth}) &= \frac{1}{4\pi G (\delta y)^2} \Psi|_{j=1}, & \sigma_{i,k}({\rm nth}) &= \frac{1}{4\pi G (\delta y)^2} \Psi|_{j=N_y},\end{aligned}$$ where $\sigma_{i,j}({\rm bot})$ denotes the screening charge on the bottom boundary, etc. Note that the screening charges have units of mass density rather than surface density, because the charge is assumed to fill a volume $\delta x \delta y \delta z$. ### Cylindrical Grid In cylindrical coordinates, one needs to deal with only four boundary surfaces: bottom (bot; $k=0$), top (top; $k=N_z+1$), inner (inn; $i=0$), and outer (out; $i=N_R+1$): the azimuthal direction is assumed periodic. Using the discrete Laplace operators given in Section \[s:equation\], one can show that the screening charges are calculated as $$\begin{aligned} \sigma_{i,j}({\rm bot}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=1}, & \sigma_{i,j}({\rm top}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=N_z},\nonumber\\ \sigma_{j,k}({\rm inn}) &= \frac{1+\delta R/(2R_{0})}{4\pi G (\delta R)^2} \Psi|_{i=1}, & \sigma_{j,k}({\rm out}) &= \frac{1-\delta R/(2R_{N_R+1})}{4\pi G (\delta R)^2} \Psi|_{i=N_R}\end{aligned}$$ in a uniform cylindrical grid, and $$\begin{aligned} \sigma_{i,j}({\rm bot}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=1}, & \sigma_{i,j}({\rm top}) &= \frac{1}{4\pi G (\delta z)^2} \Psi|_{k=N_z},\nonumber\\ \sigma_{j,k}({\rm inn}) &= \frac{1}{4\pi G (R_{0}\ln f)^2} \Psi|_{i=1}, & \sigma_{j,k}({\rm out}) &= \frac{1}{4\pi G (R_{N_R+1}\ln f)^2} \Psi|_{i=N_R}\end{aligned}$$ in a logarithmic cylindrical grid. Discrete Green’s Function and the Potential Generated by Screening Charges {#s:dgf} -------------------------------------------------------------------------- The gravitational potential $\Theta$ that results from the screening charges can be obtained by convolving $\sigma$ with the Green’s function of the operator that determines $\sigma$. Since $\sigma$ is obtained through the application of the *discrete* Laplace operator, the corresponding Green’s function should be the DGF rather than the CGF. The proper operation of the James’s method thus relies on the accurate calculation of the DGF, yet finding its analytic expressions in cylindrical coordinates is a daunting task. To our knowledge, the analytic DGF is available only in 2D Cartesian coordinates [@bune71]. In 3D Cartesian coordinates, @burk97 addressed the definition, existence, and uniqueness of the DGF and derived asymptotic expansion formulae, applicable at distances far from the source. In this section, we provide a working definition of the DGF and a numerical method to calculate $\Theta$ in Cartesian and cylindrical coordinates. We refer the reader to Appendix \[s:calc\_dgf\] for our method for the DGF. ### Cartesian Grid The DGF, ${\cal G}_{i-i',j-j',k-k'}$, in Cartesian coordinates is the gravitational potential per unit mass due to a discrete point mass at $(i',j',k')$ and ought to satisfy $$\label{eq:def_car_green} \left(\Delta_x^2 + \Delta_y^2 + \Delta_z^2\right){\cal G}_{i-i',j-j',k-k'} = 4\pi G \frac{\delta_{ii'}\delta_{jj'}\delta_{kk'}}{\cal V},$$ where the symbol $\delta_{ii'}$ is the Kronecker delta and ${\cal V}=\int_{z_{k'-1/2}}^{z_{k'+1/2}}\int_{y_{j'-1/2}}^{y_{j'+1/2}}\int_{x_{i'-1/2}}^{x_{i'+1/2}}dxdydz = \delta x\delta y\delta z$ is the volume of the $(i',j',k')$-th cell. Note that in writing the indices of ${\cal G}_{i-i',j-j',k-k'}$, we implicitly allow for the translational symmetry on a Cartesian grid. In Appendix \[s:calc\_dgf\_cart\], we follow @james77 to calculate the Cartesian DGF numerically. The gravitational potential $\Theta_{i,j,k}$ generated by the screening charges $\sigma_{i,j,k}$ is given by $$\label{eq:car_bpot_by_dgf} \Theta_{i,j,k} = \sum_{i'=0}^{N_x+1}\sum_{j'=0}^{N_y+1}\sum_{k'=0}^{N_z+1}{\cal G}_{i-i',j-j',k-k'}\sigma_{i',j',k'}{\cal V}.$$ Substituting Equation into Equation , one can easily check that $\Theta_{i,j,k}$ and $\sigma_{i,j,k}$ is a valid potential-density pair. Because Equation involves a discrete convolution, it is efficient to use FFTs for computations. Making use of certain symmetries of the problem for a hollow charge distribution, @james77 devised a formulation that uses sine and cosine transforms to expresses the potential on the each surface as the sum of seven terms. We refer the reader to Equations (4.7)–(4.20) of @james77 for the description of this formulation, which costs ${\cal O}(N^3 + N^2 \log N)$ operations. The gravitational potential $\Phi^{\rm B}$ at the domain boundary is obtained by $$\Phi^{\rm B}_{i,j,k} = \Psi^{\rm B}_{i,j,k} - \Theta^{\rm B}_{i,j,k} = -\Theta^{\rm B}_{i,j,k},$$ which provides the required Dirichlet boundary condition for the interior solver (Section \[s:interior\_solver\]). Note that $\Psi^{\rm B}_{i,j,k} = 0$ by definition. ### Cylindrical Grid The cylindrical DGF, ${\cal G}_{i,i',j-j',k-k'}$, satisfies $$\label{eq:def_cyl_green} \left(\Delta_R^2 + \Delta_\phi^2 + \Delta_z^2\right){\cal G}_{i,i',j-j',k-k'} = 4\pi G \frac{\delta_{ii'}\delta_{jj'}\delta_{kk'}}{{\cal V}_{i'}},$$ where ${\cal V}_{i'}=\int_{z_{k'-1/2}}^{z_{k'+1/2}}\int_{\phi_{j'-1/2}}^{\phi_{j'+1/2}}\int_{R_{i'-1/2}}^{R_{i'+1/2}} R\,dR d\phi dz = \tfrac{1}{2}(R_{i'+1/2}^2-R_{i'-1/2}^2)\delta\phi\delta z$ is the volume of the $(i',j',k')$-th cell. Note that the cylindrical DGF has four indices due to lack of the translational symmetry along the radial direction. In Appendix \[s:calc\_dgf\_cyl\], we present the method to calculate the cylindrical DGF and compare it with the continuous counterpart. The gravitational potential $\Theta_{i,j,k}$ generated by the screening charges $\sigma_{i,j,k}$ takes a form of $$\label{eq:gpot_by_discrete_green} \Theta_{i,j,k} = \sum_{i'=0}^{N_R+1}\sum_{j'=1}^{N_\phi}\sum_{k'=0}^{N_z+1} {\cal G}_{i,i',j-j',k-k'}\sigma_{i',j',k'}{\cal V}_{i'}.$$ One can readily verify that $\Theta_{i,j,k}$ satisfies the discrete Poisson equation in cylindrical coordinates. Since all functions are periodic in the azimuthal direction, it is natural to apply a discrete Fourier transform such that $$\label{eq:dft} \Theta^m_{ik} \equiv \sum_{j=1}^{N_\phi} \Theta_{ijk} e^{-2\pi \sqrt{-1}jm/N_\phi},$$ and similarly for ${\cal G}$ and $\sigma$. Then, Equation can be cast into a more compact form $$\label{eq:gpot_by_discrete_green_fft} \Theta^m_{ik} = \sum_{i'=0}^{N_R+1}\sum_{k'=0}^{N_z+1}{\cal G}^m_{i,i',k-k'} \sigma^m_{i'k'}{\cal V}_{i'}.$$ Since $\sigma$ is nonzero only at the ghost cells and we need to evaluate $\Theta$ also only at the ghost cells, we do not have to perform full double summations for all $m,i,k$ indices in Equation . By collecting the individual contributions of the surface charges, Equation yields the Fourier-transformed potentials at the four boundaries $$\begin{aligned} \label{eq:Theta_top} \Theta^m_i({\rm top}) &= \sum_{i'=1}^{N_R} {\cal G}^m_{i,i'}({\rm top\to top}) \sigma^m_{i'}({\rm top}){\cal V}_{i'} + \sum_{i'=1}^{N_R} {\cal G}^m_{i,i'}({\rm bot\to top}) \sigma^m_{i'}({\rm bot}){\cal V}_{i'}\nonumber\\ &+ \sum_{k'=1}^{N_z} {\cal G}^m_{i,k'}({\rm inn\to top}) \sigma^m_{k'}({\rm inn}){\cal V}_{0} + \sum_{k'=1}^{N_z} {\cal G}^m_{i,k'}({\rm out\to top}) \sigma^m_{k'}({\rm out}){\cal V}_{N_R+1},\end{aligned}$$ $$\begin{aligned} \label{eq:Theta_bot} \Theta^m_i({\rm bot}) &= \sum_{i'=1}^{N_R} {\cal G}^m_{i,i'}({\rm top\to bot}) \sigma^m_{i'}({\rm top}){\cal V}_{i'} + \sum_{i'=1}^{N_R} {\cal G}^m_{i,i'}({\rm bot\to bot}) \sigma^m_{i'}({\rm bot}){\cal V}_{i'}\nonumber\\ &+ \sum_{k'=1}^{N_z} {\cal G}^m_{i,k'}({\rm inn\to bot}) \sigma^m_{k'}({\rm inn}){\cal V}_{0} + \sum_{k'=1}^{N_z} {\cal G}^m_{i,k'}({\rm out\to bot}) \sigma^m_{k'}({\rm out}){\cal V}_{N_R+1},\end{aligned}$$ $$\begin{aligned} \label{eq:Theta_inn} \Theta^m_k({\rm inn}) &= \sum_{i'=1}^{N_R} {\cal G}^m_{k,i'}({\rm top\to inn}) \sigma^m_{i'}({\rm top}){\cal V}_{i'} + \sum_{i'=1}^{N_R} {\cal G}^m_{k,i'}({\rm bot\to inn}) \sigma^m_{i'}({\rm bot}){\cal V}_{i'}\nonumber\\ &+ \sum_{k'=1}^{N_z} {\cal G}^m_{k-k'}({\rm inn\to inn}) \sigma^m_{k'}({\rm inn}){\cal V}_{0} + \sum_{k'=1}^{N_z} {\cal G}^m_{k-k'}({\rm out\to inn}) \sigma^m_{k'}({\rm out}){\cal V}_{N_R+1},\end{aligned}$$ $$\begin{aligned} \label{eq:Theta_out} \Theta^m_k({\rm out}) &= \sum_{i'=1}^{N_R} {\cal G}^m_{k,i'}({\rm top\to out}) \sigma^m_{i'}({\rm top}){\cal V}_{i'} + \sum_{i'=1}^{N_R} {\cal G}^m_{k,i'}({\rm bot\to out}) \sigma^m_{i'}({\rm bot}){\cal V}_{i'}\nonumber\\ &+ \sum_{k'=1}^{N_z} {\cal G}^m_{k-k'}({\rm inn\to out}) \sigma^m_{k'}({\rm inn}){\cal V}_{0} + \sum_{k'=1}^{N_z} {\cal G}^m_{k-k'}({\rm out\to out}) \sigma^m_{k'}({\rm out}){\cal V}_{N_R+1},\end{aligned}$$ where we use symbolic notations such that ${\cal G}^m_{i,k'}({\rm inn\to top}) = {\cal G}^m_{i,0,N_z+1-k'}$, $\sigma^m_{i'}({\rm top}) = \sigma^m_{i',N_z+1}$, etc. Finally, we apply an inverse Fourier transform to $\Theta^m_i({\rm top}), \Theta^m_i({\rm bot}), \Theta^m_k({\rm inn})$, and $\Theta^m_k({\rm out})$ to obtain the boundary potential $\Theta_{i,j,k}^{\rm B}$ due to the surface charges. Then, the desired boundary potential $\Phi^{\rm B}$ due to the original charge $\rho$ is given by $$\label{eq:boundary_condition} \Phi^{\rm B}_{i,j,k} = \Psi^{\rm B}_{i,j,k} -\Theta_{i,j,k}^{\rm B} = -\Theta_{i,j,k}^{\rm B},$$ which gives the required Dirichlet boundary condition for the interior solver. Note that the boundary potential calculations explained above involve FFTs on 2D arrays (e.g., $\sigma^m_{i'}({\rm top})$) together with the summations of the Green’s function amounting to ${\cal O}(N^3)$ operations. Therefore, the overall computational cost of the boundary potential calculation is of order ${\cal O}(N^3 + N^2\log N)$, similarly to the case with a Cartesian grid. We note that the above formulation for the boundary potential is valid for a mass distribution under $P$-fold symmetry in $\phi$. In Appendix \[s:P-fold\_symm\], we directly demonstrate that this is really the case as long as ${\cal G}_{i,i',j-j',k-k'}$ in Equation properly accounts for the contributions to the boundary potential from all periodic images of the mass density. TEST RESULTS {#s:tests} ============ We implement our Poisson solver in [Athena++]{} which is a state-of-art astrophysical magnetohydrodynamics (MHD) code with very flexible coordinate and grid options. Using Cartesian and uniform/logarithmic cylindrical grids, we test our solver on a few test problems to check its accuracy, convergence, and parallel performance. We also run time-dependent simulations of a gravitationally-unstable isothermal ring to check if the gravity module combines well with the MHD solver of [Athena++]{} to produce the expected results of ring fragmentation. For all tests presented below, we set the gravitational constant to $G = 1$. Uniform Sphere Test {#s:staticPot} ------------------- To test the accuracy of our Poisson solver, we consider a uniform sphere with radius $r_0$ and density $\rho_0$. The analytic gravitational potential of such a sphere is given by $$\Phi_a(r) = \begin{dcases} -2\pi G \rho_0 \left(r_0^2 - \tfrac{1}{3} r^2\right), & (r < r_0),\\ -\frac{4\pi G\rho_0 r_0^3}{3r}, & (r > r_0), \end{dcases}$$ where $r$ denotes the distance from the center of the sphere. We take $r_0=0.2$ and $\rho_0=1$, and place it at an off-centered position $(x_0, y_0, z_0)$ in Cartesian coordinates and $(R_0, \phi_0, z_0)$ in cylindrical coordinates, and calculate the gravitational potential $\Phi$ numerically. Table \[tb:sphere\_test\] lists the grid dimension, resolution, and sphere position in each coordinate system adopted. As a measure of accuracy, we define the relative error between the numerical solution and the analytic solution as $${\epsilon} \equiv \left|\frac{\Phi-\Phi_a}{\Phi_a}\right|,$$ evaluated at the cell centers. Figures \[fig:car\_sphere\]–\[fig:cyl\_sphere\] plot the test results on the Cartesian, uniform cylindrical, and logarithmic cylindrical grid, respectively. Panels (a) and (b) plot the one-dimensional (1D) cut profiles of $\Phi$ and $\epsilon$ along the $x$- or $R$-direction, respectively, while panel (c) gives the 2D distribution of $\epsilon$ in the $z=0$ plane. Overall, the numerical results are exceedingly close to the analytic potential, with the mean relative error less than 0.1%. The errors are largest near the sphere boundary whose exact shape is not well resolved by any of the adopted grids. [cccccc]{}\[!t\] Cartesian & $[-0.5,0.5]\times[-0.5,0.5]\times[-0.5,0.5]$ & $64\times 64\times 64$ & $0.25$ & $0.1$ & $-0.04$\ uniform cylindrical & $[0.5,1]\times[0,2\pi]\times[-0.25,0.25]$ & $64\times 256\times 64$ & $0.72$ & $0.63$ & $-0.04$\ logarithmic cylindrical & $[10^{-2},1]\times[0,2\pi]\times[-0.25,0.25]$ & $128\times 64\times 64$ & $0.27$ & $0.38$ & $-0.04$ Convergence Test {#s:convergence} ---------------- The discrete Poisson equation used for the interior solver in Section \[s:interior\_solver\] and for the boundary condition in Section \[s:bc\] are second-order accurate by construction. If our implementation of the Poisson solver is correct, therefore, the relative errors should be inversely proportional to the square of the grid spacing. To check if this is indeed the case, we repeat the uniform sphere tests by varying the number of cells from $16^3$ to $512^3$. Figure \[fig:convergence\] plots as circles the mean relative errors $\left\langle {\epsilon} \right\rangle$ from the sphere tests as functions of $N_z$. Overall, the errors decrease roughly at a second-order rate with increasing $N_z$, but exhibit some fluctuations. @katz16 noted that these fluctuations of the errors are caused not by the truncation errors of the finite-difference scheme but by inability of an adopted grid to perfectly resolve a spherical mass distribution. This is true even for a spherical grid when the sphere center offsets from the origin. To delineate the truncation errors alone, it is thus necessary to design a solid figure whose shape is identical to the cell shape of an adopted grid. In addition, the size and mass of the solid figure should be unchanged with varying resolution. For this purpose, we consider a uniform cube with density $\rho=1$, located at $x_1 \le x \le x_2$, $y_1 \le y \le y_2$, and $z_1 \le z \le z_2$ in a Cartesian grid. In a cylindrical grid, we consider a rectangular torus with density $\rho=1$, occupying the regions with $R_1 \le R \le R_2$, $\phi_1 \le \phi \le \phi_2$, and $z_1 \le z \le z_2$. Table \[tb:convergence\_test\] lists the parameters of the solid figures that we adopt: these values ensure that the mass distribution does not change with resolution. We use our Poisson solver to calculate the gravitational potentials of the solid figures by varying resolution from $N_z=16$ to $512$, while keeping the domain sizes the same as in Section \[s:staticPot\]. As the reference potential, we take Equation (20) of @katz16 for the gravitational potential of a uniform cube. There is no algebraic expression for the potential of a rectangular torus, but @hure14 provided a closed-form expression, in terms of line integrals with smooth integrands, in their Equation (29). We use the Romberg’s method with a relative tolerance $10^{-10}$ to ensure that the numerical integrations are accurate enough to serve as a reference solution. The squares in Figure \[fig:convergence\] plot the resulting mean relative errors $\left<{\epsilon}\right>$ for the cube and rectangular torus. Note that $\left<{\epsilon}\right>$ against $N_z$ follows almost a straight line with slope of $-1.993$, $-2.003$, and $-2.003$ in the Cartesian, uniform cylindrical, and logarithmic cylindrical grid, respectively, confirming that our implementation of the Poisson solver retains a second-order accuracy. [ccccccc]{} Cartesian & 0.0625 & 0.4375 & -0.375 & 0 & -0.125 & 0.25\ uniform cylindrical & 0.625 & 0.90625 & 0 & 1.570796327 & -0.0625 & 0.1875\ logarithmic cylindrical & 0.1 & 0.7498942093 & 0 & 1.570796327 & -0.0625 & 0.1875 Performance Test ---------------- To check the parallel performance of our implementation, we conduct a weak scaling test of our Poisson solver on the [TigerCPU]{} linux cluster at Princeton University[^4]. The [TigerCPU]{} cluster consists of 408 nodes, with each node comprised of 40 $2.4\,{\rm GHz}$ intel Skylake processors. We measure the wall clock time using the [MPI\_Wtime]{} function, after a call to [MPI\_Barrier]{} to synchronize all processors. We run the job with [SLURM –exclusive]{} option to make sure that other jobs do not interfere with ours. For the weak scaling test, we divide the whole computational domain into $N_{\rm core}$ subdomains consisting of $64^3$ cells each. In the [Athena++]{} terminology, the whole domain and subdomain are referred to as [Mesh]{} and [MeshBlock]{}, respectively. We assign each [MeshBlock]{} to a single processor, while varying $N_{\rm core}$ from $1$ to $4096$. The corresponding size of [Mesh]{} varies from $64^3$ to $1024^3$. In each run, we call our Poisson solver, together with the MHD solver for comparison, 100 times, and then measure the wall clock time per cycle $t_{\rm wall}$ for various steps. We repeat the calculations five times in order to avoid unusual runs due to stale nodes. Figure \[fig:scaling\] plots the mean values $\left\langle t_{\rm wall} \right\rangle$ of the wall clock times per cycle as functions of $N_{\rm core}$ for the Cartesian (left) and cylindrical (right) grids. As noted earlier, our Poisson solver requires to run the interior solver twice and the boundary solver once. The total time taken by the Poisson solver (triangles) is dominated by the interior solver (squares) rather than the boundary solver (circles). While the time taken by the MHD solver is comparable between Cartesian and cylindrical grids, the Poisson solver is more efficient in the cylindrical grid. This is because Cartesian coordinates has no inherent periodic direction and thus requires more operations to implement the open boundary conditions. In addition, the Cartesian grid has two more boundaries than the cylindrical grid and thus needs more boundary-to-boundary interactions in the boundary solver. Notwithstanding these differences, the Poisson solver in both Cartesian and cylindrical grids takes less time than the MHD solver (stars) at least up to 4096 processors, leading us to conclude that our Poisson solver is very efficient and does not contribute much to the total computational cost of self-gravitating MHD simulations.[^5] Ring Fragmentation {#s:simulation} ------------------ As a final test of our Poisson solver, we run time-dependent simulations for fragmentation of a self-gravitating isothermal ring using [Athena++]{}. As an initial condition, we consider a rigidly-rotating ring at angular velocity $\Omega_0$ and sound speed $c_s$, surrounded by a tenuous hot external medium. The ring is initially in hydrosteady equilibrium under both self-gravity and external gravity. Assuming that the external gravity ${\bf g}_{\rm ext} = - \Omega_e^2 {\bf R}$ alone makes the ring rotate at angular frequency $\Omega_e$, the equation for such an equilibrium reads $$\label{eq:hydrosteady} c_s^2 \nabla \ln \rho + \nabla \Phi - \Omega_s^2{\bf R} = 0,$$ together with Equation , where $\Omega_s \equiv (\Omega_0^2 - \Omega_e^2)^{1/2}$ is the angular velocity due to self-gravity alone [@kim16]. One can show that the equilibrium configurations are completely specified by two dimensionless parameters: $\alpha\equiv c_s^2/(GR_A^2\rho_c)$ and $\widehat{\Omega}_s \equiv \Omega_s/(G\rho_c)^{1/2}$, where $\rho_c$ and $R_A$ denote the maximum density and the maximum radial extent of an equilibrium object, respectively. Using the self-consistent field method of @hachisu86, we solve Equations and alternatively and iteratively to find the equilibrium configuration for $\alpha=0.015$ and $\widehat{\Omega}_s=0.22$. We then boost the angular velocity of the ring to $\widehat{\Omega}_0=0.3$ to account for ${\bf g}_{\rm ext}$. The external medium is set to be hotter than the ring by two orders of magnitude. In order to check whether the configuration we set is really in equilibrium, we evolve it over time on a logarithmic cylindrical grid with size $\widehat{R}\equiv R/R_A \in[0.75,1.05]$, $\phi\in[0,2\pi]$, and $z/R_A \in[-0.15,0.15]$, without imposing any perturbations. The number of cells used is $N_R=64$, $N_\phi = 1024$, and $N_z=64$. Figure \[fig:torus\_ic\] compares the radial profiles of the dimensionless density $\widehat{\rho}\equiv\rho/\rho_c$ and pressure $\widehat{p}=\alpha\widehat{\rho}$ in the equatorial plane as functions of $\widehat{R}$, at $t/T_{\rm orb}=0$, 1, and 5, where $T_{\rm orb}=2\pi/\Omega_0$. The density and pressure profiles are almost unchanged over time except near the contact discontinuity between the ring and external medium, demonstrating that the initial steady configuration is well maintained over many orbital periods. By performing a linear-stability analysis, @kim16 found that the above equilibrium is gravitationally unstable to non-axisymmetric perturbations for a range of the azimuthal mode number $m$. The most unstable modes were found to have $m=9$ and $10$, with an almost equal growth rate ${\rm Im}(\omega)\approx 0.8(G\rho_c)^{1/2}\approx 17\,T_{\rm orb}^{-1}$ and a phase speed ${\rm Re}(\omega)/m\approx 0.3(G\rho_c)^{1/2}$, corresponding to overstability. To check if our Poisson solver can pick up these unstable modes and capture their growth, we generate random density perturbations with amplitude $10^{-5}$ and apply them to the equilibrium density $\rho$. We then evolve the system and monitor how various modes grow. Figure \[fig:gi\] plots the azimuthal profiles of the dimensionless density at $\widehat{R}=0.91$ and $\widehat{z}=0$ at a few selected epochs as well as the surface density $\Sigma=\int \rho\,dz$ at $t(G\rho_c)^{1/2}=20.2$. It is apparent that the perturbations grow as they propagate along the $\phi$-direction. In the highly nonlinear stage, the density distribution is dominated equally by the $m=9$ and $m=10$ modes, indicating that these are two fastest growing modes of the instability. To measure the growth rate and the phase speed of each mode in the numerical simulation, we calculate the Fourier transform ${\cal L}_me^{-i\vartheta_m} \equiv \int \bar{\rho}(\phi) e^{-im\phi}\,d\phi $ of the radially- and vertically-integrated density $\bar{\rho}(\phi) = \iint\rho\,dR dz$. Figure \[fig:growth\] plots the temporal variations of the Fourier amplitude ${\cal L}_{m}$ and the phase angle $\vartheta_{m}$ for $m=9$ and $m=10$. The two modes grow exponentially as they propagate at a constant phase speed. The dimensionless growth rate $d \ln {\cal L}_m/dt/(G\rho_c)^{1/2}$ and the phase speed $d {\vartheta_m}/dt/[m(G\rho_c)^{1/2}]$ are measured to be $0.80$ and $0.30$, respectively, consistent with the results of the linear stability analysis. Taken together, all the test results presented in this section demonstrate that our 3D cylindrical Poisson solver is reliable, accurate to second order, and very efficient. SUMMARY AND DISCUSSION {#s:discussion} ====================== To study dynamical evolution of self-gravitating, rotating disks, it is desirable to use a fully 3D Poisson solver in a cylindrical geometry subject to the open boundary condition. In this paper, we have presented an accurate and efficient algorithm for such a Poisson solver that works in Cartesian, uniform cylindrical, and logarithmic cylindrical coordinates. Our algorithm adopts “surface screening charge” method introduced by James, employing the DGF to calculate the boundary potential consistent with the open boundary condition (Section \[s:bc\]), and utilizes the eigenfunction expansion method to solve the interior potential (Section \[s:interior\_solver\]). The computational cost of our algorithm is of order ${\cal O}(N^3 + N^2\log N)$, with $N$ being the number of cells in one dimension. The results of the various test problems presented in Section \[s:tests\] confirm that our Poisson solver is second-order accurate and takes less computational cost than the MHD solver in [Athena++]{} up to 4096 cores. The second-order accuracy of our Poisson solver is enabled by the usage of the cylindrical DGF. Although @serafini05 and @snytnikov11 implemented the James algorithm in Cartesian and cylindrical coordinates, respectively, they simply used the CGF instead of the DGF after enlarging the computational domain. Since the difference between CGF and DGF is quite small at large distances from the source (see Appendix \[s:calc\_dgf\]), the potential based on the CGF would be similar to that with the DGF if the domain is expanded sufficiently. As mentioned in Introduction, however, domain expansion in cylindrical coordinates is very limited toward the inner radial boundary, so that the potential with the CGF would become less accurate for larger $R_{\rm max}/R_{\rm min}$, where $R_{\rm max}$ and $R_{\rm min}$ denote the outer and inner radial boundaries of the domain, respectively. To demonstrate this, we recalculate the gravitational potential of a rectangular torus on a logarithmic cylindrical grid presented in Section \[s:convergence\], but this time by using the CGF. For fare comparison, we apply domain expansion technique similar to @snytnikov11 when using CGF. Specifically, we use the same extended domain for DGF calculation defined in Appendix \[s:calc\_dgf\_cyl\] for the enlarged domain. Figures \[fig:Rmaxmin\] plots the resulting mean relative errors as red triangles against $R_{\rm max}/R_{\rm min}$, in comparison with the cases based on the DGF plotted as blue circles. The CGF works as well as the DGF for $R_{\rm max}/R_{\rm min} \lesssim 10^2$. But, it fails to give second-order convergence for systems with $R_{\rm max}/R_{\rm min} \gtrsim 10^2$, which are common in astronomical applications [e.g., @kuiper10; @seo13; @zhu12; @bae14; @ju16; @kim17]. In these cases, it is necessary to use the cylindrical DGF for accurate potential calculations. SM wishes to thank Chang-Goo Kim, Jeong-Gyu Kim and Kengo Tomida for their helpful discussions and advice. The work of SM was supported by NRF (National Research Foundation of Korea) Grant funded by the Korean Government (NRF-2017-Fostering Core Leaders of the Future Basic Science Program/Global Ph.D. Fellowship Program). The work of WTK was supported by the grant (2017R1A4A1015178) of National Research Foundation of Korea. The work of ECO on this project is supported by grant 510940 from the Simons Foundation. The computation of this work was supported by the Supercomputing Center/Korea Institute of Science and Technology Information with supercomputing resources including technical support (KSC-2018-C3-0015) and the PICSciE TIGRESS High Performance Computing Center at Princeton University. SECOND-ORDER FINITE-DIFFERENCE IN LOGARITHMIC CYLINDRICAL COORDINATES {#s:fd} ===================================================================== Here we derive a second-order finite-difference approximation to the radial part of the Laplace operator in logarithmic cylindrical coordinates. With the change of the variables $u\equiv \ln R$, the radial part of the Laplacian becomes $$\frac{1}{R} \frac{\partial}{\partial R} \left( R \frac{\partial\Phi}{\partial R} \right) = \frac{1}{R^2} \frac{\partial^2\Phi}{\partial u^2}.$$ Since the logarithmic grid in $R$ ($R_i = R_0 f^i$) corresponds to a uniform grid in $u$ ($u_i = u_0 + i\ln f$), we can apply a centered difference scheme in the $u$-space to obtain $$\label{eq:radiald_alt} \frac{1}{R^2} \frac{\partial^2\Phi}{\partial u^2} = \frac{1}{R_i^2} \frac{\Phi_{i-1} - 2\Phi_i + \Phi_{i+1}}{(\delta u)^2} + {\cal O}((\delta u)^2).$$ Noting that $\delta u = \ln f = N_R^{-1}\ln (R_{\rm max}/R_{\rm min})$, the above expression can be expressed in the $R$-space as $$\label{eq:radiald2} \frac{1}{R} \frac{\partial}{\partial R} \left( R \frac{\partial\Phi}{\partial R} \right) = \frac{\Phi_{i-1}-2\Phi_i + \Phi_{i+1}}{(R_i\ln f)^2} + {\cal O}\left( \frac{1}{N_R^2} \left(\ln\frac{R_{\rm max}}{R_{\rm min}}\right)^2 \right).$$ It is evident that the remainder decreases at a second-order rate with increasing $N_R$. We adopt Equation as our discrete Laplace operator in the logarithmic cylindrical grid (see Equation ). COMPUTATION OF THE DISCRETE GREEN’S FUNCTION {#s:calc_dgf} ============================================ The DGF is needed in order to calculate the surface potential associated with the surface screening charges. The DGF is pre-computed once at the beginning of any simulation. In this Appendix we outline the numerical method used to evaluate the DGF. Cartesian Grid {#s:calc_dgf_cart} -------------- We calculate the DGF ${\cal G}_{i-i',j-j',k-k'}$ in Cartesian coordinates by solving Equation for a point source with unit mass located in a cell at $(i',j',k')=(1,1,1)$. Once we obtain ${\cal G}_{i-1,j-1,k-1}$ for $1\le i \le N_x$, $1\le j \le N_y$, and $1\le k \le N_z$, the values for other indices can be computed from the symmetry requirement $$\label{eq:car_symm} {\cal G}_{i-i',j-j',k-k'} = {\cal G}_{|i-i'|,|j-j'|,|k-k'|},$$ for $| i-i'| \le N_x-1$, $|j-j' | \le N_y-1$, and $ | k-k' | \le N_z-1$. As we use the method presented in Section \[s:interior\_solver\] to obtain the numerical solution of Equation , we must supply an appropriate boundary condition a priori. It is reasonable to assume that far from the source, the DGF asymptotes to the CGF: $$\label{eq:car_asymptotic_green} {\cal G}_{i-1,j-1,k-1} \approx -\frac{G}{\sqrt{(x_i - x_{1})^2 + (y_j - y_{1})^2 + (z_k - z_{1})^2 }}\quad\text{(far from the source).}$$ Since high-order derivatives of the CGF are non-negligible close to the source, the near-field DGF that results from a second-order finite-difference approximation to the Poisson equation would deviate greatly from the CGF. How far should the boundary be away from the source to safely apply Equation as a proper boundary condition? The answer of @james77 to this question was 16 cells. In fact, one can compare the first two terms in the asymptotic expansion of the DGF in @burk97 to verify that the relative deviation between the DGF and CGF is $0.1\%$ at a distance of $16$ cells, regardless of the total number of cells or the grid spacing. To ensure that all boundaries are sufficiently away from the point source at $(i',j',k') = (1,1,1)$, for the calculation of the DGF only, we extend the computational domain by adding 16 additional cells to one side in each direction. The newly added cells (except for the ghost cells) are numbered as $i=-15,-14,\cdots, -1$ in the $x$-direction (and similarly in the $y$- and $z$-directions). We then apply the boundary condition (Equation ) to $i=-15$ and $N_x+1$ (and similarly for $j$ and $k$) and solve Equation in the extended domain to obtain ${\cal G}_{i-1,j-1,k-1}$ for $i=-15,-14,\cdots,N_x+1$ (and similarly for $j$ and $k$). In practice, following the method of Section \[s:interior\_solver\] we use the boundary condition to define a modified interior charge, and then employ sine transforms. After obtaining the solution, we discard the portion pertaining to the extended part of the domain, and use Equation to calculate the Cartesian DGF for whole indices. Since the DGF is calculated once and for all in the initialization step, its contribution to the computational cost of an entire simulation is almost negligible. Figure \[fig:cardgf\] plots the resulting DGF as a function of the grid distance from the source, in comparison with the CGF. The CGF diverges at the source position, whereas the DGF remains finite everywhere. Although the DGF deviates from the CGF close to the source, their difference becomes smaller as the grid distance increases, consistent with the asymptotic expansion of @burk97. Cylindrical Grid {#s:calc_dgf_cyl} ---------------- In a cylindrical grid, we need to store only 16 kinds of the DGFs, such as ${\cal G}_{i,k'}^m (\rm inn \rightarrow top)$, appearing in Equation –, each of which represents the Fourier-transformed potential at one boundary due to point masses located at the same or different boundaries. These are calculated as follows. To calculate the potential generated by point sources at the top vertical boundary, we place a point source with unit mass at $(i',j',k') = (i',1,N_z+1)$ for any $i' \in [1, N_R]$ and solve Equation numerically using the method of Section \[s:interior\_solver\]. This yields ${\cal G}_{i,i',j-1,k-N_z-1}$ and its Fourier transform ${\cal G}_{i,i',k-N_z-1}^m$ after applying Equation , but we only keep four 3D arrays ${\cal G}^m_{i,i'}({\rm top \to top}) = {\cal G}^m_{i,i',0}$, ${\cal G}^m_{i,i'}({\rm top \to bot}) = {\cal G}^m_{i,i',-N_z-1}$, ${\cal G}^m_{k,i'}({\rm top \to inn}) = {\cal G}^m_{0,i',k-N_z-1}$, and ${\cal G}^m_{k,i'}({\rm top \to out}) = {\cal G}^m_{N_R+1,i',k-N_z-1}$, corresponding to four DGFs due to the point sources at the top boundary. We repeat the above calculations by varying $i' \in [1, N_R]$ to fill all the components of the four DGFs. For the potential generated by point sources at the inner radial boundary, we place a point mass at $(i',j',k')=(0,1,k')$ for any $k' \in [1, N_z]$. We then solve Equation and apply Fourier transform to obtain ${\cal G}_{i,0,k-k'}^m$. We keep only four 3D arrays ${\cal G}^m_{i,k'}({\rm inn\to top}) = {\cal G}^m_{i,0,N_z+1-k'}$, ${\cal G}^m_{i,k'}({\rm inn\to bot}) = {\cal G}^m_{i,0,-k'}$, ${\cal G}^m_{k-k'}({\rm inn\to inn}) = {\cal G}^m_{0,0,k-k'}$, and ${\cal G}^m_{k-k'}({\rm inn\to out}) = {\cal G}^m_{N_R+1,0,k-k'}$[^6], corresponding to four DGFs due to the point sources at the inner radial boundary. We repeat the above calculations for all $k'\in[1,N_z]$ to completely fill the elements of the arrays. We follow a similar procedure to obtain the remaining eight DGFs due to points sources at the bottom and outer radial boundaries. The boundary condition in solving Equation can be obtained by requiring that the DGF at a large distance from the source is approximately equal to $$\label{eq:cyl_asymptotic_green} {\cal G}_{i,i',j-j',k-k'} \approx - \sum_{p=0}^{P-1} \frac{G}{\sqrt{R_i^2 + R_{i'}^2 - 2R_iR_{i'}\cos(\phi_j - \phi_{j'} - pL_\phi) + (z_k - z_{k'})^2 }}\quad\text{(far from the source)},$$ The summation over $p$ in Equation is to add all the contributions from the periodic images of the point mass when the mass distribution holds $P$-fold symmetry in the $\phi$-direction. As discussed in Section \[s:calc\_dgf\_cart\], Equation remains valid as long as the distance between the cells $(i,j,k)$ and $(i',j',k')$ is sufficiently large. Similarly to the Cartesian case, we extend the computational domain, only for the calculation of the DGF, by adding extra cells to each of the four boundaries. In the vertical and outer radial directions, 16 cells are wide enough to ensure that Equation is a good approximation to the DGF. In the inner radial direction, however, the 16-cell criterion based on a uniform grid spacing does not guarantee that Equation is a valid approximation especially when $f=(R_{\rm max}/R_{\rm min})^{1/N_R}$ is large. We numerically confirmed that using Equation as a Dirichlet boundary condition at the extended inner radial boundary causes large errors in the computation of the cylindrical DGF and ultimately gravitational potential returned from our Poisson solver when $R_{\rm max}/R_{\rm min} > 10^2$. It turns out that using the radial gradient of Equation as a Neumann boundary condition is a cure to this inner boundary problem. The Neumann condition on the extended inner boundary is not very accurate, either, but we empirically find that it enables the DGF to converge rapidly to the desired values and results in the accurate DGF at least in the original computational domain, if the domain is sufficiently expanded. We found that adding $N_R-16$ cells in the inner radial boundary ($16$ cells are kept for the outer radial boundary) produces enough accuracy for the DGF in the original domain. To illustrate this directly, we set up a logarithmic cylindrical grid with $64^3$ cells spanning $R\in[10^{-4},1]$, $\phi\in[0,2\pi]$, $z\in[-0.25,0.25]$. We then distribute additional $N_R=64$ cells, by adding 16 cells with $i=65, 66,\cdots,80$ to outside of the outer radial boundary and $N_R-16 = 48$ cells with $i=-47, \cdots,-1,0$ to inside of the inner radial boundary, and calculate the DGF for a point source located at $(i',j',k')=(0,1,1)$ for both Neumann and Dirichlet conditions at the extended inner boundary. Figure \[fig:dgf\](a) compares the resulting ${\cal G}_{i,0,0,0}$ as functions of $R_i$ in logarithmic scale, with crosses and squares corresponding to the cases with Neumann and Dirichlet conditions, respectively. Figure \[fig:dgf\](b) zooms in the section with $|R_i-R_{-1}| \le 10^{-4}$ into linear scale, with the inset plotting the DGF from the Neumann condition for $-9.4\times 10^{-5} \le R_i - R_{-1} \le -9.0\times 10^{-5}$. It is clear that the DGF resulting from the Dirichlet condition is not symmetric with respect to the source, and cannot thus be considered correct. On the other hand, the DGF from the Neumann condition retains symmetry with respect to the source and converges to the CGF at large distances toward the outer boundary. The test problems presented in Section \[s:tests\] verify that the DGF under the Neumann condition works very well for our purposes, yielding accurate gravitational potentials in the original domain even when $R_{\rm max}/R_{\rm min} = 10^{5}$ (see Figure \[fig:Rmaxmin\]). In the case of a uniform cylindrical grid, it is sufficient to add only $16$ cells to inside of the inner radial boundary either using Neumann or Dirichlet boundary condition[^7]. This is because $16$ cells are wide enough at both inner and outer boundary with uniform spacing. Since the radial coordinate of the extended inner boundary should be positive, this requires $R_{\rm min} > 16\delta R$, or equivalently, $R_{\rm max}/R_{\rm min} < 1+N_R/16$. This limitation is not be severe, given that the logarithmic cylindrical grid is more appropriate for large $R_{\rm max}/R_{\rm min}$. For Mass Distribution Under $P$-fold Azimuthal Symmetry {#s:P-fold_symm} ======================================================= We often meet a problem that is periodic in the $\phi$-direction, with period $2\pi / P$ for a positive integer $P$. For example, gas flows in barred galaxies may have $P=2$ symmetry (e.g., @seo13), while $P=4$ for four-armed spiral galaxies [e.g., @dobbs06; @so08]. In such situations, one can save computational time by restricting the domain to $\phi\in[0,2\pi/P]$, with a periodic boundary condition. The interior solver presented in Section \[s:interior\_solver\_cylindrical\] works without modification even when $L_\phi=2\pi/P$ does not cover a full $2\pi$ domain, because the $P$-fold symmetry is automatically taken into account in Equation . This holds true also for the boundary solver (Section \[s:bc\]) as long as the periodic boundary condition with period $L_\phi$ is imposed in solving Equation . The cylindrical DGF defined as such then represents the gravitational potential from $P$ identical point masses lying along the azimuth with a uniform angular separation of $L_\phi$. When applying the boundary condition to the cylindrical DGF, therefore, one should add all the contributions from the image masses located at $\phi\in[L_\phi, 2\pi]$ to the gravitational potential of a real point mass located in the original domain with $\phi\in[0, L_\phi]$, as in Equation . We note that the inclusion of the image masses is implicit in the DGF, so that all calculations are done in the original computational domain with size $L_\phi$, enabling a factor of $P$ reduction in the computational time as well as memory compared to the cases with a full $2\pi$-periodic domain. To verify that our method handles $P$-fold azimuthal symmetry, we consider four uniform spheres located at $\phi_0$, $\phi_0+\pi/2$, $\phi_0+\pi$, and $\phi_0+3\pi/2$ in a uniform cylindrical grid presented in Table \[tb:sphere\_test\]. We calculate the gravitational potential from the resulting mass distribution that clearly has $P=4$ symmetry over $\phi\in[0, 2\pi]$. Figure \[fig:pfold\] plots as black solid lines the mass density and the gravitational potential along the $\phi$-direction at $R=0.75$ and $z=0.0039$. We then recompute the gravitational potential of a single sphere at $\phi_0$ by reducing the computational domain to $\phi\in[0, \pi/2]$, and finally the gravitational potential of two spheres at $\phi_0+\pi$ and $\phi_0+3\pi/2$ in the domain covering $\phi\in[\pi, 2\pi]$. The resulting potentials, plotted as red and blue dashed lines, are identical within machine precision to the potential from the full $2\pi$ domain. This confirms that our Poisson solver in a restricted $\phi\in[0,2\pi/P]$ domain correctly deals with mass distributions under $P$-fold azimuthal symmetry. [^1]: <http://www.fftw.org/> [^2]: <https://www.sandia.gov/~sjplimp/docs/fft/README.html> [^3]: The radial eigenfunction ${\cal R}^l_i$ can instead be obtained numerically by solving the eigenvalue problem, and the resulting eigenfunction may be called the *discrete Bessel function*. Since it satisfies the exact discrete orthogonality relation, it may also serve as discrete kernel for the discrete Hankel transform [@john87; @baddour15]. [^4]: <https://researchcomputing.princeton.edu/systems-and-services/available-systems/tiger> [^5]: The weak scaling test shown in Figure \[fig:scaling\] hints some performance degradation from $N_{\rm core}=1$ to $64$ relative to $\langle t_{\rm wall}\rangle \propto \ln N_{\rm core}$ expected for the theoretical FFT. Our parallel FFT utilizes a “transpose algorithm” known to be efficient when a data size for communication is larger than the critical size that depends on the latency/bandwidth of the interconnecting network device and the network topology [e.g., @foster97]. An alternative “binary exchange algorithm” may work efficiently for a small data size [e.g., @muller19]. [^6]: Although ${\cal G}^m_{k-k'}({\rm inn\to inn})$ and ${\cal G}^m_{k-k'}({\rm inn\to out})$ can be stored in 2D arrays, we store them as 3D arrays for simple coding. [^7]: We use Neumann condition for consistency.

--- abstract: 'Motivated by applications to unsupervised learning, we consider the problem of measuring mutual information. Recent analysis has shown that naive kNN estimators of mutual information have serious statistical limitations motivating more refined methods. In this paper we prove that serious statistical limitations are inherent to any measurement method. More specifically, we show that any distribution-free high-confidence lower bound on mutual information cannot be larger than $O(\ln N)$ where $N$ is the size of the data sample. We also analyze the Donsker-Varadhan lower bound on KL divergence in particular and show that, when simple statistical considerations are taken into account, this bound can never produce a high-confidence value larger than $\ln N$. While large high-confidence lower bounds are impossible, in practice one can use estimators without formal guarantees. We suggest expressing mutual information as a difference of entropies and using cross-entropy as an entropy estimator. We observe that, although cross-entropy is only an upper bound on entropy, cross-entropy estimates converge to the true cross-entropy at the rate of $1/\sqrt{N}$.' author: - |       David McAllester,   Karl Stratos\ \ TTI-Chicago title: Formal Limitations on The Measurement of Mutual Information --- Introduction ============ Motivated by maximal mutual information (MMI) predictive coding [@IT-cotrain; @PartOfSpeech; @Contrastive], we consider the problem of measuring mutual information. A classical approach to this problem is based on estimating entropies by computing the average log of the distance to the $k$th nearest neighbor in a sample [@KNN-MI]. It has recently been shown that the classical kNN methods have serious statistical limitations and more refined kNN methods have been proposed [@KNN-MI2]. Here we establish serious statistical limitations on any method of estimating mutual information. More specifically, we show that any distribution-free high-confidence lower bound on mutual information cannot be larger than $O(\ln N)$ where $N$ is the size of the data sample. Prior to proving the general case, we consider the particular case of the Donsker-Varadhan lower bound on KL divergence [@DV; @MINE]. We observe that when simple statistical considerations are taken into account, this bound can never produce a high-confidence value larger than $\ln N$. Similar comments apply to lower bounds based on contrastive estimation. The contrastive estimation lower bound given in [@Contrastive] does not establish mutual information of more than $\ln k$ where $k$ is number of negative samples used in the contrastive choice. The difficulties arise in cases where the mutual information $I(x,y)$ is large. Since $I(x,y) = H(y) - H(y|x)$ we are interested in cases where $H(y)$ is large and $H(y|x)$ is small. For example consider the mutual information between an English sentence and its French translation. Sampling English and French independently will (almost) never yield two sentences where one is a plausible translation of the other. In this case the DV bound is meaningless and contrastive estimation is trivial. In this example we need a language model for estimating $H(y)$ and a translation model for estimating $H(y|x)$. Language models and translation models are both typically trained with cross-entropy loss. Cross-entropy loss can be used as an (upper bound) estimate of entropy and we get an estimate of mutual information as a difference of cross-entropy estimates. Note that the upper-bound guarantee for the cross-entropy estimator yields neither an upper bound nor a lower bound guarantee for a difference of entropies. Similar observations apply to measuring the mutual information for pairs of nearby frames of video or pairs of sound waves for utterances of the same sentence. We are motivated by the problem of maximum mutual information predictive coding [@IT-cotrain; @PartOfSpeech; @Contrastive]. One can formally define a version of MMI predictive coding by considering a population distribution on pairs $(x,y)$ where we think of $x$ as past raw sensory signals (images or sound waves) and $y$ as a future sensory signal. We consider the problem of learning stochastic coding functions $C_x$ and $C_y$ so as to maximize the mutual information $I(C_x(x),C_y(y))$ while limiting the entropies $H(C_x(x))$ and $H(C_y(y))$. The intuition is that we want learn representations $C_x(x)$ and $C_y(y)$ that preserve “signal” while removing “noise”. Here signal is simply defined to be a low entropy representation that preserves mutual information with the future. Forms of MMI predictive coding have been independently introduced in [@IT-cotrain] under the name “information-theoretic cotraining” and in [@Contrastive] under the name “contrastive predictive coding”. It is also possible to interpret the local version of DIM (DIM(L)) [@DIM] as a variant of MMI predictive coding. A closely related framework is the information bottleneck [@bottleneck]. Here one again assumes a population distribution on pairs $(x,y)$. The objective is to learn a stochastic coding function $C_x$ so as to maximize $I(C_x(x),y)$ while minimizing $I(C_x(x),x)$. Here one does not ask for a coding function on $y$ and one does not limit $H(C_x(x))$. Another related framework is INFOMAX [@linsker1988self; @bell1995information; @DIM]. Here we consider a population distribution on a single random variable $x$. The objective is to learn a stochastic coding function $C_x$ so as to maximize the mutual information $I(x,C_x(x))$ subject to some constraint or additional objective. As mentined above, in cases where $I(C_x(x),C_y(y))$ is large it seems best to train a model of the marginal distribution of $P(C_y)$ and a model of the conditional distribution $P(C_y|C_x)$ where both models are trained with cross-entropy loss. Section \[sec:cross\] gives various high confidence upper bounds on cross entropy loss for learned models. The main point is that, unlike lower bounds on entropy, high-confidence upper bounds on cross entropy loss can be guaranteed to be close to the true cross entropy. Our theoretical analyses will assume discrete distributions. However, there is no loss of generality in this assumption. Rigorous treatments of probability (measure theory) treat integrals (either Reimann or Lebesque) as limits of increasingly fine binnings. A continuous density can always be viewed as a limit of discrete distributions. Although our proofs are given for discrete case, all our formal limitations on the measurement of mutual information apply to continuous case as well. See [@diffent] for a discussion of continuous information theory. Additional comments on this point are given in section \[sec:limitations2\]. The Donsker-Varadhan Lower Bound ================================ Mutual information can be written as a KL divergence. $$I(X,Y) = KL(P_{X,Y},P_XP_Y)$$ Here $P_{X,Y}$ is a joint distribution on the random variables $X$ and $Y$ and $P_X$ and $P_Y$ are the marginal distributions on $X$ abd $Y$ respectively. The DV lower bound applies to KL-divergence generally. To derive the DV bound we start with the following observation for any distributions $P$, $Q$, and $G$ on the same support. Our theoretical analyses will assume discrete distributions. $$\begin{aligned} KL(P,Q) & = & E_{z \sim P}\;\ln \frac{P(z)}{Q(z)} \nonumber \\ \nonumber \\ & = & E_{z \sim P}\;\ln \left(\frac{G(z)}{Q(z)}\;\frac{P(z)}{G(z)} \right) \nonumber \\ \nonumber \\ & = & E_{z\sim P}\; \ln \frac{G(z)}{Q(z)} + KL(P,G) \nonumber \\ \nonumber \\ & \geq & E_{z\sim P}\; \ln \frac{G(z)}{Q(z)} \label{eq:DV1}\end{aligned}$$ Note that (\[eq:DV1\]) achieves equality for $G(z) = P(z)$ and hence we have $$\label{eq:DV2} KL(P,Q) = \sup_G\;E_{z \in P}\;\ln \frac{G(z)}{Q(z)}$$ Here we can let $G$ be a parameterized model such that $G(z)$ can be computed directly. However, we are interested in $KL(P_{X,Y},P_XP_Y)$ where our only access to the distribution $P$ is through sampling. If we draw a pair $(x,y)$ and ignore $y$ we get a sample from $P_X$. We can similarly sample from $P_Y$. So we are interested in a KL-divergence $KL(P,Q)$ where our only access to the distributions $P$ and $Q$ is through sampling. Note that we cannot evaluate (\[eq:DV1\]) by sampling from $P$ because we have no way of computing $Q(z)$. But through a change of variables we can convert this to an expression restricted to sampling from $Q$. More specifically we define $G(z)$ in terms of an unconstrained function $F(z)$ as $$G(z) = \frac{1}{Z} \;Q(z)e^{F(z)} \hspace{15mm} Z = \sum_z Q(z)e^{F(z)} = E_{z \sim Q}\;e^{F(z)} \label{eq:DV2.5}$$ Substituting (\[eq:DV2.5\]) into (\[eq:DV2\]) gives $$KL(P,Q) = \sup_F\;E_{z \sim P}\; F(z) - \ln E_{z \sim Q}\; e^{F(z)} \label{eq:DV3}$$ Equation (\[eq:DV3\]) is the Donsker-Varadhan lower bound. Applying this to mutual information we get $$\begin{aligned} I(X,Y) & = & KL(P_{X,Y},P_XP_Y) \nonumber \\ \nonumber \\ & = & \sup_{F}\; E_{x,y \sim P_{X,Y}}\; F(x,y) - \ln E_{x \sim P_X,\;y \sim P_Y} \;e^{F(x,y)} \label{eq:DV4}\end{aligned}$$ This is the equation underlying the MINE approach to maximizing mutual information [@MINE]. It would seem that we can estimate both terms in (\[eq:DV4\]) through sampling and be able to maximize $I(X,Y)$ by stochastic gradient ascent on this lower bound. Statistical Limitations of KL-Divergence Lower Bounds {#sec:limitations} ===================================================== In this section we show that the DV bound (\[eq:DV3\]) cannot be used to measure KL-divergences of more than tens of bits. In fact we will show that no high-confidence distribution-free lower bound on KL divergence can be used for this purpose. As a first observation note that (\[eq:DV3\]) involves $E_{z \sim Q}\; e^{F(z)}$. This expression has the same form as the moment generating function used in analyzing large deviation probabilities. The utility of expectations of exponentials in large deviation theory is that such expressions can be dominated by extremely rare events (large deviations). The rare events dominating the expectation will never be observed by sampling from $Q$. It should be noted that the optimal value for $F(z)$ in (\[eq:DV3\]) is $\ln (P(z)/Q(z))$ in which case the right hand side of $(\ref{eq:DV3})$ simplifies to $KL(P,Q)$. But for large KL divergence we will have that $F(z) = \ln (P(z)/Q(z))$ is typically hundreds of bits and this is exactly the case where $E_{z \sim Q}\; e^{F(z)}$ cannot be measured by sampling from $Q$. If $E_{z \sim Q} \;e^{F(z)}$ is dominated by events that will never occur in sampling from $Q$ then the optimization of $F$ through the use of (\[eq:DV3\]) and sampling from $Q$ cannot possibly lead to a function $F(z)$ that accurately models the desired function $\ln (P(z)/Q(z))$. To quantitatively analyze the risk of unseen outlier events we will make use of the following simple lemma where we write $P_{z \sim Q}(\Phi[z])$ for the probability over drawing $z$ from $Q$ that the statement $\Phi[z]$ holds. [**Outlier Risk Lemma:**]{} For a sample $S \sim Q^N$ with $N \geq 2$, and a property $\Phi[z]$ such that $P_{z \sim Q}(\Phi[z]) \leq 1/N$, the probability over the draw of $S$ that no $z \in S$ satisfies $\Phi[z]$ is at least 1/4. [**Proof:**]{} The probability that $\Phi[z]$ is unseen in the sample is at least $(1-1/N)^N$ which is at least $1/4$ for $N \geq 2$ and where we have $\lim_{N \rightarrow \infty} (1-1/N)^N = 1/e$. Q.E.D. We can use the outlier risk lemma to perform a quantitative risk analysis of the DV bound (\[eq:DV3\]). We can rewrite (\[eq:DV3\]) as $$\begin{aligned} KL(P,Q) & \geq & B(P,Q,F) \\ B(P,Q,F) & = & E_{z \sim P}\; F(z) - \ln E_{z \sim Q}\; e^{F(z)}\end{aligned}$$ We can try to estimate $B(P,Q,G)$ from samples $S_P$ and $S_Q$, each of size $N$, from the population distributions $P$ and $Q$ respectively. $$\hat{B}(S_P,S_Q,F) = \frac{1}{N} \sum_{z \in S_P}\;F(z) - \ln \frac{1}{N}\sum_{z \in S_Q}\;e^{F(z)}$$ While $B(P,Q,F)$ is a lower bound on $KL(P,Q)$, the sample estimate $\hat{B}(S_P,S_Q,F)$ is not. To get a high confidence lower bound on $KL(P,Q)$ we have to handle unseen outlier risk. For a fair comparison with our analysis of cross-entropy estimators in section \[sec:cross\], we will limit the outlier risk by bounding $F(z)$ to the interval $[0,F_{\max}]$. The largest possible value of $\hat{B}(S_P,S_q,F)$ occurs when $F(z) = F_{\max}$ for all $z \in S_P$ and $F(z) = 0$ for all $z \in S_Q$. In this case we get $\hat{B}(S_P,S_Q,F) = F_{\max}$. But by the outlier risk lemma there is still at least a 1/4 probability that $$E_{z\sim Q}\; e^{F(z)} \geq \frac{1}{N} e^{F_{\max}}.$$ Any high confidence lower bound $\tilde{B}(S_P,S_Q,F)$ must account for the unseen outlier risk. In particular we must have $$\begin{aligned} \tilde{B}(S_P,S_Q,F) & \leq & F_{\max} - \ln\frac{e^{F_{\max}}}{N} \\ \\ & = & \ln N\end{aligned}$$ Our negative results can be strengthened by considering the preliminary bound (\[eq:DV1\]) where $G(z)$ is viewed as a model of $P(z)$. We can consider the extreme case of perfect modeling of the population $P$ with a model $G(z)$ where $G(z)$ is computable. In this case we have essentially complete access to the distribution $P$. But even in this setting we have the following negative result. Let $B$ be any distribution-free high-confidence lower bound on KL(P,Q) computed with complete knowledge of $P$ but only a sample from $Q$. More specifically, let $B(P,S,\delta)$ be any real-valued function of a distribution $P$, a multiset $S$, and a confidence parameter $\delta$ such that, for any $P$, $Q$ and $\delta$, with probabililty at least $(1-\delta)$ over a draw of $S$ from $Q^N$ we have $$KL(P,Q) \geq B(P,S,\delta).$$ For any such bound, and for $N \geq 2$, with probability at least $1 -4\delta$ over the draw of $S$ from $Q^N$ we have $$B(P,S,\delta) \leq \ln N.$$ [**Proof**]{}. Consider distributions $P$ and $Q$ and $N \geq 2$. Define $\tilde{Q}$ by $$\tilde{Q}(z) = \left(1-\frac{1}{N}\right)Q(z) + \frac{1}{N}P(z).$$ We now have $KL(P,\tilde{Q}) \leq \ln N$. We will prove that from a sample $S \sim Q^N$ we cannot reliably distinguish between $Q$ and $\tilde{Q}$. We first note that by applying the high-confidence guarantee of the bound to $\tilde{Q}$ have $$P_{S \sim \tilde{Q}^N}(B(P,S,\delta) \leq KL(P,\tilde{Q})) \geq 1-\delta.$$ The distribution $\tilde{Q}$ equals the marginal on $z$ of a distribution on pairs $(s,z)$ where $s$ is the value of Bernoulli variable with bias $1/N$ such that if $s = 1$ then $z$ is drawn from $P$ and otherwise $z$ is drawn from $Q$. By the outlier risk lemma the probability that all coins are zero is at least 1/4. Conditioned on all coins being zero the distributions $\tilde{Q}^N$ and $Q^N$ are the same. Let $\mathrm{Pure}(S)$ represent the event that all coins are 0 and let $\mathrm{Small}(S)$ represent the event that $B(P,S,\delta) \leq \ln N$. We now have $$\begin{aligned} P_{S\sim Q^N}(\mathrm{Small(S)}) & = & P_{S \sim \tilde{Q}^N}(\mathrm{Small}(S)|\mathrm{Pure}(S)) \\ \\ & = & \frac{P_{S \sim \tilde{Q}^N}(\mathrm{Pure}(S) \wedge \mathrm{Small(S)})} {P_{S \sim \tilde{Q}^N}(\mathrm{Pure}(S))} \\ \\ & \geq & \frac{P_{S \sim \tilde{Q}^N}(\mathrm{Pure}(S)) -P_{S \sim \tilde{Q}^N}(\neg\mathrm{Small}(S))} {P_{S \sim \tilde{Q}^N}(\mathrm{Pure}(S))} \\ \\ & \geq & \frac{P_{S \sim \tilde{Q}^N}(\mathrm{Pure}(S)) - \delta} {P_{S \sim \tilde{Q}^N}(\mathrm{Pure}(S))} \\ \\ & = & 1 - \frac{\delta} {P_{S \sim \tilde{Q}^N}(\mathrm{Pure}(S))} \\ \\ & \geq & 1 - 4\delta.\end{aligned}$$ Statistical Limitations on Entropy Lower Bounds {#sec:limitations2} =============================================== Mutual information is a special case of KL-divergence. It is possible that tighter lower bounds can be given in this special case. In this section we show similar limitations on lower bounding mutual information. We first note that a lower bound on mutual information implies a lower bound on entropy. The mutual information between $X$ and $Y$ cannot be larger than information content of $X$ alone. $$I(X,Y) = H(X) - H(X|Y) \leq H(X)$$ So a lower bound on $I(X,Y)$ gives a lower bound on $H(X)$. We show that any distribution-free high-confidence lower bound on entropy requires a sample size exponential in the size of the bound. The above argument seems problematic for the case of continuous densities as differential entropy can be negative. However, for the continuous case we have $$I(x,y) = \sup_{C_x,C_y} \;I(C_x(x),C_y(y))$$ where $C_x$ and $C_y$ range over all maps from the underlying continuous space to discrete sets (all binnings of the continuous space). Hence an $O(\ln N)$ upper bound on the measurement of mutual information for the discrete case applies to the continuous case as well. The type of a sample $S$, denoted ${\cal T}(S)$, is defined to be a function on positive integers (counts) where ${\cal T}(S)(i)$ is the number of elements of $S$ that occur $i$ times in $S$. For a sample of $N$ draws we have $N = \sum_i\; i{\cal T}(S)(i)$. The type ${\cal T}(S)$ contains all information relevant to estimating the actual probability of the items of a given count and of estimating the entropy of the underlying distribution. The problem of estimating distributions and entropies from sample types has been investigated by various authors ([@GT; @AlwaysGT; @AlwaysGT2; @Arora]). Here we give the following negative result on lower bounding the entropy of a distribution by sampling. Let $B$ be any distribution-free high-confidence lower bound on $H(P)$ computed from a sample type ${\cal T}(S)$ with $S \sim P^N$. More specifically, let $B({\cal T},\delta)$ be any real-valued function of a type ${\cal T}$ and a confidence parameter $\delta$ such that for any $P$, with probabililty at least $(1-\delta)$ over a draw of $S$ from $P^N$, we have $$H(P) \geq B({\cal T}(S),\delta).$$ For any such bound, and for $N \geq 50$ and $k \geq 2$, with probability at least $1 -\delta - 1.01/k$ over the draw of $S$ from $P^N$ we have $$B({\cal T}(S),\delta) \leq \ln 2kN^2.$$ [**Proof:**]{} Consider a distribution $P$ and $N \geq 100$. If the support of $P$ has fewer than $2kN^2$ elements then $H(P) < \ln 2kN^2$ and by the premise of the theorem we have that, with probability at least $1-\delta$ over the draw of $S$, $B({\cal T}(S),\delta) \leq H(P)$ and the theorem follows. If the support of $P$ has at least $2kN^2$ elements then we sort the support of $P$ into a (possibly infinite) sequence $x_1,\;x_2,\;x_3,\ldots$ so that $P(x_i) \geq P(x_{i+1})$. We then define a distribution $\tilde{P}$ on the elements $x_1,\;\ldots,x_{2kN^2}$ by $$\tilde{P}(x_i) =\left(\begin{array}{ll} P(x_i) & \mbox{for $i \leq kN^2$} \\ \\ \frac{P(i> kN^2)}{kN^2} & \mbox{for $kN^2 < i \leq 2kN^2$} \end{array}\right)$$ We will let $\mathrm{Small}(S)$ denote the event that $B({\cal T}(S),\delta) \leq \ln2kN^2$ and let $\mathrm{Pure}(S)$ abbreviate the event that no element $x_i$ for $i > kN^2$ occurs twice in the sample. Since $\tilde{P}$ has a support of size $2kN^2$ we have $H(\tilde{P}) \leq \ln2kN^2$. Applying the premise of the lemma to $\tilde{P}$ gives $$\label{step1} P_{S \sim \tilde{P}^N}(\mathrm{Small}(S)) \geq 1 - \delta$$ For a type ${\cal T}$ let $P_{S \sim P^N}({\cal T})$ denote the probability over drawing $S \sim P^N$ that ${\cal T}(S) = {\cal T}$. We now have $$P_{S \sim P^N}({\cal T}|\mathrm{Pure}(S)) = P_{S \sim \tilde{P}^N}({\cal T}|\mathrm{Pure}(S)).$$ This gives the following. $$\begin{aligned} P_{S \sim P^N}(\mathrm{Small}(S)) & \geq & P_{S \sim P^N}(\mathrm{Pure}(S)\wedge \mathrm{Small}(S)) \nonumber \\ & = & P_{S \sim P^N}(\mathrm{Pure}(S)) \;P_{S \sim P^N} (\mathrm{Small}(S) \;|\;\mathrm{Pure}(S)) \nonumber \\ & = & P_{S \sim P^N}(\mathrm{Pure}(S)) \;P_{S \sim \tilde{P}^N} (\mathrm{Small}(S) \;|\;\mathrm{Pure}(S)) \nonumber \\ \label{step2} & \geq & P_{S \sim P^N}(\mathrm{Pure}(S)) \;P_{S \sim \tilde{P}^N} (\mathrm{Pure}(S) \wedge \mathrm{Small}(S)) \end{aligned}$$ For $i > kN^2$ we have $\tilde{P}(x_i) \leq 1/(kN^2)$ which gives $$P_{S\sim \tilde{P}^N}(\mathrm{Pure}(S)) \geq \prod_{j = 1}^{N-1}\;\left(1-\frac{j}{kN^2}\right)$$ Using $(1-P) \geq e^{-1.01\;P}$ for $P \leq 1/100$ we have the following birthday paradox calculation. $$\begin{aligned} \ln P_{S\sim \tilde{P}^N}(\mathrm{Pure}(S)) & \geq & - \frac{1.01}{kN^2}\sum_{j=1}^{N-1}\; j \nonumber \\ & = & - \frac{1.01}{kN^2}\;\frac{(N-1)N}{2} \nonumber \\ & \geq & - .505/k \nonumber \\ \label{step3} P_{S\sim \tilde{P}^N}(\mathrm{Pure}(S)) & \geq & e^{-.505/k} \;\geq 1- .505/k\end{aligned}$$ Applying the union bound to (\[step1\]) and (\[step3\]) gives. $$\label{step4} P_{S \sim \tilde{P}^N}(\mathrm{Pure}(S) \wedge \mathrm{Small}(S)) \geq 1 - \delta - .505/k$$ By a derivation similar to that of (\[step3\]) we get $$\label{step5} P_{S\sim P^N}(\mathrm{Pure}(S)) \geq 1 - .505/k$$ Combining (\[step2\]), (\[step4\]) and (\[step5\]) gives $$P_{S \sim P^N}(\mathrm{Small}(S)) \geq 1 - \delta - 1.01/k$$ Cross Entropy as an Entropy Estimator {#sec:cross} ===================================== Since mutual information can be expressed as a difference of entropies, the problem of measuring mutual information can be reduced to the problem of measuring entropies. In this section we show that, unlike high-confidence distribution-free lower bounds, high-confidence distribution-free upper bounds on entropy can approach the true cross-entropy at modest sample sizes even when the true cross-entropy is large. More specifically we consider the cross-entropy upper bound. $$\begin{aligned} H(P) & = & E_{x \sim P} \ln \frac{1}{P(x)} \\ & = & E_{x \sim P} \ln \left(\frac{1}{G(x)} \;\frac{G(x)}{P(x)}\right) \\ & = & H(P,G) - KL(P,G) \\ & \leq & H(P,G)\end{aligned}$$ For $G = P$ we get $H(P,G) = H(P)$ and hence we have $$H(P) = \inf_G\;H(P,G)$$ In practice $P$ is a population distribution and $G$ is model of $P$. For example $P$ might be a population distribution on paragraphs and $G$ might be an autoregressive RNN language model. In practice $G$ will be given by a network with parameters $\Phi$. In this setting we have the following upper bound entropy estimator. $$\begin{aligned} \label{eq:cross} \hat{H}(P) & = & \inf_\Phi\;H(P,G_\Phi)\end{aligned}$$ The gap between $\hat{H}(P)$ and $H(P)$ depends on the expressive power of the model class. The statistical limitations on distribution-free high-confidence lower bounds on entropy do not arise for cross-entropy upper bounds. For upper bounds we can show that naive sample estimates of the cross entropy loss produce meaningful (large entropy) results. We first define the cross-entropy estimator from a sample $S$. $$\hat{H}(S,G) = \frac{1}{|S|} \sum_{x \in S}\; -\ln\; G(x)$$ We can bound the loss of a model $G$ by ensuring a minimum probability $e^{-F_{\max}}$ where $F_{\max}$ is then the maximum possible log loss in the cross entropy objective. In language modeling a loss bound exists for any model that ultimately backs off to a uniform distribution on characters. Given a loss bound of $F_{\max}$ we have that $\hat{H}(S,G)$ is just the standard sample mean estimator of an expectation of a bounded variable. In this case we have the following standard confidence interval. For any population distribution $P$, and model distribution $G$ with $-ln\; G(x)$ bounded to the interval $[0,F_{\max}]$, with probability at least $1 -\delta$ over the draw of $S \sim P^N$ we have $$H(P,G) \in \hat{H}(S,G) \pm F_{\max}\sqrt{\frac{ \ln \frac{2}{\delta}}{2N}}$$ It is also possible to give PAC-Bayesian bounds on $H(P,G_\Phi)$ that take into account the fact that $G_\Phi$ is typically trained so as to minimize the empirical loss on the training data. The PAC-Bayesian bounds apply to“broad basin” losses and loss estimates such as the following. $$\begin{aligned} H_\sigma(S,G_\Phi) & = & E_{x \sim P} \;E_{\epsilon \sim N(0,\sigma I)}\; -\ln\; G_{\Phi + \epsilon}(x) \\ \hat{H}_\sigma(S,G_\Phi) & = & \frac{1}{|S|} \sum_{x \in S}\;E_{\epsilon \sim N(0,\sigma I)}\; -\ln\; G_{\Phi + \epsilon}(x)\end{aligned}$$ Under mild smoothness conditions on $G_\Phi(x)$ as a function of $\Phi$ we have $$\begin{aligned} \lim_{\sigma \rightarrow 0} \;H_\sigma(P,G_\Phi) & = & H(P,G_\Phi) \\ \lim_{\sigma \rightarrow 0} \;\hat{H}_\sigma(S,G_\Phi) & = & \hat{H}(S,G_\Phi)\end{aligned}$$ An L2 PAC-Bayesian generalization bound ([@PAC-Bayes]) gives that for any parameterized class of models and any bounded notion of loss, and any $\lambda > 1/2$ and $\sigma > 0$, with probability at least $1 - \delta$ over the draw of $S$ from $P^N$ we have the following simultaneously for all parameter vectors $\Phi$. $$H_\sigma(P,G_\Phi) \leq \frac{1}{1 - \frac{1}{2\lambda}}\left(\hat{H}_\sigma(S,G_\Phi) + \frac{\lambda F_{\max}}{N}\left(\frac{||\Phi||^2}{2\sigma^2} + \ln \frac{1}{\delta}\right)\right)$$ It is instructive to set $\lambda = 5$ in which case the bound becomes. $$H_\sigma(P,G_\Phi) \leq \frac{10}{9}\left(\hat{H}_\sigma(S,G_\Phi) + \frac{5 F_{\max}}{N}\left(\frac{||\Phi||^2}{2\sigma^2} + \ln \frac{1}{\delta}\right)\right)$$ While this bound is linear in $1/N$, and tighter in practice than square root bounds, note that there is a small residual gap when holding $\lambda$ fixed at 5 while taking $N \rightarrow \infty$. In practice the regularization parameter $\lambda$ can be tuned on holdout data. One point worth noting is the form of the dependence of the regularization coefficient on $F_{\max}$, $N$ and the basin parameter $\sigma$. It is also worth noting that the bound can be given in terms of “distance traveled” in parameter space from an initial (random) parameter setting $\Phi_0$. $$H_\sigma(P,G_\Phi) \leq \frac{10}{9}\left(\hat{H}_\sigma(S,G_\Phi) + \frac{5 F_{\max}}{N}\left(\frac{||\Phi-\Phi_0||^2}{2\sigma^2} + \ln \frac{1}{\delta}\right)\right)$$ Evidence is presented in ([@NonVacuous]) that the distance traveled bounds are tighter in practice than traditional L2 generalization bounds. MMI predictive Coding {#sec:coding} ===================== Recall that in MMI predictive coding we assume a population distribution on pairs $(x,y)$ where we think of $x$ as past raw sensory signals (images or sound waves) and $y$ as a future sensory signal. We then consider the problem of learning stochastic coding functions $C_x$ and $C_y$ that maximizes the mutual information $I(C_x(x),C_y(y))$ while limiting the entropies $H(C_x(x))$ and $H(C_y(y))$. Here we propose representing the mutual information as a difference of entropies. $$I(C_x(x),C_y(y)) = H(C_y(y)) - H(C_y(y)|C_x(x))$$ When the coding functions are parameterized by a function $\Psi$, the above quantities become a function of $\Psi$. We can then formulate the following nested optimization problem. $$\begin{aligned} \Psi^* & = & \operatorname*{argmax}_\Psi \; \hat{H}(C_y(y);\;\Psi) - \hat{H}(C_y(y)|C_x(x); \Psi) \\ \\ \hat{H}(C_y(y);\;\Psi) & = & \inf_\Theta \;H(C_y(y),G_\Theta;\;\Psi) \\ \\ \hat{H}(C_y(y)|C_x(x);\;\Psi) & = & \inf_\Phi \;H(C_y(y),G_\Phi|C_x(x);\;\Psi)\end{aligned}$$ The above quantities are expectations over the population distribution on pairs $(x,y)$. In practice we have only a finite sample form the population. But the preceding section presents theoretical evidence that, unlike lower bound estimators, upper bound cross-entropy estimators can meaningfully estimate large entropies from feasible samples. Conclusions =========== Maximum mutual information (MMI) predictive coding seems well motivated as a method of unsupervised pretraining of representations that maintain semantic signal while dropping uninformative noise. However, the maximization of mutual information is a difficult training objective. We have given theoretical arguments that representing mutual information as a difference of entropies, and estimating those entropies by minimizing cross entropy loss, is a more statistically justified approach than maximizing a lower bound on mutual information. Unfortunately cross entropy upper bounds on entropy fail to provide either upper or lower bounds on mutual information — mutual information is a difference of entropies. We cannot rule out the possible existence of superintelligent models, models beyond current expressive power, that dramatically reduce cross-entropy loss. Lower bounds on entropy can be viewed as proofs of the non-existence of superintelligence. We should not surprised that such proofs are infeasible. [18]{} \[1\][\#1]{} \[1\][`#1`]{} urlstyle \[1\][doi: \#1]{} Sanjeev Arora, Andrej Risteski, and Yi Zhang. Do gans learn the distribution? some theory and empirics. *ICLR*, 2018. Ishmael Belghazi, Sai Rajeswar, Aristide Baratin, R Devon Hjelm, and Aaron Courville. Mine: mutual information neural estimation. *arXiv preprint arXiv:1801.04062*, 2018. Anthony J Bell and Terrence J Sejnowski. An information-maximization approach to blind separation and blind deconvolution. *Neural computation*, 70 (6):0 1129–1159, 1995. M. Donsker and S. Varadhan. Asymptotic evaluation of certain markov process expectations for large time, iv. *Communications on Pure and Applied Mathematics*, 360 (2):0 183–212, 1983. Gintare Karolina Dziugaite and Daniel M. Roy. Computing nonvacuous generalization bounds for deep (stochastic) neural networks with many more parameters than training data. *arXiv preprint arXiv:1703.11008*, 2017. Shuyang Gao, Greg Ver Steeg, and Aram Galstyan. Efficient estimation of mutual information for strongly dependent variables. *arXiv preprint arXiv:1411.2003*, 2014. R Devon Hjelm, Alex Fedorov, Samuel Lavoie-Marchildon, Karan Grewal, Adam Trischler, and Yoshua Bengio. Learning deep representations by mutual information estimation and maximization. *arXiv preprint arXiv:1808.06670*, 2018. Alexander Kraskov, Harald Stoegbaue, and Peter Grassberger. Estimating mutual information. *arXiv preprint arXiv:cond-mat/0305641*, 2003. Ralph Linsker. Self-organization in a perceptual network. *Computer*, 210 (3):0 105–117, 1988. Charles Marsh. Introduction to continuous entropy. www.crmarsh.com/static/pdf/Charles\_Marsh\_Continuous\_Entropy.pdf, 2013. David McAllester. A pac-bayesian tutorial with a dropout bound. *arXiv preprint arXiv:1307:2118*, 2013. David [McAllester]{}. . *arXiv preprint arXiv:1802.07572*, 2018. David McAllester and Robert Schapire. On the convergence rate of good-turing estimators. *COLT*, 2000. Aaron van den Oord, Yazhe Li, and Oriol Vinyals. Representation learning with contrastive predictive coding. *arXiv preprint arXiv:1807.03748*, 2018. Alon Orlitsky and Ananda Theertha Suresh. Competitive distribution estimation: Why is good-turing good. *NIPS*, 2015. Alon Orlitsky, Narayana Santhanam, and Junan Zhang1. Always good turing: Asymptotically optimal probability estimation. *Science*, 3020 (5644), 2003. Karl Stratos. Mutual information maximization for simple and accurate part-of-speech induction. *arXiv preprint arXiv:1804.07849*, 2018. Naftali Tishby, Fernando C Pereira, and William Bialek. The information bottleneck method. *arXiv preprint physics/0004057*, 2000.

--- abstract: 'The potential weakness of the Y-00 direct encryption protocol when the encryption box ENC is not chosen properly is demonstrated in a fast correlation attack by S. Donnet et al in Phys. Lett. A 356 (2006) 406-410. In this paper, we show how this weakness can be eliminated with a proper design of ENC. In particular, we present a Y-00 configuration that is more secure than AES under known-plaintext attack. It is also shown that under any ciphertext-only attack, full information-theoretic security on the Y-00 seed key is obtained for any ENC when proper deliberate signal randomization is employed.' author: - | Horace P. Yuen [^1] and Ranjith Nair\ Center for Photonic Communication and Computing\ Department of Electrical and Computer Engineering\ Department of Physics and Astronomy\ Northwestern University, Evanston, IL 60208 title: 'On the Security of Y-00 under Fast Correlation and Other Attacks on the Key' --- Introduction ============ The quantum-noise based direct encryption protocol Y-00, called $\alpha\eta$ in our earlier papers \[1-6\], was repeatedly misrepresented in previous criticisms, but that situation has apparently changed with our recent papers \[7-9\]. For the first time, a meaningful attack on Y-00 type protocols beyond exhaustive search has been developed in [@donnet]. A fast correlation attack (FCA) was presented that was shown to succeed by simulations for moderate signal levels when the ENC box in Y-00 is a LFSR (linear feedback shift register) of a few taps and length up to 32. Even though such Y-00 is already insecure against what we call assisted brute-force search [@nair06] due to the small seed key size $|K| \leq 32$, such FCA is of interest as it brings forth the whole issue of Y-00 seed key security against similar and other attacks. The attack in [@donnet] is geared toward only the experiment reported in [@optlett03]. We have emphasized all along [@yuen04; @ptl05; @spie05] that the use of LFSR in the reported experiments was just for proof of principle demonstration, that the ENC box must be chosen appropriately in a final design, and that other techniques need to be deployed for proper security. To quote from [@spie05], “Similar to encryption based on nonlinearly combining the LFSR’s, Eve can launch a correlation attack using the following strategy: $\ldots$ many of the LFSR’s could be trivially attacked.” Thus, we were aware of the possible weakness of some ENC and in particular of FCA type attacks. Indeed, Hirota and Kurosawa [@hirota06] have already desribed a counter-measure to FCA via a “keyed mapper”, the incorporation of which in ASK-signal Y-00 [@hirota05] has been developed and is being tested. Generally speaking, it is important to study LFSR-based Y-00 despite its possible weakness, because LFSR is a practically convenient choice in various applications similar to the situation in standard cryptography. In this paper, we first briefly describe general attacks on the Y-00 seed key as a problem of decoding in real noise – a viewpoint which includes all FCA’s. For both ciphertext-only attacks (CTA) and known-plaintext attacks (KPA), we show that Y-00 may be considered as a classical stream cipher, the ENC box, with real physical noise added on top. We comment on the possible defenses involving just a properly chosen LFSR, or an added keyed mapper, or with a keyed connection polynomial for the LFSR. We describe an AES-based Y-00 that is more secure against KPA than AES (Advanced Encryption Standard) alone, in the sense that if it is broken then AES is also broken but not the other way around. The practical security advantage of such AES-based Y-00 will be indicated. Finally, for CTA, we show that Deliberate Signal Randomization (DSR) introduced in [@yuen04] provides full information-theoretic security on the Y-00 seed key for any ENC. We hope that these results would establish beyond doubt that Y-00 is an important cryptosystem to consider in theory and in practice. Attacks on Y-00 seed key ======================== Consider the original quantum-noise randomized cipher Y-00 [@prl; @yuen04] as depicted in Fig. 1. Alice encodes each data bit into a $2M-$ary phase-shifted coherent state in a qumode of energy $\alpha^2_0$. A seed key $K$ of bit length $|K|$ is used to drive a conventional stream cipher ENC to produce a running key $K'$ that is used to determine, for each qumode carrying the bit, which pair of antipodal coherent states, referred to as a basis, is to be used as a binary phase-shift keying (BPSK) signal set for Bob. With a synchronous ENC at the receiver, Bob discriminates the BPSK signals for each qumode by an appropriate receiver. With a differential (DPSK) implementation [@prl; @yuen04; @optlett03; @ptl05; @pra05; @spie05], there is no need to phase lock between Alice and Bob as is true in ordinary communications. \[htbp\] The optimum quantum receiver performance for both Bob and Eve is the same as in the non-differential case in principle, the differential implementation being a practical convenience. Even with a full copy of the quantum state granted to Eve in our KCQ approach of performance analysis [@yuen04; @pla05; @yuen05qph; @nair06], security on the data is nearly perfect when the seed key induced correlation is neglected [@prl]. Generally, it is a horrendous problem with yet no solution for meaningfully quantifying the data security of a symmetric-key cipher. In current practice, it is assumed that CTA on the data is not a problem if $|K|$ is “large”, and attention is focused on KPA on the key. For conventional or standard [@yuen05qph; @nair06] ciphers, the key is usually completely protected from CTA for uniformly random data. This is, however, not the case for the bare Y-00 [@yuen04; @pla05; @yuen05qph; @nair06]. In this paper, we address both CTA and KPA on the Y-00 seed key, the (classical) ciphertext being obtained from some quantum measurement on the qumodes assumed to be in Eve’s possession. It is seen from Fig. 1 that a CTA or KPA on the Y-00 seed key is equivalent to the corresponding attack on the standard stream cipher ENC with its output stream observed in noise resulting from the coherent state randomization of the signal phase. Thus, it is equivalent to a CTA or KPA on the ENC alone as a stream cipher but with noise on top. The connection between the running key bits $K'$ and the basis, called the “*mapper*” [@pra05; @spie05; @hirota06], a crucial component of Y-00, and the noise effect on $K'$ are described in [@spie05; @donnet]. In a FCA on a conventional stream cipher composed of, say, a nonlinear combination of the outputs of a bank of $m$ LFSR’s, one focuses on one LFSR $L_i$ at a time and looks for correlation between the final stream cipher output $K'$ and the output $k'_i$ of $L_i$. Thus, even though the complete cipher is nonrandom, $K'$ constitutes a noisy observation of $k'_i$ from which a good estimate of $k'_i$ may perhaps be obtained. Such a divide-and-conquer strategy can be repeated to yield all the keys $k_i$ for each $L_i$. For Y-00, there is real noise from the coherent states, but a similar FCA can be launched if there is a significant correlation between $K'$ and the observed $2M$-ary signal, as obtained, say, by heterodyning. In general, attack on the Y-00 seed key is *exactly* a decoding problem on a memoryless channel for both CTA and KPA. This can be seen by regarding the seed key as information bits and the observed sequence of $2M$-ary signals translated by the mapper to $K'$ as the codeword, with independent coherent-state noise for each qumode so that the memoryless channel alphabet has size $\log_2 2M$ in a CTA and $\log_2 M$ in a KPA. Note that this code from ENC, as in the case of AES, could be nonlinear with no useful linear approximation, making linear decoding not a viable attack. It is not known whether information-theoretic security may be obtained in Y-00 for a properly designed ENC, i.e., whether a (decoding) algorithm may be found that would succeed in determining the seed key with some nonvanishing probability [@yuen05qph; @nair06]. And there is the further question, if such an algorithm exists, of its complexity as the general syndrome decoding of even a linear code is exponential. In contrast, for KPA on standard “nondegenerate” nonrandom ciphers, the key is actually uniquely determined at a bit length $n_1=n_d$, the nondegeneracy distance [@yuen05qph; @nair06] which is often not very long. Thus, such cipher has no information-theoretic security against KPA, although there is still the problem of attack complexity in finding $K$ that may allow complexity-based security which can be practically as good as information-theoretic security [@yuen05qph]. The key point in this connection is that randomization introduces real noise that is otherwise absent in a nonrandom cipher, signifying its role in adding security to KPA. For standard stream ciphers built upon LFSR’s, the class of FCA described above is powerful enough to break them for sufficiently long observed length $N$ of the output. However, the complexity of all known FCA algorithms is exponential in either the memory needed or the number $t$ of tap coefficients in the LFSR \[13\]. Thus, practically there are LFSR-based stream ciphers that are not broken by any known attack when the LFSR length $|K|$ and $t$ are sufficiently large. Shorter LFSR’s or ones with long $|K|$ and with few taps are more convenient and cheaper to use in practice, but are vulnerable to computationally intensive but feasible attacks. If such LFSR is used in the ENC in Y-00, the cipher becomes vulnerable even for moderate signal level if long enough $N$ is employed when that does not lead to an undue increase in memory required. For the $|K|=32$ single LFSR case reported in [@donnet], only $N=1500$ is needed in a CTA to undermine the system at the signal level $\alpha^2_0 \sim 1.5\times 10^4$, roughly the numbers used in [@optlett03]. The convolutional-code based algorithm chosen in [@donnet] is not suited to attacking long $|K|$ LFSR with a few taps, and thus would not be able to break the $|K|=4400$ and $t=3$ LFSR used in our system in [@pra05; @corndorf04]. However, a different FCA would no doubt be able to break that system, such as those designed for small $t$. Defenses Against Fast Correlation Attacks ========================================= We have already observed that one may use practical LFSR that resists known FCA in the ENC of Y-00. There are many other ways to defeat such and even more general attacks on the Y-00 seed key, as we will discuss in the rest of this paper. First, a properly designed deterministic mapper that determines the $2M-$ary signal from the running key $K'$ would spread the noise into the different bit positions of $K'(m)$, increasing the minimum complexity of attacks. The mapper may also be keyed, e.g., the mapper function may be chosen for each qumode from the running key $K_m'$ from another ${ENC}_m$ with another seed key $K_m$. This results in a product cipher of ENC in noise and $ENC_m$, for which no obvious modification of the FCA can be made that does not involve exponential search over $K_m$. In particular, one cannot plot Fig. 3 in [@donnet] which is the basic starting point of their attack. This defense has already been proposed [@hirota06], although there is “correlation immunity” for such ciphers only under an approximation. Secondly, the connection polynomial in the LFSR can be keyed, i.e., chosen randomly from an exponential number of possibilities. The known FCA’s on LFSR all require knowledge of the LFSR connection polynomial. In a future paper, we will present information-theoretic analysis of the effect of a keyed connection polynomial. Such ciphers can clearly be implemented in software, and to a considerable extent in hardware with field programmable logic, thus retaining much of the convenience of LFSR in practical applications. We do not believe information-theoretic security can be obtained this way, but it may greatly increase the complexity of at least FCA type attacks, thus providing useful practical security in some situations. Thirdly, we now give an ENC design for Y-00 that leads to exponential complexity for CTA according to current knowledge, and more security that AES for KPA generally. Consider the ENC of Fig. 2 where a bank of $m$ parallel AES in a stream cipher mode is used to provide the $m=\log_2 M$ bits running key segment $K'(m)$ which determines, through the mapper, the basis of a qumode. Typically in our previous experimental demonstrations, $m \sim 10$ and $|K|$ is in the thousands. Thus each $K_i$ may be readily chosen to be of 256 bits. Under heterodyne or any other quantum measurement by Eve, the result is a noisy version of $K'(m)$ with independent coherent-state induced randomization for each qumode. According to the present state of knowledge, no KPA on AES is better than exhaustive (exponential) search [@stinson]. Even in a divide-and-conquer type attack as in FCA, so that a single AES is to be considered, one needs to deal with the KPA problem of artifical noise from such strategy with the addition of *real* coherent-state noise, in a CTA on the Y-00 seed key. Let $N_1$ be the length of the qumode sequence used for the attack, so that Eve may parallelize $N/{N_1}$ attacks simultaneously from the total length $N$. It is clear that even without noise, the attack complexity remains exponential for any realistic $N \leq 2^{80}$ and any $N_1$. In a KPA, the comparison is to be made with the same $N_1$ for no parallelization. Thus, Y-00 is equivalent to AES in a stream cipher mode with output observed in noise, thus harder than AES alone which does not have the decoding in noise problem. In particular, it is easily seen that if the Y-00 in the configuration of Fig. 2 can be broken, then each $AES_i$ itself can be broken. \[htbp\] [ {width="4.5in" height="2in"}]{} The question arises as to what constitutes a fair comparison between a given stream cipher ENC versus Y-00 on top of ENC. A different configuration was given for ENC in [@nair06], where a single classical stream cipher (say AES) is used without parallelization but is adjusted to give the same clock rate for encrypting each data bit. The present scheme appears simpler in principle and more secure in practice when AES is used in ENC, because the functionality of multiple AES in parallel cannot be replaced by a single AES. However, with such parallelization for maintaining the same clock rate as AES (or ENC alone), the question arises as to whether the added security from Y-00 can be obtained from, say, nonlinearly combining the parallel AES’s. This question cannot be answered until security is precisely defined and quantified. However, it may be observed in this connection that there is no known attack developed for AES observed *in noise*, and the intrinsic nonlinearity of AES renders all known decoding attacks inapplicable. The major qualitative advantage of Y-00 [@yuen05qph; @nair06] compared to a standard nonrandom cipher is that the quantum noise automatically provides high speed true randomization not available otherwise, thus giving it a different kind of protection from nonrandom ciphers. Furthermore, one has to attack such physics-based cryptosystem at the communication line with physical (measurement) equipment, which is not available to everyone at every place, whereas one only needs to sit at a computer terminal to attack conventional ciphers. In this connection, it may be mentioned that the high rate heterodyne attack needed on Y-00 is currently not quite technologically feasible, though it may be in the not-too-far future. Y-00 can be employed to realize these benefits not available otherwise. However, if it is intrinsically less secure than conventional ciphers, its utility would be in serious doubt. The configuration of Fig. 2 shows this is not the case – it can in fact be more secure that ENC or $AES_i$ by itself. There is also no known attack applicable to AES in noise. Deliberate Signal Randomization =============================== In contrast to a nondegenerate nonrandom classical cipher for which the key is completely protected in the information-theoretic sense against CTA when the data is uniformly random [@yuen05qph; @nair06], there is little distinction between CTA and KPA for the bare Y-00. Only a factor of 2 in the per qumode alphabet size is obtained in KPA versus CTA as indicated above, and expounded in [@nair06]. The question arises as to whether full information-theoretic security against CTA can be restored by modifying the bare Y-00. The authors of [@donnet] appear to be pessimistic on the possibility of achieving this. To quote: “While randomization methods might increase the security level, it remains to be seen if they will provide perfect secrecy.” In the following, we show how this is possible with Deliberate Signal Randomization (DSR) independently of the mechanism of running key generation. The reason why the seed key cannot be attacked in CTA is clear for an additive stream cipher with uniformly random data. The “channel” between the seed key and the output observation has zero capacity due to the data which acts as random noise. In particular, it is clear that no FCA can be launched. The coherent-state noise in Y-00 is not big enough for high signal level to produce a similar effect. However, further randomization may in principle be produced to achieve this end, both classically and quantum mechanically. Since the coherent-state noise in Y-00 can in principle be replaced, in an equivalent classical system, by deliberate randomization of the classical signal from Alice as we have repeatedly emphasized [@yuen04; @pla05; @yuen05qph; @nair06], we first consider this classical situation. Let $\theta_s$ be the signal point on the circle of Fig. 1, $x$ the data bit, $k'$ the running key segment that determines the basis. Before deliberate or noise randomization, $\theta_s(x,k')$ is uniquely determined by $x$ and $k'$. From $\theta_s$ one randomizes it to $\theta_r$ according to a probability density $p(\theta_r|\theta_s)$. We use continuous $\theta$’s here but the argument is identical for discrete $\theta$’s. More generally, let $\theta$ be Eve’s observed signal point, so that $\theta=\theta_r$ in a classical noiseless system with deliberate randomization. Then, $$\label{pdf} p(\theta|x,k')=\int p(\theta|\theta_r)p(\theta_r|\theta_s(x,k'))\mathrm{d} \theta_r.$$ In the classical noiseless case with just signal randomization, $p(\theta|\theta_r)=\delta(\theta-\theta_r)$, the BPSK signal may be correctly discriminated when the observed $\theta$ falls within the half-circle centred around $\theta_s$. Thus we pick $p(\theta_r|\theta_s)$ to be the uniform distribution on the half-circle with midpoint $\theta_s$. If $x$ is uniformly random, then from (1) $$\label{pdf2} p(\theta|k')=\frac{1}{2} \sum_{x=0,1} p(\theta|x,k')$$ is the uniform distribution on the full circle independent of $k'$. This proves the observation of $\theta$ to Eve yields no information at all on $k'$. In other words, Eve’s channel on $k'$ has zero capacity from DSR and uniformly random data which acts as added noise unknown to her, similar to a nondegenerate nonrandom stream cipher. For coherent-state noise described in the wedge approximation [@pla05; @nair06], whereupon a heterodyne or phase measurement the observed $\theta$ is taken to be uniformly distributed within a standard deviation around $\theta_r$ and zero outside, the same $k'-$independence for $p(\theta|k')$ obtains when $\theta_r$ is chosen in a discrete number of positions for given $\theta_s$ so that $p(\theta_r|\theta_s)$ fills out a uniform half-circle again. We have assumed an integral number of wedges would do this, which can be guaranteed by choice of the signal level $\alpha_0$. Going beyond the wedge approximation, one needs to determine the function $p(\theta_r|\theta_s)$ in (1) for a coherent state/fixed measurement $p(\theta|\theta_r)$ so that $p(\theta|x,k')$ is uniformly distributed in a half-circle, where $p(\theta|\theta_r)$ is obtained from Eve’s optimal individual qumode quantum measurement. In this case, there is the problem that the resulting error probability for Bob may be higher than the designed level even with knowledge of the seed key $K$. In principle, this problem can be handled in one of two different ways without affecting the data security as measured by the Shannon limit [@yuen05qph; @nair06]. First, one may increase $S$ and correspondingly $M$ while maintaining the same Y-00 random cipher characteristic $\Gamma = M/{\pi\sqrt{S}}$ defined in [@nair06]. Doing so will make the tail of the probability distribution that causes Bob’s error arbitrarily small. Indeed, in the classical limit $S \rightarrow \infty, M \rightarrow \infty, M/{\sqrt{S}} \rightarrow \pi\Gamma$, a constant, the error vanishes. A second way is to employ an error correcting code for Bob and randomize the entire codeword of $n$-bits in a correlated fashion in the signal space $\mathcal{C}^n$, where $\mathcal{C}$ is the coherent-state circle in $\mathbb{R}^2$. This is done by moving the $n$-bit codewords within mutually exclusive but jointly exhaustive regions that fill the entire signal space $\mathcal{C}^n$, similar to the filling of the circle $\mathcal{C}$ in the one-bit case. Detailed quantitative treatment of these will appear elsewhere. Note that Y-00 is only a random cipher for a given quantum measurement, it is not a random quantum cipher. See [@nair06]. A convenient way to make it a quantum random cipher is to randomize the parameter $\theta_s$ to $\theta_r$ that determines the quantum state $\rho(\theta_r)$ to be transmitted. The resulting output state is then, analogous to (1), $$\label{state} \rho(x,k')=\int \rho(\theta_r)p(\theta_r|\theta_s(x,k'))\mathrm{d}\theta_r.$$ It may be seen from (3) that by uniformly randomizing $\theta_s$ as above, for any state modulation $\rho(\theta_r)$, the output quantum state itself is independent of $k'$ upon averaging over $x$ as before. Thus, such quantum DSR would protect the key against CTA with the most general joint (quantum measurement) attack. In this case, there is generally a larger probability of error that Bob would decide on $x$ incorrectly as compared to no DSR, similar to the specific coherent state case under heterodyne attack. One of the above two approaches in the fixed measurement case can be similarly employed to bring the error down to any desired level. It may be noted that the deployment of full DSR just described above is practically difficult at present if only because high speed random numbers are needed. On the other hand, it may be possible to delve into the qumode sequence to take advantage of the randomization inherent in such sequence for selected deliberate randomization while providing essentially the same overall result. Detailed treatment of concrete DSR on Y-00 will be given elsewhere. Conclusion ========== We have shown that Y-00 can be designed to be secure against fast correlation attacks including that of ref. [@donnet], and that it can be configured to be more secure than AES while retaining the same high speed and its advantage as a physics-based cipher. We also prove the full information-theoretic security of Y-00 with proper deliberate signal randomization against ciphertext-only attacks. Quantitative security against known-plaintext attacks, as in the case of conventional ciphers, is a difficult, open, and important area of research. Acknowledgements ================ We would like to thank E. Corndorf, G. Kanter, P. Kumar, and C. Liang for useful discussions. This work has been supported by DARPA under grant F30602-01-2-0528 and AFOSR grant FA9550-06-1-0452. [10]{} url \#1[`#1`]{}urlprefix G. Barbosa, E. Corndorf, P. Kumar, H.P. Yuen, “Secure communication using mesoscopic coherent states”, Phys. Rev. Lett. 90 (2003) 227901. H.P. Yuen, “<span style="font-variant:small-caps;">KCQ</span>: A new approach to quantum cryptography <span style="font-variant:small-caps;">I</span>. <span style="font-variant:small-caps;">G</span>eneral principles and qumode key generation”, quant-ph/0311061. E. Corndorf, G. Barbosa, C. Liang, H. Yuen, P. Kumar, “High-speed data encryption over 25km of fiber by two-mode coherent-state quantum cryptography”, Opt. Lett. 28, 2040-2042, 2003. C. Liang, G.S. Kanter, E. Corndorf, and P. Kumar, “Quantum noise protected data encryption in a WDM network”, Photonics Tech. Lett. 17, pp. 1573-1575, 2005. E. Corndorf, C. Liang, G.S. Kanter, P. Kumar, and H.P. Yuen, “Quantum-noise–protected data encryption for WDM fiber-optic networks”, Phys. Rev. A 71 (2005) p. 062326. G.S. Kanter, E. Corndorf, C. Liang, V.S. Grigoryan, and P. Kumar, in *Fluctuation and Noise in Photonics and Quantum Optics III*, ed. P.R. Hemmer etc., Proc. of SPIE vol. 58 42 (SPIE, Bellingham, WA, 2005), pp. 74-86. H.P. Yuen, P. Kumar, E. Corndorf, R. Nair, “Comment on ‘How much security does Y-00 protocol provide us?’”, Phys. Lett. A, 346 (2005) 1-6; quant-ph/0407067. H.P. Yuen, R. Nair, E. Corndorf, G.S. Kanter, P. Kumar, To appear in *Quantum Information & Computation* Vol. 6 No. 7 (Nov 2006) 561-582; quant-ph/0509091 v. 3. R. Nair, H.P. Yuen, E. Corndorf, T. Eguchi, P. Kumar, “Quantum Noise Randomized Ciphers”, quant-ph/0603263 v. 5; To appear in Phys. Rev. A. S. Donnet, A. Thangaraj, M. Bloch, J. Cussey, J-M. Merolla, L. Larger, Phys. Lett. A, 356 (2006) 406-410. O. Hirota, K. Kurosawa, quant-ph/0604036; to appear in *Quant. Info. Proc.*. O. Hirota, M. Sohma, M. Fuse, and K. Kato, ‘Quantum stream cipher by Yuen 2000 protocol: Design and experiment by intensity modulation scheme’’, Phys. Rev. A. 72 (2005) 022335; quant-ph/0507043. F. Jonsson, Ph.D. Thesis, Lund University, Sweden, 2002; Available online at [www.pcc.lth.se/PrimePub/primepub.asp?AemneID=572&SpraakID=1&RotID=1](www.pcc.lth.se/PrimePub/primepub.asp?AemneID=572&SpraakID=1&RotID=1) E. Corndorf, G. Kanter, C. Liang, and P. Kumar, ‘Quantum-noise protected data encryption for WDM networks,’ in 2004 Conference on Lasers Electro Optics (CLEO’04), San Francisco, CA, Postdeadline CPDD8. D.R. Stinson, *Cryptography: Theory and Practice*, Chapman and Hall/CRC, 3rd ed, 2006. [^1]: yuen@ece.northwestern.edu

--- abstract: 'Deep neural language models such as BERT have enabled substantial recent advances in many natural language processing tasks. Due to the effort and computational cost involved in their pre-training, language-specific models are typically introduced only for a small number of high-resource languages such as English. While multilingual models covering large numbers of languages are available, recent work suggests monolingual training can produce better models, and our understanding of the tradeoffs between mono- and multilingual training is incomplete. In this paper, we introduce a simple, fully automated pipeline for creating language-specific BERT models from Wikipedia data and introduce 42 new such models, most for languages up to now lacking dedicated deep neural language models. We assess the merits of these models using the state-of-the-art UDify parser on Universal Dependencies data, contrasting performance with results using the multilingual BERT model. We find that UDify using WikiBERT models outperforms the parser using mBERT on average, with the language-specific models showing substantially improved performance for some languages, yet limited improvement or a decrease in performance for others. We also present preliminary results as first steps toward an understanding of the conditions under which language-specific models are most beneficial. All of the methods and models introduced in this work are available under open licenses from <https://github.com/turkunlp/wikibert>.' author: - | Sampo Pyysalo Jenna Kanerva Antti Virtanen Filip Ginter\ TurkuNLP group, Department of Future Technologies\ University of Turku, Finland\ `first.last@utu.fi` bibliography: - 'main.bib' title: 'WikiBERT models: deep transfer learning for many languages' --- Introduction {#intro} ============ Transfer learning using language models pre-trained on large unannotated corpora has allowed for substantial recent advances at a broad range of natural language processing (NLP) tasks. By contrast to earlier context-independent approaches such as word2vec [@mikolov2013efficient] and GloVe [@pennington2014glove], models such as ULMFiT [@howard2018universal], ELMo [@peters2018deep], GPT [@radford2018improving] and BERT [@devlin2018bert] create contextualized representations of meaning, capable of providing both contextualized word embeddings as well as embeddings for longer text segments than words. Recent pre-trained language models has been rapidly advancing the state of the art in a range of natural language understanding tasks [@wang2018glue; @wang2019superglue] as well as established NLP tasks such as named entity recognition and syntactic analysis [@martin2020camembert; @virtanen2019multilingual]. The transformer architecture [@vaswani2017attention] and the BERT language model of have been particularly influential, with transformer-based models in general and BERT in particular fuelling a broad range of advances in natural language processing tasks over the recent years. However, most recent work introducing new deep neural language models has focused on English, with models for other languages released later, if at all. For BERT, the original study introducing the model [@devlin2018bert] addressed only English, and Google later released a Chinese model as well as a multilingual model, mBERT,[^1] trained on text from 104 languages. A range of language-specific BERT models have since been created by various groups, for example BERTje[^2] [@de2019bertje], CamemBERT[^3] [@martin2020camembert], FinBERT[^4] [@virtanen2019multilingual], and RuBERT[^5] [@kuratov2019adaptation], demonstrating substantial improvements over the multilingual model in various language-specific downstream task evaluations. However, these efforts have so far not added up to a broad-coverage collection of consistent-quality language-specific deep transfer learning models, and we are not aware of previous efforts to introduce readily executable pipelines for creating data for pre-training deep transfer learning models. Here, we take steps towards addressing these issues by introducing both a simple, fully automated pipeline for creating language-specific BERT models from Wikipedia data as well as 42 new such models. Data ==== We next introduce the sources of unannotated data used for pre-training and annotations used for prepreprocessing and evaluation in our work. Pre-training data ----------------- The English Wikipedia was the main source of text for pre-training the original English BERT models, accounting for three-fourths of its pre-training data.[^6] The multilingual BERT models were likewise trained on Wikipedia data. To roughly mirror the original BERT pre-training data selection, we chose to pre-train our models exclusively on Wikipedias in various languages. As of this writing, the List of Wikipedias[^7] identifies Wikipedias in 309 languages. Their sizes vary widely: while the largest of the set, the English Wikipedia, contains over six million articles, the smaller half of Wikipedias (155 languages) put together only total approximately 400,000 articles. As the BERT base model has over 100 million parameters and BERT models are frequently trained on billions of words of unannotated text, it seems safe to estimate that attempting to train BERT for e.g. Old Church Slavonic, ranked 272nd with fewer than 1000 articles (under 50,000 tokens), would likely not produce a very successful model. It is nevertheless not well established how much unannotated text is required to pre-train a language-specific model, and how much the domain and quality of the pre-training data affect the model performance. An evaluation carried out by on controlling the size and text sources of the English pre-training dataset suggests that a larger pre-training dataset does not always yield better performance on downstream tasks, and even though the pure Wikipedia data rarely achieves state-of-the-art downstream performance, it gives a competitive baseline performance. However, as previously stated one must keep in mind here that the English Wikipedia is considerably larger than Wikipedias for most other languages. In order to focus our computational resources as well as best support the community, we have so far opted to exclude dead languages, i.e. languages that are not in everyday spoken use by any community, from our model pre-training pipeline. We have thus not created models for Ancient Greek, Coptic, Gothic, Latin, Old Church Slavonic, and Old French. Other than this exclusion, we have broadly proceeded to introduce preprocessing support and models for languages in decreasing order of the size of their Wikipedias an support in Universal Dependencies, discussed below. Universal Dependencies ---------------------- The Universal Dependencies (UD) is a community lead effort seeking to create cross-linguistically consistent treebank annotations for many typologically different languages. [@nivre2016universal] As of this writing, the latest release of the UD treebanks[^8] is v2.6, which includes 163 treebanks covering 92 languages. To maintain comparability with recent work on UD parsing, most importantly the study introducing the UDify parser [@kondratyuk2019], we here use the UD v2.3 treebanks[^9], with 129 treebanks in 76 languages. When assessing the WikiBERT models, we limit our evaluation to the subset of UD v2.3 treebanks that have training, development, and test sets, thus excluding e.g. the 17 parallel UD treebanks which only provide test sets. We further exclude from evaluation treebanks released without text, namely `ar_nyuad`, `fr_ftb`, `ja_bccwj` as well as the Swedish sign language treebank `swl_sslc`. Finally, we exclude `mr_ufal` `mt_mudt`, `te_mtg` and `ug_udt` as we currently do not have dedicated BERT models for these languages. Methods ======= We next briefly introduce the primary steps of the preprocessing pipeline for creating pre-training examples from Wikipedia source as well as the tools used for text processing, model pre-training, and evaluation. Preprocessing pipeline ---------------------- In order to create good quality data from raw Wikipedia dumps in the format required by BERT model training, we introduce a pipeline that performs the following primary steps: #### Data and model download The full Wikipedia database backup dump is downloaded from a mirror site[^10] and a UDPipe model for the language from the LINDAT/CLARIN repository.[^11] #### Plain text extraction WikiExtractor[^12] is used to extract plain text with document boundaries from the Wikipedia XML dump. #### Segmentation and tokenization UDPipe is used with the downloaded model to segment sentences and tokenize the plain text, producing text with document, sentence, and word boundaries. #### Document filtering A set of heuristic rules and statistical language detection[^13] are applied to optionally filter documents based on configurable criteria. #### Sampling and basic tokenization A sample of sentences is tokenized using BERT basic tokenization to produce examples for vocabulary generation that match BERT tokenization criteria. #### Vocabulary generation A subword vocabulary is generated using the SentencePiece[^14] [@kudo2018sentencepiece] implementation of byte-pair encoding [@gage1994new; @sennrich2015neural]. After generation the vocabulary is converted to the BERT WordPiece format. #### Example generation Masked language modeling and next sentence prediction examples using the full BERT tokenization specified by the generated vocabulary are created in the TensorFlow TFRecord format using BERT tools. The created vocabulary and pre-training examples can be used directly with the original BERT implementation to train new language-specific models. UDPipe ------ UDPipe [@straka2016udpipe] is a parser capable of producing segmentation, part-of-speech and morphological tags, lemmas and dependency trees. In this work we use UDPipe for sentence segmentation and tokenization. The segmentation component in UDPipe is a character-level bidirectional GRU network simultaneously predicting the end-of-token and end-of-sentence markers. Pre-training ------------ We aimed to largely mirror the original BERT process in our selection of parameters and setting for the pre-training process to create the WikiBERT models, with some adjustments made to account for differences in computational resources. Specifically, while the original BERT models were trained on TPUs, we trained on Nvidia Volta V100 GPUs with 32GB memory. We followed the original BERT processing in training for a total of 1M steps in two stages, the first 900K steps with a maximum sequence length of 128, and the last 100K steps with a maximum of 512. Due to memory limitations, each model was trained on 4 GPUs using a batch size of 140 during the 128 sequence length phase, and 8 GPUs with a batch size of 20 during the 512 phase. UDify ----- To evaluate the models, we apply the UDify parser [@kondratyuk2019] trained on Universal Dependencies data. UDify is a state-of-the-art model and can predict UD part-of-speech tags, morphological features, lemmas, and dependency trees, allowing several aspects of the models’ capabilities to be assessed straightforwardly. UDify implements a multi-task learning objective using task-specific prediction layers on top of a pre-trained BERT encoder. All prediction layers are trained simultaneously, while also fine-tuning the pre-trained encoder weights. In the following evaluation, we focus on the parsin performance using the standard Labeled Attachment Score (LAS) metric. Results ======= [max width=]{} ----------------- -------- ----------- ----------- Language (code) Tokens mBERT WikiBERT Afrikaans (af) 24M **87.85** 87.33 Arabic (ar) 184M 83.81 **85.47** Belarusian (be) 34M **81.77** 79.81 Bulgarian (bg) 71M 92.30 **92.51** Catalan (ca) 236M **92.08** 92.06 Czech (cs) 143M 90.45 **90.69** Danish (da) 65M 85.78 **85.84** German (de) 1.0B 83.16 **84.13** Greek (el) 81M 91.63 **92.35** English (en) 2.7B **88.09** 88.05 Spanish (es) 678M **90.42** 90.12 Estonian (et) 38M 85.86 **87.43** Basque (eu) 45M 82.99 **83.70** Persian (fa) 95M 86.60 **88.60** Finnish (fi) 97M 87.64 **90.81** French (fr) 858M **89.22** 88.77 Galician (gl) 58M **83.05** 82.61 Hebrew (he) 166M 88.77 **90.17** Hindi (hi) 35M 91.59 **91.86** Croatian (hr) 54M **89.46** 89.40 Hungarian (hu) 129M 83.99 **86.21** ----------------- -------- ----------- ----------- : Summary of Wikipedia training data size (Tokens) and average LAS results for UDify for Universal Dependencies treebanks in each language with mBERT and WikiBERT initialization. []{data-label="tab:results"} ----------------- -------- ----------- ----------- Language (code) Tokens mBERT WikiBERT Indonesian (id) 93M **80.40** 80.12 Italian (it) 579M 89.64 **89.77** Japanese (ja) 596M 92.78 **92.92** Korean (ko) 79M 86.19 **87.28** Lithuanian (lt) 34M **58.68** 58.40 Latvian (lv) 21M 84.29 **84.46** Dutch (nl) 300M 90.26 **91.02** Norwegian (no) 112M 91.54 **91.94** Polish (pl) 282M 94.45 **95.58** Portuguese (pt) 326M 91.91 **92.21** Romanian (ro) 85M **86.83** 86.52 Russian (ru) 565M 90.35 **91.13** Slovak (sk) 39M 91.64 **91.73** Slovenian (sl) 42M 92.83 **93.37** Serbian (sr) 96M **92.30** 91.79 Swedish (sv) 364M 86.42 **87.12** Tamil (ta) 26M **70.14** 69.63 Turkish (tr) 71M 69.33 **71.25** Ukrainian (uk) 260M 88.57 **90.41** Urdu (ur) 18M **82.66** 82.15 Vietnamese (vi) 172M 66.89 **68.87** ----------------- -------- ----------- ----------- : Summary of Wikipedia training data size (Tokens) and average LAS results for UDify for Universal Dependencies treebanks in each language with mBERT and WikiBERT initialization. []{data-label="tab:results"} ![Average relative change in LAS when replacing mBERT with a WikiBERT model for UDify initialization plotted against the WikiBERT pre-training data size in tokens. Coloring indicates language grouping by genera (black = other).[]{data-label="fig:las-comparison"}](mBERT-vs-wBERT-LAS.png){width="\textwidth"} Table \[tab:results\] summarizes the evaluation results. We find a complex, mixed picture where mBERT and WikiBERT models each appear clearly superior for different languages, for example, mBERT for Belarusian and WikiBERT for Finnish. On average across the languages, UDify initialized with WikiBERT models slightly edges out mBERT initialization, with 86.1% average for mBERT and 86.6% for WikiBERT (an approximately 4% relative decrease in LAS error). However, such averaging hides more than it reveals, and it is much more interesting to consider the various potential impacts on performance from pre-training data size, potential support from close relatives in the same language family, and other similar factors. The various UD treebanks represent very different levels of challenge, with LAS results ranging from below 60% to above 95%. To reduce the impact of the properties of the treebanks on the comparison, in the following we focus on the relative change in performance when initializing UDify with a WikiBERT model compared to the baseline approach using mBERT. Figure \[fig:las-comparison\] shows the average relative change in performance over all treebanks for a language when replacing mBERT with the relevant WikiBERT model for UDify, plotted against the number of tokens in Wikipedia for the language. While the data is very noisy due to a number of factors, we find some indication of a “sweet spot” where training a dedicated language models tends to show most benefit over using the multilingual model when at least approximately 100M tokens but fewer than 1B tokens of pre-training data are available. We also briefly note some other properties in this data: - For English, a language in the large Germanic family and the language with the largest amount of pre-training data, mBERT and WikiBERT results are effectively identical. - The greatest loss when moving from mBERT to a WikiBERT model is seen for Belarusian, a slavic language closely related to Russian, for which considerably more training data is available. - The greatest gain when moving from mBERT to a WikiBERT model is seen for Finnish, a comparatively isolated language that nevertheless has a reasonably-sized Wikipedia. Observations such as these may suggest fruitful avenues for further research into the conditions under which mono- and multilingual language model training is expected to be most successful. Discussion ========== This short manuscript has provided a first brief introduction to the WikiBERT models, a collection of dedicated language-specific BERT models covering many languages that previously lacked a dedicated deep transfer learning model of this type. We demonstrated the value of these models compared to the multilingual BERT model through evaluation on the Universal Dependencies multilingual dependency parsing data, showing that a WikiBERT model will provide better performance than multilingual BERT on average, and in multiple cases providing a more than 10% relative decrease in LAS error compared to the multilingual model. The availability of the WikiBERT collection of models opens up a broad range of potential avenues for research into the strengths, weaknesses and challenges in both mono- and multilingual language modeling. Due to scheduling constraints, this initial manuscript must necessarily leave most such questions for future work. Acknowledgements {#acknowledgements .unnumbered} ================ We gratefully acknowledge the support of the Academy of Finland, and CSC — the Finnish IT Center for Science for providing computational resources for this effort. [^1]: <https://github.com/google-research/bert/blob/master/multilingual.md>. The document explicitly states “We do not plan to release more single-language models” [^2]: <https://github.com/wietsedv/bertje> [^3]: <https://camembert-model.fr/> [^4]: <https://turkunlp.org/FinBERT/> [^5]: <https://github.com/deepmipt/deeppavlov/> [^6]: The remaining quarter of BERT pre-training data was drawn from the BooksCorpus, a unique (and now unavailable) resource for which analogous corpora in other languages are very challenging to create. [^7]: <https://en.wikipedia.org/wiki/List_of_Wikipedias> [^8]: <https://universaldependencies.org/> [^9]: <http://hdl.handle.net/11234/1-2895> [^10]: <https://dumps.wikimedia.org/> [^11]: <http://hdl.handle.net/11234/1-3131> [^12]: <https://github.com/attardi/wikiextractor> [^13]: <https://github.com/shuyo/language-detection> [^14]: <https://github.com/google/sentencepiece>

--- abstract: 'Many kinds of algebraic structures have associated dual topological spaces, among others commutative rings with $1$ (this being the paradigmatic example), various kinds of lattices, boolean algebras, $C^*$-algebras, …. These associations are functorial, and hence algebraic endomorphisms of the structures give rise to continuous selfmappings of the dual spaces, which can enjoy various dynamical properties; one then asks about the algebraic counterparts of these properties. We address this question from the point of view of algebraic logic. The datum of a set of truth-values and a “conjunction” connective on them determines a propositional logic and an equational class of algebras. The algebras in the class have dual spaces, and the duals of endomorphisms of free algebras provide dynamical models for Frege deductions in the corresponding logic.' address: | Department of Mathematics\ University of Udine\ via delle Scienze 208\ 33100 Udine, Italy author: - Giovanni Panti title: | Dynamical properties\ of logical substitutions --- [^1] Introduction ============ Everybody knows the classical truth-tables $$\begin{array}{c|cc} \land & 0 & 1 \\ \hline 0 & 0 & 0 \\ 1 & 0 & 1 \end{array} \hspace{1cm} \begin{array}{c|cc} \lor & 0 & 1 \\ \hline 0 & 0 & 1 \\ 1 & 1 & 1 \end{array} \hspace{1cm} \begin{array}{c|cc} \to & 0 & 1 \\ \hline 0 & 1 & 1 \\ 1 & 0 & 1 \end{array} \hspace{1cm} \begin{array}{c|c} &\neg \\ \hline 0 & 1 \\ 1 & 0 \end{array}$$ and uses them automatically. The [*classical propositional calculus*]{} studies the set of formulas that, when evaluated according to the truth-tables, always assume value $1$. One proceeds as follows: 1. a [*formula*]{} is a polynomial built up from the propositional variables $x_i$ using the connectives $\land,\lor,\to,\neg,0,1$; 2. a [*valuation*]{} is a function $p$, distributing over the connectives, from the set of formulas to ${\{0,1\}}$; 3. a formula $r$ is [*true*]{} if $p(r)=1$ for every valuation $p$; 4. a formula $r$ is [*deducible*]{} if: - either is an element of a certain set $\Theta$ of basic axioms, - or there exists a formula $s$ such that $s$ and $s\to r$ are deducible, - or there exists a deducible formula $s$, propositional variables ${x_1,\ldots ,x_n}$, and formulas ${t_1,\ldots ,t_n}$, such that $r$ results from $s$ by substituting every $x_i$ that occurs in $s$ with the corresponding $t_i$; 5. the completeness theorem holds: a formula is true iff it is deducible. The completeness theorem relates a semantical notion “the statement $r$ holds, regardless of the state of affairs $p$” with a computational notion “the statement $r$ can be deduced from certain statements using certain rules”. There are several computational procedures for which the completeness theorem holds: the ones sketched in (4) are known as [*substitutional Frege systems*]{}, and are the strongest —in terms of minimizing the number of steps required to prove a true statement— available proof systems [@cookrec79]. The two rules (4b) of [*Modus Ponens*]{} and (4c) of [*substitution*]{} have different flavors. The first rule is, in some sense, statical: if something is known “locally”, i.e., concerns certain propositional variables, then conclusions are drawn involving the same variables. On the other hand, the substitution rule adds dynamics to the picture: local knowledge can be moved around. This is of course just a vague heuristic, and in the course of these notes we will give a precise formal ground to it. We will work at a level of generality broader than that of classical logic, enlarging the set of truth-values to include more than *true* and *false*; such logical systems are known as [*many-valued logics*]{}. Many-valued logic is an old discipline, going back to the twenties, and has recently been relived as a founding basis for fuzzy logic and fuzzy control; see [@hajek98], [@CignoliOttavianoMundici00], [@gottwald01] for detailed presentations and further references. The key ideas of this work are the following: given a set of truth-values $M\supseteq{\{0,1\}}$, we introduce on it an algebraic structure, determined by the choice of a truth-table for the conjunction connective. We then consider the class ${\mathbf{V}}M$ of all algebras that are generated by $M$ in the sense of Universal Algebra, and we functorially associate a dual topological space to each object in ${\mathbf{V}}M$. Algebraic endomorphisms of certain objects of ${\mathbf{V}}M$ (the so-called free algebras) correspond to applications of the substitution rule in deductions in the logic determined by $M$. Moreover, such endomorphisms give rise to continuous selfmappings of the dual topological spaces. Any set $\Theta'\supseteq\Theta$ ($\Theta$ is a set of basic axioms as in (4a)) is associated to an open set $O_{\Theta'}$ in the dual, and the deduction of new formulas from $\Theta'$ corresponds to taking the union of the backwards translates of $O_{\Theta'}$ under the dynamics. Dynamical properties such as minimality or mixing have then logical consequences (see, e.g., Theorem \[ref10\], Theorem \[ref14\], and the discussion following Theorem \[ref23\]). It is worth remarking that the trade between the logical and the dynamical side may be beneficial to both: as an example, we obtain in Theorem \[ref22\] an intrinsic characterization of the differential of a piecewise-linear mapping, a concept introduced in [@Tsujii01]. A rather delicate point in our approach is the determination of the level of generality one should allow. Here we must really strike a balance: the stronger is the system (i.e., the more restrictions we put on $M$), the stronger are the results we obtain, and the more limited is the scope of the theory. The extreme case is in taking $M=\{0,1\}$, in which everything boils down to the Stone Duality. On the other extreme, one might relax the assumptions on $M$ to a bare minimum, even allowing cases in which the values $0$ and $1$ do not have a distinguished status: the only essential requirement seems to be that ${\mathbf{V}}M$ is a congruence-modular equational class. Of course, working at this level of generality requires a greater technical apparatus, and yields not easily visualizable results. We stroke our balance by forcing $M$ to be a subset of the real unit interval ${[0,1]}$, and by insisting that the conjunction connective meets some natural restrictions. In the few places where we might have wished more elbow-room, we have added some Addenda to provide references for further developments. These Addenda are meant for people having some knowledge of Universal Algebra and lattice-ordered abelian groups, and may be safely skipped by the other readers. Many-valued logic {#ref24} ================= A [*t-norm*]{} is a continuous function ${\star}$ from ${[0,1]}^2$ to ${[0,1]}$ such that $({[0,1]},{\star},1)$ is a commutative monoid for which $a\le b$ implies $c{\star}a\le c{\star}b$. We have $a{\star}0=0$ for every $a$, since $a\le 1$ implies $0{\star}a\le 0{\star}1=0$. Every t-norm induces a binary operation $\to$ on ${[0,1]}$ via $$a\to b = \sup\{c:c{\star}a\le b\}.$$ Since ${\star}$ is continuous, the defining $\sup$ is really a $\max$. We call $\to$ the [*implication*]{} (or the [*residuum*]{}) induced by ${\star}$. One checks easily that the usual lattice operations on ${[0,1]}$ are definable from ${\star}$ and $\to$ via $a\land b=a{\star}(a\to b)$ and $a\lor b=\bigl((a\to b)\to b\bigl)\land \bigl((b\to a)\to a\bigl)$. We also define $\neg a=a\to 0$. The idea underlying these definitions is that ${\star}$ is a function on truth-values representing a “conjunction” operator. Once a conjunction has been fixed, it is natural to define the truth-value of the implication $a\to b$ as the weakest value $c$ such that the truth of the conjunction of $a$ and $c$ forces the truth of $b$. Note that “weakest” means “truest”, i.e., nearest to $1$: one should regard a more implausible assertion as a stronger one. The above interrelationship of ${\star}$ and $\to$ is usually expressed by saying that they constitute an [*adjoint pair*]{}. \[ref3\] 1. $a{\star}b=a\land b$. One computes that $$a\to b= \begin{cases} 1, & \text{if $a\le b$;} \\ b, & \text{otherwise;} \end{cases} \quad \neg a= \begin{cases} 1, & \text{if $a=0$;} \\ 0, & \text{otherwise.} \end{cases}$$ This t-norm is usually called the [*Gödel-Dummett conjunction*]{}. 2. $a{\star}b=ab$ (i.e., the ordinary product of $a$ and $b$). This is the [*product conjunction*]{}, and we have $$a\to b= \begin{cases} 1, & \text{if $a\le b$;} \\ b/a, & \text{otherwise;} \end{cases} \quad \neg a= \begin{cases} 1, & \text{if $a=0$;} \\ 0, & \text{otherwise.} \end{cases}$$ 3. $a{\star}b=\max(a+b-1,0)$. Then $$a\to b= \begin{cases} 1, & \text{if $a\le b$;} \\ 1-(a-b), & \text{otherwise;} \end{cases} \quad \neg a=1-a.$$ These are the [*[Łukasiewicz]{} conjunction*]{}, [*implication*]{}, and [*negation*]{} The above examples are in some sense exhaustive: by [@MostertShields57] every t-norm is obtainable as a combination of these three basic t-norms. Fix a cardinal number $\kappa$, either finite or countable, and define the set of propositional variables to be $\{x_i:i<\kappa\}$ (then either $\kappa=n$ and the propositional variables are ${x_0,\ldots ,x_{n-1}}$, or $\kappa=\omega$ and the propositional variables are indexed by the natural numbers). Let $FORM_\kappa$ be the smallest set containing all propositional variables having index $<\kappa$, the constants $0$ and $1$, and such that, if $r,s\in FORM_\kappa$, then $(r{\star}s),(r\to s)\in FORM_\kappa$. A [*formula*]{} is an element $r$ of $FORM_\omega=\bigcup_{n<\omega}FORM_n$. We sometimes write $r(x_{i_1},\ldots,x_{i_n})$ to signify that all propositional variables occurring in $r$ are among $x_{i_1},\ldots,x_{i_n}$. We drop parentheses according to the usual conventions, and we write $r\land s$, $r\lor s$, and $\neg r$ as abbreviations for $r{\star}(r\to s)$, $\bigl((r\to s)\to s\bigl)\land \bigl((s\to r)\to r\bigl)$, and $r\to 0$, respectively. An [*algebra*]{} is a set $A$ on which two binary operations ${\star}_A,\to_A:A^2\to A$ and two elements $0_A,1_A\in A$ have been fixed. Given a formula $r({x_0,\ldots ,x_{n-1}})$ and elements ${a_0,\ldots ,a_{n-1}}\in A$, we write $r({a_0,\ldots ,a_{n-1}})$ for the element of $A$ obtained by replacing every $x_i$ with the corresponding $a_i$, and every operation symbol in $r$ with its realization in $A$ (the reader can easily supply a formal recursive definition). Given two formulas $r({x_0,\ldots ,x_{n-1}})$ and $s({x_0,\ldots ,x_{n-1}})$, we say that the identity $r=s$ is [*true*]{} in $A$, and we write $A\models r=s$, if for every ${a_0,\ldots ,a_{n-1}}\in A$ the elements $r({a_0,\ldots ,a_{n-1}})$ and $s({a_0,\ldots ,a_{n-1}})$ are equal. \[ref4\] 1. Every singleton can be given the structure of an algebra in a unique trivial way; every identity is true in such an algebra. 2. Let $A={[0,1]}$, endowed with the Gödel-Dummett conjunction and implication, as in Example \[ref3\](1). Then $A\not\models\neg\neg x_0 = x_0$, so the double negation rule fails for the Gödel-Dummett connectives (and analogously for the product connectives). On the other hand, $\neg\neg x_0 = x_0$ holds true in ${[0,1]}$ endowed with the [Łukasiewicz]{} connectives. Let $A,B$ be algebras. A mapping $\varphi:A\to B$ is a [*homomorphism*]{} if it commutes with the connectives (i.e., $\varphi(a{\star}_A b)=\varphi(a){\star}_B\varphi(b)$, $\varphi(0_A)=0_B$, and so on; in the following we will drop the subscripts). $A$ is \[isomorphic to\] a [*subalgebra*]{} of $B$ if there exists an injective homomorphism from $A$ to $B$. Let $\{A_j:j\in J\}$ be a family of algebras. The [*direct product*]{} of the family is the algebra whose base set is the cartesian product $\prod_jA_j$, and in which the operations are defined componentwise; if all factors are equal, say to $A$, then we write $A^J$. If $\varphi:A\to B$ is a homomorphism, then the [*epimorphic image*]{} $\varphi[A]$ of $A$ is a subalgebra of $B$. Let ${\mathcal{A}}$ be a class of algebras; then ${\mathbf{H}}{\mathcal{A}}$ (respectively, ${\mathbf{S}}{\mathcal{A}}$ and ${\mathbf{P}}{\mathcal{A}}$) is the class of all epimorphic images (respectively, subalgebras and direct products) of algebras in ${\mathcal{A}}$. Note that we always work up to isomorphism, so we tacitly close every class we consider under isomorphic images. A [*truth-value algebra*]{} is a subalgebra $M$ of some algebra $A$ of the form $A=({[0,1]},{\star},\to,0,1)$, where ${\star}$ and $\to$ are a t-norm and its residuum. Truth-value algebras are our basic building blocks. 1. The set ${\{0,1\}}$ is always closed under the operations, regardless of the specific t-norm we choose. Moreover, all t-norms induce the same structure on ${\{0,1\}}$, namely that of the [*two-element boolean algebra*]{}, which we denote by ${\mathbf{2}}$. 2. $M=\{0,1/m,2/m,\ldots,(m-1)/m,1\}$ endowed either with the [Łukasiewicz]{} connectives or the Gödel-Dummett ones. For any class ${\mathcal{A}}$ of algebras, let ${\mathbf{V}}{\mathcal{A}}$ be the [*equational class*]{} generated by ${\mathcal{A}}$, i.e., the class of all algebras in which are true all identities true in all algebras of ${\mathcal{A}}$. More explicitly, the algebra $B$ is in ${\mathbf{V}}{\mathcal{A}}$ iff, for every $r,s\in FORM_\omega$, if $A\models r=s$ for every $A\in{\mathcal{A}}$, then $B\models r=s$. Garrett Birkhoff’s completeness theorem [@burrissan81 Theorem II.11.9] says that ${\mathbf{V}}{\mathcal{A}}$ coincides with the class ${\mathbf{HSP}}{\mathcal{A}}$ of all epimorphic images of subalgebras of products of algebras in ${\mathcal{A}}$. We will consider classes of algebras of the form ${\mathbf{V}}M={\mathbf{HSP}}M$, where $M$ is a truth-value algebra. We shall be concerned with two main cases: - ${{\mathit{Boole}}}={\mathbf{V}}{\mathbf{2}}$. Elements of ${{\mathit{Boole}}}$ are called [*boolean algebras*]{}; - if $M={[0,1]}$ endowed with the [Łukasiewicz]{} connectives, then the elements of ${\mathbf{V}}M$ are called [*MV-algebras*]{} (MV stands for [*Many-Valued*]{}: the name is slightly misleading, since many-valued logic is not exhausted by [Łukasiewicz]{} logic, but it is firmly established; we accordingly write ${{\mathit{MV}}}$ for the equational class ${\mathbf{V}}M$). A boolean algebra can be equivalently defined as a structure $A=(A,\land,\lor,\neg,0,1)$ such that $$\begin{gathered} x\land 1=x\lor 0=x;\\ x\land\neg x=0;\quad x\lor\neg x=1;\\ \text{$\land$ and $\lor$ are commutative and mutually distributive}.\end{gathered}$$ Apart from the trivial change in the language ($\lor$ replaces $\to$), there is a theorem hidden in this equivalence, namely the fact that the above identities imply all other identities that hold in ${\mathbf{2}}$ [@Halmos63 p. 5]. An analogous alternative characterization of MV-algebras is obtained by adding a new connective $\oplus$ to the basic set $({\star},\to,0,1)$. We define $a\oplus b=\neg a\to b$, and directly compute that $\oplus$ is [*truncated addition*]{} on ${[0,1]}$, i.e., $a\oplus b=\min(a+b,1)$. Note that the basic set of connectives is equivalent to the set $(\oplus,\neg,0,1)$, since $a{\star}b=\neg(\neg a\oplus\neg b)$ and $a\to b=\neg a\oplus b$. Then, in terms of the new set, an MV-algebra is a structure $(A,\oplus,\neg,0,1)$ such that $(A,\oplus,0)$ is an abelian monoid and the identities $\neg\neg x=x$, $x\oplus 1=1$, $\neg(\neg x\oplus y)\oplus y=\neg(\neg y\oplus x)\oplus x$ are satisfied [@mundicijfa §2], [@CignoliOttavianoMundici00]. Let\[ref5\] $M$ be a truth-value algebra, $A\in{\mathbf{V}}M$. Then: - the operations $\land,\lor$ induce a lattice structure on $A$, with bottom element $0$ and top $1$; - the lattice order in (i) is given by $a\le b$ iff $a\land b=a$ iff $a\to b=1$; - $A\models r=s$ iff $A\models (r\to s)\land(s\to r)=1$. A structure $(A,\land,\lor,0,1)$ is a lattice with bottom and top iff it satisfies a certain finite set of identities (see, e.g., [@burrissan81 p. 28]). Since $M$ is totally-ordered, these identities are satisfied in $M$, and hence in $A\in{\mathbf{V}}M$. The first equivalence in (ii) is just the definition of the lattice order on $A$. By definition of $\to$ in $M$, the identity $(x_0\land x_1)\to x_1=1$ is true in $M$, and hence in $A$. Therefore, if $a\land b=a$, then $a\to b=(a\land b)\to b=1$. On the other hand, if $a\to b=1$, then $a\land b=a{\star}(a\to b)=a{\star}1=a$. This proves (ii), and (iii) is then immediate. By Lemma \[ref5\](ii) we can deal with the “less than” relation between formulas, thus writing $A\models r({x_0,\ldots ,x_{n-1}})\le s({x_0,\ldots ,x_{n-1}})$ for $A\models r\to s=1$; this just means that however we choose ${a_0,\ldots ,a_{n-1}}\in A$ we have $r({a_0,\ldots ,a_{n-1}})\le s({a_0,\ldots ,a_{n-1}})$. We then say that $r\le s$ is [*true*]{} in $A$. Under\[ref6\] the same hypothesis as in Lemma \[ref5\], the following relations are true in $A$: - $x_0{\star}x_1\le x_0\land x_1$; - $x_0\le x_1\to(x_0{\star}x_1)$; - $(x_0\to x_1){\star}(x_1\to x_2)\le x_0\to x_2$; - $(x_0\to x_1){\star}(x_2\to x_3)\le (x_0{\star}x_2)\to (x_1{\star}x_3)$; - $(x_0\to x_1){\star}(x_2\to x_3)\le (x_1\to x_2)\to (x_0\to x_3)$. One just checks that for every t-norm with residuum $\to$ the above relations are true in $({[0,1]},{\star},\to,0,1)$. Hence they are true in $M$, and therefore in every algebra in ${\mathbf{V}}M$. Fix now $\kappa$ and a truth-value algebra $M$. We want to construct an algebra $A$ in ${\mathbf{V}}M$ satisfying the following properties: - $A$ is generated by a family $\{a_i:i<\kappa\}$ of elements indexed by $\kappa$; - if $r(x_{i_1},\ldots,x_{i_n})\in FORM_\kappa$ is not true in $M$, then the element $r(a_{i_1},\ldots,a_{i_n})\in A$ is different from $1$. Essentially, this means that the $a_i$’s satisfy only those algebraic relations they cannot avoid, namely those that hold in $M$. Therefore they behave “as freely as possible”, whence the name [*free algebra in ${\mathbf{V}}M$ over $\kappa$ generators*]{} for $A$. Such an algebra is unique up to isomorphism, and can be characterized by an appropriate universal property: see, e.g., [@burrissan81 II §10]. We write ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ for $A$, and we construct it as follows: consider first $M^\kappa$, and let $a_i:M^\kappa\to M$ be the $i$-th projection. The $a_i$’s are elements of the algebra $M^{(M^\kappa)}$, and we define ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ to be the subalgebra of $M^{(M^\kappa)}$ generated by them; the first condition is then automatically met. Suppose $M\not\models r$; then there exist elements $b_{i_1},\ldots,b_{i_n}\in M$ such that $r(b_{i_1},\ldots,b_{i_n})\not=1$. Choose an element $c\in M^\kappa$ such that $a_i(c)=b_i$ for every $i\in\{{i_1,\ldots ,i_n}\}$. Then the projection of $r(a_{i_1},\ldots,a_{i_n})\in M^{(M^\kappa)}$ onto the $c$-th component has value $r(a_{i_1}(c),\ldots,a_{i_n}(c))=r(b_{i_1},\ldots,b_{i_n}) \not=1$: therefore $r(a_{i_1},\ldots,a_{i_n})$ is different from $1$ in $M^{(M^\kappa)}$, and hence in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$. Although the above construction looks baroque, it really works trivially. Suppose, e.g., we want to construct $\operatorname{Free}_3{{\mathit{Boole}}}$. We first construct ${\mathbf{2}}^3$, which contains the eight elements $c_1=(0,0,0)$, $c_2=(0,0,1)$, …, $c_8=(1,1,1)$. Then we construct ${\mathbf{2}}^{({\mathbf{2}}^3)}$, which contains $2^8$ elements; three of these elements, namely $$\begin{aligned} a_1 &= (0,0,0,0,1,1,1,1),\\ a_2 &= (0,0,1,1,0,0,1,1),\\ a_3 &= (0,1,0,1,0,1,0,1),\end{aligned}$$ correspond to the canonical projections to the first, second, and third component of the $c_j$’s (of course, the above explicit form for the $a_i$’s depends on how we listed the $c_j$’s). It is clear that if $r(x_1,x_2,x_3)$ is not $1$ in ${\mathbf{2}}$ for some choice of elements, then $r(a_1,a_2,a_3)\not=1=(1,1,1,1,1,1,1,1)$ in ${\operatorname{Free}_{3}({{\mathit{Boole}}})}$: we just tried all possible choices! Often it is not trivial to determine, for a given $M$ and $\kappa$, which are the elements of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, i.e., which functions from $M^\kappa$ to $M$ are expressible as polynomials over the projections $a_i$. The case that most concerns us is the [Łukasiewicz]{} one, which we will treat in Theorem \[ref15\]. For the classical logic case, the answer is the following: give ${\mathbf{2}}$ the discrete topology, and ${\mathbf{2}}^\kappa$ the product topology. Then: - an element $f\in{\mathbf{2}}^{({\mathbf{2}}^\kappa)}$ is in ${\operatorname{Free}_{\kappa}({{\mathit{Boole}}})}$ iff it is continuous as a function $f:{\mathbf{2}}^\kappa\to{\mathbf{2}}$. Since continuous functions from a topological space $X$ to ${\mathbf{2}}$ correspond to clopen subsets of $X$, this amounts to saying that the clopen subsets of ${\mathbf{2}}^\kappa$ are exactly the boolean combinations of the sets of the form $a_i^{-1}[1]=\{p\in{\mathbf{2}}^\kappa:p_i=1\}$. With this hint, we leave the proof of (i) as an exercise for the reader. As a corollary we obtain: - if $\kappa=n$ is finite, then ${\mathbf{2}}^n$ is a discrete space, and all functions $:{\mathbf{2}}^n\to{\mathbf{2}}$ are in ${\operatorname{Free}_{n}({{\mathit{Boole}}})}$, i.e., are expressible by $n$-variable formulas. This is sometimes called the [*functional completeness*]{} of the boolean connectives; - if $\kappa=\omega$ is countably infinite, then ${\operatorname{Free}_{\omega}({{\mathit{Boole}}})}$ is the boolean algebra of all clopen subsets of the [*Cantor space*]{} ${\mathbf{2}}^\omega$, the latter being the only compact, totally disconnected, second countable space having no isolated points [@hockingyou §2.15]. Spectral spaces {#ref13} =============== In the preceding section we have defined the equational class ${\mathbf{V}}M$ generated by a truth-value algebra $M$, and described the algebras ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$. In this section we functorially associate a dual topological space to each algebra in ${\mathbf{V}}M$; the duals of the free algebras are our main object of study. We fix a truth-value algebra $M$; all algebras we consider are elements of ${\mathbf{V}}M$. A [*filter*]{} on $A\in{\mathbf{V}}M$ is the counterimage of $1$ under some homomorphism of domain $A$: sometimes filters are called [*dual ideals*]{} since ideals, as in ring theory, are counterimages of $0$. A subset ${\mathfrak{f}}$ of $A$ is a filter iff it contains $1$ and is closed under Modus Ponens ($a,a\to b\in{\mathfrak{f}}$ implies $b\in{\mathfrak{f}}$). Every filter is closed under ${\star}$ and $\land$, and is upwards closed ($b\ge a\in{\mathfrak{f}}$ implies $b\in{\mathfrak{f}}$). Given two homomorphisms $\varphi:A\to B$ and $\psi:A\to C$, we have $\varphi^{-1}[1]=\psi^{-1}[1]$ iff there exists an isomorphism $\chi:\varphi[A]\to\psi[A]$ such that $\psi=\chi\circ\varphi$. Let ${\mathfrak{f}}=\varphi^{-1}[1]$; then clearly $1\in{\mathfrak{f}}$. If $a,a\to b\in{\mathfrak{f}}$, then $\varphi(b)\ge\varphi(a\land b)=\varphi(a{\star}(a\to b))=\varphi(a){\star}\varphi(a\to b)=1$. Conversely, assume that ${\mathfrak{f}}$ is a subset of $A$ containing $1$ and closed under Modus Ponens. Then ${\mathfrak{f}}$ is upwards closed, since $a\le b$ implies $a\to b=1\in{\mathfrak{f}}$, and hence $a\in{\mathfrak{f}}$ implies $b\in{\mathfrak{f}}$. If $a\in{\mathfrak{f}}$, then $b\to(a{\star}b)\in{\mathfrak{f}}$ (by Lemma \[ref6\](ii)); hence, if $b\in{\mathfrak{f}}$ as well, then $a{\star}b,a\land b\in{\mathfrak{f}}$ (by Lemma \[ref6\](i)). Let now ${\mathfrak{f}}$ be a subset of $A$ containing $1$ and closed under MP. Define a relation $\sim$ on $A$ by $a\sim b$ iff $a\to b,b\to a\in{\mathfrak{f}}$. Then $\sim$ is an equivalence relation (transitivity follows from Lemma \[ref6\](iii)) which respects the operations. Indeed, if $a\sim b$ and $c\sim d$, then $a{\star}c\sim b{\star}d$ (by Lemma \[ref6\](iv)) and $a\to c\sim b\to d$ (by Lemma \[ref6\](v)). We can then form the quotient algebra $A/{\mathfrak{f}}$ in the natural way. If $a/{\mathfrak{f}}$ denotes the equivalence class of $a$ w.r.t. $\sim$, then the map $\rho(a)=a/{\mathfrak{f}}$ is a surjective homomorphism from $A$ to $A/{\mathfrak{f}}$. It is then straightforward to check that the map $\tau:A/{\mathfrak{f}}\to\varphi[A]$ defined by $\tau(a/{\mathfrak{f}})=\varphi(a)$ is an isomorphism. Since $\varphi=\tau\circ\rho$, our last claim follows by composing isomorphisms. A filter ${\mathfrak{p}}$ is [*prime*]{} if it is proper (i.e., different from $A$) and for every two filters ${\mathfrak{f}},{\mathfrak{g}}$, if ${\mathfrak{p}}={\mathfrak{f}}\cap{\mathfrak{g}}$ then either ${\mathfrak{p}}={\mathfrak{f}}$ or ${\mathfrak{p}}={\mathfrak{g}}$. A filter is [*maximal*]{} if it is proper and not properly contained in any proper filter; clearly every maximal filter is prime. Although not difficult, the proof of the following lemma requires some knowledge of Universal Algebra; the reader can find a proof in [@pantigeneric Proposition 1.3]. The\[ref7\] following are equivalent: 1. ${\mathfrak{p}}$ is prime; 2. $A/{\mathfrak{p}}$ is totally-ordered; 3. ${\mathfrak{p}}=\varphi^{-1}[1]$, for some homomorphism $\varphi$ from $A$ to a totally-ordered algebra; 4. the set of filters $\supseteq{\mathfrak{p}}$ is totally-ordered by inclusion; 5. every filter $\supseteq{\mathfrak{p}}$ is prime; 6. if $a\lor b\in{\mathfrak{p}}$, then either $a\in{\mathfrak{p}}$ or $b\in{\mathfrak{p}}$. Note that the only totally-ordered boolean algebra is ${\mathbf{2}}$ (if $a$ belongs to the totally-ordered boolean algebra $A$, then either $a\le\neg a$ or $\neg a\le a$, hence either $a\to\neg a=1$ or $\neg a\to a=1$; in the first case $a=0$, and in the second $a=1$), and therefore prime filters coincide with maximal ones. This simple fact distinguishes in a crucial way boolean algebras from MV-algebras and other algebras related to many-valued logics, as we will see later. Let $A$ be an algebra, and let $\operatorname{Spec}A$ be the set of all prime filters of $A$. For every $a\in A$, let $O_a$ be the set of all ${\mathfrak{p}}\in\operatorname{Spec}A$ such that $a$ does not belong to ${\mathfrak{p}}$. Impose on $\operatorname{Spec}A$ the weakest topology in which all $O_a$’s are open (i.e., take the family of all $O_a$’s as an open subbasis). This is called the hull-kernel topology on $\operatorname{Spec}A$, and the resulting space is the [*spectral*]{} (or [*dual*]{}) [*space*]{} of $A$. For every subset $D$ of $A$, let $O_D=\bigcup\{O_a:a\in D\}= \{{\mathfrak{p}}:D\not\subseteq{\mathfrak{p}}\}$: it is an open set, and we will see in Theorem \[ref2\](ii) that every open set has this form. We write $F_a=(\operatorname{Spec}A)\setminus O_a$ and $F_D=(\operatorname{Spec}A)\setminus O_D$ for the corresponding closed sets. The mapping $A\mapsto\operatorname{Spec}A$ is functorial. Indeed, let $\varphi:A\to B$ be any homomorphism, and define $\varphi^*:\operatorname{Spec}B\to\operatorname{Spec}A$ by $\varphi^*({\mathfrak{p}})=\varphi^{-1}[{\mathfrak{p}}]$. $\varphi^*({\mathfrak{p}})$ is a prime filter because, if ${\mathfrak{p}}=\psi^{-1}[1]$ for some homomorphism $\psi$ from $B$ to a totally-ordered algebra $C$, then $\varphi^*({\mathfrak{p}})$ is the kernel of $\psi\circ\varphi$, and the epimorphic image $(\psi\circ\varphi)[A]$ is totally-ordered, since it is a subalgebra of $C$. We have $(\varphi^*)^{-1}[O_D]=\{{\mathfrak{p}}\in\operatorname{Spec}B:\varphi^*({\mathfrak{p}})\in O_D\}=\{{\mathfrak{p}}:D\not\subseteq\varphi^{-1}[{\mathfrak{p}}]\}=\{{\mathfrak{p}}:\varphi[D]\not\subseteq{\mathfrak{p}}\}=O_{\varphi[D]}$, and hence $\varphi^*$ is continuous. Moreover, $(\psi\circ\varphi)^*=\varphi^*\circ\psi^*$, so $\operatorname{Spec}$ is a contravariant functor from ${\mathbf{V}}M$ (viewed as a category with the homomorphisms as arrows) to the category of topological spaces and continuous mappings. Let\[ref2\] $A$ be an algebra. - The open sets in $\operatorname{Spec}A$ are in 1–1 correspondence with the filters of $A$, and this correspondence is an isomorphism w.r.t. the $\subseteq$ relation. - We have $O_a\cap O_b=O_{a\lor b}$ and $O_a\cup O_b=O_{a\land b}$. The defining subbasis is intersection-closed, and an open set is compact iff it is of the form $O_a$. $\operatorname{Spec}A$ is second countable iff $A$ is countable. - $\operatorname{Spec}A$ is $T_0$, compact, and every closed irreducible set is the closure of a point. The key point is that every filter ${\mathfrak{f}}$ is the intersection of all prime filters $\supseteq{\mathfrak{f}}$; this fact follows from a standard application of the Zorn Lemma. As a consequence, for every $D\subseteq A$, the intersection of all filters containing $D$ coincides with the intersection of all prime filters containing $D$. This intersection, namely $\bigcap F_D$, is the smallest filter containing $D$, and we denote it by ${\mathfrak{f}}(D)$. One verifies easily that ${\mathfrak{f}}(D)$ is the set of all $a\in A$ such that there exist ${a_1,\ldots ,a_r}\in D$ satisfying $a\ge a_1{\star}\cdots{\star}a_r$. Consider the mappings $$\begin{aligned} A\supseteq D &\longmapsto F_D\in\text{sets closed in $\operatorname{Spec}A$} \\ \text{filters of $A$}\ni\textstyle{\bigcap} P &\longleftarrow\!\mapstochar P\subseteq\operatorname{Spec}A\end{aligned}$$ They both reverse the $\subseteq$ relation. Their composition gives, on the left side, the mapping $D\mapsto \bigcap F_D={\mathfrak{f}}(D)$ that associates to a set the filter it generates, and on the right side the topological closure mapping $P\mapsto F_{\bigcap P}$ (Proof: ${\mathfrak{p}}$ belongs to the topological closure of $P$ iff $\forall a({\mathfrak{p}}\in O_a{\Rightarrow}O_a\cap P\not=\emptyset)$ iff $\forall a(P\subseteq F_a{\Rightarrow}a\in{\mathfrak{p}})$ iff ${\mathfrak{p}}\supseteq\bigcap P$). As a consequence, they induce an antiisomorphism between the lattice of filters of $A$ and the lattice of closed sets of $\operatorname{Spec}A$; this proves (i). We leave (ii) as an exercise, and prove (iii). The $T_0$ property is clear, because the closure of ${\mathfrak{p}}$ is $F_{\bigcap\{{\mathfrak{p}}\}}=F_{\mathfrak{p}}=\{{\mathfrak{q}}:{\mathfrak{q}}\supseteq{\mathfrak{p}}\}$. Compactness follows from (ii) and the fact that $\operatorname{Spec}A=O_0$. Let $F_{\mathfrak{f}}$ be a closed irreducible set, i.e., a closed set that cannot be expressed nontrivially as the union of two closed sets; we must show that ${\mathfrak{f}}$ is prime, i.e., that $a\lor b\in{\mathfrak{f}}$ implies ($a\in{\mathfrak{f}}$ or $b\in{\mathfrak{f}}$). Assume $a\lor b\in{\mathfrak{f}}$; then $F_{a\lor b}\supseteq F_{\mathfrak{f}}$. Since $F_{a\lor b}=F_a\cup F_b$, we have $F_{\mathfrak{f}}=(F_{\mathfrak{f}}\cap F_a)\cup(F_{\mathfrak{f}}\cap F_b)$, which must be a trivial decomposition. Hence either $F_a\supseteq F_{\mathfrak{f}}$ or $F_b\supseteq F_{\mathfrak{f}}$, i.e., either $a\in{\mathfrak{f}}$ or $b\in{\mathfrak{f}}$. A [*spectral space*]{} is a topological space in which the compact open sets form a basis closed under finite intersections, and such that the conditions in Theorem \[ref2\](iii) hold. By [@Hochster69], these are exactly the prime ideal spaces of commutative rings with $1$. The spectral spaces of boolean algebras have a further property: the compact open sets are exactly the clopen sets (i.e., the sets which are both closed and open). Indeed, as we observed after Lemma \[ref7\], if $A\in{{\mathit{Boole}}}$ then every ${\mathfrak{p}}\in\operatorname{Spec}A$ is of the form ${\mathfrak{p}}=\varphi^{-1}[1]$ for some homomorphism $\varphi:A\to{\mathbf{2}}$. This immediately implies that $a\in{\mathfrak{p}}$ iff $\neg a\notin{\mathfrak{p}}$, i.e., $F_a=O_{\neg a}$. Therefore the $O_a$’s are clopen, and since every clopen is compact, there are no other clopens. Thus $A$ is in bijection with the clopen sets of $X=\operatorname{Spec}A$ via $a\mapsto F_a$, and since $F_{a\land b}=F_a\cap F_b$, $F_{\neg a}=X\setminus F_a$, $F_0=\emptyset$, and $F_1=X$, this bijection is a boolean algebra isomorphism. We have thus proved the [*Stone Representation Theorem*]{} [@Halmos63 §18]: every boolean algebra is isomorphic to the algebra of clopen subsets of its spectrum. Spectral\[ref11\] spaces can be functorially introduced in any congruence-modular equational class. In general, filters should be substituted by congruences (unless the class turns out to be ideal-determined [@GummUrsini84]), and one introduces the notion of prime congruence by using the commutator product; the construction carries on smoothly [@agliano93]. The main trouble is with Theorem \[ref2\](i): the open sets of $\operatorname{Spec}A$ will now be in 1-1 correspondence only with the radical congruences of $A$, i.e., those congruences $\Phi$ such that, for every congruence $\Psi$, if $\Phi$ contains the commutator product of $\Psi$ with itself, then $\Phi$ already contains $\Psi$ (the spectrum of ${\mathbb{Z}}$ as a commutative ring is a typical example, the radical ideals being those generated by a squarefree integer). In our case there are no such problems, since equational classes are generated by truth-value algebras, in which a lattice structure is term-definable. All our classes are therefore congruence-distributive, and all congruences are radical. Proofs and dynamics =================== We can now make precise the heuristic in the Introduction about the dynamics in Frege proof systems. A [*substitution*]{} on $FORM_\kappa$ is any mapping $\sigma:FORM_\kappa\to FORM_\kappa$ which distributes over the connectives (i.e., $\sigma(0)=0$, $\sigma(1)=1$, and $\sigma(r\circ s)=\sigma(r)\circ \sigma(s)$, for $\circ \in\{{\star},\to\}$). A substitution is therefore determined by an arbitrary assignment of formulas to propositional variables. Note that all variables must be substituted at the same time: e.g., if $r=x_1\to x_2$, $\sigma(x_1)=x_3{\star}x_2$, and $\sigma(x_2)=x_1$, then $\sigma(r)=(x_3{\star}x_2)\to x_1$. Given a formula $r$ and a set of formulas $\Theta$, a [*deduction*]{} of $r$ from $\Theta$ is a finite sequence of formulas ${r_1,\ldots ,r_h}$ such that $r_h=r$ and for every $1\le j\le h$ we have: - either $r_j\in\Theta$; - or there exist $1\le k,m<j$ such that $r_m$ has the form $r_k\to r_j$ (we then say that $r_j$ follows from $r_k$ and $r_k\to r_j$ via Modus Ponens); - or there exists $1\le k<j$ and a substitution $\sigma$ such that $r_j=\sigma(r_k)$. An [*MP-deduction*]{} is a deduction in which the substitution rule (c) is never applied. For a fixed truth-value algebra $M$, it is often possible —although sometimes difficult— to find effectively a set of formulas $\Theta$ such that the formulas deducible from $\Theta$ are exactly the formulas which are true in $M$. If this happens, then we say that $\Theta$ provides an [*axiomatization*]{} of ${\mathbf{V}}M$. Let $\Theta$ be the set of the following eight formulas: $$\begin{gathered} \bigl((x_0\to x_1){\star}(x_1\to x_2)\bigr)\to(x_0\to x_2); \\ (x_0{\star}x_1)\to x_0; \quad 0\to x_0; \\ (x_0{\star}x_1)\to(x_1{\star}x_0); \quad (x_0\land x_1)\to(x_1\land x_0); \\ \bigl(x_0\to(x_1\to x_2)\bigr)\to\bigl((x_0{\star}x_1)\to x_2\bigr); \quad \bigl((x_0{\star}x_1)\to x_2\bigr)\to\bigl(x_0\to(x_1\to x_2)\bigr); \\ \bigl[\bigl((x_0\to x_1)\to x_2\bigr){\star}\bigl((x_1\to x_0)\to x_2\bigr)\bigr]\to x_2.\end{gathered}$$ All of them —except perhaps the last one— have rather transparent meanings: the first one expresses transitivity of implication, the third is *ex falso quodlibet*, the fourth expresses commutativity of conjunction, and so on. We have [@hajek98]: - $\Theta\cup\{\neg\neg x_0\to x_0\}$ axiomatizes ${{\mathit{MV}}}$; - $\Theta\cup\{\neg\neg x_0\to\bigl((x_1{\star}x_0\to x_2{\star}x_0)\to (x_1\to x_2)\bigr), \neg(x_0\land\neg x_0)\}$ axiomatizes the equational class generated by $M={[0,1]}$ endowed with the product connectives in Example \[ref3\](2); - $\Theta\cup\{x_0\to(x_0{\star}x_0)\}$ axiomatizes the equational class generated by $M={[0,1]}$ endowed with the Gödel-Dummett connectives; - $\Theta\cup\{x_0\lor\neg x_0\}$ axiomatizes ${{\mathit{Boole}}}$. Fix a truth-value algebra $M$ and a cardinal $\kappa$. Let $\Theta_\kappa$ be the set of all formulas in $FORM_\kappa$ which are true in $M$. Hence $r(x_{i_1},\ldots,x_{i_n})\in\Theta_\kappa$ iff $r(a_{i_1},\ldots,a_{i_n})=1$ in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, where the $a_i$’s are the free generators. It is a customary abuse of notation to write $x_i$ for $a_i$, so the symbol $r$ may denote either an element of $FORM_\kappa$ or an element of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$. This slight ambiguity is really sought for: it is exactly the ambiguity that results in working modulo $\Theta_\kappa$, or in identifying a formula with the function it induces on truth-values. Formally stated: $r,s\in FORM_\kappa$ are equal in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ iff $M\models r=s$ iff $r\to s,s\to r\in\Theta_\kappa$. The set $\Theta_\omega$ is closed under Modus Ponens and substitution, so nothing new can be deduced from it. We let now $\Delta$ be any set of formulas, and raise two questions: 1. What can be deduced from $\Theta_\omega\cup\Delta$? 2. Which substitutions are needed for such a deduction? If\[ref8\] $r$ is deducible from $\Theta_\omega\cup\Delta$, then there is a deduction involving only variables already appearing in $\{r\}\cup\Delta$. By renaming variables we may assume that the variables appearing in $\{r\}\cup\Delta$ are exactly those variables with index $<\kappa$, for a certain $\kappa$. Let a deduction of $r$ from $\Theta_\omega\cup\Delta$ be given. By a standard argument [@church p. 149], we may transform the given deduction into an MP-deduction ${r_1,\ldots ,r_h}=r$ of $r$ from $\Theta_\omega\cup\{\sigma(t):\sigma\text{ is a substitution and }t\in\Delta\}$. Let $\tau$ be the substitution given by $\tau(x_j)=x_j$ if $j<\kappa$, and $\tau(x_j)=1$ otherwise. Then $\tau(r_1),\ldots,\tau(r_h)=r$ is an MP-deduction of $r$ from $\Theta_\kappa\cup\{\sigma(t):\sigma\text{ is a substitution, }t\in\Delta,\text{ and }\sigma(t)\in FORM_\kappa\}$, and hence a deduction of $r$ from $\Theta_\kappa\cup\Delta$ in which all formulas and all substitutions involve only variables with index $<\kappa$. Identifying $FORM_\kappa$ with ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, a substitution is nothing more than an endomorphism of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ (i.e., a homomorphism from ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ to itself). The freeness of the generators says that however we choose elements $\{b_i:i<\kappa\}$ in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ there is precisely one endomorphism that maps $x_i$ to $b_i$. We denote by $\Sigma_\kappa$ the monoid of all endomorphisms of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, and by $\Xi_\kappa\subseteq\Sigma_\kappa$ the group of all automorphisms of ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ (i.e., of the invertible elements of $\Sigma_\kappa$). As explained before Theorem \[ref2\], to every $\sigma\in\Sigma_\kappa$ there corresponds a continuous selfmapping $\sigma^*$ of $X_\kappa=\operatorname{Spec}{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$, which we call the [*dual*]{} of $\sigma$. Let $\Pi\subseteq\Sigma_\kappa$, and let $O$ be an open subset of $X_\kappa$. We define $(\Pi,O)$ to be the union of all backwards translates of $O$ under iteration of the substitutions in $\Pi$. Explicitly stated, $$(\Pi,O)=\bigcup\{(\sigma^*)^{-1}[O]:\sigma\text{ is in the submonoid of }\Sigma_\kappa\text{ generated by }\Pi\}.$$ If $(\Pi,O)=X_\kappa$ for every $O\not=\emptyset$, then we say that $\Pi$ acts [*minimally*]{} on $X_\kappa$: this is equivalent to saying that every point of $X_\kappa$ has a dense orbit under $\Pi$. Let\[ref9\] $\{r\}\cup\Delta\subseteq FORM_\kappa$. Then: - $r$ can be MP-deduced from $\Theta_\omega\cup\Delta$ iff $O_r\subseteq O_\Delta$ in $X_\kappa$; - if $\Delta'\subseteq FORM_\kappa$ is another set of formulas, then $O_\Delta=O_{\Delta'}$ iff $\Theta_\omega\cup\Delta$ and $\Theta_\omega\cup\Delta'$ MP-deduce the same formulas; - $r$ can be deduced from $\Theta_\omega\cup\Delta$ iff $O_r\subseteq(\Sigma_\kappa,O_\Delta)$. \(i) follows from Theorem \[ref2\](i) and the proof of Lemma \[ref8\]: $r$ can be MP-deduced from $\Theta_\omega\cup\Delta$ iff $r$ can be MP-deduced from $\Theta_\kappa\cup\Delta$ iff $r$ belongs to the filter ${\mathfrak{f}}(\Delta)$ in ${\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ iff ${\mathfrak{f}}(r)\subseteq{\mathfrak{f}}(\Delta)$ iff $O_r=O_{{\mathfrak{f}}(r)}\subseteq O_{{\mathfrak{f}}(\Delta)}=O_\Delta$. (ii) is clear: the sets $\Theta_\omega\cup\Delta$ and $\Theta_\omega\cup\Delta'$ MP-deduce the same formulas iff ${\mathfrak{f}}(\Delta)={\mathfrak{f}}(\Delta')$ iff $O_\Delta=O_{{\mathfrak{f}}(\Delta)}=O_{{\mathfrak{f}}(\Delta')}=O_{\Delta'}$. We prove (iii): assume that $r$ can be deduced from $\Theta_\omega\cup\Delta$. By Lemma \[ref8\], there exists a deduction ${r_1,\ldots ,r_h}=r$ such that every $r_j$ is in $\Theta_\kappa\cup\Delta$ and all substitutions applied are in $\Sigma_\kappa$. Working by induction on $h$ we assume that $$O_{r_1}\cup\cdots\cup O_{r_{h-1}}\subseteq (\Sigma_\kappa,O_\Delta).$$ If $r_h\in\Delta$, then of course we are through (note that $s\in\Theta_\kappa$ is equivalent to $O_s=\emptyset$). For every $s,t\in{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ we have $O_t\subseteq O_s\cup O_{s\to t}$. Indeed, if ${\mathfrak{p}}\notin O_s$ and ${\mathfrak{p}}\notin O_{s\to t}$, then $s,s\to t\in{\mathfrak{p}}$. Since filters are closed under MP, we have $t\in{\mathfrak{p}}$ and ${\mathfrak{p}}\notin O_t$. Therefore, if $r_h$ has been obtained via MP, then the induction hypothesis guarantees that $O_{r_h}\subseteq(\Sigma_\kappa,O_\Delta)$. Finally, if $r_h=\sigma(r_j)$ for some $\sigma\in\Sigma_\kappa$ and $1\le j<h$, then $O_{r_h}=O_{\sigma(r_j)}= (\sigma^*)^{-1}[O_{r_j}]\subseteq(\sigma^*)^{-1}[(\Sigma_\kappa, O_\Delta)]\subseteq (\Sigma_\kappa,O_\Delta)$. We leave the reverse implication as an exercise for the reader (Hint: $O_r$ is compact). We say that $\Theta\subseteq FORM_\omega$ is [*equationally complete*]{} if, however we choose $r,s\in FORM_\omega$ with $r\notin\Theta$, the formula $s$ is deducible from $\Theta\cup\{r\}$. The\[ref10\] set $\Theta_\omega$ is equationally complete iff $\Sigma_\omega$ acts minimally on $X_\omega$. Let $O$ be a nonvoid open subset of $X_\omega$, and assume that $\Theta_\omega$ is equationally complete: we want to show that $(\Sigma_\omega,O)=X_\omega$. Let $r\in{\operatorname{Free}_{\omega}({\mathbf{V}}M)}$ be such that $\emptyset\not=O_r\subseteq O$; we then have $r\notin\Theta_\omega$. By assumption, the formula $0$ is deducible from $\Theta_\omega\cup\{r\}$, and hence we get $X_\omega=O_0\subseteq(\Sigma_\omega,O_r)\subseteq(\Sigma_\omega,O)$ from Lemma \[ref9\](iii). The same argument yields the reverse implication. The equational completeness of $\Theta_\omega$ amounts to the lack of nontrivial equational subclasses of ${\mathbf{V}}M$. Among classes generated by truth-value algebras, the only one fulfilling this property is ${{\mathit{Boole}}}$ (easy proof, resting on the fact that ${\mathbf{2}}$ is a subalgebra of any $M$). Theorem \[ref10\] then implies that $\Sigma_\omega$ acts minimally on $X_\omega$ only in the case of boolean algebras. There are other cases in which an algebra $M$ (not a truth-value algebra in our sense) generates a congruence-distributive equational class having no nontrivial subclasses. A particularly interesting case is when $M$ is the set of integers equipped with its natural structure $({\mathbb{Z}},+,-,0,\lor,\land)$ of lattice-ordered group [@bkw], [@andersonfei]. The resulting equational class is the class of all lattice-ordered groups, and the above properties are fulfilled. Theorem \[ref10\] says then that the endomorphisms of the free lattice-ordered groups act minimally on the relative spectra: see [@pantiprime] for a description of such spaces. For every $p=(\ldots,p_i,\ldots)\in M^\kappa$, the evaluation mapping at $p$, given by $r(x_{i_1},\ldots,x_{i_n})\mapsto r(p_{i_1},\ldots,p_{i_n})$, is a homomorphisms $\varphi:{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}\to M$. Since $M$ is totally-ordered, the kernel $\varphi^{-1}[1]$ is a prime filter ${\mathfrak{p}}$, hence an element of $X_\kappa$. We thus get a mapping $p\mapsto{\mathfrak{p}}$ from $M^\kappa$ to $X_\kappa$, which we denote by $\pi$. The map $\pi$ has dense range ($\emptyset\not= O_r$ ${\Rightarrow}$ $r\notin\Theta_\kappa$ ${\Rightarrow}$ $\exists p\in M^\kappa\;r(p)\not=1$ ${\Rightarrow}$ $\exists p\; r\notin\pi(p)$ ${\Rightarrow}$ $\exists p\;\pi(p)\in O_r$), but it is not necessarily continuous; it is continuous in the two cases that most concern us, namely in classical logic (see Lemma \[ref25\]) and in [Łukasiewicz]{} logic (see the next section). Let $\sigma:{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}\to{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}$ be a substitution, $s_i=\sigma(x_i)$. Then the $\kappa$-tuple $(\ldots,s_i,\ldots)$ determines a function $S:M^\kappa\to M^\kappa$ via $p\mapsto(\ldots,s_i(p),\ldots)$. Since $\pi(S(p))=\{r\in{\operatorname{Free}_{\kappa}({\mathbf{V}}M)}: r(\ldots,s_i(p),\ldots)=1\}=\{r:[\sigma(r)](p)=1\}= \sigma^{-1}(\{t:t(p)=1\})=\sigma^*(\pi(p))$, the diagram $$\tag{$*$} \begin{split} \begin{xy} \xymatrix{ {M^\kappa} \ar[r]^S \ar[d]_{\pi} & {M^\kappa} \ar[d]^{\pi} \\ {X_\kappa} \ar[r]_{\sigma^*} & {X_\kappa} } \end{xy} \end{split}$$ commutes. As $\pi$ has dense range, $S$ determines the continuous function $\sigma^*$. We call $\sigma^*$ the [*dual*]{} of $\sigma$, and $S$ the [*mapping on truth-values*]{} induced by $\sigma$. In\[ref25\] classical logic, the map $\pi:{\mathbf{2}}^\kappa \to X_\kappa=\operatorname{Spec}{\operatorname{Free}_{\kappa}({{\mathit{Boole}}})}$ is a homeomorphism. Recall from the end of Section \[ref24\] that ${\operatorname{Free}_{\kappa}({{\mathit{Boole}}})}$ is the boolean algebra of all clopen subsets of ${\mathbf{2}}^\kappa$, where the latter space is given the product topology. If $p$ and $q$ are distinct points of ${\mathbf{2}}^\kappa$, then there exists a clopen set $C$ containing $p$ and not containing $q$. Hence $C\in\pi(p)\setminus\pi(q)$ and $\pi$ is injective (as usual, we are identifying clopen subsets with their characteristic functions). Let ${\mathfrak{p}}\in X_\kappa$. Since ${\mathfrak{p}}$ is a proper filter, $\emptyset\notin{\mathfrak{p}}$ and in particular the intersection of any finite family of elements of ${\mathfrak{p}}$ is nonempty. By compactness, $\bigcap{\mathfrak{p}}$ contains a point $p\in{\mathbf{2}}^\kappa$. Since $C\in{\mathfrak{p}}$ implies $p\in C$, we have ${\mathfrak{p}}\subseteq\pi(p)$. But in a boolean algebra every prime filter is maximal, and hence ${\mathfrak{p}}=\pi(p)$. So $\pi$ is a bijection. Both in ${\mathbf{2}}^\kappa$ and in $X_\kappa$ the clopen sets generate the topology; moreover, as shown before Addendum \[ref11\], the mapping $C\mapsto F_C$ is a bijection between the two families of clopen sets. Since $p\in C$ iff $C\in\pi(p)$ iff $\pi(p)\in F_C$, the map $\pi$ is a homeomorphism. Given a point $p$ and a nonvoid open subset $O$ of the Cantor space ${\mathbf{2}}^\omega$, one easily constructs a homeomorphism $S:{\mathbf{2}}^\omega\to{\mathbf{2}}^\omega$ such that $S(p)\in O$ (if $p=(p_0,p_1,\ldots)$ and $[{a_0,\ldots ,a_n}]=\{q\in{\mathbf{2}}^\omega:q_i=a_i\text{ for }i=0,\ldots,n\}$ is a block contained in $O$, then the mapping that exchanges $0$ with $1$ in those indices $i$ for which $p_i\not=a_i$ is such a homeomorphism). Hence not only $\Sigma_\omega$, but even $\Xi_\omega$ acts minimally on $\operatorname{Spec}{\operatorname{Free}_{\omega}({{\mathit{Boole}}})}$. Of course we can do better than that, because there exist many minimal homeomorphisms of the Cantor space. The simplest example is obtained by identifying ${\mathbf{2}}^\omega$ with the topological group of $2$-adic integers ${\mathbb{Z}}_2$, and letting $S$ be the translation by $1$: $S(p)=p+1$. Let us compute the substitution $\sigma$ on ${\operatorname{Free}_{\omega}({{\mathit{Boole}}})}$ for which $S=\sigma^*$. If $x_i$ is the $i$-th free generator, then $F_{x_i}=\{p\in{\mathbf{2}}^\omega:p_i=1\}$, and $F_{\sigma(x_i)}=(\sigma^*)^{-1}[F_{x_i}]=\{p:S(p)\in F_{x_i}\}=\{p:(p+1)_i=1\}$. Since addition in ${\mathbb{Z}}_2$ is just addition in base $2$ with carry, we have that $p\in F_{\sigma(x_i)}$ iff - either $p_i=1$ and $p_j=0$ for some $j<i$; - or $p_i=0$ and $p_j=1$ for every $j<i$. Therefore $$\begin{aligned} F_{\sigma(x_i)} &= \bigl[F_{x_i}\cap(O_{x_0}\cup\cdots\cup O_{x_{i-1}})\bigr] \\ &\quad \cup\bigl[O_{x_i}\cap F_{x_0}\cap\cdots\cap F_{x_{i-1}}\bigr] \\ &= \bigl[F_{x_i}\cap(F_{\neg x_0}\cup\cdots\cup F_{\neg x_{i-1}})\bigr] \\ &\quad \cup\bigl[F_{\neg x_i}\cap F_{x_0}\cap\cdots\cap F_{x_{i-1}}\bigr].\end{aligned}$$ Consider the following formulas: $$\begin{aligned} s_0 &= \neg x_0 \\ s_i &= \bigl[x_i\land(\neg x_0\lor\cdots\lor\neg x_{i-1})\bigr] \lor\bigl[\neg x_i\land x_0\land\cdots\land x_{i-1}\bigr] \\ &= x_i\triangle(x_0\land\cdots\land x_{i-1}) \quad\text{(for $i>0$)}\end{aligned}$$ ($\triangle$ is the boolean symmetric difference: $a\triangle b=(a\land\neg b)\lor(\neg a\land b)$). Then $F_{\sigma(x_i)}=F_{s_i}$ by the isomorphism cited before Addendum \[ref11\]. The required substitution is therefore the one defined by $\sigma(x_i)=s_i$. From the point of view of proof systems, we have thus obtained the following result. From\[ref14\] the set of boolean tautologies plus any given non-tautology we can derive every formula using only Modus Ponens and the substitution $\sigma$ given above. [Łukasiewicz]{} logic ===================== In the rest of this paper we will concentrate on [Łukasiewicz]{} logic; we therefore fix $M={[0,1]}$ endowed with the connectives in Example \[ref3\](3). A key distinguishing feature of the [Łukasiewicz]{} connectives is their continuity with respect to the standard topology of ${[0,1]}$. As a matter of fact [Łukasiewicz]{} logic is the only t-norm based logic in which all connectives are continuous [@menupav]. A [*\[rational\] cellular complex over $M^n={[0,1]}^n$*]{} is a finite set $W$ of [*cells*]{} (i.e., compact convex polyhedrons), whose union is ${[0,1]}^n$, and such that: 1. every vertex of every cell of $W$ has rational coordinates; 2. if $C\in W$ and $D$ is a face of $C$, then $D\in W$; 3. every two cells intersect in a common face. A [*McNaughton function*]{} is a continuous function $f:{[0,1]}^n\to{[0,1]}$ for which there exists a complex as above and affine linear functions with integer coefficients $F_j(\bar x)=a_j^1x_1+\cdots+a_j^nx_n+a_j^{n+1}$, in 1-1 correspondence with the $n$-dimensional cells $C_j$ of the complex, such that $f\restriction C_j=F_j$ for each $j$. The\[ref15\] elements of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$ (i.e., the functions from ${[0,1]}^n$ to ${[0,1]}$ induced by a formula of [Łukasiewicz]{} logic) are exactly the McNaughton functions. Here are typical McNaughton functions, for $n=1$ and $n=2$: they are induced by the formulas $\neg x_0\lor\bigl((x_0\land\neg x_0)\oplus (x_0\land\neg x_0)\bigr)$ and $(x_0\to x_1)\land(x_0\oplus x_0\oplus x_1\oplus x_1)$, respectively. {width="3cm" height="3cm"} {width="3cm" height="3cm"} Given a substitution $\sigma:{\operatorname{Free}_{n}({{\mathit{MV}}})}\to{\operatorname{Free}_{n}({{\mathit{MV}}})}$, to each function $s_i=\sigma(x_i)$ there corresponds a cellular complex $W_i$ such that $s_i$ is affine linear on each cell of $W_i$. Let $W$ be a complex that is a common refinement of ${W_0,\ldots ,W_{n-1}}$. Then on each cell $C_j$ of $W$ the function $S:{[0,1]}^n\to{[0,1]}^n$ defined before Lemma \[ref25\] is given by $$\tag{$**$} \begin{pmatrix} p_0 \\ \vdots \\ p_{n-1} \end{pmatrix} \mapsto A_j\begin{pmatrix} p_0 \\ \vdots \\ p_{n-1} \end{pmatrix} +B_j,$$ where $A_j$ is an $n\times n$ matrix and $B_j$ a column vector, both having integer coefficients. Conversely, every continuous selfmapping $S$ of ${[0,1]}^n$ which is piecewise affine linear with integer coefficients (i.e., is locally expressible in the form $(**)$, using finitely many $A_j$’s and $B_j$’s) is induced by some endomorphism $\sigma$ of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$. We call such an $S$ a [*McNaughton mapping*]{}; if moreover $S$ is invertible we call it a [*McNaughton homeomorphism*]{}; McNaughton homeomorphisms on ${[0,1]}^n$ are exactly the mappings on truth-values induced by the automorphisms of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$. Apart from its relevance in [Łukasiewicz]{} logic, the class of McNaughton mappings is quite interesting *per se*. Consider the following complexes over ${[0,1]}^2$; they are both symmetric under a $\pi$ rotation about the centre of the square. {height="3cm" width="3cm"} {height="3cm" width="3cm"} The vertices of the lower inner triangle are $p_0=(1/4,1/4)$, $p_1=(1/2,1/4)$, $p_2=(1/4,1/2)$; for $0\le i\le 2$, let $p_i'$ be the vertex symmetric to $p_i$. Then there exists a unique homeomorphism $S$ such that: 1. $S(p_i)=p_{i+1\pmod{3}}$, and $S(p'_i)=p'_{i+1\pmod{3}}$; 2. every other vertex is fixed; 3. $S$ is affine linear on each cell. In short, the first complex is mapped onto the second by “rotating counterclockwise” the two inner triangles, and distorting accordingly the border triangles. As a matter of fact, $S$ is topologically conjugate to the union of two twists [@pantibernoulli §5]. The data above determine the matrix $A_j$ and the column vector $B_j$ on each triangle $C_j$. One checks directly that all these matrices and vectors have integer entries; hence $S$ is a McNaughton homeomorphism. In doing computations, it is expedient to write $p=({p_0,\ldots ,p_{n-1}})\in{[0,1]}^n$ using projective coordinates $({a_0:\ldots :a_n})\sim({p_0:\ldots :p_{n-1}}:1)$. For example, if $C_1$ is the triangle ${\langle p_0,(1,0),p_1 \rangle}$, which is mapped to ${\langle p_1,(1,0),p_2 \rangle}$, then $A_1$ and $B_1$ are the upper left $2\times 2$ matrix and upper right $2\times 1$ column vector in the matrix $$\begin{pmatrix} -1 & -5 & 2 \\ 1 & 4 & -1 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 & 1 \\ 1 & 0 & 2 \\ 4 & 1 & 4 \end{pmatrix} \begin{pmatrix} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 4 & 1 & 4 \end{pmatrix}^{-1}.$$ If $S$ is induced by $\sigma$, then the action of $S$ on ${[0,1]}^n$ is just the surface of the action of the full dual $\sigma^*$ on $X_n$. Indeed, as proved in [@mundicijfa Proposition 8.1], in the case of [Łukasiewicz]{} logic the map $\pi$ in the diagram $(*)$ is a homeomorphic embedding of ${[0,1]}^n$ onto the subspace of maximal filters. By Lemma \[ref7\], the points of $X_n$ (indeed, of any spectrum) form a forest under the [*specialization order*]{}: ${\mathfrak{p}}\le{\mathfrak{q}}$ iff ${\mathfrak{q}}$ is in the closure of $\{{\mathfrak{p}}\}$ iff ${\mathfrak{p}}\subseteq{\mathfrak{q}}$ (a [*forest*]{}, sometimes called a [*root system*]{}, is a poset in which the elements greater than any given element form a chain). A full description of $X_n$ is given in [@pantiprime]. Since it is rather involved, here we limit ourselves to the cases $n=1$ and $n=2$. Assume $n=1$. If $p\in(0,1)$ is rational, then there are two incomparable prime filters $\pi(p)^+$ and $\pi(p)^-$ properly contained in the maximal $\pi(p)$. Namely, $\pi(p)^-$ is the filter of all McNaughton functions $:{[0,1]}\to{[0,1]}$ that are $1$ in a left neighborhood of $p$, and analogously for $\pi(p)^+$ w.r.t. right neighborhoods. The only prime filter contained in $\pi(0)$ (respectively, in $\pi(1)$) is $\pi(0)^+$ (respectively, $\pi(1)^-$). If $p$ is irrational, then $\pi(p)$ is minimal in the specialization order. Now assume $n=2$, $p=(p_0,p_1)\in(0,1)^2$. If $p_0,p_1,1$ are linearly independent over ${\mathbb{Q}}$, then $\pi(p)$ is a minimal —as well as maximal— prime filter. If $p_0,p_1,1$ satisfy exactly one (up to scalar multiples) nontrivial linear dependence over ${\mathbb{Q}}$, then there are two incomparable prime filters below $\pi(p)$, and both of them are minimal. Otherwise, consider the unit circle $S^1$ in the tangent space to $p$. For every $u\in S^1$ there is a prime filter ${\mathfrak{p}}_u$ contained in $\pi(p)$. If the line in the tangent space connecting the origin with $u$ does not hit any point with rational coordinates, then ${\mathfrak{p}}_u$ is minimal. Otherwise, ${\mathfrak{p}}_u$ contains two minimal prime filters ${\mathfrak{p}}_u^+$ and ${\mathfrak{p}}_u^-$. For $p$ along the border of the unit square, this description gets modified in the obvious way. The following picture may clarify the situation: {width="4cm" height="2cm"} Let $n$ (not necessarily $n=1,2$) be given, ${\mathfrak{p}}\in X_n$. Let ${\mathfrak{p}}={\mathfrak{p}}_0\subset {\mathfrak{p}}_1\subset\cdots \subset{\mathfrak{p}}_t$ be the chain, of length $t$, of elements above ${\mathfrak{p}}$ in the specialization order. Given an endomorphism $\sigma$ of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$, we have $$\frac{{\operatorname{Free}_{n}({{\mathit{MV}}})}}{\sigma^{-1}[{\mathfrak{p}}]} \simeq \frac{\sigma[{\operatorname{Free}_{n}({{\mathit{MV}}})}]}{{\mathfrak{p}}\cap\sigma[{\operatorname{Free}_{n}({{\mathit{MV}}})}]} \subseteq \frac{{\operatorname{Free}_{n}({{\mathit{MV}}})}}{{\mathfrak{p}}},$$ and hence the MV-algebra ${\operatorname{Free}_{n}({{\mathit{MV}}})}/\sigma^*({\mathfrak{p}})$ is a subalgebra of ${\operatorname{Free}_{n}({{\mathit{MV}}})}/{\mathfrak{p}}$. By [@pantiprime Theorem 4.7(i) and Corollary 4.9], this implies that the length of the chain above $\sigma^*({\mathfrak{p}})$ is less than or equal to $t$. Since ${[0,1]}^n$ is dense in $X_n$, in principle we might reduce the study of $\sigma^*$ to the study of $S$. However, taking into consideration the action of $\sigma^*$ on the full spectrum gives us deeper insight. For example, it is possible to provide an intrinsic (i.e., coordinate-free) characterization of the differentials $T_pS$ of a McNaughton mapping $S$. Differentials of piecewise-linear maps have been constructed by Tsujii in [@Tsujii01]; we show here how Tsujii’s construction can be intrinsically described in purely algebraic terms. It may be helpful for the reader to recall the coordinate-free description of the differentials of a morphism $S:X\to Y$ of differentiable varieties. Let $p$ be a point of $X$, $q=S(p)$, ${\mathcal{O}}_p$ and ${\mathcal{O}}_q$ the rings of germs of differentiable functions at $p$ and $q$, respectively. ${\mathcal{O}}_p$ and ${\mathcal{O}}_q$ are local rings: let ${\mathfrak{m}}$ and ${\mathfrak{n}}$ be the respective maximal ideals (i.e., ${\mathfrak{m}}=\{f\in{\mathcal{O}}_p:f(p)=0\}$, and analogously for ${\mathfrak{n}}$). The mapping $\sigma:{\mathcal{O}}_q\to{\mathcal{O}}_p$ defined by $\sigma(g)=g\circ S$ is a well-defined ring homomorphism, and $\sigma[{\mathfrak{n}}]\subseteq{\mathfrak{m}}$. Therefore, $\sigma$ induces a vector space homomorphism from ${\mathfrak{n}}/{\mathfrak{n}}^2$ to ${\mathfrak{m}}/{\mathfrak{m}}^2$, which we denote by $\bar{\sigma}$. The tangent spaces $T_pX$ and $T_qY$ are canonically isomorphic to the dual vector spaces $({\mathfrak{m}}/{\mathfrak{m}}^2)'$ and $({\mathfrak{n}}/{\mathfrak{n}}^2)'$, respectively, and under these isomorphisms the differential $T_pS$ corresponds to the dual mapping $\bar{\sigma}':({\mathfrak{m}}/{\mathfrak{m}}^2)'\to({\mathfrak{n}}/{\mathfrak{n}}^2)'$. Explicitly, if $T_pX\ni v:{\mathfrak{m}}/{\mathfrak{m}}^2\to{\mathbb{R}}$ is a tangent vector at $p$, then $(T_pS)(v)$ is the tangent vector at $q$ defined by $[(T_pS)(v)](g/{\mathfrak{n}}^2)=(v\circ\bar{\sigma})(g/{\mathfrak{n}}^2)= v((g\circ S)/{\mathfrak{m}}^2)$. We will develop an analogous description for piecewise-linear maps. Before doing so, we need a few more preliminaries; see [@bkw], [@andersonfei] for more details and unproved claims. A [*lattice-ordered abelian group*]{} ([*[$\ell$-group]{}*]{} for short) is a structure $(G,+,-,0,\land,\lor)$ such that $(G,+,-,0)$ is an abelian group, $(G,\land,\lor)$ is a lattice, and $+$ distributes over the lattice operations. [$\ell$-homomorphisms]{} of [$\ell$-groups]{} are groups homomorphisms that are also lattice homomorphisms. The class of all [$\ell$-groups]{} is equational and, as such, contains free objects. The free [$\ell$-group]{} over $n$ generators, $\operatorname{F\ell}(n)$, is the [$\ell$-group]{} (under pointwise operations) of all functions $g:{\mathbb{R}}^n\to{\mathbb{R}}$ that are continuous and piecewise-linear with integer coefficients (i.e., there exist finitely many homogeneous linear polynomials ${g_1,\ldots ,g_m}\in{\mathbb{Z}}[{x_1,\ldots ,x_n}]$ such that, for every $w\in{\mathbb{R}}^n$, $g(w)=g_j(w)$ for some $1\le j\le m$). The set $\operatorname{\ell-Hom}(\operatorname{F\ell}(n),{\mathbb{R}})$ of all [$\ell$-homomorphisms]{} from $\operatorname{F\ell}(n)$ to ${\mathbb{R}}$ is in 1-1 correspondence with ${\mathbb{R}}^n$, via the map that associates to $w\in{\mathbb{R}}^n$ the evaluation mapping $\varphi_w:g\mapsto g(w)$. A [*strong unit*]{} of the [$\ell$-group]{} $G$ is an element $0\le u\in G$ such that, for every $g\in G$, $g\le nu$ for some positive integer $n$. If $u$ is a strong unit of $G$, then the interval $[0,u]= \{g\in G:0\le g\le u\}$ can be given the structure of an MV-algebra $\Gamma(G,u)=([0,u],\oplus,\neg,0,1)$ by setting $g\oplus h=(g+h)\land u$, $\neg g=u-g$, $0=0_G$, $1=u$. The mapping $(G,u)\mapsto\Gamma(G,u)$ is functorial, and determines a categorical equivalence between the category of [$\ell$-groups]{} with strong unit and the category of MV-algebras [@mundicijfa]. In particular, the filters of $\Gamma(G,u)$ are in natural 1-1 correspondence with the kernels of [$\ell$-homomorphisms]{} of domain $G$. The preliminaries being over, let $S:{[0,1]}^n\to{[0,1]}^n$ be a McNaughton mapping, $p\in{[0,1]}^n$, $q=S(p)$. Then ${\mathfrak{m}}=\pi(p)$ and ${\mathfrak{n}}=\pi(q)$ are maximal filters of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$. Given ${\mathfrak{p}}\in X_n$, the [*germinal filter*]{} corresponding to ${\mathfrak{p}}$ is the filter ${\mathfrak{g}}_{\mathfrak{p}}=\bigcap\{{\mathfrak{q}}\in X_n:{\mathfrak{q}}\subseteq{\mathfrak{p}}\}$. By [@bkw Proposition 10.5.3 and Definition 10.5.6], and using the properties of the $\Gamma$ functor, the quotient $A_p={\operatorname{Free}_{n}({{\mathit{MV}}})}/{\mathfrak{g}}_{\mathfrak{m}}$ is the MV-algebra of germs at $p$ of McNaughton functions; analogously for $A_q={\operatorname{Free}_{n}({{\mathit{MV}}})}/{\mathfrak{g}}_{\mathfrak{n}}$. The MV-algebra $A_p$ is [*local*]{}, i.e., has a unique maximal filter ${\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}}$. Let $\sigma$ be the endomorphism of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$ that induces $S$. Notation being as above, $\sigma[{\mathfrak{n}}]\subseteq{\mathfrak{m}}$ and $\sigma[{\mathfrak{g}}_{\mathfrak{n}}]\subseteq{\mathfrak{g}}_{\mathfrak{m}}$. By the commutativity of the diagram $(*)$, $\sigma^{-1}[{\mathfrak{m}}]= \sigma^*({\mathfrak{m}})={\mathfrak{n}}$, so the first statement is immediate. Let ${\mathfrak{p}}$ be a prime filter below ${\mathfrak{m}}$ in the specialization order. Since $\sigma^*$ is continuous and ${\mathfrak{m}}$ is in the closure of $\{{\mathfrak{p}}\}$, the maximal filter ${\mathfrak{n}}=\sigma^*({\mathfrak{m}})$ must be in the closure of $\{\sigma^*({\mathfrak{p}})\}$. Therefore $\sigma^*({\mathfrak{p}})$ is below ${\mathfrak{n}}$, hence ${\mathfrak{g}}_{\mathfrak{n}}\subseteq\sigma^*({\mathfrak{p}})= \sigma^{-1}[{\mathfrak{p}}]$ and $\sigma[{\mathfrak{g}}_{\mathfrak{n}}]\subseteq{\mathfrak{p}}$. As a consequence, $\sigma$ determines a homomorphism of [$\ell$-groups]{}$$\bar{\sigma}:{\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}\to {\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}},$$ which plays the rôle of the codifferential on cotangent spaces. Since the composition of [$\ell$-homomorphisms]{} is an [$\ell$-homomorphism]{}, $\bar{\sigma}$ induces a dual mapping $\bar{\sigma}':\operatorname{\ell-Hom}({\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}},{\mathbb{R}})\to \operatorname{\ell-Hom}({\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}},{\mathbb{R}})$ by $\bar{\sigma}'(\varphi)=\varphi\circ\bar{\sigma}$. Under\[ref22\] the identification $w\mapsto\varphi_w$ of ${\mathbb{R}}^n$ with $\operatorname{\ell-Hom}(\operatorname{F\ell}(n),{\mathbb{R}})$ described above, the map $\bar{\sigma}'$ corresponds to Tsujii’s differential. For simplicity’s sake, we assume that $p$ and $q=S(p)$ have rational coordinates and are in the topological interior of the $n$-cube: we will discuss in Addendum \[ref21\] how these assumptions can be discarded. Write $q$ in projective coordinates $({a_0:\ldots :a_n})$, with $a_n>0$, and let $Q=({a_0,\ldots ,a_n})\in{\mathbb{R}}^{n+1}$. Let ${\mathfrak{N}}$ be the kernel of $\varphi_Q$, and let ${\mathfrak{G}}_{\mathfrak{N}}$ be the germinal kernel associated to ${\mathfrak{N}}$ [@bkw Proposition 10.5.3]. Since $q$ has rational coordinates, $Q$ has rank $1$ according to [@pantiprime p. 188]. By [@pantiprime Theorem 4.8], the quotient ${\mathfrak{N}}/{\mathfrak{G}}_{\mathfrak{N}}$ is an [$\ell$-group]{}, which is [$\ell$-isomorphic]{} to $\operatorname{F\ell}(n)$ under the map $D_Q$ defined in [@pantiprime Definition 2.2]. By the properties of the $\Gamma$ functor, the [$\ell$-groups]{} ${\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}$ and ${\mathfrak{N}}/{\mathfrak{G}}_{\mathfrak{N}}$ are [$\ell$-isomorphic]{} as well. We therefore obtain an [$\ell$-isomorphism]{} $D_q:{\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}\to \operatorname{F\ell}(n)$ which, by explicit computation, has the form $$\bigl[D_q(r/{\mathfrak{g}}_{\mathfrak{n}})\bigr](w)= \lim_{h\to0^+}\frac{r(q+hw)-r(q)}{h}.$$ We have of course an analogous [$\ell$-isomorphism]{}  $D_p:{\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}}\to\operatorname{F\ell}(n)$. Let $\mathcal{D}_pS$ denote the Tsujii differential of $S$ at $p$, and let $v\in{\mathbb{R}}^n$. Since $S$ is continuous and defined everywhere on ${[0,1]}^n$, the definition in [@Tsujii01 Eq. (13)] simplifies to $$(\mathcal{D}_pS)(v)= \lim_{h\to0^+}\frac{S(p+hv)-S(p)}{h}.$$ We want to show that $\bar{\sigma}'(\varphi_v\circ D_p)= \varphi_{(\mathcal{D}_pS)(v)}\circ D_q$. Setting $(\mathcal{D}_pS)(v)=w\in{\mathbb{R}}^n$, this amounts to the commutativity of the diagram $$\begin{xy} \xymatrix{ {{\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}} \ar[0,2]^{\bar{\sigma}} \ar[d]_{D_q} & & {{\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}}} \ar[d]^{D_p} \\ {\operatorname{F\ell}(n)} \ar[r]_{\varphi_w} & {{\mathbb{R}}} & {\operatorname{F\ell}(n)} \ar[l]^{\varphi_v} } \end{xy}$$ Choose $r/{\mathfrak{g}}_{\mathfrak{n}}\in{\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}$. As remarked in [@Tsujii01 p. 358], the terms $h^{-1}\bigl(r(q+hw)-r(q)\bigr)$, $h^{-1}\bigl((r\circ S)(p+hv)-(r\circ S)(p)\bigr)$, and $h^{-1}\bigl(S(p+hv)-S(p)\bigr)$ take constant values for sufficiently small $h>0$. If $h_0$ is such an $h$, we get $$\begin{aligned} \varphi_w\bigl(D_q(r/{\mathfrak{g}}_{\mathfrak{n}})\bigr) &= \bigl(D_q(r/{\mathfrak{g}}_{\mathfrak{n}})\bigr)(w) \\ &= h_0^{-1}\bigl(r(q+h_0w)-r(q)\bigr) \\ &= h_0^{-1}\bigl(r(q+S(p+h_0v)-S(p))-r(q)\bigr) \\ &= h_0^{-1}\bigl((r\circ S)(p+h_0v)-(r\circ S)(p)\bigr) \\ &= \bigl(D_p((r\circ S)/{\mathfrak{g}}_{\mathfrak{m}})\bigr)(v) \\ &= \varphi_v\bigl(D_p(\bar{\sigma}(r/{\mathfrak{g}}_{\mathfrak{n}}))\bigr),\end{aligned}$$ as required. In\[ref21\] the proof of Theorem \[ref22\] we assumed that $p$ and $q$ have rational coordinates and are in the topological interior of ${[0,1]}^n$. The first assumption is motivated by the fact that, by definition, McNaughton mappings have integer coefficients. This implies that the only tangent vectors, say at $q$, that can be algebraically recognized are those in the ${\mathbb{R}}$-span $V$ of the set of all $v\in{\mathbb{R}}^n$ such that the affine line through $q$ and $q+v$ is the intersection of affine hyperspaces having integer (equivalently, rational) coefficients. The [$\ell$-group]{} ${\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}$ is then [$\ell$-isomorphic]{} to $\operatorname{F\ell}(k)$, for $k=\dim(V)\le n$, and the algebraic tangent space $({\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}})'$ is isomorphic to ${\mathbb{R}}^k$. According to the personal interests, one may either accept these underdimensional tangent spaces, or treat them drastically, by tensoring everything with ${\mathbb{R}}$. This means considering piecewise-linear functions with arbitrary real coefficients, so dropping the assumption that the cellular complexes involved have rational vertices, and passing from [$\ell$-groups]{} and MV-algebras to real vector lattices [@baker68] and their $\Gamma$ images. All quotients ${\mathfrak{n}}/{\mathfrak{g}}_{\mathfrak{n}}$ are then isomorphic to the free vector lattice over $n$ generators $\operatorname{FVL}(n)$ [@pantiprime Theorem 3.8]. The dual of $\operatorname{FVL}(n)$, i.e., the set of all ${\mathbb{R}}$-linear [$\ell$-homomorphisms]{} from $\operatorname{FVL}(n)$ to ${\mathbb{R}}$ is still in bijection with ${\mathbb{R}}^n$ via the evaluation mapping, and all dimensionality problems disappear. About the other assumption: if $p$ or $q$ (say $p$) is on the boundary of ${[0,1]}^n$, then the quotient ${\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}}$ is [$\ell$-isomorphic]{}not to $\operatorname{F\ell}(n)$, but to a quotient of $\operatorname{F\ell}(n)$ by a principal kernel or, equivalently, to the [$\ell$-group]{} of restrictions of the elements of $\operatorname{F\ell}(n)$ to a polyhedral cone $W$. The dual $({\mathfrak{m}}/{\mathfrak{g}}_{\mathfrak{m}})'$ is then in bijection with $W$, in agreement with [@Tsujii01 Eq. (14)], and the proof of Theorem \[ref22\] carries on. Chaotic actions =============== Let $p=(\ldots,p_i,\ldots)\in{[0,1]}^\kappa$. We say that $p$ has [*finite denominator*]{} if all $p_i$’s are rational numbers, and there exists $0<d\in{\mathbb{Z}}$ such that $dp\in{\mathbb{Z}}^\kappa$. The least such $d$ is the [*denominator*]{} of $p$, written $\operatorname{den}(p)$. Let\[ref16\] $p,q\in{[0,1]}^\kappa$. - If $p$ has finite denominator and $\sigma\in\Sigma_\kappa$, then $S(p)$ has finite denominator and $\operatorname{den}(S(p))\mid\operatorname{den}(p)$. - If $\operatorname{den}(q)\mid\operatorname{den}(p)$, then $S(p)=q$ for some $\sigma\in\Sigma_\kappa$. - If $p$ does not have finite denominator, then the $\Sigma_\kappa$-orbit of $p$ is dense. \(i) Let $d=\operatorname{den}(p)$, $S(p)=(\ldots,q_i,\ldots)$, $\sigma(x_i)= s_i({x_{j_1},\ldots ,x_{j_n}})$. Then $q_i=s_i({p_{j_1},\ldots ,p_{j_n}})= a^1p_{j_1}+\cdots+a^np_{j_n}+a^{n+1}$, for some $a^1,\ldots,a^{n+1}\in{\mathbb{Z}}$. Therefore $dq_i\in{\mathbb{Z}}$ and $dS(p)\in{\mathbb{Z}}^\kappa$, and our claim easily follows.(ii) Let $d=\operatorname{den}(p)$. The integers $\{dp_i:i<\kappa\}\cup\{d\}$ must be relatively prime (otherwise $\operatorname{den}(p)$ would be smaller than $d$). Therefore there exist indices ${j_1,\ldots ,j_m}$ and integer numbers $a^1,\ldots,a^m,a^{m+1}$ such that $d(a^1p_{j_1}+\cdots+ a^mp_{j_m}+a^{m+1})=1$. Let $n\le\kappa$ be greater than ${j_1,\ldots ,j_m}$. Then the affine linear polynomial $f_i=q_id(a^1x_{j_1}+\cdots+a^mx_{j_m}+a^{m+1})$ has integer coefficients, since $\operatorname{den}(q)\mid d$. The function $(f_i\lor 0)\land 1: {[0,1]}^n\to{[0,1]}$ is a McNaughton function, hence by Theorem \[ref15\] it is expressible via a formula $s_i\in{\operatorname{Free}_{n}({{\mathit{MV}}})}$. Since $s_i({p_{j_1},\ldots ,p_{j_n}})=q_i$, the substitution $\sigma\in\Sigma_\kappa$ defined by $\sigma(x_i)=s_i$ satisfies our requirements.(iii) It suffices to show that, for every $0\le a<b\le 1$, there exists an element $s\in{\operatorname{Free}_{\kappa}({{\mathit{MV}}})}$ such that $a<s(p)<b$. Let $G$ be the additive subgroup of ${\mathbb{R}}$ generated by $\{p_i:i<\kappa\}\cup\{1\}$. Since $p$ does not have finite denominator, $G$ is dense in ${\mathbb{R}}$, and therefore there exist indices ${j_1,\ldots ,j_m}$ and integer numbers $a^1,\ldots,a^m,a^{m+1}$ such that $a<a^1p_{j_1}+\cdots+ a^mp_{j_m}+a^{m+1}<b$. One then argues as in (ii) above. In the following we will tacitly identify via $\pi$ the $\kappa$-cube ${[0,1]}^\kappa$ with the subspace of $X_\kappa$ whose elements are the maximal filters. Let\[ref17\] $\operatorname{Rat}_n$ be the set of rational points in ${[0,1]}^n$. Then $\operatorname{Rat}_n$ is a dense subset both of ${[0,1]}^n$ and of $X_n$. All the elements of $\operatorname{Rat}_n$ have a finite $\Sigma_n$-orbit. No point of $X_\omega$ has a finite $\Sigma_\omega$-orbit. $\operatorname{Rat}_n$ coincides with the set of points in ${[0,1]}^n$ having finite denominator, and is dense in ${[0,1]}^n$. As the $n$-cube is dense in $X_n$, $\operatorname{Rat}_n$ is dense in $X_n$ as well. If $p\in\operatorname{Rat}_n$, then by Lemma \[ref16\] the $\Sigma_n$-orbit of $p$ is the set of points whose denominator divides $\operatorname{den}(p)$, and this set is finite. Since the submonoid of $\Sigma_\omega$ whose elements are all the substitutions $\sigma$ such that $\sigma(x_i)\in{\{0,1\}}$ has the cardinality of the continuum, our last claim is immediate. By Lemma \[ref16\](i) $\Sigma_\kappa$ does not act minimally on $X_\kappa$. We are therefore lead to weaken the requirement of minimality to that of topological transitivity: we say that $\Pi\subseteq\Sigma_\kappa$ is [*topologically transitive*]{} on $X_\kappa$ if $(\Pi,O)$ is dense in $X_\kappa$, for every nonempty open set $O$. Using the fact that ${[0,1]}^\kappa$ is dense in $X_\kappa$, one shows easily that $\Pi$ is topologically transitive on $X_\kappa$ iff it is topologically transitive on the $\kappa$-cube. By standard arguments [@Walters82 Theorem 5.9], this amounts to the existence of a point in ${[0,1]}^\kappa$ (or a $G_\delta$ dense set of such points) whose $\Pi$-orbit is dense. If $\Pi$ is topologically transitive and the set of points whose $\Pi$-orbit is finite is dense, then we say that $\Pi$ acts [*chaotically*]{}. By Lemma \[ref16\](iii) and Corollary \[ref17\], $\Sigma_n$ acts chaotically both on $X_n$ and on ${[0,1]}^n$. It is well known that a chaotic action on a space such as ${[0,1]}^n$ implies sensitive dependence on initial conditions, hence chaotic behaviour in the sense of Devaney [@Banksetal92]. One constructs easily chaotic elements of $\Sigma_n$. Indeed, the standard tent map on ${[0,1]}$ is a McNaughton function, expressible by the formula $s(x_0)=(x_0\land\neg x_0)\oplus(x_0\land\neg x_0)$. The substitution $\sigma:x_i\mapsto s(x_i)$, for $i<n$, induces therefore on ${[0,1]}^n$ the direct product of $n$ tent maps, which is mixing w.r.t. Lebesgue measure. Hence $\sigma$ acts in a topologically transitive and chaotic way. It is not so easy to construct elements of $\Xi_n$ which are chaotic; we will obtain such mappings for even $n$ in Corollary \[ref20\]. Let $\sigma\in\Xi_n$, let $S$ be the induced McNaughton homeomorphism of ${[0,1]}^n$, and let $W$ be a complex over the $n$-cube such that $S$ has the form $(**)$ on each $n$-dimensional cell $C_j$ of $W$. Then all the matrices $A_j$ have the same determinant, which is either $+1$ or $-1$. Since the inverse of $S$ is expressible as in $(**)$ via matrices and vectors having integer entries, it is clear that all matrices $A_j$ are invertible and their inverses have integer entries. Therefore all $A_j$’s have determinant $\pm1$. Suppose by contradiction that the $n$-dimensional cells of $W$ are ${C_1,\ldots ,C_k}$ and that there is some $1\le r<k$ such that $\det(A_j)=+1$ for $1\le j\le r$, and $\det(A_j)=-1$ for $r<j\le k$. If $1\le j'\le r$ and $r<j''\le k$, then $C_{j'}$ and $C_{j''}$ cannot intersect in an $(n-1)$-dimensional face, because this would contradict the injectivity of $S$. Let $D$ be the topological interior of the $n$-cube, $E=D\cap(C_1\cup\cdots\cup C_r)$, $F=D\cap(C_{r+1}\cup\cdots\cup C_k)$. Then $E$ and $F$ are nonempty, and closed in the relative topology of $D$. By [@kelley Theorem 1.17], $(E\setminus F)\cup(F\setminus E)=D\setminus(E\cap F)$ is not connected, and this contradicts [@hockingyou Theorem 3.61], since $E\cap F$ has topological dimension $\le n-2$. For\[ref18\] every $n$ and every $\sigma\in\Xi_n$, the homeomorphism $S$ preserves the Lebesgue measure on ${[0,1]}^n$. The only invertible substitutions on ${\operatorname{Free}_{1}({{\mathit{MV}}})}$ are the identity and the flip $x_0\mapsto \neg x_0$. If $S$ is induced by $\sigma\in\Xi_1$, then either $S$ has the form $x_0+k_p$ (with $k_p\in{\mathbb{Z}}$) in every $p$ in which $S$ is differentiable, or the form $-x_0+k_p$. Since the range of $S$ is ${[0,1]}$, it must be $k_p=0$ in the first case, or $k_p=1$ in the second. The following is the main result of [@pantibernoulli]. There\[ref19\] is an explicitly constructible family $\{\sigma_{lm}:1\le l,m\in{\mathbb{N}}\}$ of elements of $\Xi_2$ such that, for every $\sigma_{lm}$ in the family, the induced McNaughton homeomorphism $S_{lm}$ has the following properties: - $S_{lm}$ fixes pointwise the boundary of ${[0,1]}^2$; - $S_{lm}$ is ergodic with respect to the Lebesgue measure of the unit square; - $S_{lm}$ is non-uniformly hyperbolic and Bernoulli. We recall that a measure-preserving bijection is [*Bernoulli*]{} if it is measure-theoretically isomorphic to a Bernoulli $2$-sided full shift. For \[ref20\] every even $n$ there exist Bernoulli McNaughton homeomorphisms of ${[0,1]}^n$. These mappings are mixing w.r.t. Lebesgue measure, topologically transitive, and chaotic in the sense of Devaney. The Bernoulli property implies mixing, and is preserved under direct products [@CornfeldFomSi82 Ch. 10 §1]. Since nonvoid open subsets of ${[0,1]}^n$ have positive Lebesgue measure, mixing homeomorphisms are topologically transitive, and hence chaotic by Corollary \[ref17\]. It would be very interesting to construct a McNaughton homeomorphism of ${[0,1]}^3$ having the Bernoulli property: this would allow to extend Corollary \[ref20\] to all $n\ge 2$. Up to now, we can only prove the following result about the action of $\Xi_n$ as a group. For\[ref23\] every $n\ge2$, $\Xi_n$ acts chaotically on ${[0,1]}^n$. As discussed above, we have a stronger result for even $n$, so we assume $n$ odd. Let $S$ and $T$ be topologically transitive McNaughton homeomorphisms of ${[0,1]}^{n-1}$ and ${[0,1]}^2$, respectively. Consider the following direct products: $$\begin{aligned} Q&=S\times(\text{identity map on the last coordinate});\\ R&=(\text{identity map on the first $n-2$ coordinates}) \times T.\end{aligned}$$ Both $Q$ and $R$ are McNaughton homeomorphisms of ${[0,1]}^n$, induced by elements of $\Xi_n$. For every $i<n$, let $0\le a_i<a'_i\le1$ and $0\le b_i<b'_i\le1$: we want to show that an appropriate composition of $Q$ and $R$ maps some point of the open box $A=\prod_{i<n}(a_i,a'_i)$ in the open box $B=\prod_{i<n}(b_i,b'_i)$. Since $S$ is topologically transitive, there exists $h\ge0$ such that the open set $$U=Q^h[A]\cap\bigl((b_0,b'_0)\times\cdots\times(b_{n-2},b'_{n-2}) \times(a_{n-1},a'_{n-1})\bigr)$$ is nonempty. Let $0\le c_i<c'_i\le1$ be such that the box $C=\prod_{i<n}(c_i,c'_i)$ is contained in $U$. The open box $D=\bigl(\prod_{i<n-1}(c_i,c'_i)\bigr)\times(b_{n_1},b'_{n-1})$ is contained in $B$ and, since $T$ is topologically transitive, there exists $k\ge0$ such that $R^k[C]\cap D\not=\emptyset$. Therefore $(R^k\circ Q^h)[A]\cap B\not=\emptyset$. For every Borel probability measure $\mu$ on ${[0,1]}^n$, let $f_\mu(r)$ denote the integral of the formula $r({x_0,\ldots ,x_{n-1}})$ (viewed as a function $r:{[0,1]}^n\to{[0,1]}$) with respect to $\mu$. The number $f_\mu(r)$ may be thought of as the “average truth-value” of $r$ w.r.t. $\mu$. It is natural to restrict attention to measures which are [*faithful*]{} ($r\not=0$ implies $f_\mu(r)\not=0$) and [*automorphism-invariant*]{} ($f_\mu(r)=f_\mu(\sigma(r))$, for every $\sigma\in\Xi_n$). This latter property is particularly relevant: it says that the average truth-value of a formula should be intrinsic to the formula, and not depending on the particular embedding of it in ${\operatorname{Free}_{n}({{\mathit{MV}}})}$. Lebesgue measure $\lambda$ is faithful (clearly) and automorphism-invariant (by Corollary \[ref18\]). Let $\mu$ be another probability measure on ${[0,1]}^n$, absolutely continuous with respect to $\lambda$. We may assume that $n$ is even, possibly introducing a dummy variable. By Corollary \[ref20\], there exists an automorphism $\sigma$ of ${\operatorname{Free}_{n}({{\mathit{MV}}})}$ that induces a mixing homeomorphism $S$ on ${[0,1]}^n$. Therefore the push forward $S_*^k\mu$ of $\mu$ by $S^k$ converges to $\lambda$ in the weak${}^*$ topology [@Walters82 §4.9 and Theorem 6.12(ii)]. In particular, for $r$ as above we get $$\lim_{k\to\infty}f_\mu(\sigma^k(r))= \lim_{k\to\infty}\int r\circ S^k\,d\mu= \lim_{k\to\infty}\int r\,d(S_*^k)\mu= \int r\,d\lambda=f_\lambda(r).$$ Hence the existence of mixing McNaughton homeomorphisms gives a distinguished status to $\lambda$. It appears plausible that the only ergodic $\Xi_n$-invariant measures on ${[0,1]}^n$ are $\lambda$ and the measures supported on finite orbits. We leave this as an open problem: since measures supported on finite orbits are not faithful, a positive answer would imply that the only reasonable averaging measure on truth-values in [Łukasiewicz]{} logic is Lebesgue measure. [10]{} P. Aglian[ò]{}. Prime spectra in modular varieties. , 30(4):581–597, 1993. M. Anderson and T. Feil. . Reidel, Dordrecht, 1988. K. Baker. Free vector lattices. , 20:58–66, 1968. J. Banks, J. Brooks, G. Cairns, G. Davis, and P. Stacey. On [D]{}evaney’s definition of chaos. , 99(4):332–334, 1992. A. Bigard, K. Keimel, and S. Wolfenstein. , volume 608 of [ *Lecture Notes in Math.*]{} Springer, 1977. S. Burris and H. P. Sankappanavar. , volume 78 of [*Graduate Texts in Mathematics*]{}. Springer, 1981. A. Church. . Princeton University Press, 1956. R. Cignoli, I. D’Ottaviano, and D. Mundici. , volume 7 of [*Trends in logic*]{}. Kluwer, 2000. S. A. Cook and R. A. Reckhow. The relative efficiency of propositional proof systems. , 44(1):36–50, 1979. I. P. Cornfeld, S. V. Fomin, and Ya. G. Sinaĭ. , volume 245 of [*Grundlehren der Mathematischen Wissenschaften*]{}. Springer, 1982. S. Gottwald. , volume 9 of [*Studies in Logic and Computation*]{}. Research Studies Press Ltd., Baldock, 2001. H. P. Gumm and A. Ursini. Ideals in universal algebras. , 19:45–54, 1984. P. H[á]{}jek. , volume 4 of [*Trends in logic*]{}. Kluwer, 1998. P. R. Halmos. . Van Nostrand Mathematical Studies, No. 1. D. Van Nostrand Co., Inc., Princeton, N.J., 1963. M. Hochster. Prime ideal structure in commutative rings. , 142:43–60, 1969. J. G. Hocking and G. S. Young. . Dover, 1988. First published in 1961 by Addison-Wesley. J. L. Kelley. . Van Nostrand, 1955. R. McNaughton. A theorem about infinite-valued sentential logic. , 16:1–13, 1951. J. Menu and J. Pavelka. A note on tensor products on the unit interval. , 17:71–83, 1976. P. S. Mostert and A. L. Shields. On the structure of semigroups on a compact manifold with boundary. , 65:117–143, 1957. D. Mundici. Interpretation of [AF $C^*$]{}-algebras in [[Ł]{}]{}ukasiewicz sentential calculus. , 65:15–63, 1986. D. Mundici. A constructive proof of [M]{}c[N]{}aughton’s theorem in infinite-valued logic. , 59:596–602, 1994. G. Panti. Prime ideals in free $\ell$-groups and free vector lattices. , 219(1):173–200, 1999. G. Panti. Generic substitutions. Available at `http://arXiv.org/abs/math/0308177`, to appear in the *J. of Symbolic Logic*, 2003. G. Panti. Bernoulli automorphisms in many-valued logic. Available at `http://arxiv.org/abs/math/0408370`, submitted for publication, 2004. M. Tsujii. Absolutely continuous invariant measures for expanding piecewise linear maps. , 143(2):349–373, 2001. P. Walters. , volume 79 of [*Graduate Texts in Mathematics*]{}. Springer, 1982. [^1]: *2000 Math. Subj. Class.*: 03B50; 37A05

--- title: 'H.E.S.S. observations of $\gamma$-ray bursts in 2003–2007' --- [Very-high-energy (VHE; $\ga$100 GeV) $\gamma$-rays are expected from $\gamma$-ray bursts (GRBs) in some scenarios. Exploring this photon energy regime is necessary for understanding the energetics and properties of GRBs. ]{} [GRBs have been one of the prime targets for the H.E.S.S. experiment, which makes use of four Imaging Atmospheric Cherenkov Telescopes (IACTs) to detect VHE $\gamma$-rays. Dedicated observations of 32 GRB positions were made in the years 2003–2007 and a search for VHE $\gamma$-ray counterparts of these GRBs was made. Depending on the visibility and observing conditions, the observations mostly start minutes to hours after the burst and typically last two hours.]{} [Results from observations of 22 GRB positions are presented and evidence of a VHE signal was found neither in observations of any individual GRBs, nor from stacking data from subsets of GRBs with higher expected VHE flux according to a model-independent ranking scheme. Upper limits for the VHE $\gamma$-ray flux from the GRB positions were derived. For those GRBs with measured redshifts, differential upper limits at the energy threshold after correcting for absorption due to extra-galactic background light are also presented.]{} Introduction ============ Gamma-ray bursts (GRBs) are the most energetic events in the $\gamma$-ray regime. Depending on their duration (e.g. $T_{90}$), GRBs are categorized into long GRBs ($T_{90}>2$ s) and short GRBs ($T_{90}<2$ s). First detected in late 1960s [@klebe73], GRBs remained mysterious for three decades. Breakthroughs in understanding GRBs came only after the discovery of longer-wavelength afterglows with the launch of *BeppoSAX* in 1997 [@paradijs00]. Multi-wavelength (MWL) observations have proved to be crucial in our understanding of GRBs, and provide valuable information about their physical properties. These MWL afterglow observations are generally explained by synchrotron emission from shocked electrons in the relativistic *fireball* model [@piran99; @zhang04]. A plateau phase is revealed in many of the *Swift*/XRT light curves, the origin of which is still not clear [@zhang06]. Observations of GRBs at energies $>$10 GeV may test some of the ideas that have been suggested to explain the X-ray observations [@fan08]. In the framework of the relativistic *fireball* model, photons with energies up to $\sim$10 TeV or higher are expected from the GRB afterglow phase [@zhang04; @fan08b]. Possible leptonic radiation mechanisms include forward-shocked electrons up-scattering self-emitted synchrotron photons [SSC processes; @dermer00; @zhang01; @fan08] or photons from other shocked regions [@wang01]. Physical parameters, such as the ambient density of the surrounding material ($n$), magnetic field equipartition fraction ($\epsilon_B$), and bulk Lorentz factor ($\Gamma_{\rm bulk}$) of the outflow, may be constrained by observations at these energies [@wang01; @peer05]. A possible additional contribution to VHE emission relates to the X-ray flare phenomenon. X-ray flares are found in more than 50% of the *Swift* GRBs during the afterglow phase [@chincarini07]. The energy fluence of some of them (e.g. GRB 050502B) is comparable to that of the prompt emission. Most of them are clustered at $\sim$10$^2$–10$^3$s after the GRB [see Figure 2 in @chincarini07], while late X-ray flares ($>$10$^4$s) are also observed; when these happen they can cause an increase in the X-ray flux of an order of magnitude or more over the power-law temporal decay [@curran08]. The cause of X-ray flares is still a subject of debate, but corresponding VHE $\gamma$-ray flares from inverse-Compton (IC) processes are predicted [@wang06; @galli07; @fan08]. The accompanying external-Compton flare may be weak if the flare originated behind the external shock, e.g. from prolonged central engine activity [@fan08]. However, in the external shock model, the expected SSC flare is very strong at GeV energies and can be readily detected using a VHE instrument with an energy threshold of $\sim$100 GeV [@galli08], such as the H.E.S.S. array, for a typical GRB at z$\sim$1. Therefore, VHE $\gamma$-ray data taken during an X-ray flare may help for distinguishing the internal/external shock origin of the X-ray flares, and may be used as a diagnostic tool for the late central engine activity. @waxman00 and @Murase08 suggest that GRBs may be sources of ultra-high-energy cosmic rays (UHECRs). In this case, $\pi$-decays from proton-$\gamma$ interaction may generate VHE emission. The VHE $\gamma$-ray emission produced from such a hadronic component is generally expected to decay more slowly than the leptonic sub-MeV radiation [@Boettcher98]. @dermer07 suggests a combined leptonic/hadronic scenario to explain the rapidly-decaying phase and plateau phase seen in many of the *Swift*/XRT light curves. This model can be tested with VHE observations taken minutes to hours after the burst. Most searches for VHE $\gamma$-rays from GRBs have obtained negative results [@connaughton97; @atkins05]. There may be indications of excess photon events from some observations, but these results are not conclusive [@amenomori96; @padilla98; @atkins00; @poirier03]. Currently, the most sensitive detectors in the VHE $\gamma$-ray regime are IACTs. @horan07 presented upper limits from 7 GRBs observed with the Whipple Telescope during the pre-*Swift* era. Upper limits for 9 GRBs with redshifts that were either unknown or $>$3.5 were also reported by the MAGIC collaboration [@albert07]. In general, these limits do not violate a power-law extrapolation of the keV spectra obtained with satellite-based instruments. However, most GRBs are now believed to originate at cosmological distances, therefore absorption of VHE $\gamma$-rays by the EBL [@nikishov62] must be considered when interpreting these limits. In this paper, observations of 22 $\gamma$-ray bursts made with H.E.S.S. during the years 2003–2007 are reported. They represent the largest sample of GRB afterglow observations made by an IACT array and result in the most stringent upper limits obtained in the VHE band. The prompt phase of GRB 060602B was observed serendipitously with H.E.S.S. The results of observations before, during, and after this burst are presented in @aha09. The H.E.S.S. experiment and GRB observation strategy ==================================================== The H.E.S.S. array[^1] is a system of four 13m-diameter IACTs located at 1800 m above sea level in the Khomas Highland of Namibia ($23\degr16\arcmin18\arcsec$ S, $16\degr30\arcmin00\arcsec$ E). Each of the four telescopes is located at a corner of a square with a side length of 120 m. This configuration was optimized for maximum sensitivity to $\sim$100 GeV photons. The effective collection area increases from $\sim$10$^3\mathrm{m}^2$ at 100 GeV to more than $10^5\mathrm{m}^2$ at 1 TeV for observations at a zenith angle (Z.A.) of 20$\degr$. The system has a point source sensitivity above 100 GeV of $\sim$1.4$\times$10$^{-11} \mathrm{erg}\,\mathrm{cm}^{-2}\,\mathrm{s}^{-1}$ ($3.5\%$ of the flux from the Crab nebula) for a $5 \sigma $ detection in a 2 h observation. Each H.E.S.S. camera consists of 960 photomultiplier tubes (PMTs), which in total provide a field of view (FoV) of $\sim$5$\degr$. This relatively large FoV allows for the simultaneous determination of the background events from off-source positions, so that no dedicated off run is needed [@aha06c]. The slew rate of the array is $\sim$100$\degr$ per minute, enabling it to point to any sky position within $\sim$2 minutes. The H.E.S.S. array is currently the only IACT array in the Southern Hemisphere used for an active GRB observing programme[^2]. The trigger system of the H.E.S.S. array is described in @funk04. The stereoscopic technique is used, i.e. a coincidence of at least two telescopes triggering within a window of (normally) 80 nanoseconds is required. This largely rejects background events caused by local muons that trigger only a single telescope. The observations reported here were obtained over the period March 2003 to October 2007. The observations of two GRBs in 2003 were made using two telescopes while the system was under construction. Before 2003 July, each of the two telescopes took data separately. Stereo analysis was then performed on the data, which requires coincidence of events to be determined offline using GPS time stamps. After the installation of the central trigger system in 2003 July, the stereo multiplicity requirement was determined on-line. All observations since 2004 have made use of the completed four-telescope array and the stereo technique [@aha06c]. Most of the data were taken in 28 minute runs using *wobble* mode, i.e. the GRB position is placed at an offset, $\theta_\mathrm{offset}$, of $\pm0\fdg5$ or $0\fdg7$ (in R.A. and Decl.) relative to the centre of the camera FoV during observations. Onboard GRB triggers distributed by the [*Swift*]{} satellite, as well as triggers from *INTEGRAL* and *HETE-II* confirmed by ground-based analysis, are followed by H.E.S.S. observations. Upon the reception of a GCN[^3] notice from one of these satellites (with appropriate indications[^4] that the source is a genuine GRB), the burst position is observed if Z.A.$\la$45$\degr$ (to ensure a reasonably low-energy threshold) during H.E.S.S. dark time[^5]. An automated program is running on site to keep the shift crew alerted of any new detected GRBs in real time. Depending on the observational constraints and the measured redshifts of the GRBs reported through GCN circulars[^6], observations of the burst positions are started up to $\sim$24 hours after the burst time, typically with an exposure time of $\approx$120 minutes in *wobble* mode. The remarkably nearby, bright GRB 030329 was an exceptional case. It was not observed until 11.5 days after the burst because of poor weather, which prohibited observation any earlier. The GRB observations ==================== Thirty-two GRBs were observed with H.E.S.S. during the period from March 2003 to October 2007. After applying a set of data-quality criteria that rejects observation runs with non-optimal weather conditions and hardware status, 22 GRB observations were selected for analysis and are described in this section. -------------- ----------- --------- ------------------------------------------- ------------------------------- ------------- ------------- ------------------------- ------------------- ---------- ---------- ---------- ------------- ----------- GRB Satellite Trigger R.A.[^7] Decl.$^a$ Error$^a$ Energy band Fluence[^8] $T_\mathrm{90}^b$ X[^9] O$^c$ R$^c$ $z$ Rank[^10] number ($\arcsec$) (keV) ($10^{-8}$ergcm$^{-2}$) (s) 071003 *Swift* 292934 $20^\mathrm{h}07^\mathrm{m}24\fs25$ +$10\degr56\arcmin48\farcs8$ 5.7 15–150 830 $\sim$150 $\surd$ $\surd$ $\surd$ 1.604[^11] 5 070808 *Swift* 287260 $00^\mathrm{h}27^\mathrm{m}03\fs36$ +$01\degr10\arcmin34\farcs8$ 1.9 15–150 120 $\sim$32 $\surd$ $\surd$ $.$ $\cdots$ 9 070724A *Swift* 285948 $01^\mathrm{h}51^\mathrm{m}13\fs96$ -$18\degr35\arcmin40\farcs1$ 2.2 15–150 3 $\sim$0.4 $\surd$ $\times$ $\times$ 0.457[^12] 21 070721B *Swift* 285654 $02^\mathrm{h}12^\mathrm{m}32\fs95$ -$02\degr11\arcmin40\farcs6$ 0.9 15–150 360 $\sim$340 $\surd$ $\surd$ $\times$ 3.626[^13] 10 070721A *Swift* 285653 $00^\mathrm{h}12^\mathrm{m}39\fs24$ -$28\degr22\arcmin00\farcs6$ 2.3 15–150 7.1 3.868 $\surd$ $\surd$ $.$ $\cdots$ 20 070621 *Swift* 282808 $21^\mathrm{h}35^\mathrm{m}10\fs14$ -$24\degr49\arcmin03\farcs1$ 2 15–150 430 33 $\surd$ $\times$ $.$ $\cdots$ 1 070612B *Swift* 282073 $17^\mathrm{h}26^\mathrm{m}54\fs4$ -$08\degr45\arcmin08\farcs7$ 4.7 15–150 168 13.5 $\surd$ $\times$ $.$ $\cdots$ 15 070429A *Swift* 277571 $19^\mathrm{h}50^\mathrm{m}48\fs8$ -$32\degr24\arcmin17\farcs9$ 2.4 15–150 91 163.3 $\surd$ $\surd$ $.$ $\cdots$ 3 070419B *Swift* 276212 $21^\mathrm{h}02^\mathrm{m}49\fs57$ -$31\degr15\arcmin49\farcs7$ 3.5 15–150 736 236.4 $\surd$ $\surd$ $.$ $\cdots$ 7 070209 *Swift* 259803 $03^\mathrm{h}04^\mathrm{m}50^\mathrm{s}$ -$47\degr22\arcmin30\arcsec$ 168 15–150 2.2 0.09 $\times$ $\times$ $.$ 0.314?[^14] 22 061110A *Swift* 238108 $22^\mathrm{h}25^\mathrm{m}09\fs9$ -$02\degr15\arcmin30\farcs7$ 3.7 15–150 106 40.7 $\surd$ $\surd$ $.$ 0.758[^15] 11 060526 *Swift* 211957 $15^\mathrm{h}31^\mathrm{m}18\fs4$ +$00\degr17\arcmin11\farcs0$ 6.8 15–150 126 298.2 $\surd$ $\surd$ $.$ 3.21[^16] 8 060505 *Swift* 208654 $22^\mathrm{h}07^\mathrm{m}04\fs50$ -$27\degr49\arcmin57\farcs8$ 4.7 15–150 94.4 $\sim$4 $\surd$ $\surd$ $.$ 0.0889[^17] 18 060403 *Swift* 203755 $18^\mathrm{h}49^\mathrm{m}21\fs80$ +$08\degr19\arcmin45\farcs3$ 5.5 15–150 135 30.1 $\surd$ $\times$ $.$ $\cdots$ 16 050801 *Swift* 148522 $13^\mathrm{h}36^\mathrm{m}35^\mathrm{s}$ -$21\degr55\arcmin41\arcsec$ 1 15–150 31 19.4 $\surd$ $\surd$ $\times$ 1.56[^18] 2 050726 *Swift* 147788 $13^\mathrm{h}20^\mathrm{m}12\fs30$ -$32\degr03\arcmin50\farcs8$ 6 15–150 194 49.9 $\surd$ $\surd$ $.$ $\cdots$ 13 050509C *HETE-II* H3751 $12^\mathrm{h}52^\mathrm{m}53\fs94$ -$44\degr50\arcmin04\farcs1$ 1 2–30 60 25 $\surd$ $\surd$ $\surd$ $\cdots$ 19 050209 *HETE-II* U11568 $08^\mathrm{h}26^\mathrm{m}$ +$19\degr41\arcmin$ 420 30–400 200 46 $.$ $\times$ $.$ $\cdots$ 14 041211B[^19] *HETE-II* H3622 $06^\mathrm{h}43^\mathrm{m}12^\mathrm{s}$ +$20\degr23\arcmin42\arcsec$ 80 30–400 1000 $>$100 $.$ $\times$ $.$ $\cdots$ 4 041006 *HETE-II* H3570 $00^\mathrm{h}54^\mathrm{m}50\fs23$ +$01\degr14\arcmin04\farcs9$ 0.1 30–400 713 $\sim$20 $\surd$ $\surd$ $\surd$ 0.716[^20] 6 030821 *HETE-II* H2814 $21^\mathrm{h}42^\mathrm{m}$ -$44\degr52$ [^21] 30–400 280 23 $.$ $.$ $.$ $\cdots$ 17 030329 *HETE-II* H2652 $10^\mathrm{h}44^\mathrm{m}49\fs96$ +$21\degr31\arcmin17\farcs44$ $10^{-3}$ 30–400 10760 33 $\surd$ $\surd$ $\surd$ 0.1687[^22] 12 -------------- ----------- --------- ------------------------------------------- ------------------------------- ------------- ------------- ------------------------- ------------------- ---------- ---------- ---------- ------------- ----------- Properties of the GRBs ---------------------- For each burst, the observational properties as obtained from the triggering satellite are shown in Table \[GRBtable1\]. These include trigger number, energy band, fluence in that energy band, and the duration of the burst ($T_{90}$). Whenever there were follow-up observations in the X-ray, optical, or radio bands, whether a detection has occurred (denoted by a tick $\surd$) or not (denoted by a cross $\times$) is also shown. If no observation at a given wavelength was reported, a dot ($.$) is shown. The reported redshifts ($z$) of 10 GRBs are also presented, of which 6 are lower than one. Two observed bursts, GRB 070209 and GRB 070724A, are short GRBs while the rest are long GRBs. The population of short GRBs has a redshift distribution [@Berger07] significantly less than that of the long GRBs [@Jakobsson06]. Therefore, on average they are likely to suffer from a lower level of EBL absorption. X-ray flares were detected from three of the GRBs in the H.E.S.S. sample. They occurred at 273s after the burst for GRB 050726, 284s for GRB 050801, and $2.6\times10^5$s for GRB 070429A [@curran08]. Unfortunately, the flares occurred outside the time windows of the H.E.S.S. observations. H.E.S.S. observations --------------------- For each burst, the start time, $T_{\rm start}$, of the H.E.S.S. observations after the burst is shown in Table \[GRB\_stat\]. Since an observing strategy to start observing the burst position up to $\sim$24 hours after the burst time is applied, the mean $T_{\rm start}$ is of the order of 10 hours. The (good-quality) exposure time of the observations using $N_{\rm tel}$ telescopes for each burst is included. The mean Z.A. of the observations is also presented. The ranking scheme {#sect_rank} ------------------ As mentioned in the introduction, there is no lack of models predicting VHE emission from GRBs. However, the evolution of the possible VHE $\gamma$-ray emission with time is model-dependent. To give an empirical, model-independent estimate of the relative expected VHE flux of each GRB (which also depends on $T_{\rm start}$), it is assumed that: (1) the relative VHE signal scales as the energy released in the prompt emission, taken as a typical energy measure of a GRB. Hence $F_{\rm VHE} \propto F_{15-150 \rm \,keV}$ where $F_{15-150 \rm \,keV}$ is the fluence in the [*Swift*]{}/BAT band. For bursts not triggered by BAT, the measured fluence is extrapolated into this energy band; (2) the possible VHE signal fades as time goes on, as observed in longer wavelength (e.g. X-ray) data. In particular, the VHE flux follows the average decay of the X-ray flux and therefore $F_{\rm VHE} \propto F_{15-150 \rm \,keV}\times t^{-1.3}$ where $t$ denotes the time after the burst and 1.3 is the average X-ray afterglow late-time power-law decay index [@nousek06]. Since in most cases the exposure time of the observations is much shorter than $T_{\rm start}$ (the start time of the corresponding H.E.S.S. observations after the trigger), the expected flux at $T_{\rm start}$ can be used as a measure of the strength of the VHE signal, and therefore of the relative possibility of detecting a VHE signal from that GRB. By setting $t$ to $T_{\rm start}$, we have $$\label{rank_eqn} F_{\rm VHE} \propto F_{15-150 \rm \,keV}\times T_{\rm start}^{-1.3}$$ The rank of each GRB according to equation \[rank\_eqn\] is shown in the last column in Table \[GRBtable1\]. Note that redshift information (available for only a few GRBs), and thus the corresponding EBL absorption, is not taken into account in the ranking scheme. Data Analysis ============= Calibration of data, event reconstruction and rejection of the cosmic-ray background (i.e. $\gamma$-ray event selection criteria) were performed as described in @aha06c, which employs the techniques described by @hillas96. Gamma-like events were then taken from a circular region (on-source) of radius $\theta_{\rm cut}$ centred at the burst position given in Table \[GRBtable1\]. The background was estimated using the reflected-region background model as described in @berge07, in which the number of background events in the on-source region ($N_\mathrm{off}$) is estimated from $n_\mathrm{region}$ off-source regions located at the same $\theta_\mathrm{offset}$ as the on-source region during the same observation. The number of $\gamma$-like events is given by $N_{\rm on}-\alpha N_{\rm off}$ where $N_{\rm on}$ is the total number of events detected in the on-source region and $\alpha=1/n_\mathrm{region}$ the normalization factor. Independent analyses of various GRBs using different methods and background estimates [@berge07] yielded consistent results. Analysis technique {#sample_analysis} ------------------ Two sets of analysis cuts were applied to search for a VHE $\gamma$-ray signal from observational data taken with three or four telescopes. These are ‘standard’ cuts [@aha06c] and ‘soft’ cuts[^23] [the latter have lower energy thresholds, as described in @aha06a]. For standard (soft) cuts, $\theta_{\rm cut}=0.11\degr$ ($\theta_{\rm cut}=0.14\degr$). While standard cuts are optimized for a source with a power-law spectrum of photon index $\Gamma=2.6$, soft cuts are optimized for a source with a steep spectrum ($\Gamma=5.0$), and have better sensitivity at lower energies. Since EBL absorption is less severe for lower energy photons, the soft-cut analysis is useful in searching for VHE $\gamma$-rays from GRBs which are at cosmological distances. For example, the photon indices of two blazars PKS 2005-489 [@aha05] and PG 1553+113 [@aha08] were measured to be $\Gamma\ga4$. An exception to this analysis scheme is GRB 030329. As the central trigger system had yet to be installed when this observation was made, a slightly different analysis technique was used. The description of the image and analysis cuts used for the data from GRB 030329 can be found in @aha05. For GRB 030821, only the standard-cut analysis (for two-telescope data) was performed (see Sect. \[sect\_030821\]). The positional error circle of most GRBs, with the exceptions of GRB 030821, GRB 050209, and GRB 070209, is small compared to the H.E.S.S. point spread function (PSF). The 68% $\gamma$-ray containment radius, $\theta_{68}$, of the H.E.S.S. PSF can be as small as $\sim$3$\arcmin$, depending on the Z.A. and $\theta_\mathrm{offset}$ of the observations, and the analysis cuts applied. The 68% containment radius, $\theta_{68}$, of the observations of GRB 050209 and GRB 070209 is about 9$\arcmin$ using standard-cut analysis[^24], slightly larger than the corresponding error circles. Therefore, point-source analyses were performed for all GRBs except GRB 030821, the error box of which is much bigger than the H.E.S.S. PSF (see Sect. \[sect\_030821\] for its treatment). Energy threshold ---------------- The energy threshold, $E_{\rm th}$, is conventionally defined as the peak in the differential $\gamma$-ray rate versus energy curve of a fictitious source with photon index $\Gamma$ [@konopelko99]. This curve is a convolution of the effective area with the expected energy spectrum of the source as seen on Earth. Such energy thresholds, obtained by the standard-cut analysis and the soft-cut analysis for each GRB observation, are shown in Table \[GRB\_stat\], assuming $\Gamma=2.6$. The energy threshold depends on the Z.A. of the observations and the analysis used. The larger the Z.A., the higher is the energy threshold. Moreover, soft-cut analysis gives a lower value of $E_{\rm th}$ than that of standard-cut analysis. Note that $\gamma$-ray photons with energies below $E_{\rm th}$ can be detected by the telescopes. Optical efficiency of the instrument ------------------------------------ The data presented were also corrected for the long-term changes in the optical efficiency of the instrument. The optical efficiency has decreased over a period of a few years. This has changed the effective area and energy threshold of the instrument. Specifically, the energy threshold has increased with time. Using images of local muons in the FoV, this effect in the calculation of flux upper limits is corrected [c.f. @aha06c]. Results ======= No evidence of a significant excess of VHE $\gamma$-ray events from any of the GRB positions given in Table \[GRBtable1\] during the period covered by the H.E.S.S. observations was found. The number of on-source ($N_{\rm on}$) and off-source events ($N_{\rm off}$), normalization factor ($\alpha$), excess, and statistical significance[^25] of the excess in standard deviations ($\sigma$) are given for each of the 21 GRBs in Table \[GRB\_stat\]. The results for GRB 030821 are given in Sect. \[sect\_030821\]. Figure \[soft\_sigma\_dist\] shows the distribution of the significance obtained from the soft-cut analysis of the observations of each of the 21 GRBs. A Gaussian distribution with mean zero and standard deviation one, which is expected in the case of no detection, is shown for comparison. The distribution of the statistical significance is consistent with this Gaussian distribution. Thus no significant signal was found from any of the individual GRBs. A search for serendipitous source discoveries in the H.E.S.S. FoV during observations of the GRBs also resulted in no significant detection. The 99.9% confidence level (c.l.) flux upper limits (above $E_\mathrm{th}$) have been calculated using the method of @feldman98 for both standard cuts (assuming $\Gamma=2.6$) and soft cuts (assuming $\Gamma=5$), and are included in Table \[GRB\_stat\]. The limits are as observed on Earth, i.e. the EBL absorption factor was not taken into account. The systematic error on a H.E.S.S. integral flux measurement is estimated to be $\sim$20%, and it was not included in the calculation of the upper limits. For those GRBs with reported redshifts, the effect of the EBL on the H.E.S.S. limits can be estimated. Using the EBL model P0.45 described in @Aha06_EBL_nature, differential upper limits (again assuming $\Gamma=5$) at the energy threshold were calculated from the integral upper limits obtained using soft-cut analysis. These upper limits, as well as those calculated without taking the EBL into account, are shown in Table \[GRB\_differential\_flux\_ULs\]. ------------- ----------------- ---------- --------------- ----------- ----------------- ------------------ ---------- -------- --------- -------------- ---------------------- ----------------- ------------------ ---------- -------- ---------- -------------- ---------------------- ----------------- ------------- GRB[^26] $T_{\rm start}$ Exposure $N_{\rm tel}$ Z.A. $N_\mathrm{ON}$ $N_\mathrm{OFF}$ $\alpha$ Excess Signi- $E_{\rm th}$ Flux ULs $N_\mathrm{ON}$ $N_\mathrm{OFF}$ $\alpha$ Excess Signi- $E_{\rm th}$ Flux ULs $\chi^2$/d.o.f. P($\chi^2$) (min) (min) ($\degr$) ficance (GeV) (cm$^{-2}$ s$^{-1}$) ficance (GeV) (cm$^{-2}$ s$^{-1}$) 070621 6.5 234.6 4 16 204 2273 0.091 -2.6 -0.18 250 $2.8\times10^{-12}$ 731 5903 0.13 -6.9 -0.24 190 $5.6\times10^{-12}$ 19.2/28 0.89 050801 15.0 28.2 4 43 13 173 0.091 -2.7 -0.68 400 $3.2\times10^{-12}$ 46 442 0.13 -9.3 -1.2 310 $1.6\times10^{-11}$ 0.168/3 0.98 070429A 64 28.2 4 23 4 78 0.091 -3.1 -1.2 290 $2.4\times10^{-12}$ 20 203 0.13 -5.4 -1.0 220 $1.0\times10^{-11}$ 6.39/3 0.094 567.1 14.2 3 64 9 87 0.11 -0.67 -0.21 1850 $6.8\times10^{-12}$ 27 236 0.17 -12 -1.9 1360 $2.6\times10^{-11}$ 742.3 112.3 4 44 76 1247 0.063 -1.9 -0.21 380 $3.7\times10^{-12}$ 317 4353 0.083 -46 -2.4 280 $1.8\times10^{-11}$ 623.3 56.2 4 35 16 272 0.10 -11 -2.2 390 $1.0\times10^{-12}$ 97 785 0.14 -15 -1.4 280 $1.4\times10^{-11}$ 691.1 56.2 3 41 25 204 0.10 4.6 0.93 480 $5.6\times10^{-12}$ 79 547 0.14 0.86 0.091 340 $1.5\times10^{-11}$ 041006 626.1 81.9 4 27 80 770 0.10 3 0.32 200 $1.1\times10^{-11}$ 302 1974 0.14 20 1.1 150 $6.8\times10^{-11}$ 8.89/9 0.45 070419B 907 56.4 4 47 28 391 0.091 -7.5 -1.3 700 $2.4\times10^{-12}$ 121 1069 0.13 -13 -1.0 520 $7.5\times10^{-12}$ 11.9/6 0.064 060526 284.2 112.8 4 25 93 1068 0.10 -13.8 -1.3 280 $2.9\times10^{-12}$ 492 3711 0.14 -38 -1.6 220 $9.2\times10^{-12}$ 19.8/12 0.072 070808 306.2 112.8 4 34 49 659 0.091 -11 -1.4 310 $3.2\times10^{-12}$ 209 1733 0.13 -7.6 -0.49 260 $7.5\times10^{-12}$ 15.8/12 0.20 070721B 925.7 103.8 4 40 59 984 0.063 -2.5 -0.31 440 $1.4\times10^{-12}$ 237 2676 0.083 14 0.89 320 $8.8\times10^{-12}$ 15.5/11 0.16 061110A 407.68 112.8 4 25 76 838 0.093 -1.9 -0.21 280 $4.3\times10^{-12}$ 314 2671 0.13 -20 -1.0 200 $8.4\times10^{-12}$ 4.66/11 0.95 030329[^27] 16493.5 28.0 2 60 4 26 0.14 0.27 0.13 1360 $2.6\times10^{-12}$ $\cdots$ 5.93/3 0.12 050726 772.7 112.8 4 40 107 1031 0.083 21 2.1 320 $7.1\times10^{-12}$ 333 2619 0.11 42 2.3 260 $3.4\times10^{-11}$ 14.7/12 0.26 050209 1208.5 168.6 4 48 104 1096 0.11 -18 -1.6 480 $4.4\times10^{-12}$ 528 4204 0.14 -73 -2.8 340 $1.5\times10^{-11}$ 36.3/18 0.0065 070612B 901.7 112.8 4 18 104 1190 0.091 -4.2 -0.39 240 $4.1\times10^{-12}$ 415 3233 0.13 11 0.51 180 $1.5\times10^{-11}$ 4.87/12 0.96 060403 820.4 52.8 4 39 33 252 0.091 10 1.9 440 $4.8\times10^{-12}$ 128 875 0.13 19 1.6 320 $1.3\times10^{-11}$ 10.4/6 0.11 060505 1163 111 4 42 99 837 0.091 23 2.4 520 $5.6\times10^{-12}$ 339 2740 0.13 -3.5 -0.18 400 $3.9\times10^{-12}$ 22.1/12 0.036 050509C 1289 28.2 4 22 31 344 0.083 2.3 0.41 200 $1.7\times10^{-11}$ 112 965 0.11 4.8 0.43 150 $1.5\times10^{-10}$ 0.301/3 0.96 070721A 893.5 112.8 4 30 90 1436 0.059 5.5 0.58 320 $6.5\times10^{-12}$ 280 3837 0.077 -15 -0.86 260 $1.3\times10^{-11}$ 6.78/12 0.87 070724A 927.5 84.6 4 23 73 720 0.091 7.5 0.88 260 $7.3\times10^{-12}$ 246 2042 0.13 -9.3 -0.55 200 $1.0\times10^{-11}$ 14.3/9 0.11 070209 926.7 56.4 4 41 37 444 0.091 -3.4 -0.51 480 $2.3\times10^{-12}$ 185 1442 0.13 4.8 0.33 370 $1.1\times10^{-11}$ 5.35/6 0.50 ------------- ----------------- ---------- --------------- ----------- ----------------- ------------------ ---------- -------- --------- -------------- ---------------------- ----------------- ------------------ ---------- -------- ---------- -------------- ---------------------- ----------------- ------------- : H.E.S.S. observations of GRBs from March 2003 to October 2007.[]{data-label="GRB_stat"} ![Distribution of the statistical significance (*histogram*) as derived from the observations of 20 GRBs using soft-cut analysis. The mean is $-$0.4 and the standard deviation is 1.4. Each entry corresponds to one GRB. The *solid line* is a Gaussian function with mean zero and standard deviation unity.[]{data-label="soft_sigma_dist"}](1072fig1.eps){width="9.5cm"} GRB Redshift $E_\mathrm{th}$ (GeV) $F_\mathrm{UL}$[^28] $F_\mathrm{corrected}$$^a$ ------------- ---------- ----------------------- ---------------------- ---------------------------- 060505 0.0889 400 3.9$\times10^{-14}$ 5.8$\times10^{-14}$ 030329 0.1687 1360 7.6$\times10^{-15}$ 9.7$\times10^{-14}$ 070209 0.314 370 1.2$\times10^{-13}$ 8.7$\times10^{-13}$ 070724A 0.457 200 2.1$\times10^{-13}$ 1.0$\times10^{-12}$ 041006 0.716 150 1.8$\times10^{-12}$ 2.7$\times10^{-11}$ 061110A 0.758 200 1.7$\times10^{-13}$ 1.7$\times10^{-11}$ 050801 1.56 310 2.1$\times10^{-13}$ [^29] 071003[^30] 1.604 280 2.0$\times10^{-13}$ $^b$ 060526 3.21 220 1.7$\times10^{-13}$ $^b$ 070721B 3.626 320 1.1$\times10^{-13}$ $^b$ : Differential flux upper limits at the energy thresholds from the H.E.S.S. observations of GRBs with reported redshifts.[]{data-label="GRB_differential_flux_ULs"} ---------- --------- ---------- -------------- Number Soft-cut Standard-cut of GRBs analysis analysis Sample A 10 -2.13 -1.81 Sample B 6 -0.20 1.45 Sample C 11 -0.53 0.48 all GRBs 21 -1.98 -0.18 ---------- --------- ---------- -------------- : Combined significance of 3 subsets of GRBs selected based on the requirements listed in Sect. \[sect\_stacking\][]{data-label="subset"} Stacking analysis {#sect_stacking} ----------------- Although no significant excess was found from any individual GRB, co-adding the excess events from the observations of a number of GRBs may reveal a signal that is too weak to be seen in the data from one GRB, provided that the PSFs of the H.E.S.S. observations are bigger than the error box of the GRB positions (which is the case, see Sect. \[sample\_analysis\]). Firstly, stacking of all GRBs (except GRB 030821, which has a high positional uncertainty) in the sample was performed. This yielded a total of $-$157 excess events and a statistical significance of $-$1.98 using the soft-cut analysis. Use of standard cuts produced a similar result (see Table \[subset\]). Secondly, combining the significance of the results from three selected subsets extracted from the whole sample was performed. The *a priori* selection criteria were to choose those GRBs with a higher expected VHE flux or a lower level of EBL absorption. The following requirements were used to select three subsets: Sample A: : the first 10 in the ranking described in Sect. \[sect\_rank\]; Sample B: : all GRBs with a measured redshift $z<1$; Sample C: : all GRBs with a soft-cut energy threshold lower than 300 GeV and with either a measured redshift $z<1$ or with an unknown redshift. The result is shown in Table \[subset\]. As can be seen, there is no significant evidence of emission in any of these subsets. Temporal analysis ----------------- As possible VHE radiation from GRBs is expected to vary with time, a temporal analysis to search for deviation from zero excess in the observed data was performed. Soft-cut analysis was used for all GRBs (except GRB 030329) since this analysis has a lower energy threshold and a better acceptance of $\gamma$-rays and cosmic rays and therefore increases the statistics. The $\gamma$-like excess events were binned in 10-minute time intervals for each GRB data set and were compared to the assumption of no excess throughout the observed period. The $\chi^2$/d.o.f. value and the corresponding probability are shown in Table \[GRB\_stat\] for each GRB. Within the whole sample, the lowest probability that the hypothesis that the excess was zero throughout the observation period is correct is $1.2\times10^{-3}$ (for GRB 071003) and no significant deviation from zero within any of the GRB temporal data was found. Standard-cut analysis produced consistent results. GRB 070621: Observations of a GRB with the fastest reaction and the longest exposure time {#sect_070621} ----------------------------------------------------------------------------------------- GRB 070621 is the highest-ranked GRB in the sample (Sect. \[sect\_rank\]), i.e. it has the highest relative expected VHE flux at the start time of the observations. The duration of the *Swift* burst was $T_{90}\sim33$s, thus clearly classifying the burst as a long GRB. The fluence in the 15–150 keV band was $\sim$4.3$\times10^{-6}$ erg cm$^{-2}$. The XRT light curve is represented by an initial rapidly-decaying phase and a shallow phase, with the transition happening around $t_0 + 380$s where $t_0$ denotes the trigger time [@sbarufatti07]. Despite extensive optical monitoring, no fading optical counterpart was found. The H.E.S.S. observations started at $t_0+420$s and lasted for $\sim$5 hours, largely coincident with the X-ray shallow phase. These observations were both the most prompt and the longest among those presented. Figure \[GRB070621\_lc\] shows the 99.9% H.E.S.S. energy flux upper limits above $200$ GeV (using soft-cut analysis), together with the XRT results [@evans07]. As seen, the limits for this period are at levels comparable to the X-ray energy flux during the same period. Unfortunately the lack of redshift information for this burst prevents further interpretation of the limits. ![The 99.9% confidence level energy flux upper limits (in red) at energies $>$200 GeV derived from H.E.S.S. observations at the position of GRB 070621. The ends of the horizontal lines indicate the start and end times of the observations from which the upper limits were derived. The XRT energy flux in the 0.3–10 keV band is shown in black for comparison [@evans07].[]{data-label="GRB070621_lc"}](1072fig2.eps){width="9.0cm"} GRB 030821: Observations of a GRB with a high positional uncertainty {#sect_030821} -------------------------------------------------------------------- Some GRBs, such as GRB 030821, have a high uncertainty in position; with a relatively large camera FoV ($\sim$5$\degr$), the H.E.S.S. telescopes are able to cover the whole positional error box of such GRBs. Observations of GRB 030821 started 18 hours after the burst and lasted for a live-time of 55.5 minutes, with a mean Z.A. of 28$\degr$. The observations were taken when the array was under construction and only two telescopes were operating, resulting in an energy threshold of 260 GeV. The GRB has a relatively high uncertainty in position as determined from *IPN* (the third Interplanetary Network) triangulation [@hurley2359], and its error box is bigger than the PSF of H.E.S.S. However, because of the relatively large FoV of the camera, the whole error box, and thus the possible GRB position, is within the H.E.S.S. FoV. The sky excess map overlaid with the error box is shown in Fig. \[030821overlay\]. As can be seen, there is no significant excess at any position within the error box. The sky region with the largest number of peak excess events is located in the south-eastern part of the error box. Using a point-source analysis centred at this peak, a flux upper limit (above 260 GeV) of $\sim$1.7$\times10^{-11}$ cm$^{-2}$ s$^{-1}$ was derived. Since an upper limit derived for any location in the error box with fewer excess events is *lower* than this value[^31], it may be regarded as a *conservative* upper limit of the VHE flux associated with GRB 030821 during the period of the H.E.S.S. observations. Discussion ========== The upper limits presented in this paper are among the most stringent ever derived from VHE $\gamma$-ray observations of GRBs during the afterglow period. In fact, the 99.9% confidence level limits (in energy flux) are at levels comparable to the X-ray energy flux as observed by *Swift*/XRT during the same period (see, e.g. Fig. \[GRB070621\_lc\]). Unless most of the GRBs are located at high redshifts and thus their VHE flux is severely absorbed by the EBL (this possibility is discussed below), one expects detection of the predicted VHE component with energy flux levels comparable to those in X-rays in some scenarios [@dermer00; @wang01; @zhang01; @peer05; @fan08]. On the other hand, the unknown redshifts of many of the GRBs in the sample (including GRB 070621, the highest-ranking, which is discussed in Sect. \[sect\_rank\]) complicate the physical interpretation of the data, because EBL absorption at VHE energies is severe for a GRB with $z>1$. The mean and median redshift of the 10 GRBs with reported redshifts is 1.3 and 0.7, respectively. If the 12 GRBs without redshift have the same redshift distribution, one would expect $\sim$40% of them ($\sim$5 GRBs) to have $z<0.5$. In this case, the EBL absorption may not preclude the detection of the predicted VHE $\gamma$-rays for the GRB sample presented here[^32]. There is no reported X-ray flare during the H.E.S.S. observational time windows, therefore no conclusion on whether or not X-ray flares are accompanied by VHE flares, as well as the origin of X-ray flares, can be drawn. If UHECRs are generated in nearby GRB sources, as suggested by some authors, a detectable VHE flux is expected from nearby GRBs. Therefore, although the unknown redshifts of a significant fraction of GRBs in our sample (12 out of 22) and the uncertainty in the modeled VHE temporal evolution are surely in play, the results presented here do not indicate (but also not exclude) that GRBs are dominant sources of UHECRs. Outlook ======= The data from our sample of 22 GRBs do not provide any evidence for a strong VHE $\gamma$-ray component from GRBs during the afterglow phase. EBL absorption can explain the lack of detection in our sample. However, this does not exclude a population of GRBs that exhibit a strong VHE component. While the EGRET experiment did not detect MeV–GeV photons from most BATSE GRBs in its FoV, some strong bursts (e.g. GRB 940217) have proved to emit delayed emission, $\sim$1.5 hours after the burst, at energies as high as $\sim$20 GeV [@hurley94]. With *Fermi*’s observations of GRBs having started in mid-2008, it is likely that our knowledge of the high-energy emission of GRBs will be improved in the near future. The future prospects for detection at VHE energies rely on the likelihood of observing a GRB with low redshift (e.g. $z<0.5$) early enough. In the cases where there is no detection, sensitive and early upper limits on the intrinsic VHE luminosity of these nearby GRBs will still improve our understanding of the radiation mechanisms of GRBs. The existence of a distinct population of low-luminosity (LL) GRBs was suggested based on the high detection rate of low-redshift LL GRBs such as GRB 980425 and GRB 060218 [e.g. @Soderberg04_nat]. Due to their proximity, they are good targets for VHE observations. Since they are sub-energetic compared to other GRBs, they may be accompanied by a lower VHE luminosity. On the other hand, if most radiation are emitted at high energies, the detection probability would be much higher. @Franceschini08 claimed a very small $\gamma$-ray opacity due to EBL absorption. The optical depth is about a factor of three less than the one we used, depending on the energies [@Aha06_EBL_nature]. Therefore, on-going GRB observations with H.E.S.S., as well as other ground-based VHE detectors, are crucial to test this model. Conclusions =========== During 5 years of operation (2003–2007), 32 GRBs were observed during the afterglow phase using the H.E.S.S. experiment. Those 22 GRBs with high-quality data were analysed and the results presented in this paper. Depending on the visibility and observing conditions, the start time of the observations varied from minutes to hours after the burst. There is no evidence of VHE emission from any individual GRB during the period covered by the H.E.S.S. observations, nor from stacking analysis using the whole sample and *a priori* selected sub-sets of GRBs. Fine-binned temporal data revealed no short-term variability from any observation and no indication of VHE signal from any of these time bins was found. Upper limits of VHE $\gamma$-ray flux during the observations from the GRBs were derived. These 99.9% confidence level energy flux upper limits are at levels comparable to the contemporary X-ray energy flux. For those GRBs with reported redshifts, differential upper limits at the energy threshold after correcting for EBL absorption are presented. H.E.S.S. phase II will have an energy threshold of about 30 GeV. With much less absorption by the EBL at such low energies, it is hoped that the H.E.S.S. experiment will enable the detection of VHE $\gamma$-ray counterparts of GRBs. The support of the Namibian authorities and of the University of Namibia in facilitating the construction and operation of H.E.S.S. is gratefully acknowledged, as is the support by the German Ministry for Education and Research (BMBF), the Max Planck Society, the French Ministry for Research, the CNRS-IN2P3 and the Astroparticle Interdisciplinary Programme of the CNRS, the U.K. Science and Technology Facilities Council (STFC), the IPNP of the Charles University, the Polish Ministry of Science and Higher Education, the South African Department of Science and Technology and National Research Foundation, and by the University of Namibia. We appreciate the excellent work of the technical support staff in Berlin, Durham, Hamburg, Heidelberg, Palaiseau, Paris, Saclay, and in Namibia in the construction and operation of the equipment. P.H. Tam acknowledges support from IMPRS-HD. This work has made use of the GCN Notices and Circulars provided by NASA’s Goddard Space Flight Center, as well as data supplied by the UK *Swift* Science Data Centre at the University of Leicester. [^1]: http://www.mpi-hd.mpg.de/hfm/HESS/HESS.html [^2]: http://www.lsw.uni-heidelberg.de/projects/hess/HESS/grbs.phtml [^3]: The Gamma ray bursts Coordinates Network, http://gcn.gsfc.nasa.gov/ [^4]: which include, e.g., burst position incompatible with known sources, and a high signal-to-noise ratio of the burst [^5]: H.E.S.S. observations are taken in darkness and when the moon is below the horizon. The fraction of H.E.S.S. dark time is about 0.2 [^6]: http://gcn.gsfc.nasa.gov/gcn3\_archive.html [^7]: R.A., Decl., and the positional errors (90% containment) were taken from GCN Reports (http://gcn.gsfc.nasa.gov/report\_archive.html) for GRB 061110A – GRB 071003 and GCN Circulars otherwise. [^8]: Fluence and $T_\mathrm{90}$ data for GRB 050726 – GRB 070612B were taken from @sakamoto08 except that $T_\mathrm{90}$ of GRB 060505 was taken from @Palmer06_GCN5076. Fluence and $T_\mathrm{90}$ data of GRB 030329 and GRB 030821 were taken from @sakamoto05, and those of GRB 041006 from @Shirasaki08. Other data were taken from GCN Circulars and *HETE* pages (http://space.mit.edu/HETE/Bursts). [^9]: X: X-ray, O: optical, R: radio; “$\surd$" indicates the detection of a counterpart, “$\times$" a null detection, and “$.$" that no measurement was reported in the corresponding energy range, from http://grad40.as.utexas.edu/grblog.php [^10]: The relative expected VHE flux for each GRB is ranked according to the empirical scheme described in Sect. \[sect\_rank\] [^11]: @perley08 [^12]: @Cucchiara07_GCN6665 [^13]: @Malesani07_GCN6651 [^14]: Redshift of a candidate host galaxy @berger6101. [^15]: @Fynbo07_GCN6759 [^16]: @berger5170 [^17]: @ofek06 [^18]: Redshift according to @pasquale07, based on afterglow modelling [^19]: Although this burst was referred to as GRB 041211 in various GCN Circulars, the proper name GRB 041211B [e.g., in @pelangeon06] should be used to distinguish it from another burst, GRB 041211A (=H3621) which occurred earlier on the same day (Pélangeon, A., private communication). [^20]: @soderberg06 [^21]: The position error of this burst is large, see Fig. \[030821overlay\] [^22]: @stanek03 [^23]: ‘Soft’ cuts were called ‘spectrum’ cuts in @aha06a. [^24]: $\theta_{68}$ is larger using soft-cut analysis [^25]: calculated by eq. (17) in @LiMa83 [^26]: The GRBs are listed in the order of the ranking scheme described in Sect. \[sect\_rank\]. GRB 030821 is not listed, the results of which are given in Sect. \[sect\_030821\]. [^27]: A slightly different analysis technique was used, see Sect. \[sample\_analysis\]. Soft-cut analysis is not available for this observation. [^28]: Limits are given in units of $\mathrm{cm}^{-2}\,\mathrm{s}^{-1}\,\mathrm{GeV}^{-1}$. [^29]: The limits corrected for EBL absorption are $>$10 orders of magnitude larger than that observed. [^30]: Only 4-telescope data were used. [^31]: A larger excess implies a higher value of the upper limit, since the integrated exposure, that depends on Z.A. and $\theta_\mathrm{offset}$ of the observations, is largely the same over the whole error box. [^32]: The optical depth of EBL absorption for a $\sim$100 GeV photon is $\sim$3 at $z=1$, according to the P0.45 model demonstrated in @Aha06_EBL_nature.

--- abstract: 'While most Bayesian nonparametric models in machine learning have focused on the Dirichlet process, the beta process, or their variants, the gamma process has recently emerged as a useful nonparametric prior in its own right. Current inference schemes for models involving the gamma process are restricted to MCMC-based methods, which limits their scalability. In this paper, we present a variational inference framework for models involving gamma process priors. Our approach is based on a novel stick-breaking constructive definition of the gamma process. We prove correctness of this stick-breaking process by using the characterization of the gamma process as a completely random measure (CRM), and we explicitly derive the rate measure of our construction using Poisson process machinery. We also derive error bounds on the truncation of the infinite process required for variational inference, similar to the truncation analyses for other nonparametric models based on the Dirichlet and beta processes. Our representation is then used to derive a variational inference algorithm for a particular Bayesian nonparametric latent structure formulation known as the infinite Gamma-Poisson model, where the latent variables are drawn from a gamma process prior with Poisson likelihoods. Finally, we present results for our algorithms on nonnegative matrix factorization tasks on document corpora, and show that we compare favorably to both sampling-based techniques and variational approaches based on beta-Bernoulli priors.' author: - | Anirban Roychowdhury, Brian Kulis\ Department of Computer Science and Engineering\ The Ohio State University\ `roychowdhury.7@osu.edu`,kulis@cse.ohio-state.edu bibliography: - 'refpaper.bib' title: 'Gamma Processes, Stick-Breaking, and Variational Inference' --- Introduction ============ The gamma process is a versatile pure-jump Lévy process with widespread applications in various fields of science. Of late it is emerging as an increasingly popular prior in the Bayesian nonparametric literature within the machine learning community; it has recently been applied to exchangeable models of sparse graphs [@graphs_gp] as well as for nonparametric ranking models [@pl_luce_gp]. It also has been used as a prior for infinite-dimensional latent indicator matrices [@Gampois]. This latter application is one of the earliest Bayesian nonparametric approaches to count modeling, and as such can be thought of as an extension of the venerable Indian Buffet Process to modeling latent structures where each feature can occur multiple times for a datapoint, instead of being simply binary. The flexibility of gamma process models allows them to be applied in a wide variety of Bayesian nonparametric settings, but their relative complexity makes principled inference nontrivial. In particular, all direct applications of the gamma process in the Bayesian nonparametric literature use Markov chain Monte Carlo samplers (typically Gibbs sampling) for posterior inference, which often suffers from poor scalability. For other Bayesian nonparametric models—in particular those involving the Dirichlet process or beta process—a successful thread of research has considered variational alternatives to standard sampling methods [@v_dp; @cv_hdp; @ov_hdp]. One first derives an explicit construction of the underlying “weights" of the atomic measure component of the random measures underlying the infinite priors; so-called “stick-breaking" processes for the Dirichlet and beta processes yield such a construction. Then these weights are truncated and integrated into a mean-field variational inference algorithm. For instance, stick-breaking was derived for the Dirichlet process in the seminal paper by Sethuraman [@stick], which was in turn used for variational inference in Dirichlet process models [@v_dp]. Similar stick-breaking representations for a special case of the Indian Buffet Process [@ibp_stick] and the beta process [@beta_st] have been constructed, and have naturally led to mean-field variational inference algorithms for nonparametric models involving these priors [@ibp_vi_fdv; @beta_st_vi]. Such variational inference algorithms have been shown to be more scalable than the sampling-based inference techniques normally used; moreover they work with the full model posterior without marginalizing out any variables. In this paper we propose a variational inference framework for gamma process priors using a novel stick-breaking construction of the process. We use the characterization of the gamma process as a *completely random measure* (CRM), which allows us to leverage Poisson process properties to arrive at a simple derivation of the rate measure of our stick-breaking construction, and show that it is indeed equal to the Lévy measure of the gamma process. We also use the Poisson process formulation to derive a bound on the error of the truncated version compared to the full process, analogous to the bounds derived for the Dirichlet process [@ish_james_2001], the Indian Buffet Process [@ibp_vi_fdv] and the beta process [@beta_st_vi]. We then, as a particular example, focus on the infinite Gamma-Poisson model of [@Gampois] (note that variational inference need not be limited to this model). This model is a prior on infinitely wide latent indicator matrices with non-negative integer-valued entries; each column has an associated parameter independently drawn from a gamma distribution, and the matrix values are independently drawn from Poisson distributions with these parameters as means. We develop a mean-field variational technique using a truncated version of our stick-breaking construction, and a sampling algorithm that uses Monte Carlo integration for parameter marginalization, similar to [@beta_st], as a baseline inference algorithm for comparison. Finally we compare the two algorithms on a non-negative matrix factorization task involving the Psychological Review, NIPS, KOS and New York Times document corpora. **Related Work.** To our knowledge there has been no previous exposition of an explicit recursive “stick-breaking"-like construction of the gamma CRM, and by extension no instance of variational algorithms for such priors. The very general inverse Lévy measure algorithm of [@wolp] requires inversion of the exponential integral, as does the generalized CRM construction technique of [@orbanz_w] when applied to the gamma process; since the closed form solution of the inverse of an exponential integral is not known, these techniques do not give us an analytic construction of the weights, and hence cannot be adapted to variational techniques in a straightforward manner. Other constructive definitions of the gamma process include [@thithesis], who discusses a sampling-based scheme for the weights of a gamma process by sampling from a Poisson process. Further, the characterization of the Dirichlet process as a normalized gamma process may possibly be utilized for sampling gamma process weights, but to our knowledge no existing methods for variational inference employ these approaches. As an alternative to gamma process-based models for count modeling, recent research has examined the negative binomial-beta process and its variants [@zhou_1; @zhou_2; @tab_bnb]; the stick-breaking construction of [@beta_st] readily extends to such models since they have beta process priors. The beta stick-breaking construction has also been used for variational inference in beta-Bernoulli process priors [@beta_st_vi], though they have scalability issues when applied to the count modeling problems addressed in this work, as we show in the experimental section. Background ========== Completely random measures -------------------------- A completely random measure [@crmorig; @crmjord] $\mathbb{G}$ on a space $(\Omega, \mathcal{F})$ is defined as a stochastic process on $\mathcal{F}$ such that for any two disjoint Borel subsets $\mathcal{A}_{1} \text{ and } \mathcal{A}_{2}$ in $\mathcal{F}$, the random variables $\mathbb{G}(\mathcal{A}_{1})\text{ and }\mathbb{G}(\mathcal{A}_{2})$ are independent. The canonical way of constructing a completely random measure $\mathbb{G}$ is to first take a $\sigma$-finite product measure $H\text{ on }\Omega\otimes\mathbb{R}^{+}$, then draw a countable set of points $\{(\omega_{k},p_{k})\}$ from a Poisson process on a Borel $\sigma$-algebra on $\Omega\otimes\mathbb{R}^{+}$ with $H$ as the rate measure. Then the CRM is constructed as $\mathbb{G}=\sum_{k=0}^{\infty}p_{k}\delta_{\omega_{k}}$, where the measure given to a measurable Borel set $B\subset \Omega\text{ is }\mathbb{G}(B) = \sum\limits_{k:\omega_{k}\in B}p_{k}$. In this notation $p_{k}$ are referred to as weights and the $\omega_{k}$ as atoms. If the rate measure is defined on $\Omega\otimes[0,1]$ as $H(d\omega, dp) = cp^{-1}(1-p)^{c-1}B_{0}(d\omega)dp$, where $B_{0}$ is an arbitrary finite continuous measure on $\Omega$ and $c$ is some constant (or function of $\omega$), then the corresponding CRM constructed as above is known as a beta process. If the rate measure is defined as $H(d\omega, dp) = cp^{-1}e^{-cp}G_{0}(d\omega)dp$, with the same restrictions on $c$ and $G_{0}$, then the corresponding CRM constructed as above is known as the gamma process. The total mass of the gamma process $G, G(\Omega)$, is distributed as $\text{Gamma}(cG_{0}(\Omega),c)$. The improper distributions in these rate measures integrate to infinity over their respective domains, ensuring a countably infinite set of points in a draw from the Poisson process. For the beta process, the weights $p_{k}$ are in \[0,1\], whereas for the gamma process they are in $[0,\infty)$. In both cases however the sum of the weights is finite, as can be seen from Campbell’s theorem [@crmorig], and is governed by $c$ and the total mass of the base measure on $\Omega$. For completeness we note that completely random measures as defined in [@crmorig] have three components: a set of fixed atoms, a deterministic measure (usually assumed absent), and a random discrete measure. It is this third component that is explicitly generated using a Poisson process, though the fixed component can be readily incorporated into this construction [@kingman_pp]. If we create an atomic measure by normalizing the weights $\{p_{k}\}$ from the gamma process, i.e. $D=\sum_{k=0}^{\infty}\pi_{k}\delta_{\omega_{k}}$ where $\pi_{k}=p_{k}/\sum_{i=0}^{\infty}p_{i}$, then $D$ is known as a *Dirichlet process* [@fergs_dp], denoted as $D\sim \text{DP}(\alpha_{0},H_{0})$ where $\alpha_{0}=G_{0}(\Omega)\text{ and }H_{0}=G_{0}/\alpha_{0}$. It is not a CRM as the random variables induced on disjoint sets lack independence because of the normalization; it belongs to the class of normalized random measures with independent increments (NRMIs). Stick-breaking for the Dirichlet and Beta Processes --------------------------------------------------- A recursive way to generate the weights of random measures is given by stick-breaking, where a unit interval is subdivided into fragments based on draws from suitably chosen distributions. For example, the sick-breaking construction of the Dirichlet process [@stick] is given by $$D = \sum\limits_{i=1}^{\infty}V_{i}\prod_{j=1}^{i-1}(1-V_{j})\delta_{\omega_{i}},$$ where $V_{i}\overset{iid}{\sim}\text{Beta}(1,\alpha), \quad \omega_{i}\overset{iid}{\sim}H_{0}$. Here the length of the first break from a unit-length stick is given by $V_{1}$. In the next round, a fraction $V_{2}$ of the remaining stick of length $1-V_{1}$ is broken off, and we are left with a piece of length $(1-V_{2})(1-V_{1})$. The length of the piece in the next round is therefore given by $V_{3}(1-V_{2})(1-V_{1})$, and so on. Note that the weights belong to (0,1), and since this is a normalized measure, the weights sum to 1 almost surely. This is consistent with the use of the Dirichlet process as a prior on probability distributions. This construction was generalized in [@beta_st] to yield stick-breaking for the beta process: $$\label{beta_st} B = \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{C_{i}}V_{ij}^{(i)}\prod_{l=1}^{i-1}(1-V_{ij}^{(l)})\delta_{\omega_{ij}},$$ where $V_{ij}^{(i)}\overset{iid}{\sim}\text{Beta}(1,\alpha),\quad C_{i} \overset{iid}{\sim}\text{Poisson}(\gamma), \quad \omega_{ij}\overset{iid}{\sim}\frac{1}{\gamma}B_{0}$. We use this representation as the basis for our stick breaking-like construction of the Gamma CRM, and use Poisson process-based proof techniques similar to [@beta_st_pp] to derive the rate measure. The Stick-breaking Construction of the Gamma Process ==================================================== Constructions and proof of correctness -------------------------------------- We propose a simple recursive construction of the gamma process CRM, based on the stick-breaking construction for the beta process proposed in [@beta_st; @beta_st_pp]. In particular, we augment (or ‘mark’) a slightly modified stick-breaking beta process with an independent gamma-distributed random measure and show that the resultant Poisson process has the rate measure $H(d\omega, dp) = cp^{-1}e^{-cp}G_{0}(d\omega)dp$ as defined above. We show this by directly deriving the rate measure of the marked Poisson process using product distribution formulae. Our proposed stick-breaking construction is as follows: $$\label{gam_st} G = \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{C_{i}}G_{ij}^{(i)}V_{ij}^{(i)}\prod_{l=1}^{i}(1-V_{ij}^{(l)})\delta_{\omega_{ij}},$$ where $G_{ij}^{(i)}\overset{iid}{\sim}\text{Gamma}(\alpha+1,c),\quad V_{ij}^{(i)}\overset{iid}{\sim}\text{Beta}(1,\alpha),\quad C_{i} \overset{iid}{\sim}\text{Poisson}(\gamma), \quad \omega_{ij}\overset{iid}{\sim}\frac{1}{\gamma}H_{0}$. As with the beta process stick-breaking construction, the product of beta random variables allows us to interpret each $j$ as corresponding to a stick that is being broken into an infinite number of pieces. Note that the expected weight on an atom in round $i$ is $\alpha^{i}/c(1+\alpha)^i$. The parameter $c$ can therefore be used to control the weight decay cadence along with $\alpha$. The above representation provides the clearest view of the construction, but is somewhat cumbersome to deal with in practice, mostly due to the introduction of the additional gamma random variable. We reduce the number of random variables by noting that the product of a $\text{Beta}(1,\alpha)\text{ and a }\text{Gamma}(\alpha+1,c)$ random variable has an $\text{Exp}(c)$ distribution; we also perform a change of variables on the product of the $(1-V_{ij})$s to arrive at the following equivalent construction, for which we now prove its correctness: A gamma CRM with positive concentration parameters $\alpha\text{ and }c$ and finite base measure $H_{0}$ may be constructed as $$\label{gam_st2} G = \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{C_{i}}E_{ij}e^{-T_{ij}}\delta_{\omega_{ij}}$$ where $E_{ij}\overset{iid}{\sim}\text{Exp}(c),\quad T_{ij}\overset{ind}{\sim}\text{Gamma}(i,\alpha), \quad C_{i} \overset{iid}{\sim}\text{Poisson}(\gamma), \quad \omega_{ij}\overset{iid}{\sim}\frac{1}{\gamma}H_{0}$. Note that, by construction, in each round $i$ in , each set of weighted atoms $\{\left(\omega_{ij},E_{ij}e^{-T_{ij}}\right)\}_{j=1}^{C_{i}}$ forms a Poisson point process since the $C_{i}$ are drawn from a Poisson($\gamma$) distribution. In particular, each of these sets is a *marked* Poisson process [@kingman_pp], where the atoms $\omega_{ij}$ of the Poisson process on $\Omega$ are marked with the random variables $E_{ij}e^{-T_{ij}}$ that have a probability measure on $(0,\infty)$. The superposition theorem of [@kingman_pp] tells us that the countable union of Poisson process is itself a Poisson process on the same measure space; therefore denoting $\mathlarger{G_{i} = \sum\limits_{j=1}^{C_{i}}E_{ij}e^{-T_{ij}}\delta_{\omega_{ij}}}$, we can say $\mathlarger{G=\bigcup_{i=1}^{\infty}G_{i}}$ is a Poisson process on $\Omega\times[0,\infty)$. We show below that the rate measure of this process equals that of the Gamma CRM. Now, we note that the random variable $E_{ij}e^{-T_{ij}}$ has a probability measure on $[0,\infty)$; denote this by $q_{ij}$. We are going to mark the underlying Poisson process with this measure. The density corresponding to this measure can be readily derived using product distribution formulae. To that end, ignoring indices, if we denote $W = \exp{(-T)}$, then we can derive its distribution by a change of variable. Then, denoting $Q=E\times W\text{ where }E\sim \text{Exp}(c),$ we can use the product distribution formula to write the density of $Q$ as $$f_{Q}(q) = \int\limits_0^1\frac{\alpha^{i}}{\Gamma(i)}\left(-\log{w}\right)^{i-1}w^{\alpha-2}ce^{-c\frac{q}{w}}\mathrm{d}w,$$ where $T\sim\text{Gamma}(i,\alpha)$. Formally speaking, this is the Radon-Nikodym density corresponding to the measure $q$, since it is absolutely continuous with respect to the Lebesgue measure on $[0,\infty)$ and $\sigma$-finite by virtue of being a probability measure. Furthermore, these conditions hold for all the measures that we have in our union of marked Poisson processes; this allows us to write the density of the combined measure as $$\begin{aligned} f(p) &= \sum\limits_{i=1}^{\infty}\int\limits_0^1\frac{\alpha^{i}}{\Gamma(i)}\left(-\log{w}\right)^{i-1}w^{\alpha-2}ce^{-c\frac{p}{w}}\mathrm{d}w \\ &= \int\limits_0^1\sum\limits_{i=1}^{\infty}\frac{\alpha^{i}}{\Gamma(i)}\left(-\log{w}\right)^{i-1}w^{\alpha-2}ce^{-c\frac{p}{w}}\mathrm{d}w &\text{ by monotone convergence} \\ &= \int\limits_0^1 \alpha w^{-2}ce^{-c\frac{p}{w}}\mathrm{d}w \\ &= \alpha p^{-1}e^{-cp} \\ &= cp^{-1}e^{-cp}\frac{\alpha}{c} \end{aligned}$$. Note that the measure defined on $\mathcal{B}([0,\infty))$ by the “improper" gamma distribution $p^{-1}e^{-cp}$ is $\sigma$-finite, in the sense that we can decompose $[0,\infty)$ into the countable union of disjoint intervals $[1/k, 1/(k-1)),\quad k=1,2,\ldots\infty$, each of which has finite measure. In particular, the measure of the interval $[1,\infty)$ is given by the exponential integral. Therefore the rate measure of the process G as constructed here is $G(d\omega, dp) = cp^{-1}e^{-cp}G_{0}(d\omega)dp$ where $G_{0}$ is the same as $H_{0}$ up to the multiplicative constant $\frac{\alpha}{c}$, and therefore satisfies the finiteness assumption imposed on $H_{0}$. We use the form specified in the theorem above in our variational inference algorithm since the variational distributions on almost all the parameters and variables in this construction lend themselves to simple closed-form exponential family updates. As an aside, we note that the random variables $(1-V_{ij})$ have a $\text{Beta}(\alpha,1)$ distribution; therefore if we denote $U_{ij}=1-V_{ij}$ then the construction in is equivalent to $$G = \sum\limits_{i=1}^{\infty}\sum\limits_{j=1}^{C_{i}}E_{ij}^{(i)}\prod_{l=1}^{i}U_{ij}^{(l)}\delta_{\omega_{ij}},$$ where $E_{ij}^{(i)}\overset{iid}{\sim}\text{Exp}(c),\quad U_{ij}^{(i)}\overset{iid}{\sim}\text{Beta}(\alpha,1),\quad C_{i} \overset{iid}{\sim}\text{Poisson}(\gamma), \quad \omega_{ij}\overset{iid}{\sim}\frac{1}{\gamma}H_{0}$. This notation therefore relates our construction to the stick-breaking construction of the Indian Buffet Process [@ibp_stick], where the Bernoulli probabilities $\pi_{k}$ are generated as products of iid $\text{Beta}(\alpha,1)$ random variables : $\pi_{1}=\nu_{1}, \quad \pi_{k}=\prod\limits_{i=1}^{k}\nu_{i}\quad \text{where }\nu_{i}\overset{iid}{\sim}\text{Beta}(\alpha,1)$. In particular, we can view our construction as a generalization of the IBP stick-breaking, where the stick-breaking weights are multiplied with independent Exp($c$) random variables, with the summation over $j$ providing an explicit Poissonization. Truncation analysis {#sec:trunc} ------------------- The variational algorithm requires a truncation level for the number of atoms for tractability. Therefore we need to analyze the closeness between the marginal distributions of the data drawn from the full prior and the truncated prior, with the stick-breaking prior weights integrated out. Our construction leads to a simpler truncation analysis if we truncate the number of rounds (indexed by $i$ in the outer sum), which automatically truncates the atoms to a finite number. For this analysis, we will use the stick-breaking gamma process as the base measure of a Poisson likelihood process, which we denote by $PP$; this is precisely the model for which we develop variational inference in the next section. If we denote the gamma process as $G=\sum_{k=0}^{\infty}g_{k}\delta_{\omega_{k}}$, with $g_{k}$ as the recursively constructed weights, then $PP$ can be written as $PP=\sum_{k=0}^{\infty}p_{k}\delta_{\omega_{k}}\text{ where }p_{k} = \text{Poisson}(g_k)$. Under this model, we can obtain the following result, which is analogous to error bounds derived for other nonparametric models [@ish_james_2001; @ibp_vi_fdv; @beta_st_vi] in the literature. Let N samples $\textbf{X}=(X_{1},..,X_{N})$ be drawn from $PP(G)$. If $G\sim \Gamma\text{P}(c,G_{0})$, the full gamma process, then denote the marginal density of $\textbf{X}\text{ as }\textbf{m}_{\infty}(\textbf{X})$. If $G$ is a gamma process truncated after $R$ rounds, denote the marginal density of $\textbf{X}\text{ as }\textbf{m}_{R}(\textbf{X})$. Then $$\frac{1}{4}\int|\textbf{m}_{\infty}(\textbf{X})-\textbf{m}_{R}(\textbf{X})|d\textbf{X} \leq 1-\exp\left\lbrace-N\gamma\frac{\alpha}{c}\left(\frac{\alpha}{1+\alpha}\right)^{R}\right\rbrace.$$ The starting intuition is that if we truncate the process after R rounds, then the error in the marginal distribution of the data will depend on the probability of positive indicator values appearing for atoms after the $\text{R}^{th}$ round in the infinite version. Combining this with ideas analogous to those in [@ish_james_2000] and [@ish_james_2001], we get the following bound for the difference between the marginal distributions: $$\frac{1}{4}\int|\textbf{m}_{\infty}(\textbf{X})-\textbf{m}_{R}(\textbf{X})|d\textbf{X} \leq \mathbb{P}\left\lbrace\exists (k,j), k>\sum_{r=1}^{R}C_{r}, 1\leq n\leq N \text{ s.t. }X_n(\omega_{kj})>0 \right\rbrace.$$ Since we have a Poisson likelihood on the underlying gamma process, this probability can be written as $$\mathbb{P}(\cdot) = 1-\mathbb{E}\left[\mathbb{E}\left\lbrace\left(\prod\limits_{r=R+1}^{\infty}\prod\limits_{j=1}^{C_{r}}e^{-\pi_{rj}}\right)^{N} \Bigg |C_{r}\right\rbrace\right],$$ where $\pi_{rj}=G_{rj}^{(r)}V_{rj}^{(r)}\prod_{l=1}^{r}(1-V_{rj}^{(l)})$. We may then use Jensen’s inequality to bound it as follows: $$\begin{aligned} \mathbb{P}(\cdot) &\leq 1-\exp\left[N\sum\limits_{r=R+1}^{\infty}\mathbb{E}\left\lbrace\sum\limits_{j=1}^{C_{r}}\log(e^{-\pi_{rj}})\right\rbrace\right] \\ &= 1-\exp\left[N\gamma\frac{1}{c}\sum\limits_{r=R+1}^{\infty}\left(\frac{\alpha}{1+\alpha}\right)^{r}\right] \\ &= 1-\exp\left\lbrace-N\gamma\frac{\alpha}{c}\left(\frac{\alpha}{1+\alpha}\right)^{R}\right\rbrace. \end{aligned}$$ Variational Inference ===================== As discussed in Section \[sec:trunc\], we will focus on the infinite Gamma-Poisson model, where a gamma process prior is used in conjunction with a Poisson likelihood function. When integrating out the weights of the gamma process, this process is known to yield a nonparametric prior for sparse, infinite count matrices [@Gampois]. We note that our approach should easily be applicable to other models involving gamma process priors. The Model --------- To effectively perform variational inference, we re-write $G$ as a single sum of weighted atoms, using indicator variables $\{d_{k}\}$ for the rounds in which the atoms occur, similar to [@beta_st]: $$\label{gp_eqn} G = \sum\limits_{k=1}^{\infty}E_{k}e^{-T_{k}}\delta_{\omega_{k}},$$ where $E_{k}\overset{iid}{\sim}\text{Exp}(c),\quad T_{k}\overset{ind}{\sim}\text{Gamma}(d_{k},\alpha), \quad \sum\limits_{k=1}^{\infty}\mathbbm{1}_{(d_{k}=r)} \overset{iid}{\sim}\text{Poisson}(\gamma), \quad \omega_{k}\overset{iid}{\sim}\frac{1}{\gamma}H_{0}$. We also place gamma priors on $\alpha, \gamma\text{ and }c: \alpha\sim\text{Gamma}(a_{1},a_{2}), \gamma\sim\text{Gamma}(b_{1},b_{2}),c\sim\text{Gamma}(c_{1},c_{2})$. Denoting the data, the latent prior variables and the model hyperparameters by $\mathcal{D}, \Pi\text{ and }\Lambda$ respectively, the full likelihood may be written as $P(\mathcal{D}, \Pi|\Lambda)=P(\mathcal{D}, \Pi_{-G}|\Pi_{G},\Lambda)\cdot P(\Pi_{G}|\Lambda)\text{ where }P(\Pi_{G}|\Lambda)=P(\alpha)\cdot P(\gamma)\cdot P(c)\cdot P(\mathbf{d}|\gamma)\cdot \prod\limits_{k=1}^{K}P(E_{k}|c)\cdot P(T_{k}|d_{k},\alpha)\cdot \prod\limits_{n=1}^{N}P(z_{nk}|E_{k},T_{k})$. We truncate the infinite gamma process to $K$ atoms, and take $N$ to be the total number of datapoints. $\Pi_{-G}$ denotes the set of the latent variables excluding those from the Poisson-Gamma prior; for instance, in factor analysis for topic models, this contains the Dirichlet-distributed factor variables (or topics). From the Poisson likelihood, we have $z_{nk}|E_{k},T_{k}\sim\text{Poisson}(E_{k}e^{-T_{k}})$, independently for each $n$. The distributions of $T_{k}\text{ and }\mathbf{d}$ involve the indicator functions on the round indicator variables $d_{k}$: $$P(T_{k}|d_{k},\alpha)=\frac{\alpha^{\nu_{k}(0)}}{\prod\limits_{r\geq 1}\Gamma(r)^{\mathbbm{1}_{(d_{k}=r)}}}T_{k}^{\nu_{k}(1)}e^{-\alpha T_{k}},$$ where $\nu_{k}(s) = \sum\limits_{r\geq 1}(r-s)\mathbbm{1}_{(d_{k}=r)}$, and $$P(\mathbf{d}|\gamma)=\prod\limits_{r=1}^{\infty}\frac{\gamma^{\sum_{k}\mathbbm{1}_{(d_{k}=r)}}}{\left(\sum_{k}\mathbbm{1}_{(d_{k}=r)}\right)!}\cdot \exp\left\lbrace-\gamma\mathbb{I}\left(\sum\limits_{r^{'}=r}^{\infty}\sum\limits_{k=1}^{\infty}\mathbbm{1}_{(d_{k}=r^{'})} > 0\right)\right\rbrace.$$See [@beta_st_vi] for a discussions on how to approximate these factors in the variational algorithm. The Variational Prior Distribution ---------------------------------- Mean-field variational inference involves minimizing the KL divergence between the model posterior, and a suitably constructed *variational* distribution which is used as a more tractable alternative to the actual posterior distribution. To that end, we propose a fully-factorized variational distribution on the Poisson-Gamma prior as follows: $$Q=q(\alpha)\cdot q(\gamma)\cdot q(c)\cdot\prod\limits_{k=1}^{K}q(E_{k})\cdot q(T_{k})\cdot q(d_{k})\cdot\prod\limits_{n=1}^{N}q(z_{nk}),$$ where $q(E_{k})\sim \text{Gamma}(\acute{\xi_{k}},\acute{\epsilon_{k}}),\quad q(T_{k})\sim \text{Gamma}(\acute{u_{k}},\acute{\upsilon_{k}}),\quad q(\alpha)\sim \text{Gamma}(\kappa_{1},\kappa_{2}),\quad q(\gamma)\sim \text{Gamma}(\tau_{1},\tau_{2}),\quad q(c)\sim \text{Gamma}(\rho_{1},\rho_{2}),\quad q(z_{nk})\sim\text{Poisson}(\lambda_{nk}),\quad q(d_{k})\sim\text{Mult}(\varphi_{k})$. Instead of working with the actual KL divergence between the full posterior and the factorized proxy distribution, variational inference maximizes what is canonically known as the *evidence lower bound* (ELBO), a function that is the same as the KL divergence up to a constant. In our case it may be written as $\mathcal{L}=\mathbb{E}_{Q}\log P(\mathcal{D}, \Pi|\Lambda) - \mathbb{E}_{Q}\log Q$. We omit the full representation here for brevity. The Variational Parameter Updates --------------------------------- Since we are using exponential family variational distributions, we leverage the closed form variational updates for exponential families wherever we can, and perform gradient ascent on the ELBO for the parameters of those distributions which do not have closed form updates. We list the updates on the distributions of the prior below. The closed-form updates for the hyperparameters in $q(E_{k}), q(\alpha), q(c)\text{ and }q(\gamma)$ are as follows: $$\begin{aligned} \acute{\xi_{k}}=\sum_{n=1}^{N}\mathbb{E}_{Q}(z_{nk})+1,\quad\acute{\epsilon_{k}}=\mathbb{E}(c)+N\times\mathbb{E}_{Q}\left[e^{-T_{k}}\right], \quad \kappa_{1}=\sum_{k=1}^{K}\sum\limits_{r\geq 1}r\varphi_{k}(r)+a_{1}, \\ \kappa_{2}=\sum_{k=1}^{K}\mathbb{E}_{Q}(T_{k})+a_{2}, \quad \rho_{1}=c_{1}+K,\quad\rho_{2}=\sum_{k=1}^{K}\mathbb{E}_{Q}(E_{k})+c_{2},\\ \tau_{1}=b_{1}+K,\quad\tau_{2}=\sum\limits_{r\geq 1}\left\lbrace1-\prod\limits_{k=1}^{K}\sum\limits_{\acute{r}=1}^{r-1}\varphi_{k}(\acute{r})\right\rbrace+b_{2}.\end{aligned}$$ The updates for the multinomial probabilities in $q(d_{k})$ are given by: $$\begin{split} \varphi_{k}(r)\propto \exp\{ r\mathbb{E}_{Q}(\log\alpha)-\log\Gamma(r)+(r-1)\mathbb{E}_{Q}(\log T_{k})-\zeta\cdot\sum\limits_{i\neq k}\varphi_{i}(r)\\-\mathbb{E}_{Q}(\gamma)\sum\limits_{j=2}^{r}\prod\limits_{k^{'}\neq k}\sum\limits_{r^{'}=1}^{j-1}\varphi_{k^{'}}(r^{'})\}. \end{split}$$ In addition to these updates, our variational algorithm requires gradient ascent updates on $q(T_{k})$ and updates on $q(\Pi_{-G})\text{ and }q(z_{nk})$ as follows: The gradients for the two variational parameters in $q(T_{k})$ are: $$\begin{split} \frac{\partial\mathcal{L}}{\partial\acute{u_{k}}}=\sum\limits_{r\geq 1}(r-1)\varphi_{k}(r)\psi^{'}(\acute{u_{k}})-\frac{\mathbb{E}_{Q}(\alpha)}{\acute{\upsilon_{k}}}-\sum\limits_{n=1}^{N}\mathbb{E}_{Q}(E_{k})\left(\frac{\acute{\upsilon_{k}}}{\acute{\upsilon_{k}}+1}\right)^{\acute{u_{k}}}\cdot\log\left(\frac{\acute{\upsilon_{k}}}{\acute{\upsilon_{k}}+1}\right)\\-\sum\limits_{n=1}^{N}\mathbb{E}_{Q}(z_{nk})\frac{1}{\acute{\upsilon_{k}}}-(\acute{u_{k}}-1)\psi^{'}(\acute{u_{k}}) - 1 \end{split}$$ $$\begin{split} \frac{\partial\mathcal{L}}{\partial\acute{\upsilon_{k}}}=-\sum\limits_{r\geq 1}(r-1)\varphi_{k}(r)\frac{1}{\acute{\upsilon_{k}}}+\mathbb{E}_{Q}(\alpha)\frac{\acute{u_{k}}}{\left(\acute{\upsilon_{k}}\right)^{2}}-\sum\limits_{n=1}^{N}\mathbb{E}_{Q}(E_{k})\acute{u_{k}}\frac{\acute{\upsilon_{k}}^{\acute{u_{k}}-1}}{(\acute{\upsilon_{k}}+1)^{\acute{u_{k}}+1}}\\+\sum\limits_{n=1}^{N}\mathbb{E}_{Q}(z_{nk})\frac{\acute{u_{k}}}{\left(\acute{\upsilon_{k}}\right)^{2}}- \frac{1}{\acute{\upsilon_{k}}}. \end{split}$$ For the topic modeling problems, we model the observed vocabulary-vs-document corpus count matrix $D$ as $D\sim \text{Poi}(\Phi Z)$, where the $V\times K$ matrix $\Phi$ models the factor loadings, and the $K\times N\text{ matrix }Z$ models the actual factor counts in the documents. We put the $K-$truncated Poisson-Gamma prior on $Z$, and put a Dirichlet$(\beta_{1},\ldots,\beta_{V})$ prior on the columns of $\Phi$. The variational distribution $Q$ consequently gets a Dirichlet$(\Phi|\{\textbf{b}\}_{k})$ distribution multiplied to it, where $\textbf{b}=(b_{1},\ldots,b_{V})$ are the variational Dirichlet hyperparameters. This setup does not immediately lend itself to closed form updates for the $b$-s, so we resort to gradient ascent. The gradient of the ELBO with respect to each variational hyperparameter is $$\begin{split} \frac{\partial\mathcal{L}}{\partial b_{vk}} = - \mathbb{E}_{Q}(z_{nk})\cdot\frac{\sum_{v}b_{vk}- b_{vk}}{\left(\sum_{v}b_{vk}\right)^{2}} + \psi^{'}(b_{vk})\cdot\left(\beta_{v}-b_{vk}+\sum_{n}d_{vn}\right)+\psi^{'}(\sum_{v}b_{vk})\times\\\left(\sum_{v}b_{vk}-V-\beta_{v}-\sum_{n}d_{vn}+1\right). \end{split}$$ In practice however we found a closed-form update facilitated by a simple lower bound on the ELBO to converge faster. We describe the update here. First note that the part of the ELBO relevant to a potential closed form variational update of $\phi_{vk}$ can be written as $$\mathcal{L}= -\phi_{vk}\cdot\sum_{n}\mathbb{E}_{Q}(z_{nk}) + \sum_{n}d_{vn}\cdot\log\phi_{vk} + \cdots,$$ which can then be lower bounded as $$\mathcal{L} \geq \log\phi_{vk}\cdot\left(-\sum_{n}\mathbb{E}_{Q}(z_{nk}) + \sum_{n}d_{vn}\right)+\cdots.$$ This allows us to analytically update $b_{vk}$ as $b_{vk}=-\sum_{n}\mathbb{E}_{Q}(z_{nk}) + \sum_{n}d_{vn} + \beta_{v}$. This frees us from having to choose appropriate corpus-specific initializations and learning rates for the $\Phi$s. A similar lower bound on the ELBO allows us to update the variational parameters of $q(z_{nk})$ as $\lambda_{nk} = -1 - \sum_{v}d_{vn} + \mathbb{E}_{Q}(\log E_{k}) + \mathbb{E}_{Q}(T_{k})$. The MCMC Sampler ================ As a baseline, we also derive and compare with a standard MCMC sampler for this model. We use the construction in for sampling from the model. To avoid inferring the latent variables in all the atom weights of the Poisson-Gamma prior, we use Monte Carlo techniques to integrate them out, as in [@beta_st]. This affects posterior inference for the indicators $z_{nk}$, the round indicators $\mathbf{d}$ and the hyperparameters $c\text{ and }\alpha$. The posterior distribution for $\gamma$ is closed form, as are those for the likelihood latent variables in $\Phi_{-G}$. We re-write the construction of the Poisson-Gamma prior: $$G = \sum\limits_{k=1}^{\infty}E_{k}e^{-T_{k}}\delta_{\omega_{k}},$$ $E_{k}\overset{iid}{\sim}\text{Exp}(c),\quad T_{k}\overset{ind}{\sim}\text{Gamma}(d_{k},\alpha), \quad \sum\limits_{k=1}^{\infty}\mathbbm{1}_{(d_{k}=r)} \overset{iid}{\sim}\text{Pois}(\gamma), \quad \omega_{k}\overset{iid}{\sim}\frac{1}{\gamma}H_{0}$. We put improper priors on $\alpha$ and $c$, and a noninformative Gamma prior on $\gamma$. The indicator counts are given by $Z_{nk}\sim\text{Pois}(g_{k}),\text{ where } g_{k}=E_{k}e^{-T_{k}}$. To avoid sampling the atom weights $E_{k}\text{ and }T_{k}$, we integrate them out using Monte Carlo techniques in the sampling steps for the prior. Sampling the round indicators ----------------------------- The conditional posterior for the round indicators $\mathbf{d}=\left\lbrace d_{k}\right\rbrace_{k=1}^{K}$ can be written as $$p\left(d_{k}=i|\lbrace d_{l}\rbrace_{l=1}^{k-1},\lbrace Z_{nk}\rbrace_{n=1}^{N},\alpha,c,\gamma\right)\propto p\left(\lbrace Z_{nk}\rbrace_{n=1}^{N}|d_{k}=i,\alpha,c\right)p\left(d_{k}=i|\lbrace d_{l}\rbrace_{l=1}^{k-1}\right).$$ For the first factor, we collapse out the stick-breaking weights and approximate the resulting integral using Monte-Carlo techniques as follows: $$\begin{aligned} p\left(\lbrace Z_{nk}\rbrace_{n=1}^{N}|d_{k}=i,\alpha,c\right) &= \int_{[0,\infty]^{i}}\prod\limits_{n=1}^{N}\text{Pois}(Z_{nk}|g_{k})\mathrm{d}G\\ &\approx \frac{1}{S}\sum\limits_{s=1}^{S}\prod\limits_{n=1}^{N}\text{Pois}(Z_{nk}|g_{k}^{(s)}),\end{aligned}$$ where $g_{k}^{(s)} = E_{k}^{(s)}e^{-T_{k}^{(s)}}\overset{d}{=}V_{k,d_{k}}^{(s)}\prod_{l=1}^{d_{k}}(1-V_{kl}^{(s)})$. Here $S$ is the number of simulated samples from the integral over the stick-breaking weights. We take $S=1000$ in our experiments. The second factor is the same as [@beta_st]: $$p(d_{k}=d|\gamma,\lbrace d_{l}\rbrace_{l=1}^{k-1}) = \left\{ \begin{array}{lll} 0 & \mbox{if } d < d_{k-1} \\ \frac{1-\sum_{t=1}^{D_{k-1}}\text{Pois}(t|\gamma)}{1-\sum_{t=1}^{D_{k-1}-1}\text{Pois}(t|\gamma)} & \mbox{if } d = d_{k-1} \\ \left(1-\frac{1-\sum_{t=1}^{D_{k-1}}\text{Pois}(t|\gamma)}{1-\sum_{t=1}^{D_{k-1}-1}\text{Pois}(t|\gamma)}\right)(1-\text{Pois}(0|\gamma))\text{Pois}(0|\gamma)^{h-1} & \mbox{if } d = d_{k-1}+h. \end{array} \right.$$ Here $D_{k}\overset{\Delta}{=}\sum\limits_{j=1}^{k}\mathbb{I}(d_{j}=d_{k})$. Normalizing the product of these two factors over all $i$ is infeasible, so we evaluate this product for increasing $i$ till it drops below $10^{-2}$, and normalize over the gathered values. Sampling the factor variables ----------------------------- Here we consider the Poisson factor modeling scenario that we use to model vocabulary-document count matrices. Recall that a $V\times N$ count matrix $D$ is modeled as $D=\text{Poi}(\Phi Z)$, where the $V\times K$ matrix $\Phi$ models the factor loadings, and the $K\times N\text{ matrix }Z$ models the actual factor counts in the documents.. We put the Poisson-Gamma prior on $Z$ and symmetric Dirichlet$(\beta_{1},\ldots,\beta_{V})$ priors on the columns of $\Phi$. The sampling steps for $\Phi$ and $Z$ are described next. ### Sampling $\Phi$ First note that the elements of the count matrix are modeled as $d_{vn}=\text{Poi}\left(\sum_{k=1}^{K}\phi_{vk}z_{kn}\right)$, which can be equivalently written as $d_{vn}=\sum_{k=1}^{K}d_{vkn},\quad d_{vkn}=\text{Poi}(\phi_{vk}z_{kn})$. Standard manipulations then allow us to sample the $d_{vkn}$’s from $\text{Mult}(d_{vn};p_{v1n},\ldots,p_{vKn})$ where $p_{vkn}=\phi_{vk}z_{kn}/\sum_{k}^{K}\phi_{vk}z_{kn}$. Now we have $\phi_{k}\sim\text{Dirichlet}(\beta_{1},\ldots,\beta_{V})$. Using standard relationships between Poisson and multinomial distributions, we can derive the posterior distribution of the $\phi_{k}$’s as $\text{Dirichlet}(\beta_{1}+d_{1k},\ldots,\beta_{V}+d_{Vk}),\text{ where }d_{vk}=\sum_{n=1}^{N}d_{vkn}$. ### Sampling Z In our algorithm we sample each $z_{nk}$ conditioned on all the other variables in the model; therefore the conditional posterior distribution can be written as $$\begin{aligned} p(z_{nk}|D, \Phi,Z_{n,-k},\mathbf{d},\alpha,c,\gamma) &= p(D|Z_{n},\Phi)p(z_{nk}|\mathbf{d},\alpha,c,Z_{n,-k}) \\ &= \prod\limits_{v=1}^{V}\text{Poi}\left(d_{vn}|\sum\limits_{k=1}^{K}\phi_{vk}z_{kn}\right)\frac{p(Z_{n}|\mathbf{d},\alpha,c)}{p(Z_{n,-k}|\mathbf{d},\alpha,c)}.\end{aligned}$$ The distributions in both the numerator and denominator of the second factor can be sampled from using the Monte Carlo techniques described above, by integrating out the stick-breaking weights. Sampling hyperparameters ------------------------ As mentioned above, we put a noninformative Gamma prior on $\gamma$ and improper (1) priors on $\alpha$ and $c$. The posterior sampling steps are described below: ### Sampling $\gamma$ Given the round indicators $\mathbf{d}=\left\lbrace d_{k}\right\rbrace$, we can recover the round-specific atom counts as described above. Then the conjugacy between the Gamma prior on $\gamma$ and the Poisson distribution of $C_{i}$ gives us a closed form posterior distribution for $\gamma$: $p(\gamma|\mathbf{d},Z,\alpha,c) = \text{Gamma}(\gamma|a+\sum_{i=1}^{K}C_{i},b+d_{K})$. ### Sampling $\alpha$ The conditional posterior distribution of $\alpha$ may be written as: $$p(\alpha|Z,\mathbf{d},c)\propto p(\alpha)\prod_{n=1}^{N}\prod_{k=1}^{K}p(Z|\mathbf{d},\alpha,c).$$ We calculate the posterior distribution of $Z$ using Monte Carlo techniques as described above. Then we discretize the search space for $\alpha$ around its current values as $\left(\alpha_{cur}+t\Delta\alpha\right)_{t=L}^{U}$, where the lower and upper bounds $L$ and $U$ are chosen so that the unnormalized posterior falls below $10^{-2}$. The search space is also clipped below at 0. $\alpha$ is then drawn from a multinomial distribution on the search values after normalization. ### Sampling c We sample $c$ in exactly the same way as $\alpha$. We first write the conditional posterior as $$p(c|Z,\mathbf{d},\alpha)\propto p(c)\prod_{n=1}^{N}\prod_{k=1}^{K}p(Z|\mathbf{d},\alpha,c).$$ The search space $(c>0)$ is then discretized using appropriate upper and lower bounds as above, and $Z$ is sampled using Monte Carlo techniques. $c$ is then drawn from a multinomial distribution on the search values after normalization. Experiments =========== We consider the problem of learning latent topics in document corpora. Given an observed set of counts of vocabulary words in a set of documents, represented by say a $V\times N$ count matrix, where $V$ is the vocabulary size and $N$ the number of documents, we aim to learn $K$ latent factors and their vocabulary realizations using Poisson factor analysis. In particular, we model the observed corpus count matrix $D$ as $D\sim \text{Poi}(\Phi\mathbf{I})$, where the $V\times K$ matrix $\Phi$ models the factor loadings, and the $K\times N\text{ matrix }\mathbf{I}$ models the actual factor counts in the documents. As a baseline, we also derive and compare with a standard MCMC sampler for this model. We use the construction in for sampling from the model. To avoid inferring the latent variables in all the atom weights of the Poisson-Gamma prior, we use Monte Carlo techniques to integrate them out, as in [@beta_st]. This affects posterior inference for the indicators $z_{nk}$, the round indicators $\mathbf{d}$ and the hyperparameters $c\text{ and }\alpha$. The posterior distribution for $\gamma$ is closed form, as are those for the likelihood latent variables in $\Pi_{-G}$. The complete updates are described in the supplementary. We implemented and analyzed the performance of three variational algorithms corresponding to three different priors on $\mathbf{I}$: the Poisson-gamma process prior from this paper (abbreviated hereafter as VGP), the Bernoulli-beta prior from [@beta_st_vi] (VBP) and the IBP prior from [@ibp_vi_fdv] (VIBP), along with the MCMC sampler mentioned above (SGP). For the Bernoulli-beta priors we modeled $\mathbf{I}$ as $\mathbf{I}=W\circ Z$ as in [@beta_st_vi], where the nonparametric priors are put on $Z$ and a vague Gamma prior is put on $W$. For the VGP and SGP models we set $\mathbf{I}=Z$. In addition, for all four algorithms, we put a symmetric Dirichlet$(\beta_{1},\ldots,\beta_{V})$ prior on the columns of $\Phi$. We added corresponding variational distributions for the variables in the collection denoted as $\Pi_{-G}$ above. We use held-out per-word test log-likelihoods and times required to update all variables in $\Pi$ in each iteration as our comparison metrics, with 80% of the data used for training. We used the same likelihood metric as [@zhou_1], with the samples replaced by the expectations of the variational distributions. **Synthetic Data.** As a warm-up, we consider the performances of VGP and SGP on some synthetic data generated from this model. We generate 200 weighted atoms from the gamma prior using the stick-breaking construction, and use the Poisson likelihood to generate 3000 values for each atom to yield the indicator matrix $Z$. We simulated a vocabulary of 200 terms, generated a 200$\times$200 factor-loading matrix $\Phi$ using symmetric Dirichlet priors, and then generated $D=\text{Poi}(\Phi Z)$. For the VGP, we measure the test likelihood after every iteration and average the results across 10 random restarts. These measurements are plotted in fig.\[figr:synt\]. As shown, VGP’s measured heldout likelihood converges within 10 iterations. The SGP traceplot shows the first thirty heldout likelihoods measured after burn-in. Per-iteration times were 15 seconds and 2.36 minutes for VGP (with $K$=125) and SGP respectively. The SGP learned $K$ online, with values oscillating around 50. SNBP refers to the Poisson-Gamma mixture (“NB process") sampler from [@zhou_1]. Its traceplot shows the first 30 likelihoods measured after 1000 burn-in iterations. We see that it performed similarly to our algorithms, though slightly worse. **Real data.** We used a similar framework to model the count data from the Psychological Review (PsyRev)[^1], NIPS[^2], KOS[^3] and New York Timescorpora. The vocabulary sizes are 2566, 13649, 6906 and 100872 respectively, while the document counts are 1281, 1740, 3430 and 300000 respectively. For each dataset, we ran all three variational algorithms with 10 random restarts each, measuring the held-out log-likelihoods and per-iteration runtimes for different values of the truncation factor $K$. The learning rates for gradient ascent updates were kept on the order of $10^{-4}$ for both VGP and VBP, with 5 gradient steps per iteration. A representative subset of results is shown in figs.\[figr:psypp\] through \[figr:nytt\]. We used vague gamma priors on the hyperparameters $\alpha, \gamma\text{ and }c$ in the variational algorithms, and improper (1) priors for the sampler. We found the test likelihoods to be independent of these initializations. The results for the variational algorithms were dependent on the Dirichlet prior $\beta$ on $\Phi$, as noted in fig.\[figr:psypp\]. We therefore used the learned test likelihood after 100 iterations as a heuristic to select $\beta$. We found the three variational algorithms to attain very similar test likelihoods across all four datasets after a few hours of CPU time, with the VGP and VBP having a slight edge over the VIBP. The sampler somewhat unexpectedly did not attain a competitive score for any dataset, unlike the synthetic case. For instance, as shown in fig.\[figr:psypitn\], it oscillated around -7.45 for the PsyRev dataset, whereas the variational algorithms attained -7.23. For comparison, the NB process sampler from [@zhou_1] attains -7.25 each iteration after 1000 iterations of burn-in. Also as seen in fig.\[figr:psypitn\], VGP was faster to convergence (in less than 10 iterations in $\sim$5 seconds) than VIBP and VBP ($\sim$50 iterations each). The test log-likelihoods after a few hours of runtime were largely independent of the truncation $K$ for the three variational algorithms. Behavior for the other datasets was similar. Among the three variational algorithms, the VIBP scaled best for small to medium datasets as a function of the truncation factor due to all updates being closed-form, in spite of having to learn the additional weight matrix $W$. The VGP running times were competitive for small values of $K$ for these datasets. However, in the large NYT dataset, VGP was orders of magnitude faster than the Bernoulli-beta algorithms (note the log-scale in fig.\[figr:nytt\]). For example, with a truncation of 100 atoms, VGP took around 45 seconds per iteration, whereas both VIBP and VBP took more than 3 minutes. The VBP scaled poorly for all datasets, as seen in figs.\[figr:psyt\] through \[figr:nytt\]. The reason for this is three-fold: learning the parameters for the additional matrix $W$ which is directly affected by dimensionality (also the reason for VIBP being slow for NYT dataset), gradient updates for two variables (as opposed to one for VGP) and a Taylor approximation required for these gradient updates (see [@beta_st_vi]). The sampler SGP required around 7 minutes per iteration for the small datasets and an hour and 40 minutes on average for NYT. To summarize, we found the VGP to post running times that are competitive with the fastest algorithm (VIBP) in small to medium datasets, and outperform the other methods completely in the large NYT dataset, all the while providing similar accuracy compared to variational algorithms for similar models, as measured by held-out likelihood. It was also the fastest to converge, typically taking less than 15 iterations. Compared with SGP, our variational method is substantially faster (particularly on large-scale data) and produces higher likelihood scores on real data. [0.3]{} ![Plots of held-out test likelihoods and per-iteration running times. Plots (d), (e) and (f) are for PsyRev, KOS, and NYT respectively. Plots (b) and (c) are for the PsyRev dataset. Algorithm trace colors are common to all plots. See text for full details.[]{data-label="resfig"}](synres "fig:"){width="\textwidth"} [0.3]{} ![Plots of held-out test likelihoods and per-iteration running times. Plots (d), (e) and (f) are for PsyRev, KOS, and NYT respectively. Plots (b) and (c) are for the PsyRev dataset. Algorithm trace colors are common to all plots. See text for full details.[]{data-label="resfig"}](psypp "fig:"){width="\textwidth"} [0.3]{} ![Plots of held-out test likelihoods and per-iteration running times. Plots (d), (e) and (f) are for PsyRev, KOS, and NYT respectively. Plots (b) and (c) are for the PsyRev dataset. Algorithm trace colors are common to all plots. See text for full details.[]{data-label="resfig"}](psypitn "fig:"){width="\textwidth"} [0.3]{} ![Plots of held-out test likelihoods and per-iteration running times. Plots (d), (e) and (f) are for PsyRev, KOS, and NYT respectively. Plots (b) and (c) are for the PsyRev dataset. Algorithm trace colors are common to all plots. See text for full details.[]{data-label="resfig"}](psytimes "fig:"){width="\textwidth"} [0.3]{} ![Plots of held-out test likelihoods and per-iteration running times. Plots (d), (e) and (f) are for PsyRev, KOS, and NYT respectively. Plots (b) and (c) are for the PsyRev dataset. Algorithm trace colors are common to all plots. See text for full details.[]{data-label="resfig"}](kostimes "fig:"){width="\textwidth"} [0.3]{} ![Plots of held-out test likelihoods and per-iteration running times. Plots (d), (e) and (f) are for PsyRev, KOS, and NYT respectively. Plots (b) and (c) are for the PsyRev dataset. Algorithm trace colors are common to all plots. See text for full details.[]{data-label="resfig"}](nytimes "fig:"){width="\textwidth"} Conclusion ========== We have described a novel stick-breaking representation for gamma processes and used it to derive a variational inference algorithm. This algorithm has been shown to be far more scalable for large datasets than variational algorithms for related models while attaining similar accuracy, and also outperforms sampling-based methods. We expect that recent improvements to variational techniques can also be applied to our algorithm, potentially yielding even further scalability. [^1]: http://psiexp.ss.uci.edu/research/programs\_data/toolbox.htm [^2]: http://www.stats.ox.ac.uk/ teh/data.html [^3]: \[uci\]https://archive.ics.uci.edu/ml/datasets/Bag+of+Words

--- abstract: | We consider [*monotone*]{} embeddings of a finite metric space into low dimensional normed space. That is, embeddings that respect the order among the distances in the original space. Our main interest is in embeddings into Euclidean spaces. We observe that any metric on $n$ points can be embedded into $l_2^n$, while, (in a sense to be made precise later), for almost every $n$-point metric space, every monotone map must be into a space of dimension $\Omega(n)$ (Lemma \[momad2\]).\ It becomes natural, then, to seek explicit constructions of metric spaces that cannot be monotonically embedded into spaces of sublinear dimension. To this end, we employ known results on [*sphericity*]{} of graphs, which suggest one example of such a metric space - that defined by a complete bipartite graph. We prove that an $\delta n$-regular graph of order $n$, with bounded diameter has sphericity $\Omega(n/(\lambda_2+1))$, where $\lambda_2$ is the second largest eigenvalue of the adjacency matrix of the graph, and $0 < \delta \leq {\frac 1 2}$ is constant (Theorem \[our-bound\]). We also show that while random graphs have linear sphericity, there are [*quasi-random*]{} graphs of logarithmic sphericity (Lemma \[alex\]).\ For the above bound to be linear, $\lambda_2$ must be constant. We show that if the second eigenvalue of an $n/2$-regular graph is bounded by a constant, then the graph is close to being complete bipartite. Namely, its adjacency matrix differs from that of a complete bipartite graph in only $o(n^2)$ entries (Theorem \[main1\]). Furthermore, for any $0 < \delta < {\frac 1 2}$, and $\lambda_2$, there are only finitely many $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$ (Corollary \[no-graphs\]). author: - 'Yonatan Bilu and Nati Linial [^1]' bibliography: - 'bib.bib' title: 'Monotone Maps, Sphericity and Bounded Second Eigenvalue' --- 0.2cm \[section\] \[section\] \[section\] \[section\] \[section\] \[THEOREM\][Problem]{} =msbm10 [*Keywords:*]{} Embedding, Finite Metric Space, Graphs, Sphericity, Eigenvalues, Bipartite Graphs, Second Eigenvalue. Introduction ============ Euclidean embeddings of finite metric spaces have been extensively studied, with the aim of finding an embedding that doesn’t distort the metric too much. We refer the reader to the survey papers of Indyk ([@Indyk]) and Linial ([@Nati]), as well as chapter 15 of Matou[š]{}ek’s Discrete Geometry book [@Matousek]. Here we focus on a different type of embeddings. Namely, those that preserve the order relation of the distances. We call such embeddings [*monotone*]{}. There are quite a few applications that make this concept natural and interesting, since there are numerous algorithmic problems whose solution depends only on the order among the distances. Specifically, questions that concern nearest neighbors. The notion of monotone embeddings suggests the following general strategy toward the resolution of such problems. Namely, embed the metric space at hand monotonically into a “nice” space, for which good algorithms are known to solve the problem. Solve the problem in the “nice” space - the same solution applies as well for the original space. “Nice” often means a low dimensional normed space. Thus, we focus on the minimal dimension which permits a monotone embedding.\ In section \[dope\] we observe that any metric on $n$ points can be monotonically embedded into an $n$-dimensional Euclidean space, and that the bound on the dimension is asymptotically tight. The embedding clearly depends only on the order of the distances (Lemma \[dope-lemma\]). We show that for almost every ordering of the ${n \choose 2}$ distances among $n$ points, the host space of a monotone embedding must be $\Omega(n)$-dimensional. Similar bounds are given for embeddings into $l_\infty$, and some bounds are also deduced for other norms.\ Next we consider embeddings that are even less constrained. Given a metric space $(X,\delta)$ and some threshold $t$, we seek a mapping $f$ that only respects this threshold. Namely, $||f(x)-f(y)||<1$ iff $\delta(x,y)<t$. The input to this problem can thus be thought of as a graph (adjacency indicating distances below the threshold $t$). The minimal dimension $d$, such that a graph $G$ can be mapped this way into $l_2^d$ is known as the [*sphericity*]{} of G, and denoted $Sph(G)$. Reiterman, R[ö]{}dl and [Š]{}i[ň]{}ajov[á]{} ([@RRS89a]) show that the sphericity of $K_{n,n}$ is $n$. This is, then, an explicit example of a metric space which requires linear dimension to be monotonically embedded into $l_2$. Other than that, the best lower bounds previously known to us are logarithmic. In section \[prox-graph-sec\] we prove a novel lower bound, namely that for $0 < \delta \leq {\frac 1 2}$, $Sph(G) = \Omega(\frac n {\lambda_2 + 1})$, for any $n$-vertex $\delta n$-regular graph, with bounded diameter. Here $\lambda_2$ is the second largest eigenvalue of the graph. We also show examples of quasi-random graphs of logarithmic sphericity. This is somewhat surprising since quasi-random graphs tend to behave like random graphs, yet the latter have linear sphericity.\ In our search for further examples of graphs of linear sphericity, we investigate in section \[lambda2-cons\] families of graphs whose second eigenvalue is bounded by a constant (for which the aforementioned lower bound is linear). We show that such graphs are close to being complete bipartite, in the sense that one needs to modify only $o(n^2)$ entries in the adjacency matrix to get the latter from the former. As a corollary, we get that for $0 < \delta < {\frac 1 2}$, and $\lambda_2$ there are only finitely many $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$. {#dope} Definitions ----------- Let $X = ([n],\delta)$ be a metric space on $n$ points, such that all pairwise distances are distinct. Let $||\;||$ be a norm on $\R^d$. We say that $\phi:X \rightarrow (\R^d,||\;||)$ is a [**]{} if for every $w,x,y,z \in X$, $\delta(x,y) < \delta(w,z) \Leftrightarrow ||\phi (x) - \phi (y)|| < ||\phi (w) - \phi (z)||$. We denote by $d(X,||\;||)$ the minimal $t$ such that there exists a from $X$ to $(\R^t,||\;||)$. We denote by $d(n,||\;||) = \max_X d(X,||\;||)$, the smallest dimension to which every $n$ point metric can be mapped monotonically. The first thing to notice is that we are actually concerned only with the [*order*]{} among the distances between the points in the metric space, and not with the actual distances. Let $(X,\delta)$ be a finite metric space, and let $\rho$ be a linear order on $X \choose 2$. We say that $\rho$ and $(X,\delta)$ are [*consistent*]{} if for every $w,x,y,z \in X$, $\delta(x,y) < \delta(w,z) \Leftrightarrow (x,y) <_\rho (w,z)$. We start with an easy, but useful observation. \[dope-lemma\] Let $X$ be a finite set. For every strict order relation $\rho$ on $X \choose 2$, there exists a distance function $\delta$ on $X$, that is consistent with $\rho$. Let $\{\epsilon_{ij}\}_{(i,j)\in {X \choose 2}}$ be small, non-negative numbers, ordered as per $\rho$. Define $\delta(i,j) = 1 + \epsilon_{ij}$. It is obvious that $\delta$ induces the desired order on the distances of $X$, and, that if the $\epsilon$’s are small, the triangle inequality holds. When we later (Section \[l2\]) use this observation, we refer to it as a [*standard $\epsilon$-construction*]{}, where $\epsilon = \max \epsilon_{ij}$. It is not hard to see that this metric is Euclidean, that is, the resulting metric can be isometrically embedded into $l_2$, see Lemma \[momad2\] below. We say that an order relation $\rho$ on $[n] \choose 2$ is [*realizable*]{} in $(\R^d,||\;||)$ if there exists a metric space $(X,\delta)$ on $n$ points which is consistent with $\rho$, and a $\phi:X \rightarrow \R^d$. We say that $\phi$ is a realization of $\rho$. (In other words, $d(n,||\;||)$ is the minimal $d$ such that any linear order on $[n] \choose 2$ is realizable in $(\R^d,||\;||)$.) We denote by $J = J_n$ the $n \times n$ all ones matrix, and by $PSD_n$ the cone of real symmetric $n \times n$ positive semidefinite matrices. We omit the subscript $n$ when it is clear from the context. Finally, for a graph $G$, and $U, V$ subsets of its vertices, we denote by $e(U,V) = |\{(u,v) \in E(G): u \in U, v\in V\}|$, and $e(U) = |\{(u,u') \in E(G): u,u' \in U\}|$. into $l_\infty$. ---------------- $\frac n 2 - 1 \leq d(n,l_\infty) \leq n$ It is well known that any metric $X$ on $n$ points can be embedded into $l_\infty^n$ isometrically, hence $d(n,l_\infty) \leq n$. For the lower bound, we define a metric space $(X,\delta)$ with $2n+2$ points that cannot be realized in $l_\infty^n$. By lemma \[dope-lemma\], it suffices to define an ordering on the distances. In fact, we define only a partial order, any linear extension of which will do. The $2n+2$ points come in $n+1$ pairs, $\{x_i,y_i\}_{i=1,\ldots,n+1}$. If $z \notin \{x_i,y_i\}$, we let $\delta(x_i,y_i) > \delta(x_i,z), \delta(y_i,z)$. Assume for contradiction that a $\phi$ into $l_\infty^n$ does exist. For each pair $(x,y)$ define $j(x,y)$ to be some index $i$ for which $|\phi(x)_i - \phi(y)_i|$ is maximized, that is, an index $i$ for which $|\phi(x)_i - \phi(y)_i|=\|\phi(x)-\phi(y)\|_{\infty}$. By the pigeonhole principle there exist two pairs, say $(x_1,y_1)$ and $(x_2,y_2)$, for which $j(x_1,y_1) = j(x_2,y_2)=j$. It is easy to verify that our assumptions on the four real numbers $\phi(x_1)_j$, $\phi(x_2)_j$, $\phi(y_1)_j$, $\phi(y_2)_j$, are contradictory. Thus $d(n,l_\infty) \geq \frac n 2 - 1$. into $l_2$. {#l2} ----------- \[momad2\] $\frac n 2 \leq d(n,l_2) \leq n$. Furthermore, for every $\delta > 0$, and every large enough $n$, almost no linear orders $\rho$ on ${[n] \choose 2}$ can be realized in dimension less than $\frac n {2+\delta}$. The upper bound is apparently folklore. As we could not find a reference for it, we give a proof here. Let $\rho$ be a linear order on ${[n] \choose 2}$. Let $\epsilon$ be a real symmetric matrix with the following properties: - $\epsilon_{ii} = 0$ for all $i$. - $\frac{1}{n} > \epsilon_{ij} > 0$, for all $i \neq j$. - The numbers $\epsilon_{i,j}$ are consistent with the order $\rho$. Since the sum of each row is strictly less than one, all eigenvalues of $\epsilon$ are in the open interval $(-1,1)$. It follows that the matrix $I - \epsilon$ is positive definite. Therefore, there exists a matrix $V$ such that $V V^t = I - \epsilon$. Denote the $i$’th row of $V$ by $v_i$. Clearly, the $v_i$’s are unit vectors, and $<v_i,v_j> = - \epsilon_{i,j}$ for $i \neq j$. Therefore, $||v_i - v_j||_2^2 = <v_i,v_i> + <v_j,v_j> - 2<v_i,v_j> = 2 + 2 \epsilon_{i,j}$. It follows that the map $\phi(i) = v_i$ is a realization of $\rho$, and the upper bound is proved. In fact, one can add another point without increasing the dimension, by mapping it to $0$, and perturbing the diagonal.\ For the lower bound, it is essentially known that if $X$ is the metric induced by $K_{n,n}$, then $d(X,l_2) \geq n$. We discuss this in more detail in the next section. For the second part of the lemma we need a bound on the number of [*sign-patterns*]{} of a sequence of real polynomials. Let $p_1,...,p_m$ be real polynomials in $l$ variables of (total) degree $d$, and let $x \in \R^l$ be a point where none of them vanish. The sign-pattern at $x$ is $(\sgn(p_1(x)),...,\sgn(p_m(x)))$. Denote the total number of different sign-patterns that can be obtained from $p_1,...,p_m$ by $s(p_1,...,p_m)$. A variation of the Milnor-Thom theorem [@Milnor] due to Alon, Frankl and Rödl [@AFR85] shows: \[Milnor\][@AFR85] Let $p_1,...,p_m$ be real polynomials as above. Then for any integer $k$ between 1 and $m$: $$\begin{aligned} s(p_1,...,p_m) \leq 2kd \cdot (4kd - 1) ^ {l + \frac m k - 1}\end{aligned}$$ Set $n = c \cdot d$, for some constant $c$, and $l = n \cdot d$. Consider a point $x \in R^l$, and think of it as an $n \times d$ matrix. Denote the $i$th row of this matrix by $x_i$. As before, $x$ [*realizes*]{} an order $\rho$ on ${[n] \choose 2}$ if the distances $||x_i - x_j||$ are consistent with $\rho$. For two different pairs, $(i_1,j_1)$ and $(i_2,j_2)$, define the polynomial $$\begin{aligned} p_{(i_1,j_1),(i_2,j_2)}(x) = ||x_{i_1} - x_{j_1}||^2 - ||x_{i_2} - x_{j_2}||^2.\end{aligned}$$ The list contains $m = {{n \choose 2} \choose 2}$ polynomials of degree 2. Note that there is a $1:1$ correspondence between orders on ${[n]} \choose {2}$ and sign-patterns of $p_1,...,p_m$, thus no more than $s=s(p_1,...,p_m)$ orders may be realized in $l_2^d$. Take $k=\mu n^2$, for some large constant $\mu$. Then $\log s$ is approximately $2 c d^2 \log d$. By contrast, that total number of orders is ${n \choose 2}!$, so its log is about $c^2d^2 \log d$. If $c$ is bigger than 2, almost all order relations can not be realized. In fact, the same proof shows that for any positive integer $t$, almost all orders on $n \choose 2$ require linear dimension to be realized, and in particular that $d(n,l_{2t}) = \Omega(n)$ (where the constant of proportionality depends only t): Simply repeat the argument above with polynomials of degree $2t$ rather than quadratic polynomials. Other Norms ----------- We conclude this section with two easy observations about into other normed spaces. The first gives an upper bounds on the dimension required for embedding into $l_p$: $d(n,l_p) \leq {n \choose 2}$. By Lemma \[momad2\], any metric space on $n$ points can be mapped monotonically into $l_2$. It is known (see [@DeLa] and also chapter 15 of [@Matousek]) that any $l_2$ metric on $n$ points can be isometrically embedded into ${n \choose 2}$-dimensional $l_p$. The composition of these mappings is a monotone mapping of the metric space into ${n \choose 2}$-dimensional $l_p$. The second observation gives a lower bound for arbitrary norms. We first note the following: Let $||\;||$ be an arbitrary $n$-dimensional norm and let $x_1,...,x_{5^n}$ be points in $\R^n$, such that $||x_i-x_j||>1$ for all $i \neq j$. Then there exits a pair $(x_i,x_j)$ that $||x_i-x_j|| \geq 2$ Denote by $v$ the volume of $B$, the unit ball in $(\R^n,||\;||)$. The translates $x_i + {\frac 1 2}B$ are obviously non intersecting, so the volume of their union is $(\frac{5}{2})^n v$. Assume for contradiction that all pairwise distances are less than $2$, then all these balls are contained in a single ball of radius less than $\frac 5 2$. But this is impossible, since the volume of this ball is less than $(\frac{5}{2})^n v$. Note that the $l_\infty$ norm shows that indeed an exponential number of points is required for the lemma to follow. We do not know, however, the smallest base of the exponent for which the claim holds. The determination of this number seems to be of some interest. There exists an $n$-point metric spaces $(X,\delta)$ such that for any norm $||\;||$, $d(n,||\;||) = \Omega(\log n)$. We construct a distance function on $5^n+1$ points which can not be realized in any $n$-dimensional norm. By lemma \[dope-lemma\] it suffices to define a partial order on the distances. Denote the points in the metric space $0,\ldots,5^n$. Let the distance between $0$ and any other point be smaller than any distance between any two points $i \neq j > 0$. Consider a monotone map $\phi$ of the metric space into $n$-dimensional normed space. Assume, w.l.o.g., that $\min_{i,j=1,\ldots,5^n}||\phi(i)-\phi(j)|| = 1$. By the previous lemma there exists a pair of points, $i,j \neq 0$, such that $||\phi(i)-\phi(j)||>2$. But for $\phi$ to be monotone it must satisfy $||\phi(0)-\phi(i)||<1$ and $||\phi(0)-\phi(j)||<1$, contradicting the triangle inequality. Sphericity {#prox-graph-sec} ========== So far we have concentrated on embeddings of a metric space into a normed space, that preserve the order relations between distances. However, in the examples that gave us the lower bounds for $l_\infty$ and for arbitrary norms, we actually only needed to distinguish between “long” and “short” distances. This motivates the introduction of a broader class of maps, that need only respect the distinction between short and long distances. More formally, let $X=([n],\delta)$ be a metric space. Its [*proximity graph*]{} with respect to some threshold $\tau$, is a graph on $n$ vertices, with an edge between $i$ and $j$ iff $\delta(i,j) \leq \tau$. An embedding of a proximity graph, is a mapping $\phi$ of its vertices into normed space, such that $||\phi (i) - \phi (j) || < 1$ iff $(i,j)$ is an edge in the proximity graph (We assume that no distance is exactly 1). The minimal dimension in which a graph can be so embedded (in Euclidean space) was first studied by Maehara in [@Maehara] under the name [*sphericity*]{}, and denoted $Sph(G)$. Following this terminology, we call such an embedding [*spherical*]{}.\ The sphericity of graphs was further studied by Maehara and Frankl in [@FraMa], and then by Reiterman, R[ö]{}dl and [Š]{}i[ň]{}ajov[á]{} in [@RRS89a], [@RRS89b], [@RRS92]. Breu and Kirkpatrick have shown in [@BK93] that it is NP-hard to recognize graphs of sphericity 2 (also known as [*unit disk graphs*]{}) and graphs of sphericity 3. We refer the reader to [@RRS89b] for a survey of most known results regarding this parameter, and mention only a few of them here.\ \[list\] Let $G$ be graph on $n$ vertices with minimal degree $\delta$. Let $\lambda_n$ the least eigenvalue of its adjacency matrix. 1. $Sph(K_{m,n}) \leq m + \frac n 2 - 1$ [@Maehara]. 2. $Sph(G) = O(\lambda_n^2 \log n)$ [@FraMa].\[it-lam\] 3. $Sph(G) = O((n-\delta) \log (n-\delta))$ [@RRS89b]. 4. $Sph(K_{n,n}) \geq n$ [@RRS89a]. 5. All but a $\frac 1 n$ fraction of graphs on $n > 37$ vertices have sphericity at least $\frac n {15} - 1$ [@RRS89b]. 6. $Sph(G) \geq \frac {\log \alpha(G)} {\log (2r(G)+1)}$, where $\alpha(G)$ is the independence number of $G$, and $r(G)$ is its radius [@RRS89a]. The first thing to notice is that any lower bower on the sphericity of some graph on $n$ vertices is also a lower bound on $d(n,l_2)$. In particular, the fact that $Sph(K_{n,n}) \geq n$ proves the lower bound in Lemma \[momad2\]. (Similarly, any upper bound on the former also applies to the latter.)\ In this section we are interested in graphs of large sphericity. The above results tell us that they exist in abundance, yet that graphs of very small or very large degree have small sphericity (the maximal degree is an upper bound on $|\lambda_n|$, hence by (\[it-lam\]) the sphericity is small if all degrees are small). Other than the complete bipartite graph, the above results do not point out an explicit graph with super-logarithmic sphericity. Upper Bound on Margin --------------------- Following Frankl and Maehara [@FraMa], consider an embedding of a proximity graph where there is a large margin between short and long distances. In such a situation, the Johnson-Lindenstrauss Lemma ([@JoLi84]) would yield a spherical embedding into lower dimension: It allows reducing the dimension at the cost of some distortion. If the distortion is small with respect to the margin, the short and long distances remain separated. Alas, we show that for most regular graphs this margin is not large enough for the method to be useful: Let $G$ be a $\delta n$-regular graph, with second eigenvalue $\lambda_2 > \frac 2 n$. Let $\phi$ be an embedding of $G$ as a proximity graph. Denote $a=\max_{u \sim v}||\phi(u) - \phi(v)||_2^2$, and $b=\min_{u \not \sim v}||\phi(u) - \phi(v)||_2^2$. Then $b-a = O(\frac {\lambda_2 + \delta} {\delta n})$. Denote $m=\min\{1-a, b-1\}$, and for a vertex $i$, denote $v_i = \phi(i)$. The largest value $m$ can attain, over all embeddings $\phi$, is given by the following quadratic semidefinite program: $$\begin{aligned} & \max m & \\ & s.t. \forall (i,j) \in E(G) & ||v_i - v_j||^2 \leq 1-m\\ & \forall (i,j) \notin E(G) & ||v_i - v_j||^2 \geq 1+m\end{aligned}$$ Its dual turns out to be: $$\begin{aligned} & \min {\frac 1 2}tr A &\\ & s.t. & A \in PSD \\ & \forall (i,j) \in E(G) & A_{ij} \leq 0 \\ & \forall (i,j) \notin E(G), i\neq j & A_{ij} \geq 0 \\ & \forall i & \sum_{j=1,...,n} A_{ij} = 0 \\ & & \sum_{i \neq j} |A_{ij}| = 1\end{aligned}$$ Equivalently, we can drop the last constraint, and change the objective function to $ \min \frac {tr A} {\sum_{i \neq j} |A_{ij}|} $. Next we construct an explicit feasible solution for the dual program, and conclude from it a bound on m. Let $M$ be the adjacency matrix of $G$. Define $A = I + \alpha J - \beta M$. To satisfy the constraints we need: $$\begin{aligned} & & A \in PSD \\ & & \beta \geq \alpha \geq 0\\ & & 1 + \alpha n - \beta \delta n = 0\\\end{aligned}$$ The last condition implies $\alpha = \beta \delta - \frac 1 n$, so it follows that $\beta \geq \alpha$, and the constraint on $\beta$ is $\beta \geq \frac {1} {\delta n}$. Now, since we assume that the graph is $\delta n$-regular, its Perron eigenvector is $\vec{1}$, corresponding to eigenvalue $\delta n$. Therefore, we can consider the eigenvectors of $M$ to be eigenvectors of $J$ and $I$ as well, and hence also eigenvectors of $A$. If $\lambda \neq \delta n$ is an eigenvalue of $M$, then $1-\beta \lambda$ is an eigenvalue of $A$, corresponding to the same eigenvector. Denote by $\lambda_2$ the second largest eigenvalue of $M$, then in order to satisfy the condition $ A \in PSD$ it is enough to set $\beta = \frac {1} {\lambda_2}$, in which case all the constraints are fulfilled. We conclude that: $$\begin{aligned} m & \leq & \frac {tr A} {\sum_{i \neq j} |A_{ij}|} = \frac {n(1+\alpha)} {\delta n^2 (\beta - \alpha) + ((1-\delta) n^2 - n) \alpha} \\ & = & \frac {n+ \frac {\delta n} {\lambda_2} - 1} {\delta n (\frac {n+\delta n} {\lambda_2} - 1) + ((1-\delta) n - 1) (\frac {\delta n} {\lambda_2} - 1)} < 4 \frac {1 + \frac {\delta} {\lambda_2}} {\frac {\delta n} {\lambda_2}} = 4 \frac {\lambda_2 + \delta} {\delta n}.\end{aligned}$$ In particular, $b-a = O(\frac {\lambda_2 + \delta} {\delta n})$. In order to derive a non trivial result from Johnson-Lindenstrauss lemma, we need that $\frac 1 {m^2} \log n = o(n)$, and in particular that $\lambda_2=\Omega(\delta \sqrt{n \log n})$. The above shows that this can happen only if $\lambda_2 = \omega(\delta \sqrt{n \log n})$. On the other hand, Frankl and Maehara show that their method does give a non trivial bound when $\lambda_n = o(\sqrt{\frac n {\log n}})$. Consequently, we get that a $\delta n$-regular graph (think of $\delta$ as constant) can’t have both $\lambda_2=o(\sqrt{n \log n})$ and $\lambda_n = o(\sqrt{\frac n {\log n}})$. This is a bit more subtle than what one gets from the second moment argument, namely, that the graph can’t have both $\lambda_2=o(\sqrt{n})$ and $\lambda_n = o(\sqrt{n})$. Lower Bound on Sphericity ------------------------- \[our-bound\] Let $G$ be a $d$-regular graph with diameter $D$ and $\lambda_2$, the second largest eigenvalue of $G$’s adjacency matrix, is at least $d - {\frac 1 2}n$. Then $Sph(G) = \Omega(\frac {d - \lambda_2} {D^2(\lambda_2 + O(1))})$. In the interesting range where $d \leq \frac n 2$, and $\lambda_2 \geq 1$ the bound is $Sph(G) = \Omega(\frac {d - \lambda_2} {D^2 \lambda_2})$. It will be useful to consider the following operation on matrices. Let $A$ be an $n \times n$ symmetric matrix, and denote by ${\vec{a}}$ the vector whose $i$-th coordinate is $A_{ii}$. Define $R(A)$ to be the $n \times n$ matrix with all rows equal to ${\vec{a}}$, and $C(A) = R(A)^t$. Define: $$\begin{aligned} \breve{A} = 2A - C(A) - R(A) + J\end{aligned}$$ First note that the rank of $\breve{A}$ and that of $A$ can differ by at most 3. Now, consider the case where $A$ is the Gram matrix of some vectors $v_1,...,v_n \in R^d$. Then all diagonal entries of $\breve{A}$ equal one, and the $(i,j)$ entry is 2$<v_i,v_j> - <v_i,v_i> - <v_j,v_j> + 1 = 1 - ||v_i-v_j||^2$. We will need the following lemma (see [@HoJo], p.175): \[d-lemma\] Let $X$ be a real symmetric matrix, then $rank(X) \geq \frac {(tr X)^2} {\sum_{i,j}X_{i,j}^2}$ Applying this to $\breve{A}$, we conclude that: $$\begin{aligned} \label{d-bound} rank(\breve{A}) \geq \frac {n^2} {n + \sum_{i \neq j} (1 - ||v_i - v_j||^2)^2 }\end{aligned}$$ Let $v_1,...,v_n \in \R^d$ be an embedding of $G$. By the discussion above it is enough to show that $$\begin{aligned} \label{denom} \sum_{i \neq j} (1-||v_i - v_j||^2)^2 = O(D^2 n^2 \frac {\lambda_2} {d - \lambda_2}).\end{aligned}$$ By the triangle inequality $||v_i - v_j|| \leq D$ for any two vertices. So the LHS of (\[denom\]) is bigger by at most a factor of $D^2$ than: $$\begin{aligned} && \sum_{(i,j) \notin E} (||v_i - v_j||^2 - 1) + \sum_{(i,j) \in E} (1 - ||v_i - v_j||^2) =\end{aligned}$$ $$\begin{aligned} \label{hs-norm} && \sum_{(i,j) \notin E} ||v_i - v_j||^2 - \sum_{(i,j) \in E} ||v_i - v_j||^2 - {n \choose 2} + nd\end{aligned}$$ We can bound this sum from above, by solving the following SDP: $$\begin{aligned} & \max & \sum_{(i,j) \notin E} (V_{ii} + V_{jj} - 2V_{ij}) + \sum_{(i,j) \in E} (- V_{ii} - V_{jj} + 2V_{ij}) - {n \choose 2} + nd\\ & s.t. & V \in PSD \\ & \forall (i,j) \in E & V_{ii} + V_{jj} - 2V_{ij} \leq 1 \\ & \forall (i,j) \notin E & V_{ii} + V_{jj} - 2V_{ij} \geq 1\end{aligned}$$ The dual problem is: $$\begin{aligned} & \min & {\frac 1 2}tr A \\ & s.t. & A \in PSD \\ & \forall (i,j) \in E & A_{ij} \leq -1 \\ & \forall (i,j) \notin E, i\neq j & A_{ij} \geq 1 \\ & \forall i \in [n] & \sum_{j=1,...,n} A_{ij} = 0\end{aligned}$$ Let $M$ by the adjacency matrix of the graph, and set $A = (\alpha d - n)I + J - \alpha M$, where $\alpha \geq 2$ will be determined shortly. This takes care of the all constraints except for $A \in PSD$. Note that since $M$ is regular, its eigenvectors are also eigenvectors of $A$. Moreover, if $M u = \lambda u$ for a non constant $u$, then $A u = \alpha d - n - \alpha \lambda$ (and $A \vec{1} = 0$). So take $\alpha = \frac {n} {d - \lambda_2}$, and by our assumption on $\lambda_2$, $\alpha \geq 2$. Now $A$ gives an upper bound on (\[hs-norm\]): $$\begin{aligned} {\frac 1 2}tr A = {\frac 1 2}n(\alpha d - n + 1) = {\frac 1 2}n^2 \frac d {d-\lambda_2} - {\frac 1 2}n^2 + {\frac 1 2}n= {\frac 1 2}n^2 \frac {\lambda_2} {d-\lambda_2} + {\frac 1 2}n.\end{aligned}$$ This, by (\[d-bound\]), shows that the dimension of the embedding is $\Omega\left(\frac {d - \lambda_2} {D^2(\lambda_2 + O(1))} \right)$. A Quasi-random Graph of logarithmic Sphericity ---------------------------------------------- It is an intriguing problem to construct new examples of graphs of linear sphericity. Since random graphs have this property, it is natural to search among quasi-random graphs. There are several equivalent definitions for such graphs (see [@AlSp]). The one we adopt here is: A family of graphs is called [*quasi-random*]{} if the graphs in the family are $(1+o(1))\frac n 2$-regular, and all their eigenvalues except the largest one are (in absolute value) $o(n)$ . Counter-intuitively, perhaps, quasi-random graphs may have very small sphericity. \[alex\] Let $\G$ be the family of graphs with vertex set $\{0,1\}^k$, and edges connecting vertices that are at Hamming distance at most $\frac k 2$. Then $\G$ is a family of quasi-random graphs of logarithmic sphericity. The fact that the sphericity is logarithmic is obvious - simply map each vertex to the vector in $\{0,1\}^n$ associated with it. To show that all eigenvalues except the largest one are $o(2^k)$ we need the following facts about Krawtchouk polynomials (see [@vL]). Denote by $K_s^{(k)}(i) = \sum_{j=0}^s(-1)^j{i \choose j}{{k-i} \choose {s-j}}$ the Krawtchouk polynomial of order $s$ over $\Z_2^k$. For simplicity we assume that $k$ is odd. 1. For any $x \in \Z_2^k$ with $|x|=i$, $\sum_{z \in \Z_2^k\\|z|=s}(-1)^{<x,z>} = K_s^{(k)}(i)$. 2. $\sum_{s=0}^l K_s^{(k)}(i) = K_l^{(k-1)}(i-1)$. 3. For any $s$ and $k$, $\max_{i=0,\dots,n} |K_s^{(k)}(i)| = K_s^{(k)}(0) = {k \choose s}$. Observe that $G$ is a Cayely graph for the group $\Z_2^k$ with generator set $\{g \in \Z_2^k : |g| \leq \frac k 2\}$. Since $\Z_2^k$ is abelian, the eigenvectors of the graphs are independent of the generators, and are simply the characters of the group written as the vector of their values. Namely, corresponding each $y \in \Z_2^k$ we have an eigenvector $v^y$, such that $v^y_x = (-1)^{<x,y>}$. For every $y$, $v^y_0 = 1$, so to figure out the eigenvalue corresponding to $v^y$, we simply need to sum the value of $v^y$ on the neighbors of $0$. Note that for $y=0$ we get the all $1$s vector, which corresponds to the largest eigenvalue. So we are interested in $y$’s such that $|y| > 0$. By the first two facts above we have: $$\lambda_y = \sum_{g \in \Z_2^k, |g| \leq \frac k 2}(-1)^{<y,g>} = \sum_{s=0}^{\frac {k-1} 2} K_s^{(k)}(|y|) = K_{\frac {k-1} 2}^{(k-1)}(|y|-1).$$ By the third fact, this is at most ${{k-1} \choose {\frac {k-1} 2}} \approx \frac {2^{k-1}} {\sqrt{k-1}} = o(2^{k-1})$. Graphs with bounded $\lambda_2$ {#lambda2-cons} =============================== Theorem \[our-bound\] suggests families of graphs that have linear sphericity. Namely, for $0 < \delta \leq {\frac 1 2}$, and $\lambda_2 > 0$, the theorem says that $\delta n$-regular graphs with second eigenvalue at most $\lambda_2$ have linear sphericity. In this section we characterize such graphs. We prove that for $\delta = {\frac 1 2}$ such graphs are nearly complete bipartite, and that for other values, only finitely many graphs exist.\ It is worth noting that graphs with bounded second eigenvalue have been previously studied. The apex of these works is probably that of Cameron, Goethals, Seidel and Shult, who characterize in [@CGSS] graphs with second eigenvalue at most 2. $n/2$-regular graphs -------------------- In this section we consider the family $\G$ of $n/2$-regular graphs, and second largest eigenvalue $\lambda_2$ bounded by a constant. We prove that, asymptotically, they are nearly complete bipartite. Let $G$ and $H$ be two graphs on $n$ vertices. We say that $G$ and $H$ are [*close*]{}, if there is a labeling of their vertices such that $|E(G) \bigtriangleup E(H)| = o(n^2)$. \[main1\] Every $G \in \G$ is close to $K_{n/2,n/2}$, where $n$ is the number of vertices in $G$. By passing to the complement graph, if $\lambda_n = O(1)$, then $G$ is close to the disjoint union of two cliques, $K_{n/2} \dot{\cup} K_{n/2}$. We need several lemmas. The first is the well-known expander mixing lemma [@FriPi]. The second is a special case of Simonovitz’s stability theorem ([@Sim]), for which we give a simple proof here. The third is a commonly used corollary of Szemeredi’s Regularity Lemma. We shall also make use of the Regularity Lemma itself (see e.g. [@Diestal]). \[no-clique\] Let $G$ be an $\frac n 2$-regular graph on $n$ vertices with second largest eigenvalue $\lambda_2$. Then every subset of vertices with $k$ vertices has at most $\frac 1 4 k^2 + {\frac 1 2}\lambda_2 k$ internal edges. \[close-bi\] Let $R$ be a triangle-free graph on $n$ vertices, with $n^2/4 - o(n^2)$ edges. Then $R$ is close to $K_{n/2,n/2}$. Furthermore, all but $o(n)$ of the vertices have degree $\frac n 2 \pm o(n)$. Denote by $d_i$ the degree of the $i$th vertex in $R$, and by $m$ the number of edges. Then: $$\sum_{(i,j) \in E(R)}(d_i+d_j) = \sum_{i \in V(R)} d_i^2 \geq \frac 1 n (\sum_{i \in V(R)} d_i)^2 = \frac {4m^2} n.$$ Thus, there is some edge $(i,j) \in E(R)$ such that $d_i + d_j \geq \frac {4m} n = n - o(n)$. Let $\Gamma_i$ and $\Gamma_j$ be the neighbor sets of $i$ and $j$. Since $i$ and $j$ are adjacent, and $R$ has no triangles, the sets $\Gamma_i$ and $\Gamma_j$ are disjoint and independent. If we delete the $o(n)$ of vertices in $V\backslash (\Gamma_i \cup \Gamma_j)$ we obtain a bipartite graph. We have deleted only $o(n^2)$ edges, so the remaining graph still has $n^2/4 - o(n^2)$ edges. But this means that $|\Gamma_i|, |\Gamma_j| = \frac n 2 - o(n)$, and that the degree of each vertex in these sets is $\frac n 2 \pm o(n)$ Recall that the Regularity Lemma states that for every $\epsilon > 0$ and $m \in \N$ there’s an $M$, such that the vertex set of every large enough graph can be partitioned into $k$ subsets, for some $m \leq k \leq M$ with the following properties: All subsets except one, the “exceptional” subset, are of the same size. The exceptional subset contains less than an $\epsilon$-fraction of the vertices. All but an $\epsilon$-fraction of the pairs of subsets are $\epsilon$-regular.\ The regularity graph with respect to such a partition and a threshold $d$, has the $k$ subsets as vertices. Two subsets, $U_1$ and $U_2$ are adjacent, if they are $\epsilon$-regular, and $e(U_1,U_2) > dn^2$. \[[@Diestal], Lemma 7.3.2\] Let $G$ be a graph on $n$ vertices, $d \in (0,1]$, $\epsilon = d^{-4}$. Let $R$ be an $\epsilon$-regularity graph of $G$, with (non exceptional) sets of size at least $\frac s {\epsilon}$, and threshold $d$. If $R$ contains a triangle, then $G$ contains a complete tripartite subgraph, with each side of size $s$. If $G \in \G$, and $R$ is as in the lemma, with $s = 10 \lambda_2$, then $R$ is triangle free. In this case, if $R$ has $\frac {k^2} 4 - o(k^2)$ edges, then $R$ is close to complete bipartite. If $R$ contains a triangle, then $G$ contains a complete tripartite subgraph, with $s$ vertices on each side. Let $U$ be the set vertices in this subgraph. Then $e(U) = 3s^2 = 300 \lambda_2^2$, but by lemma \[no-clique\] $e(U) \leq 50 \lambda_2^2$ - a contradiction. The second part now follows from Lemma \[close-bi\]. (Theorem) We would like to apply the Regularity Lemma to graphs in $\G$, and have $\epsilon = o(1)$, and $k = \omega(1)$ as well as $k = o(n)$. Indeed, this can be done. Since $M$ depends only on $m$ and $\epsilon$, choose $d=o(1)$, and $m = \omega(1)$, such that the $M$ given by the lemma satisfies $\frac n {(M+1)} \geq \frac s {\epsilon}$. As $M$ depends only on $m$ and $\epsilon$, $\frac M {\epsilon}$ can be made small enough, even with the requirements on $d$ and $m$. Let $R$ be the regularity graph for the partition given by the Regularity Lemma, with threshold $d$ as above. Denote by $k$ the number of sets in the partition, and their size by $l$ (so $k\cdot l = n(1-\eta)$, for some $\eta \leq \epsilon$). We shall show that $R$ is close to complete bipartite, and that $G$ is close the graph obtained by replacing each vertex in $R$ with $l$ vertices, and replacing each edge in $R$ by a $K_{l,l}$. Call an edge in $G$ $(i)$ “irregular” if it belongs to an irregular pair; $(ii)$ “internal” if it connects two vertices within the same part; $(iii)$ “redundant” if it belongs to a pair of edge density smaller than $d$, or touches a vertex in the exceptional set. Otherwise $(iv)$, call it “good”.\ Recall that $\epsilon = o(1)$, so only $o(k^2)$ pairs of sets are not $\epsilon$-regular. Thus, $G$ can have only $o(l^2k^2) = o(n^2)$ irregular edges. Also, $d = o(1)$, so the number of redundant edges is $k^2 \cdot o(l^2) + o(l) \frac n 2 = o(n^2)$. Finally, the number of internal edges is at most ${\frac 1 2}l^2 k$, hence there are $\frac {n^2} 4 - o(n^2)$ good edges.\ The number of edges between two sets is at most $l^2$, so $R$ must have at least $$\frac {n^2 - o(n^2)} {4l^2} = \frac {k^2} 4 - o(k^2)$$ edges. The corollary implies that it is close to complete bipartite. By lemma \[close-bi\], the valency of all but $o(k)$ of the vertices in $R$ is indeed $\frac k 2 \pm o(k)$. This means that every edge in $R$ corresponds to $l^2 - o(l^2)$ good edges in $G$ (as the number of edges in $R$ is also no more than $\frac {k^2} 4 + o(k^2)$).\ To see that $G$ is close to complete bipartite, let’s count how many edges need to be modified. First, delete $o(n^2)$ edges that are not “good”. Next add all possible $o(n^2)$ new edges between pairs of sets that have “good” edges between them. As $R$ is close to complete bipartite, we need to delete or add all edges between $o(k^2)$ pairs. Each such step modifies $l^2$ edges, altogether $o(l^2k^2) = o(n^2)$ modifications. Finally, divide the $o(n)$ vertices of the exceptional set evenly between the two sides of the bipartite graph, and add all the required edges, and the tally remains $o(n^2)$. In essence, the proof shows that a graph with no dense induced subgraphs is close to complete bipartite. This claim is similar in flavor to Bruce Reed’s [*Mangoes and Blueberries*]{} theorem [@Reed99]. Namely, that if every induced subgraph $G'$ of $G$ has an independent set of size ${\frac 1 2}|G'| - O(1)$, then $G$ is close to being bipartite. The conclusion in Reed’s theorem is stronger in that only a [*linear*]{} number of edges need to be deleted to get a bipartite graph. In fact, the proof gives something a bit stronger. Let $t_r(n)$ be the number of edges in an $n$-vertex complete $r$-partite graph, with parts of equal size. Using the general Stability Theorem ([@Sim]) instead of Lemma \[close-bi\], the same proof shows that if a graph has $t_n - o(n^2)$ edges and no dense induced subgraphs, then it is close to being complete $r$-partite. $\delta n$-regular graphs ------------------------- In Theorem \[main1\] we required that the degree is $n/2$. We can deduce from the theorem that this requirement can be relaxed: Let $\G$ be a family of $d$-regular graphs, with $d \leq \frac n 2$, ($n$ being the number of vertices in the graph) and bounded second eigenvalue, then every $G \in \G$ is close to a complete bipartite graph. Let $M \in \M_n$ be the adjacency matrix of such a $d$-regular graph, and denote $\bar{M} = J - M$, where $J$ is the all ones matrix. Consider the graph $H$ corresponding to the following matrix: $$\begin{aligned} N = \left( \begin{array}{cc} M & \bar{M} \\ \bar{M}^t & M \end{array} \right)\end{aligned}$$ Clearly $H$ is an $n$-regular graph on $2n$ vertices. Denote by $(x, y)$ the concatenation of two $n$-dimensional vectors, $x$, $y$, into a $2n$ dimensional vector. Let $v$ be an eigenvector of $M$ corresponding to eigenvalue $\lambda$. It is easy to see that $v$ is also an eigenvalue of $\bar{M}$: If $v = \vec{1}$ (and thus $\lambda = d$) it corresponds to eigenvalue $n - \lambda$, otherwise to $(-\lambda)$.\ Thus, $(v, v)$ and $(v, -v)$ are both eigenvectors of $N$. If $v = \vec{1}$ they correspond to eigenvalues $n$, $2d-n$, respectively, otherwise to $0$, $2\lambda$. Since the $v$’s are linearly independent, so are the $2n$ vectors of the form $(v, v)$ and $(v, -v)$: Consider a linear combination of these vectors that gives $0$. Both the sum and the difference of the coefficients of each pair have to be $0$, and thus both are 0. So we know the entire spectrum of $N$, and see, since $d \leq \frac n 2$, that theorem \[main1\] holds for it.\ Let $H'$ be a complete bipartite graph that is close to $H$. Since $H$ differs from $H'$ by $o(n^2)$ edges, the same holds for subgraphs over the same set of vertices. In particular, $G$ is close to the subgraph of $H'$ spanned by the first $n$ vertices. Obviously, every such subgraph is itself complete bipartite. \[no-graphs\] For every $0 < \delta < {\frac 1 2}$ and $c$, there are only finitely many $\delta n$-regular graphs with $\lambda_2 < c$. Consider such a graph with $n$ large. By the previous corollary it is close to complete bipartite. Since it is also regular, it must be close to $K_{\frac n 2, \frac n 2}$, which contradicts the constraint $\delta < {\frac 1 2}$. Graphs with both $\lambda_2$ and $\lambda_{n-1}$ bounded by a constant ---------------------------------------------------------------------- Theorem \[main1\] can loosely be stated as follows: A regular graph with spectrum similar to that of a bipartite graph ($\lambda_1$ being close to $n/2$ and $\lambda_2$ being close to $0$) is close to being complete bipartite. We conclude this section by noting that if we strengthen the assumption on how close the spectrum of a graph is to that of a bipartite graph, we get a stronger result as to how close it is to a complete bipartite graph. \[main2\] Let $\G$ be a family of $\frac n 2$-regular graphs on $n$ vertices, with both $\lambda_2$ and $\lambda_{n-1}$ bounded by a constant. Then every $G \in \G$ is close to a $K_{\frac n 2,\frac n 2 }$, in the sense that such a graph can be obtained from $G$ by modifying a linear number of edges for $O(\sqrt{n})$ vertices of $G$, and $O(\sqrt{n})$ edges for the rest. First note that it follows that $\lambda_n(G) = -\frac n 2 + O(1)$. Take $G \in \G$, and let $A$ be its adjacency matrix. Clearly $tr(A^2) = \frac {n^2} 2$. If $\lambda_{n-1}(G) = - O(1)$, then $$\frac {n^2} 2 = tr(A^2) = \lambda_1^2 + \lambda_n^2 + \sum_{i=2,...,n-1}\lambda_i^2$$ Since $\lambda_1 = \frac n 2$ $$\lambda_n^2 = \frac {n^2} 2 - \left(\frac n 2\right)^2 - \sum_{i=2,...,n-1}\lambda_i^2$$ As $\lambda_2,\ldots,\lambda_{n-1} = O(1)$ we have $$\lambda_n^2 = \frac {n^2} 4 + O(n)$$ And since $\lambda_n$ is negative, and is smaller than $\lambda_1$ in absolute value: $$\lambda_n = -\frac n 2 + O(1).$$ Let $x$ be an eigenvector corresponding to $\lambda_n$. Suppose, w.l.o.g that $||x||_{\infty} = 1$ and that $x_v = 1$. Denote $A = \{u : x_u \leq -(1-\frac 1 {\sqrt{n}})\}$, and $B = \{w : x_w \geq (1-\frac 1 {\sqrt{n}})\}$. The eigenvalue condition on $v$ entails: $$\begin{aligned} \frac n 2 - O(1) = -\sum_{u:(u,v) \in E} x_u.\end{aligned}$$ Thus, there is a vertex $u$ such that $x_u \leq -(1-O(\frac 1 n))$. It is not hard to verify that $v$ must have $\frac n 2 - O(\sqrt{n})$ neighbors in $A$, and that $u$ must have $\frac n 2 - O(\sqrt{n})$ neighbors in $B$. Now denote $A' = \{u : x_u \leq -{\frac 1 2}\}$, and $B' = \{w : x_w \geq {\frac 1 2}\}$. Again, it is not hard to check that each vertex in $A$ must have $\frac n 2 - O(\sqrt{n})$ neighbors in $B'$, and vice versa. Thus, delete the $O(\sqrt{n})$ vertices that are neither in $A$ nor in $B$. For each remaining vertex in $A$ (similarly in $B$), its degree is at most $\frac n 2$, and at least $\frac n 2 - O(\sqrt{n})$. It has $\frac n 2 - O(\sqrt{n})$ neighbors in $B$, so the number of its neighbors in $A$, and the number of its non-neighbors in $B$ is $O(\sqrt{n})$. By deleting and adding $O(\sqrt{n})$ edges to each vertex, we get a complete bipartite graph. Alternatively, we could have defined $\G$ as a family of $\frac n 2$-regular graphs with $\lambda_2$ bounded, and $\lambda_n(G) = -\frac n 2 + O(1)$. It’s interesting to note that in this case it follows that $\lambda_{n-1}$ is bounded. For $G \in G$, if $G$ is bipartite, then it is complete bipartite, and $\lambda_{n-1}(G) = 0$. Otherwise, $\chi(G) > 2$, and by a theorem of Hoffman ([@Hoff]) $\lambda_n(G) + \lambda_{n-1}(G) + \lambda_1(G) \geq 0$. By our assumption, $\lambda_n(G) + \lambda_1(G) = O(1)$, and since $\lambda_{n-1}(G) < 0$ (otherwise the eigenvalues won’t sum up to 0), it follows that $\lambda_{n-1}(G) = -O(1)$. Conclusion and Open Problems ============================ The only explicit examples known so far for graphs that have linear sphericity are $K_{n,n}$ and small modifications of it. We conjecture that more complicated graphs, such as the Paley graph, also have linear sphericity. Note that the lower bound presented here only shows a bound of $\Omega(\sqrt{n})$. It is also interesting to know if the bound can be improved, either as a pure spectral bound, or with some further assumptions on the structure of the graph.\ What is the largest sphericity, $d=d(n)$, of an $n$-vertex graph? We know that $\frac n 2 \leq d \leq n-1$. Can this gap be closed? For a seemingly related question, the smallest dimension required to realize a sign matrix (see [@AFR85]) the answer is known to be $\frac n 2 \pm o(n)$. We have also seen a similar gap for $d(n,l_2)$ and $d(n,l_\infty)$. Can this be closed? Can some kind of interpolation arguments generalize the bounds we know for these two numbers to bounds on $d(n,l_p)$ for $p>2$?\ Our interest in sphericity arose from a search for a lower bound on $d(n,l_2)$. But why limit the discussion to Euclidean space? What can be said of spherical embeddings into $l_1$ or $l_{\infty}$? The former may be particularly interesting, as it will give a non-trivial lower bound on $d(n,l_1)$.\ We have seen that $\frac n 2$-regular graphs with bounded second eigenvalue are $o(n^2)$-close to complete bipartite. However, the only example we know of such graphs are constructed by taking a complete bipartite graph, and changing a constant number of edges for each vertex. These graphs are $O(n)$-close to being complete bipartite. Are there examples of such families which are further from complete bipartite graphs, or can a stonger notion of closeness be proved? Acknowledgments =============== We would like to thank Alex Samorodnitsky for showing us how to prove Lemma \[alex\] and Noga Alon for sending us Lemma \[d-lemma\]. [^1]: Institute of Computer Science, Hebrew University Jerusalem 91904 Israel . This research is supported by the Israeli Ministry of Science and the Israel Science Foundation.

--- author: - 'Kory M. Stiffler' title: 'A WALK THROUGH SUPERSTRING THEORY WITH AN APPLICATION TO YANG-MILLS THEORY: K-STRINGS AND D-BRANES AS GAUGE/GRAVITY DUAL OBJECTS' ---

--- abstract: 'We consider the five-dimensional bulk spacetime with negative $\Lambda$ described by the Nariai metric (which is not conformally flat) and match it with a vacuum brane satisfying the proper boundary conditions. It is shown that the brane metric corresponds to a cloud of string dust of constant energy density.' address: | $^a$Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411 007, India.\ $^b$Bogolyubov Institute for Theoretical Physics, Kiev 03143, Ukraine author: - 'Naresh Dadhich$^{a}$ and Yuri Shtanov$^{b}$' title: Brane corresponding to the Nariai bulk --- PACS: 04.50.+h, 11.10.Kk, 98.80.Hw A fresh impetus to the old paradigm of spacetime with extra dimensions was recently given in [@dvali], where it was suggested that compact extra dimensions may be macroscopic while our space-time is described as a lower-dimensional domain wall (brane) where all the matter is concentrated. A novel approach to higher-dimensional braneworld cosmology emerged after Randall and Sundrum postulated the existence of a [*noncompact*]{} spacelike fifth dimension [@RS]. According to this world-view, our perception of ‘normal’ four-dimensional gravity arises because we live on a domain wall (brane) embedded in or bounding a ‘bulk’ anti-de Sitter space (AdS). The metric describing the full (4+1)-dimensional space-time is non-factorizable, and the small value of the true five-dimensional Planck mass is related to its large effective four-dimensional value by the extremely large warp of the five-dimensional space. The novelty of the Randall–Sundrum (RS) model is to use the curvature of the bulk spacetime (with the $Z_2$ symmetry of reflection relative to the brane) to keep zero-mass gravitons localized on the brane. This theory was studied in detail in the case of 5D anti de Sitter (AdS) bulk with flat or Schwarzschild vacuum brane and in the cosmological context. The bulk and brane solutions are matched by the Israel boundary conditions. The effective Einstein equation on the brane can be written [@SMS] by using the Gauss–Codazzi relations. It would additionally involve square of the stress-energy tensor and projection of the bulk Weyl curvature tensor to the brane. The latter is trace-free and is known as the Weyl dark energy/radiation. In this sense, the system of equations on the brane is obviously not closed. It is therefore very difficult to find exact complete solutions with both bulk and brane metrics satisfying the proper boundary conditions. There exist only a few examples of complete solutions, among which the AdS bulk with flat or Schwarzschild brane and Schwarzschild–AdS bulk with FRW brane. Most of other solutions including black hole [@DMPZ] and collapse [@BGM; @GD] are solutions of only the brane equations without the corresponding solution in the bulk. The purpose of this paper is to give one more simple example of complete solution. Specifically, the Nariai metric [@Nariai] offers an interesting case of the Einstein space which is not conformally flat. After the generalization of this metric to 5D case with negative $\Lambda$, the question of graviton confinement was studied for this conformally non-flat bulk spacetime [@param], and it was shown that there exist no normalized modes for massless graviton (once again, there is a pointer to fine tuning of parameters inherent in the Randall–Sundrum (RS) model [@PS; @DK]). However, this was done with the bulk metric alone without any reference to the brane spacetime. In this paper, we complete the solution by finding the corresponding brane satisfying the proper boundary conditions. The theory that we consider here is described by the following general action (see [@CHS]): $$\label{action} S = M^3 \sum \left[\int_{\rm bulk} \left( {\cal R} - 2 \Lambda \right) - 2 \int_{\rm brane} K \right] + \int_{\rm brane} \left( m^2 R - 2 \sigma \right) + \int_{\rm brane} L \left( h_{\alpha\beta}, \phi \right)$$ in the standard notation, where the sum is taken over the bulk components bounded by the brane. We use the signature and sign conventions of [@Wald]. The lagrangian $L \left( h_{\alpha\beta}, \phi \right)$ corresponds to the presence of matter fields $\phi$ on the brane and describes their dynamics, and the extrinsic curvature $K_{\alpha\beta}$ of the brane is defined with respect to the inner normal $n^a$, as it is done in [@Shtanov]. Note that we have included the curvature term in the action for the brane which arises when one incorporates quantum effects generated by matter fields residing on the brane. The equations in the bulk and on the brane are obtained from the variation of Eq. (\[action\]), which gives $$\label{bulk} {\cal G}_{ab} + \Lambda g_{ab} = 0 \, ,$$ $$\label{brane} m^2 G_{\alpha\beta} + \sigma h_{\alpha\beta} = \tau_{\alpha\beta} + M^3 \sum \left(K_{\alpha\beta} - h_{\alpha\beta} K \right) \, ,$$ where $h_{\alpha\beta}$ is the induced metric on the brane, $\tau_{\alpha\beta}$ is the stress-energy tensor resulting from the Lagrangian $L \left( h_{\alpha\beta}, \phi \right)$, and the sum of the extrinsic curvatures on either side of the brane is taken. The Nariai metric in the bulk as given in Ref. [@param] reads as $$\label{nariai} d s^2_5 = e^{- 2k|y|} \left( - dt^2 + dr^2 \right) + dy^2 + \frac{1}{2 k^2} \left(d \th^2 + \sinh^2 \th d \phi^2 \right) \, .$$ It is a solution of the bulk equation (\[bulk\]) with $\Lambda = -3k^2$. Since the Weyl curvature is non-zero for this metric and hence its projection, $E_{\mu \nu} := C_{\mu a \nu b} \, n^{a} \, n^{b}$ on the brane $y = \mbox{const}$, would be non-zero. Now we consider the brane located at $y = 0$ which has induced metric $h_{\alpha\beta}$ given by the line element $$ds^2_4 = - dt^2 + dr^2 + \frac{1}{2 k^2} \left(d \th^2 + \sinh^2 \th d \phi^2 \right) \, .$$ This is a spacetime having the structure of the product of a flat 2-dimensional space and a 2-sphere of constant curvature [@nd]. Further, it can be shown that the stress-energy tensor corresponding to this metric describes a cloud of string dust [@let; @bose]. The stress-energy tensor for a string-dust distribution is given by [@let; @bose], $$T_{\rm string}^{\mu\nu} = \rho \Sigma^{\mu\beta} \Sigma^{\nu}_{\beta} \, ,$$ where $\rho$ is the proper energy density of the cloud, and $\Sigma^{\mu \nu}$ is the bivector associated with this world-sheet: $\Sigma^{\mu\nu} = \displaystyle \epsilon^{AB} {\partial x^\mu\over \partial \xi^A} {\partial x^\nu\over \partial \xi^B}$. Here, $\epsilon^{AB}$ is the 2D Levi-Civita tensor (normalized so that $\epsilon^{AB} \epsilon_{AB} = 2$) and $\xi^A = (\xi^0, \xi^1)$ are the coordinates on the string world-sheet. Following Refs. [@let; @bose], we readily conclude that the stress-energy tensor corresponding to the above brane metric accord with the string-dust stress-energy tensor, which satisfies the equation of state $T^0{}_0 + T^i{}_i = 0$, a typical of topological defects like cosmic string and global monopole. The components of the Einstein tensor of this metric in the coordinates $(t, r, \theta, \phi)$, are given by $$G^\alpha{}_\beta = {\rm diag} \left( 2k^2, 2k^2, 0, 0 \right) \, .$$ Clearly, the above-mentioned equation of state for the string-dust distribution is satisfied and the string dust has constant negative energy density $\rho = -2k^2$. The extrinsic curvature on either side of the brane is given by $$K^\alpha{}_\beta = {\rm diag} ( -k, -k, 0, 0 ) \, , \quad K = - 2k \, .$$ Substituting it into Eq. (\[brane\]), we obtain the system of equations for the vacuum brane $\left( \tau_{\alpha\beta} = 0 \right)$ $$2 m^2 k^2 + \sigma = 2 M^3 k \, , \quad \sigma = 4 M^3 k \, ,$$ whence we get the conditions of “fine tuning” $$\label{fine} M^3 = - m^2 k < 0 \, , \quad \sigma = - 4 m^2 k^2 < 0 \, .$$ The brane metric thus describes a cloud of string dust of constant negative energy density. Furthermore, we see that the solution (\[nariai\]) with brane located at $y =0$ requires both the five-dimensional Planck mass $M$ and brane tension $\sigma$ also to be [*negative*]{}. According to the fine-tuning conditions (\[fine\]), only two of the four fundamental constants in this theory are independent; for example, the constants $M$ and $\sigma$ can be expressed in terms of $m$ and $\Lambda$ via Eq. (\[fine\]). In contrast to the original RS model, our solution requires $m \ne 0$, i.e., it requires the presence of the induced curvature term in the action for the brane. The Nariai bulk and the corresponding string-dust brane may not appear to be of much cosmological and astrophysical importance. However, they undoubtedly represent an interesting spacetime solution, and the Nariai metric has already seen a good bit of application in the context of black hole and quantum-gravity considerations [@odin]. Our aim was, in view of the paucity of complete exact solutions for the bulk-brane system, to present one more example involving very simple spacetimes. [*Acknowledgments:*]{} Yu. S. would like to thank IUCAA for warm hospitality and acknowledges partial support from the INTAS grant for project No. 2000-334. [99]{} nkd@iucaa.ernet.in shtanov@bitp.kiev.ua N. Arkani-Hamed, S. Divopolous, and G. Dvali, Phys. Lett. B [**429**]{} (1998) 263, hep-ph/9803315;  I. Antoniadis, N. Arkani-Hamed, S. Divopolous, and G. Dvali, Phys. Lett. B [**436**]{} (1998) 257, hep-ph/9804398. L. Randall and R. Sundrum, Phys. Rev. Lett. [**83**]{} (1999) 3370, hep-ph/9905221;  L. Randall and R. Sundrum, Phys. Rev. Lett. [**83**]{} (1999) 4690, hep-th/9906064. T. Shiromizu, K. Maeda, and M. Sasaki, Phys. Rev. D [**62**]{} (2001) 024012, hep-th/9910076. N. Dadhich, R. Maartens, P. Papadopoulos, and V. Rezania, Phys. Lett. B [**487**]{} (2000) 1, hep-th/0003061. M. Bruni, C. Germani, and R. Maartens, Phys. Rev. Lett. [**87**]{} (2001) 231302, gr-qc/0108013. M. Govender and N. Dadhich, Phys. Lett. B [**538**]{} (2002) 233, hep-th/0109086. H. Nariai, Sci. Rep. Tohoku Univ. [**34**]{} (1950) 160. P. Singh and N. Dadhich, Phys. Lett. B [**511**]{} (2001) 291, hep-th/0104174. T. Padmanabhan and S. Shankaranarayanan, Phys. Rev. D [**63**]{} (2001) 105021, hep-th/0011159. N. Deruelle and J. Katz, Phys. Rev. D [**64**]{} (2001) 083515, gr-qc/0104007. H. Collins and B. Holdom, Phys. Rev. D [**62**]{} (2000) 105009, hep-ph/0003173;  Yu. V. Shtanov, [*On brane-world cosmology*]{}, hep-th/0005193. R. M. Wald, [*General Relativity*]{}, The University of Chicago Press, Chicago (1984). Yu. V. Shtanov, Phys. Lett. B [**541**]{} (2002) 177, hep-th/0108153; Yu. V. Shtanov, Phys. Lett. B [**543**]{} (2002) 121, hep-th/0108211. N. Dadhich, [*On product spacetime with 2-sphere of constant curvature*]{}, gr-qc/0003026. P. S. Letelier, Phys. Rev. D [**20**]{} (1979) 1294. S. Bose and N. Dadhich, Phys. Lett. B [**488**]{} (2000) 1;  Erratum—[*ibid.*]{} [**494**]{} (2000) 346, hep-th/0001119. S. Nojiri and S. D. Odintsov, Phys. Rev. D [**59**]{} (1999) 044026, hep-th/9804033;  A. A. Bytsenko, S. Nojiri, and S. D. Odintsov, Phys. Lett. B [**443**]{} (1998) 121, hep-th/9808109;  O. B. Zaslavskii, Class. Q. Grav. [**17**]{} (2000) 497, gr-qc/9812052;  S. Nojiri and S. D. Odintsov, Phys. Lett. B [**523**]{} (2001) 165, hep-th/0110064;  M. Ortaggio, Phys. Rev. D [**65**]{} (2002) 084046, gr-qc/0110126.

--- abstract: | We generalise and improve a result of Stoll, Walsh and Yuan by showing that there are at most two solutions in coprime positive integers of the equation in the title when $b=p^{m}$ where $m$ is a non-negative integer, $p$ is prime, $(a,p)=1$, $a^{2}+p^{2m}$ not a perfect square and $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has an integer solution. This result is best possible. We also obtain best possible results for all positive integer solutions when $m=1$ and $2$. When $b$ is an arbitrary square with $(a,b)=1$ and $a^{2}+b^{2}$ not a perfect square, we are able to prove there are at most three solutions in coprime positive integers provided $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution and $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-b^{2}$ has only one family of solutions. Our proof is based on a novel use of the hypergeometric method that may also be useful for other problems. address: 'London, UK' author: - Paul M Voutier title: 'Sharp bounds on the number of solutions of $X^{2}-\left( a^{2}+b^{2} \right) Y^{4}=-b^{2}$' --- Introduction ============ Diophantine equations of the form $aX^{2}-bY^{4}=c$ are linked to several important areas in number theory. They are a quartic model of elliptic curves, for example. They are also associated with squares in binary recurrence sequences too. Ljunggren (see [@L1; @L2; @L3; @L4] for some of his many results) made significant contributions to the study of the integer solutions of such equations, especially when $a$, $b$ are positive integers and $c=\pm 1, \pm 2, \pm 4$. They have been the subject of much attention since then too (see, for example, Akhtari’s result [@Akh] and the references there). For other values of $c$, the study of such equations appears to be much more difficult. In 2009, Stoll, Walsh and Yuan [@SWY] showed that for any non-negative integer $m$, there are at most three solutions in odd positive integers to $$X^{2} - \left( 1+2^{2m} \right) Y^{4} = -2^{2m}.$$ Here we generalise, and improve, their result to the equation $$\label{eq:2} X^{2} - \left( a^{2}+b^{2} \right) Y^{4} = -b^{2},$$ under the conditions stated in our theorems below. \[thm:1.1\] Let $a$, $m$ and $p$ be non-negative integers with $a \geq 1$, $p$ a prime, $\gcd \left( a,p^{m} \right)=1$ and $a+p^{2m}$ not a perfect square. Suppose $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has a solution. Then $\eqref{eq:2}$ has at most two coprime positive integer solutions. Note that the conditions in Theorem \[thm:1.1\] are always satisfied for $a=1$ and $p=2$, so the results here to include, and improve, the results in [@SWY]. \[rem:1\] Theorem \[thm:1.1\] is best possible. One can find infinitely many examples of $a,m$ and $p$ such that there are two solutions in coprime positive integers. Example 1: let $b$ be any odd positive integer not divisible by $5$ and $a=\left( b^{2}-5 \right)/4$. Then we have the obvious solution, $(a,1)$, of . The fundamental solution of the negative Pell equation here is $\left( a+2, 1 \right)$, so $\left( a+\sqrt{a^{2}+b^{2}} \right) \left( (a+2) + \sqrt{a^{2}+b^{2}} \right)^{2}$ gives rise, after simplifying, to another solution, $\left( \left( b^{6}+5b^{4}+15b^{2}-5 \right)/16, \left( b^{2}+1 \right) /2 \right)$, of . Example 2: let $b$ be any odd positive integer and $a=\left( 5b^{2}-1 \right)/4$. also has two solutions. In addition to the obvious solution, $(a,1)$, of , we also have the following solution, $\left( \left( 3125b^{6}+625b^{4}+75b^{2}-1 \right)/16, \left( 25b^{2}+1 \right)/2 \right)$. Of course, it would be satisfying to remove the condition that the coordinates of the integer solutions be coprime. We have not been able to do that in the same generality as in Theorems \[thm:1.1\], but we have been able to prove the following. \[cor:1.1\] Let $a$, $m$ and $p$ be positive integers with $a \geq 1$, $m=1,2$, $p$ a prime, $\gcd \left( a,p \right)=1$ and $a^{2}+p^{2m}$ not a perfect square. Suppose $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has a solution. Then $\eqref{eq:2}$ has at most three positive integer solutions. (of Corollary \[cor:1.1\]) From Theorem \[thm:1.1\], we know there are at most two coprime solutions. If there is a solution with $\gcd(x,y) \neq 1$, then for both $m=1$ and $m=2$, we can remove the common factors to get $-1$ on the right-hand side. We can now appeal to Theorem D of [@Chen1] to show there is at most one such solution. Corollary \[cor:1.1\] is also best possible. We can use Example 1 in Remark \[rem:1\] to see this. Suppose $b$ there is a perfect square, $b=b_{1}^{2}$. In addition to the two solutions given in Remark \[rem:1\], we also have the solution $\left( \left( b^{3}+3b \right)/4, b_{1} \right)$. We only found one example with $b$ prime and three solutions, namely $a=31$, $b=5$ with the solutions $(31, 1)$, $(785, 5)$, $(3076289, 313)$. It is natural to wonder what happens when $p^{m}$ is replaced by any positive integer $b$. Our technique here can be used to show that Theorem \[thm:1.1\] and Corollary \[cor:1.1\] are both true if we replace $p^{m}$ with $2p^{m}$. The proof is nearly identical to what follows, so we have not pursued this here. We are also able to prove the following result. \[thm:1.2\] Let $a$ and $b$ be relatively prime positive integers such that $a^{2}+b^{2}$ is not a perfect square. Suppose $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has a solution and that all coprime integer solutions $(x,y)$ to the quadratic equation $$\label{eq:quad-eqnc} x^{2} - \left( a^{2}+b^{2} \right)y^{2}=-b^{2}$$ are given by $$\label{eq:14c} x+y \sqrt{a^{2}+b^{2}} = \pm \left( \pm a + \sqrt{a^{2}+b^{2}} \right) \alpha^{2k}, \hspace{1.0mm} k \in {\mathbb{Z}},$$ where $\alpha = \left( T_{1}+U_{1} \sqrt{a^{2}+b^{2}} \right)/2$ and $\left( T_{1},U_{1} \right)$ is the minimum solution of the equation $x^{2}-\left( a^{2}+b^{2} \right)y^{2}=-4$ in positive integers. Then $\eqref{eq:2}$ has at most three coprime positive integer solutions. We have not been able to find any equations satisfying these conditions that have three solutions, so we believe that there are at most two coprime solutions of such equations too. It would also be of interest to eliminate the condition that $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has a solution. However, we have not been able to do so. The obstacle is that Lemma \[lem:3.2aa\] is no longer true without that condition. An example where this fails is provided in the remark after Lemma \[lem:3.2aa\]. Diophantine Approximation via Hypergeometric Functions ====================================================== Recall that by an [*effective irrational measure*]{} for an irrational number, $\alpha$, we mean an inequality of the form $$\left| \alpha - \frac{p}{q} \right|>\frac{c}{|q|^{\mu}},$$ for all $p/q \in {\mathbb{Q}}$ with $\gcd(p,q)=1$ and $|q|>Q$, where $c$, $Q$ and $\mu$ are all effectively computable. By Liouville’s famous result [@Liou], where he constructed the first examples of numbers proven to be transcendental, we have such effective irrational measures for algebraic numbers of degree $n$, with $\mu=n$. But for most applications we require $\mu<n$. We can use the hypergeometric method to obtain effective irrationality measures that improve on Liouville’s result for the algebraic numbers that arise here. However, that does not suffice for us to prove our theorem. The problem here arises not because of the exponent, $\mu$, in the effective irrationality measure, but because the constant, $c(\alpha)$, is too large. Upon investigating this further, we found that we can complete the proof of Theorem \[thm:1.1\] if we use not the effective irrationality measures from the hypergeometric method, but rather consider more carefully the actual results that we obtain from the use of hypergeometric functions. The means of doing so is the following lemma. \[lem:2.1\] Let $\theta \in {\mathbb{C}}$ and let ${\mathbb{K}}$ be an imaginary quadratic field. Suppose that there exist $k_{0},\ell_{0} > 0$ and $E,Q > 1$ such that for all non-negative integers $r$, there are algebraic integers $p_{r}$ and $q_{r}$ in ${\mathbb{K}}$ with $\left| q_{r} \right| < k_{0}Q^{r}$ and $\left| q_{r} \theta - p_{r} \right| \leq \ell_{0}E^{-r}$ satisfying $p_{r}q_{r+1} \neq p_{r+1}q_{r}$. For any algebraic integers $p$ and $q$ in ${\mathbb{K}}$, let $r_{0}$ be the smallest positive integer such that $|q|<E^{r_{0}}/ \left( 2 \ell_{0} \right)$. [(a)]{} We have $$\left| q\theta - p \right| > \frac{1}{2k_{0}Q^{r_{0}+1}}.$$ [(b)]{} When $p/q \neq p_{r}/q_{r}$, we have $$\left| q\theta - p \right| > \frac{1}{2k_{0}Q^{r_{0}}}.$$ We can improve the constants here somewhat, replacing $1/\left( 2k_{0} \right)$ in both parts by $\left( 1-1/E \right)/k_{0}$ and defining $r_{0}$ by $\left( Q-1/E \right)/\left( Q-1 \right)\ell_{0}|q|<E^{r_{0}}$. This would be helpful when reducing the size of $c$ here is important. This would have reduced the size of the bound on $a^{2}+b^{2}$ in Case 3 of Lemma \[lem:thm\] below. But as the remaining calculation to finish the proof of Lemma \[lem:thm\] is so quick, we have not pursued this here. The proof is identical to that of Lemma 6.1 of [@V2] except at the end of the proof we do not convert the lower bounds into ones involving $|q|^{-(\kappa+1)}$. Construction of Approximations ------------------------------ \ Let $t, u_{1}$ and $u_{2}$ be rational integers with $t<0$. We let $u=\left( u_{1}+u_{2}\sqrt{t} \right)/2$ be an algebraic integer in ${\mathbb{K}}={\mathbb{Q}}\left( \sqrt{t} \right)$ with $\sigma(u)=\left( u_{1}-u_{2}\sqrt{t} \right)/2$ as its algebraic (and complex) conjugate. Put $\omega = u/\sigma(u)$ and write $\omega=e^{i\varphi}$, where $-\pi<\varphi \leq \pi$. For any real number $\nu$, we shall put $\omega^{\nu}= e^{i\nu\varphi}$. Suppose that $\alpha, \beta$ and $\gamma$ are complex numbers and $\gamma$ is not a non-positive integer, ${}_{2}F_{1}(\alpha, \beta, \gamma, z)$ shall denote the classical (or Gauss) hypergeometric function of the complex variable $z$. For positive integers $m$ and $n$ with $0 < m < n$, $(m,n) = 1$ and $r$ a non-negative integer, we put $\nu=m/n$ and $$X_{m,n,r}(z)={}_{2}F_{1}(-r-\nu, -r, 1-\nu, z), \quad Y_{m,n,r}=z^{r}X_{m,n,r} \left(z^{-1} \right)$$ and $$\begin{aligned} R_{m,n,r}(z) &= \frac{\Gamma(r+1+\nu)}{r!\Gamma(\nu)} \int_{1}^{z} (1-t)^{r}(t-z)^{r}t^{-r-1+\nu}dt \\ &= (z-1)^{2r+1} \frac{\nu \cdots (r+\nu)}{(r+1) \cdots (2r+1)} {} _{2}F_{1} \left( r+1-\nu, r+1; 2r+2; 1-z \right),\end{aligned}$$ where $0$ is not on the path of integration from $1$ to $z$. We collect here some facts about these functions that we will require. \[lem:2.2\] [(a)]{} Suppose that $|\omega-1|<1$. We have $$\omega^{\nu}Y_{m,n,r}(\omega)-X_{m,n,r}(\omega)=R_{m,n,r}(\omega).$$ [(b)]{} We have $$X_{m,n,r}(\omega)Y_{m,n,r+1}(\omega) \neq X_{m,n,r+1}(\omega)Y_{m,n,r}(\omega).$$ [(c)]{} If $|\omega|=1$ and $|\omega-1|<1$, then $$\left| R_{m,n,r}(\omega) \right| \leq \frac{\Gamma(r+1+\nu)}{r!\Gamma(\nu)} |\varphi| \left| 1-\sqrt{\omega} \right|^{2r}.$$ [(d)]{} If $|\omega|=1$ and $|\omega-1|<1$, then $$\left| X_{m,n,r}(\omega) \right| = \left| Y_{m,n,r}(\omega) \right| < 1.072\frac{r!\Gamma(1-\nu)}{\Gamma(r+1-\nu)} \left| 1 + \sqrt{\omega} \right|^{2r}.$$ [(e)]{} For $|\omega|=1$ and ${\operatorname{Re}}(\omega) \geq 0$, we have $$\left| {} _{2}F_{1} \left( r+1-\nu, r+1; 2r+2; 1-\omega \right) \right| \geq 1,$$ with the minimum value occurring at $\omega=1$. Part (a) is established in the proof of Lemma 2.3 of [@Chen1]. Part (b) is Lemma 4 of [@Baker]. Part (c) is Lemma 2.5 of [@Chen1]. Part (d) is a slight refinement of Lemma 7.3(a) of [@V2]. In the proof of that lemma, we showed that in our notation here $$\left| X_{m,n,r}(\omega) \right| \leq \frac{4}{\left| 1 + \sqrt{w} \right|^{2}} \frac{\Gamma(1-m/n) \, r!}{\Gamma(r+1-m/n)} \left| 1+\sqrt{\omega} \right|^{2r}.$$ Since $\omega$ is on the unit circle, we can write $1+\sqrt{\omega}=1+w_{1} \pm \sqrt{1-w_{1}^{2}}i$, where $0 \leq w_{1} \leq 1$. Here we have $|\theta|<\pi/3$ in order that $|\omega-1|<1$ holds. Hence $w_{1}=\cos(\theta/2)>\cos(\pi/6)$, and so $$\frac{4}{\left| 1 + \sqrt{w} \right|^{2}}<1.072.$$ For part (e), we use Pochammer’s integral (see equation (1.6.6) of [@Sl]), along with the transformation $t=1/s$, to write $$\begin{aligned} {}_{2} F_{1} \left( a,b; c; z \right) &=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_{1}^{\infty} (s-1)^{c-b-1}s^{a-c}(s-z)^{-a}ds \\ &=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_{0}^{\infty} s^{c-b-1}(s+1)^{a-c}(s+1-z)^{-a}ds.\end{aligned}$$ Thus $${}_{2} F_{1} \left( a,b; c; 1-z \right) =\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_{0}^{\infty} s^{c-b}(s+1)^{a-c}(s+z)^{-a}ds/s$$ and our problem becomes one of showing that the absolute value of the function $$\int_{0}^{\infty} t^{\alpha} (t+1)^{-\beta}(t+z)^{-\gamma}\frac{dt}{t}$$ with $\alpha, \beta, \gamma>0$ and $\beta+\gamma>\alpha$ attains its minimum in $\Omega=\{z:|z|=1, z\notin[-1,0]\}$ at $z=1$. Note that here we have $\alpha=c-b$, $\beta=c-a$ and $\gamma=a$. We can change the integration path to any path that avoids the singularities of the integrand, i.e., any path that stays in the open angle bounded by the rays $\{-\tau z:\tau>0\}$ and $\{-\tau:\tau>0\}$ containing the positive semi-axis. So we will change it to the ray $\{\tau\sqrt{z}: \tau>0\}$. Thus out integral becomes $$\int_{0}^{\infty} \left( \sqrt{z} t \right)^{\alpha} \left( \sqrt{z}t+1 \right)^{-\beta} \left( \sqrt{z}t+z \right)^{-\gamma} \frac{dt}{t} = z^{(\alpha-\beta-\gamma)/2} \int_{0}^{\infty} t^{\alpha} \left( t+1/\sqrt{z} \right)^{-\beta} \left( t+z/\sqrt{z} \right)^{-\gamma} \frac{dt}{t}.$$ Putting $w=1/\sqrt{z}$ and recalling that $|z|=1$, we have $$\left| z^{(\alpha-\beta-\gamma)/2} \int_{0}^{\infty} t^{\alpha} \left( t+1/\sqrt{z} \right)^{-\beta} \left( t+z/\sqrt{z} \right)^{-\gamma} \frac{dt}{t} \right| = \left| \int_{0}^{\infty} t^{\alpha} \left( t+w \right)^{-\beta} \left( t+wz \right)^{-\gamma} \frac{dt}{t} \right|,$$ so the problem is reduced to establishing the following:\ let $w,z' \in {\mathbb{C}}$, ${\operatorname{Re}}(w), {\operatorname{Re}}(z')>0$ and $z'w \in {\mathbb{R}}_{+}$. Then $$\left|\int_0^\infty t^\alpha(t+w)^{-\beta}(t+z')^{-\gamma}\frac {dt}t\right|\ge \left|\int_0^\infty t^\alpha(t+|w|)^{-\beta}(t+|z'|)^{-\gamma}\frac {dt}t\right|\,.$$ Since $c=2a$, $|z|=1$ and our definition of $w$, here we have $\beta=\gamma$ and $z'=z/\sqrt{z}=\sqrt{\bar{z}}$, this is immediate because the integrand on the left is then positive and obviously greater than the one on the right. Since $|w|=|z'|=1$, we have $$\left|\int_0^\infty t^\alpha(t+|w|)^{-\beta}(t+|z'|)^{-\gamma}\frac {dt}t\right| =\left|\int_0^\infty t^\alpha(t+1)^{-\beta}(t+1)^{-\gamma}\frac {dt}t\right|,$$ which shows the integral attains its minimum at $z=1$, as stated. We let $D_{n,r}$ denote the smallest positive integer such that $D_{n,r} X_{m,n,r}(x) \in {\mathbb{Z}}[x]$ for all $m$ as above. For $d \in {\mathbb{Z}}$, we define $N_{d,n,r}$ to be the largest integer such that $\left( D_{n,r}/ N_{d,n,r} \right)X_{m,n,r}\left( 1-\sqrt{d}\,x \right) \in {\mathbb{Z}}\left[ \sqrt{d} \right] [x]$, again for all $m$ as above. We will use $v_{p}(x)$ to denote the largest power of a prime $p$ which divides into the rational number $x$. We put $$\label{eq:ndn-defn} {\mathcal{N}}_{d,n} =\prod_{p|n} p^{\min(v_{p}(d)/2, v_{p}(n)+1/(p-1))}.$$ In what follows, we shall restrict our attention to $m=1$, $n=4$ (so $\nu=1/4$) and $t=-1$. \[lem:2.4\] We have $$\label{eq:cndn-defn} \frac{\Gamma(3/4) \, r!}{\Gamma(r+3/4)} \frac{D_{4,r}}{N_{d,4,r}} <{\mathcal{C}}_{4,1} \left( \frac{e^{1.68}}{{\mathcal{N}}_{d,4}} \right)^{r} \text{ and } \hspace*{1.0mm} \frac{\Gamma(r+5/4)}{\Gamma(1/4)r!} \frac{D_{4,r}}{N_{d,4,r}} < {\mathcal{C}}_{4,2} \left( \frac{e^{1.68}}{{\mathcal{N}}_{d,4}} \right)^{r}$$ for all non-negative integers $r$, where ${\mathcal{C}}_{4,1}=0.83$, ${\mathcal{C}}_{4,2}=0.2$ and ${\mathcal{D}}_{4}=e^{1.68}$. From Lemma 7.4(c) of [@V2], we have $$\max \left( 1, \frac{\Gamma(3/4) \, r!}{\Gamma(r+3/4)}, 4\frac{\Gamma(r+5/4)}{\Gamma(1/4)r!} \right) \frac{D_{4,r}}{N_{d,4,r}} < 100 \left( \frac{e^{1.64}}{{\mathcal{N}}_{d,4}} \right)^{r}$$ However, the value $100$ results in us requiring a lot of computation to complete the proof of our theorem here. Therefore, we seek a smaller value at the expense of replacing $1.64$ by a larger value, whose value has less of an impact on our proof. For $r \geq 156$, $100\exp(1.64r)<0.2\exp(1.68r)$, so we compute directly the left-hand sides of for $r \leq 155$. We find that the maximum values of the left-hand sides of divided by $\exp(1.68r)$ both occur for $r=3$.The lemma follows. Put $$p_{r}'=\frac{D_{4,r}}{N_{d,4,r}} X_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2} \right) ^{r}, \quad q_{r}'=\frac{D_{4,r}}{N_{d,4,r}} Y_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2} \right)^{r},$$ and $$R_{r}'=\frac{D_{4,r}}{N_{d,4,r}} R_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2} \right)^{r},$$ where $r$ is any non-negative integer and $d$ will be determined below. By the definitions of $D_{4,r}$ and $N_{d,4,r}$, we easily observe that $p_{r}'$ and $q_{r}'$ are algebraic integers of ${\mathbb{Q}}(i)$. We can see that as follows. $X_{1,4,r}(z)$ is a polynomial of degree $r$ and $$p_{r}' = \frac{D_{1,4,r}}{N_{d,4,r}} X_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2} \right)^{r} = \frac{D_{1,4,r}}{N_{d,4,r}} X_{r} \left( 1-u_{2}i \frac{2}{u_{1}-u_{2}i} \right) \left( \frac{u_{1}-u_{2}i}{2} \right)^{r}.$$ So with $d=u_{2}^{2}$, we have $p_{r}'$ is an algebraic integer of ${\mathbb{Q}}(i)$. The analogous expression for $q_{r}'$ shows that it is also an algebraic integer. In fact, there may be some further common factors. As in [@V3], put $$\begin{aligned} g_{1} & = \gcd \left( u_{1}, u_{2} \right), \\ g_{2} & = \gcd \left( u_{1}/g_{1}, t \right)=1,\\ g_{3} & = \left\{ \begin{array}{ll} 1 & \mbox{if $t \equiv 1 \bmod 4$ and $\left( u_{1}-u_{2} \right)/g_{1} \equiv 0 \bmod 2$}, \\ 2 & \mbox{if $t \equiv 3 \bmod 4$ and $\left( u_{1}-u_{2} \right)/g_{1} \equiv 0 \bmod 2$},\\ 4 & \mbox{otherwise,} \end{array} \right. \\ & = \left\{ \begin{array}{ll} 2 & \mbox{if $\left( u_{1}-u_{2} \right)/g_{1} \equiv 0 \bmod 2$},\\ 4 & \mbox{otherwise,} \end{array} \right. \\ g & = g_{1}\sqrt{g_{2}/g_{3}}.\end{aligned}$$ Then we can put $$\label{eq:7} p_{r}=\frac{D_{4,r}}{N_{d,4,r}} X_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right) ^{r}, \quad q_{r}=\frac{D_{4,r}}{N_{d,4,r}} Y_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)^{r},$$ and $$R_{r}=\frac{D_{4,r}}{N_{d,4,r}} R_{1,4,r}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)^{r},$$ where $$d=\left( u-\sigma(u) \right)^{2}/g^{2} = u_{2}^{2}t/g^{2} = -u_{2}^{2}/g^{2}.$$ Since $\left( u_{1}-u_{2}i \right)/g$ is an algebraic integer, the argument above still applies to show that $p_{r}$ and $q_{r}$ are algebraic integers. So in Lemma \[lem:2.4\], we can take $$Q = \frac{{\mathcal{D}}_{4} \left| u_{1} + \sqrt{u_{1}^{2}+u_{2}^{2}} \right|}{|g|{\mathcal{N}}_{d,4}}$$ and $$\label{eq:k-UB} k_{0}<1.072{\mathcal{C}}_{4,1}<0.89.$$ We also have $$E = \frac{|g|{\mathcal{N}}_{d,4} \left| u_{1} + \sqrt{u_{1}^{2}+u_{2}^{2}} \right|}{{\mathcal{D}}_{4}u_{2}^{2}}$$ and $$\ell_{0}={\mathcal{C}}_{4,2}|\varphi|=0.2|\varphi|.$$ Other Preliminary Lemmas ======================== \[lem:3.1\] Let $a$, $m$ and $p$ be positive integers with $a \geq 1$ and $p$ a prime. Put $b=p^{m}$ or $b=2p^{m}$ and suppose that $\gcd \left( a,b \right)=1$ and $a^{2}+b^{2}$ not a perfect square. Furthermore, suppose that $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution. All coprime integer solutions $(x,y)$ to the quadratic equation $$\label{eq:quad-eqnc-dup} x^{2} - \left( a^{2}+b^{2} \right)y^{2}=-b^{2}$$ are given by $$\label{eq:14c-dup} x+y \sqrt{a^{2}+b^{2}} = \pm \left( \pm a + \sqrt{a^{2}+b^{2}} \right) \alpha^{2k}, \hspace{1.0mm} k \in {\mathbb{Z}},$$ where $\alpha = \left( T_{1}+U_{1} \sqrt{a^{2}+b^{2}} \right)/2$ and $\left( T_{1},U_{1} \right)$ is the minimum solution of the equation $x^{2}-\left( a^{2}+b^{2} \right)y^{2}=-4$ in positive integers. The condition that $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution is not required here, but will be needed required for Lemma \[lem:3.2aa\]. Here we can replace $\alpha^{2}$ by $T+U \sqrt{a^{2}+b^{2}}$ where $\left( T,U \right)$ is the minimum solution of the equation $x^{2}-\left( a^{2}+b^{2} \right)y^{2}=1$ in positive integers. The proof uses the fact that for $\beta, \gamma \in {\mathcal{O}}_{{\mathbb{K}}}$ for some number field, ${\mathbb{K}}$, we have $(\beta)=(\gamma)$ if and only if $\beta=\gamma\epsilon$ where $\epsilon$ is a unit in ${\mathcal{O}}_{{\mathbb{K}}}$. In what follows, we let ${\mathbb{K}}= {\mathbb{Q}}\left( \sqrt{a^{2}+b^{2}} \right)$. First suppose that $b=p^{m}$ with $p \neq 2$. Since the Legendre symbol $\left( a^{2}+b^{2} / p \right)=\left( a^{2} / p \right)=1$ if $p \neq 2$ and $a^{2}+b^{2} \equiv 1 \bmod 8$ if $p=2$ (since $b^{2} \geq 8$ by our assumption), we know there is a prime ideal, ${\mathfrak{p}}$ in ${\mathcal{O}}_{{\mathbb{K}}}$, such that $(p)={\mathfrak{p}}\bar{{\mathfrak{p}}}$, where $\bar{{\mathfrak{p}}}=\left\{ a_{1}-b_{1}\sqrt{a^{2}+b^{2}}: a_{1}+b_{1}\sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}\right\}$. Let $(x,y)$ be any relatively prime solution of and consider $x+y \sqrt{a^{2}+b^{2}}$. It has norm $-b^{2}=-p^{2m}$, so it is a member of $(p)^{2m}$. Since $x$ and $y$ are relatively prime, we must have either $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m}$ or $x+y \sqrt{a^{2}+b^{2}} \in \bar{{\mathfrak{p}}}^{2m}$. Note that it cannot be a member of ${\mathfrak{p}}^{m_{1}} \bar{{\mathfrak{p}}}^{2m-m_{1}}$ for $1 \leq m_{1}<2m$, as such an ideal would have a power of $(p)$ as a factor and hence $x$ and $y$ would no longer be relatively prime – it is here where we need the assumption that $p \neq 2$. Without loss of generality, let us suppose that $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m}$ and also that $a+\sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m}$. The proofs for the other cases follow by the fact that the other factor of $(p)$ is $\bar{{\mathfrak{p}}}$, the conjugate of ${\mathfrak{p}}$. Since $\left( a+\sqrt{a^{2}+b^{2}} \right)$, $\left( x+y\sqrt{a^{2}+b^{2}} \right)$ and ${\mathfrak{p}}^{2m}$ all have norm $b^{2}$ and $a+\sqrt{a^{2}+b^{2}}, x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m}$, it follows that ${\mathfrak{p}}^{2m} = \left( a+\sqrt{a^{2}+b^{2}} \right) = \left( a+\sqrt{a^{2}+b^{2}} \right)$. Therefore, $x+y \sqrt{a^{2}+b^{2}}$ must be a unit of norm $1$ times $a+\sqrt{a^{2}+b^{2}}$ and the result follows. If $p=2$, we also need to consider the possibility that it is a member of $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}\bar{{\mathfrak{p}}}^{2m-1}$ or $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}^{2m-1} \bar{{\mathfrak{p}}}$. As above, we may suppose that $x+y \sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}\bar{{\mathfrak{p}}}^{2m-1}$ and also that $a+\sqrt{a^{2}+b^{2}} \in {\mathfrak{p}}\bar{{\mathfrak{p}}}^{2m-1}$. The same argument as above now holds to show that the lemma holds in this case too. We now consider $b=2p^{m}$. We may assume that $p \neq 2$, since the case of $p=2$ is covered above. Here $b^{2} \equiv 4 \bmod 8$ and hence $a^{2}+b^{2} \equiv 5 \bmod 8$. Here $(2)$ is a prime ideal in ${\mathcal{O}}_{{\mathbb{K}}}$ and, as shown above, $(p)$ splits into the product of two prime ideals, ${\mathfrak{p}}$ and $\bar{{\mathfrak{p}}}$, which are conjugates of each other. Let $(x,y)$ be any relatively prime solution of and consider $x+y \sqrt{a^{2}+b^{2}}$. It has norm $-b^{2}=-4p^{2m}$. Since $x$ and $y$ are relatively prime, we must have either $x+y \sqrt{a^{2}+b^{2}} \in (2){\mathfrak{p}}^{m}$ or $x+y \sqrt{a^{2}+b^{2}} \in (2) \bar{{\mathfrak{p}}}^{m}$. As above, it cannot be a member of $(2){\mathfrak{p}}^{m_{1}} \bar{{\mathfrak{p}}}^{2m-m_{1}}$ for $1 \leq m_{1}<2m$. Arguing as above, the result now follows. \[lem:3.2aa\] Let $a$ and $b$ be relatively prime positive integers such that $a^{2}+b^{2}$ is not a perfect square. Suppose that all coprime positive integer solutions of $\eqref{eq:quad-eqnc}$ are given by $\eqref{eq:14c}$ and that $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution. If $(X,Y) \neq (a,1)$ is a coprime positive integer solution to $$X^{2} - \left( a^{2}+b^{2} \right)Y^{4}=-b^{2},$$ then $$\pm X \pm bi = \left( a+bi \right) \left( r \pm si \right)^{4}, \quad Y=r^{2}+s^{2},$$ where $r,s \in {\mathbb{Z}}$ with $\gcd(r,s)=1$ and $s>r>0$. Note that we can also express the solution $(X,Y)=(a,1)$ in this form, but with $r=1$ and $s=0$ (i.e., we remove the condition that $s>0$). The condition that $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution is required here. It arises in the proof as it implies that any solution $(X,Y)$ of $X^{2} - \left( a^{2}+b^{2} \right) Y^{4}=-b^{2}$ comes from an even power of $\alpha$. This provides us with the representations in that play a key role in obtaining the desired representation. For $k \geq 0$, we define $T_{k}$ and $U_{k}$ by $$\alpha^{k} = \frac{T_{k} + U_{k} \sqrt{a^{2}+b^{2}}}{2}.$$ Note that $T_{k}, U_{k} \in {\mathbb{Z}}$ and $T_{k} \equiv U_{k} \bmod 2$, with $T_{k} \equiv U_{k} \equiv 1 \bmod 2$ only possible if $a^{2}+b^{2} \equiv 1 \bmod 4$. Therefore, by expanding , a solution in coprime positive integers $(X,Y) \neq (a,1)$ to $X^{2}-\left( a^{2}+b^{2} \right)Y^{4}=-b^{2}$ arises from $$X+Y^{2}\sqrt{a^{2}+b^{2}} = \left( \pm a + \sqrt{a^{2}+b^{2}} \right) \left( \frac{T_{2k}+U_{2k}\sqrt{a^{2}+b^{2}}}{2} \right)$$ and so such a solution is equivalent to $$\label{eq:15} 2X= \left( a^{2}+b^{2} \right) U_{2k} \pm aT_{2k}, \quad 2Y^{2}=T_{2k} \pm aU_{2k}$$ for some $k \geq 1$. Note we need $k \neq 0$ since $(X,Y) \neq (a,1)$. We now show that the expressions for $X$ and $Y^{2}$ in are actually positive. Since $T_{2k}^{2}-\left( a^{2}+b^{2} \right)U_{2k}^{2}=1$, we have $T_{2k}>\sqrt{a^{2}+b^{2}}U_{2k}>aU_{2k}$. Hence $Y^{2}>0$ is satisfied. From $T_{2k}^{2}-\left( a^{2}+b^{2} \right)U_{2k}^{2}=1$, we have $\left( a^{2}+b^{2} \right)U_{2k}-T_{2k}^{2}/U_{2k}=-1/U_{2k}$ and $T_{2k}>\sqrt{a^{2}+b^{2}}U_{2k}$. So we have $$\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k} >\left( a^{2}+b^{2} \right)U_{2k}-T_{2k}\sqrt{a^{2}+b^{2}} >\left( a^{2}+b^{2} \right)U_{2k}-T_{2k}^{2}/U_{2k}=-1/U_{2k}.$$ Since $U_{2k} \geq 1$ and $\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k} \in {\mathbb{Z}}$, it follows that $\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k} \geq 0$. If $\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k}=0$ held, then since $\gcd(a,b)=1$, any prime divisor of $a^{2}+b^{2}$ must divide $T_{2k}$. But then $T_{2k}^{2}-\left( a^{2}+b^{2} \right)U_{2k}^{2}=1$ would be impossible. Therefore $\left( a^{2}+b^{2} \right)U_{2k}-aT_{2k}>0$, and as a result $\left( a^{2}+b^{2} \right) U_{2k} \pm aT_{2k}>0$ holds. So $X$ is also positive, as required. Notice that this tells us that $$2X+2Y^{2}\sqrt{a^{2}+b^{2}} = -\left( \pm a + \sqrt{a^{2}+b^{2}} \right) \left( T_{2k}+U_{2k}\sqrt{a^{2}+b^{2}} \right)$$ is not possible. Also, corresponding to $k<0$, $$2X+2Y^{2}\sqrt{a^{2}+b^{2}} = \left( \pm a + \sqrt{a^{2}+b^{2}} \right) \left( T_{2k}-U_{2k}\sqrt{a^{2}+b^{2}} \right)$$ gives us $2X=\pm aT_{2k}-\left( a^{2}+b^{2} \right) U_{2k}$ and $2Y^{2}=T_{2k} \mp aU_{2k}$. But from our argument above, we see that this value of $X$ can never be positive. Hence all the solutions must come from . Now we use the expressions arising from to prove the lemma. Note that $$\begin{aligned} \alpha^{2k} & = & \frac{T_{2k} + U_{2k} \sqrt{a^{2}+b^{2}}}{2} = \left( \frac{T_{k} + U_{k} \sqrt{a^{2}+b^{2}}}{2} \right)^{2} \\ & = & \frac{T_{k}^{2}+\left(a^{2}+b^{2} \right) U_{k}^{2} +2T_{k}U_{k} \sqrt{a^{2}+b^{2}}}{4}.\end{aligned}$$ Thus $$\label{eq:my-2} T_{2k}=\frac{T_{k}^{2}+\left(a^{2}+b^{2} \right) U_{k}^{2}}{2} \quad \text{ and } \quad U_{2k}=T_{k}U_{k}.$$ Since $T_{k} \equiv U_{k} \bmod 2$ and $T_{k} \equiv U_{k} \equiv 1 \bmod 2$ only if $a^{2}+b^{2} \equiv 1 \bmod 4$, we see that $T_{2k}, U_{2k} \in {\mathbb{Z}}$, $T_{2k} \equiv U_{2k} \bmod 2$ and $T_{2k} \equiv U_{2k} \equiv 1 \bmod 2$ only if $a^{2}+b^{2} \equiv 1 \bmod 4$. Also notice from the expression for $2X$ in that if $a^{2}+b^{2} \equiv 1 \bmod 4$ and $T_{2k} \equiv U_{2k} \equiv 1 \bmod 2$, then we must have $a$ odd and hence $b$ even. Otherwise, the right-hand side of the expression for $2X$ is odd. By the expressions in for $T_{2k}$ and $U_{2k}$, implies that $$(2Y)^{2}=T_{k}^{2}+\left( a^{2}+b^{2} \right)U_{k}^{2} \pm 2aT_{k}U_{k} = \left( T_{k} \pm aU_{k} \right)^{2} + \left( bU_{k} \right)^{2}.$$ Our statements above about the parity of $a$, $b$, $T_{k}$ and $U_{k}$ imply that both $T_{k} \pm aU_{k}$ and $bU_{k}$ are always even. Therefore, $$\label{eq:my-3aa} Y^{2}= \left( \frac{T_{k} \pm aU_{k}}{2} \right)^{2} + \left( \frac{bU_{k}}{2} \right)^{2}.$$ Observe that $\gcd \left( Y+\left( T_{k} \pm aU_{k} \right)/2, Y-\left( T_{k} \pm aU_{k} \right)/2 \right)$ divides $\left( T_{k} \pm aU_{k} \right)$ and $bU_{k}/2$. Since $\left( T_{k}+aU_{k} \right) \left( T_{k}-aU_{k} \right)/4= \left( bU_{k}/2 \right)^{2} \pm 1$, and $T_{k}+aU_{k}$ and $T_{k}-aU_{k}$ have the same parity, it follows that $\gcd \left( T_{k} \pm aU_{k}, bU_{k}/2 \right)$ divides $2$. Thus $\gcd \left( Y+\left( T_{k} \pm aU_{k} \right)/2, Y-\left( T_{k} \pm aU_{k} \right)/2 \right)$ divides $2$. We consider each of the possibilities for this gcd now. If $\gcd \left( Y+\left( T_{k} \pm aU_{k} \right)/2, Y-\left( T_{k} \pm aU_{k} \right)/2 \right)=1$, then there are positive integers $b_{1}, b_{2}$, $r_{1}$ and $s_{1}$ satisfying $\gcd \left( b_{1}, b_{2} \right)=\gcd \left( r_{1}, s_{1} \right)=1$ such that $b_{1}b_{2}=b$, $r_{1}s_{1}=U_{k}$ and $$\label{eq:16a} Y+\left( T_{k} \pm aU_{k} \right)/2=b_{1}^{2}s_{1}^{2}, \quad Y-\left( T_{k} \pm aU_{k} \right)/2=b_{2}^{2}r_{1}^{2} \quad \text{ and } \quad b_{1}s_{1}>b_{2}r_{1}.$$ If $\gcd \left( Y+\left( T_{k} \pm aU_{k} \right)/2, Y-\left( T_{k} \pm aU_{k} \right)/2 \right)=2$, then both $Y+\left( T_{k} \pm aU_{k} \right)/2$ and $Y-\left( T_{k} \pm aU_{k} \right)/2$ are twice a square. So we have $$\label{eq:16b} Y+\left( T_{k} \pm aU_{k} \right)/2=2b_{1}^{2}s_{1}^{2}, \quad Y-\left( T_{k} \pm aU_{k} \right)/2=2b_{2}^{2}r_{1}^{2} \quad \text{ and } \quad b_{1}s_{1}>b_{2}r_{1},$$ where $\gcd \left( b_{1}, b_{2} \right)=\gcd \left( r_{1}, s_{1} \right)=1$, and either $2b_{1}b_{2}=b$, $r_{1}s_{1}=U_{k}$; or $b_{1}b_{2}=b$ and $2r_{1}s_{1}=U_{k}$. In the first case, subtracting the two expressions in and substituting for $U_{k}$, we obtain $$U_{k}=r_{1}s_{1} \quad \text{ and } \quad T_{k}=b_{1}^{2}s_{1}^{2}-b_{2}^{2}r_{1}^{2} \mp ar_{1}s_{1}.$$ We have $T_{k}^{2}- \left( a^{2}+b^{2} \right) U_{k}^{2}=\pm 4$. However, the proof is the same for both cases, so we consider only $T_{k}^{2}- \left( a^{2}+b^{2} \right) U_{k}^{2}=4$ here. Substituting the above expressions for $T_{k}$ and $U_{k}$ into $T_{k}^{2}- \left( a^{2}+b^{2} \right) U_{k}^{2}=4$ and then simplifying leads to the equation $$b_{2}^{4}r_{1}^{4} \pm 2ab_{2}^{2}r_{1}^{3}s_{1}-3b^{2}r_{1}^{2}s_{1}^{2} \mp 2ab_{1}^{2}r_{1}s_{1}^{3}+b_{1}^{4}s_{1}^{4}=4.$$ Multiplying both sides by $2bi$, we obtain $$\label{eq:my-3} (a+bi) \left( b_{2}r_{1} \pm b_{1}s_{1}i \right)^{4} -(a-bi) \left( b_{2}r_{1} \mp b_{1}s_{1}i \right)^{4}=32bi.$$ We can write $$\begin{aligned} & & (a+bi) \left( b_{2}r_{1} \pm b_{1}s_{1}i \right)^{4} +(a-bi) \left( b_{2}r_{1} \mp b_{1}s_{1}i \right)^{4} \\ & = & 8a \left( T_{k}^{2}+\left( a^{2}+b^{2} \right)U_{k}^{2} \right) -16 \left( a^{2}+b^{2} \right)T_{k}U_{k} = 32X,\end{aligned}$$ the last equality follows from applying to our expression for $X$ from with the signs there all positive. Combining this with yields $$16X+16bi= (a+bi) \left( b_{2}r_{1} \pm b_{1}s_{1}i \right)^{4}.$$ Had we considered $T_{k}^{2}- \left( a^{2}+b^{2} \right) U_{k}^{2}=-4$ above, we would have found that $$16X-16bi= (a-bi) \left( b_{2}r_{1} \pm b_{1}s_{1}i \right)^{4}.$$ We noted above that $b_{2}r_{1}$ and $b_{1}s_{1}$ have the same parity. Therefore, $b_{2}r_{1} \pm b_{1}s_{1}i$ is divisible by $1 \pm i$, say $b_{2}r_{1} \pm b_{1}s_{1}i=(1 \pm i)(r+si)$ for some integers $r$ and $s$ with $\gcd(r,s)=1$ and $s>r>0$ (since $b_{1}s_{1}>b_{2}r_{1}$ from ). The expression in the lemma for both $4X+4bi$ and $4X-4bi$ follow. From the expression for $4X+4bi$, we obtain $$16 \left( X^{2}+b^{2} \right) = 16\left( a^{2}+b^{2} \right) \left( r^{2}+s^{2} \right)^{4},$$ but we also have $16 \left( X^{2}+b^{2} \right)=16\left( a^{2}+b^{2} \right)Y^{4}$. The expression in the lemma for $2Y$ follows. When the signs appearing in are negative, a nearly identical argument to the above leads to $$-4X\pm 4bi= (a+bi) \left( b_{2}r_{1}-b_{1}s_{1}i \right)^{4}, \quad b_{1}b_{2}=b, \quad \gcd \left( r_{1},s_{1} \right)=1.$$ As above, this completes the proof of this lemma. Next, in Lemma \[lem:3.3\] below, we establish a gap principle separating possible solutions of . We need a few additional lemmas to help us first. Lemma \[lem:3.4\](b) will also play a key role in the proof of Theorem \[thm:1.1\] too. \[lem:3.4\] Suppose that $a$ and $b$ are relatively prime positive integers and $(X,Y)$ is a coprime positive integer solution of $x^{2}- \left( a^{2}+b^{2} \right)y^{4}=-b^{2}$ with $Y>1$. [(a)]{} $Y$ is only divisible by primes, $p \equiv 1 \bmod 4$. As a consequence, if $Y>1$, then $Y \geq 5$ and if $Y>5$, then $Y \geq 13$. [(b)]{} Let $a$ and $b$ be as in the statement of Theorem $\ref{thm:1.2}$. Then $Y>b/2$. [(c)]{} Let $a$, $m$ and $p$ be as in the statement of Theorem $\ref{thm:1.1}$ and put $b=p^{m}$, then $Y>b^{2}/4$. If $p \neq 2$, then $Y>b^{2}/2$. The additional conditions in parts (b) and (c) are required. Without them, the lower bound for $Y$ in part (b) does not hold in general. E.g., if $b$ is an element of the recurrence sequence, $b_{0}=-3$, $b_{1}=4$, $b_{n+2}=50b_{n+1}-b_{n}$ for $n \geq 0$, then $\left( 1+b^{2} \right) 5^{4}-b^{2}=624b^{2}+625$ is a perfect square and $(X,Y)=\left( \sqrt{624b^{2}+625}, 5 \right)$ is a solution of the diophantine equation for $a=1$ and such values of $b$. Hence $Y$ stays fixed as $b$ grows. Also note that the bound in part (b) is nearly best-possible. For any odd $b' \equiv 0,2,8 \bmod 10$ with $b'>2$, put $b=(b')^{2}-1$, $a=(b')^{3}/4-3b'/2$, then $(x,y)= \left( b' \left( b'^6+4b'^{4}+5b'^{2}+10 \right)/4, b+2 \right)$ is a solution of . So it appears that the correct bound is $Y>b$. The bound in part (c) is best-possible for $b$ odd, as can be seen by considering Example 1 in Remark \[rem:1\]. \(a) If $Y=2k$, then we are seeking solutions of $X^{2}=16k^{4}a^{2} + \left( 16k^{4}-1 \right)b^{2}$. Since $\gcd(a,b)=1$, if $a$ is even, then $b$ is odd and $16k^{4}a^{2} + \left( 16k^{4}-1 \right)b^{2} \equiv 3 \bmod 4$. This implies that $X^{2} \equiv 3 \bmod 4$, which is not possible, so there are no solutions with $Y$ even in this case. If $b$ is even, then $X$ is even, but that violates our assumption that $\gcd(X,Y)=1$. Suppose $Y=pk$, where $p$ is an odd prime. Then we have $X^{2}=a^{2}p^{4}k^{4}+ \left( p^{4}k^{4}-1 \right) b^{2}$. Thus $X^{2} \equiv -b^{2} \bmod p$. If $b \equiv 0 \bmod p$, then $X \equiv 0 \bmod p$, which is not allowed by our assumption that $X$ and $Y$ are relatively prime. For such $b$, $X^{2} \equiv -b^{2} \bmod p$ is not solvable if $p \equiv 3 \bmod 4$, as required. \(b) and (c) Recall from the proof of Lemma \[lem:3.2aa\] that we have either $$Y+\left( T_{k} \pm aU_{k} \right)/2=b_{1}^{2}s_{1}^{2} \quad \text{ and } \quad Y-\left( T_{k} \pm aU_{k} \right)/2=b_{2}^{2}r_{1}^{2}$$ with $\gcd \left( b_{1}, b_{2} \right)=1$, $\gcd \left( r_{1}, s_{1} \right)=1$ and $b_{1}b_{2}=b$; or $$Y+\left( T_{k} \pm aU_{k} \right)/2=2b_{1}^{2}s_{1}^{2} \quad \text{ and } \quad Y-\left( T_{k} \pm aU_{k} \right)/2=2b_{2}^{2}r_{1}^{2}$$ where $\gcd \left( b_{1}, b_{2} \right)=\gcd \left( r_{1}, s_{1} \right)=1$ and $2b_{1}b_{2}=b$. These relations are and . If $k=1$, then adding these expressions we have three possibilities. First, if $$Y+\left( T_{k} \pm aU_{k} \right)/2=b_{1}^{2}s_{1}^{2} \quad \text{ and } \quad Y-\left( T_{k} \pm aU_{k} \right)/2=b_{2}^{2}r_{1}^{2}$$ with $\gcd \left( b_{1}, b_{2} \right)=1$ and $b_{1}b_{2}=b$, then $$2Y=b_{1}^{2}s_{1}^{2}+b_{2}^{2}r_{1}^{2} = b_{1}^{2}s_{1}^{2}+\left( b/b_{1} \right)^{2}r_{1}^{2}.$$ Differentiating this expression with respect to $b_{1}$, we find that the derivative is only zero for $b_{1}$ positive if $b_{1}=\sqrt{r_{1}b/s_{1}}$. Here we have $2Y=2r_{1}s_{1}b \geq 2b$. If $b_{1}=1$, then we have $2Y=s_{1}^{2}+b^{2}r_{1}^{2}$. So $2Y \geq b^{2}+1 \geq 2b$. Similarly, if $b_{1}=b$, then we also have $2Y \geq 2b$. Furthermore, if $b=p^{m}$ is a prime power, then either $b_{1}=b$ or $b_{2}=b$. Here we obtain $2Y>b^{2}$. Second, if $$Y+\left( T_{k} \pm aU_{k} \right)/2=2b_{1}^{2}s_{1}^{2} \quad \text{ and } \quad Y-\left( T_{k} \pm aU_{k} \right)/2=2b_{2}^{2}r_{1}^{2}$$ with $\gcd \left( b_{1}, b_{2} \right)=1$ and $2b_{1}b_{2}=b$, then $$2Y=2b_{1}^{2}s_{1}^{2}+2b_{2}^{2}r_{1}^{2}=2b_{1}^{2}s_{1}^{2}+2 \left( b/ \left( 2b_{1} \right) \right)^{2}r_{1}^{2}.$$ Differentiating this expression with respect to $b_{1}$, we find that the derivative is only zero for $b_{1}$ positive if $b_{1}=\sqrt{r_{1}b/\left( 2s_{1} \right)}$. Here we have $2Y=2r_{1}s_{1}b \geq 2b$. If $b_{1}=1$, then we have $2Y=2s_{1}^{2}+b^{2}r_{1}^{2}/2$. Since $b$ is even, we have $b \geq 2$ and so $2Y \geq b^{2}/2+2 \geq 2b$. Similarly, if $b_{1}=b/2$, then we also have $2Y \geq 2b$. Furthermore, if $b=p^{m}$ is a prime power, then either $b_{1}=b/2$ or $b_{2}=b/2$. Here we obtain $2Y>2b^{2}/4$. This completes the proof for $k=1$. Now we consider $k>1$. If $a=U_{1}=1$, then we have $T_{1}^{2}-\left( 1+b^{2} \right) =-4$. This is only possible if $T_{1}=1$ and $b=2$. Using Magma, we find that there are no integer solutions of $X^{2}-5Y^{4}=-4$ with $Y>1$. Hence we can ignore this case and assume that $aU_{1} \geq 2$. Here we have $T_{1}^{2} \geq b^{2}$. Combining this with , we have $$Y^{2}= \left( T_{k} \pm aU_{k} \right)^{2}4 + \left( bU_{k}/2 \right)^{2} \geq \left( bU_{2}/2 \right)^{2}=b^{2}T_{1}^{2}U_{1}^{2} \geq b^{4}.$$ This argument holds since $\left\{ U_{k} \right\}$ satisfies the recurrence sequence $U_{k+2}=2T_{1}U_{k+1}+U_{k}$ and $T_{1} \geq 1$, from the minimal polynomial for $\alpha$, so $U_{k} \geq U_{2}$ for $k \geq 2$. Thus $Y \geq b^{2}$. \[lem:3.2bb\] Let $\omega=e^{i\theta}$ with $-\pi < \theta \leq \pi$ and put $\omega^{1/4}=e^{i\theta/4}$. If $$0 < \left| \omega^{1/4} - z \right| <c_{1},$$ for some $z \in {\mathbb{C}}$ with $|z|=1$ and $0 < c_{1} < 1$, then $$\left| \omega - z^{4} \right| > c_{2} \left| \omega^{1/4} - z \right|,$$ where $c_{2}=\left( 2-c_{1}^{2} \right) \sqrt{4-c_{1}^{2}}$. We can write $$\left| \omega - z^{4} \right| = \left| \omega^{1/4} - z \right| \times \prod_{k=1}^{3} \left| \omega^{1/4} - e^{2\pi ik/4}z \right|.$$ Multiplying by $\omega^{-1/4}$ and expanding the resulting expression, the product above equals $$\prod_{k=1}^{3} \left| e^{2\pi ik/4-i\theta/4}z-1 \right| = \left| e^{3i\varphi}+e^{2i\varphi}+e^{i\varphi}+1 \right|,$$ for some $-\pi < \varphi \leq \pi$. Squaring this quantity and simplifying, we obtain $$8\cos^{2}(\varphi) \left( \cos(\varphi)+1 \right).$$ If $\left| \omega^{1/4} - z \right|=c_{1}$, we have $2-2\cos(\varphi)=c_{1}^{2}$ and the lemma follows under this assumption by a routine substitution. Since $c_{2}$ is a decreasing function of $c_{1}$, the lemma also holds in general. \[lem:3.3\] Under the same assumptions as in Theorem $\ref{thm:1.1}$, suppose that $\left( X_{1},Y_{1} \right)$ and $\left( X_{2},Y_{2} \right)$ are two solutions in coprime positive integers to with $Y_{2}>Y_{1}>1$. Then $$Y_{2}>7.98\frac{a^{2}+b^{2}}{b^{2}} Y_{1}^{3}.$$ By Lemma \[lem:3.2aa\], there are integers $r_{1}, s_{1}, r_{2}, s_{2}$ such that $$\pm 4X_{j} \pm 4bi = \left( a+bi \right) \left( r_{j} \pm s_{j} i \right)^{4}, \quad 2Y_{j}=r_{j}^{2}+s_{j}^{2}, \hspace{1.0mm} \text{$j=1,2$}.$$ We will assume that $$\label{eq:3.2} 4X_{1} \pm 4bi = \left( a+bi \right) \left( r_{1} + s_{1} i \right)^{4}, \quad 4X_{2} \pm 4bi = \left( a+bi \right) \left( r_{2} + s_{2} i \right)^{4},$$ as the argument for the other cases is exactly the same. It follows that $$\label{eq:3.3} \left( a+bi \right) \left( r_{j}+s_{j}i \right)^{4} - \left( a-bi \right) \left( r_{j}-s_{j}i \right)^{4} = \pm 8bi, \quad \text{($j=1,2$)}.$$ Putting $\omega = \left( a-bi \right) / \left( a+bi \right)$, by Lemma \[lem:3.4\](a) and , we have $$\label{eq:3.4} \left| \omega - \left( \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right)^{4} \right| =\left| \frac{\pm 8bi}{\left( a+bi \right)\left( r_{j}-s_{j}i \right)^{4}} \right| = \frac{8b}{\sqrt{a^{2}+b^{2}} 4Y_{j}^{2}} < 2/25.$$ For $j=1,2$, let $\zeta_{4}^{(j)}$ be the $4$-th root of unity such that $$\left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right| = \min_{0 \leq k \leq 3} \left| \omega^{1/4} - \zeta_{4}^{k} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right|.$$ From , we immediately have $$\left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right| < (2/25)^{1/4},$$ for $j=1,2$. In fact, we will show that this quantity is even smaller. Applying Lemma \[lem:3.2bb\] with $c_{1}=(2/25)^{1/4}$, we can take $c_{2}=3.31$. Combining this with , we obtain $$\left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right| < (2/25)/3.31<0.025.$$ Applying Lemma \[lem:3.2bb\] again, now with $c_{1}=0.025$, we obtain $$\label{eq:20} \left| \omega - \left( \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right)^{4} \right| > 3.998 \left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right|$$ and $$\label{eq:3.6} \left| \omega^{1/4} - \zeta_{4}^{(j)} \frac{r_{j}+s_{j}i}{r_{j}-s_{j}i} \right| <0.5003 \frac{b}{\sqrt{a^{2}+b^{2}} Y_{j}^{2}}.$$ Hence $$\begin{aligned} \label{eq:3.7} \left| \zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} - \zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i} \right| & \leq \left| \omega^{1/4} - \zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} \right| + \left| \omega^{1/4} - \zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i} \right| \nonumber \\ & < 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}} Y_{1}^{2}} + 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}}.\end{aligned}$$ Next we obtain a lower bound for this same quantity. If $$\zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} =\zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i},$$ then from our expression for $Y_{j}$ in Lemma \[lem:3.2aa\] $$\zeta_{4}^{(1)}\frac{\left( r_{1}+s_{1}i \right)^{2}}{2Y_{1}} = \zeta_{4}^{(2)} \frac{\left( r_{2}+s_{2}i \right)^{2}}{2Y_{2}},$$ so $$\frac{\left( r_{1}+s_{1}i \right)^{4}}{4Y_{1}^{2}} = \pm \frac{\left( r_{2}+s_{2}i \right)^{4}}{4Y_{2}^{2}}.$$ From , it follows that $$\left( X_{1} \pm bi \right) Y_{2}^{2} = \pm \left( X_{2} \pm bi \right) Y_{1}^{2}.$$ Comparing the imaginary parts of both sides of this equation, we find that $Y_{1}=Y_{2}$, but this contradicts our assumption that $Y_{2}>Y_{1}$. Let $x+yi= \left( r_{1}-s_{1}i \right) \left( r_{2}+s_{2}i \right)$. We can write $$\left| \zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} - \zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i} \right| = \frac{2\zeta_{4}^{(1)}x-\left( \zeta_{4}^{(1)}+\zeta_{4}^{(2)} \right)( x+yi)} {\left( r_{1}-s_{1}i \right) \left( r_{2}-s_{2}i \right)}.$$ Regardless of the values of $\zeta_{4}^{(1)}$ and $\zeta_{4}^{(2)}$, we always have $(1+i)| \left( \zeta_{4}^{(1)} + \zeta_{4}^{(2)} \right)$. Hence $1+i$ always divides the numerator of the above expression. Also notice that since $2Y_{j}=r_{j}^{2}+s_{j}^{2}$ is even, we have $r_{j} \equiv s_{j} \bmod 2$, so $x$ and $y$ must both be even and $$\left| \zeta_{4}^{(1)} \frac{r_{1}+s_{1}i}{r_{1}-s_{1}i} - \zeta_{4}^{(2)} \frac{r_{2}+s_{2}i}{r_{2}-s_{2}i} \right| \geq \frac{2|1+i|}{\left| {\left( r_{1}-s_{1}i \right) \left( r_{2}-s_{2}i \right)} \right|} =\frac{\sqrt{2}}{\sqrt{Y_{1}Y_{2}}}.$$ Combining this with , we have $$\frac{\sqrt{2}}{\sqrt{Y_{1}Y_{2}}} < 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}}} \left( \frac{1}{Y_{1}^{2}} + \frac{1}{Y_{2}^{2}} \right).$$ From $Y_{2}>Y_{1}$, this immediately gives us $$\frac{\sqrt{2}}{\sqrt{Y_{1}Y_{2}}} < 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}}}\frac{2}{Y_{1}^{2}},$$ so $$Y_{2}>1.997\frac{a^{2}+b^{2}}{b^{2}} Y_{1}^{3}>1.997Y_{1}^{3}.$$ We can use this gap principle to improve its constant term. Applying Lemma \[lem:3.4\](a), $$Y_{2}^{2}>3.988Y_{1}^{6} \geq 2490Y_{1}^{2}$$ yielding $$\frac{\sqrt{2}}{\sqrt{Y_{1}Y_{2}}} < 0.5003 \frac{b}{\sqrt{a^{2}+b^{2}}} \frac{1.0005}{Y_{1}^{2}}$$ and finally $$Y_{2}>7.98\frac{a^{2}+b^{2}}{b^{2}} Y_{1}^{3}.$$ completing our proof. Proof of Theorem \[thm:1.1\] ============================ \[lem:thm\] Let $a$ and $b$ be relatively prime positive integers such that $a^{2}+b^{2}$ is not a perfect square. Suppose $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has a solution and that all coprime integer solutions $(x,y)$ to the quadratic equation $\eqref{eq:quad-eqnc}$ are given by $\eqref{eq:14c}$. Then $\eqref{eq:2}$ has at most one solution, $\left( X, Y \right)$, in coprime positive integers solutions with $Y>b^{2}/2$. Suppose that $\left( X_{1}, Y_{1} \right)$ and $\left( X_{2}, Y_{2} \right)$ are two coprime positive integer solutions to with $Y_{2}>Y_{1}>b^{2}/2$ and that the assumptions in the statement of the lemma hold. From Lemma \[lem:3.4\](a), we obtain $$\label{eq:25} X_{1}^{2}= \left( a^{2}+b^{2} \right) Y_{1}^{4}-b^{2} > \left( a^{2}+b^{2} \right) \left( Y_{1}^{4}-1 \right) > 0.9984 \left( a^{2}+b^{2} \right) Y_{1}^{4},$$ and so $$\label{eq:26} \sqrt{X_{1}^{2}+b^{2}}= \sqrt{\left(a^{2}+b^{2}\right) Y_{1}^{4}} < 1.001 X_{1}.$$ By Lemma \[lem:3.2aa\], there are integers $r_{1},s_{1},r_{2},s_{2}$ such that $$\pm X_{j} \pm bi = (a+bi) \left( r_{j} \pm s_{j} i \right)^{4}, \hspace{1.0mm} Y_{j}=r_{j}^{2}+s_{j}^{2}, \hspace{1.0mm} \gcd \left( r_{j}, s_{j} \right)=1, \hspace{1.0mm} s_{j}>r_{j}>0,$$ for $j=1,2$. We will assume that $$X_{1} \pm bi = (a+bi) \left( r_{1} + s_{1} i \right)^{4}, \quad X_{2} \pm bi = (a+bi) \left( r_{2} + s_{2} i \right)^{4},$$ as the argument is identical in the other three cases. Thus $$(a+bi) \left( r_{j} + s_{j} i \right)^{4} - (a-bi) \left( r_{j} - s_{j} i \right)^{4} =2i{\operatorname{Im}}\left( X_{j} \pm bi \right) = \pm 2bi$$ for $j=1,2$. Applying this for $j=2$ and using our expressions above for $X_{1} \pm bi$ and $Y_{1}$, we have $$\begin{aligned} & & \left( X_{1} \pm bi \right) \left( r_{1} - s_{1} i \right)^{4}\left( r_{2} + s_{2} i \right)^{4} - \left( X_{1} \mp bi \right) \left( r_{1} + s_{1} i \right)^{4}\left( r_{2} - s_{2} i \right)^{4} \\ & = & (a+bi) \left( r_{1}^{2} + s_{1}^{2} \right)^{4} \left( r_{2} + s_{2} i \right)^{4} - (a-bi) \left( r_{1}^{2} + s_{1}^{2} \right)^{4} \left( r_{2} - s_{2} i \right)^{4} \\ & = & \left( r_{1}^{2} + s_{1}^{2} \right)^{4} \left( (a+bi) \left( r_{2} + s_{2} i \right)^{4} - (a-bi) \left( r_{2} - s_{2} i \right)^{4} \right) = 2i \left( r_{1}^{2} + s_{1}^{2} \right)^{4} {\operatorname{Im}}\left( X_{2} \pm bi \right) \\ & = & \pm 2bY_{1}^{4}i.\end{aligned}$$ Letting $$x+yi= \left( r_{1}-s_{1}i \right) \left( r_{2}+s_{2}i \right),$$ we have $$\label{eq:27} |f(x,y)|=\left| \left( X_{1} \pm bi \right) (x+yi)^{4} - \left( X_{1} \mp bi \right)(x-yi)^{4} \right| =2bY_{1}^{4}.$$ Put $$\omega = \frac{X_{1} \pm bi}{X_{1} \mp bi}$$ and let $\zeta_{4}$ be the $4$-th root of unity such that $$\left| \omega^{1/4} - \zeta_{4} \frac{x-yi}{x+yi} \right| = \min_{0 \leq k \leq 3} \left| \omega^{1/4} - e^{2k\pi i/4} \frac{x-yi}{x+yi} \right|.$$ From , our expressions for $x+yi$, $X_{1}$ and $Y_{1}$ and (which implies that $\left| X_{1} \pm bi \right|^{2} =X_{1}^{2}+b^{2}=\left( a^{2}+b^{2} \right)Y_{1}^{4}$), we have $$\left| \omega - \left( \frac{x-yi}{x+yi} \right)^{4} \right| = \frac{2bY_{1}^{4}}{\left| X_{1} \mp bi \right| \left| r_{1} \mp s_{1}i \right|^{4}\left| r_{2} \mp s_{2}i \right|^{4}} =\frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}}$$ By Lemma \[lem:3.4\](a) and Lemma \[lem:3.3\], $Y_{2} >7.98Y_{1}^{3}>7.98 \cdot 5^{3}$. So $Y_{2} \geq 998$. Then $$\left| \omega^{1/4} - \zeta_{4} \frac{x-yi}{x+yi} \right| < \left( 2/Y_{2}^{2} \right)^{1/4} <0.04.$$ Thus we can apply Lemma \[lem:3.2bb\] with $c_{1}=0.04$ to find that $$\label{eq:29} \frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}} = \left| \omega - \left( \frac{x-yi}{x+yi} \right)^{4} \right| > 3.99 \left| \omega^{1/4} - \zeta_{4} \frac{x-yi}{x+yi} \right|.$$ In what follows, we shall require a lower bound for this last quantity. To derive such a bound we shall use the lower bounds in Lemma \[lem:2.1\] with a sequence of good approximations $p_{r}/q_{r}$ obtained from the hypergeometric functions. So we collect here the required quantities. Let $u_{1}=2X_{1}$ and $u_{2}=\pm 2b$. Since $\gcd(a,b)=\gcd \left( X_{1}, Y_{1} \right)=1$, we have $g_{1}=2$. If $ab \equiv 1 \bmod 2$, then $g_{3}=2$, otherwise $g_{3}=4$. Therefore, $g=\sqrt{2}$ and $d=2b^{2}$ if $ab \equiv 1 \bmod 2$, while $g=1$ and $d=4b^{2}$ otherwise. Therefore, $${\mathcal{N}}_{d,n}=2^{\min \left( v_{2}(d)/2, v_{2}(4)+1 \right)} =2^{\min \left( v_{2}(d)/2, 3 \right)}.$$ If $ab$ is odd, then ${\mathcal{N}}_{d,4}=\sqrt{2}$. If $a$ is even and $b$ is odd, then ${\mathcal{N}}_{d,4}=2$. If $b=2$, then ${\mathcal{N}}_{d,4}=4$. If $b=2^{m}$ with $m \geq 2$, then ${\mathcal{N}}_{d,4}=8$. Thus $8 \geq |g|{\mathcal{N}}_{d,4} \geq 2$ and from Lemma \[lem:2.4\] we have $$Q=\frac{e^{1.68} \left| 2X_{1}+2\sqrt{X_{1}^{2}+b^{2}}\right|}{|g|{\mathcal{N}}_{d,4}} < e^{1.68} \left| X_{1}+\sqrt{X_{1}^{2}+b^{2}}\right|.$$ From , we also have $X_{1}<\sqrt{X_{1}^{2}+b^{2}}=\sqrt{a^{2}+b^{2}} Y_{1}^{2}$, so $$\label{eq:Q-UB2} Q<e^{1.68} \cdot 2 \sqrt{a^{2}+b^{2}} Y_{1}^{2} <10.74\sqrt{a^{2}+b^{2}} Y_{1}^{2}.$$ Similarly, we have $$E = \frac{|g|{\mathcal{N}}_{d,4} \left| u_{1} + \sqrt{u_{1}^{2}+u_{2}^{2}} \right|}{{\mathcal{D}}_{4}u_{2}^{2}} > \frac{2 \left| 2X_{1}+2\sqrt{X_{1}^{2}+b^{2}} \right|}{e^{1.68} \cdot 4b^{2}}.$$ From , $$\label{eq:E-LB} E > \frac{\left( 1+\sqrt{0.9984} \right)\sqrt{a^{2}+b^{2}} Y_{1}^{2}}{e^{1.68}b^{2}} > \frac{0.372\sqrt{a^{2}+b^{2}} Y_{1}^{2}}{b^{2}}.$$ By Lemma \[lem:3.4\](a), $Y_{1} \geq 5$, so $E>1$, as required for its use with Lemma \[lem:2.1\]. Also $Q\geq e^{1.68}\left| X_{1}+\sqrt{X_{1}^{2}+b^{2}} \right|/4>1$, again as needed for Lemma \[lem:2.1\]. Recall from that we take $k_{0}=0.89$. Since $\omega = \left( X_{1} \pm bi \right)^{2}/ \left( X_{1}^{2}+b^{2} \right)$, we have $\left| \tan \left( \varphi \right) \right| = 2b/X_{1}$. From $|\varphi| \leq |\tan (\varphi)|$, we can take $$\label{eq:ell-UB} \ell_{0}={\mathcal{C}}_{4,2}|\varphi|=0.4b/X_{1}.$$ From Lemma \[lem:3.4\](a) we have $$X_{1}^{2}=\left( a^{2}+b^{2} \right)Y_{1}^{4}-b^{2} >\left( Y_{1}^{4}-1 \right) b^{2} \geq 624b^{2},$$ which yields $|\varphi| \leq \left| \tan(\varphi) \right|=2b/X_{1}<2\sqrt{1/624}<0.081$. Therefore the condition $|\omega-1|<1$ in Lemma \[lem:2.2\] is satisfied too. Let $p=x-yi$ and $q=x+yi=\left( r_{1}-s_{1}i \right) \left( r_{2}+s_{2}i \right)$. We are now ready to deduce the required contradiction from the assumption that there are two coprime solutions $\left( X_{1}, Y_{1} \right)$ and $\left( X_{2}, Y_{2} \right)$ to with $Y_{2}>Y_{1}>b^{2}/2$. We will consider three cases according to the value of $r_{0}$ defined in Lemma \[lem:2.1\]. [**Case 1:**]{} $r_{0}=1$ and $q_{1}\zeta_{4}p \neq qp_{1}$. In this case, by , we have $$\begin{aligned} \frac{2b}{\sqrt{a^{2}+b^{2}} \, Y_{2}^{2}} &= \left| \omega - \left( \frac{p}{q} \right)^{4} \right| > 3.99 \left| \omega^{1/4} - \zeta_{4} \frac{p}{q} \right| \\ &> \frac{3.99}{2k_{0}Q\sqrt{Y_{1}Y_{2}}} > \frac{3.99}{2 \cdot 0.89 \cdot 10.74 \sqrt{a^{2}+b^{2}} Y_{1}^{2}\sqrt{Y_{1}Y_{2}}},\end{aligned}$$ where the second inequality comes from Lemma \[lem:2.1\](b) applied with $r_{0}=1$ and $|p|=\sqrt{Y_{1}Y_{2}}$, and the last inequality comes and . Thus we obtain $$Y_{2}^{3} < 92b^{2}Y_{1}^{5}.$$ On the other hand, we have $Y_{2}^{3}>508Y_{1}^{9}$ by Lemma \[lem:3.3\], thus we get $Y_{1}^{4}<b^{2}/5$. But Lemma \[lem:3.4\](a) and our assumption that $Y_{1}>b^{2}/2$ imply that $Y_{1}^{4} > 5^{3}b^{2}/2$. Thus we cannot have two solutions with $Y_{2}>Y_{1}>1$ in this case. [**Case 2:**]{} $r_{0}=1$ and $q_{1}\zeta_{4}p= qp_{1}$. From the definitions of $p_{1}$ and $q_{1}$ in , along with Lemma \[lem:2.2\] parts (a) and (e) and $Y_{11,4,1}(\omega)=(3\omega+5)/3$, we have $$\begin{aligned} \left| \omega^{1/4} - \zeta_{4} \frac{p}{q} \right| &= \frac{1}{q_{1}} \left| q_{1}\omega^{1/4} - p_{1} \right| = \left| \frac{N_{d,4,1}}{D_{1,r} Y_{1,4,1}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)} \right| \left| \frac{D_{4,1}}{N_{d,4,1}} R_{1,4,1}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right) \right| \\ &= \left| \frac{N_{d,4,1}}{D_{1,r} Y_{1,4,1}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)} \right| \left| \frac{D_{4,1}}{N_{d,4,1}} (\omega-1)^{3} \frac{(1/4)(5/4)}{2 \cdot 3} {} _{2}F_{1} \left( 7/4, 2; 4; 1-\omega \right) \left( \frac{u_{1}-u_{2}i}{2g} \right) \right| \\ & \geq \left| \frac{N_{d,4,1}}{D_{1,r} Y_{1,4,1}(\omega) \left( \frac{u_{1}-u_{2}i}{2g} \right)} \right| \left| \frac{D_{4,1}}{N_{d,4,1}} (\omega-1)^{3} \frac{(1/4)(5/4)}{2 \cdot 3} \left( \frac{u_{1}-u_{2}i}{2g} \right) \right| \\ & = \left| \frac{5(\omega-1)^{3}}{96Y_{1,4,1}(\omega)} \right| = \left| \frac{5(\omega-1)^{3}}{32(3\omega+5)} \right|.\end{aligned}$$ Since $3\omega+5=\left( 8X_{1} \mp 2bi \right)/\left( X_{1} \mp bi \right)$, we have $$\frac{(\omega-1)^{3}}{(3\omega+5)} =-\frac{4b^{3}i}{\left( 4X_{1} \mp bi \right)\left( X_{1} \mp bi \right)^{2}}.$$ so by , $$\left| \frac{(\omega-1)^{3}}{(3\omega+5)} \right| =\frac{4b^{3}}{\sqrt{16X_{1}^{2}+b^{2}}\left( X_{1}^{2}+b^{2} \right)} > \frac{b^{3}}{\left( a^{2}+b^{2} \right)^{3/2}Y_{1}^{6}}.$$ Therefore, $$\left| \omega^{1/4} - \zeta_{4} \frac{p}{q} \right| \geq \frac{0.156b^{3}}{\left( a^{2}+b^{2} \right)^{3/2}Y_{1}^{6}},$$ so $$\frac{2b}{\sqrt{a^{2}+b^{2}} \, Y_{2}^{2}} > 3.7\frac{0.156b^{3}}{\left( a^{2}+b^{2} \right)^{3/2}Y_{1}^{6}} > \frac{0.62b^{3}}{\left( a^{2}+b^{2} \right)^{3/2}Y_{1}^{6}}.$$ This inequality, along with our gap principle in Lemma \[lem:3.3\] implies that $$\left( \frac{2}{0.62} \right)^{2} \frac{\left( a^{2}+b^{2} \right)^{2}}{b^{4}} Y_{1}^{12}>Y_{2}^{4} >7.98^{4} \frac{\left( a^{2}+b^{2} \right)^{4}}{b^{8}} Y_{1}^{12}.$$ This implies that $$0.0027>\left( \frac{2}{0.62 \cdot 63} \right)^{2} >\frac{\left( a^{2}+b^{2} \right)^{2}}{b^{4}}>1,$$ which is impossible. Hence we cannot have two coprime solutions with $Y_{2}>Y_{1}>b^{2}/2$ in Case 2. [**Case 3:**]{} $r_{0}>1$. Here we establish a stronger gap principle here for $Y_{1}$ and $Y_{2}$ than the one in Lemma \[lem:3.3\]. We then use this to obtain a contradiction with Lemma \[lem:2.1\](a). Here the gap principle is simpler to obtain as we can appeal to the definition of $r_{0}$ in Lemma \[lem:2.1\]. From that definition we have $$|q| \geq E^{r_{0}-1} / \left( 2\ell_{0} \right).$$ Recall too that $q=x+yi$ so that $|q|=\sqrt{Y_{1}Y_{2}}$. Thus $$\sqrt{Y_{1}Y_{2}} \geq E^{r_{0}-1} / \left( 2\ell_{0} \right).$$ From , and , $$\sqrt{Y_{1}Y_{2}} > \frac{X_{1}}{0.8b} \left( \frac{0.372\sqrt{a^{2}+b^{2}} Y_{1}^{2}}{b^{2}} \right)^{r_{0}-1} > \frac{1.248\sqrt{a^{2}+b^{2}}Y_{1}^{2}}{b} \left( \frac{0.372\sqrt{a^{2}+b^{2}} Y_{1}^{2}}{b^{2}} \right)^{r_{0}-1}.$$ Therefore, $$\label{eq:y2UB-case3} Y_{2} > 11.25 \cdot 0.138^{r_{0}} \left( a^{2}+b^{2} \right)^{r_{0}} b^{2-4r_{0}} Y_{1}^{4r_{0}-1}.$$ From equation , Lemma \[lem:2.1\](a) and $|x+yi|=\sqrt{Y_{1}Y_{2}}$, we have $$\label{eq:32} \frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}} > 3.99 \left| \omega^{1/4} - \zeta_{4} \frac{x-yi}{x+yi} \right| > \frac{3.99}{2k_{0}Q^{r_{0}+1}\sqrt{Y_{1}Y_{2}}}.$$ Applying and , we obtain $$\frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}} > \frac{3.99}{2 \cdot 0.89 \left( 10.74\sqrt{a^{2}+b^{2}} Y_{1}^{2} \right)^{r_{0}+1}\sqrt{Y_{1}Y_{2}}},$$ so $$\frac{(3.56b)^{2}}{3.99^{2}} 10.74^{2r_{0}+2} \left( a^{2}+b^{2} \right)^{r_{0}} Y_{1}^{4r_{0}+5} >Y_{2}^{3}.$$ We will simplify this to $$\label{eq:y2LB-case3} 92b^{2} \cdot 116^{r_{0}} \left( a^{2}+b^{2} \right)^{r_{0}} Y_{1}^{4r_{0}+5} >Y_{2}^{3}.$$ We now combine and , obtaining $$92b^{2} \cdot 116^{r_{0}} \left( a^{2}+b^{2} \right)^{r_{0}} Y_{1}^{4r_{0}+5} > 1420 \cdot b^{6-12r_{0}} \cdot 0.0026^{r_{0}} \left( a^{2}+b^{2} \right)^{3r_{0}} Y_{1}^{12r_{0}-3}.$$ Applying the assumption that $Y_{1}>b^{2}/2$ and simplifying, we have $$0.0646 \left( \frac{44,620}{\left( a^{2}+b^{2} \right)^{2}} \right)^{r_{0}} > b^{4-12r_{0}} Y_{1}^{8r_{0}-8} > b^{4r_{0}-12} 2^{-8r_{0}+8}.$$ That is, $$0.000253 \left( \frac{11,423,000}{\left( a^{2}+b^{2} \right)^{2}} \right)^{r_{0}} > b^{4r_{0}-12}.$$ For $r_{0} \geq 2$, we have $$0.000253 \frac{b^{4}}{\left( a^{2}+b^{2} \right)^{4}} \left( 11,423,000 \right)^{2} > 1.$$ This is never satisfied for $a^{2}+b^{2} \geq 181,700$. We could attempt to address the outstanding values of $a$ and $b$ by using the `IntegralQuarticPoints()` function within MAGMA, but the number of equations is quite large. Instead we proceed as follows. Suppose that $Y_{1} \geq \max \left( 1700, b^{2}/2 \right)$. From , $$X_{1}^{2}>0.9984 \left( a^{2}+b^{2} \right) Y_{1}^{4} \geq 0.9984 \left( a^{2}+b^{2} \right) 1700 \left( b^{2}/2 \right)^{3} > 212b^{6}.$$ For such $X_{1}$, we have $$E^{2}> \left( \frac{2 \left| 2X_{1}+2\sqrt{X_{1}^{2}+b^{2}} \right|}{e^{1.68} \cdot 4b^{2}} \right)^{2} >\frac{64X_{1}^{2}}{e^{3.36} \cdot 16b^{4}} >\frac{64 \cdot 212b^{6}}{e^{3.36} \cdot 16b^{4}}>29.4b^{2}.$$ In addition, we have $$\frac{Q}{E}<\left( 10.74\sqrt{a^{2}+b^{2}} Y_{1}^{2} \right) \frac{b^{2}}{0.372\sqrt{a^{2}+b^{2}} Y_{1}^{2}} < 29b^{2}.$$ Thus $$E^{3}>Q.$$ Therefore, $$Q^{r_{0}-2}<E^{3(r_{0}-2)} = E^{-3}E^{3(r_{0}-1)} \leq E^{-3} \left( 2\ell_{0}|q| \right)^{3} =E^{-3} \left( \frac{0.8b}{X_{1}} \sqrt{Y_{1}Y_{2}} \right)^{3}$$ From , we have $$\frac{2b}{\sqrt{a^{2}+b^{2}} Y_{2}^{2}} > \frac{3.99}{2k_{0}Q^{r_{0}+1}\sqrt{Y_{1}Y_{2}}}. =\frac{3.99X_{1}^{3}}{1.78Q^{3}E^{-3} \left( 0.8b \sqrt{Y_{1}Y_{2}} \right)^{3}\sqrt{Y_{1}Y_{2}}}.$$ We saw above that $Q/E<29b^{2}$, so $$\frac{2b}{\sqrt{a^{2}+b^{2}}} >\frac{3.99X_{1}^{3}}{1.78 \cdot 29^{3}b^{6} \left( 0.8b \sqrt{Y_{1}} \right)^{3}\sqrt{Y_{1}}}.$$ Combining this with , we find that $$1.25 \cdot 10^{8} b^{20} Y_{1}^{4}> \left( a^{2}+b^{2} \right) X_{1}^{6} > \left( a^{2}+b^{2} \right) \left( 0.9984 \left( a^{2}+b^{2} \right) Y_{1}^{4} \right)^{3}.$$ Thus $$\label{eq:y1-UB} 1.26 \cdot 10^{8} \frac{b^{20}}{\left( a^{2}+b^{2}\right)^{4}}>Y_{1}^{8}.$$ We then calculated all coprime pairs $(a,b)$ such that $a^{2}+b^{2}<181,700$ and that had a solution $(X,Y)$ with $Y \geq 2$ and either $Y<1700$ or holding. We found 35 such pairs, $(a,b)$. Of these, for the following 12 the negative Pell equation is solvable and there is only one family of solutions of the associated quadratic equation: $(a,b)=(1,1),(1,3),(3,7),(9,7),(11,3)$, $(11,7),(18,43),(19,9)$, $(29,11)$, $(29,17),(31,5),(41,13)$. We solved each of these $12$ equations using MAGMA (version V2.23-9) and its\ `IntegralQuarticPoints()` function. No further coprime solutions were found for any of these equations. Proof of Theorem \[thm:1.1\] ---------------------------- If $\left( X_{1},Y_{1} \right)$ is a coprime positive integer solution of $x^{2} - \left( a^{2}+b^{2} \right)y^{4}=-b^{2}$ with $Y_{1}>1$, then $Y_{1}>b^{2}/2$ by Lemma \[lem:3.4\](b). Therefore, we can apply Lemma \[lem:thm\] to show that there is no other solution $(X,Y)$ with $Y>b^{2}/2$. Thus this theorem holds. Proof of Theorem \[thm:1.2\] ---------------------------- If $\left( X_{1},Y_{1} \right)$ and $\left( X_{2},Y_{2} \right)$ are coprime positive integer solutions of $x^{2} - \left( a^{2}+b^{2} \right)y^{4}=-b^{2}$ with $Y_{2}>Y_{1}>1$, then $Y_{2}>7.98b^{3}$ by Lemma \[lem:3.3\]. Therefore, we can apply Lemma \[lem:thm\] to show that there is no solution $(X,Y)$, other than $\left( X_{2},Y_{2} \right)$, with $Y>b^{2}/2$. Thus this theorem holds. [10]{} S. Akhtari, The Diophantine equation $aX^{4}-bY^{2} = 1$, [*Journal für die reine und angewandte Mathematik*]{} [**630**]{} (2009), 33–57. A. Baker, Rational approximations to $\sqrt[3]{2}$ and other algebraic numbers, [*Quarterly J. Math. Oxford*]{} [**15**]{} (1964), 375–383. Chen Jian Hua, P. Voutier, Complete solution of the diophantine equation $X^{2}+1=dY^{4}$ and a related family of quartic Thue equations, [*J. Number Theory*]{} [**62**]{} (1997), 71–99. J. Liouville, Sur des classes très-étendues de quantités dont la valeur n’est ni algébrique, ni même réductible à des irrationnelles algébriques, [*C. R. Acad. Sci. Paris*]{}, Sér. A 18 (1844) 883–885. W. Ljunggren, Zur Theorie der Gleichung $x^{2}+1=Dy^{4}$, [*Avh. Norsk. Vid. Akad. Oslo*]{} (1942) 1–27. W. Ljunggren, Über die Gleichung $x^{4}-Dy^{2}=1$, [*Arch. Math. Naturv.*]{} 45 (1942), no. 5. W. Ljunggren, Ein Satz über die Diophantische Gleichung $Ax^{2}-By^{4}=C$ ($C=1,2,4$), [*Tolfte Skand. Matemheikerkongressen, Lund, 1953*]{}, 188–194 (1954). W. Ljunggren, On the Diophantine equation $Ax^{4}-By^{2}=C$ ($C=1,4$), [*Math. Scand.*]{} 21 (1967) 149–158. L. J. Slater, [*Generalized Hypgergeometric Functions*]{}, Cambridge Univ. Press, 1966. M. Stoll, P. G. Walsh and P. Yuan, The Diophantine equation $X^{2}-\left( 2^{2m}+1 \right) Y^{4} = -2^{2m}$ II, [*Acta Arith.*]{} [**139**]{} (2009), 57–63. P. M. Voutier, Thue’s Fundamentaltheorem, I: The General Case, [*Acta Arith.*]{} [**143**]{} (2010), 101–144. P. M. Voutier, Thue’s Fundamentaltheorem, II: Further Refinements and Examples, [*J. Number Theory*]{} [**160**]{} (2016) 215–236.

--- abstract: 'We summarize the present status of the predictions of massive star models for the evolution of their surface properties. After discussing luminosity, temperature and chemical composition, we focus on the question whether massive stars may arrive at critical rotation during their evolution, either on the main sequence or in later stages. We find both cases to be possible and briefly discuss observable consequences.' author: - Norbert Langer - Alexander Heger title: The Evolution of Surface Parameters of Rotating Massive Stars --- Introduction ============ Massive main sequence stars are rapid rotators, with equatorial rotation velocities in the range of $100 ... 400\kms$ (Fukuda 1982, Penny 1996, Howarth et al. 1997). It is known since a long time that rotation can affect the stellar interior in several ways. [*Rapid rotation*]{} can reduce the effective gravity in the star, and it produces large scale flows (Eddington 1925). During the evolution, [*differential rotation*]{} occurs in all stars, with the possibility of the occurrence of various local hydrodynamic instabilities (cf. Endal & Sofia 1978, Zahn 1983) and corresponding mixing of chemical elements and angular momentum. Of relevance for massive stars are the shear instability (cf. Maeder 1997), the baroclinic instability (Zahn 1983, Spruit & Knobloch 1984), and the Solberg-H[ø]{}iland and Goldreich-Schubert-Fricke instabilities (cf. Korycansky 1991). Time dependent evolutionary models for massive stars including rotation have been constructed in the past in one dimension, using various degrees of approximation (e.g. Endal & Sofia 1978, Maeder 1987, Langer 1991, Talon et al. 1997, Langer 1998). Today, it is beyond reasonable doubts that the evolution of massive stars is influenced by rotation due to the physical mechanisms mentioned above (cf. also Fliegner et al. 1996). While the principle effects of rotation in the interior of massive stars during their evolution all the way to iron core collapse are described elsewhere (Langer et al. 1997a, Heger et al. 1998a), we concentrate here on observable surface parameters, i.e. (latitudal averages of) luminosity, effective temperature and surface abundances (Sect. 2), and equatorial rotation velocity. In particular, we discuss the question whether massive stars have the potential to evolve their surface to critical rotation, either during core hydrogen burning (Sect. 3) or beyond (Sect. 4). Evolution of luminosity, surface temperature, and abundances ============================================================ Fig. 1 displays the main effects of rotation on the initial position and evolution of massive stars in the HR diagram at the example of 10$\mso$ tracks for various degrees of rotation (cf. Fliegner et al. 1996). First, the centrifugal force reduces the effective gravity in the stellar interior, i.e. the star appears to be less massive. Its luminosity and surface temperature are reduced (von Zeipel 1924, Kippenhahn 1977). The order of magnitude of this effect can be seen comparing the ZAMS positions of the 10$\mso$ tracks. However, during the further evolution of core hydrogen burning the effect of chemical mixing becomes dominant. Shear instability and Eddington-Sweet currents transport chemical elements synthesized in the stellar interior outwards, while the baroclinic instability smoothes out chemical gradients on equipotential surfaces. Due to the transport of helium into the envelope the average mean molecular weight of the star is increased compared to the non-rotating case, leading to much higher luminosities (Kippenhahn and Weigert 1990). The effect on the surface temperature depends on the amount of mixing, i.e. on the degree of rotation. In the extreme case of chemically homogeneous evolution, the stars would evolve to the left of the ZAMS directly towards the helium main sequence (cf. Maeder 1987). However, more typical may be the case of moderate rotation in Fig. 1, which brings the star to cooler surface temperatures than the non-rotating models (cf. also Langer 1991). I.e., the main sequence band may be considerably widened due to rotationally induced mixing, which may make the requirement of “convective core overshooting” (Stothers & Chin 1992, Schroeder et al. 1997) obsolete. In any case, Fig. 1 shows that even on the main sequence the stellar evolutionary track in the HR diagram depends on the initial rotation rate. I.e., rotation does not only have quantitative effects but qualitatively alters fundamental stellar characteristics as isochrones, the initial mass function, and mass-luminosity relations (Langer et al. 1997b). A similar statement holds for the surface composition of massive stars: it is altered stronger for larger initial masses but also for larger initial rotation rates. In principle, all chemical species which are affected by proton captures at core hydrogen burning temperatures can show variations at the surface of rotating stars. However, as shown by Fliegner et al. (1996), the variations of different species do occur at different times. For example, boron is depleted very early during the main sequence evolution, while nitrogen and helium enrichments are achieved only much later. Fliegner et al. use B and N observations in B stars to show that the abundance pattern in massive early type stars (cf. Venn et al. 1996, and references therein) is in fact produced by rotational mixing and not by close binary interaction. =0.8 The time sequence of element abundance alteration is boron depletion, nitrogen enhancement together with carbon depletion, oxygen depletion, helium enhancement, and possibly sodium enhancement. The radionuclide $^{26}$Al may also be transported to the surface of rotating massive main sequence stars. For the effect of rotation on isotopic chemical yields of massive stars see Langer et al. (1997a). Evolution of the rotational velocity during core hydrogen burning ================================================================= The evolution of the surface rotation rate of stars depends on three processes: the expansion or contraction of the star during its evolution, angular momentum redistribution due to the physical processes mentioned in Sect. 1, and the loss of angular momentum at the stellar surface. =0.4 During the main sequence evolution, the radius of massive stars increases by a factor of 2...3. In case the specific angular momentum would remain constant at the surface, the rotational velocity would decrease by that factor. However, according to Zahn (1994), rigid rotation is a good approximation for the angular momentum distribution of massive main sequence stars (however, see Maeder, this volume). In that case, the transport of angular momentum out of the convective core — which increases its density by a factor of 2...3 during core hydrogen burning — supplies angular momentum for the surface layers such that, as net effect, their rotational velocity remains roughly constant (e.g., Packet et al. 1980). However, massive main sequence stars can lose angular momentum through a stellar wind, even in the absence of magnetic fields. The mechanism of this angular momentum loss is sketched in Fig. 2 for the case of rigid rotation; it works in the same way for differentially rotating stars provided that the time scale for angular momentum transport from the core to the surface is shorter than the mass loss time scale. Note that the effect of chemical evolution of the star, which leads to an increase of the stellar radius with time, is neglected in Fig. 2. Since stars of 10...20$\mso$ lose only small amounts of their total mass during core hydrogen burning, they could be spun down only through magnetic winds. However, main sequence mass loss may be substantial for higher initial masses. Examining the evolution of 60$\mso$ stars, Langer (1998) finds that massive main sequence stars may reach the $\Omega$-limit, i.e. the state of critical rotation, with the critical rotational velocity defined as to include the effect of radiation pressure (cf. Langer 1997). The considered stars may reach critical rotation not by spinning up but by a reduction of their critical rotational velocity as they evolve closer to the Eddington limit. It is shown by Langer (1998) that massive main sequence stars may reach the $\Omega$-limit without catastrophic consequences. Only the mass loss rate is increased such that the corresponding angular momentum loss rate (cf. Fig. 2) ensures that the $\Omega$-limit is never exceeded. For a 60$\mso$ star, mass loss rates of the order of $10^{-5}\msoy$ are achieved at the $\Omega$-limit, resulting in a considerable spin-down. As the mass loss will not occur in a spherically symmetric wind but rather in a disk, and since it is unclear whether the stellar radiation can push all lost material to infinity (cf. Owocki & Gayley 1997), stars at the $\Omega$-limit might appear peculiar, perhaps like B\[e\] stars (Zickgraf et al. 1996). Evolution of the rotational velocity beyond core hydrogen exhaustion ==================================================================== During the post main sequence evolution, strong chemical composition and entropy gradients at the location of the hydrogen burning shell source inhibit efficient mixing of angular momentum from the core into the hydrogen-rich envelope. Therefore, the angular momentum evolution of the latter can be — as first approximation — considered as independent of the core evolution (Heger et al. 1998ab). Very massive stars may reach the $\Omega$-limit again immediately after core hydrogen exhaustion. While the opacity peak which brought them close to the Eddington limit on the main sequence is due to metal opacities, the peak due to helium ionisation becomes relevant for $T_{\rm eff} \simle 25\, 000\,$K. Since the stellar evolution proceeds more than hundred times faster in this phase, correspondingly higher mass loss rates have to be expected, with the result of a more eruptive phenomenon, perhaps resembling Luminous Blue Variables (García-Segura et al. 1997). =0.7 Heger & Langer (1998) found that also stars with masses below $\sim 20\mso$ may arrive at the $\Omega$-limit during their post-main sequence evolution. Stars in that mass range may undergo so called blue loops during core helium burning, i.e. excursions from the red supergiant branch into the B star regime. Since blue loops are connected to a decrease of the stellar envelope by roughly a factor of 10, a corresponding spin-up of the star may be expected. However, at the same time the structure of the hydrogen-rich envelope changes from convective to radiative. Heger & Langer showed that the assumption of rapid angular momentum transport in convective regions results in the fact that most of the angular momentum is retained in the convective outer part of the envelope as the mass of this convective part decreases with time while retaining its spatial extent. Consequently, it spins up much more than if angular momentum were conserved locally. Fig. 3 gives an impression of the order of magnitude of the effect. For the example of a 12$\mso$ star, the rotational velocity is increased by a factor of 100 during the blue loop, where only a factor of 10 would be expected from the mere contraction. This is enough to bring the star to critical rotation, and only a mass loss enhancement and a correspondingly high angular momentum loss rate prevents the star from exceeding the $\Omega$-limit. The consequence of this spin-up of contracting red supergiant envelopes is an episode of highly aspherical mass loss. Its consideration may be relevant for the interpretation of B\[e\] stars of rather low luminosity ($\sim 10^4\lso$, cf. Gummersbach et al. 1995), for the circumstellar material around blue supergiants (cf. Brandner et al. 1997) in general, and Supernova 1987A in particular (Woosley et al. 1997), and for post-AGB stars and the formation of bipolar planetary nebulae (García-Segura et al. 1998). This work was supported in part through the Deutsche Forschungsgemeinschaft through grant La 587/15-1. Brandner W., Grebel E.K., Chu Y.-H., Weis, K., 1997, ApJ 475, L45 Eddington A.S., 1925, Observatory 48, 73 Endal A.S., Sofia S., 1978, ApJ 220, 279 Fliegner J., Langer N. & Venn K., 1996, A&A 308, L13 Fukuda I., 1982, PASP 94, 271 García-Segura G., Langer N., Mac Low M.-M., 1997 in [*Luminous Blue Variables: Massive Stars in Transition*]{}, A. Nota, H.J.G.L.M. Lamers, eds, ASP Conf. Ser., p. 332 García-Segura G., Langer N., Różyczka N., Mac Low M.-M., Franco J., 1997, in proc. 6$^{th}$ Texas-Mexico Conference on Astrophysics on [*Astrophysical Plasmas — Near and Far*]{}, eds. S. Torres-Peimbert, R. Dufour, RevMexAA, in press Gummersbach C.A., Zickgraf F.-J., Wolf B., 1995, A&A 302, 409 Heger A., Langer N., 1998, A&A, in preparation Heger A., Woosley S.E., Langer N., 1998a, ApJ, in preparation Heger A., Jeannin L., Langer N., Baraffe I., 1998b, A&A, in press Howarth I.D., Siebert K.W., Hussain G.A.J., Prinja R.K., 1997, M.N.R.A.S. 284, 265 Kippenhahn R., 1977, A&A 58. 267 Kippenhahn R., Weigert A., 1990, Stellar Structure and Evolution, Springer, Berlin Korycansky D.G., 1991, ApJ 381, 515 Langer N., 1991, A&A 243, 155 Langer N., 1997, in [*Luminous Blue Variables: Massive Stars in Transition*]{}, A. Nota, H.J.G.L.M. Lamers, eds, ASP Conf. Ser., p. 83 Langer N., 1998, A&A, in press Langer N., Fliegner J., Heger A., Woosley S.E., 1997a, in [*Nuclei in the Cosmos IV*]{}, ed. M. Wiescher, Nucl. Phys. A, in press Langer N., Heger A., Fliegner J., 1997b, in [Fundamental Stellar Properties]{}, proc. IAU-Symp. 189, T.R. Bedding et al., eds., Kluwer, Dordrecht, p. 343 Maeder A., 1987, A&A 178, 159 Maeder A., 1997, A&A 321, 134 Owocki S.P., Gayley K.G., 1997, in [*Luminous Blue Variables: Massive Stars in Transition*]{}, A. Nota, H.J.G.L.M. Lamers, eds, ASP Conf. Ser., p. 121 Packet W., Vanbeveren D., De Greve J.-P., de Loore C., Sreenivasan S.R., 1980, A&A 82, 73 Penny L.R., 1996, ApJ 463, 737 Schroeder K.-P., Pols O.R., Eggleton P.P., 1997, M.N.R.A.S. 285, 696 Spruit H.C., Knobloch E., 1984, A&A 132, 89 Stothers R.B., Chin C.-W., 1991, ApJ 383, 820 Talon S., Zahn J.-P., Maeder A., Meynet G., 1997, A&A 322, 209 Venn K.A., Lambert D.L., Lemke M., 1996, A&A 307, 849 von Zeipel H., 1924, M.N.R.A.S. 84, 665 Woosley, S.E., Heger, A., Weaver, T.A., Langer, N., 1997, in: [*SN 1987A: Ten years after*]{}, Eds. M.M. Phillips, N.B. Suntzeff, PASP, in press Zahn J.-P., 1983, in proc. 13th Saas-Fee Course, A. Maeder et al., eds., Geneva Observatory Zahn J.-P., 1994, in: [*Evolution of Massive Stars: A Confrontation between Theory and Observations*]{}, D. Vanbeveren et al., eds., Kluwer, Dordrecht, p. 285 Zickgraf F.J., Humphreys R.M., Lamers H.J.G.L.M., Smolinski J., Wolf B., Stahl O., 1996, A&A 315, 510

--- abstract: 'Given a polynomial $\phi$ over a global function field $K/\mathbb{F}_q(t)$ and a wandering base point $b\in K$, we give a geometric condition on $\phi$ ensuring the existence of primitive prime divisors for almost all points in the orbit $\mathcal{O}_\phi(b):=\{\phi^n(b)\}_{n\geq0}$. As an application, we prove that the Galois groups (over $K$) of the iterates of many quadratic polynomials are large and use this to compute the density of prime divisors of $\mathcal{O}_\phi(b)$.' author: - Wade Hindes title: Prime divisors in polynomial orbits over function fields --- [^1] [^2] [Introduction]{} From elliptic divisibility sequences to the Fibonacci numbers, it is an important problem in number theory to prove the existence of “new" prime divisors of an arithmetically defined sequence. For example, such ideas have applications ranging from the undecidability of Hilbert’s $10$th problem [@Poonen], to the classification of certain families of subgroups of finite linear groups [@Feit; @primcycl; @Linear; @trans; @alg]. In this paper, we study the set of prime divisors of polynomial recurrence sequences defined by iteration over global function fields. To wit, let $K/\mathbb{F}_q(t)$ be a finite extension, let $V_K$ be a complete set of valuations on $K$, and let $\mathcal{B}=\{b_n\}\subseteq K$ be any sequence. We say that $v\in V_K$ is a *primitive prime divisor* of $b_n$ if $$v(b_n)>0\;\;\;\text{and}\;\;\; v(b_m)=0\;\; \text{for all}\; 1\leq m\leq n-1.$$ Likewise, we define the *Zsigmondy set of* $\mathcal{B}$ to be $$\mathcal{Z}(\mathcal{B}):=\big\{n\geq1\;\big\vert\; b_n\;\text{has no primitive prime divisors}\big\}.$$ Over number fields, there are numerous results regarding the finiteness (and size) of $\mathcal{Z}(\mathcal{B})$; for example, see [@PrimDiv; @Tucker; @Silv-Ing; @Krieger; @Silv-Vojta; @SilvPrimDiv]. In this paper, we are interested in studying the finiteness of $\mathcal{Z}(\phi,b):=\mathcal{Z}(\mathcal{O}_\phi(b))$, where $\mathcal{O}_\phi(b):=\{\phi^n(b)\}_{n\geq0}$ is the *orbit* of $b\in K$ for $\phi\in K[x]$; here the superscript $n$ denotes iteration (of $\phi$). The key geometric notion, allowing us to use techniques in the theory of rational points on curves over $K$ to study $\mathcal{Z}(\phi,b)$, is the following: Let $\phi\in K(x)$ and let $\ell\geq2$ be an integer. Then we say that $\phi$ is *dynamically $\ell$-power non-isotrivial* if there exists an integer $m\geq1$ such that, $${\label{Curve}} C_{m,\ell}(\phi):=\big\{(X,Y)\in \mathbb{A}^2(\bar{K})\;\big\vert\;Y^\ell=\phi^m(X)=(\underbracket{\phi\circ\phi\dots \circ\phi}_m)(X)\big\}$$ is a non-isotrivial curve [@isotrivial] of genus at least $2$. Similarly, we have the following refined notions of primitive prime divisors and Zsigmondy sets: Let $\phi\in K(x)$, let $b\in K$, and let $\ell$ be an integer. We say that a place $v\in V_K$ is an *$\ell$-primitive prime divisor* for $\phi^n(b)$ if all of the following conditions are satisfied: 1. $v(\phi^n(b))>0$, 2. $v(\phi^m(b))=0$ for all $1\leq m\leq n-1$ such that $\phi^m(b)\neq0$, 3. $v(\phi^n(b))\not\equiv0{\ (\textup{mod}\ \ell)}$. Moreover, we call $${\label{Zig}} \mathcal{Z}(\phi,b,\ell):=\big\{n\;\big\vert\; \phi^n(b)\; \text{has no $\ell$-primitive prime divisors}\big\}$$ the *$\ell$-th Zsigmondy set* for $\phi$ and $b$. Note that $\mathcal{Z}(\phi,b)\subseteq\mathcal{Z}(\phi,b,\ell)$ for all $\ell$. Hence, it suffices to show that $\mathcal{Z}(\phi,b,\ell)$ is finite for a single $\ell$ to ensure that all but finitely many elements of $\mathcal{O}_\phi(b)$ have primitive prime divisors. Moreover, we use the notions of height $h_K$ and canonical height $\hat{h}_\phi$ found in [@Baker]. [\[PrimDivThm\]]{} Suppose that $\phi\in K[x]$, $b\in K$, and $\ell\geq2$ satisfy the following conditions: 1. $\phi$ is dynamically $\ell$-power non-isotrivial, 2. $b$ is wandering (i.e. $\hat{h}_\phi(b)>0$). Then $\mathcal{Z}(\phi,b,\ell)$ and $\mathcal{Z}(\phi,b)$ are finite. In particular, all but finitely many elements of $\mathcal{O}_\phi(b)$ have primitive prime divisors. In addition to determining whether or not a sequence has primitive prime divisors, it is interesting to compute the “size" of its set of prime divisors (in terms of density) [@Hasse; @Lagarias], a problem which has applications to the dynamical Mordell-Lang conjecture [@Mordell-Lang] and to questions regarding the size of hyperbolic Mandelbrot sets [@RafeThesis]. To do this, let $\mathcal{O}_K$ be the integral closure of $\mathbb{F}_q[t]$ in $K$ and let $\mathfrak{q}\subseteq \mathcal{O}_K$ be a prime ideal, determining a valuation $v_\mathfrak{q}$ on $K$. For such $\mathfrak{q}$, define the *norm* of $\mathfrak{q}$ to be $N(\mathfrak{q}):=\#(\mathcal{O}_K/\mathfrak{q}\mathcal{O}_K)$, and let $\delta(\mathcal{P})$ be the *Dirchlet density* of a set of primes $\mathcal{P}$ of $K$: $$\delta(\mathcal{P}):=\lim_{s\rightarrow 1^+}\frac{\sum_{\mathfrak{q}\in\mathcal{P}}N(\mathfrak{q})^{-s}}{\sum_{\mathfrak{q}}N(\mathfrak{q})^{-s}}$$ We use Theorem \[PrimDivThm\] and ideas from the Galois theory of iterates to compute the density of $$\mathcal{P}_\phi(b):=\big\{\mathfrak{q}\in{\operatorname{Spec}}(\mathcal{O}_K)\;\big\vert\; v_\mathfrak{q}(\phi^n(b))>0\;\text{for some $n\geq0$}\big\},$$ the set of prime divisors of the orbit $\mathcal{O}_\phi(b)$. In particular, we establish a version of [@R.Jones Conj. 3.11]; see [@uniformity Theorem 1] for the corresponding statement in characteristic zero (with uniform bounds) and [@B-J; @R.Jones] for introductions to dynamical Galois theory. [\[Galois\]]{} Let $K/\mathbb{F}_q(t)$ for some odd $q$ and let $\phi\in K[x]$ be a quadratic polynomial.\ Write $\phi(x)=(x-\gamma)^2+c$ and suppose that $\phi$ satisfies the following conditions: 1. $\phi$ is not post-critically finite (i.e. $\gamma$ is wandering), 2. the adjusted critical orbit $\widebar{\mathcal{O}}_\phi(\gamma)=\{-\phi(\gamma),\phi^n(\gamma)\}_{n\geq2}$ contains no squares in K, 3. the $j$-invariant of the elliptic cure $E_\phi: Y^2=(X-c)\cdot\phi(X)$ is non-constant. Then all of the following statements hold: 1. $\mathcal{Z}(\phi,b,2)$ is finite for all wandering points $b\in K$, 2. $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup, 3. $\delta(\mathcal{P}_\phi(b))=0$ for all $b\in K$. One expects similar statements to hold for $\phi(x)=x^\ell+c$ and $\ell$ a prime. Namely, if one can show that $\phi$ is dynamically $\ell$-power non-isotrivial, then $\mathcal{Z}(\phi,0,\ell)$ is finite by Theorem \[PrimDivThm\] and $G_\infty(\phi)$ is a finite index subgroup of ${\operatorname{Aut}}(T(\phi))$ by [@Eventually Theorem 25]. In particular, we apply Theorem \[PrimDivThm\] and Corollary \[Galois\] to the explicit family $$\phi_{(f,g)}(X)=\big(X-f(t)\cdot g(t)^d\big)^2+g(t)\;\;\;\;\text{for}\;\;\; f\in\mathbb{F}_q(t),-g\notin(\mathbb{F}_q(t))^2,\;\text{and}\;d\geq1.$$ Note that by letting $f=0$ and $g=t$, we recover the main result of [@RafeThesis]. [\[eg\]]{} Let $K=\mathbb{F}_q(t)$ and let $\phi:=\phi_{(f,g)}$ be such that the $j$-invariant of the elliptic curve $$E_{\phi_{(f,g)}}: Y^2=(X-g(t))\cdot \phi_{(f,g)}(X)$$ is non-constant. Then $\mathcal{Z}(\phi,b,2)$ is finite for all wandering base points, $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup, and $\delta(\mathcal{P}_\phi(b))=0$ for all $b\in K$. [Primitive Prime Divisors and Superelliptic Curves]{} To prove Theorem \[PrimDivThm\], we build on our techniques from the characteristic zero setting [@uniformity]. There, among other things, we prove the uniform bound $$\#\mathcal{Z}(\phi,\gamma,2)\leq17$$ for all $\phi(x)=(x-\gamma(t))^2+c(t)$ satisfying $\deg(\gamma)\neq \deg(c)$; here $\gamma$ and $c$ are polynomials with coefficients in any field of characteristic zero. Additionally, we adapt ideas from [@Tucker] and use linear height bounds for rational points on non-isotrivial curves [@Kim] to prove our results. Throughout the proof, it will be useful to consider only $\ell$-primitive prime divisors avoiding some finite subset $S\subseteq V_K$. Therefore, we make the following convention: $$\mathcal{Z}(\phi,b,S,\ell):=\big\{n\;\big\vert\; \phi^n(b)\; \text{has no $\ell$-primitive prime divisors}\; v\in V_K{\mathbin{\fgebackslash}}S\big\}.$$ Clearly $\mathcal{Z}(\phi,b,\ell)\subseteq\mathcal{Z}(\phi,b,\ell,S)$ for all $S$, and so it suffices to show that $\mathcal{Z}(\phi,b,\ell,S)$ is finite for some $S\subseteq V_K$. Note that there is no harm in enlarging $S$. In particular, we may assume that $$\text{(a)}.\;\;\;b\in\mathcal{O}_{K,S}\;\;\;\;\;\; \text{(b)}.\;\;\;\phi\in\mathcal{O}_{K,S}[x]\;\;\;\;\;\;\text{(c)}.\;\;\; v(a_d)=0\;\;\text{for all}\; v\notin S\;\;\;\;\;\; \text{(d)}.\;\;\; \mathcal{O}_{K,S}\;\text{is a UFD,}$$ where $a_d$ is the leading term of $\phi$. Note that condition (d) is made possible by [@Rosen Prop. 14.2 ]. Similarly, we see that $$\mathcal{Z}(\phi,b,S,\ell)\subseteq\mathcal{Z}(\phi,\phi^n(b),S,\ell)\cup\{t\in\mathbb{Z}\;\vert\; 1\leq t\leq n\}\;\;\,\text{for all}\;n\geq0.$$ Therefore, after replacing $b$ with $\phi^n(b)$ for some $n$, we may assume that $0\notin\mathcal{O}_\phi(b)$. By the assumptions on $S$ above, we see that $\phi^n(b)\in\mathcal{O}_{K,S}$ for all $n$, permitting us to write $${\label{decomp}} \phi^n(b)=u_n\cdot d_n\cdot y_n^\ell,\;\;\text{for some}\;\;\; d_n,y_n\in\mathcal{O}_{K,S}\;,\;u_n\in\mathcal{O}_{K,S}^*.$$ However, since $\mathcal{O}_{K,S}^*$ is a finitely generated group [@Rosen Prop. 14.2 ], we may write $u_n=\textbf{u}_1^{r_1}\cdot \textbf{u}_2^{r_2}\dots \textbf{u}_t^{r_t}$ for some basis $\{\textbf{u}_i\}$ of $\mathcal{O}_{K,S}^*$ and some integers $0\leq r_i\leq\ell-1$. In particular, the height $h_K(u_n)$ is bounded independently of $n\geq0$. Similarly, we may assume that $0\leq v(d_n)\leq\ell-1$ for all $v\notin S$. To see this, we use the correspondence $V_K{\mathbin{\fgebackslash}}S\longleftrightarrow{\operatorname{Spec}}(\mathcal{O}_{K,S})$ discussed in [@Rosen Ch. 14] and the fact that $\mathcal{O}_{K,S}$ is a UFD to write $$d_n=p_1^{e_1}\cdot p_2^{e_2}\cdots p_s^{e_s}\big(p_1^{q_1}\cdot p_2^{q_2}\cdots p_s^{q_s}\big)^\ell,\;\;\;\;\; p_i\in{\operatorname{Spec}}(\mathcal{O}_{K,S})$$ for some integers $e_i, q_i$ satisfying $v_{p_i}(d_n)=q_i\cdot\ell+e_i$ and $0\leq e_i<\ell$. In particular, by replacing $d_n$ with $\big(p_1^{e_1}\cdot p_2^{e_2}\cdots p_s^{e_s}\big)$ and $y_n$ with $\big(y_n\cdot p_1^{q_1}\cdot p_2^{q_2}\cdots p_s^{q_s}\big)$, we may assume that $0\leq v(d_n)\leq\ell-1$ for all $v\in V_K{\mathbin{\fgebackslash}}S$ as claimed. Now suppose that $n\in Z(\phi,b,S,\ell)$. It is our goal to show that $n$ is bounded. To do this, first note that conditions (b) and (c) imply that $\phi$ has good reduction (see [@Silv-Dyn Thm. 2.15]) modulo the primes in ${\operatorname{Spec}}(\mathcal{O}_{K,S})$. In particular, if $p$ is such that $v_{p}(d_n)>0$ and $n\in Z(\phi,b,S,\ell)$, then $v_p(\phi^m(b))>0$ for some $1\leq m\leq n-1$. Moreover, $$\phi^{n-m}(0)\equiv\phi^{n-m}(\phi^{m}(b))\equiv\phi^n(b)\equiv0{\ (\textup{mod}\ p)};$$ see [@Silv-Dyn Thm. 2.18]. Therefore, we have the refinement $${\label{refinement}} d_n=\prod p_i^{e_i},\;\;\text{where}\;\; p_i\big\vert\phi^{t_i}(b)\;\text{or}\; p_i\big\vert\phi^{t_i}(0)\;\; \text{for some}\; 1\leq t_i\leq\Big\lfloor \frac{n}{2}\Big\rfloor.$$ Moreover, as noted above, we may assume that $0\leq e_i\leq\ell-1$. Hence $${\label{htestimate}} \boxed{h_K(d_n)\leq (\ell-1)\cdot\bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^i(b))+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^j(0))\bigg)}$$ Now, choose (and fix) an integer $m\geq1$ such that the curve $$C_{m,\ell}(\phi): Y^\ell=\phi^m(X)$$ is nonsingular and of genus at least two - possible, since $\phi$ is dynamically $\ell$-power non-isotrivial. If $n\leq m$ for all $n\in\mathcal{Z}(\phi,b,\ell,S)$, then we are done. Otherwise, we may assume that $n>m$ so that (\[decomp\]) implies that $$P_n:=\big(\phi^{n-m}(b)\;,\;y_n\cdot\sqrt[\ell]{u_n\cdot d_n}\, \big)\in C_m(\phi)\big(K\big(\sqrt[\ell]{u_n\cdot d_n}\big)\big)\;\;.$$ It follows from any of the bounds (suitable to positive characteristic) discussed in the introductions of [@Kim] or [@htineq] that there exist constants $A_1, A_2>0$ such that $${\label{Szpiro}} h_{\kappa(\phi,m)}(P_n)\leq A_1\cdot d(P_n)+A_2,$$ where $\kappa(\phi,m)$ is the canonical divisor of $C_{m,\ell}(\phi)$, $$d(P_n):=\frac{2\cdot {\operatorname{genus}}\big(K_n\big)-2}{\big[K_n:K\big]},\;\;\;\;\text{and}\;\;\;K_n:=K\big(\sqrt[\ell]{u_n\cdot d_n}\,\big).$$ Crucially, the bounds $A_i=A_i(\phi,\ell,m)$ are independent of both $b$ and $n$. We note that the bounds on (\[Szpiro\]) have been improved by Kim [@Kim Theorem 1], although we do not need them here. On the other hand, it follows from [@ffields Prop. III.7.3 and Remark III.7.5] that $d(P_n)\leq B_1\cdot h_K(u_n\cdot d_n)+B_2$ for some positive constants $B_i=B_i(K)$. Likewise, $h_K(u_n\cdot d_n)\leq h_K(d_n)+B_3(K,S,\ell)$, since the height of $u_n$ is absolutely bounded. In particular, after combining these bounds with those on (\[Szpiro\]), we see that $${\label{combin1}} h_{\kappa(\phi,m)}(P_n)\leq D_1\cdot h_K(d_n)+D_2, \;\;\;\;\;\;\; D_i=D_i(K,\phi,S,\ell,m)>0.$$ However, if $\mathcal{D}_1$ is an ample divisor on $C_{m,\ell}(\phi)$ and $\mathcal{D}_2$ is an arbitrary divisor, then $${\label{Divisor}} \lim_{h_{\mathcal{D}_1}(P)\rightarrow\infty}\frac{h_{\mathcal{D}_2}(P)}{h_{\mathcal{D}_1}(P)}=\frac{\deg\mathcal{D}_2}{\deg{\mathcal{D}_1}},\;\;\;\;\;\;P\in C_{m}(\phi);$$ see [@SilvA Thm III.10.2]. In particular, if $\pi:C_{m,\ell}(\phi)\rightarrow\mathbb{P}^1$ is the covering $\pi(X,Y)=X$, then a degree one divisor on $\mathbb{P}^1$ (giving the usual height $h_K$ on projective space) pulls back to a $\deg(\pi)$ divisor $\mathcal{D}_2$ on $C$ satisfying $h_{\mathcal{D}_2}(P)=h_K(\pi(P))$. We deduce from (\[Divisor\]) that there exist constants $\beta$ and $E=E(\phi,m,\ell)$ such that $${\label{limit}} h_{\kappa(\phi,m)}(P)>\beta\;\;\;\;\;\text{implies}\;\;\;\;\; h_K(\pi(P))\leq E\cdot h_{\kappa(\phi,m)}(P)+1$$ for all $P\in C_m(\phi)(\bar{K})$. However, note that $${\label{finiteness}} T:=\big\{P_n\;\big\vert\; h_{\kappa(\phi,m)}(P_n)\leq\beta\}\subseteq\big\{P\in C_m(\phi)(\bar{K})\;\big\vert\;\; h_{\kappa(\phi,m)}(P)\leq\beta\;\;\,\text{and}\,\;\; [K(P):K]\leq\ell\big\},$$ and the latter set is finite, since $\kappa(\phi,m)$ is ample; see [@SilvA Thm. 10.3]. Hence, (\[htestimate\]),(\[combin1\]), (\[limit\]), and (\[finiteness\]) together imply that $${\label{combin2}} h_K(\phi^{n-m}(b))\leq F_1\cdot \bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^i(b))+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}h_K(\phi^j(0))\bigg)+F_2$$ for all but finitely many $n\in Z(\phi,b,S,\ell)$ and some positive constants $F_i=F_i(K,\phi,S,\ell,m)$. On the other hand, it is well known that the canonical height $\hat{h}_\phi$ satisfies: $$\text{(a).}\;\;\;\hat{h}_\phi=h_K+O(1)\;\;\;\;\;\;\;\;\;\;\;\; \text{(b).}\;\;\;\hat{h}_\phi(\phi^s(\alpha))=d^s\cdot \hat{h}_\phi(\alpha)$$ for all $\alpha\in K$ and all integers $s\geq0$; see [@Silv-Dyn Thm. 3.20]. In particular, (\[combin2\]) implies that $${\label{combin3}} d^{n-m}\cdot\hat{h}_\phi(b)\leq G_1\cdot\bigg(\sum_{i=1}^{\lfloor\frac{n}{2}\rfloor}d^i\cdot \hat{h}_\phi(b)+ \sum_{j=1}^{\lfloor\frac{n}{2}\rfloor}d^j\cdot\hat{h}_\phi(0) \bigg)+G_2\cdot n+G_3$$ for almost all $n\in Z(\phi,b,S,\ell)$ (those $n$ such that $P_n\notin T$) and some constants $G_i=G_i(K,\phi,S,\ell,m)$. In particular, for almost all $n\in Z(\phi,b,S,\ell)$, $$d^{n-m}\leq G\cdot \big(d^{\lfloor\frac{n}{2}\rfloor+1}+n+1\big),\;\;\;\;\text{where}\;\;G:=\max\bigg\{G_1,\frac{G_1\cdot \hat{h}_\phi(0)}{\hat{h}_\phi(b)},\frac{G_2}{\hat{h}_\phi(b)},\frac{G_2}{\hat{h}_\phi(b)}\bigg\}.$$ However, since $m$ is fixed, such $n$ are bounded. [Dynamical Galois Groups]{} Our main interest in proving the finiteness of $\mathcal{Z}(\phi,b,\ell)$ comes from the Galois theory of iterates. In particular, if $\phi(x)=(x-\gamma)^2+c$ is a quadratic polynomial and $\mathcal{Z}(\phi,\gamma,2)$ is finite, then the Galois groups of $\phi^n$ are large [@RafeThesis Theorem 3.3], enabling us to compute the density of prime divisors $\mathcal{P}_\phi(b)$ (for all $b\in K$) via a suitable Chebotarev density theorem [@Eventually; @Jones]. To define the relevant dynamical Galois groups, let $\phi$ be a polynomial and assume that $\phi^n$ is separable for all $n\geq1$; hence, the set $T_n(\phi)$ of roots of $\phi, \phi^2,\dots ,\phi^n$ together with $0$, carries a natural $\deg(\phi)$-ary rooted tree structure: $\alpha,\beta\in T_n(\phi)$ share an edge if and only if $\phi(\alpha) =\beta$ or $\phi(\beta)=\alpha$. Furthermore, let $K_n:=K(T_n(\phi))$ and $G_n(\phi):={\operatorname{Gal}}(K_n/K)$. Finally, we set $${\label{Arboreal}} T_\infty(\phi):=\bigcup _{n \geq 0} T_n(\phi)\;\;\text{and}\;\; G_\infty(\phi)=\varprojlim G_n(\phi).$$ Since $\phi$ is a polynomial with coefficients in $K$, it follows that $G_n(\phi)$ acts via graph automorphisms on $T_n(\phi)$. Hence, we have injections $G_n(\phi) \hookrightarrow {\operatorname{Aut}}(T_n(\phi))$ and $G_\infty(\phi) \hookrightarrow {\operatorname{Aut}}(T_\infty(\phi))$ called the *arboreal representations* associated to $\phi$. A major problem in dynamical Galois theory, especially over global fields, is to understand the size of $G_\infty(\phi)$ in ${\operatorname{Aut}}(T_\infty(\phi))$; see [@B-J; @Me; @R.Jones; @Odoni; @Stoll]. We now use Theorem \[PrimDivThm\] to prove a finite index result for many quadratic polynomials (c.f. [@uniformity Theorem 1]), including the family defined in Corollary \[eg\], and provide an outline for further examples. Let $\phi(x)=(x-\gamma)^2+c$ and let $m\geq2$. Then we have a map $${\label{map}} \Phi_m:C_{2,m}(\phi)\rightarrow E_\phi,\;\;\;\; \Phi(x,y)=\big(\phi^{m-1}(x),\;y\cdot(\phi^{m-2}(x)-\gamma)\big).$$ It follows from Proposition \[prop\] below that $C_{2,m}$ is non-isotrivial. On the other hand, since $\widebar{\mathcal{O}}_\phi(\gamma)$ contains no squares in $K$, [@Jones Proposition 4.2] implies that $\phi^n$ is an irreducible polynomial over $K$ for all $n$; hence, $C_{2,m}(\phi)$ and $E_\phi$ are non-singular; see [@Jones Lemma 2.6]. Therefore, we may choose $m$ so that $C_{2,m}(\phi)$ is a non-isotrivial curve of genus at least $2$. In particular, $\phi$ is dynamically 2-power non-isotrivial and Theorem \[PrimDivThm\] implies that $\mathcal{Z}(\phi,b,2)$ is finite for all wandering $b\in K$. For the second claim, we apply this fact to $b=\gamma$ and use [@RafeThesis Theorem 3.3] to deduce that $G_\infty(\phi)\leq{\operatorname{Aut}}(T(\phi))$ is a finite index subgroup. Finally, [@R.Jones Theorem 4.2] an [@RafeThesis Theorem 1.3] imply that the density of $\mathcal{P}_\phi(b)$ is zero for all $b\in K$. We now apply Corollary \[Galois\] to the family $\phi_{(f,g)}$ defined in Corollary \[eg\]. Let $K=\mathbb{F}_q(t)$ and let $\phi(x)=\phi_{(f,g)}=(x-f(t)\cdot g(t)^d)^2+g(t)$. It suffices to check conditions (a) and (b) of Corollary \[Galois\] hold to prove Corollary \[eg\]. We first show that the adjusted critical orbit of $\phi$, the set $\{-\phi(f\cdot g^d),\phi^2(f\cdot g^d),\phi^3(f\cdot g^d),\dots\}$, contains no perfect squares in $K$; in particular, $\phi^n$ is an irreducible polynomial over $K$ for all $n\geq1$; see [@Jones Proposition 4.2]. Note that $-\phi(f\cdot g^d)=-g$ is not a square in $K$ by assumption. On the other hand, we let $h:=g-f\cdot g^d$ and suppose that $${\label{iterate}} j^2=\phi^n(f\cdot g^d)=((((h^2+h)^2+h)^2+h)^2+\dots+h)^2+g$$ for some polynomial $j\in \mathbb{F}_q[t]$ and some $n\geq2$. Hence, $j^2=g^2\cdot k^2+g=g\cdot(g\cdot k^2+1)$ for some $k\in \mathbb{F}_q[t]$, since $g\vert h$. However, because $\mathbb{F}_q[t]$ is a UFD and $g$ and $g\cdot k^2+1$ are coprime, it follows that $g=l^2$ and $g\cdot h^2+1=m^2$ for some $l,m\in \mathbb{F}_q[t]$. In particular, $1=(m+l\cdot h)(m-l\cdot h)$ and both factors are constant. Hence, $2m=(m+l\cdot h)+(m-l\cdot h)$ is also constant. Finally, since $m^2-1=g\cdot h^2$ and $h=g-f\cdot g^d$, it follows that $g, h$, and $f$ are all constant. In particular, the $j$-invariant of $E_\phi$ is constant, a contradiction. On the other hand, the right hand side of (\[iterate\]) implies that $$\deg(\phi^n(f\cdot g^d))\leq\max\big\{2^{n-1}\cdot\deg(h),\deg(g)\big\}=\max\big\{2^{n-1}\cdot[\deg(g)+\deg(1-f\cdot g^{d-1})],\deg(g)\big\},$$ with equality if the terms are unequal. In particular, if $\deg(h)\neq 0$, then we may choose $n$ large enough so that $\deg(\phi^n(f\cdot g^d))=2^{n-1}\cdot\deg(h)\geq 2^{n-1}$, and $\phi$ is post-critically infinite. Otherwise, we may assume that $h$ is constant. Consequently, since $h=g\cdot(1-f\cdot g^{d-1})$, we deduce that either $g$ is constant or $1-f\cdot g^{d-1}=0$. However, $g$ and $h$ constant implies that $f$ is constant, a contradiction. We deduce that $1-f\cdot g^{d-1}=0$. In particular, $g=f\cdot g^d$ and $\phi(x)=(x-g)^2+g$. In this case the elliptic curve $$E_\phi:\;Y^2=(X-g)\cdot\phi(X)=(X-g)\cdot\big((X-g)^2+g)$$ has $j$-invariant $1728$, contradicting our assumption that $E_\phi$ have non-constant $j$-invariant. We expect that most $\phi\in K[x]$ are dynamically $\ell$-power non-isotrivial for some $\ell$, and we state a conjecture along these lines: Suppose that $\phi\in K[x]{\mathbin{\fgebackslash}}\widebar{\mathbb{F}}_q[x]$ satisfies the following conditions: 1. $\deg(\phi)\geq2$, 2. $\phi$ is not conjugate to $x^d$ (as a function on $\mathbb{P}^1$) 3. $\gcd(\deg(\phi),q)=1$, 4. $0\notin \mathcal{O}_\phi(\gamma)$ for all critical points $\gamma\in\bar{K}$ of $\phi$. Then there exists $\ell\geq2$ such that $\phi$ is dynamically $\ell$-power non-isotrivial and $\gcd(\ell,q)=1$. Conditions (3) and (4) imply that the discriminant of $\phi^n$ is non-zero for all $n$; see [@Jones Lemma 2.6]. In particular, the curves $C_{m,\ell}(\phi)$ are non-singular and eventually of large genus, since $\deg(\phi^m)$ grows exponentially by condition (1). We finish by proving an auxiliary result used in the proof of Corollary \[eg\]. The argument below was suggested by Bjorn Poonen. [\[prop\]]{} Let $\Phi: C_1\rightarrow C_2$ be a non-constant morphism. If $C_1$ is isotrivial, then $C_2$ is isotrivial. Suppose not. That is, suppose that $C_1$ is isotrivial but $C_2$ is not. By replacing $K$ by a finite extension, we may assume that $C_1$ is constant. We may also assume that $\Phi$ is separable (if not, then it factors as a power of Frobenius composed with a separable morphism, say $C_1 \rightarrow C_1' \rightarrow C_2$, and then (maybe after a finite extension) $C_1'$ is isomorphic to the curve obtained by taking the $p^n$-roots of all the coefficients of $C_1$, so $C_1'$ is constant too, and is separable over $C_2$; rename $C_1'$ as $C_1$). We fix some notation. Let $g_i$ be the genus of $C_i$, and let $J_i$ be the Jacobian of $C_i$. Since $C_2$ is non-isotrivial, $g_2>0$. Let $\Omega$ be an uncountable algebraically closed field containing $F_q$. Specializing (by choosing an $F_q$-homomorphism $\sigma: K \rightarrow \Omega$) gives separable maps over $\Omega$ from the same $C_1$ to uncountably many non-isomorphic curves $C_2^\sigma$. Each isogeny class of an elliptic curve over $\Omega$ consists of at most countably many elliptic curves, so if $g_2=1$, this would imply that in the decomposition of $J_i$ up to isogeny, uncountably many isogeny factors occur, which is impossible. If $g_2>1$, then the infinitely many maps from $C_1$ to various curves $C_2^\sigma$ contradict [@Samuel Theorem 2]. With some additional hypothesis, Proposition \[prop\] holds for projective varieties of arbitrary dimension [@isotrivial]. Such a result is necessary if one hopes to generalize Theorem \[PrimDivThm\] as in [@Silv-Vojta] **Acknowledgements:** It is a pleasure to thank Joseph Silverman, Bjorn Poonen, Felipe Voloch, and Rafe Jones for the discussions related to the work in this paper. [13]{} M. Baker. *A finiteness theorem for canonical heights attached to rational maps over function fields.* Journal für die reine und angewandte Mathematik (Crelles Journal) 2009.626 (2009): 205-233. R. Benedetto, D. Ghioca, P. Kurlberg, and T. Tucker. *A case of the dynamical Mordell-Lang conjecture*. Math. Ann. 352(1): 1-26, (2012). A. Bhatnagar, *Projective varieties covered by isotrivial families,* Proceedings of the American Mathematical Society 142.5 (2014): 1561-1566. Y. Bilu, G. Hanrot, P.M. Voutier, *Existence of primitive divisors of Lucas and Lehmer numbers*, J. Reine Angew. Math. 539: 75-122 (2001). W. Bosma, J. Cannon and C. Playoust, *The Magma algebra system 1: The user language*, J. Symbolic Comput, **24** (1997): 235-265. N. Boston and R. Jones, *The image of an arboreal Galois representation.* Pure and Applied Mathematics Quarterly 5(1) (Special Issue: in honor of Jean-Pierre Serre, Part 2 of 2): 213-225, (2009). W. Feit. *On large Zsigmondy primes*. Proc. Amer. Math. Soc. 102: 29–36, (1988). S.P. Glasby, F. Lübeck, A. Niemeyer, and C. Praeger, *Primitive prime divisors and the $n$-th cyclotomic polynomial*, J. Aust. Math. Soc. 0: 1-13, (2015). R. Guralnick, T. Penttila, C. Praeger and J. Saxl. *Linear groups with orders having certain large prime divisors*. Proc. London Math. Soc. 78 (1999), 167–214. C. Gratton, K. Nguyen, and T. Tucker. *ABC implies primitive prime divisors in arithmetic dynamics.* Bull. London Math. Soc. 45: 1194-1208, (2013). S. Hamblen, R. Jones, and K. Madhu, *The density of primes in orbits of $z^ d+ c$*. Int. Math. Res. (2014): rnt349. H. Hasse, *Über die Dichte der Primzahlenp, für die eine vorgegebene ganzrationale Zahl $a\neq0$ von gerader bzw. ungerader Ordnung mod. p ist,* Math. Annalen 166.1 (1966): 19-23. C. Hering, *Transitive linear groups and linear groups which contain irreducible subgroups of prime order*, Geometriae Dedicata 2 (1974), 425–460. W. Hindes, *Galois uniformity in quadratic dynamics over $k(t)$*, J. Number Theory, 148 (2015): 372-383. W.Hindes, *The Vojta conjecture implies Galois rigidity in dynamical families*, arXiv:1412.8206. P. Ingram, J. Silverman, *Uniform estimates for primitive divisors in elliptic divisibility sequences*, Number theory, Analysis and Geometry, Springer-Verlag, 2010, 233-263. R. Jones, *The Density of prime divisors in the arithmetic dynamics of quadratic polynomials.* J. Lond. Math. Soc. 78.2 (2008): 523-544. R. Jones, *Galois representations from pre-image trees: an arboreal survey,* Pub. Math. Besancon, 107-136, (2013). R. Jones, *Galois towers, their associated martingales, and the p-adic Mandelbrot set,* Compos. Math., 143.5 (2007): 1108-1126. M. Kim, *Geometric height inequalities and the Kodaira-Spencer map*. Compositio Math. 105.1: 43-54 (1997). H. Krieger. *Primitive Prime Divisors in the Critical Orbit of $z^d+ c$.* International Mathematics Research Notices (2012): rns213. J. Lagarias, *The set of primes dividing the Lucas numbers has density 2/3.* Pacific J. Math 118.2 (1985): 449-461. A. Moriwaki, *Geometric height inequality on varieties with ample cotangent bundles*, J. Algebraic Geom., 4.2: 385-396, (1995). A. Niemeyer and C. Praeger, *A recognition algorithm for classical groups over finite fields*, Proc. London Math. Soc. 77 (1998), 117–169. R. W. K. Odoni. *The Galois theory of Iterates and Composites of Polynomials*. Proc. London Math. Soc., 51.3 (1985): 385-414. B. Poonen. *Using elliptic curves of rank one towards the undecidability of Hilbert’s Tenth Problem over rings of algebraic integers.* Algorithmic number theory. Springer Berlin Heidelberg, 2002. 33-42. M. Rosen. *Number theory in function fields*. Vol. 210. Springer-Verlag, (2002). P. Samuel, *Compléments á un article de Hans Grauert sur la conjecture de Mordell*, Publications Mathématiques de l’HÉS (1966), Volume: 29, page 55-62. J. Silverman, *Advanced Topics in the Arithmetic of Elliptic Curves*. Graduate Texts in Math. 151, Springer (1994). J. Silverman, *Primitive prime divisors, dynamical Zsigmondy sets, and Vojta’s conjecture.* J. Number Theory, 133: 2948-2963, (2013). J. Silverman, *The arithmetic of dynamical systems*, Vol. 241, Springer-Verlag, (2007). J. Silverman, *The arithmetic of elliptic curves.* Graduate Texts in Mathematics. Springer-Verlag, GTM 106, 1986. Expanded 2nd Edition, 2009. J. Silverman, *Wieferich’s criterion and the abc-conjecture*, J. Number Theory 30: 226-237 (1988). H. Stichtenoth. *Algebraic function fields and codes*. Vol. 254. Springer-Verlag, (1993). M. Stoll, *Galois groups over $\mathbb{Q}$ of some iterated polynomials.* Arch. Math. 59 (1992): 239-244. [^1]: 2010 *Mathematics Subject Classification*: Primary: 14G05, 37P55. Secondary: 11R32. [^2]: *Key words and phrases*: Arithmetic Dynamics, Rational Points on Curves, Galois Theory.

--- abstract: 'Unsupervised bilingual word embedding (BWE) methods learn a linear transformation matrix that maps two monolingual embedding spaces that are separately trained with monolingual corpora. This method assumes that the two embedding spaces are structurally similar, which does not necessarily hold true in general. In this paper, we propose using a pseudo-parallel corpus generated by an unsupervised machine translation model to facilitate structural similarity of the two embedding spaces and improve the quality of BWEs in the mapping method. We show that our approach substantially outperforms baselines and other alternative approaches given the same amount of data, and, through detailed analysis, we argue that data augmentation with the pseudo data from unsupervised machine translation is especially effective for BWEs because (1) the pseudo data makes the source and target corpora (partially) parallel; (2) the pseudo data reflects some nature of the original language that helps learning similar embedding spaces between the source and target languages.' author: - | Sosuke Nishikawa, Ryokan Ri and Yoshimasa Tsuruoka\ The University of Tokyo\ 7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan\ [sosuke-nishikawa@nii.ac.jp]{}\ [{li0123,tsuruoka}@logos.t.u-tokyo.ac.jp]{}\ bibliography: - 'coling2020.bib' title: Data Augmentation for Learning Bilingual Word Embeddings with Unsupervised Machine Translation ---

--- author: - Mohammad Reza Ahmadzadeh Raji bibliography: - 'thesis1.bib' date: | A Thesis Presented in Partial Fulfilment of the Requirements for the Degree of Master of Engineering in Computer Engineering\ Razi University\ 2014 title: '**Refactoring Software Packages via Community Detection from Stability Point of View**' --- \ **Faculty of Engineering**\ **Department of Computer Engineering**\ **M.Sc.Thesis**\ \[6mm\] 2.5cm 1.5cm 1.5cm **June 2014** Acknowledgements {#acknowledgements .unnumbered} ================ ##### First and foremost, I would like to express my appreciation and thanks to my supervisor, Dr. Behzad Montazeri. I would like to thank him for his encouragement, motivation and priceless advices throughout my research. ##### I would also like to express my deepest appreciation to my father, Dr. Mehrdad Ahmadzadeh, my mother and my wife, Sara who have been the most supportive, through thick and thin and have helped me in the difficulties of the paths I have taken. **Dedicated to my beloved wife, for her endless support and encouragement** Abstract {#abstract .unnumbered} ======== ##### As the complexity and size of software projects increases in real-world environments, maintaining and creating maintainable and dependable code becomes harder and more costly. Refactoring is considered as a method for enhancing the internal structure of code for improving many software properties such as maintainability. ##### In this thesis, the subject of refactoring software packages using community detection algorithms is discussed, with a focus on the notion of package stability. The proposed algorithm starts by extracting a package dependency network from Java byte code and a community detection algorithm is used to find possible changes in package structures. In this work, the reasons for the importance of considering dependency directions while modeling package dependencies with graphs are also discussed, and a proof for the relationship between package stability and the modularity of package dependency graphs is presented that shows how modularity is in favor of package stability. ##### For evaluating the proposed algorithm, a tool for live analysis of software packages is implemented, and two software systems are tested. Results show that modeling package dependencies with directed graphs and applying the presented refactoring method, leads to a higher increase in package stability than undirected graph modeling approaches that have been studied in the literature.\ \ **Keywords:** Graph clustering, Community detection, Package refactoring, Software metrics, Stability, Coupling, Cohesion. Introduction to code refactoring ================================ ##### There are many properties that can be associated with good code. Sommerville describes good code as one that is highly maintainable, dependable, efficient and usable [@sommerville2004]. Truly reusable code is considered gold in the software industry as it significantly effects productivity and thus lowers costs [@lim1994] and without a doubt, good code is backed by a good design. A professional software engineer must first design a software and then implement the code based on the design. However, in real-world scenarios, the great attributes of a good software might fade away as the project grows. Tight schedules, high customer demands and the high number of programmers involved in large projects are considered as some of the reasons that make efficient and engineered implementations change into a mess. A mess that is not easily maintained, reused, changed or depended upon. Refactoring is considered the cure for this infiltration of the project. ##### Refactoring is a common word for a day-to-day programmer with its origins in mathematics and ultimately in the Latin language. The root *factor* has the meaning of *maker* and hence refactoring is known as re-making something. In mathematics, when you factor an expression, you re-make it and provide a more cleaner version. The exact origin of the word, refactoring, in computer science is somewhat unknown, however the Forth language community is known to have been the first people to have used this expression [@fowler1999]. Chapter six of Leo Brodie’s book, Thinking Forth is dedicated to the subject of refactoring [@brodie1984]. ##### Martin Fowler, the author of one of the most canonical books on refactoring [@fowler1999], describes it as *“the process of changing a software system in such a way that it does not alter the external behavior of the code, yet improves its internal structure.”* ##### This thesis solely focuses on refactoring methods that involve the use of graph clustering methods, however to better understand the reasons and effects of proposed methods, a brief and concise explanation of known refactoring techniques is given. Well-known refactoring techniques --------------------------------- - **Rename method.** This technique may be the most simple refactoring method one can use. Simply renaming identifiers and variables will make the code clearer, more understandable and can reduce the need for comments. An appropriate name for a method, variable or a class is one that is descriptive so that a new programmer can understand its work just by a glance. - **Inline temp.** Temporary variables can make methods longer and more complicated. It is suggested that temporary variables that are being used only once or are a result of a method call be completely removed and the value assigned to them be used in the code. An example is provided below. **Incorrect:** ``` {language="python"} def add_something(): return 1 + 2 def foo(): temp_variable = add_something() print "The result is " + temp_variable ``` **Correct:** ``` {language="python"} def foo(): print "The result is " + add_something() ``` - **Extract method.** Known as arguably the most important refactoring technique, Extract method aims at reducing the size of long methods by breaking them into smaller methods with descriptive variables. Many refactoring and simplifying techniques in software engineering involve breaking code and algorithms into smaller, more understandable chunks. This method is one of them. ``` {language="python"} class Foo: username def __init__(self): # Some initialization code self.username = "Some username" def func1(self): print "Welcome" print "You have logged in as " + self.username print "Something else" def func2(self): print "Welcome" print "You have logged in as " + self.username print "Some reports" ``` ##### In the provided example, lines 8 and 9 are equal to lines 13 and 14 and can be extracted into a new method that greets the user. Extract method is considered as an important and basic refactoring technique that highly effects the cohesion of classes from which methods have been extracted. Extract method suggests the extraction of pieces of code that are used more than once (duplicate code). If this condition is met while extracting piece of code A from methods B and C, then after refactoring, both B and C will be using A and thus reducing the cohesion in their class. However, one must realize that if appropriate interfaces are not used in the code and other classes in a package use method A, then instead of reducing cohesion, coupling will be increased. A thorough study on this issue and a metric for finding appropriate pieces of code for extracting while considering the notion of cohesion is provided in [@goto2013extract]. Considering our focus on graph clustering methods in refactoring, it is worth noting that some work has been done in detemining the class a method belongs to, with the help of community detection techniques [@pan2009class]. However, introducing new methods and extracting them with community detection is still in need of attention. - **Inline method.** In some cases, the opposite of Extract method should be applied. Suppose method A is simple, clear and is being used only once, possibly in a stable class whose content is not likely to change. In this case, using an identifier for the code in method A only results in an extra call for no benefit. This method can be removed and its content can be used inline. - **Replace method with Method object.** This technique can be considered as an aid, in situations where Extract method becomes difficult because of the high number of temporary variables in a long method. In a case where the number of temporary variables is high, Extract method can become a cumbersome task because passing around all the temporary variables between the extracted methods can prove to be messy and finding the needed temporary variables for a piece of extracted code can take a lot of time. To resolve this issue, one approach is to move the long method into a new class, set the local temporary variables as class attributes and then apply Extract method. This method provides a better state, from which we can continue our refactoring using Extract method or other techniques. - **Pull up method.** Imagine a scenario in which a piece of code is duplicated in two different classes, it is best to pull that code up into a super class of those two classes. **Before refactoring:** ``` {language="python"} class Person: firstname = None lastname = None def __init__(self): # Some initialization code class Student(Person): studentNo = None def __init__(self): # Some initialization code def makeFullName(self): return self.firstname + " " + self.lastname def getStudentNo(self): return self.studentNo class Employee(Person): salary = None def __init__(self): # Some initialization code def makeFullName(self): return self.firstname + " " + self.lastname def getSalary(self): return self.salary ``` **After refactoring:** ``` {language="python"} class Person: firstname = None lastname = None def __init__(self): # Some initialization code def makeFullName(self): return self.firstname + " " + self.lastname class Student(Person): studentNo = None def __init__(self): # Some initialization code def getStudentNo(self): return self.studentNo class Employee(Person): salary = None def __init__(self): # Some initialization code def getSalary(self): return self.salary ``` - **Extract surrounding method.** Imagine a case in which several different methods are almost identical but with a slight difference in the middle of each one. In some languages, one can pull up the duplicated code into a new method and pass the middle section to the method which it yields to. This ability is provided in some languages like Ruby and can be simulated in some other languages by passing callback functions. A Ruby example is given below. ``` {language="ruby"} def testMethod puts "Something printed from inside testMethod" yield puts "Something printed from inside testMethod" end testMethod {puts "Something printed from the block"} ``` - **Replace conditional with polymorphism.** This method of refactoring can help remove the complexity and code smell of conditional logic and demonstrate the principle of true object-oriented design. Code quality and software metrics ================================= ##### Code quality is one of the most important factors that directly effects a project’s maintainability phase. The quality of a good code determines how flexible a project is and determines the limits of the resuability of its components. Good code can be changed more quickly and newcomers to the project can understand it more easily. Another important feature that code quality can greatly effect, is how much we can trust that new changes will not cause unexpected effects and will not introduce bugs to the project [@baggen2012standardized]. ##### Although code quality is considered as an important subject in software projects, many developers and projects simply neglect its importance and features. This is normally due to tightened schedules, high customer demands and low budgets. ##### Code quality has been discussed in the literature for a long time and more specifically, quantitative measurement of quality factors and providing metrics has been greatly studied. However, the study of software metrics has been diverse and the need for a strong and refined approach is felt [@kitchenham2010s]. In this thesis we focus on three well-known object-oriented quality factors, namely coupling, cohesion and package stability. Coupling and cohesion --------------------- ##### Coupling is one of the most famous internal product attributes. Generally, two pieces of code are said to be coupled if changes in one causes the other to change. In the object-oriented paradigm, coupling between two classes is considered a bad and unwanted attribute, however a system with no coupling between its classes would mean that interaction is not occurring between the classes and therefore it would simply fail to function. ##### Cohesion, which almost always comes with coupling, is another important internal product attribute. In an object-oriented system, a class is said to have a high cohesion if its internal structures and methods have high connectivity with themselves. The goal in a good design is high cohesion and low coupling, meaning that classes should be cohesive and therefore fully related to their responsibility while they have a low coupling with other classes so that they can change without causing too many changes in other parts of the system. Designs with high cohesion and low coupling make the system more reliable and maintainable [@fenton1998software], [@troy1981measuring]. ##### The notions of coupling and cohesion have been excessively studied in the literature and many metrics have been proposed for measuring them. This thesis briefly surveys different approaches in the literature. ### Basic definitions by Myers ##### Myers, Stevens and Constantine introduced the concept of coupling in procedural programming. Based on this, Fenton defined six different levels of coupling [@alghamdi2008measuring]. These levels of coupling are shown below from worst to best. - **Content coupling.** If one element branches into or changes the internal statements of another element, they are said to have content coupling. - **Common coupling.** If two elements refer to the same global variable, they are said to have common coupling. - **Control coupling.** If the data that one element sends to the other controls its behavior, then control coupling is implied. - **Stamp coupling.** Two elements are stamp coupled if they send more information to each other than necessary. - **Data coupling.** If two elements communicate with each other by parameters with no control coupling, then they are data coupled. - **No coupling.** If two elements have no communication with each other then they are not coupled. ### Fenton and melton’s metric ##### Fenton and melton proposed a metric for coupling which is expressed as $$\label{eq:fenton} C(x, y) = i + n / (n+1)$$ where $n$ is the number of interactions between the two components $x$ and $y$ and $i$ is the level of the worst coupling found between $x$ and $y$. In their metric, the coupling level is based on Myer’s classification. No coupling is given a coupling level of 0 and the next levels have higher numeric values. ##### Alghamdi discusses several important points about this metric [@alghamdi2008measuring]. - All types of interconnections are considered equal, with equal effects on coupling. - The Fenton and Melton metric is an example of a inter-modular metric, meaning that it calculates the coupling between a pair of components in contrast with intrinsic metrics that measure the coupling of a component individually. - Coupling values approach the next level when the interconnections between two components increases. ##### Alghamdi also proposes a new coupling metric based on a description matrix of the system [@alghamdi2008measuring]. ### Chidamber and Kemerer’s suite ##### Chidamber and Kemerer [@chidamber1991towards] gives the first formal definition of coupling by defining coupling as any evidence that a method from one class uses a method or variable of another class. In their proposed suite, known as the CK suite, Chidamber and Kemerer give provide different metrics. The six metrics are as follows. - Weighted Method per Class (WMC) - Number Of Children (NOC) - Depth of Inheritence Tree (DIT) - Coupling Between Objects (CBO) - Lack of Cohesion in Methods (LCOM) - Response For a Class (RFC) ##### Among their six metrics, CBO (Coupling Between Objects) is proportional to the number of non-inheritance related couples with other classes. For measuring coupling, CBO aggregates the total number of couples a class has to another class, which implies that different couples have the same strength and effect. Hitz and Montazeri [@hitz1995measuring] argue that the CK suite does not fully conform to measurment theory. ### Alghamdi’s coupling metric ##### Alghamdi’s approach is based on the idea of generating a description matrix of all the factors that effect coupling and then calculating a coupling matrix based on the collected data. An overview of this approach is depicted in Fig. \[alghamdi\] ![An overview of Alghamdi’s coupling metric[]{data-label="alghamdi"}](alghamdi.png){width=".7\columnwidth"} ##### The description matrix is an $m$ by $n$ matrix where $m$ is the number of system components and $n$ is the number of component members. In an object-oriented system, components are represented by classes and members are class variables and methods. An example of a description matrix is depicted in Table \[desc\_matrix\]. **Component** $E_1$ $E_2$ ... $E_n$ --------------- ---------- ---------- ----- ---------- $C_1$ $d_{11}$ $d_{12}$   $d_{1n}$ $C_2$ $d_{21}$ $d_{22}$   $d_{2n}$ ...         $C_m$ $d_{m1}$ $d_{m2}$   $d_{mn}$ : An example of Alghamdi’s description matrix[]{data-label="desc_matrix"} ### A qualitative approach to coupling and cohesion ##### While many quantitative approaches for measuring coupling and cohesion have been proposed in the literature, few qualitative approaches have been discussed. Kelsen [@representational2003] proposes an interesting information-based method for analyzing coupling and cohesion and finding refactoring suggestions. Kelsen’s approach considers a special type of coupling, namely representational coupling. When an object calls a method of another object, some information about the callee is exposed. If the information is about the low level implementation of the callee, then representational coupling is high. If the call exposes higher level information, then representational coupling is low. Many metrics in the literature, including the works of Chidamber and Kemerer [@chidamber1991towards] can not capture representational coupling [@representational2003]. The main reason behind this issue is that many works simply count different types of interactions and assign ordinal numbers to these interactions. Kelsen, also presents a minimum for the representational coupling inherently contained in a system, which is known as intrinsic representational coupling. ##### Kelsen’s approach is based on the idea that if one can find two states in a system, namely *witness states*, that yield different messages between objects but cannot affect the states of other objects, then this indicates that coupling can be improved and representational coupling is higher than it should. The elevator example below is borrowed from [@representational2003]. ##### Suppose that the behavior of some elevators in a building is modeled using two classes, ElevatorControl and Elevator. Every elevator has two methods, *direction()* and *position()*, which return the direction and position of the elevator. The ElevatorControl class is responsible for handling requests and checks every elevator’s distance and position for finding the closest elevator for a request. Two different implementations for ElevatorControl’s *handleRequest* method can be written. $minDist \gets Infinity$ Elevator $ec \gets null$ //reference to closes elevator $d \gets $ Compute $distance(e, r)$ using $e.direction()$ and $e.position()$ $ec \gets e$ $minDist \gets d$ $ec.addRequest(r)$ $minDist \gets Infinity$ Elevator $ec \gets null$ //reference to closes elevator $d \gets e.computeDistance(r)$ $ec \gets e$ $minDist \gets d$ $ec.addRequest(r)$ ##### In the second implementation, the task of computing en elevator’s distance is given to each elevator with the $computeDistance$ method. It is clear that the $ElevatorControl$ class does not need to know the distance and position of every elevator and only needs their distances, therefore less information from the $Elevator$ class is being exposed in the second implementation and representational coupling is decreased. ##### Kelsen’s qualitative approach may be considered a precise and great method for measuring representational coupling, however, because of its non-quantitative nature it is not clear if it can be applied to large, real-life software systems with many classes, and its utilization in real-life scenarios is currently considered as an open problem. Stability --------- ##### Stability is the amount of likeliness, that a class or a package will not change. Stability is inherently difficult to measure because the future changes and needs of a project are not well known, however some metrics exist that try to measure stability. The importance of this stability in software metrics was first mentioned by Hitz and Montazeri [@hitz1995measuring]. ##### Some methods for measuring the stability of a software package, utilize the history of the class’s changes in the past and try to predict its future. The changes of a class or a package is typically accessed through version control systems such as Git[^1] and Subversion[^2], however these approaches can not be used in early stages of software design because of the lack of change history available at the time. ##### Robert Martin [@martin2003agile] takes a different approach to measuring the stability of a software package. He believes that stability is proportional to responsibility and a package is said to be responsible and independent if many other entities depend on it, while it doesn’t depend on others itself. A package $p$ is said to be irresponsible and thus unstable, if it depends on many other entities, meaning that if they change they cause $p$ to change as well. By Martin’s definition, in Fig. \[stable\_package\], X is an example of a stable package and in Fig. \[unstable\_package\], Y resembles an unstable package. ![An example of a stable package[]{data-label="stable_package"}](stable_package.png) ![An example of an unstable package[]{data-label="unstable_package"}](unstable_package.png) ##### As a metric for stability, Martin defines the instability of a package as given in Eq. \[eq:Instability\] where $I$ is instability, $C_a$ is afferent couplings and $C_e$ represents the number of efferent couplings. Afferent couplings is the number of classes outside the package that depend on classes within the package and efferent couplings is the number of classes within the package that depend on outside classes. $$\label{eq:Instability} I=\frac{C_e}{C_a + C_e}$$ ##### If a package $p$ has an *instability* of 0, then the $p$ has maximum stability and if the package holds a value of 1 for *instability*, then it would mean that the number of afferent couplings is 0 and therefore $p$ depends on other packages while no other package depends on $p$ and this would make it an extremely unstable package. ##### Martin also proposes the Stable Dependencies Principle (SDP) that helps the software design process by ensuring that modules that should be easily changeable not depend on modules that are harder to change [@martin2003agile]. In this case, packages should always have a higher $I$ metric than the ones they depend on. Concenting to this principle, one would be able to see a tree of packages, in which stable ones are placed at the bottom and the most unstable ones are at the top. The benefit of this approach is that packages that are violating SDP can be easily spotted. Any package depending on a package above it, would mean a violation of the principle. ##### It is important to note that not all packages should or could be fully stable, as this would cause an unchangeable and inflexible system. Also, not all packages can be unstable as this would create an irresponsible system with a large number of connections and a high coupling. It is clear that pieces of code that are likely to change should be placed into unstable packages and pieces of code that are not very likely to change in the future should be placed in stable packages. Martin argues that high level design can not be placed in unstable packages because it resembles the architectural decisions of the projects, however if high level code is placed in stable packages then it would almost be impossible to change it after the project becomes more mature and more pieces of code start depending on it. The solution to this dilemma is the use of abstract classes that can introduce the flexibility and flow of stability that is needed. The basic idea behind the Stable Abstraction Principle (SAP) is that a package has to be as abstract as it is stable. This principle ensures that the stability of a package does not contradict its flexibility. The SAP proposes a metric for measuring the abstractness of a package which is a simple ratio and is shown in Eq. \[eq:abstractness\] in which $N_a$ is the number of abstract classes inside the package and $N_c$ is the number of classes inside the package. ![The relationship between asbtractness and instability[]{data-label="abstract_stable_graph"}](abstract_stable_graph.png) ##### Martin defines three important areas in the relationship between abstractness and stability. If we set abstractness (A) as the vertical axis and instability (I) as the horizontal axis in a cartesian graph, then three spots depicted in Fig. \[abstract\_stable\_graph\] are as follows. - **Zone of pain.** The zone of pain is where a package is highly stable and yet its abstractness is zero. Such a package is hardly changeable. - **Zone of uselessness.** A package in this zone is highly abstract and also highly unstable and not depended on. This means that its abstractness is useless. - **The main sequence.** This is the ideal point for a package. A package near the main sequence is a package that conforms to the SAP and is as abstract as it is stable. The sequence is ideal and thus not many packages can truly be placed on this line, however the distance of a package from this ideal line can be measured. $$\label{eq:abstractness} A = \frac{N_a}{N_c}$$ $$\label{eq:abstract_stable_distance} D = \frac{|A + I - 1|}{\sqrt{2}} D' = |A + I - 1|$$ ##### In Eq. \[eq:abstract\_stable\_distance\], $D$ is the distance from the main sequence and $D'$ is its normalized version that ranges from $[0, 1]$. Community detection and applications ==================================== ##### Community detection is defined as the process of finding communities of nodes in networks, such that the nodes inside a community have a higher property resemblance to one another compared to nodes in another community. A network is a graph with a pair of sets, $V$ and $E$, where $V$ is the set of all vertices and $E$ is the set of all edges in the network. Every community in a network is considered as a partition of the set $V$. Typically, the process of detecting communities in a network consists of the following steps. 1. Specifying a quality measure that defines the quality of a partitioned graph. 2. Using a specific method to assign nodes to different communities (clusters) in a way that increases the quality measure in step 1. There are normally three different terms, related to the subject of community detection that are sometimes used interchangeably by mistake, thereforee distinguishing between these terms is needed before going into the details of each method. - **Graph partitioning vs Community detection.** The most important difference is that the problem of graph partitioning is universally defined as a problem where the number and sizes of the clusters are specified a priori. This is not the case in graph clustering or cluster analysis in general. The second, less important difference between these two terms is that clustering excludes the possibility of overlap by convention, so that it is still possible to speak of an overlapping clustering, whereas a partition or partitioning excludes the possibility of overlap by definition. - **Graph clustering vs Community detection.** Graph clustering and community detection are normally used interchangeably in the litrature and in this thesis. ![The Zachary club[]{data-label="zachary_club"}](zachary.png) ##### Many different community detection and graph partitioning algorithms have been proposed in the literature, some of which will be briefly discussed in this thesis. Classification of clustering methods ------------------------------------ ##### Graph clustering methods are normally difficult to classify, however Wiggerts [@wiggerts1997using] believes that they can generally be divided into the following methods. - **Hierarchical methods.** Hierarchical approaches are known as some of the early solutions to the problem. These methods provide a hierarchy of partitions like a tree, known as a dendrogram. A sample dendrogram is depicted in Fig. \[zachary\_club\_dendrogram\]. Hierarchical methods are themselves divided into the two groups of agglomerative approaches and divisive approaches. In agglomerative approaches, the algorithm starts with placing every node inside a separate cluster. Then the algorithm starts merging the clusters based on their similarity. It is important to note that the algorithm will not stop unless told to, thereforee knowing the number of wanted partitions in the result is crucial. In divisive hierarchical approaches, the algorithm starts with a single cluster that contains all the nodes of the graph. The algorithm then splits the cluster based on the similarity between the nodes, keeping the similar ones in the same cluster. Different hierarchical algorithms are distinguished by their distance function which is responsible for determining the similarity between two given nodes. - **Optimization based methods.** These algorithms generally take an initial inaccurate clustering and with the help of a quality measure, try to enhance and improve the cluster and optimize the quality. One of the most common and famous quality measures in the literature is the modularity measure proposed by Girvan and Newman [@chen2013new]. Various kinds of optimization techniques are applicable in this category of graph clustering algorithms, such as genetic algorithm based optimization methods, particle swarm methods, etc. A simple genetic algorithm approach can be like the following [@shtern2012clustering]. 1. Select a random population of partitions 2. Generate a new population by selecting the best according to a quality measure, such as Newman’s modularity 3. Repeating step 2 until a certain criteria is met - **Graph theoretical based methods.** Graph theoretical algorithms are methods that utilize the formal descriptions and properties of graphs and their respective subgraphs. In these methods, various subgraphs and properties are used to extract meaningful clusters from the original graph. Two important and common types of graph theoretical algorithms exist, namely aggregation algorithms and minimal spanning tree algorithms. Aggregation algorithms use the function of reduction on different nodes and merge them in each step. Different potential nodes for merging are chosen using different techniques, such as neighbourhoodness, strong connections and etc. Minimal spanning tree algorithms use the minimal spanning tree of the graph. These algorithms are normally not considered accurate as they tend to create large clusters, however some enhanced versions of these algorithms have been suggested in the literature [@shtern2012clustering]. - **Construction algorithms.** These algorithms assign nodes into clusters in one pass. The bisection algorithm and density search techniques are considered as examples of such methods. ![A dendrogram for the Zachary club[]{data-label="zachary_club_dendrogram"}](zachary_dendrogram.eps) ##### The minimum cut approach is the most obvious and the most easiest way of tackling the problem of community detection. In this method, one tries to find two groups/partitions in a graph for which the edges connecting the two is the least. This approach mostly falls in the area of graph partitioning, because the number of partitions in the end result must be known a priori so that one can know how many times the algorithm should be applied. It is worth noting that if the minimum cut approach were to be used with no constraint, then a trivial solution to the problem would be to place all vertices in one partition only, thus minimizing the number of edges between partitions. Clearly this solution would not give any information on the *communities* in a network. In the software engineering sense, the result of such a method would be a system with zero coupling and maximum cohesion, which seems the goal. However many important aspects of the software such as reusability, separation of concerns, object orientedness, flexibility, etc. will be lost. This raises the idea that maybe another measurement apart from coupling and cohesion is needed that can help find an optimum point for the two. This measure must be able to truly model and represent different objects in a software dependency network. In the subject of graph theory, a measure that can model the *goodness* of a partition is known as a quality measure. Using a community quality measure in the field of software engineering has only recently been discussed in the literature [@pan2009class], [@pan2013refactoring]. Quality measures ---------------- ##### The quality of a partition found by a community detection algorithm is determined with a quality measure. This measure should show how good a partition is. Many algorithms provide many partitions without equal goodness, therefore it is absolutely necessary to measure the quality of the provided partitions and detect the best. Quality functions give a number to each partition so that the partitions can be ranked and compared to one another. Arguably, the most common and famous quality function is Newman and Girvan’s Modularity [@newman2004finding]. ##### Modularity is based on the idea that a random graph contains no meaningful community. Based on this idea, if one can make a similar graph to the one being analyzed with the same number of vertices, edges and degrees but with edges placed at random, then by comparing it to the original graph one can find the major differences that have created communities. To understand the notion of modularity, we start by another measure for the goodness of a partition and build on it. Let $G$ be a graph with elements of its adjacency matrix presented as $A_{vw}$, where $A_{vw}$ is 1 if nodes $v$ and $w$ are connected and 0 otherwise, and $C_v$ being the community in which vertex $v$ belongs to. The following measure shows the fraction of edges in graph $G$, that fall within communities. $$\label{eq:Fraction of edges in the same community} \frac{\sum_{vw} A_{vw}\delta(C_v, C_w) } {\sum_{vw}(A_{vw}) } = \frac{1}{2m} \sum_{vw}A_{vw}\delta(C_v, C_w)$$ where $\delta$ is the Kronecker delta function and $m$ is the number of edges in the graph. ##### This fraction takes the value of 1 when all edges fall in one community and hence is not a good enough measure. ##### The idea behind modularity is that a random graph does not have a meaningful community structure and thus, if generated carefully, should provide a good point of comparison. Carefully generating a random graph that can depict the features and properties of the original graph but with no meaningful community is known as providing a null model in the area of complex systems. In this case, one can provide a graph which has the same amount of vertices, edges and vertex degrees while its edges are rewired randomly, so that the graph looses its community structure. In such a graph, the probability of an edge being in between vertices $v$ and $w$, if connections are made at random is calculated as below. $$\label{eq:The probability of an edge between v and w} \frac{k_vk_w}{2m}$$ where $k_v$ and $k_w$ are the degrees of vertex $v$ and $w$ respectively. Now, by using equations \[eq:Fraction of edges in the same community\] and \[eq:The probability of an edge between v and w\], one can calculate the modularity measure as $$\label{eq:Modularity} Q=\frac{1}{2m}\sum_{vw}[A_{vw} - \frac{k_vk_w}{2m}] \delta(C_v, C_w).$$ ##### By looking at Eq. \[eq:Modularity\], one can see some important aspects of this measure. The Kronecker delta function makes sure that a connection between two graph nodes in two different communities makes no contribution to modularity. Two connected nodes inside a community, make a positive contribution to modularity and the contribution is inversely proportional to the degrees of the two nodes. Also two nodes that are not connected, yet still reside in one community provide a negative contribution to the overall modularity of the clustering. A brief discussion of well known clustering methods --------------------------------------------------- In this section, several common graph clustering methods are briefly studied. ### The fast greedy method ##### A typical greedy method for clustering a graph while utilizing Newman’s modularity consists of the following steps. 1. Start with each vertex in its own community, thus having $n$ communities for $n$ vertices. 2. In each step, merge two communities whose join makes the highest increase in modularity $Q$. 3. After $n-1$ joins, one community remains and a dendrogram can be created. 4. Take the clustered solution that has the highest Q. ##### The simple greedy method, can waste a good deal of time when dealing with sparse graphs. In the implementation of the simple greedy approach, one has to merge many columns and rows of the sparse adjacency matrix and consequently time and space is wasted on merging elements with the value of 0. For this reason, Clauset and Newman have presented an enhanced version of the greedy method, namely the fast greedy method [@clauset2004finding] which performs much better than many other algorithms in the literature. In the fast greedy method, some data structures such as max heaps and balanced binary trees are used with some alterations in the algorithm that results in the runtime of $|V||E|log(|V|)$. ### The edge-betweenness based method ##### The edge betweenness based method, proposed by Girvan and Newman [@girvan2002community] before presenting the modularity measure, is a graph clustering algorithm that focuses on the edges that are between communities in contrast to many other older algorithms that focus on the connections inside a community. Edge betweenness is described as the number of shortest paths between pairs of vertices that run along it. The algorithm for this method is as follows. 1. Calculate edge betweeenness for all edges 2. Remove the edge with the highest betweenness value 3. Recalculate edge betweenness for the rest of the edges 4. Repeat step 2 until no edges remain ##### Calculating betweenness for all $m$ edges and $n$ vertices of a graph can be calculated using Newman’s algorithm for betweenness [@girvan2002community] which can be calculated in time $O(mn)$. Edge betweenness has to be recalculated for every edge removal and thus the algorithm can work in time $|N||M|^2$. ### The walktrap based method ##### The walktrap method is based on the notion of random walks [@pons2005computing]. The main idea behind the walktrap method is that random walks in a graph tend to get trapped in dense parts of the graph which could represent communities. In the walktrap method, a distance $r$ between communities is calculated based on the properties of random walks. After this step, typically an agglomerative algorithm is used to merge communities and create a dendrogram, much like other methods. This algorithm has a runtime of $|E||V|^2$. ### The leading eigenvector based method ##### The leading eigenvector algorithm utilizes the eigenvalues of the modularity matrix. In this algorithm one determines the eigenvector corresponding to the most positive eigenvalue of the modularity matrix and divide the network into two groups according to the signs of the elements of this vector. Community detection for directed graphs --------------------------------------- ##### Community detection in directed networks is a difficult task [@malliaros2013clustering]. Various algorithms for community detection in undirected graphs have been presented in the literature, however methods for directed approaches have been less common. A comprehensive survey of community detection methods for directed graphs can be found in [@malliaros2013clustering] by Malliaros et al. They propose the following classification for community detection approaches in directed graph. 1. **Naive graph transformation approach.** In this method, directions are simply removed from the graph and undirected community detection techniques are applied. 2. **Transformations maintaining directionality.** In this category of methods, the graph is transformed to an undirected version while directionality is maintained using other methods. The original graph can be tranformed to a unipartite weighted graph or a bipartite graph for this approach. An overview of such transformations is depicted in Fig. \[transformation\]. 3. **Extending objective functions and methodologies in directed graphs.** Many objective functions and quality measures used in undirected graphs can be extended to directed versions, i.e. modularity, spectral clustering, page rank and random walk methods, local density clustering. 4. **Alternative approaches.** Other methods that can not be placed in the first three categories also exist. Such as information theoretic approaches and blockmodeling approaches. ![An example of a transformation that preserves directionality. []{data-label="transformation"}](transform.png){width=".8\columnwidth"} ##### Although some algorithms exist for this purpose, many clustering algorithms for undirected graphs can be extended for directed graphs with the help of a direction-compliant quality measure. Several extensions of modularity for directed graphs have been proposed in the literature. Arenas et al [@arenas2007size] proposed an extension of modularity. Their idea is based on the fact that in a directed graph $G$, if vertex $i$ exists with more out-links and vertex $j$ exists with more in-links, then it is more probable that in a random rewiring a link be found from $i$ to $j$ rather than the opposite. Considering the original idea of modularity, this suggests that if an edge is found from $j$ to $i$, then this edge is contributing to a community structure more than $i$ to $j$ would, simply because it is more suprising and less random. By this definition, modularity can be altered for directed networks by changing the null model to a graph with the same number of vertices, edges, out-links and in-links as the original graph. The equation for modularity $Q$ in a graph with the adjacency matrix $A$ and $m$ number of edges can then be expressed as $$\label{eq:Modularity_directed} Q=\frac{1}{m}\sum_{ij}[A_{ij} - \frac{k_i^{out}k_j^{in}}{m}] \delta(C_i, C_j)$$ where $\delta$ is the Kronecker delta function, $C_i$ and $C_j$ denote the communities that nodes $i$ and $j$ belong to, and $k_i^{out}$ and $k_j^{in}$ are the number of vertex $i$ and $j$’s out-links and in-links respectively. Applications of community detection in software engineering ----------------------------------------------------------- ##### Graph clustering is widely used in the literature as a method for finding meaning in a structure. This need for finding meaning in a complex system is generally used in four main areas of software engineering. ### Reflexion ##### Reflexion is the art of bridging the gap between software and humans, when it comes to analyzing a legacy system. Reflexion analysis tries to build an understandable high level abstraction of a large system, given the source code. In the process, the source code is analyzed and mapped to a new higher level model. This cumbersome task is typically done manually, however graph clustering can be used in semi automated mappings of source code to entities with the help the user’s knowledge about the system. Some related work has been presented in the literature [@murphy1995software], [@mancoridis1999bunch]. ### Refactoring ##### There are many properties that can be associated with good code. Sommerville describes good code as one that is highly maintainable, dependable, efficient and usable [@sommerville2004]. Truly reusable code is considered gold in the software industry as it significantly effects productivity and thus lowers costs [@lim1994] and without a doubt, good code is backed by a good design. Refactoring is the art of improving the internal structure of code while leaving the outer side intact [@fowler1999]. One of the problems that has been tackled in the literature is refactoring large and complicated legacy systems and also analyzing the structure of new code. Graph clustering techniques can be considered a good method for finding the correct structure and packages of a large system by analyzing the relationships in a software’s dependency graph. Some work has been done in the area of refactoring at the class level, using graph clustering algorithms [@pan2009class]. Recently, some work has also been presented in the package level [@pan2013refactoring], however the lack of an accurate package analysis tool that considers important object oriented aspects, such as stability and reusability is strongly felt in the literature. ### Parallel computing ##### Task to processor mappings is considered an important problem in parallel environments. The two general strategies used in such problems is placing tasks that can run concurrently on different processors, while keeping tasks that need many communications on the same processors, in order to increase locality. Graph partitioning tools have been used in some cases to map tasks to hypercube structures [@sadayappan1990cluster]. ### Ontologies and concept grouping ##### One of the areas that highly utilizes graph clustering methods is ontologies and the semantic web. Various applications have been presented in the literature. One important application is extracting new concepts and taxonomies from ontologies. Extracting more generalized concepts and relations is one of the outputs of an ontology clustering. Tang et al. presents a great survey on such methods [@tang2012survey]. Modularization is also considered important for the problem of ever growing and over grown ontologies. The works in [@ghafourian2013modularization] is one of the most recent methods in this specific area. Partition stability ------------------- ##### In some works the notion of partition stability, also known as robustness is considered as an important property of a good clustering algorithm. The idea is that a stable partition is one that can be recovered even if the structure of the graph is modified, as long as the change in the graph is not too extensive. It important to stress that this thesis only studies stability in the software package sense of the word and does not cover cluster stability. Refactoring packages using community detection ============================================== ##### Several studies have attempted to use community detection methods or cluster analysis in order to find refactoring opportunities [@pan2009class], [@pan2013refactoring], [@melton2006identifying]. These methods have analyzed the code in three main levels. - Method level - Class level - Package level ##### This thesis focuses on refactoring at the package level for which there has been little discussions in the literature. Pan et al. [@pan2013refactoring] proposes a novel method for refactoring using the notion of modularity, however neglects the use of a directed clustering approach. In this chapter, the importance of a directed graph model is discussed with regard to the notion of package stability and an improved version of a package refactoring method using community detection is presented. Basics of modeling packages with graphs --------------------------------------- ##### As discussed in previous chapters, many metrics have been proposed for different software properties at the class level. At the package level, which is in a higher level in the abstraction hierarchy compared to a class, the most important property in the literature is the dependency between two packages. When a class inside a package depends on a class from another package, the former package is said to depend on the latter. ##### Let $G$ be a graph with the adjacency matrix $A$. Vertices in $G$ represent classes and edge $E_{ij}$ between vertex $i$ and vertex $j$ resembles a dependency between the two classes. Communities in this graph represent package structure. A dependency between two classes can be any usage of methods or variables or inheritance. Classes are being modeled to graph vertices for the sole purpose of using community detection methods for finding appropriate clusters which represent packages and different relationships between classes are not considered different. ##### A thorough metric for package dependencies has been proposed in [@gupta2009package] by Gupta et al, which takes into account the different types of connections between packages when sub-packages also exist in the software. The metric is validated using Briand’s evaluation criteria [@briand1996property]. Gupta et al consider two classes of two packages connected if any of the following relationships are found between them. - Aggregation relationships between two classes, i.e., one class’s attribute has the type of another class - Class inheritence or implementing interfaces - Method invocation of one class by the method of another class - A class’s method referencing an attribute from another class - A class’s method has a parameter of the type of another class - A class’s method has a local variable of the type of another class - A class’s method invoking a method having a parameter of the type of another class ##### By Gupta et al’s metric, coupling between two packages $p_a^i$ and $p_b^i$, where $i$ denotes the hierarchical level, is expressed as $$\begin{aligned} \label{eq:coup} Coup(p_a^i, p_b^i) &=& \left\{\begin{matrix} 0, & (n=0, m=0)\\ \sum_{x=1}{n}\sum_{y=1}^{m}r(e_x^{i+1}, e_y^{i+1}) \\ + \sum_{y=1}^{m}\sum_{x=1}^{n} r(e_y^{i+1}, e_x^{i+1}), & (n \geq 1 \wedge m \geq 1) \end{matrix}\right.\end{aligned}$$ where $n$ and $m$ are the number of elements of package $p_a^i$ and $p_b^i$ respectively at hierarchy level $i+1$, and $r$ is the binary connection between elements. An example of different hierarchical levels given in [@gupta2009package] is depicted in Fig. \[hierarchy\_levels\]. The binary connection between elements ($r$) can be calculated as $$\begin{aligned} \label{eq:r_gupta} r(e_x^{i+1}, e_y^{i+1}) &=& \left\{\begin{matrix} 1, & e_x^{i+1} \rightarrow e_y^{i+1};\\ 0, & otherwise; \end{matrix}\right.\end{aligned}$$ where $e_x^{i+1} \rightarrow e_y^{i+1}$ denotes that element $x$ depends on element $y$. ![An example of different hierarchical levels[]{data-label="hierarchy_levels"}](hierarchy_levels.png){width=".5\columnwidth"} Basics of refactoring with community detection ---------------------------------------------- ##### The use of community detection methods for refactoring packages has only recently been studied in the literature by Pan et al [@pan2013refactoring]. An overview of their method is as follows. 1. Gather software information and dependencies from Java classes and jar files. 2. Construct an undirected weighted dependency network based on the information gathered in the first step. 3. Apply community detection to the dependency network to find the optimal placement of classes in packages. 4. Compare the optimized clustering with the original packages structure of the code and suggest a list of possible refactoring candidates. ##### In the first step of their algorithm Pan et al take into account two types of dependencies between code attributes, method accessing attribute dependency and method call dependency. Any of the two mentioned dependencies between two classes implies a dependency between the two classes. ##### Pan et al model package structure with the help of two different networks, namely the undirected Feature Dependency Network (uFDN) and the undirected Weighted Class Dependency Network (uWCDN). Nodes in uFDN represent features inside the software and edges represent dependencies between features. By this definition, uFDN can be expressed as $$\label{eq:ufdn} uFDN = (V_f, E_f, W_f)$$ where $V_f$ and $E_f$ represent the set of vertices and edges in uFDN respectively and $W_f$ is the adjacency matrix for the network. The subscript $f$ shows that the two sets and the adjacency matrix are at the feature level. An example of a uFDN presented in [@pan2013refactoring], consisting of two communities, is shown in Fig. \[ufdn\]. ![A sample uFDN[]{data-label="ufdn"}](ufdn.png){width=".7\columnwidth"} The code resembeling the network in \[ufdn\] is given below. ``` {language="java"} public class X { private int a; public void c() {} public void b() {c();} public void d() {a++; b(); c();} } public class Y { public void f() { X x = new X(); x.c(); } public void e() {f();} } ``` ##### In uWCDN, only the relationship among the classes are shown. A weight is used for every class dependency that represents the number of connections between the the attributes and methods of the two classes involved in the relationship. uWCDN can be defined as $$\label{eq:uwcdn} uWCDN = (V_c, E_c, W_c)$$ where $V_c$ denotes the set of all vertices at the class level, $E_c$ denotes the set of all edges and $W_c$ is the weighted adjacency matrix of the network. Every entry in $W_c$ can be shown as $w_c(i, j)$ which is the weight between the two elements $i$ and $j$ and is used to denote the strength of a dependency between nodes $i$ and $j$. This weight can be calculated as $$\label{eq:uwcdn_weights} w_c(i, j) = |\bigcup_{n_i\in F_i}R_{i1} \cap F_j|$$ ##### The difference between uFDN and uWCDN is shown in Fig. \[ufdn\_uwcdn\]. ![A sample uWCDN compared to its respective uFDN[]{data-label="ufdn_uwcdn"}](ufdn_uwcdn.png){width="0.6\columnwidth"} where $R_{ik}$ denotes the set of all nodes reachable from $i$ within a distance of $k$ and $Fi$ is the set of all the features of class $i$. It is important to note that $w_c(i, j)$ is equal to $w_c(j, i)$. ##### The community detection algorithm used by Pan et al utilizes an older definition of modularity [@newman2004fast]. The importance of directed graphs in modeling package relationships ------------------------------------------------------------------- ##### Many studies in the literature have utilized undirected community detection methods for various applications. Fortunato [@fortunato2010community] presents a comprehensive review on undirected community detection methods. Many studies that include a directed model of a problem simply discard the information that the directions in the graph provides, and use a naive graph transformation approach. In the naive tranformation approach, graph directions are simply discarded and normal undirected community detection methods are applied to the graph. This can cause many important information to be discarded. We briefly discuss three main problems that an undirected approach can cause and how it effects refactoring and package stability. ### Citation based cluster models ##### Using naive transformation approaches for undirected community detection, introduces inaccuracy in certain graphs such as the citation based model that is depicted in Fig. \[citation\_graph\]. In this graph, the two middle vertices can clearly form a meaningful community. The two vertices have in-links from the the same set of vertices while the vertices that they have out-links to are also the same. In the package sense, the middle community resembles a package that is more stable than the package containing the vertices from the left. Many utility packages and libraries contain packages with a similar structure. There is little or no connection between the vertices inside the package, yet they belong to the same community as they are used in similar situations. ![Citation based cluster[]{data-label="citation_graph"}](citation.png){width="0.2\columnwidth"} ##### After applying naive transformation and trying to find optimal communities in the graph in Fig. \[citation\_graph\], the output simply looses the intended community structure. The output is given in Fig. \[citation\_modular\]. Black vertices have been put into one community by the algorithm and white vertices have been placed in another community. In this clustering, it is clear that SDP (Stable Dependencies Principle) is violated and both communities depend on each other. Using a community detection algorithm intended for undirected graphs has changed the SDP compliant structure that the programmer had intended. ![Citation based cluster after naive transformation[]{data-label="citation_modular"}](citation_modular.png) ### Bidirected graphs and loss of information ##### As discussed in [@malliaros2013clustering], the information needed for correct community detection is simply lost in certain graphs such as the bidirected graph shown in Fig. \[bidirected\_graph\]. ![An example of a bidirected graph with two communities[]{data-label="bidirected_graph"}](bidirected.eps) ![An example of a bidirected graph after naive transformation[]{data-label="bidirected_no_direction"}](bidirected_no_direction.eps) From a stability perspective, the dependency graph in Fig. \[bidirected\_graph\] shows a two packages that fully conform to SDP. The community created by the four vertices on the right represent a very stable package that the left community is depending upon. By performing naive transformation the graph would look like Fig. \[bidirected\_no\_direction\]. This graph has lost its community structure and the two left most vertices and the two right most vertices will be treated in the same way when it is given to a community detection method. Fig. \[bidirected\_no\_direction\_modular\] shows this graph after applying community detection while optimizing Newman’s modularity. ![A clustered version of the graph in Fig \[bidirected\_no\_direction\][]{data-label="bidirected_no_direction_modular"}](bidirected_no_direction_modular.eps) Stability and modularity ------------------------ In this section, the relationship between the directed version of modularity and the Stability Dependencies Principle (SDP) in refactoring packages is discussed. In a scenrio where a class is chosen to be moved from one package to another using community detection methods, we show that modularity is in favor of SDP and hiding dependencies that violate SDP inside packages has a higher contribution to modularity than hiding non-violating dependencies. To show this behavior, some prior definitions are needed. A movement of class $i$ from package $p_1$ to package $p_2$ is shown as the tuple $(i, p_1, p_2)$. A border node in a package is defined as a node that has connections with nodes in other packages and thus directly effects the package’s instability metric. ##### SDP is generally satisfied in a case where no stable package depends on an unstable package. When considering the movement of only two border classes, while all other classes and packages are left intact, then the only dependencies effecting the two package’s instability metric are the dependencies of the two border nodes. If a border node $i$ from stable package $p_1$ depends on a node $j$ from unstable package $p_2$, then clearly SDP is violated. \[sdp\_satisfaction\] Let $k_i^{out}$ and $k_j^{out}$ be the out-link degree of vertices $i$ and $j$ respectively, and $k_i^{in}$ and $k_j^{in}$ be the in-link degree of vertices $i$ and $j$. If $k_i^{out} > k_i^{in}$ and $k_j^{out} < k_j^{in}$ and node $i$ and node $j$ are border nodes, then SDP is satisfied. \[sdp\_dissatisfaction\] Let $k_i^{out}$ and $k_j^{out}$ be the out-link degree of vertices $i$ and $j$ respectively, and $k_i^{in}$ and $k_j^{in}$ be the in-link degree of vertices $i$ and $j$. If $k_i^{out} < k_i^{in}$ and $k_j^{out} > k_j^{in}$ and node $i$ and node $j$ are border nodes, then SDP is not satisfied. \[prop:sdp\_mod\] Let $i$ and $j$ be two classes in dependency graph $G$. If a movement $(i, c_i, c_j)$ exists and the conditions of remark \[sdp\_satisfaction\] holds, then the increase in modularity $Q$ is more, compared to the situation in which the conditions of remark \[sdp\_dissatisfaction\] holds true. Let $Q$ denote modularity while the conditions in remark \[sdp\_satisfaction\] holds true and $\bar{Q}$ denote modularity while the conditions in remark \[sdp\_dissatisfaction\] holds true. $Q$ and $\bar{Q}$ can be calculated using Eq. \[eq:Modularity\_directed\] as $$\begin{aligned} Q&=&\frac{1}{m}\sum_{ij}[A_{ij} - \frac{k_i^{out}k_j^{in}}{m}] \delta(C_i, C_j), \\ \bar{Q}&=&\frac{1}{m}\sum_{ij}[A_{ij} - \frac{\bar{k}_i^{out}\bar{k}_j^{in}}{m}] \delta(C_i, C_j).\end{aligned}$$ The bar on in-link or out-link $k$ denotes that it is being calculated in the scenario of remark \[sdp\_dissatisfaction\], and is therefore equivelant to the out-link and in-link in the scenario of remark \[sdp\_satisfaction\] respectively. Thus one can write $$\begin{aligned} \label{eq:kij_bar_relation} \bar{k}_i^{out} &=& k_j^{out} \\ \bar{k}_j^{in} &=& k_i^{in}.\end{aligned}$$ By looking at the conditions of remark \[sdp\_satisfaction\] and remark \[sdp\_dissatisfaction\] it is clear that $$\begin{aligned} \label{eq:kij_bar_relation2} \bar{k}_i^{out}\bar{k}_j^{in} &<& k_i^{out}k_j^{in} \\ \frac{\bar{k}_i^{out}\bar{k}_j^{in}}{m} &<& \frac{k_i^{out}k_j^{in}}{m} \\ A_{ij} - \frac{\bar{k}_i^{out}\bar{k}_j^{in}}{m} &>& A_{ij} - \frac{k_i^{out}k_j^{in}}{m} \\ \bar{Q} &>& Q.\end{aligned}$$ ##### The above proposition shows how modularity is compatible with the notion of SDP. Modularity is in favor of non-random structure in a network. Violating SDP would mean that a stable package is depending on an unstable package. In this scenario, the above proof shows that keeping two nodes that have violated SDP before, inside a single package is better for $Q$ than keeping two nodes that did not violate SDP. It is also important to note that if $i$ and $j$ belong to two different packages, then the condition will have no contribution to modularity and therefore is not discussed. ##### As an example for the proved proposition, suppose that a system contains two packages $C_1$ and $C_2$, where $C_1$ is unstable and $C_2$ is a stable package. Two slighly different versions of this system is depicted in Fig. \[example1\]. In both of these versions, vertices 1, 2, 3 and 4 are members of $C_2$ and vertices 5, 6, 7 and 8 belong to $C_1$. It is clear that in condition (b), edge $(1,5)$ is violating SDP. Based on Proposition \[prop:sdp\_mod\], we show that moving node 1 from $C_2$ to $C_1$ has more positive contribution for package modularity, than in the case of condition (a). If movement $(1, C_2, C_1)$ happens, then four new edges positively contribute to the overall modularity of the dependency graph while one edge’s contribution is eliminated. The reason for this is that edges between two communities provide no contribution to modularity because the kronecker delta function in Eq. \[eq:Modularity\_directed\] becomes zero. therefore edges $(5, 1)$, $(6, 1)$, $(7, 1)$ and $(8, 1)$ will have new contributions to modularity and edge $(1, 3)$ will no longer have any contribution. The changes in modularity $Q$ for condition (b) can be calculated using Eq. \[eq:Modularity\_directed\] as $$\begin{aligned} \label{eq:example1} \Delta Q &=& \overbrace{4(1 - \frac{1 \times 1}{2m})}^{\mathclap{\text{Contribution of the 4 new edges}}} - \underbrace{(1 - \frac{1 \times 1}{2m})}_{\mathclap{\text{Contribution of edge }(1, 3)}} = 3(1 - \frac{1 \times 1}{2m}).\end{aligned}$$ ##### By replacing $m$ with the number of edges, we have $$\begin{aligned} \label{eq:example1b} \Delta Q &=& \frac{57}{20} = 2.85.\end{aligned}$$ ##### Changes in modularity for condition (a) can be calculated the same way as follows. $$\begin{aligned} \label{eq:example1c} \Delta Q &=& \overbrace{(1 - \frac{4 \times 4}{2m})}^{\mathclap{\text{Contribution of edge }(5, 1)}} + 3(1-\frac{1 \times 4}{2m}) - (1 - \frac{1 \times 1}{2m}) = \frac{33}{20} = 1.65.\end{aligned}$$ ##### The results clearly indicate that the graph gained more modularity when trying to suppress an SDP violation than when it is not. ![Two different graph dependency conditions.[]{data-label="example1"}](example1.eps) Proposed refactoring method --------------------------- ##### By considering the discussed importance of directed graphs in refactoring software packages and the package coupling metric proposed by [@gupta2009package], we present a package refactoring algorithm. ##### For calculating the dependencies, we use the package coupling metric provided by Gupta et al [@gupta2009package] at hierarchy level $i+1$. This is a crucial point that must be noted. Hierarchy level $i+1$ is being used because it gives access to elements inside packages at level $i$. The classes and sub-pakages in this level of hierarchy are the ones that will be analyzed for possible refactorings. In this study, only one package level is analyzed for refactoring, as deeper levels cause many open problems that need to be tackled. The most basic problem with optimizing software metrics such as coupling and cohesion in many levels of abstractness simultaneously is that cohesion inside one level can be considered as coupling in a deeper level, thus the problem of minimizing coupling contradicts with the problem of maximizing cohesion in a higher level of abstractness, i.e., the package level $i$. therefore, in this work, only packages at level $i$ and their respective elements at level $i+1$ are considered. ##### For calculating the package dependency graph’s modularity, we use the directed and weighted version of modularity [@arenas2007size] expressed as $$\begin{aligned} \label{eq:directed_weighted_mod} Q&=&\frac{1}{2w}\sum_{ij}[w_{ij} - \frac{w_i^{out}w_j^{in}}{2w}] \delta(C_i, C_j). \\\end{aligned}$$ where $w_i^{out}$ and $w_j^{in}$ are respectively the output and input weights of nodes $i$ and $j$ and $$\begin{aligned} \label{eq:wiout} w_i^{out} &=& \sum_{j}w_{ij} \\ w_j^{in} &=& \sum_{i}w_{ij} \\ 2w &=& \sum_{i} w_i^{out} = \sum_{j} w_j^{in} = \sum_{i} \sum_{j} w_{ij}.\end{aligned}$$ The weights for an edge is equal to the edge’s coupling metric given in Eq. \[eq:r\_gupta\]. These weights are used in the package dependency network, similar to the weights in uWCDN (Eq. \[eq:uwcdn\]) provided by Pan et al [@pan2013refactoring]. Considering the directedness of the network we can define an enhanced version of uWCDN, namely DWPDN (Directed, Weighted Package Dependency Network) that can be expressed as $$\label{eq:dwpdn} DWPDN = (V_{i+1}, E_{i+1}, W_{i+1})$$ where $V_{i+1}$ denotes the set of all vertices at hierarchy level $i+1$, $E_{i+1}$ denotes the set of all edges at hierarchy level $i+1$ and $W_{i+1}$ is the assymetric and weighted adjacency matrix of the network at hierarchy level $i+1$. Every element of $W_{i+1}$ can calculated as $$\label{eq:dwpdn_weights} w_{i+1}(j, k) = Coup(p_j^{i+1}, p_k^{i+1})$$ where $j$ and $k$ are two elements and $Coup$ is the coupling function from Eq. \[eq:coup\]. ##### The main phases of the proposed package refactoring algorithm are presented in Alg. \[alg:proposed\_refactoring\]. **Input:** A DWPDN\ **Output:** A list of package movement suggestions and the optimal $Q$ that can be gained $suggestedMovements \gets \varnothing$ $Q' \gets -1$ $Q \gets \text{modularity based on Eq. \ref{eq:directed_weighted_mod}}$ $selectedCommunity \gets 0$ $C_i \gets $ node $i$’s community $C_j \gets $ node $j$’s community $tempQ \gets \text{modularity, while considering node } i \text{ in }C_j$ $Q' \gets tempQ$ $selectedCommunity \gets C_j$ Add movement $(i, C_i, selectedCommunity)$ to suggestedMovements Move node $i$ to $selectedCommunity$ $Q \gets Q'$ $i \gets 1$ **return** $Q'$ and $suggestedMovements$ Evaluation ========== ##### This chapter evaluates the proposed algorithm with two case studies using our implemented package refactoring tool. The two subjects which the algorithm was applied on are the same open source subjects used in [@pan2013refactoring]. In the remaining sections of this chapter, the two subjects are briefly introduced and analyzed by the implemented tool. The proposed refactoring algorithm is applied on the two subjects and the results are evaluated. For simplicity, the first version of the implemented tool does not apply weights and considers all weights between classes to be one. Subjects -------- ##### The two subjects being analyzed in this chapter are the same as the subjects in [@pan2013refactoring], namely Trama[^3] and FrontEndForMySQL[^4]. ##### Trama is a graphical tool for manipulating and working with matrices. FrontEndForMySQL is a graphical front end for the MySQL database system and provides an easier and more user friendly environment for working with MySQL queries. Some details of the two subjects are shown in Table \[subjects\]. **System** Version Number of packages Number of classes ------------------ --------- -------------------- ------------------- -- Trama 1.0 6 58 FrontEndForMySQL 1.0 10 56 : Details of the systems analyzed[]{data-label="subjects"} ##### The original packaging structure for Trama is depicted in Fig. \[original\_trama\]. The original modularity calculated for the default packaging of Trama is calculated as 0.28 and the list of its packages is as follows. - visao - visao.renderizador - persistencia - negocio - negocio.leitor.Interface - negocio.leitor ![Original packaging structure of Trama[]{data-label="original_trama"}](trama_original.png) ##### FrontEndForMySQL is a larger system compared to Trama, with an initial package modularity of 0.21. The system’s default packaging structure is depicted in Fig. \[front\_original\] and it contains the following packages. - frontendformysql - frontendformysql.domain.BackEnd - frontendformysql.domain.BackEndData - frontendformysql.domain.BackEndComponent.Editor - frontendformysql.domain.BackEndInterfaces - frontendformysql.domain.BackEnd.System - frontendformysql.domain.BackEndComponent.DriverModule - frontendformysql.domain.BackEndData - frontendformysql.domain.BackEndComponent.XMLutil - frontendformysql.domain.BackEndComponent.IO - frontendformysql.domain.BackEndComponent.DataStructures - frontendformysql.domain.BackEndComponent.Editor - frontendformysql.domain.BackEndInterfaces - frontendformysql.domain.BackEnd.System - frontendformysql.domain.BackEndComponent.DriverModule - frontendformysql.domain.BackEndComponent.XMLutil - frontendformysql.domain.BackEndComponent.IO - frontendformysql.domain.BackEndComponent.DataStructure ![Original packaging structure of FrontEndForMySQL[]{data-label="front_original"}](trama_original.png) Case studies and results ------------------------ ##### After applying the proposed refactoring algorithm, with considering the importance of edge directions, the clustering of Trama changes to the depicted structure in Fig. \[refactor\_trama\] and the suggested movements are given in Table \[trama\_suggestions\]. The new packaging of Trama has a directed modularity of 0.43 and shows an improvement over the original 0.28. It is important to note that not all movements are acceptable and the suggestions should be given to a programmer for final analysis. **Order** **Class name** **Old package** **Suggested package** ----------- ---------------------- ----------------- ----------------------- -- 1 Main negocio visao 2 Matriz negocio persistencia 3 ModeloTabela visao persistencia 4 JTableCustomizado visao visao.renderizador 5 JTableCustomizado\$1 visao visao.renderizador 6 JTableCustomizado\$2 visao visao.renderizador 7 LeitorDeModelo negocio.leitor negocio 8 Tela\$23 visao persistencia 9 Tela\$22 visao persistencia 10 Tela\$24 visao visao.renderizador 11 Tela\$3\$1 visao visao.renderizador : Suggested movements for Trama classes[]{data-label="trama_suggestions"} ![New packaging of the Trama system after refactoring[]{data-label="refactor_trama"}](trama_directed.png) ##### As a comparison, an undirected version of the algorithm, using naive transformation, was applied on the Trama system. The produced clustering is shown in Fig. \[trama\_undirected\]. In this clustering, modularity gets a value of 0.41. It is important to note that comparing the modularity of the two approaches would not be correct, as the formula for the two quality measures are inherently different. However, a comparison on package instability is shown in Table \[trama\_instability\_comparison\], in which $OI$ is the original instability of a package, $DI$ is the instability of a package after the proposed refactoring algorithm with edge directions, is applied and $UI$ is the instability of a package after applying the undirected version of the algorithm. ![New packaging of the Trama system after refactoring with naive transformation[]{data-label="trama_undirected"}](trama_undirected.png) **Package name** **OI** **DI** **UI** --------------------------- -------- -------- -------- -- negocio 0.478 0.529 0.6 persistencia 0 0.368 0.409 visao.renderizador 0.428 0.538 0 negocio.leitor 0 0 0 visao 0.64 0.578 0.5 negacio.leitor.Intergface 0 0 0 : Comparison of Trama’s instability metric for different approaches[]{data-label="trama_instability_comparison"} ##### Table \[trama\_instability\_comparison\] shows how two packages became more stable after applying the proposed, directed clustering algorithm, while the stability of package *visao* decreased by 0.078. From Fig. \[trama\_undirected\], it is also clear that the *visao.renderizador* is merged with other packages and thus is not taken into account for comparison. ##### The implementation of the proposed algorithms was also applied to the FrontEndForMySQL system. The original package structure for FrontEndForMySQL and its structure after refactoring are depicted in Fig. \[original\_frontendformysql\] and Fig. \[refactor\_frontendformysql\] respectively. The original modularity for FrontEndForMySQL is calculated as 0.21. ![Original packaging of the FrontEndForMySQL system[]{data-label="original_frontendformysql"}](front_original.png) ![New packaging of the FrontEndForMySQL system after refactoring[]{data-label="refactor_frontendformysql"}](front_directed.png) ##### Similar to the previous case study, an undirected version of the algorithm, using a naive transformation for removing edge directions was applied to FrontEndForMySQL and the clustering result is depicted in Fig. \[front\_undirected\]. The comparison table for this package instability measures is given in Table \[front\_instability\_comparison\]. ![New packaging of the FrontEndForMySQL system after refactoring with naive transformation[]{data-label="front_undirected"}](front_undirected.png) **Package name** **OI** **DI** **UI** --------------------------------- -------- -------- -------- -- BackEndInterfaces 0 0 0.375 BackEnd 0.969 1 0.714 BackEnd.System 0.2 0 0 BackEndComponent.IO 0 0.2 0 BackEndComponent.XMLutil 0 0 0 BackEndComponent.Editor 0 0 0 BackEndComponent.DriverModule 0.818 0.25 0.25 BackEndComponent.DataStructures 0 0 0 frontendformysql 0.666 0 0.6 BackEndData 0.238 0.125 0.5 : Comparison of FrontEndForMySQL’s instability metric for different approaches[]{data-label="front_instability_comparison"} ##### Table \[front\_instability\_comparison\] clearly shows that the overall instability of packages is higher when edge directions are not taken into account in the refactoring algorithm. Live analysis of graph clusters =============================== ##### Considering the vast number of graph clustering applications in software engineering, a need for a tool that can import different graph modeled structures, perform graph clustering algorithms and provide a rich client for tweaking the properties of the model, is clearly felt. This need has motivated us to create a tool, namely Picasso, with such capabilities. Fig. \[picasso\_overview\] shows an overall view of this tool while analyzing a software package. Colors are used to show the different communities inside the graph. The server-side and client-side codes of this tool are given in Appendices A and B respectively. Picasso overview ---------------- ![Picasso: A tool for live package dependendy analysis[]{data-label="picasso_overview"}](picasso_overview.png) ##### Picasso applies the proposed refactoring algorithm on software packages and provides a list of class moving suggestions. An example of the suggestions that Picasso presents is depicted in Fig. \[picasso\_suggestions\]. Every suggestion is a class movement from a source package to a target package. ![An example of some suggestions provided by Picasso[]{data-label="picasso_suggestions"}](picasso_suggestions.png) ##### Picasso provides many extra features that are as follows. - Import Java jar files and class files. - Import UML structures. - Provides an option to choose famous graphs such as the Zachary club network. - Calculates modularity and provides a refactored solution for a software system using Alg. \[alg:proposed\_refactoring\]. - Calculates Martin’s instability metric for software packages. - Hierarchically provides cluster graphs of a graph. - Provides an extendible messaging system for future works. - Provides an edited version of JSNetworkX’s force layout graph visualization algorithm. - Provides functions for adding and removing graph edges and nodes. - Provides the ability to lock graph nodes in one position for better viewing. ##### Picasso’s top menu provides the main functionalities of the tool. The menu bar is depicted in Fig. \[picasso\_menu\] and shows that the tool is in working mode and awaits a response from the Picasso server. The gray section of the top bar shows some information such as the modularity measure of the current clustering and the name of the current selected class in the dependency graph. The top buttons consist of two main groups. The left, green buttons provide directed refactoring, undirected refactoring and the original clustering of the software system being analyzed. The right, blue buttons provide the options for viewing the graph’s clustering graph, viewing the movement suggestions after refactoring and viewing instability measures for different packages. An example of the instability measure window is shown in Fig. \[picasso\_inst\]. ![Picasso’s top menu bar[]{data-label="picasso_menu"}](features_bar.png) ![An example of Picasso’s instabilities window[]{data-label="picasso_inst"}](picasso_inst.png) ![Picasso’s sequence diagram[]{data-label="seq_diagram"}](seq.eps) Picasso’s 3rd party dependencies -------------------------------- ##### Picasso utilizes many diverse 3rd party libraries. Some of these libraries have been customized and tweaked specially for Picasso. The following list contains some brief information on these libraries. - **Coffea[^5] java analysis tool.** Coffea is an open source static code analyzer for Java byte code that can export package dependency graphs in various graph file formats. Coffea is written in Python and therefore can be integrated well with Picasso. - **D3[^6] visualization library.** D3 stands for Data-Driven Documents, and is arguably one of the best Javascript data visualization tools that utilizes HTML5, SVG (Scalable Vector Graphics), CSS3 and Javascript capabilities and provides an extremely flexible platform for data visualization. - **JSNetworkX[^7] network visualization library.** This library is a port of the popular NetworkX Python graph library and is build upon the D3 platform. - **Python’s igraph[^8] library.** Python’s igraph library is used in Picasso for creating and manipulating graphs on the server side. - **Python’s Socks-js[^9] library.** The Socks-JS library is used by Picasso for creating a web socket messaging system that can pass graph and graph cluster information between the server and client sides of the program. ##### The sequence diagram in Fig. \[seq\_diagram\] shows how Picasso interacts with these dependencies. Conclusion and future works =========================== ##### The fast expansion of software systems and their complexities, makes large software projects difficult to maintain, and their components hard to reuse. The focus of this work is to use the benefits of graph clustering algorithms and present a refactoring technique for software packages while considering several important software metrics such as coupling, cohesion and stability. ##### This work presents a proposition and proof that the cluster quality metric provided by Newman [@newman2004finding] is in favor of Martin’s Stable Dependencies Principle [@martin2003agile] and provides examples that show how directed graphs are important when a system is being modeled with dependency graphs. ##### For evaluating our proposed algorithm and to test it in a real life scenario, we implemented a tool, namely Picasso for refactoring software packages and visualizing their directed dependency graphs. The provides tool takes a software system written in the Java language and gives a list of suggested movements for classes. ##### Some ideas are presented in the following sections as future works. These possible works are divided into two main categories; refactoring and tool improvements. Refactoring ----------- ##### The refactoring method presented in this work utilizes a directed and weighted version of Newman’s modularity. This requires modularity to be calculated in every step of the proposed algorithm and thus performs slower than the algorithm of Pan et al [@pan2013refactoring]. This may be considered as one of the problems that can be tackled in future works. Also, some rare problems have been found with the directed version of modularity [@malliaros2013clustering] and alternative approaches should also be considered, i.e. random walk based mathods such as LinkRank. ##### The importance of directed dependency graphs can also be analyzed in the class level, while using an appropriate metric for class couplings and cohesion. Tool improvements ----------------- ##### Some improvements can be applied on the tool proposed in this work. Currently a force directed layout is used for visualizing graphs. A force directed layouts simulate physical forces between nodes and edges to aesthetically draw a graph. Spring like attractive forces that are based on Hooke’s law are typically used. The force directed layout can be enhanced with collision detection algorithms, so that nodes that are members of the same community can be grouped together instead of being mixed in with nodes from other communities. Also several problems with force directed layouts in large graphs have been pointed out in the literature [@yakovlev2009cluster] and radial tree layouts have been proposed as alternatives. Radial tree layouts can be considered in future implementations of the tool. An example of a radial tree layout from a tool named Barrio, provided in [@yakovlev2009cluster] is depicted in Fig. \[radial\_tree\]. ![An example of a radial tree layout[]{data-label="radial_tree"}](radial_tree.png) ##### Being able to force a node to be a member of a certain community while calculating the resulting modularity of the graph cluster can be considered as one of the important options in future versions of the application. Some library classes might need to be kept in their original package even though modularity is decreased by doing so. Server side code for Picasso ============================ ``` {language="python"} # -*- coding: utf-8 -*- """ Picasso, ver 0.1 Author: Mohammad A.Raji Depends on: -sockjs-tornado for the asynchronus python server -D3.js for visualizing graphs -JSNetworkX for visualizing graphs -igraph for community detection algorithms -Coffea for extracting java dependencies """ from __future__ import division import os import tornado.ioloop import tornado.web import sockjs.tornado import igraph import time import json import hashlib from igraph import * # Request handles class for the index page class IndexHandler(tornado.web.RequestHandler): def get(self): self.render('picasso.html') # Connection class: responsible for all the client/server connections class Connection(sockjs.tornado.SockJSConnection): participants = set() def on_open(self, info): # Add client to the clients list self.participants.add(self) if len(sys.argv) > 1: if len(sys.argv) > 2: if sys.argv[1] == "--famous": g = Graph.Famous(sys.argv[2]) #g.to_undirected() self.broadcast(self.participants, "graph/" + refactoring.graphToString()) else: refactoring.parseCode(sys.argv[1]); self.broadcast(self.participants, "graph/" + refactoring.graphToString()) self.broadcast(self.participants, "labels/" + refactoring.getVertexLabels()) def on_message(self, message): # Take appropriate action when a message arrives from the client self.parseAndApplyMessage(message) def on_close(self): # Remove client from the clients list and broadcast leave message self.participants.remove(self) def parseAndApplyMessage(self, msg): global refactoring message = msg.split("/") command = message[0] if (len(message) > 1): argument = message[1]; if command == "clusters": refactoring.parseGraph(argument) refactoring = Refactoring(refactoring.detectCommunities().cluster_graph()) self.broadcast(self.participants, "graph/" + refactoring.graphToString()) elif command in ["addnode", "removenode", "addedge", "removeedge"]: refactoring.parseChange(command, argument) elif command == "getoriginal": self.broadcast(self.participants, "membership/" + str(refactoring.original_membership).strip("[]")) self.broadcast(self.participants, "measures/" + Refactoring.formatMeasures(refactoring.original_modularity)); self.broadcast(self.participants, "instability/" + Refactoring.formatInstability(refactoring.original_package_instability)); elif command == "refactor": refactored_results = refactoring.refactor() go_membership = refactored_results[1] self.broadcast(self.participants, "membership/" + str(go_membership).strip("[]")) self.broadcast(self.participants, "measures/" + Refactoring.formatMeasures(refactored_results[0])); self.broadcast(self.participants, "suggestions/" + Refactoring.formatSuggestions(refactored_results[2])); self.broadcast(self.participants, "instability/" + Refactoring.formatInstability(refactoring.getInstabilityForEachPackage(refactoring.g, go_membership, refactoring.packages))); elif command == "urefactor": refactored_results = refactoring.refactor(False) go_membership = refactored_results[1] self.broadcast(self.participants, "membership/" + str(go_membership).strip("[]")) self.broadcast(self.participants, "measures/" + Refactoring.formatMeasures(refactored_results[0])); self.broadcast(self.participants, "suggestions/" + Refactoring.formatSuggestions(refactored_results[2])); self.broadcast(self.participants, "instability/" + Refactoring.formatInstability(refactoring.getInstabilityForEachPackage(refactoring.g, go_membership, refactoring.packages))); elif command == "fastgreedy": go_membership = refactoring.detectCommunities().membership self.broadcast(self.participants, "membership/" + str(go_membership).strip("[]")) self.broadcast(self.participants, "measures/" + refactoring.getClusterMeasures()); else: refactoring.parseGraph(command) go_membership = refactoring.detectCommunities().membership self.broadcast(self.participants, "membership/" + str(go_membership).strip("[]")) self.broadcast(self.participants, "measures/" + refactoring.getClusterMeasures()); # All refactoring and graph related capabilities class Refactoring(): g = None; gc = None; parsed_code_filename = None; original_membership = None; original_modularity = None; def __init__(self, graph=None): self.packages = dict() self.original_package_instability = dict() self.g = graph def parseChange(self, command, arg): if command == "addnode": self.g.add_vertex(arg) elif command == "removenode": self.g.delete_vertices(arg) elif command == "addedge": from_edge = int(arg.split(",")[0]) to_edge = int(arg.split(",")[1]) self.g.add_edge(from_edge, to_edge) elif command == "removeedge": from_edge = int(arg.split(",")[0]) to_edge = int(arg.split(",")[1]) self.g.delete_edges((from_edge, to_edge)) def parseGraph(self, st): self.g = Graph() st_graph = st.split("|") vertices = st_graph[0].split(";") for v in vertices: self.g.add_vertex(v) edges = st_graph[1].split(";") for e in edges: from_edge = e.split(",")[0] to_edge = e.split(",")[1] self.g.add_edge(from_edge, to_edge) return self.g def graphToString(self): vertexlist = [] for v in self.g.vs: vertexlist.append(v.index) vertex_str = str(vertexlist).strip("[]"); vertex_str = vertex_str.replace(" ", ""); s = str(self.g.get_edgelist()).strip("[]"); s = s.replace("(", ""); s = s.replace("),", ";"); s = s.replace(")", ""); s = s.replace(" ", ""); s = vertex_str + "|" + s return s @staticmethod def formatMeasures(measure): measures = ""; measures += "modularity:" + str(round(measure, 2)) # + "," return measures; @staticmethod def formatSuggestions(suggestions): return json.dumps(suggestions) @staticmethod def formatInstability(package_instability): return json.dumps(package_instability.items()) def getClusterMeasures(self): if (self.gc == None): self.gc = self.detectCommunities() measures = ""; measures += "modularity:" + str(round(self.gc.modularity, 2)) # + "," return measures; def getVertexLabels(self): msg = ""; for v in self.g.vs: msg = msg + v['label'] + ","; msg = msg.strip(",") return msg def detectCommunities(self): self.g = self.g.simplify(loops='False', multiple='False') gc = self.g.as_undirected().community_fastgreedy() gc = gc.as_clustering() self.gc = gc return gc # This function works independently from local graph g def makeDwpdnMembership(self, graph): # If this graph has no label attribute at all if "label" not in graph.vs.attribute_names(): custom_package_index = 0 for v in graph.vs: v['label'] = str(custom_package_index) + "." custom_package_index += 1 self.packages = dict() membership = [] recent_package = 0; for v in graph.vs: if v['label'] == None: # Make a random package name if this package is a new isolated # node with no name random_package_name = hashlib.md5(str(time.time())).hexdigest()[0:5] + "." v['label'] = random_package_name package_name = v['label'].rsplit(".", 1)[0]; if (self.packages.has_key(package_name)): membership.append(self.packages[package_name]); else: self.packages[package_name] = recent_package; membership.append(recent_package); recent_package += 1; return membership # This function works independently from local graph g def calculateQ(self, graph, membership): Q = 0.0; graph = graph.simplify(loops='False', multiple='False') m = graph.ecount(); edge_count_factor = 2*m; if graph.is_directed() == True: edge_count_factor = m for i in graph.vs: for j in graph.vs: if membership[i.index] != membership[j.index]: continue else: Aij = 0 if graph.are_connected(i, j): Aij = 1 wi_out = i.outdegree() wj_in = j.indegree() Q += Aij - (wi_out*wj_in) / edge_count_factor Q *= 1/edge_count_factor return Q; def getInstabilityForEachPackage(self, graph, membership, packages): package_in = packages.fromkeys(packages.iterkeys(), 0); package_out = packages.fromkeys(packages.iterkeys(), 0); package_instability = packages.fromkeys(packages.iterkeys(), 0); for e in graph.es: if (membership[e.target] != membership[e.source]): package_in[self.getPackageNameFromIndex(membership[e.target])] += 1; package_out[self.getPackageNameFromIndex(membership[e.source])] += 1; print package_out print package_in for package in packages: if package_out[package] + package_in[package] != 0: package_instability[package] = package_out[package] / (package_out[package] + package_in[package]) else: package_instability[package] = 0; return package_instability def getPackageNameFromIndex(self, index): for name, i in self.packages.iteritems(): if i == index: return name def refactor(self, directed=True): if directed == True: graph = self.g; else: graph = self.g.as_undirected() suggested_movements = []; Q_prime = -1; membership = self.makeDwpdnMembership(graph); # Check if there is only one package if membership.count(0) == len(membership): membership = range(len(membership)) Q = self.calculateQ(graph, membership); selected_community = 0; v_range = range(graph.vcount()) while True: restart_loop = False for index in v_range: i = graph.vs[index] for j in graph.vs: temp_membership = list(membership) temp_membership[i.index] = temp_membership[j.index]; temp_Q = self.calculateQ(graph, temp_membership); if (temp_Q > Q_prime): Q_prime = temp_Q; selected_community = membership[j.index]; if (Q_prime > Q): suggested_movements.append((i['label'], self.getPackageNameFromIndex(membership[i.index]), self.getPackageNameFromIndex(selected_community))); membership[i.index] = selected_community; Q = Q_prime; restart_loop = True break; if not restart_loop: break; print "Done refactoring" return [Q_prime, membership, suggested_movements] def parseCode(self, filename): os.system("coffea -R -i " + filename + " -f gml -o temp.gml") self.parsed_code_filename = filename self.g = read('temp.gml') self.original_membership = self.makeDwpdnMembership(self.g) self.original_modularity = self.calculateQ(self.g, self.original_membership); self.original_package_instability = self.getInstabilityForEachPackage(self.g, self.original_membership, self.packages) if __name__ == "__main__": import logging logging.getLogger().setLevel(logging.DEBUG) # Instantiate the main refactoring object refactoring = Refactoring(); # Create the router Router = sockjs.tornado.SockJSRouter(Connection, '/picasso') # Create Tornado application app = tornado.web.Application( [(r"/", IndexHandler)] + Router.urls ) # Make Tornado app and listen on port 8081 port = 8081 app.listen(port) print "Listening on port " + str(port); # Start IOLoop tornado.ioloop.IOLoop.instance().start() ``` Client side Javascript of Picasso ================================= ``` {language="javascript"} last = 1; conn = null; labels = []; $(function() { colors = ['#FF7F0E', '#AEC7E8', '#2CA02C', '#D62728', '#1F77B4'] color = window.d3.scale.category20(); function log(msg) { console.log(msg); } function parseAndApplyMessage(msg) { var message = msg.split("/"); var command = message[0]; if (message.length > 1) { var arguments = message[1]; if (command == "membership") { applyMembership(arguments); } else if (command == "graph") { applyGraph(arguments); } else if (command == "labels") { saveLabels(arguments); } else if (command == "measures") { updateMeasures(arguments); } else if (command == "suggestions") { setSuggestions(arguments); $("#refactor-btn").button("reset"); $("#urefactor-btn").button("reset"); } else if (command == "instability") { setInstabilities(arguments); } } } function setSuggestions(msg) { var suggestions = JSON.parse(msg); txt = "<ol>"; for (i in suggestions) { txt += "<li>Move class <span class='class-name'>" + suggestions[i][0] + "</span> from package <span class='package-name'>" + suggestions[i][1] + "</span> to package <span class='package-name'>" + suggestions[i][2] + "</span></li>"; } txt += "</ol>" $(".modal-body").html(txt); } function setInstabilities(msg) { var instabilities = JSON.parse(msg); txt = "<ol>"; for (i in instabilities) { txt += "<li>" + instabilities[i][0] + ": " + instabilities[i][1] + "</li>"; } txt += "</ol>" $("#instability .modal-body").html(txt); } function updateMeasures(msg) { $("#measures").text(msg); } function saveLabels(msg) { labels = msg.split(","); for (var i = 0;i<G.nodes().length;i++) { d3.select("#node" + i.toString()).attr("data-label", labels[i]); } } function applyGraph(msg) { G.clear(); var splitted_str = msg.split("|") var vertex_str = splitted_str[0]; var edges_str = splitted_str[1]; var vertices = vertex_str.split(","); for (key in vertices) { var vertex = parseInt(vertices[key]); G.add_node(vertex); } var edges = edges_str.split(";"); for (key in edges) { var edge = edges[key]; var from = parseInt(edge.split(",")[0]); var to = parseInt(edge.split(",")[1]); G.add_edge(from, to); } window.d3.selectAll(".node").on("mouseover", function(){ jQuery("#label").text(d3.select(this).attr("data-label")); }); } function applyMembership(msg) { membership = msg.split(", "); iteration = G.nodes_iter(); for (key in membership) { node = iteration.next(); G.node.get(node).color = color(membership[key]); window.d3.select("#node" + node.toString() + " circle" ).style("fill", color(membership[key])); } } function connect() { disconnect(); var transports = $('#protocols input:checked').map(function(){ return $(this).attr('id'); }).get(); conn = new SockJS('http://' + window.location.host + '/picasso', transports); log('Connecting...'); conn.onopen = function() { log('Connected.'); update_ui(); }; conn.onmessage = function(e) { //log('Received: ' + e.data); parseAndApplyMessage(e.data); }; conn.onclose = function() { log('Disconnected.'); conn = null; update_ui(); }; } function disconnect() { if (conn != null) { log('Disconnecting...'); conn.close(); conn = null; update_ui(); } } function update_ui() { var msg = ''; if (conn == null || conn.readyState != SockJS.OPEN) { $('#status').text('disconnected'); $('#connect').text('Connect'); } else { $('#status').text('connected (' + conn.protocol + ')'); $('#connect').text('Disconnect'); } } $('#connect').click(function() { if (conn == null) { connect(); } else { disconnect(); } update_ui(); return false; }); connect(); addNode = function(node_name) { G.add_node(node_name); conn.send(G.nodes().join(";") + "|" + G.edges().join(";")); } addEdge = function(from, to) { G.add_edge(from, to); conn.send(G.nodes().join(";") + "|" + G.edges().join(";")); } removeEdge = function(from, to) { G.remove_edge(from, to); conn.send(G.nodes().join(";") + "|" + G.edges().join(";")); } drawGraph = function() { jsnx.draw(G, { element: '#canvas', with_labels: true, pan_zoom: { enabled: false }, layout_attr: { 'charge': -420, 'linkDistance': 100 }, node_style: { fill: function(d) { return d.data.color || '#AAA'; }, stroke: 'none' }, edge_style: { fill: '#999' }, label_style: { fill: 'white', 'font-size': '12px' } }, true); } G = jsnx.DiGraph(); drawGraph(); }); ``` [^1]: <http://git-scm.com> [^2]: <http://subversion.apache.org> [^3]: <http://trama.sourceforge.net> [^4]: <http://frontend4mysql.sourceforge.net> [^5]: <https://github.com/sbilinski/coffea> [^6]: <http://d3js.org> [^7]: <http://felix-kling.de/JSNetworkX> [^8]: <http://igraph.org> [^9]: <https://github.com/sockjs/sockjs-client>

--- address: - 'Rice University, Houston, Texas, 77005-1892' - 'SUNY at Buffalo, Buffalo,N.Y.' author: - 'Tim D. Cochran' - Joseph Masters title: 'The Growth Rate of the First Betti Number in Abelian Covers of $3$-Manifolds ' --- [^1] Abstract {#abstract .unnumbered} ======== We give examples of closed hyperbolic 3-manifolds with first Betti number $2$ and $3$ for which no sequence of finite abelian covering spaces increases the first Betti number. For $3$-manifolds $M$ with first Betti number $2$ we give a characterization in terms of some generalized self-linking numbers of $M$, for there to exist a family of $\mathbb{Z}_n$ covering spaces, $M_n$, in which $\beta _1(M_n)$ increases linearly with $n$. The latter generalizes work of M. Katz and C. Lescop \[KL\], by showing that the non-vanishing of any one of these invariants of $M$ is sufficient to guarantee certain optimal systolic inequalities for $M$ (by work of Ivanov and Katz \[IK\]). Introduction {#introduction .unnumbered} ============ Motivated by Waldhausen’s work on Haken manifolds, and by W. Thurston’s [**Geometrization Conjecture**]{}, it has been variously conjectured that, if $M$ is an orientable, irreducible closed $3$-manifold with infinite fundamental group, then: $M$ is finitely covered by a Haken manifold; Some finite cover of $M$ has positive first Betti number; Either $\pi_1(M)$ is virtually solvable or $M$ has finite covers with arbitrarily large first Betti number; $M$ has a finite cover that fibers over the circle. There are easy implications VIBNC$\Longrightarrow$VPBNC$\Longrightarrow$VHC and VFC$\Longrightarrow$VPBNC $\Longrightarrow$VHC. Each implies, if $M$ is atoroidal, the long-standing conjecture of Thurston that such a manifold admits a geometric structure. It is interesting to note that even if $M$ is [**assumed**]{} to be hyperbolic, the conjectures above are open. In this paper, we restrict our attention to VIBNC. (We note in passing that the alternative “$\pi_1(M)$ is virtually solvable” is sometimes replaced by the a priori stronger alternative that “$M$ is finitely covered by the 3-torus, a nilmanifold or a solvmanifold.”) One rich source of finite covering spaces is the set of iterated (regular) finite [**abelian**]{} covering spaces. Thus specifically, in this paper we consider the question: Does there exist an integer $m$, such that, if $M$ is any closed, atoroidal $3$-manifold with $\beta_1(M) \geq m$ then $\b_1(M)$ can be increased in a finite abelian covering space? Note that some condition on $H_1(M)$ is necessary, for if $H_1(M)=0$, then $M$ admits no non-trivial abelian covering spaces. Counter-examples also exist for many manifolds with $\b_1(M)=1$. For if $M$ is zero-framed surgery on a knot in $S^3$, then it is easy to show that $H_1(\wt M;\BQ)\cong\BQ\op Q[t,t^{-1}]/\<\Delta_k,t^n-1\>$ where $\wt M$ is the $n$-fold cyclic cover and $\Delta_k$ is the Alexander polynomial of $K$. Thus $\b_1(\wt M)=\b_1(M)=1$ except when $\Delta_k$ has a cyclotomic factor. We begin this paper by observing that counter-examples also exist in the cases $\b_1(M)=2$ and $\b_1(M)=3$. \[mainthm1\] *There exist closed hyperbolic $3$-manifolds $M$ with $\b_1(M)=2$ (respectively $3$) for which no sequence of finite abelian covers increases the first Betti number. More generally, if a sequence of regular covers of M increases the first Betti number, then one of the covering groups contains a non-trivial perfect subgroup.* It is noteworthy that Question A is still open. If $\b_1 > 0$, then there is an epimorphism $\pi_1(M)\to \mathbb{Z}$, and a corresponding sequence of finite cyclic covers of $M$. Our second contribution is, in the case $\b_1(M)=2$, to give necessary and sufficient conditions, of a somewhat geometric flavor, for the Betti number of these covers to increase linearly with the covering degree. This is the content of Section 2. On abelian covers of hyperbolic 3-manifolds with $\b_1(M)=2$ and $3$ {#failure} ==================================================================== In this section, we observe that, if Question A has an affirmative answer, then the integer $m$ must be at least 4. \[failure23\] There exist closed hyperbolic $3$-manifolds $M$ with $\b_1(M)=2$ (respectively $3$) such that if $\wt M$ is obtained from $M$ by taking a sequence of finite abelian covering spaces, then $\b_1(\wt M)=\b_1(M)$. More generally, if a sequence of regular covers of M increases the first Betti number, then one of the covering groups contains a non-trivial perfect subgroup. Begin with a “seed” manifold $N$ whose fundamental group is nilpotent. Recall that the [**Heisenberg manifold**]{} with Euler class $e$ is the circle bundle over the torus with Euler class $e$. The fundamental group of such a $3$-manifold is the nilpotent group $\langle x,y,t : [x,y]=t^e, [x,t], [y,t]\rangle$, called the [**Heisenberg group**]{} of Euler class $e$. For our seed manifold with $\b_1(N)=2$, we shall take $N$ to be the Heisenberg manifold with Euler class $1$, that can also be described as $0$-framed surgery on a Whitehead link. Thus in this case $\pi_1(N)\cong F/F_3$ where $F$ is the free group of rank 2 and $F_3$ is the third term of the lower-central series of $F$. When $\b_1(N)=3$, we take our seed manifold $N$ to be $S^1\x S^1\x S^1$, the Heisenberg manifold of Euler class $0$. Note that each of the Heisenberg groups of non-zero Euler class has $\beta_1=2$ while the Heisenberg group of Euler class $0$ has $\beta_1=3$. First we claim that *no* finite cover of $N$ will increase the first Betti number, which follows immediately from the Lemma below (which surely is well-known to experts). \[nilpotent\] Suppose $A$ is a Heisenberg group with non-zero (respectively zero) Euler class. If $\wt A$ is any finite index subgroup of $A$, then $\wt A$ is a Heisenberg group of non-zero (respectively, zero) Euler class. Hence in all cases $\beta_1(\wt A) = \beta_1(A)$. The result is obvious for $A=\mathbb{Z}\x \mathbb{Z}\x \mathbb{Z}$ so we assume that $A$ is a Heisenberg group of non-zero Euler class $e$. Then $A$ is a central extension as shown below. $$1\lra \mathbb{Z}\overset{i}{\lra} A \overset{\pi}{\lra} \mathbb{Z}\times\mathbb{Z}\lra 1$$ Since $\wt A$ is a finite index subgroup of $A$, $\pi(\wt A)$ is a finite index subgroup of $\mathbb{Z}\times\mathbb{Z}$ which is hence isomorphic to $\mathbb{Z}\times\mathbb{Z}$. Moreover the kernel of the map $\pi:\wt A\to \pi(\wt A)$ is a finite index subgroup of kernel($\pi$)$=\mathbb{Z}$ which is contained in the center of $A$. It follows that $\wt A$ is also a central extension of the above form and hence is also a Heisenberg group. We claim that $\wt A$ has non-zero Euler class. Suppose not. Then $\wt A$ is abelian. But $A\cong \langle x,y,t : [x,y]=t^e, [x,t], [y,t]\rangle$ where $e\neq 0$. Consider the elements $\{x,y\}$. There is some positive integer $n$ such that both $x^n$ and $y^n$ lie in the subgroup $\wt A$ where they commute. Thus $[x^n,y^n]=1$ in $A$. However since $[x,y]=t^e$, and $t$ commutes with $x$ and $y$, it is easy to see that $x^ny^n=t^ky^nx^n$ where $k=n^2e$ and so $1=[x^n,y^n]=t^k$. This implies that $t$ is of finite order. However, any Heisenberg group is the fundamental group of a circle bundle over the torus, which is an aspherical 3-manifold. Thus $A$ has geometric dimension $3$ and cannot have torsion, for a contradiction. Next alter the seed manifold in a subtle way using the following result of A. Kawauchi \[Ka1 p. 450-452 , and Ka2 Corollary 4.3\] (see also Boileau-Wang \[BW section 4\]). \[hyperbolic\] (Kawauchi) For any closed $3$-manifold $N$, there exists a hyperbolic $3$-manifold $M$ and a degree 1 map $f:M\to N$ that induces an isomorphism on homology groups with local coefficients in $\pi_1(N)$. Equivalently, if $\wt N$ is any covering space of $N$ and $\wt f:\wt M \to \wt N$ is the pull-back, then $\wt f$ induces isomorphisms on homology groups. To the best of our knowledge, this result was first established by Kawauchi using his theory of [**almost identical imitations**]{}. We sketch a proof using the approach of Boileau and Wang (which overlaps substantially with Kawauchi’s approach). Recall that any $3$-manifold $N$ contains a knot $J$ whose exterior is hyperbolic. With more work, Boileau and Wang ensure that there exists such a knot $J$ which is “totally null-homotopic”, i.e., bounds a map of a 2-disk, $\phi:D^2\to N$, such that the inclusion map $\pi_1({\operatorname{image}}\phi)\to\pi_1(N)$ is trivial. Let $M_n$ be the result of $1/n$-Dehn surgery on $N$ along $J$. By work of W. Thurston, for almost all $n$, $M_n$ is hyperbolic. Choose such an $M_n$ and denote it by $M$. Since $J$ is null-homotopic there is a degree one map $f:M\to N$ that induces an isomorphism on $H_1$. Let $\wt N$ be a cover of $N$. Since $J$ is null-homotopic, it lifts to $\wt N$, and there is an induced cover $\wt M$ and an induced map $\tl f:\wt M\to\wt N$. Since $J$ is totally null-homotopic, the pre-images of $J$ bound disjoint Seifert surfaces in $\wt M$, and so $\tl f:\wt M\to\wt N$ is an isomorphism on homology. For any map $f: M \to N$ satisfying the conclusion of Proposition \[hyperbolic\], $ker(f_*)$ is a perfect group. Indeed, Proposition \[hyperbolic\] states that for [**any**]{} covering space $\wt M$ of $M$ that is “induced” from a cover $\wt N$ of $N$, the induced map $\tl f:\wt M\to\wt N$ is an isomorphism on homology, so $\b_1(\wt M)=\b_1(\wt N)$. Specifically, letting $\wt N$ be the universal cover, $H_1(\wt M)\cong H_1(\wt N)=0$ showing that that $\pi_1(\wt M)$ is a perfect group. But $\pi_1(\wt M)$ is kernel$(f_*)$. (Indeed, the condition that $f:M\to N$ induce an isomorphism on first homology with local coefficients in $\pi_1(N)$ is equivalent to the condition that the kernel of $f_*:\pi_1(M)\to\pi_1(N)$ be a perfect group). Returning to the proof of our theorem, recall that $N$ is our seed Heisenberg manifold, and let $M$ be the manifold guaranteed by Proposition \[hyperbolic\]. We claim that the manifold $M$ satisfies the conclusion of the theorem. For suppose $\wt M\overset{p}{\lra}M$ is a regular finite covering space of $M$ corresponding to a surjection $\psi:\pi_1(M)\to F$, where $F$ is a finite group that contains no nontrivial perfect subgroup (for example if $F$ is abelian). Then, since the kernel of $f_*:\pi_1(M)\to\pi_1(N)$ is a perfect group $P$, and the perfect subgroup $\psi(P)\subset F$ must be trivial, $\psi$ factors through $f_*:\pi_1(M)\to\pi_1(N)$ via a surjection $\phi:\pi_1(N)\to F$. Therefore there is a finite regular cover $\wt N$ of $N$ and a lift $\tl f:\wt M\to\wt N$. Notice that the only property of $M$ and $N$ needed for this argument is that the kernel of $f_*:\pi_1(M)\to\pi_1(N)$ is a perfect group. Proceeding, by Proposition \[hyperbolic\] $H_1(\wt M)\cong H_1(\wt N)$ and by Lemma \[nilpotent\], $\beta_1(\wt N)=\beta_1(N)$. Since $\beta_1(M)=\beta_1(N)$ we conclude that $\beta_1(\wt M)=\beta_1(M)$. This shows that the first Betti number of $M$ cannot be increased by a *single* regular $F$-cover unless $F$ contains a nontrivial perfect subgroup. In particular, it shows that the first Betti number of $M$ cannot be increased by a single *abelian* cover. Now suppose that $M_k \to ... \to M_0 = M$ is a sequence of regular covers, with covering groups $F_1, ..., F_k$, where no $F_i$ contains a nontrivial perfect subgroup. In the last paragraph we showed that the cover $M_1 \to M_0$ is the pull-back of a corresponding cover $N_1 \to N_0$. We claim that the kernel, $P_1$, of the lift $(f_1)*:\pi_1(M_1)\to\pi_1(N_1)$ is *equal to* the kernel, $P_0$, of $f_*:\pi_1(M_0)\to\pi_1(N_0)$ (here we view $\pi_1(M_1)$ as a subgroup of $\pi_1(M_0)$). For, obviously $P_1 \subset P_0$ and since $F_0$ contains no perfect subgroups, $P_0\subset P_1$. Thus $P_1$ is a perfect group and thus $\wt f_1$ induces an isomorphism on homology (even with twisted coefficients). Thus we have recovered the inductive hypothesis of the previous paragraph and continuing inductively, we get a sequence of finite covers $N_k \to ... \to N_1$, with $\b_1(M_k) = \b_1(N_k)$. Therefore, to finish the proof we only need to observe that $\b_1(N)$ cannot be increased by any sequence of finite covers, which was shown in Lemma \[nilpotent\]. Linear Growth of Betti Numbers in Cyclic Covering Spaces {#lineargrowth} ======================================================== In this section we ask whether or not it is possible to increase the first Betti number with *linear growth rate* in some *compatible family* of cyclic covering spaces. If $M_\infty$ is a fixed infinite cyclic covering space corresponding to an epimorphism $\psi :\pi_1(M)\to \mathbb{Z}$ then by a *compatible family* we mean the usual family of finite cyclic covers $M_n$ associated to $\pi_1(M)\to \mathbb{Z}\to \mathbb{Z}_n$. By a *linear growth rate* we mean $\varinjlim (\beta_1(M_n)/n)$ is positive. It was already known that linear growth occurs precisely when $H_1(M_\infty)$ has positive rank as a $\mathbb{Z}[t,t^{-1}]$-module [@Lu2 Theorem 0.1][@Lu1 pg.35 Lemma 1.34,pg.453]. Therefore our contribution is to offer a more geometric way of viewing this criterion. We also point out an application to certain optimal systolic inequalities for such $3$-manifolds as have appeared in work of Katz \[IK\]\[KL\]. One should note from the outset that if $\pi_1(M)$ admits an epimorphism to $\mathbb{Z}\ast\mathbb{Z}$, then it is an easy exercise to show that $\beta_1(M)$ can be increased linearly in finite cyclic covers since the same is patently true of the wedge of two circles. Such manifolds arise, for example, as $0$-framed surgery on $2$-component boundary links. This condition is not necessary, however, as we shall see in Example \[example3\] below. Suppose $M$ is a closed, oriented $3$-manifold with $\b_1(M)=2$. Given any basis $\{x,y\}$ of $H^1(M,\BZ)$ we shall define a sequence of higher-order invariants $\b^n(x,y)$; $n\ge1$ taking values in sets of rational numbers. The invariants can be interpreted as certain Massey products in $M$. The invariant $\b^1(x,y)$ is always defined, is independent of basis, and essentially coincides with the invariant $\la$, an extension of Casson’s invariant, due to Christine Lescop \[Les\]. If $\b^i$ is defined for all $i<n$ and is zero, then $\b^n$ is defined (this is why the invariants are called higher-order). If $H_1(M)$ has no torsion, so that $M$ can be viewed as $0$-framed surgery on a 2-component link in a homology sphere (with Seifert surfaces dual to $\{x,y\}$) then $\b^n$, when defined, is the same as the sequence of link concordance invariants of the same name due to the first author \[C1\]. In this case $\b^1$ was previously known as the Sato-Levine invariant. After defining the invariants $\b^n(x,y)$, we show that their vanishing is equivalent to the linear growth of Betti numbers in the family corresponding to the infinite cyclic cover associated to $x$. \[linear\] Let $M$ be a closed oriented $3$-manifold with $\b_1(M)=2$. The following are equivalent. 1. There exists a compatible family $\{M_n|n\ge1\}$ of finite cyclic covers of $M$ such that $\b_1(M_n)$ grows linearly with $n$. 2. There exists a primitive class $x\in H^1(M;\BZ)$ such that for $\textbf{any}$ basis $\{x,y\}$ of $H^1(M;\BZ)$, $\b^n(x,y)=0$ for all $n\ge1$. 3. There exists a primitive class $x\in H^1(M;\BZ)$ such that for $\textbf{some}$ basis $\{x,y\}$ of $H^1(M;\BZ)$, $\b^n(x,y)$ can be defined and contains $0$ for each $n\ge1$. \[systole\] Let $M$ be a closed oriented $3$-manifold with $\b_1(M)=2$. Let $\wt M$ denote the universal torsion-free abelian ($\textbf{Z}\oplus \textbf{Z}$) cover of $M$. Let $[F]$ denote the class in $H_1(\wt M)$ of a lift of a typical fiber of the Abel-Jacobi map of $M$ (represented by a lift of the circle we called $c(x,y)$ below). If, for **some** $\{x,y\}$, and **some** $n$, $\b^n(x,y)\neq 0$ then $[F]$ is non-zero. The above Corollary generalizes an (independent) result of A. Marin (see Prop.12.1 of \[KL\]), which dealt with only the case $n=1$. The significance of this Corollary is that it has been previously shown by Ivanov and Katz (\[IK, Theorem 9.2 and Cor.9.3\]) that the conclusion of Corollary \[systole\] is sufficient to guarantee a certain optimal systolic inequality for $M$. The interested reader is referred to those works. Suppose $c$ and $d$ are disjointly embedded oriented circles in $M$ that are zero in $H_1(M;\BQ)$. Then the [**linking number of $c$ with $d$**]{}, ${\ensuremath{\ell k}}(c,d)\in\BQ$ is defined as follows. Choose an embedded oriented surface $V_d$ whose boundary is “$m$ times $d$” (i.e. a circle in a regular neighborhood $N$ of $d$ that is homotopic in $N$ to $md$) for some positive integer $m$, and set: $${\ensuremath{\ell k}}(c,d) = \f1m(V_d\cd c).$$ Given this, the invariants $\b^n(x,y)$ are defined as follows. Let $\{V_x,V_y\}$ be embedded, oriented connected surfaces that are Poincaré Dual to $\{x,y\}$ and meet transversely in an oriented circle that we call $c(x,y)$ (by the proof of \[C1, Theorem 4.1\] we may assume that $c(x,y)$ is connected). Let $c^+(x,y)$ denote a parallel of $c(x,y)$ in the direction given by $V_y$. Note that $\{V_x,V_y\}$ induce two maps $\psi_x$, $\psi_y$ from $M$ to $S^1$ wherein the surfaces arise as inverse images of a regular value. The product of these maps yields a map $\psi:M\to S^1\x S^1$ that induces an isomorphism on $H_1$/torsion. Since $c(x,y)$ and $c^+(x,y)$ are mapped to points under $\psi$, they represent the zero class in $H_1(M;\BQ)$. Therefore we may define $\b^1(x,y)={\ensuremath{\ell k}}(c(x,y)$, $c^+(x,y))$. In fact, $-\b^1(x,y)\cd|{\operatorname{Tor}}H_1(M;\BZ)|$ is precisely Lescop’s invariant of $M$ \[Les; p.90-94\]. An example is shown in Figure \[satolevine\] of a manifold with $\b^1(x,y)= -k$. (105,92) (10,10)[![Example of $\b^1(x,y)= -k$[]{data-label="satolevine"}](satolevine.eps "fig:")]{} (11,63)[$0$]{} (92,20)[$0$]{} (76,61)[$k$]{} (39,0)[$M$]{} The idea of the higher invariants is to iterate this process as long as possible (compare \[C1\]). Since $c^+(x,y)$ is rationally null-homologous, there is a surface $V_{c(x,y)}$ whose boundary is “$k$ times $c^+(x,y)$” (in the sense above). We could then define $c(x,x,y)$ to be $V_x\cap V_{c(x,y)}$, an embedded oriented circle on $V_x$. If $c(x,x,y)$ is rationally null-homologous, then $\b^2(x,y)$ is defined as ${\ensuremath{\ell k}}(c(x,x,y),c^+(x,x,y))$ and we may also continue and define $c(x,x,x,y)$. In general $c(x,x\dots,x,y)=c(x^n,y)$ will be able to be defined using the chosen surfaces if $c(x^{n-1},y)$ is defined and is also rationally null-homologous (but to do so involves one more choice of a bounding surface). Once $c(x^{n},y)$ is defined and is rationally null-homologous, we may define $\b^n(x,y)$. In general, we do not claim that the value of $\b^n(x,y)$ is independent of the choices of surfaces. Therefore the invariants can be thought of as taking values in a set, just like Massey products. This indeterminacy will not concern us here, for we are only interested in the first non-vanishing value (if it exists) and we shall see that this is independent of the surfaces. Much of the time it is convenient to abbreviate $c(\overbrace{x\dots x}^n,y)$ as $c(n)$ so $c(x,y)=c(1)$. If $c(n)$ is defined and rationally null-homologous then $\b^n(x,y)$ is defined to be the set of rational numbers ${\ensuremath{\ell k}}(c(n),c^+(n))$, ranging over all possible ways of defining such a $c(n)$. If no such $c(n)$ exists then $\b^n(x,y)$ is undefined. \[Example3\] Consider the manifold $M$, shown in Figure \[example3\], obtained from $0$-framed surgery on a two component link $\{ L_x,L_y\}$. Use a genus one Seifert surface for $L_y$ obtained from the obvious twice-punctured disk and a tube that goes up to avoid $L_x$. Let $V_y$ be this surface capped-off in $M$. Similarly use the fairly obvious Seifert surface for $L_x$ in the complement of $L_y$. Then $c^+(x,y)$ is shown. Since it has self-linking zero with respect to $V_x$, $\b^1(x,y)=\b^1(y,x)=0$. Furthermore $\b^2(x,y)=-1$ (note the link $\{c^+(x,y),L_x\}$ is very similar to that of Figure \[satolevine\]). This means that $\pi_1(M)$ does **not** admit an epimorphism to $\mathbb{Z}\ast \mathbb{Z}$ since that would imply that $\{ L_x,L_y\}$ were a homology boundary link. But $\b^2(x,y)=-1$ precludes this by [@C2]. Nonetheless, further $c(yy...y,x)$ may be taken to be empty since $c^+(x,y)$ and $L_y$ form a boundary link in the complement of $L_x$. Thus $\b^n(y,x)= 0$ for all $n$, indicating, by Theorem \[linear\], that the first Betti numbers will grow linearly in the family of finite cyclic covers corresponding to the map $\pi_1(M)\to \mathbb{Z}$ that sends a meridian of $L_x$ to zero and a meridian of $L_y$ to one. (123,73) (10,10)[![Example with linear growth in cyclic covers but no map to $\mathbb{Z}\ast\mathbb{Z}$[]{data-label="example3"}](example3.eps "fig:")]{} (72,39)[$c^+(x,y)$]{} (-5,39)[$L_y$]{} (117,17)[$L_x$]{} (65,0)[$M$]{} (36,46)[$0$]{} (119,68)[$0$]{} \[Example2\] Consider the family of manifolds $M_k$, shown in Figure \[example2\] and Figure \[example2b\], obtained from $0$-framed surgery on a two component link. (135,98) (10,10)[![Example with $\b^1(x,y)=0$,$\b^2(x,y)=-k$, $\b^2(y,x)= -1$[]{data-label="example2"}](example2.eps "fig:")]{} (83,93)[$0$]{} (41,15)[$0$]{} (110,61)[$k$]{} (65,0)[$M$]{} (135,98) (10,10)[![The circle $c(x,y)$[]{data-label="example2b"}](example2b.eps "fig:")]{} (83,93)[$0$]{} (41,15)[$0$]{} (110,61)[$k$]{} (69,61)[$c(x,y)$]{} (65,0)[$M$]{} If $V_x$ denotes the capped-off Seifert surface (obtained using Seifert’s algorithm) for the link component, $L_x$, on the right-hand side and $V_y$ denotes the capped-off Seifert surface for the link component, $L_y$, on the left-hand side, then the dashed circle in Figure \[example2b\] is $c(x,y)=V_x\cap V_y$. The circle $c(y,x)$ is merely this circle with opposite orientation. Since it lies on an untwisted band of $V_x$, $\b^1(x,y)=0=\b^1(y,x)$. Therefore the Lescop invariant of $M$ vanishes. But the link $\{c(x,y),L_x\}$ is the link of Figure \[satolevine\] so $\b^2(x,y)=-k$, whereas the link $\{c(y,x),L_y\}$ is a Whitehead link so $\b^2(y,x)= -1$. We claim further that, as long as $k\neq 0$, for $\textbf{any}$ basis $\{X,Y\}$ of $H^1(M)$, $\b^2(X,Y)\neq 0$. It will then follow from Theorem \[linear\] that the first Betti number of $M$ will grow sub-linearly in **any** family of finite cyclic covers. A general basis, $\{V_X,V_Y\}$, of $H_2(M)$ can be represented as follows. Represent $V_X$ by $p$ parallel copies of $V_x$ together with $q$ parallel copies $V_y$, and represent $V_Y$ by $r$ parallel copies of $V_x$ together with $s$ parallel copies $V_y$, where $ps-qr=\pm 1$. Thus $c(X,Y)=-c(Y,X)$ is represented by $ps-qr$ parallel copies of $c(x,y)$. It follows that $\b^1(X,Y)=\b^1(x,y)=0$, reinforcing our above claim that $\b^1$ is independent of basis. Hence $V_{c(X,Y)}=\pm V_{c(x,y)}$ so $c(X,X,Y)$ is represented by $\pm pc(x,x,y)\mp qc(y,y,x)$. Since $\b^2(X,Y)$ is the self-linking number of this class, it can be evaluated to be $$p^2\b^2(x,y) + q^2\b^2(y,x) - 2pq{\ensuremath{\ell k}}(c(x,x,y),c(y,y,x))$$ but the latter mixed linking number is easily seen to be zero in this case. Hence $\b^2(X,Y)=-kp^2-q^2$ which is non-zero if $k$ is non-zero. \[equivalence\] Suppose $c(1),\dots,c(n)$ have been defined as embedded oriented curves on $V_x$ arising as $c(1)=V_x\cap V_y$ and $$c(j) = V_x\cap V_{c(j-1)}\qquad2\le j\le n$$ where $V_{c(j)}$, $1\le j\le n-1$, is an embedded, oriented connected surface whose boundary is a positive multiple $k_j$ of $c^+(j)$ (in the sense above). Then $\b^j$ is defined for $1\le j\le n-1$ and the following are equivalent: 1. $\b^1,\dots,\b^n$ are defined using the given system of surfaces. 2. $\b^j$ is defined for $1\le j\le n$ and is [**zero**]{} for $1\le j\le\[\f n2\]$ 3. $c(n+1)$ exists 4. For all $s$, $t$ such that $1\le s\le t$ and $s+t\le n$ , ${\ensuremath{\ell k}}(c(s),c^+(t))=0$. Assume $1\le j\le n-1$. The hypotheses imply that a positive multiple of $c^+(j)$ is (homotopic to) the boundary of a surface so $c^+(j)$ and $c(j)$ are rationally null-homologous. Thus their linking number is well-defined, establishing the first claim. [**B1$\Longleftrightarrow$B3**]{}: $\b^n$ is defined precisely when $[c(n)]=0$ in $H_1(M;\BQ)$ which is precisely the condition under which $c(n+1)$ can be defined. [**B1$\Longrightarrow$B4**]{}: If $n=1$ the implication is true since B4 is vacuous. Thus assume by induction that the implication is true for $n-1$, that is our inductive assumption is that, for all $s+t<n$, ${\ensuremath{\ell k}}(c(s),c^+(t))=0$. Now consider the case that $s+t=n$. Since $\b^n$ is defined $[c(n)]=0$ in $H_1(M;\BQ)$. We claim this is true precisely when $c(n)\cd c(1)=0$ (here we refer to oriented intersection number on the surface $V_x$). For suppose $\psi_x:M\to S^1$ and $\psi_y:M\to S^1$ are maps such that $\psi^{-1}_x(*)=V_x$ and $\psi^{-1}_y(*)=V_y$. Then $(\psi_x)_*([c(n)])=0$ since $c(n)\subset V_x$; and $(\psi_y)_*([c(n)])=0$ precisely when $c(n)\cd V_y=c(n)\cd c(1)=0$. But the map $\psi_x \times \psi_y$ completely detects $H_1(M)$/Torsion. Therefore, once $c(n)$ exists, $\b^n$ is defined if and only if: $$\begin{aligned} 0 = c(n)\cd c(1) &= \pm(c(1)\cd V_{c(n-1)})\\ &= \pm k_{n-1}{\ensuremath{\ell k}}(c(1),c^+(n-1))\end{aligned}$$ which establishes B4 in the case $s=1$. But we claim that, if B4 is true for $s+t<n$, then for $s+t=n$ and $s< t$, $$k_{t-1}{\ensuremath{\ell k}}(c(s+1),c^+(t-1))=k_s{\ensuremath{\ell k}}(c(s),c^+(t)).$$ This equality can then be applied, successively decreasing $s$, to establish B4 in generality. This claimed equality is established as follows. $$\begin{aligned} \pm k_{t-1}{\ensuremath{\ell k}}(c(s+1),c^+(t-1)) &= \pm V_{c(t-1)}\cd c(s+1)\\ &= \pm V_{c(t-1)}\cd(V_x\cap V_{c(s)})\\ &= V_{c(s)}\cd(V_x\cap V_{c(t-1)})\\ &= V_{c(s)}\cd c(t)\\ &= k_s{\ensuremath{\ell k}}(c(t),c^+(s))\\ &=k_s{\ensuremath{\ell k}}(c(s),c^+(t)).\end{aligned}$$ The last step is justified by verifying that $c(s)\cd c(t)=0$ if $s< t$. For $$\begin{aligned} c(s)\cd c(t) &= c(s)\cd V_{c(t-1)}\\ &= \pm k_{t-1}{\ensuremath{\ell k}}(c(s),c^+(t-1))\end{aligned}$$ which vanishes by our inductive assumption since $s+(t-1)<n$. [**B4$\Longrightarrow$B1**]{}: Since $\b^1$ is always defined we may assume $n>1$. It follows from B4 that ${\ensuremath{\ell k}}(c(1),c^+(n-1))=0$ if $n>1$. But we saw in the proof of B1$\Longrightarrow$B4 that once $c(n)$ was defined, this was equivalent to $\b^n$ being defined. [**B2$\Longrightarrow$B1**]{}: This is obvious. [**B4$\Longrightarrow$B2**]{}: Since B4$\Longrightarrow$B1, we have $\b^j$ defined for $j\le n$. Now suppose $1\le j\le\[\f n2\]$. Since $\b^j={\ensuremath{\ell k}}(c(j),c^+(j)$ and $2j\le n$, this vanishes by B4. This completes the proof of Lemma \[equivalence\]. The proof shows slightly more, namely that there is a correspondence between the infinite cyclic cover implicit in part $A$ and the class $x$ in parts $B$ and $C$. Suppose $\{M_n\}$ is a family of $n$-fold cyclic covers of $M$ corresponding to the infinite cyclic cover $M_\infty$. Note that $H_1(M_\infty;\BQ)$ is a finitely generated $\La=\BQ[t,t^{-1}]$ module (this involves a choice of generator of the infinite cyclic group of deck translations of $M_\infty$). Throughout this proof, homology will be taken with rational coefficients unless specified otherwise. [**Step 1**]{}: $\b_1(M_n)$ grows linearly $\Longleftrightarrow H_1(M_\infty;\BQ)$ has positive rank as a $\La$-module. As remarked above, this fact was previously known. We present a quick proof for the convenience of the reader. We are indebted to Shelly Harvey for showing us this elementary proof. Since $\La$ is a PID, $$H_1(M_\infty)\cong\La^{r_1}\oplus_j\f\La{\<p_j(t)\>}$$ where $p_j(t)\neq0$. By examining the “Wang sequence” with $\BQ$-coefficients $$H_2(M_\infty)\lra H_2(M_n)\overset{\p_*}{\lra}H_1(M_\infty) \overset{t^n-1}{\lra}H_1(M_\infty)\overset{\pi}{\lra}H_1(M_n) \overset{\p_*}{\lra}H_0(M_\infty)\overset{t^n-1}{\lra}$$ it is easily seen that $$\begin{aligned} H_1(M_n) &\cong\f{H_1(M_\infty)}{\<t^n-1\>}\op\BQ\\ &\cong\(\f\La{\<t^n-1\>}\)^{r_1}\oplus_j\f\La{\<p_j(t),t^n-1\>}\op\BQ.\end{aligned}$$ The first summand contributes $nr_1$ to $\b_1(M_n)$. The $\BQ$-rank of the second summand is bounded above by the sum of the degrees of the $p_j$, a number that is [**independent**]{} of $n$. Therefore $\b_1(M_n)$ grows linearly with $n$ if $r_1\neq0$ and otherwise is bounded above by a constant (independent of $n$). [**Step 2**]{}: $H_1(M_\infty)$ has positive $\La$-rank $\Longleftrightarrow H_1(M_\infty)$ has no $(t-1)$-torsion (equivalently $t-1$ acts injectively). To verify Step 2, consider the “Wang sequence” with $\BQ$-coefficients $$H_2(M_\infty)\lra H_2(M)\overset{\p_*}{\lra}H_1(M_\infty) \overset{t-1}{\lra}H_1(M_\infty)\overset{\pi}{\lra}H_1(M) \overset{\p_*}{\lra}H_0(M_\infty)\overset{t-1}{\lra}H_0(M_\infty)$$ associated to the exact sequence of chain complexes $$0\lra C_*(M_\infty;\BQ)\overset{t-1}{\lra} C_*(M_\infty;\BQ)\overset{\pi}{\lra}C_*(M;\BQ)\lra0.$$ Since $H_0(M_\infty)\cong\BQ$, ${\operatorname{image}}\p_*\cong\BQ$ on $H_1(M)$. If $\b_1(M)=2$ then it follows that $\BQ\cong{\operatorname{ker}}\p_*={\operatorname{image}}(\pi)\cong{\operatorname{cokernel}}(t-1)$. It follows that $H_1(M_\infty)$ contains at most one summand of the form $\La/\<(t-1)^m\>$ since each such summand contributes precisely one $\BQ$ to ${\operatorname{cokernel}}(t-1)$. Similarly each $\La$ summand of $H_1(M_\infty)$ contributes one $\BQ$ to the cokernel. Therefore $H_1(M_\infty)$ has positive $\La$ rank if and only if it has no summand of the form $\La/\<(t-1)^m\>$. The latter is equivalent to saying that it has no $(t-1)$-torsion, or that $t-1$ acting on $H_1(M_\infty)$ is injective. This completes Step 2. [**Step 3**]{}: $(t-1):H_1(M_\infty)\to H_1(M_\infty)$ is injective $\Longleftrightarrow$ For any surface $V_x$, dual to $x$, and for each surface $V_y$ such that $\{[V_y],[V_x]\}$ generates $H_2(M;\BZ)$, the class $[\tl c(x,y)]\in H_1(M_\infty;\BQ)$ is zero. Moreover the latter statement is equivalent to one where “for each” is replaced by “for some”. Suppose that $t-1$ is injective. Note that the injectivity of $t-1$ is equivalent to $\p_*:H_2(M)\to H_1(M_\infty)$ being the zero map. Then, for [**any**]{} $[V_y]$ as above, $\p_*([V_y])=0$. But we claim that $\p_*([V_y])$ is represented by $[\tl c(x,y)]$, since $V_x$ is Poincaré Dual to the class $x$ defining $M_\infty$. For if $Y=M-{\operatorname{int}}(V_x\x[-1,1])$ then a copy of $Y$, denoted $\wt Y$, can be viewed as a fundamental domain in $M_\infty$, as shown in Figure \[cover\]. (177,84) (10,10)[![Fundamental Domain of $M_\infty$[]{data-label="cover"}](cover.eps "fig:")]{} (15,62)[$\wt c(x,y)$]{} (80,33)[$\wt V_y$]{} (0,12)[$\wt V_x$]{} (173,12)[$t\wt V_x$]{} (179,59)[$t\wt c(x,y)$]{} (87,-5)[$\wt Y$]{} Moreover if $\wt V_y$ denotes $p^{-1}(V_y)\cap \wt Y$ then $\wt V_y$ is a compact surface in $\wt M$ whose boundary is $t_*(\tl c(x,y))-\tl c(x,y)$. Thus $\wt V_y$ is a 2-chain in $M_\infty$ such that $\pi_\#(\wt V_y)$ gives the chain representing $[V_y]$. Since $\p\wt V_y$ is $(t-1)\tl c(x,y)$ in $C_*(M_\infty;\BQ)$, it follows from the explicit construction of $\p_*$ in the proof of the Zig-Zag Lemma \[Mu,Section 24\] that $\p_*([V_y])=[\tl c(x,y)]$. Conversely, if $\p_*([V_y])=0$ for [**some**]{} $[V_y]$ then $\p_*$ is the zero map (note that since $V_x$ lifts to $M_\infty$, $[V_x]$ lies in the image of $H_2(M_\infty)\lra H_2(M)$ so $\p_*([V_x])=0$ t). Therefore the injectivity statement implies the “for each” statement which clearly implies the “for some” statement. Conversely, the “for some” statement implies the injectivity statement. [**Step 4**]{}: The class $[\tl c(x,y)]$ from Step 3 is 0 if and only if it is divisible by $(t-1)^k$ for every positive $k$. In fact it suffices that it be divisible by $(t-1)^N$ where $N$ is the largest nonnegative integer such that $\La/\<(t-1)^N\>$ is a summand of $H_1(M_\infty,\BQ)$. One implication is immediate, so assume that there exists a class $[V_y]$ as in Step 3 such that $\p_*([V_y])=[\tl c_{1}]=(t-1)^N\b$ for some $\b\in H_1(M_\infty)$. Since $[\tl c_{1}]\in{\operatorname{image}}\p_*$, it is $(t-1)$-torsion so $\b$ is $(t-1)^{N+1}$-torsion. Moreover $\b$ lies in the submodule $A\subset H_1(M_\infty,\BQ)$ consisting of elements annihilated by some power of $t-1$, so, by choice of $N$, $(t-1)^N\b=0=[\tl c_{1}]=0$ as desired. This completes the verification of Step 4. [**Step 5**]{}: C$\Rightarrow$A Let $\{x,y\}$ be as in the hypotheses of C and let $M_\infty$ correspond to the class $x$. Let $N$ be the positive integer as above. If $\b^{(N+1}$ can be defined, we know in particular that there exists some system of surfaces $\{V_x,V_y,...,V_{c(N)}\}$ that defines $\{c(j)\}$, $1\leq j\leq (N+1)$. Choose a preferred lift $\wt V_x$, of $V_x$ to $M_\infty$ and a preferred fundamental domain $\wt Y$ as above lying on the positive side of $\wt V_x$. Consider any $m, 1\leq m \leq N$. Since $c(m)$ and $c(m+1)$ lie on $V_x$, they lift to oriented 1-manifolds $\tl c(m)$ and $\tl c(m+1)$ in $\wt V_x$. Similarly $c^+(m)$ lifts to $\tl c^+(m)$, which is a push-off of $\tl c(m)$ lying in $\wt Y$. Recall that $c(m+1)=V_{c(m)}\cap V_x$ where $\p V_{c(m)}=k_mc^+_{(m)}$ for some positive integer $k_m$. Letting $\wt V_{c(m)}$ be $V_{c(m)}$ cut open along $c(m+1)$ we observe that $\wt V_{c(m)}$ can be lifted to $\wt Y$ and viewed as a 2-chain showing that $k_m[\tl c^+(m)]=(t-1)[\tl c(m+1)]$ in $H_1(M_\infty;\BQ)$, as in Figure \[stepfive\]. (177,84) (10,10)[![[]{data-label="stepfive"}](stepfive.eps "fig:")]{} (15,64)[$\wt c(m+1)$]{} (108,72)[$k_m\wt c^+(m)$]{} (65,34)[$\wt V_{c(m)}$]{} (0,12)[$\wt V_x$]{} (173,12)[$t\wt V_x$]{} (179,59)[$t\wt c(m+1)$]{} (83,-5)[$\wt Y$]{} Thus $[\tl c^+(1)]=(t-1)(1/k_1)[\tl c(2)]=(t-1)^2(1/k_1)(1/k_2)[\tl c(3)]$, et cetera, showing that $[\tl c(1)]$ is divisible by $(t-1)^N$. By Steps 1 through 4, this implies A of Theorem \[linear\], completing Step 5. [**Step 6**]{}: A$\Rightarrow$B We assume that there is a primitive class $x\in H^1(M;\BZ)$ corresponding to $M_\infty$ and $\{M_n\}$ where $\b_1(M_n)$ grows linearly. By Steps 1, 2, and 3, for any $\{x,y\}$ generating $H^1(M;\BZ)$, $H_1(M_\infty;\BQ)$ has no $(t-1)$-torsion, and for any surfaces dual to ${x,y}$, $[\tl c(1)]=0$. Recall that $c(1)$ and $c(2)$ are always defined. We shall establish inductively that for all $m\ge2$, $c(m)$ is defined and that for *any* system of surfaces used to define $c(m-1)$, $[\tl c(m-1)]=0$ in $H_1(M_\infty;\BQ)$. This has already been shown for $m=2$. Suppose it has been established for $m$ (and all lesser values). We now establish it for $m+1$. Since $c(m)$ and $c(m-1)$ exist, the argument in Step 5 (Figure \[stepfive\]) shows that $k_{m-1}[\tl c(m-1)]=(t-1)[\tl c(m)]$ in $H_1(M_\infty;\BQ)$. But $[\tl c(m-1)]=0$ so $[\tl c(m)]$ is $(t-1)$-torsion. Since there is no non-trivial $(t-1)$-torsion, $[\tl c(m)]=0$ in $H_1(M_\infty;\BQ)$. Hence $[c(m)]=0$ in $H_1(M;\BQ)$ so $c(m+1)$ is defined. Since this holds for any system of defining surfaces, this completes the inductive step. Since $c(m)$ is defined for all $m\ge1$, by Lemma \[equivalence\], $\b^n(x,y)=0$ for all $n\ge1$. This completes the proof of Step 6. Since B clearly implies C, this completes the proof of Theorem \[linear\]. Assume some $\b^m(x,y)\neq 0$. If $[F]$ were zero then certainly, for the fixed infinite cyclic cover, $M_\infty$, corresponding to $x$, $[\wt c(x,y)]=0$ in $H_1(M_\infty;\mathbb{Q})$ so by Steps $1-3$ of the above proof, $\b_1(M_n)$ grows linearly with $n$. By Theorem \[linear\], this would imply that $\b^m(x,y)= 0$ for all $m$, a contradiction. [Str]{} T. Cochran, [*Geometric Invariants of Link Cobordism*]{}, Comment. Math. Helvetici, [**60**]{} (1985), 291-311. T. Cochran, [*Derivatives of links: Milnor’s concordance invariants and Massey’s products*]{}, Memoirs of AMS \#427, American Math. Soc., Providence, RI, 1990. M. Boileau and S. Wang, [*Non-zero degree maps and surface bundles over $S^1$*]{}, J. Diff. Geometry [**43**]{} (1996), 789–806. S. Ivanov and M. Katz, *Generalized degree and optimal Loewner-type inequalities*, Israel J. Math., 141 (2004),221-233. M. Katz and C. Lescop, [*Filling Area Conjecture, Optimal Systolic Inequalities, and the fiber class in Abelian covers*]{}, **Contemporary Math.** vol. 387, preprint math.DG/0412011. Akio Kawauchi, [*An imitation theory of manifolds*]{}, Osaka J. Math. [**26**]{} (1989), 447-464. , [*Almost identical imitations of $(3,1)$-manifold pairs*]{}, Osaka Journal Math. [**26**]{} (1989), 743-758. D. Kraines, [*Massey Higher Products*]{}, Trans. Amer. Math. Soc. [**124**]{} (1966), 431-449. C. Lescop, **Global Surgery Formula for the Casson-Walker Invariant**, Annals of Math Studies 140, Princeton Univ. Press, Princeton, N.J., 1996. W. Lück, **$L^2$-invariants: Theory and Applications to Geometry and K-theory**, A series of Modern Surveys in Mathematics volume[**44**]{}, Springer-Verlag, Berlin Heidelberg New York,(2002). , [*Approximating $L^2$-invariants by their finite-dimensional analogues*]{}, Geom.Funct.Anal. [**4**]{}(4) (1994), 455-481. [^1]: The first author was partially supported by the National Science Foundation. The second author was supported by a National Science Foundation Postdoctoral Research Fellowship

--- abstract: 'In a recent paper (Phys. Rev. Lett. 109, 160501 (2012). arXiv:1201.0849), it is claimed that any quantum protocol for classical two-sided computation between Alice and Bob can be proven completely insecure for Alice if it is secure against Bob. Here we show that the proof is not sufficiently general, because the security definition it based on is only a sufficient condition but not a necessary condition.' author: - Guang Ping He title: 'Comment on Complete insecurity of quantum protocols for classical two-party computation' --- Let us first look at the security definition in [@qbc61]. As stated in the paragraph below its FIG. 1, let $\varepsilon \geq 0$ and write $\rho \simeq _{\varepsilon }\sigma $ (i.e., $\rho $ is $\varepsilon $-close to $\sigma $) if the purified distance $\sqrt{1-(tr\sqrt{\sqrt{\rho }% \sigma \sqrt{\rho }})^{2}}$ between the density matrices $\rho $ and $% \sigma $ is not greater than $\varepsilon $. Then a two-party quantum protocol corresponding to a completely positive trace-preserving (CPTP) map $\pi $ is defined as $\varepsilon $-secure against dishonest Bob if for any real adversary $B^{\prime }$ there exists an ideal adversary $\hat{B}% ^{\prime }$ such that $[id_{R}\otimes \pi _{A,B^{\prime }}](\rho _{UVR})\simeq _{\varepsilon }[id_{R}\otimes \mathcal{F}_{\hat{A},\hat{B}% ^{\prime }}](\rho _{UVR})$. Here $A$ denotes the real honest Alice, $% B^{\prime }$ the dishonest Bob, and $\hat{A}$, $\hat{B}^{\prime }$ the ideal versions. Both parties obtain an input (Alice’s $u$ in register $U$ and Bob’s $v$ in register $V$) drawn from the distribution $p(u,v)$. $% [id_{R}\otimes \pi _{A,B^{\prime }}](\rho _{UVR})$ is the output state of the protocol augmented by the reference $R$, where $\rho _{UVR}$ is a purification of $\sum\nolimits_{u,v}p(u,v)\left\vert u\right\rangle \left\langle u\right\vert _{U}\left\vert v\right\rangle \left\langle v\right\vert _{V}$. And $\mathcal{F}$ is an ideal functionality which measures the inputs and outputs orthogonal states that correspond to the function values of the classical two-sided computation. Please see [qbc61]{} for more detailed explanations of the notations. In simple words, as can be seen from Sec. 1.6 of [@qi1087] (i.e., Ref. \[12\] of [@qbc61]), the meaning of this definition can be understood as follows. Let $\alpha $ and $\beta $ be the physical systems accessible to Alice and Bob, respectively. Denote the density matrices of $\alpha $, $% \beta $ as $\rho _{\alpha }$, $\rho _{\beta }$ when Bob plays honestly, or as $\rho _{\alpha }^{\prime }$, $\rho _{\beta }^{\prime }$ when he applies a certain cheating strategy. If there is $\rho _{\alpha }^{\prime }\simeq _{\varepsilon }\rho _{\alpha }$, the cheating strategy will be nearly undetectable to Alice so that Bob can pass the security checks in the protocol successfully, while if there is $\rho _{\beta }^{\prime }\simeq _{\varepsilon }\rho _{\beta }$, a dishonest Bob can hardly gain any extra information other than what is accessible to an honest Bob. Then the above security definition means that a protocol is secure against Bob if for any cheating strategy, there is always $\rho _{\beta }^{\prime }\simeq _{\varepsilon }\rho _{\beta }$. For simplicity, we call such a cheating strategy as a type I strategy. Obviously, if *any* cheating strategy currently known or potentially exists in the world belongs to type I, then the corresponding protocol is surely secure. Thus it is a sufficient condition for guaranteeing the security of a protocol. But it is important to question whether the reversed statement is also true. That is, if a protocol is secure, does it necessarily guarantee that *all* cheating strategies have to be type I strategies? In fact, if there is a cheating strategy which does not satisfy $\rho _{\alpha }^{\prime }\simeq _{\varepsilon }\rho _{\alpha }$, then it will be detectable to Alice, so that the protocol can remain secure against Bob no matter $\rho _{\beta }^{\prime }\simeq _{\varepsilon }\rho _{\beta }$ is satisfied or not. We call strategies satisfying neither $\rho _{\alpha }^{\prime }\simeq _{\varepsilon }\rho _{\alpha }$ nor $\rho _{\beta }^{\prime }\simeq _{\varepsilon }\rho _{\beta }$ as type II strategies. Actually, they are no strangers to quantum cryptography. In many existing protocols, there are security checks in which the parties agree to continue with the protocols only when some conditions are met. Otherwise they can choose to abort in the middle of the process, and the protocols output failinstead of the output obtained by honest players. This implies that the protocols are designed against type II strategies. Thus it is clear that the existence of type II strategies does not necessarily hurt the security of protocols. If a protocol is secure, then both types I and II strategies are possible. That is, all cheating strategies belong to type I is not the necessary condition for a protocol to be secure. Therefore, while the security definition in [@qbc61] is a true statement, it cannot be used as a two-party quantum protocol is $\varepsilon $-secure against Bob *if and only if* for any real adversary $B^{\prime }$ there exists an ideal adversary $\hat{B}^{\prime }$ such that $[id_{R}\otimes \pi _{A,B^{\prime }}](\rho _{UVR})\simeq _{\varepsilon }[id_{R}\otimes \mathcal{F% }_{\hat{A},\hat{B}^{\prime }}](\rho _{UVR})$, since the reversed statement for any real adversary $B^{\prime }$, there exists an ideal adversary $\hat{B}^{\prime }$ such that $[id_{R}\otimes \pi _{A,B^{\prime }}](\rho _{UVR})\simeq _{\varepsilon }[id_{R}\otimes \mathcal{F% }_{\hat{A},\hat{B}^{\prime }}](\rho _{UVR})$ if the protocol is $\varepsilon $-secure against Bob is not true. There can be type II strategies which are not $\varepsilon $-close to any ideal adversary. Now back to the no-go proof for two-sided computation in [@qbc61]. In brief, the key starting points of the proof are as follows. Suppose that there is a quantum protocol for classical two-sided computation which is already assumed to be secure against a dishonest Bob. To prove that it must be insecure against Alice, in the paragraph before Eq. (1) of [@qbc61], the following cheating strategy of Bob is considered. He plays the honest but purified strategy and outputs the purification of the protocol (register $Y_{1}^{\prime }$) and the output values $f(u,v)$ (register $Y$). We call it strategy $B_{0}^{\prime }$ hereafter. Since the protocol is $\varepsilon $-secure against Bob, in the opinion of [@qbc61] there exists a secure state $\sigma _{RX\tilde{V}Y^{\prime }}$ satisfying $\sigma _{RXY^{\prime }}\simeq _{\varepsilon }\rho _{RXY^{\prime }}$, where $Y^{\prime }=Y_{1}^{\prime }Y$. Applying Uhlmann’s theorem on $\sigma _{RXY^{\prime }}\simeq _{\varepsilon }\rho _{RXY^{\prime }}$, Eq. (1) of [@qbc61] can be obtained, which further leads to the rest part of the no-go proof. However, according to our above discussion on the security definition, the protocol is $\varepsilon $-secure against Bob does not necessarily guarantees that all cheating strategies (including strategy $B_{0}^{\prime }$) must be type I strategies, because the latter statement is not the necessary condition of the former. If $B_{0}^{\prime }$ belongs to type II, then the protocol can still be secure against Bob, while the equation $\sigma _{RXY^{\prime }}\simeq _{\varepsilon }\rho _{RXY^{\prime }}$ no longer holds. Consequently, Eq. (1) does not necessarily remain valid so that the no-go proof will lose its base. Thus we can see that the proof in [@qbc61] may apply to a protocol for which $B_{0}^{\prime }$ can be proven to be a type I strategy (given that all other features of the protocols studied in [@qbc61] are also met). But it is not sufficient general to cover all protocols, since there is no evidence (at least not provided in [@qbc61]) showing that $B_{0}^{\prime }$ always has to be a type I strategy for any protocol potentially exists. By designing proper security checks which can make $B_{0}^{\prime }$ appear as a type II strategy, it is possible to build protocols not covered by the proof in [@qbc61]. Therefore, the door for finding secure quantum protocols for classical two-party computation is not closed completely. The work was supported in part by the NSF of China under grant No. 10975198, the NSF of Guangdong province, and the Foundation of Zhongshan University Advanced Research Center. [9]{} H. Buhrman, M. Christandl, and C. Schaffner, *Phys. Rev. Lett.* **109**, 160501 (2012). *arXiv:1201.0849*. Complete insecurity of quantum protocols for classical two-party computation D. Unruh, *quant-ph/0409125*. Simulatable security for quantum protocols

--- abstract: 'A model of global magnetic reconnection rate in relativistic collisionless plasmas is developed and validated by the fully kinetic simulation. Through considering the force balance at the upstream and downstream of the diffusion region, we show that the global rate is bounded by a value $\sim 0.3$ even when the local rate goes up to $\sim O(1)$ and the local inflow speed approaches the speed of light in strongly magnetized plasmas. The derived model is general and can be applied to magnetic reconnection under widely different circumstances.' author: - 'Yi-Hsin Liu' - Michael Hesse - Fan Guo - William Daughton - Hui Li title: A model of global magnetic reconnection rate in relativistic collisionless plasmas --- [*Introduction–*]{} Magnetic fields often serve as the major energy reservoirs in high energy astrophysical systems, such as pulsar wind nebulae [@coroniti90a; @arons12a; @lyubarsky01a; @devore15a], gamma-ray bursters [@thompson94a; @zhangB11a; @mckinney12a] and jets from active galactic nuclei [@beckwith08a; @giannios10a; @jaroschek04a], where relativistic cosmic rays and gamma rays of energies up to TeV are generated explosively [@abdo11a; @bottcher13a]. Among the proposed physics processes (e.g.,[@sironi15a; @uhm14a; @zweibel09a]) that could unleash the magnetic energy, magnetic reconnection is considered to be a promising mechanism. For comparison, collisionless shocks, regarded to be efficient for particle acceleration in weakly magnetized plasmas, are inefficient in dissipating energy and accelerating non-thermal particles in magnetically dominated flows [@sironi15a]. Hence the study of magnetic reconnection in these exotic systems continues to be an interesting topic in high energy astrophysics. One of the most important issues in relativistic reconnection studies is how fast magnetic energy can be dissipated in the reconnection layer, which determines the time scale of the explosive energy release events. Another related problem is the mechanism of non-thermal particle acceleration [@YYuan16a; @werner16a; @FGuo16a; @FGuo15a; @FGuo14a; @melzani14b; @sironi14a; @cerutti12a; @bessho12a; @zenitani01a]. Proposed mechanisms include the direct acceleration by the reconnection electric field at the diffusion region [@zenitani01a; @uzdensky11a], the Fermi mechanism at the outflow regions that involves particles bouncing back and forth between reconnection outflows emanated from different x-lines [@FGuo14a; @dahlin14a; @drake06a], and many other ideas (e.g., [@zank14a; @drury12a; @pino05a]). In collisionless plasmas, the energy gain of a particle must come from the work done by the electric field $\sim q\int{{\bf E}\cdot {\bf v} dt}$. Thus, determining the reconnection electric field in the relativistic limit is crucial to determine the acceleration rate and efficiency. In such magnetically-dominated plasmas, the magnetic energy density is much larger than the rest mass energy density and the Alfvén speed approaches the speed of light. Early theoretical work suggested that the magnetic reconnection rate in the relativistic limit may increase compared to the non-relativistic case due to the enhanced inflow arising from the Lorentz contraction of plasma passing through the diffusion region [@blackman94a; @lyutikov03a]. However, it was later pointed out that the thermal pressure within a pressure-balanced current sheet will constrain the outflow to mildly relativistic conditions, where the Lorentz contraction is negligible [@lyubarsky05a] and a relativistic inflow is therefore impossible. Recently, fully kinetic simulations by Liu et al. [@yhliu15a] showed that the local inflow speed approaches the speed of light, and the reconnection rate normalized to the immediately upstream condition of the diffusion region can be enhanced to $\sim O(1)$ in strongly magnetized plasmas. However, the global reconnection rate normalized to the far upstream asymptotic value remains $\lesssim 0.3$ [@FGuo15a; @yhliu15a; @melzani14a; @sironi14a; @bessho12a; @sironi16a] and this discrepancy is not understood. While the relativistic resistive-Petschek model [@petschek64a] suggests a similar value for the global rate [@lyubarsky05a], to realize a Petschek solution requires an [*ad hoc*]{} localized resistivity [@biskamp86a; @sato79a], otherwise, the current sheet collapses to the long Sweet-Parker layer [@sweet58a; @parker57a]. A mechanism for the localized diffusion region is therefore essential to model the reconnection rate. In this Letter, we derive the relation between the global rate and the degree of localization through considering the force balance at the upstream and downstream of the diffusion region. We then propose a mechanism that naturally leads to the localization in such collisionless plasmas. [*Simulation setup–*]{} The kinetic simulation is performed using a Particle-in-Cell code- VPIC [@bowers09a], which solves the fully relativistic dynamics of particles and electromagnetic fields. The relativistic Harris sheet [@yhliu15a; @kirk03a; @zenitani07a; @wliu11a; @bessho12a; @melzani14a] is employed as the initial condition. The initial magnetic field ${\bf B}=B_{x0} \mbox{tanh}(z/\lambda) \hat{\bf x}$ corresponds to a layer of half-thickness $\lambda$. Each species has a distribution $f_h \propto \mbox{sech}^2(z/\lambda)\mbox{exp}[-\gamma_d(\gamma_Lmc^2+ mV_d u_y)/T']$ in the simulation frame, which is a component with a peak density $n'_0$ and temperature $T'$ boosted by a drift velocity $\pm V_d$ in the y-direction for positrons and electrons, respectively. In addition, a non-drifting background component $f_b \propto \mbox{exp}(-\gamma_L m c^2/T_b)$ with a uniform density $n_b$ is included. Here ${\bf u}=\gamma_L {\bf v}$ is the the space-like components of 4-velocity, $\gamma_L=1/[1-(v/c)^2]^{1/2}$ is the Lorentz factor of a particle, and $\gamma_d \equiv 1/[1-(V_d/c)^2]^{1/2}$. The drift velocity is determined by Ampére’s law $cB_{x0}/(4\pi\lambda)=2 e\gamma_d n'_0 V_d $. The temperature is determined by the pressure balance $B_{x0}^2/(8\pi)=2 n'_0 T'$. The resulting density in the simulation frame is $n_0=\gamma_d n'_0$. In this Letter, the primed quantities are measured in the fluid rest (proper) frame, while the unprimed quantities are measured in the simulation frame unless otherwise specified. Densities are normalized by the initial background density $n_b$, time is normalized by the plasma frequency $\omega_{pe}\equiv(4\pi n_b e^2/m_e)^{1/2}$, velocities are normalized by the light speed $c$, and spatial scales are normalized by the inertial length $d_e\equiv c/\omega_{pe}$. The domain size is $L_x\times L_z=384d_e \times 384d_e$ and is resolved by $3072\times6144$ cells. We load 100 macro-particles per cell for each species. The boundary conditions are periodic in the x-direction, while in the z-direction the field boundary condition is conducting and the particles are reflected at the boundaries. The half-thickness of the initial sheet is $\lambda=d_e$, $n_b=n'_0$, $T_b/m_ec^2=0.5$ and $\omega_{pe}/\Omega_{ce}=0.05$ where $\Omega_{ce}\equiv eB_{x0}/(m_e c)$ is a cyclotron frequency. The upstream magnetization parameter is $\sigma_{x0}=B_{x0}^2/(4\pi w)$ with enthalpy $w=2n'_b m_ec^2+[\Gamma/(\Gamma-1)]P'$. Here $\Gamma$ is the ratio of specific heats and $P'\equiv 2n'_b T'_b$ the total thermal pressure. For $\Gamma=5/3$ [@weinberg72a; @synge57a], $\sigma_{x0}=89$ in this run. A localized perturbation with amplitude $B_z=0.03B_{x0}$ is used to induce a dominant x-line at the center of simulation domain. ![The evolution of measured global reconnection rate $R_G$, local rate $R_L$, local inflow speed $V_{in,L}/c$ and $B_{xL}/B_{x0}$ in a plasma of $\sigma_{x0}=89$. The blue circle marks the deviation of $R_L$ from $R_G$. The grey dashed line at value $0.0$ is for reference. The orange vertical line marks the time for the analyses in Fig. \[feature\] and \[force\_balance\]. []{data-label="rate"}](rate){width="8.5cm"} [ *Simulation results–*]{} In this Letter, we define the global reconnection rate as $R_G \equiv cE_y/(B_{x0}V_{A0})$ and the local reconnection rate as $R_L\equiv cE_y/(B_{xL}V_{AL})$. Subscripts “0” and “L” indicate quantities far from, and immediately upstream of, the diffusion region where the frozen-in condition ${\bf E}+{\bf V_e}\times {\bf B}=0$ breaks ($|z|\lesssim3.5d_e$ [@yhliu15a]). $E_y$ is the reconnection electric field at the x-line and the Alfvén speed in the relativistic limit [@sakai80a; @anile89a; @lichnerowicz67a; @yhliu14a] is $V_{A0}=c[\sigma_{x0}/(1+\sigma_{x0})]^{0.5}$ and $V_{AL}=c[\sigma_{xL}/(1+\sigma_{xL})]^{0.5}$ with $\sigma_{xL}\simeq (B_{xL}/B_{x0})^2\sigma_{x0}$. The evolution of reconnection rates are plotted in Fig. \[rate\], along with the local electron inflow speed, $V_{in,L}$, and the ratio of magnetic fields $B_{xL}/B_{x0}$. Before a quasi-steady state is reached, both the local and global rates increase as the simulation progresses. The deviation of the local rate from the global rate occurs at time $t\simeq 250/\omega_{pe}$ and $B_{xL}/B_{x0}\simeq 0.8$. $R_G$ reaches a plateau of value $\simeq 0.15$ at $t \gtrsim 300/\omega_{pe}$ while $R_L$ continues to grow and $B_{xL}/B_{x0}$ continues to drop. The local rate $R_L$ eventually reaches a plateau of value $\simeq 0.6$ and $B_{xL}/B_{x0}$ reaches a plateau of value $\simeq 0.22$ at time $t\gtrsim 600/\omega_{pe}$. The local inflow speed basically traces the local rate because of the frozen-in condition $E_y\simeq V_{in,L}B_{xL}/c$ and $V_{AL}\simeq c$ in this case, which leads $R_L=V_{in,L}/V_{AL}\simeq V_{in,L}/c$ . The values of these two quantities can approach $\sim O(1)$ with a larger $\sigma_{x0}$, as reported before [@yhliu15a; @lyutikov16a; @zenitani09a]. ![The morphology of relativistic magnetic reconnection at $t=600/\omega_{pe}$. In (a), the $V_{ez}$ and a cut at $x=0$; In (b), the $|B_x|$ and a cut of $B_x$ at $x=0$. The white contour is the in-plane magnetic flux. To better illustrate the variation of the upstream field in (b), we have put an upper limit $B_{x0}$ in the color scale, which artificially reduces the $|B_x|$ around the magnetic islands at outflow exhausts.[]{data-label="feature"}](feature){width="9cm"} To get a better idea of the spatial variation of the inflow velocity and magnetic fields at the quasi-steady state, the $V_{ez}$ and $B_x$ at time $t=600/\omega_{pe}$ are shown in Fig. \[feature\] with the in-plane magnetic flux overlaid. Immediately upstream of the intense thin current sheet, the $|V_{ez}|$ peaks at $|z|\simeq d_e$ with value $\simeq 0.65$, where $B_x$ drops to a value $\simeq 3$. Because of the thin current sheet, $d_e$-scale secondary tearing modes [@yhliu15a] are generated repeatedly, which can be seen in Fig. \[feature\]. Note that $R_G$ reaches the plateau in Fig. \[rate\] long before the generation of secondary tearing modes. The enhancement of $V_{ez}/c$ closer to the diffusion region is anti-correlated with the reduction of $B_x$ because $E_y\simeq V_{ez} B_x/c$ should be spatially uniform in a quasi-steady state under the 2D constraint, per Faraday’s law. ![In (a), the force balance in the z-direction along $x=0$ in Fig. \[feature\]; In (b), the pressure balance along $x=0$.[]{data-label="force_balance"}](force_balance){width="8.5cm"} To get a clue of how the $B_{xL}$ drops from $B_{x0}$, we examine the force balance across the x-line at $x=0$. By combining the momentum equations for electrons and positrons [@yhliu15a; @hesse07a], the equation of force balance can be derived as $$\sum_j^{e,p}mn_j{\bf V}_j\cdot\nabla{\bf U}_j+\nabla\frac{B^2}{8\pi}+\nabla\cdot {\bf P}-\frac{{\bf B}\cdot\nabla {\bf B}}{4\pi}=-\sum_j^{e,p}mn_j\frac{\partial}{\partial t}{\bf U}_j \label{force}$$ Here the pressure tensor ${\bf P} \equiv \sum_j^{e,p} \int d^3u {\bf v u}f_j-n_j{\bf V}_j{\bf U}_j$, and subscripts “e” and “p” stand for electrons and positrons respectively. ${\bf U} \equiv (1/n)\int d^3u {\bf u} f$ is the first moment of the space-like components of 4-velocity, and ${\bf V} \equiv (1/n)\int d^3u{\bf v} f$ as usual. On the left hand side of Eq. (\[force\]), the terms represent the inertial force, magnetic pressure gradient force, plasma thermal gradient force and magnetic tension, respectively. In the upstream region the magnetic pressure is balanced by the tension force as shown in Fig. \[force\_balance\](a). The thermal pressure is negligible because of the small plasma $\beta\equiv P/(B^2/8\pi)\simeq 0.005$. The time-derivative of inertia is negligible in the quasi-steady state. Therefore, the force balance results in a significant reduction of $B_x$ from the value far upstream at $|z| \gtrsim 150d_e$ to the value immediately upstream of the diffusion region at $|z|\simeq 3.5 d_e$ as shown in the profile of $B^2/8\pi$ in Fig. \[force\_balance\](b). [ *Simple model–*]{} When the current sheet pinches locally, it implies a curved upstream magnetic field as illustrated in Fig. \[model\](a). The local magnetic field immediately upstream of the diffusion region, $B_{xL}$, becomes smaller than $B_{x0}$, so that the magnetic pressure gradient force balances the magnetic tension. A larger degree of localization implies a larger curvature, and a smaller $B_{xL}$, as indicated by the “line-density” of the in-plane flux in both Fig. \[model\](a), and the upstream region in Fig. \[feature\]. Hence, even though the local reconnection rate can be enhanced significantly due to the normalization, the global reconnection rate may not increase much. To estimate this effect in the $\beta \ll 1$ limit, we analyze the force balance, $\nabla B^2/8\pi \simeq {\bf B}\cdot\nabla {\bf B}/4\pi$, at point 1 marked in Fig. \[model\](b): $$\frac{B_{x0}^2-B_{xL}^2}{8\pi\Delta z}\simeq \left(\frac{B_{x0}+B_{xL}}{2}\right)\frac{2B_z}{4\pi\Delta x}. \label{up_force}$$ Note that $\nabla\cdot {\bf B}=0$ is also satisfied at point 1. The $B_x$ at point 1 is linearly interpolated from $B_{x0}$ and $B_{xL}$. The upstream inertial force can be formally ordered out, and it is also negligible in Fig. \[force\_balance\]. A curved upstream magnetic field naturally implies an flaring angle, and that is measured by $\Delta z/\Delta x\simeq B_z/[(B_{x0}+B_{xL})/2]$. For the proof of principle, we match it to the opening angle of the reconnection exhaust just outside of the diffusion region: $\Delta z/\Delta x\simeq B_{zL}/B_{xL}$. We obtain the relation, $$\frac{B_{zL}}{B_{xL}}\simeq \sqrt{\frac{1-B_{xL}/B_{x0}}{1+B_{xL}/B_{x0}}}. \label{BzL_BxL}$$ This expression suggests that a larger opening angle requires a further reduction of $B_{xL}/B_{x0}$. In this sense, $B_{xL}/B_{x0}$ gauges the localization of sheet pinch. When $B_{xL}/B_{x0} \rightarrow 0$, the opening angle approaches $45^\circ$ in this model. ![The cartoons of magnetic field lines upstream of the diffusion region ($z >0$) in (a), and the geometry of reconnection in (b). The dimension of the diffusion region in (c). The predictions with $\sigma_{x0}=89$ in (d), the dashed lines use $V_{out,L}=V_{AL}$. The orange vertical line corresponds to that in Fig. \[rate\].[]{data-label="model"}](model_all){width="8cm"} Combined with $E_y\simeq B_{zL} V_{out,L}/c$, the reconnection rates are $$R_G\simeq \left(\frac{B_{xL}}{B_{x0}}\right)\left(\frac{B_{zL}}{B_{xL}}\right)\left(\frac{V_{out,L}}{V_{A0}}\right);\ R_L\simeq \left(\frac{B_{zL}}{B_{xL}}\right)\left(\frac{V_{out,L}}{V_{AL}}\right) \label{RG}$$ and the local inflow speed is $$V_{in,L}\simeq R_L V_{AL}. \label{vin}$$ Using Eq. (\[BzL\_BxL\]) and the outflow speed $V_{out,L}\sim V_{AL}$, the predicted $R_G$, $R_L$ and $V_{in,L}/c$ as functions of $B_{xL}/B_{x0}$ are plotted in Fig. \[model\](d) as dashed-lines. If $B_{xL}/B_{x0}=1$, the opening angle is zero and reconnection is not expected. In the limit of $B_{xL}/B_{x0}\rightarrow 0$, the reconnecting component vanishes and reconnection ceases (i.e., $R_G=0$). However, the geometrical constrain can reduce the outflow speed from $V_{AL}$ when the opening angle approaches $45^\circ$. This correction can be modeled through analyzing the force-balance in the x-direction at point 2 of Fig. \[model\](c): $n'mU_{out}^2/L+B_{zL}^2/8\pi L\simeq (B_{zL}/2)2(B_{xL}/2)/4\pi\delta$, where the inertial force becomes important. The outflow can be relativistic, $U_{out}\sim \gamma_{out}V_{out,L}\sim V_{out,L}^2/(1-V_{out,L}^2/c^2)$. Assuming the incompressibility of plasmas, then the aspect ratio of the diffusion region $\delta/L\sim B_{zL}/B_{xL}$, and the outflow speed becomes $$V_{out,L}\simeq c\sqrt{\frac{(1-B_{zL}^2/B_{xL}^2)\sigma_{xL}}{1+(1-B_{zL}^2/B_{xL}^2)\sigma_{xL}}}. \label{vout}$$ This expression suggests that when $\delta/L \ll 1$ (i.e., $B_{zL}/B_{xL} \ll 1$) then $V_{out,L}\sim V_{AL}$. When $\delta/L\rightarrow 1$ (i.e., $45^\circ$), the outflow tension is balanced by the magnetic pressure and the outflow vanishes. Plugging Eqs.(\[BzL\_BxL\]) and (\[vout\]) back to Eqs. (\[RG\])-(\[vin\]), we get the solid curves in Fig. \[model\](d). This correction further constrains the reconnection rate when the opening angle is larger and $B_{xL}/B_{x0}$ is smaller. This model suggests that during the pinching of the current sheet, a weak localization with $B_{xL}/B_{x0} \lesssim 0.9$ is enough to lead $R_G$ to $\sim 0.2$, then it varies slowly over a wide range of $B_{xL}/B_{x0}$. The local rate $R_L$ and local inflow speed $V_{in,L}/c$ can reach $\sim O(1)$ under stronger localization. The evolution of reconnection rates in Fig. \[rate\] can be qualitatively described by this model through decreasing $B_{xL}/B_{x0}$. The rates in the quasi-steady state at time $t=600/\omega_{pe}$ of Fig. \[rate\] also compares well with the prediction at $B_{xL}/B_{x0}\simeq 0.22$ with the predicted $R_G \simeq 0.14$, $R_L \simeq 0.69$ and $V_{in,L} \simeq 0.62c$. Given the simplicity of this model, this agreement is quite remarkable. While the localization mechanism may vary in different systems, we point out a natural tendency that can lead to the $B_{xL}/B_{x0}$ reduction in such plasmas: A diffusion region sandwiched by a large $B_{xL}\simeq B_{x0}$ at $d_e$-scale (i.e., where the frozen-in condition is broken) requires the current sheet plasma to have a huge thermal pressure to balance the magnetic pressure, and a high drift speed to support the current. For instance, the initial $d_e$-scale current sheet has $T'=100m_ec^2$, $n_0\simeq 10$ and $\gamma_dV_d\simeq 10$. However, the maximum possible reconnection electric field may not be efficient enough in heating and accelerating the cold non-drifting inflowing plasma before they exit the diffusion region [@hesse11a], hence the $B_{xL}$ drops significantly until the $d_e$-scale current sheet becomes sustainable in the quasi-steady state. If this drop continues with a larger $\sigma_{x0}$, reconnection in the more extreme limit is prone to choke itself off in the quasi-steady state. [*Discussion–*]{} Knowing the magnitude of electric field is essential for estimating the acceleration of super-thermal particles in highly magnetized astrophysical systems. This study suggests that the magnitude of the reconnection electric field is bounded by $\sim 30\%$ of the reconnecting component of magnetic field, even in the large-$\sigma_{x0}$ limit. While a weak localization of the diffusion region is required, the global reconnection rate $R_G \sim 0.1-0.3$ is not sensitive to a further increase of localization over a wide range of $B_{xL}/B_{x0}$, but the local rate and local inflow speed are. This explains the large difference between the local and global reconnection rates observed in the simulation. In this model, a larger $\sigma_{x0}$ has little effect on the profile of the global rate $R_G$, but it could make the local inflow speed closer to the speed of light [@yhliu15a]. In addition, the effect of a guide field can be included by making the relevant Alfvén speed $V_A=c[\sigma_x/(1+\sigma_x+\sigma_g)]^{0.5}$ with $\sigma_g\equiv (B_g/B_{x0})^2\sigma_{x0}$ accounting for the effect a guide field $B_g$. This expression is basically the projection of the total Alfvén speed in the outflow direction [@yhliu14a; @melzani14a; @yhliu15a]. A guide field also has little effect on $R_G$, but it significantly reduces the local inflow speed and the magnitude of the reconnection electric field through reducing the speed of Alfvénic outflows, as observed in Liu et al. [@yhliu15a]. The prediction in the non-relativistic and low-$\beta$ limit can be obtained by taking $\sigma_x \ll 1$, and the resulting $R_G$ has a slightly smaller amplitude.\ Y.-H. Liu thanks for helpful discussions with J. Dorelli. This research was supported by an appointment to the NASA Postdoctoral Program at the NASA-GSFC, administered by Universities Space Research Association through a contract with NASA. Simulations were performed with LANL institutional computing, NASA Advanced Supercomputing and NERSC Advanced Supercomputing. 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--- abstract: 'We present ALMA maps of the starless molecular cloud core Ophiuchus/H-MM1 in the lines of deuterated ammonia (ortho-$\dammo$), methanol ($\meth$), and sulphur monoxide (SO). While the dense core is outlined by $\dammo$ emission, the $\meth$ and SO distributions form a halo surrounding the core. Because methanol is formed on grain surfaces, its emission highlights regions where desorption from grains is particularly efficient. Methanol and sulphur monoxide are most abundant in a narrow zone that follows one side of the core. The region of the brightest emission has a wavy structure that rolls up at one end. This is the signature of Kelvin-Helmholtz instability occurring in sheared flows. We suggest that in this zone, methanol and sulphur are released as a result of grain-grain collisions induced by shear vorticity.' author: - Jorma Harju - 'Jaime E. Pineda' - 'Anton I. Vasyunin' - Paola Caselli - 'Stella S.R. Offner' - 'Alyssa A. Goodman' - Mika Juvela - 'Olli Sipil[ä]{}' - Alexandre Faure - Romane Le Gal - 'Pierre Hily-Blant' - 'Jo[ã]{}o Alves' - Luca Bizzocchi - Andreas Burkert - Hope Chen - 'Rachel K. Friesen' - 'Rolf G[ü]{}sten' - 'Philip C. Myers' - Anna Punanova - Claire Rist - Erik Rosolowsky - Stephan Schlemmer - Yancy Shirley - Silvia Spezzano - Charlotte Vastel - Laurent Wiesenfield bibliography: - 'hmm1.bib' title: Efficient methanol desorption in shear instability --- Introduction {#sec:intro} ============ Unexpectedly high abundances of gaseous methanol ($\meth$) have been found in the outer parts of cold starless cores (; @2014ApJ...795L...2V; ; @2016ApJ...830L...6J; @2018ApJ...855..112P). The fractional abundances ($\sim 10^{-9}$ relative to $\htwo$) exceed the predictions from pure gas-phase chemical models by orders of magnitude (@2006FaDi..133...51G; ). Methanol is believed to form almost exclusively on the surfaces of dust grains via hydrogenation of frozen carbon monoxide (CO; @2002ApJ...571L.173W; @2006FaDi..133..177G), and it is a common constituent of interstellar ices. To be detectable in the gas phase, methanol must be released from grains as a result of heating or some non-thermal mechanism. In cold, starless cores, non-thermal processes are required, and of these the so-called reactive desorption, desorption caused by exothermic surface reactions, is currently the strongest candidate (@2006FaDi..133...51G; ; @2013ApJ...769...34V; @2015MNRAS.449L..16B; @2016ApJ...830L...6J; @2017ApJ...842...33V). The efficiency of desorption after the molecule formation is, however, uncertain, and it is given as a free parameter in chemistry models (; @2018ApJ...853..102C). While reactive desorption provides a plausible explanation for observed methanol distributions, it should be noted that core boundaries, where methanol is usually found, are also subject to dynamical effects, such as accretion, velocity shears, and turbulence. Several cores show a sharp transition from supersonic to subsonic turbulence in a thin layer surrounding the core (@1998ApJ...504..223G; @2010ApJ...712L.116P; @2017ApJ...843...63F; @2019ApJ...872..207A). In this region, the scaling relation between the velocity dispersion, $\sigma_v$, and the scale length $l$, $\sigma_v \propto l^a$, seems to break [@1998ApJ...504..223G], suggesting that part of the turbulent energy of the surrounding gas is dissipated, while at the same time the exterior turbulence compresses the core. Also gravitational accretion from the surrounding cloud and collisions between cores can lead to conversion of kinetic energy into heat. These effects can contribute to the evaporation of the ice coatings of dust grains. Finally, the chemical composition of the outer parts of dense cores can be affected by the external radiation field, especially in the vicinity of young massive stars, and when the core lies near the edge of a cloud. Radiation is the main source of heat at the core boundaries, and can also directly influence the chemistry during the diffuse contraction phase (e.g., ). Here we present maps of a nearby prestellar core in the spectral lines of methanol, deuterated ammonia ($\dammo$), and sulphur monoxide (SO), obtained using the Atacama Large (Sub)millimeter Array (ALMA). The spatial resolution of these observations is $\sim 500$ au. We discuss the origin of gas-phase methanol based on the observed molecular distributions and the physical conditions of the emission regions. In addition to reactive desorption, we identify low-velocity grain-grain collisions induced by turbulent grain acceleration as a possible mechanism enhancing methanol abundance in the envelopes of prestellar cores. The present observations suggest that vorticity associated with shear instability is a particularly effective way of inducing grain-grain collisions that lead to the removal methanol from grains. Observations {#sec:observations} ============ The target of the present observations is the nearby prestellar core Ophiuchus/H-MM1 located on the eastern side of the L1688 cloud (@2004ApJ...611L..45J; ; ). The dimensions of the kidney-shaped dense core are approximately $60\arcsec\times 90\arcsec$. The target is prominent in the $850\,\mu$m dust continuum maps of Ophiuchus with SCUBA-2 [@2015MNRAS.450.1094P], and in the $\ammo$ map of L1688 from the Greenbank Ammonia Survey [@2017ApJ...843...63F]. The present data were taken during the ALMA cycle 4 (project 2016.1.00035.S). Here we discuss the $J_k=2_k - 1_k$ rotational lines of $\meth$ at 96.7 GHz and the $J_{K_a,K_c}=1_{11}^{\rm s}-1_{01}^{\rm a}$ rotation-inversion line of ortho-$\dammo$ at 85.9 GHz, which were observed simultaneously with the 3 millimeter continuum in the ALMA Band 3. The ’continuum’ spectral window included the $J_N=3_2-2_1$ rotation line of SO at 99.3GHz. This line is unresolved because the channel width in this spectral cube is $\sim 4.9$MHz ($\sim 1.5\,\kms$). An area of $50\arcsec\times80\arcsec$ covering the densest part of the H-MM1 core was imaged using the ALMA 12m array (40 antennas) in one of its most compact configurations, and the ALMA Compact Array (ACA) with 10 7m antennas. The total power (TP) antennas were not used. With the 12m array, the mapping was carried out by a five-point mosaic, whereas with the 7m array, a single point was measured. The data were calibrated and imaged using the CASA version 4.7.2. The angular resolution of the final images is $4\arcsec$, corresponding to 480 au (assuming a distance of 120 pc; ). The integrated intensity maps of the ortho-$\dammo$, $\meth$ and SO lines at 85.9, 96.7 and 99.3GHz, respectively, are shown in Figures \[figure:line\_maps\]a, b and c. The fourth map, shown in panel d of this figure, is the $\htwo$ column density map derived from 8$\mum$ extinction. For this we have used the 8$\mum$ surface brightness map measured by the InfraRed Array Camera (IRAC) of the Spitzer Space Telescope, smoothed to a $4\arcsec$ resolution (the original resolution is $\sim 2\arcsec$). The method used for deriving the $N(\htwo)$ map is described in Appendix \[sec:column\]. The 3mm continuum is weak, and the map from the present ALMA observations misses extented emission. Consequently, only the emission peak in the center of the core is detected by ALMA. The 3mm continuum map using only the ACA data is shown in Figure \[figure:line\_maps\]d, as a contour plot superposed on the $N(\htwo)$ image. (160,145)(0,0) (-2,70) (0,0) {width="9.0cm"} (90,70) (0,0) {width="9.0cm"} (-2,-2) (0,0) {width="9.0cm"} (90,-3) (0,0) {width="9.0cm"} (70,138)[(0,0)[ **a)**]{}]{} (160,138)[(0,0)[ **b)**]{}]{} (70,65)[(0,0)[ **c)**]{}]{} (160,65)[(0,0)[ **d)**]{}]{} The $\dammo$ map resembles the $\htwo$ column density map shown in Figure \[figure:line\_maps\]d, and the far-infrared dust emission maps of the core, e.g., the 850$\mu{\rm m}$ SCUBA-2 map shown in Figure \[figure:taumidfar\]. The $\meth$ and SO distributions closely resemble each other, and they are almost complementary to the $\dammo$ distribution; methanol and sulphur monoxide follow the edges of the elongated core but are stronger on the left, concave side of the core (the eastern side in the sky) than elsewhere. Analysis of the line data {#sec:analysis} ========================= Four lines of the $(2_k-1_k)$ group of $\meth$ were included in the spectral window around 96.7 GHz. Three of them belong to the $E$ symmetry species and one to the $A$ species. The rotational temperatures, $T_{\rm rot}$, and the column densities, $N$, of $E$-type methanol were derived adopting the procedure described by [@2000ApJS..128..213N], where no assumption about the optical thickness of the lines is made. The rotational temperatures are in the range $T_{\rm rot} \sim 8 - 11$K. The peak value of the $E$-methanol column density is $N(E-\meth) \sim 1.5\times 10^{14}$ cm$^{-2}$. In the region with bright $\meth$ emission, bordering the eastern side of the core, $N(E-\meth) \geq 6\times10^{13}$ cm$^{-2}$. The $A$-methanol column density, $N(A-\meth)$, was estimated using the single $A$-line in the spectrum, assuming that the rotational temperatures of the $A$ and $E$ symmetry species are the same. The $A/E$ ratios in the bright region are in the range $1.2-1.5$ (the high-temperature statistical limit is 1). This indicates that $N({\meth}) \sim 2\times N(E-\meth)$. The data set contains only one spectrally unresolved SO line, $J_N=3_2-2_1$, at 99.3 GHz. Therefore, we only can derive the lower limit of the SO column density assuming optically thin emission. In this approximation, the upper state column density is directly proportional to the integrated brightness temperature, $\int\, T_{\rm B}\, dv$, of the line. A rotation temperature, assumed to be the same for all rotational transitions, needs to be adopted for the calculation of the partition function. Assuming $T_{\rm rot}=9$K, we get $N({\rm SO})=1.2\times10^{13}\,{\rm cm}^{-2} \times \int\, T_{\rm B}\, dv$, when the integral is in K$\kms$. For $T_{\rm rot}=5$K, the numerical factor before the integral is $1.7\times10^{13}\,{\rm cm}^{-2}$. The fractional $\meth$ and SO abundance distributions were determined by dividing the column density maps by the $N(\htwo)$ map shown in Figure \[figure:line\_maps\]d. The fractional $\meth$ abundance map is shown in Figure \[figure:xmeth\]. The estimates are limited to the region with bright methanol emission, where the S/N ratio exceeds 3. The highest methanol abundances, $X(\meth)\sim 1\times10^{-8}$ (assuming equal abundances for the $E$ and $A$ types) are found at the eastern border of the core. The SO abundances are highest south of the methanol peak, with $X({\rm SO}) \sim 5-8\times 10^{-10}$, depending on the assumed $T_{\rm rot}$ (9 or 5 K). (80,63)(0,0) (0,-2) (0,0) ![Fractional $E-\meth$ abundance distribution. The contours represent the $\htwo$ column density. They go from $1\times10^{22}$cm$^{-2}$ to $10\times10^{22}$cm$^{-2}$.[]{data-label="figure:xmeth"}](hmm1_xmeth.png "fig:"){width="8.0cm"} In this paper, we only use the kinematic information from the $\dammo$ lines; the $\dammo$ column densities and fractional abundances will be discussed elsewhere. The ortho-$\dammo$ spectral cube was analyzed by performing multicomponent Gaussian fits to the hyperfine structure of the $1_{11}-1_{01}$ line. The model used for the hyperfine structure takes into account the splittings owing to the electric quadrupole moments of both N and D nuclei . The fits were made to positions where the signal-to-noise ratio exceeds 3. These fits give accurate estimates for the line velocity and width. The distributions of the one-dimensional velocity dispersion of the ortho-$\dammo$ and $\meth$ lines are shown in Figure \[figure:sigma\_histos\]. (80,55)(0,0) (0,0) (0,0) ![One-dimensional velocity dispersions of the ortho-$\dammo$ and $\meth$ lines in Ophiuchus/H-MM1. These are obtained from Gaussian fits to the line profiles. The blue and red vertical lines indicate the thermal velocity dispersions of $\dammo$ and $\meth$ at 10 K. The green vertical line indicates the sound speed at this temperature. This Figure shows that methanol at the core interface has broader (non-thermal) line widths that $\dammo$.[]{data-label="figure:sigma_histos"}](sigmav_histos.png "fig:"){width="8.0cm"} Discussion ========== Reactive desorption {#sec:reactdes} ------------------- In accordance with previous mappings, we find relatively high abundances of methanol in a prestellar core, offset from the density maximum. Thanks to the high spatial resolution achieved with ALMA, methanol emission in H-MM1 is found to be confined to a layer that follows the core boundaries. The projected thickness of this layer ranges from $\sim 500$ au to $\sim 1500$ au. The critical density of the coexistent SO$(3_2-2_1)$ line is $3\times 10^5$cm$^{-3}$ at 10K, whereas for the methanol lines this is $3\times10^ 4\,{\rm cm^{-3}}$. We assume that the higher of these values, $\sim 10^5$cm$^{-3}$, is characteristic of the gas component detected in $\meth$ and SO. This value agrees with the densities derived by for methanol emission regions in prestellar cores. At still higher densities, traced by $\dammo$, both $\meth$ and SO are apparently frozen out. The behaviour of $\meth$ and SO conform, at least qualitatively, with the predictions of the reactive desorption model (@2017ApJ...842...33V; their Figures 4 and 8). In particular, the formation of methanol should be most efficient in the ’CO freeze-out zone’ at densities of $10^4 - 10^5$ cm$^{−3}$. This zone is characterized by the balance between CO depletion and the production of fresh CO in the gas phase. Reactive desorption is favored in this region by the fact that the surface of the icy mantles is dominated by the apolar, CO-rich component [@2017ApJ...842...33V]. The reactive desorption model therefore naturally explains why $\meth$ seems to avoid the densest parts, and sometimes shows a shell-like distribution (; ; @2018ApJ...855..112P). Sulphur monoxide can form both in the gas phase and on grain surfaces. Without efficient desorption, gaseous SO depletes quickly at high densities. On the other hand, as soon as atomic sulphur is available in the gas phase, SO and other sulphur-bearing molecules are thought to form quickly in the gas phase through reactions with O, $\otwo$ and OH (; @2017MNRAS.469..435V). Sulphur is probably released from ice mantles either as $\htwos$ or in the atomic form. Methanol and sulphur monoxide are not directly related. Their spatial coincidence could possibly be explained by reactive codesorption of $\meth$ and SO or sulphur in some other form. Alternatively, gas-phase reactions forming SO may be efficient in the same conditions where $\meth$ is desorbed. In the model of [@2017ApJ...842...33V], gaseous $\meth$ is abundant at a depth where water is partially photodissociated (their Figuge 6). The presence of hydrogen atoms and hydroxyl radicals in this layer can also promote the formation of SO. The strong asymmetry observed in H-MM1, and previously in the prestellar core L1544 (; @2018ApJ...855..112P) is, however, difficult to explain without invoking an external agent. suggested that the methanol distribution in L1544 reflects asymmetric illumination which hinders CO production on the more exposed side of the cloud. Intense radiation could also inhibit SO formation by keeping sulphur atoms ionized. The ionization energy of S is relatively low, 10.36eV; the limiting wavelength of ionizing radiation, $\lambda < 120$nm, falls in the range where photons also can dissociate $\meth$ efficiently ($114\,{\rm nm} < \lambda < 180\,{\rm nm}$; ). The ambient cloud around H-MM1 continues towards the east, giving a reason to believe that the interstellar radiation field is indeed stronger on the side where $\meth$ and SO are weaker. This side faces the Upper Sco-Cen (USC) subgroup of the Scorpius-Centaurus OB association, including the luminous B-type double stars $\rho$ Oph and HD 147889 (; @2008hsf2.book..235P; @2018arXiv180711884D). HD 147889, about 1.2pc west of H-MM1, is the dominant UV source in the region (; @2013MNRAS.428.2617R). The H-MM1 core is, however, embedded in the molecular cloud and the UV field at the core boundaries is probably weak. Methanol and SO emissions are confined within the column density contour $N(\htwo)=10^{22}\,{\rm cm^{-2}}$, which corresponds to a total visual extintion of $A_{\rm V}\sim 10^{\rm mag}$ through the cloud. One can reasonably assume that the visual extinction from the cloud surface to the western edge of the core with weak $\meth$ emission is $A_{\rm V}\sim5$ or higher. This implies that less than $0.2\%$ of UV radiation with $\lambda < 180$nm will reach this region[^1], and that the scarcity of $\meth$ on the western side is not caused by photodissociation. Nevertheless, dust grains are primarily heated by absorption of starlight, and the asymmetry in the radiation field can have caused a temperature difference between the two sides of the core, which in turn can influence on the abundances of $\meth$ and SO. We examined this hypothesis using mid- to far-infrared maps from Spitzer, Herschel and SCUBA-2, and found that the dust temperature reaches its minimum on the eastern side of the core, close to the $\meth$ maximum. The analysis of the dust continuum maps is presented in Appendix \[sec:tdust\]. The result of the analysis suggests that a temperature drop from about $14-15$K to about $11-12$K (see Fig. \[figure:taumidfar\]) favours methanol formation on grains. This is likely to be caused by effective accretion and hydrogenation of CO on grains at low temperatures. It should be noted, however, that the cooler region does not cover the whole eastern side of the core where strong $\meth$ emission is found, and it does not extend to the SO maximum. At the northern and southern ends of the integral-shaped $\meth$ and SO emission region, the dust temperature is similar to that on western side where these species show only weak emission. Shocks {#sec:shocks} ------ Both $\meth$ and SO are known to increase in shocks, and they have been used to probe outflows and the accretion process associated with star formation (@1997ApJ...487L..93B; ; @2016ApJ...824...88O). Strong enhancement of SO and $\sotwo$ is predicted by models of magnetized molecular C-shocks, as a result of neutral-neutral reactions in the shock-heated gas and the erosion of S-rich icy grain mantles owing to bombardment by heated gas particles (@1993MNRAS.262..915P; @1994MNRAS.268..724F). Sputtering of the grain mantles associated with shocks can also increase the $\meth$ abundance substantially in the gas phase . Previous observations of mid-$J$ CO rotational lines toward Perseus and Taurus complexes suggest low-velocity shocks associated with dissipation of turbulence and core formation in molecular clouds (@2014MNRAS.445.1508P; @2015ApJ...806...70L). Judging from the fact that the non-thermal velocity dispersion experiences an abrupt change at the core boundary (@2019ApJ...872..207A; see Section \[sec:vgrain\]), low-velocity shocks caused by accreting material are also possible in the case of H-MM1. However, the present data consisting of low-lying rotational lines $\meth$, SO, and $\dammo$ do not show any evidence of shock heating or velocity gradients that would be large enough ($\Delta v \ga 10\,\kms$) to give rise to significant shock-induced sputtering. Grain-grain collisions in the turbulent envelope {#sec:vgrain} ------------------------------------------------ The interior parts of the H-MM1 core have a very low level of non-thermal motions, as evinced by the velocity dispersion of the ortho-$\dammo$ lines (Figure \[figure:sigma\_histos\]). In a study based on $\ammo(1,1)$ and $(2,2)$ inversion line observations from the GreenBank Ammonia Survey [@2017ApJ...843...63F], [@2019ApJ...872..207A] show that the non-thermal velocity dispersion increases suddenly at the boundaries of this and several other cores in Ophiuchus. Hydrodynamic drag and magnetic fields accelerate grains in a turbulent environment, and the acceleration is more effective for large grains than for small grains (@1985prpl.conf..621D; @2002ApJ...566L.105L; @2004ApJ...616..895Y). This gives rise to velocity differences between small and large grains, and to an enhanced rate of grain-grain collisions. As discussed by , low-velocity collisions between grains can lead to grain heating above the temperature threshold ($\sim 30$K) that triggers explosive radical-radical reactions and the partial disruption of the grain mantle. Assuming that the grains consist of a silicate core and a mantle of water ice constituting $\sim 15\%$ of the grain mass (corresponding to a fractional water abundance of $\sim 10^{-4}$; @2013ChRv..113.9043V), one finds that the enthalpy change needed to raise the grain temperature from 10K to 30K corresponds to a collision velocity of $\sim 30$ms$^{-1}$. Here we have used the Debye model for the heat capacity of the silicate core as a function of temperature (adopting the characteristics of olivine), and the formula from (their Eq.4) for the heat capacity of water ice. In Figure \[figure:vgrain\] we show relative grain velocities as a function of grain radius according to the models of [@1985prpl.conf..621D] and [@2002ApJ...566L.105L], for conditions characteristic of the outer envelope of a cold dense core ($T=10$K, $n(\htwo)=10^5$cm$^{-3}$, $B=100\,\mu$G). Here it is assumed that the turbulent velocity field has a Kolmogorov-like spectrum, $v \propto l^{1/3}$. The absolute scale is set by assuming that the turbulent velocity is $0.8\,\kms$ on the scale 0.05pc, which corresponds to the observed velocity gradient across the core ($\Omega \sim v/l$; for discussion about the connection between velocity gradients and turbulence see @2000ApJ...543..822B). Except for the smallest grains, the grain velocities are determined by turbulent velocity fluctuations occurring on a time scale comparable to that of the hydrodynamical drag. In Kolmogorov turbulence, the velocity is proportional to the square-root of the eddy turn-over time. Because the drag time is directly proportional to the grain radius, $a$, the velocity distribution of large grains has a square-root dependence on the grain radius, $v \propto a^{1/2}$. The smallest grains are also coupled to the magnetic field (a great majority of grains is negatively charged at visual extinctions above $A_{\rm V} \sim 3^{\rm mag}$; @2015ApJ...812..135I), and their velocity dispersion is determined by turbulent fluctuations on the time scale comparable to the Larmor time. This causes a $v \propto a^{3/2}$ dependence for the smallest grains. Assuming the MRN grain size distribution [@1977ApJ...217..425M], the small, slow grains with velocities below $10\,{\ms}$ comprise $\sim 50\%$ of the total surface area of the grains, whereas the share of large, fast grains with $v > 30\,\ms$ is only $\sim 5\%$ of the surface area. (80,61)(0,0) (0,0) (0,0) ![Relative speed of grains as a function of the grain radius according to the turbulent acceleration model (@1985prpl.conf..621D; @2002ApJ...566L.105L) in dense dark cloud conditions (see text). The regimes of “high” and “low” velocity grains are indicated with red and blue, respectively. The discontinuity in the gradient is caused by the coupling of the smallest grains to the magnetic field.[]{data-label="figure:vgrain"}](vgrain.png "fig:"){width="8cm"} Assuming that a given molecule X is being formed solely on the grains, and the number density of these molecules on the slow grains is $n_{\rm X}^{\rm surf, slow}$ (corresponding in our example to $\sim 50\%$ of the total number density, $n_{\rm X}^{\rm surf}$), and that every collision between a slow and a fast grain leads to the complete desorption of molecules X from the surface of the small grain, the equilibrium gas-phase abundance of the molecule X, $n_{\rm X}^{\rm gas}$, can be written as $$n_{\rm X}^{\rm gas} = \frac{ n_{\rm X}^{\rm surf,slow} \, n_{\rm g}^{\rm fast} \; \pi ({\bar a}^{\rm fast} + {\bar a}^{\rm slow})^2 \, ( {\bar v}_{\rm g}^{\rm fast} - {\bar v}_{\rm g}^{\rm slow} ) } {\Sigma_{\rm g} \, {\bar v}_{\rm therm}} \; ,$$ where $n_{\rm g}^{\rm fast}$ is the number density of fast grains, ${\bar a}^{\rm fast}$ and ${\bar a}^{\rm slow}$ are the average radii of the fast and slow grains, respectively, ${\bar v}_{\rm g}^{\rm fast}$ and ${\bar v}_{\rm g}^{\rm slow}$ are their average speeds, $\Sigma_{\rm g}$ is the total surface area of the grains per volume element, and ${\bar v}_{\rm therm}$ is the average thermal speed of the gas particles (; ). The denominator is the depletion rate per molecule. When both particles are negatively charged, the kinetic energy overcomes the Coulomb barrier in collisions between fast and slow grains. The observed peak fractional abundance of methanol, $\sim 1\times 10^{-8}$ can be reproduced if the total methanol abundance on grains is $n_{\meth}^{\rm surf}/n(\htwo) \sim 5\times10^{-6}$. The grain speeds decrease rapidly below the critical value ($30\,\ms$) toward higher densities. On the other hand, at lower densities, the formation of methanol is inhibited because CO does not freeze onto grains efficiently enough. So, like in reactive desorption, the abundance of gaseous methanol owing to grain-grain collisions should peak at densities characteristic of core boundaries. The process should affect all atoms and molecules residing in the CO-rich outer layers of grain mantles, and this would explain why $\meth$ and SO have similar distributions. In this scenario, the asymmetric distributions of $\meth$ and SO around H-MM1 would either indicate that turbulence is stronger on the eastern side of the core or that dust grain surfaces are richer in CO on that side. The latter suggestion is supported by the fact that the $\meth$ peak coincides with the dust temperature minimum (Appendix \[sec:tdust\]). The former condition cannot be properly tested using the present data because the boundaries are not probed with any other lines than methanol and SO; the $\dammo$ line emission is confined to the inner regions where the lines are narrow. The angular resolution of the GreenBank $\ammo$ maps ($32\arcsec$; @2017ApJ...843...63F), which cover both the core and the ambient cloud, is not sufficient for detailed comparison of the velocity dispersions at the eastern and western boundaries. Nevertheless, a similar analysis as performed by [@2019ApJ...872..207A], but averaging over semicircles, shows that the non-thermal velocity dispersion (measured along the line of sight) grows more slowly on the eastern side than on the western side, contradicting the supposition of a more vigorous turbulence on that side. A diagram illustrating this difference is shown in Figure \[figure:sigmas\]. Grain-grain collisions in a shear layer {#sec:shear} --------------------------------------- One remarkable feature of the brightest $\meth$ emission region is that it closely follows the eastern boundary of the core as marked by $\dammo$ emission. The wavy shape and the rolling up seen in the north (Figure \[figure:xmeth\]) are the hallmarks of Kelvin-Helmholtz instability (KHI), which can occur in sheared flows with density stratification. The methanol distribution gives the impression that the more tenuous gas at the eastern boundary flows to the north, more or less in the plane of the sky, or that the denser material of the core flows to the south. The presence of shear in a gas flow with a high Reynolds number implies small-scale vorticity, and the undulating methanol emission region probably consists of a chain of secondary billows that are unresolved in the present map. As described in standard textbooks, such as [@Batchelor1967], advection of vorticity amplifies the dominant sinusoidal disturbance in a vortex sheet and makes it roll up. The evolution of vortex sheets has been studied numerically and semi-analytically by [@1976JFM....73..215P] and [@1976JFM....73..241C]. According to their results, vorticity reaches its maximum values at places they call “braids” (near the troughs) and “cores” or cat’s eyes (near the crests), separated by approximately half the wavelength of the dominant disturbance. By comparing the methanol map to the computed images of [@1976JFM....73..215P], one can identify the methanol peak with a braid and the overturning billow in the north with a core. The apparent wavelength of the methanol feature is approximately $60\arcsec$, corresponding to $7,200$ au or $0.035$pc In what follows, we attempt to estimate the flow properties, assuming that, in analogy with atmospheric billow clouds, the largest wavelength correspond to the so called internal gravity waves, which oscillate at the Brunt-V[ä]{}is[ä]{}l[ä]{} frequency. This frequency, also known as the buoyancy frequency, is defined by $N=\sqrt{\frac{-g}{\rho}\frac{\partial{\rho}}{\partial{z}}}$, where $g$ is the gravitational acceleration (directed to the negative $z$ direction) and $\rho$ is the gas density. We estimated the density and the density gradient in the supposed shear layer by fitting a Plummer-type function to the column density profile from the IRAC $8\,\mu$m absorption. The method is explained by . The fit was made to the cross-sectional profile along an axis going through the methanol peak at RA 16:27:59.5, Dec. -24:33:30 (J2000). The axis is tilted with respect to the R.A. axis by $25^\circ$. According to this fit, the number densities at methanol peak and at the spine of cloud are $\sim 4\times10^5\,{\rm cm}^{-3}$ and $\sim 1\times 10^6\,{\rm cm}^{-3}$, respectively. The separation between these points is approximately $10\arcsec$ (1,200au). We assumed that the gravitational field at the methanol peak is dominated by the mass contained in a circular region of a radius of $10\arcsec$, centered at the crossing of the cloud spine and the cross-sectional axis. This mass is $0.08\,M_\odot$. The Brunt-V[ä]{}is[ä]{}l[ä]{} frquency obtained is $N \sim 1.4\times10^{-12}$Hz (period 140,000 yr). The multiplication $N$ by the apparent wavelength gives a phase velocity of $\sim 240\,{\rm m\,s^{-1}}$. This should correspond to the average speed of the sheared flow. The fact that the flow is subject to KHI imposes a condition to the velocity shear: the Richardson number, defined by ${\rm Ri} = \frac{N^2}{(\partial{v}/\partial{z})^2}$, is less than $1/4$. The implied minimum shear is $\sim 2.8\times 10^{-12}\,{\rm s}^{-1}$ or $\sim 90\, \kms\,{\rm pc}^{-1}$. This value exceeds the north-south velocity gradient derived from the molecular line maps by a factor of 4-5. Assuming that the velocity at the outer boundary of the shear layer does not exceed $400\,{\rm m\,s^{-1}}$, which is the typical non-thermal velocity far from the core, the maximum thickness of the layer is approximately 1000au. This estimate agrees with the thickness of the methanol layer. We conjecture that vorticity in the shear layer can accelerate dust grains to velocities deviating from the mean flow, and induce grain-grain sollisions, in the same manner as Kolmogorov-type turbulence discussed in the previous section. However, the energy spectrum in a vortex sheet differs from that in fully developed turbulence which has $E(k) \propto k^{-5/3}$, where $k$ is the wave number, and $\int_0^\infty E(k) dk$ is the average kinetic energy per unit mass. Kraichnan suggested that, because of the conservation of the mean-square vorticity for two-dimensional turbulence in an invisced fluid, the energy spectrum at the smallest scales should have the form $E(k) \propto k^{-3} \left\{\ln(k/k_1)\right\}^{-1/3}$ for $k\gg k_1$, where $k_1$ is the wavenumber at which vorticity is pumped in (@1967PhFl...10.1417K; @1971JFM....47..525K). Numerical studies of two-dimensional turbulence give $E(k) \propto k^{-\gamma}$, where $3 < \gamma < 4$ (e.g., @1988JFM...193..475G; @2011PhRvE..84b6318A). These energy spectra correspond to steeper velocity scaling laws than the Kolmogorov spectrum; the exponent $\gamma = 3$ gives $v \propto l$, and $\gamma = 4$ is equivalent to $v \propto l^{3/2}$. The scaling law $v \propto l$ implies one single time-scale in the energy cascade. Following the argumentaion of [@1985prpl.conf..621D], this would mean that grains with the drag time (corresponding to certain radius $a$) comparable to this time-scale can be accelerated to any velocities present in the vortex sheet. Smaller grains would be advected by the swirling flow, whereas larger grains would not respond to the velocity fluctuations. The maximum speed is likely to be comparable with the free-stream velocity, estimated above to be of the order of $200\,{\rm m\,s^{-1}}$. The time-scale depends on the structure of the vortex sheet. For a regular structure, such as the one given by the stream function in Eq. (3.9) of [@1976JFM....73..241C], the time-scale is $\sim \lambda/U$, where $\lambda$ is the vortex size and $U$ is the free-stream velocity. The maximum drag time for grains in the MRN size distribution is $\sim 2,000$yr in conditions described here. Using the speed mentioned above, we get for the vortex size $\lambda < 80$au. Kraichnan’s energy spectrum with the logarithmic correction and any simple power law with $\gamma > 3$ would imply that the smallest grains are accelerated to the highest speeds. For example, the formula of [@1971JFM....47..525K] gives approximately $v \propto a^{-1}$, and the power $\gamma=4$ gives $v \propto a^{-3}$. In view of the uncertainties concerning the “vertical” scale and the structure of the vorticity, we do not attempt to estimate methanol production rate in the suggested vortex sheet. We merely state that the present observations indicate that the formation methanol is more efficient in this layer than elsewhere in the core envelope. We suggest, based on the theoretical and numerical studies quoted above, that this is caused by the fact that two-dimensional turbulence can have larger velocity fluctuations than full three-dimensional turbulence on a time scale comparable to the drag time of dust grains. Conclusions {#sec:concl} =========== Gaseous methanol and sulphur monoxide coexist on the outskirts of the prestellar core H-MM1 in Ophiuchus. They are confined to a halo which follows the core boundaries and which is particularly prominent on the eastern side of the core. Because methanol is mainly produced on grains, the emission indicates regions of enhanced desorption. Sulphur monoxide may have co-desorbed with $\meth$ or formed in the gas phase following the release of $\htwos$. We suggest that in addition to desorption caused by spontaneous exothermic surface reactions [@2017ApJ...842...33V], also grain-grain collisions can efficiently release molecules from grains in the core envelope. The latter mechanism is likely to proceed through mild heating that triggers explosive radical-radical reactions and partial disruption of grain mantles . Collisions between grains can be induced by Kolmogorov turbulence (@1985prpl.conf..621D; @2002ApJ...566L.105L), or by shear vorticity. The brightest methanol emission region at the eastern boundary of the core shows signatures of Kelvin-Helmholtz instability, indicating strong velocity shear. On the other hand, the non-thermal velocity dispersion along the line of sight, as traced by $\ammo$ lines, grows more steeply on the opposite side of the core, where both $\meth$ and SO emissions are weak. We interpret this so that at the eastern boundary, laminar flow occurring in the plane of the sky is currently transitioning into turbulence through shear instability, whereas on the western side, the flow has already developed into full three-dimensional turbulence. The asymmetries of the $\meth$ and SO distributions suggest that shear vorticity induces more energetic or more frequent grain-grain collisions than Kolmogorov turbulence. This is likely to be related to the fact that two-dimensional turbulence has steeper energy spectrum [@1971JFM....47..525K], implying larger velocity fluctuations on the time scale needed to accelerate grains through hydrodynamic drag. We thank Tom Hartquist, David Williams, Kalevi Mattila, and Hannu Savij[ä]{}rvi for helpful and enjoyable conversations. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2016.1.00035.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. This work is based in part on observations made with the Spitzer Space Telescope, and made use of the NASA/IPAC Infrared Science Archive. These facilities are operated by the Jet Propulsion Laboratory, California Institute of Technology under a contract with National Aeronautics and Space Administration. This work was supported by the Academy of Finland (grants 285769 and 307157). Column density map from 8$\mum$ extinction {#sec:column} ========================================== The core is obscured from ultraviolet radiation and it serves as an absorbing component at 8$\mum$. The surface brightness in the core region can therefore written as $I^{8\,\mum} = I_{\rm bg}^{8\,\mum}\,e^{-\tau^{8\,\mum}} + I_{\rm fg}^{8\,\mum}$, where $\tau^{8\,\mum}$ is the optical thickness at 8$\mum$, $I_{\rm bg}$ is the surface brightness at the background of the core, and $I_{\rm fg}$ is the surface brightness of the foreground component (see, e.g., @2009ApJ...696..484B). The latter term may also contain a zero point offset. The optical thickness map can be obtained from $$\tau^{8\,\mum} = -\ln \left \{ \frac{I^{8\,\mum}-I_{\rm fg}^{8\,\mum}} { I_{\rm bg}^{8\,\mum}} \right \} \; .$$ In order to estimate the background emission, $I_{\rm bg}^{8\,\mum}$, we made a fit to the surface brightness distribution with the core masked out. For masking we used the SCUBA-2 $850\,\mum$ map. The foreground component, $I_{\rm fg}^{8\,\mum}$ (including possible zero-point offset), was assumed to be constant in the mapped region. Its value was estimated by requiring that the ratio of the peak optical thicknesses is $\tau^{8\,\mum}/\tau^{850\,\mum}=780$, corresponding to the adopted dust opacity model, which was the model for unprocessed dust grains with thin ice mantles by . The comparison was done using an IRAC map smoothed to the resolutions of the SCUBA map. At $850\,\mum$, the core is optically thin and it is seen in thermal dust emission. The observing method filters out extended emission. The surface brightness is therefore $I^{850\,\mum} = B^{850\,\mum}(T_{\rm d})\,\tau^{850\,\mum}$, where $T_{\rm d}$ is the dust temperature and $B$ is the Planck function. The opacity map at $850\,\mum$, $\tau^{850\,\mum}$, was calculated by combining the $850\,\mum$ surface brightness map with the dust color temperature map, $T_{\rm C}$, derived from Herschel/SPIRE maps . The angular resolution of the $T_{\rm C}$ map is $\sim 40\arcsec$, whereas for SCUBA it is approximately $14\arcsec$. We think this discrepancy in the angular resolutions is acceptable because the $850\,\mum$ surface brightness is not very sensitive to the dust temperaure (see below). Finally, $\htwo$ column density map shown in Fig. \[figure:line\_maps\]d) was calculated by dividing the $8\,\mum$ optical thickness map by the absorption cross-section of dust per $\htwo$ molecule, which according to the adopted dust model is $4.1\times10^{-23}\,{\rm cm^2\, \htwo\, molecule^{-1}}$ ($8.86\,{\rm cm^{2}\,g^{-1}}$). Dust temperature distribution {#sec:tdust} ============================= At $70\,\mum$, a cloud can be seen either in absoption or emission depending on the dust temperature and the brightness of the background. At this wavelength, the Planck function is very sensitive to small changes in the temperature. For example, a temperature change from 12 to 14K for a given dust column increases the emission by an order of magnitude, while at $850\,\mum$, the corresponding change is $\sim 20\%$. The surface brightness at $70\,\mum$ takes the form $$I^{70\,\mum} = I_{\rm bg}^{70\,\mum}\,e^{-\tau^{70\,\mum}} + I_{\rm fg}^{70\,\mum} + B^{70\,\mum}(T_{\rm d})(1-e^{-\tau^{70\,\mum}}) \; .$$ We assumed that the $\tau^{70\,\mum}$ distribution is identical to that at $8\,\mum$ save a constant factor depending on the adopted dust opacity model (giving $\tau^{8\,\mum}/\tau^{70\,\mum}=1.2$). The foreground level (including a possible zero-point offset) was adjusted so that the dust temperatures on the outskirts of the core are similar to $T_{\rm C}$ values derived from Herschel/SPIRE. The obtained dust temperature distribution in the core region is shown in the bottom right panel of Figure \[figure:taumidfar\]. For the $T_{\rm d}$ calculation, the $8\,\mum $ and $70\,\mum$ maps were smoothed to the $14\arcsec$ resolution of the SCUBA-2 map. This map indicates that the dust temperature minimum is shifted east from the column density maximum, and is nearly coincident with the $\meth$ maximum. We tested this result by constructing a core model resembling H-MM1. Here it was assumed that the line-of-sight density distribution has the same shape as the $\tau^{850\,\mum}$ profile along horisontal cuts across the core. The core was illuminated by an isotropic radiation field plus a point source located on its western side. The dust opacity model was the same as used above. The calculations were done using the dust continuum radiative transfer program described in . The strenghts of the isotropic component and the point source were adjusted until the $850\,\mum$ and $70\,\mum$ surface brightness maps agreed with the observations. The dust temperature distribution derived from the simulated $70\,\mum$ map in the same manner as described above shows a similar shift of the minimum as seen in Figure \[figure:taumidfar\]f. In the “true”, three-dimensional distribution, the dust temperature minimum is shifted to the same direction, but lies a little closer to the density peak. This experiment shows that dust temperatures derived from te $70\,\mum$ surface brightness temperature map can correctly reflect, although exaggerating slightly, the displacement of the dust temperature minimum from the density maximum (and the $850\,\mum$ peak) caused by anisotropic illumination. (160,80)(0,0) (10,0) (0,0) {width="16.0cm"} (28,74)[(0,0)[ **a)**]{}]{} (28,38)[(0,0)[ **b)**]{}]{} (78,74)[(0,0)[ **c)**]{}]{} (78,38)[(0,0)[ **d)**]{}]{} (128,74)[(0,0)[ **e)**]{}]{} (128,38)[(0,0)[ **f)**]{}]{} Velocity dispersion profiles {#sec:sigmav} ============================ Thermal and non-thermal velocity dispersion profiles on the eastern and western sides of the core were calculated using $\ammo(1,1)$ and $(2,2)$ inversion line data from the GreenBank Ammonia Survey (GAS; @2017ApJ...843...63F). The method is described in detail by [@2019ApJ...872..207A]. The spectra were first aligned using the Greenbank pipeline-reduced LSR velocity maps, and then averaged in concentric semiannular regions. The stacked $\ammo(1,1)$ and $(2,2)$ were analyzed using the standard method described, for example, by . This analysis gives estimates for the kinetic temperature and total velocity dispersion (along the line of sight). The radial distributions of thermal and non-thermal velocity dispersions were calculated from these data. The results for the two hemispheres of H-MM1 are shown in Figure \[figure:sigmas\]. The non-thermal dispersion is approximately half the sound speed near the center, and reaches a value of $\sim 400\,{\rm m\,s^{-1}}$, that is, twice the sound speed far from the center of the core. The transition from subsonic to supersonic regimes occurs, however, closer the center on the western side of the core than the eastern side, where the strongest methanol emssion comes from. (160,130)(0,0) (5,0) (0,0) {width="16.0cm"} [^1]: Estimated using the extiction law and formula from [@1989ApJ...345..245C], and assuming the extinction parameter value $R_{\rm V}=A_{\rm V}/E_{B-V}=4.0$ which is suggested to be appropriate for the $\rho$ Ophiuchi cloud (@1993AJ....105.1010V; @2001ApJ...547..872W).

--- abstract: 'Exact solutions of Schrödinger equation for PT-/non-PT-symmetric and non-Hermitian Morse and Pöschl-Teller potentials are obtained with the position-dependent effective mass by applying a point canonical transformation method. Three kinds of mass distributions are used in order to construct exactly solvable target potentials and obtain energy spectrum and corresponding wave functions.' author: - | Özlem Yeşiltaş $^1$ and Ramazan Sever $^2$ [^1]\ $^1$ Turkish Atomic Energy Authority, Istanbul Road, 30 km Kazan 06983, Ankara, Turkey\ $^2$ Department of Physics, Middle East Technical University, 06531, Ankara, Turkey title: ' Exact solutions of Schrödinger equation for PT-/non-PT-symmetric and non-Hermitian Exponential Type Potentials with the position-dependent effective mass ' --- PACS Nos: 03.65.Db, 03.65.Ge\ Keywords:Schrödinger equation; PT-symmetry; Morse potential; Pöschl-Teller potential; Point canonical transformation; Position dependent mass Introduction ============ In the past few years, theoretical researches on great variety of non-Hermitian Hamiltonians have received an important increase. Because many of these systems are invariant under combined parity and time reversal (PT) transformation which lead to either real (in case of broken PT symmetry) or pairs of complex conjugate energy eigenvalues (in case of spontaneously broken PT symmetry) \[1,2\]. This property of energy eigenvalues in non-Hermitian PT invariant systems can be related to the pseudo-hermiticity \[3\] or anti-unitary symmetry \[4,5\] of the corresponding Hamiltonians. In ref. \[6\] it was proposed a new class of non-Hermitian Hamiltonians with real spectra which are obtained using pseudo-symmetry. Moreover, completeness and orthonormality conditions for eigenstates of such potentials are proposed \[7\]. In the study of PT-invariant potentials various techniques have been applied to a great variety of quantum mechanical fields as variational methods, numerical approaches, Fourier analysis, semi-classical estimates, quantum field theory and Lie group theoretical approaches \[7-16\]. In additional, PT-symmetric and non-PT symmetric and also non-Hermitian potential cases such as oscillator type potentials and a variety of potentials within the framework of SUSYQM \[17-21\], exponential type screened potentials \[22\], quasi/conditionally exactly solvable ones \[23\], PT-symmetric and non-PT symmetric and also non-Hermitian potential cases within the framework of SUSYQM via Hamiltonian Hierarchy Method \[24\] and some others are studied \[25-27\].\ On the other hand, there has been respected interest in a position-dependent mass, which is generally written as (PDM) $M(r)=m_{0}m(r)$, problems associated with a quantum mechanical particle forms an effective model for the study of many physical problems \[28-39\] due to considerable applications in condensed matter physics and material science. The model applied to wide variety of physical systems such as quantum dots \[40\], liquid crystals \[41\], kinetics of evolution of microstructures and atomic displacements in the string \[42\], He cluster \[43\] semiconductor heterostructures \[44\] and nuclei \[45\]. Generally, those works are concentrated in obtaining the energy eigenvalues and the potential function for the given quantum system with the PDM. In the mapping of nonconstant mass Schrödinger equation, point canonical transformations (PCTs) are employed \[46-49\]. During the process, it is needed to transform non-constant mass, which is known as “effective mass” characterizes the curvature of the dispersion relation, to a constant one so that the latter equation can be solved. Hence, energy spectra and corresponding wave functions of the target problem are produced easily. Various potentials, which satisfy the concept of exactly solvability, such as oscillator, Coulomb, Morse \[50\], hard-core potential \[51\], trigonometric type \[52\] and conditionally exactly solvable potentials \[53\] as well as the Scarf and Rosen-Morse type \[54\] ones including the PT-symmetry are considered for the construction of exact solution via PCT. The aim of this work is to apply PCT to the exact solutions of the nonconstant mass Schrödinger equation for Pöschl-Teller and Morse potentials which are complex and/or PT/non-PT symmetric, non-Hermitian and the exponential type systems.\ \ The contents of the present paper is as follows: In section II, it is shown how to construct effective mass Schrödinger equation by using PCT method. In section III, IV and V, using three different type mass distributions, PCT method is applied to general Morse and non-Hermitian, PT/Non-PT symmetric Morse potentials. In section VI, VII and VIII, the general form of Pöschl-Teller potential and non-Hermitian, PT/Non-PT symmetric Pöschl-Teller potentials are studied by using PCT method within three different mass functions in order to construct the target problem including energy eigenvalues and corresponding wavefunctions within PT symmetry.\ Effective Mass Schrödinger equation ==================================== As is well known, the general form of one dimensional time independent position-dependent mass Schrödinger equation (PDMSE) gives rise to $$\begin{aligned} -\frac{1}{2}\left[\nabla_{x}\frac{1}{M(x)}\nabla_{x}\right]\psi(x)- \left[E-V(x)\right]\psi(x)=0,\end{aligned}$$ where $M(x)=m_{0}m(x)$. So the Eq. (1) reads $$\begin{aligned} \psi^{''}(x)-\left(\frac{m^{'}}{m}\right)\psi^{'}(x)+2m\left[E-V(x)\right]\psi(x)=0,\end{aligned}$$ where $\hbar=1$ amd $m_{0}$ is a constant. The one dimensional Schrödinger equation with a constant mass is $$\begin{aligned} \Phi^{''}(y)+2\left[\varepsilon-V(y)\right]\Phi(y)=0.\end{aligned}$$ A transformation is defined as $y\rightarrow x$ and for a mapping $y=f(x)$, we rewrite the wave functions in the form $$\begin{aligned} \Phi(y)=g(x)\psi(x)\end{aligned}$$ The transformed Schrödinger equation reads $$\begin{aligned} \psi^{''}(x)+2\left(\frac{g^{'}}{g}-\frac{f^{''}}{f^{'}}\frac{g^{'}}{g}\right)\psi^{'}(x)+ \left(\left(\frac{g^{''}}{g}-\frac{f^{''}}{f^{'}}\frac{g^{'}}{g}\right)+ 2(f^{'})^{2}\left[V(f(x)-\varepsilon)\right]\right) \psi(x)=0.\end{aligned}$$ Comparing Eqs. (2) and (5), we get the following identities $$\begin{aligned} g(x)=\sqrt{\frac{f^{'}(x)}{m(x)}}\end{aligned}$$ and $$\begin{aligned} V(x)-E=\frac{(f^{'})^{2}}{m}\left[V(f(x)-\varepsilon)\right]-\frac{1}{2m}F(f,g)\end{aligned}$$ where $F(f,g)=\left(\frac{g^{''}}{g}-\frac{f^{''}}{f^{'}} \frac{g^{'}}{g}\right)$. As it is seen from Eqs. (2) and (5), if we substitute $(f^{'})^{2}=m$ in Eq. (7), then the reference problem is transformed to the target problem including the energy spectra of the bound states, potential and wave function as $$\begin{aligned} E_{n}=\varepsilon_{n}\end{aligned}$$ $$\begin{aligned} V(x)=V(f(x))-\frac{1}{8m}\left[\frac{m^{''}}{m}- \frac{7}{4}\left(\frac{m^{'}}{m}\right)^{2}\right]\end{aligned}$$ $$\begin{aligned} \psi(x)= [m(x)]^{1/4} \Phi_{n}(f(x)).\end{aligned}$$ The PCT method can be applied to a problem which has an exact solution by using the procedure given below. Generalized Morse Potential =========================== Consider the Morse potential as the reference problem \[19,22\] $$\begin{aligned} V(y)=V_{1}e^{-2\alpha y}-V_{2}e^{-\alpha y}\end{aligned}$$ The energy eigenvalues and eigenfunctions of the our source potential are given as $$\begin{aligned} \varepsilon_{n}=-\frac{\alpha^{2}}{4}\left[\frac{V_{2}}{\alpha \sqrt{V_{1}}}-(2n+1)\right]^{2}\end{aligned}$$ $$\begin{aligned} \Phi_{n}(y)=C_{n}s^{2\epsilon}e^{-\gamma s}L^{4\epsilon}_{n}(2\gamma s)\end{aligned}$$ where $s=\sqrt{V_{1}}e^{-\alpha y}$. Asymptotically vanishing mass distribution ------------------------------------------ In this section, we use asymptotically vanishing type mass distribution as given below in order to get some target potentials providing us the exact solutions $$\begin{aligned} m(x)=\frac{\alpha^{2}}{x^{2}+q}\end{aligned}$$ The mapping function becomes $$\begin{aligned} y=f(x)=\int^{x} \sqrt{m(x)}dx=\alpha ln \left(x+\sqrt{x^{2}+q}\right)\end{aligned}$$ and $$\begin{aligned} x=sinh_{q}\left(\frac{y}{\alpha}\right), \alpha\neq0.\end{aligned}$$ Using Eqs. (12-16), the new potential is obtained as $$\begin{aligned} V(x)=V_{1}\left(x+\sqrt{x^{2}+q}\right)^{-2\alpha^{2}}- V_{2}\left(x+\sqrt{x^{2}+q}\right)^{-\alpha^{2}}- \frac{1}{8\alpha^{2}}\left(1+\frac{q}{x^{2}+q}\right)\end{aligned}$$ Hence, the energy eigenvalues and corresponding wave functions for the general Morse potential are obtained as $$\begin{aligned} E_{n}=\varepsilon_{n}\end{aligned}$$ and $$\begin{aligned} \psi_{n}(x)=C_{n}\frac{\sqrt{\alpha}}{\left(x^{2}+ q\right)^{1/4}}(f(x))^{2\epsilon} e^{-\gamma f(x)} L^{4\epsilon}_{n}(2\gamma f(x)).\end{aligned}$$ where $\epsilon^{2}=-\frac{E}{2\alpha^{2}}$, $\gamma=\frac{1}{\alpha^{2}}$. Mass Distribution $m(x)=\frac{\alpha^{2}}{(b+x^{2})^{2}}$ --------------------------------------------------------- In the second example of mass distribution, the mapping function becomes $$\begin{aligned} y=f(x)=\alpha tan^{-1}\frac{x}{q}\end{aligned}$$ and $$\begin{aligned} x=\sqrt{q}tan(\frac{y}{\alpha})\end{aligned}$$ The mapping function leads to the following target system having the same energy spectra $$\begin{aligned} V(x)=V_{1}e^{-2\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}}- V_{2}e^{-\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}}- \frac{73x^{2}-16q}{32\alpha^{2}}\end{aligned}$$ and $$\begin{aligned} \psi_{n}(x)=\sqrt{\frac{q+x^{2}}{\alpha}}(f(x))^{2\epsilon} e^{-\gamma f(x)} L^{4\epsilon}_{n}(2\gamma f(x)).\end{aligned}$$ Exponential Type Mass Distribution ---------------------------------- If we consider the third type exponential mass function given as $$\begin{aligned} m(x)=e^{-\alpha x}\end{aligned}$$ $$\begin{aligned} y=f(x)=-\frac{2}{\alpha}e^{-\frac{\alpha x}{2}}\end{aligned}$$ and $$\begin{aligned} x=-\frac{2}{\alpha}ln(-\frac{\alpha y}{2})\end{aligned}$$ The mapping yields to potential with the same energy spectra $$\begin{aligned} V(x)=V_{1}e^{4e^{-\alpha x/2}}-V_{2}e^{2e^{-\alpha x/2}}+\frac{3\alpha^{2}e^{-\alpha x}}{32}.\end{aligned}$$ and corresponding wave function is $$\begin{aligned} \psi_{n}(x)=(f(x))^{2\epsilon}e^{-\alpha x/4-\gamma f(x)}L^{4\epsilon}_{n}(2\gamma f(x))\end{aligned}$$ Non-PT symmetric and non-Hermitian Morse Potential ================================================== In the equation (11), if the potential parameters are defined as $V_{1}=(A+iB)^{2}$, $V_{2}=(2C+1)(A+iB)$ and $\alpha=1$, then the potential becomes $$\begin{aligned} V(y)=(A+iB)^{2}e^{-2y}-(2C+1)(A+iB)e^{-y}\end{aligned}$$ where $A$, $B$ and $C$ are arbitrary real parameters and $i=\sqrt{-1}$. Similarly, the energy eigenvalues for the reference potential is given as \[19,22\] $$\begin{aligned} \varepsilon_{n}=-(n-C)^{2}\end{aligned}$$ Asymptotically vanishing mass distribution ------------------------------------------ Following the same procedure as in above, we get the target system $$\begin{aligned} E_{n}=\varepsilon_{n}\end{aligned}$$ $$\begin{aligned} V(x)=(A+iB)^{2}\left(x+\sqrt{x^{2}+q}\right)^{-2 \alpha}- (2C+1)(A+iB)\left(x+\sqrt{x^{2}+q}\right)^{- \alpha}- \frac{1}{8\alpha^{2}}\left(1+\frac{q}{x^{2}+q}\right).\end{aligned}$$ Mass Distribution $m(x)=\frac{\alpha^{2}}{(b+x^{2})^{2}}$ --------------------------------------------------------- Following the same procedure, we obtain the target potential with same energy spectra and $$\begin{aligned} V(x)=(A+iB)^{2}e^{-2\alpha tan^{-1}\frac{x}{\sqrt{q}}}- (2C+1)(A+iB)e^{-\alpha tan^{-1}\frac{x}{\sqrt{q}}}- \frac{73x^{2}-16q}{32\alpha^{2}}\end{aligned}$$ Exponential Type Mass Distribution ---------------------------------- We obtain the target potential with same energy spectrum for exponential type mapping function as $$\begin{aligned} V(x)=(A+iB)^{2}e^{\frac{4e^{-\frac{\alpha x}{2}}}{\alpha}}- (2C+1)(A+iB)e^{\frac{2e^{-\frac{\alpha x}{2}}}{\alpha}}+\frac{3\alpha^{2}e^{-\alpha x}}{32}.\end{aligned}$$ PT symmetric and non-Hermitian Morse Potential ============================================== When $\alpha=i\alpha$ and $V_{1}, V_{2}$ are real, the Morse potential becomes $$\begin{aligned} V(y)=V_{1}e^{-2i \alpha y}-V_{2}e^{-i\alpha y}\end{aligned}$$ The energy eigenvalues are given for this potential as \[19,22\] $$\begin{aligned} \varepsilon_{n}=\alpha^{4}\left[(n+\frac{1}{2})+\frac{V_{2}}{2\alpha \sqrt{|-V_{1}|}}\right]^{2}\end{aligned}$$ If we take the parameters of Eq.(25) as $V_{1}=-\omega^{2}$, $V_{2}=D$ and $\alpha=2$ then, corresponding energy eigenvalues for any n-th state are, $$\begin{aligned} \varepsilon_{n}=(2n+1+\frac{D}{2\omega})^{2}\end{aligned}$$ which ,is studied by Znojil and Bagchi and Quesne. \[10-11,19\]. Asymptotically vanishing mass distribution ------------------------------------------ Thus, the target system with asymptotically vanishing mass distribution are given as $$\begin{aligned} E_{n}=\varepsilon_{n}\end{aligned}$$ $$\begin{aligned} V(x)=V_{1}\left(x+\sqrt{x^{2}+q}\right)^{-2i \alpha^{2}}- V_{2}\left(x+\sqrt{x^{2}+q}\right)^{-i \alpha^{2}}- \frac{1}{8\alpha^{2}}\left(1+\frac{q}{x^{2}+q}\right).\end{aligned}$$ Mass Distribution $m(x)=\frac{\alpha^{2}}{(b+x^{2})^{2}}$ --------------------------------------------------------- In the PT symmetric and non-Hermitian case, new potential is given by $$\begin{aligned} V(x)=V_{1}\sqrt{\frac{1-2\alpha^{2}\frac{x}{\sqrt{q}}}{1+2\alpha^{2}\frac{x}{\sqrt{q}}}}- V_{2}\sqrt{\frac{1-\alpha^{2}\frac{x}{\sqrt{q}}}{1+\alpha^{2}\frac{x}{\sqrt{q}}}}- \frac{73x^{2}-16q}{32\alpha^{2}}.\end{aligned}$$ Exponential Type Mass Distribution ---------------------------------- The new potential with the same energy spectra is $$\begin{aligned} V(x)=V_{1}e^{4ie^{-\alpha x/2}}-V_{2}e^{2ie^{-\alpha x/2}}+\frac{3\alpha^{2}e^{-\alpha x}}{32}.\end{aligned}$$ Pöschl-Teller Potential ======================= The general form of the Pöschl-Teller potential is \[19,22\] $$\begin{aligned} V(y)=-4V_{0}\frac{e^{-2\alpha y}}{(1+qe^{-2\alpha y})^{2}}\end{aligned}$$ Its energy spectra and corresponding wavefunctions are $$\begin{aligned} \varepsilon_{n}=-\frac{\alpha^{2}}{4} \left(-(2n+1)+\sqrt{1+\frac{8V_{0}}{q\alpha^{2}}}\right)^{2}\end{aligned}$$ $$\begin{aligned} \psi_{n}(y)=s^{-\epsilon}(1-s)^{\nu/2}P^{(2\epsilon,\nu-1)}_{n}(1-2qs).\end{aligned}$$ where $s=-e^{-2\alpha y }$, $P^{-\nu_{2}-\frac{1}{2},\nu_{2}-\frac{1}{2}}_{n}(y)$ stands for Jacobi polynomials and $\nu_{1}=\sqrt{1+\frac{8V_{0}}{q\alpha^{2}}}$, $\nu_{2}=\sqrt{\frac{8V_{0}}{q\alpha^{2}}}$. Asymptotically vanishing mass distribution ------------------------------------------ The target system is obtained as with the mass function $$\begin{aligned} E_{n}=\varepsilon_{n}\end{aligned}$$ $$\begin{aligned} V(x)=-4V_{0}\frac{\left(x+\sqrt{x^{2}+q}\right)^{-2\alpha^{2}}}{\left[1+q(\left(x+ \sqrt{x^{2}+q}\right)^{-2\alpha^{2}}\right]^{2}}- \frac{1}{8\alpha^{2}}\left(1+\frac{q}{x^{2}+q}\right)\end{aligned}$$ $$\begin{aligned} \psi_{n}(x)=\frac{\left(x^{2}+ q\right)^{1/4}}{\sqrt{\alpha}}(f(x))^{-\epsilon}(1-f(x))^{\nu/2}P^{(2\epsilon,\nu-1)}_{n}(1-2qf(x))\end{aligned}$$ Mass Distribution $m(x)=\frac{\alpha^{2}}{(q+x^{2})^{2}}$ --------------------------------------------------------- If we consider to obtain a target potential for the Pöschl-Teller Potential, it can be obtain as $$\begin{aligned} V(x)=-4V_{0}\frac{e^{-2\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}}}{\left(1+ qe^{-2\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}}\right)^{2}}- \frac{73x^{2}-16q}{32\alpha^{2}}.\end{aligned}$$ $$\begin{aligned} \psi_{n}(x)=\frac{\left(x^{2}+ q\right)^{1/2}}{\sqrt{\alpha}}(f(x))^{-\epsilon}(1-f(x))^{\nu/2}P^{(2\epsilon,\nu-1)}_{n}(1-2qf(x))\end{aligned}$$ Exponential Type Mass Distribution ---------------------------------- With the exponential Type Mass Distribution, it can be obtained with the same energy spectra as $$\begin{aligned} V(x)=-4V_{0}\frac{e^{4e^{-\alpha x/2}}}{[1+qe^{4e^{-\alpha x/2}}]^{2}}+\frac{3\alpha^{2}e^{-\alpha x}}{32}.\end{aligned}$$ and $$\begin{aligned} \psi_{n}(x)=e^{\alpha x/4}(f(x))^{-\epsilon}(1-f(x))^{\nu/2}P^{(2\epsilon,\nu-1)}_{n}(1-2qf(x))\end{aligned}$$ Non-PT symmetric and non-Hermitian Pöschl-Teller cases ====================================================== In this case, $V_{0}$ and $q$ are complex parameters $V_{0}=V_{0R}+iV_{0I}$ and $q=q_{R}+iq_{I}$ but $\alpha$ is a real parameter. Although the potential is complex and the corresponding Hamiltonian is non-Hermitian and also non-PT symmetric, there may be real spectra if and only if $V_{0I}q_{R}=V_{0R}q_{I}$. When both parameters $V_{0}$ and $q$ are taken pure imaginary, the potential turns out to be \[19,22\], $$\begin{aligned} V(y)=-4V_{0}\frac{2qe^{-4\alpha y}+i(1-q^{2}e^{-4\alpha y})}{(1+q^{2}e^{-4\alpha y})^{2}}\end{aligned}$$ For simplicity, we use the notation $V_{0}$ and $q$ instead of $V_{0I}$ and $q_{I}$. In this case, the same energy eigenvalues are obtained as in the Eq.(30). Asymptotically vanishing mass distribution ------------------------------------------ The new potential is $$\begin{aligned} V(x)=-4V_{0}\frac{2q\left(x+\sqrt{x^{2}+q}\right)^{-4\alpha^{2}}+ i\left(1-q^{2}\left(x+\sqrt{x^{2}+q}\right)^{-4\alpha^{2}}\right)}{\left[1+q^{2}\left(x+\sqrt{x^{2}+q}\right)^ {-4\alpha^{2}}\right]^{2}}- \frac{1}{8\alpha^{2}}\left(1+\frac{q}{x^{2}+q}\right).\end{aligned}$$ Mass Distribution $m(x)=\frac{\alpha^{2}}{(b+x^{2})^{2}}$ --------------------------------------------------------- The target potential for this case can be obtained as $$\begin{aligned} V(x)=-4V_{0} \frac{2qe^{-4\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}}+ i\left(1-q^{2}e^{-4\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}}\right)}{\left(1+q^{2} e^{-4\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}}\right)^{2}}- \frac{73x^{2}-16q}{32\alpha^{2}}.\end{aligned}$$ Exponential Type Mass Distribution ---------------------------------- The target potential for this case can be obtained with same the energy eigenvalues as $$\begin{aligned} V(x)=-4V_{0} \frac{2qe^{8e^{-\alpha x/2}}+i(1-q^{2})e^{8e^{-\alpha x/2}}}{(1+q^{2}e^{8e^{-\alpha x/2}})^{2}}+\frac{3\alpha^{2}e^{-\alpha x}}{32}.\end{aligned}$$ PT symmetric and non-Hermitian Pöschl-Teller cases ================================================== We choose the parameters $V_{0}$ and$q$ are real and also $\alpha=i \alpha$. Then, the potential turns into $$\begin{aligned} V(x)=-4V_{0}\frac{(1+q^{2})cos2\alpha x+2q+i(q^{2}-1)sin2\alpha x}{(1+q^{2})^{2}+4q cos2\alpha x((1+q cos2\alpha x+q^{2})}\end{aligned}$$ and corresponding energy eigenvalue is given as \[19,22\] $$\begin{aligned} \varepsilon_{n}=-\frac{\alpha^{2}}{4}\left[2n+1+\sqrt{1+\frac{16V_{0}}{\alpha^{2}}}\right]^{2}\end{aligned}$$ Asymptotically vanishing mass distribution ------------------------------------------ The new system is $$\begin{aligned} E_{n}=\varepsilon_{n}\end{aligned}$$ $$\begin{aligned} V(x) &=-4V_{0}\frac{\left[q\left(x+\sqrt{q+x^{2}}\right)^{i\alpha^{2}}- \left(x+\sqrt{q+x^{2}}\right)^{-i\alpha^{2}}\right]^{2}}{\left(1+q^{2}\right)^{2}+ 4qcos\left[2\alpha^{2}ln\left(x+\sqrt{q+x^{2}}\right)\right] \left(1+qcos\left[2\alpha^{2}ln\left(x+\sqrt{q+x^{2}}\right)\right]+q^{2}\right)}- \frac{1}{8\alpha^{2}}\left(1+\frac{q}{x^{2}+q}\right)\end{aligned}$$ Mass Distribution $m(x)=\frac{\alpha^{2}}{(b+x^{2})^{2}}$ --------------------------------------------------------- The new potential is given as $$\begin{aligned} V(x) =-4V_{0}\frac{(qe^{i\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}}+ e^{-i\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}})^{2}}{(1+q^{2})^{2}+4qcos(2 \alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}))(1+qcos2\alpha^{2}tan^{-1}\frac{x}{\sqrt{q}}+q^{2})}- \frac{73x^{2}-16q}{32\alpha^{2}}.\end{aligned}$$ Exponential Type Mass Distribution ---------------------------------- The target potential can be obtained with having the same energy spectra for this case as $$\begin{aligned} V(x) =-4V_{0}\frac{(qe^{-2ie^{-\alpha x/2}}+e^{2ie^{-\alpha x/2}})^{2}}{(1+q^{2})^{2}+4qcos(4e^{-\alpha x/2})(1+q cos(4e^{-\alpha x/2})+q^{2})}+\frac{3\alpha^{2}e^{-\alpha x}}{32}.\end{aligned}$$ Conclusions =========== In this article we have explored the PCT approach to a class of exponential type PT/non-PT symmetric and nonhermitian Hamiltonians such as Morse and Pöschl-Teller potentials with some spatially dependent effective masses. For each type of potentials we have obtained a set of exactly solvable target potentials by using three position dependent mass distributions. It is pointed out that the importance of the mapping function which aids to construct of closed forms of the energy spectrum and corresponding wavefunctions. Specially, the reference system with the source potential and the new system with the target potential have the same bound state energy eigenvalues. It was shown that the exact solvability depends not only on the form of the potential, but also on the spatial dependence on the mass within the PT symmetric framework. Acknowledgements ================ This research was partially supported by the Scientific and Technological Research Council of Turkey. [99]{} C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80 (1998) 5243; C. M. Bender , J. Math. Phys. 40 (1999) 2201; C. M. Bender and D. C. Brody and H. F. Jones, Phys. Rev. Lett. 89 (2002) 270401; C. M. Bender et all, J. Phys. A 36 (2003) 1029. F.M. Fernandez, R. Guardiola, J. Ros and M. Znojil, J. Phys. A, 32 (1999) 3105. A. Mostafazadeh, J. Math. Phys., 43 (2002) 205. S. Weigert, quant-ph/0209054 S. Weigert, Phys. Rev. A, 68 (2003) 062111. C. M. Bender,F. Cooper and P. N. Meisinger, Phys. Lett. A, 259 (1999) 229. V. Buslaev and V. Grecchi, J. Phys. A, 36 (1993) 5541. E. Delabaere and F. Bham, Phys. Lett. A, 250 (1998) 25. C. M. Bender, K. A. Milton and V. M. Savage, Phys.Rev.D, 62 (2000) 085001; C. M. Bender, S. Boettcher and H. F. Jones and P. N. Meisinger, J.Math. Phys., 42 (2001) 1960. B. Bagchi and C. Quesne, Phys. Lett. A, 273 (2000) 285. B. Bagchi and C. Quesne, Phys. Lett. A, 300 (2000) 18. G. Levai, F. Cannata and A. Ventura, J. Phys. A, 34 (2001) 839. G. Levai, F. Cannata and A. Ventura A, J. Phys. A, 35 (2002) 5041. F. M. Fernandez, R. Guardiola, J. Ros Znojil M, J. of Phys. A, 31 (1998) 10105; G. A. Merzinescu, J. Phys. A, 33 (2000) 4911; O. Mustafa and M. Znojil, J. Phys. A, 35 (2002) 8929; M. Znojil, F. Gemperle and O. Mustafa, J. Phys. A, 35 (2002) 5781. V. Milanovic and Z. Ikonic, Phys. Lett. A, 293 (2002) 29. C. S. Jia, X. L. Zeng and L. T. Sun, Phys. Lett. A, 294 (2002) 185. M. Znojil, J. Phys. A, 36 (2003) 7639. M. Znojil, J. Phys. A, 36 (2003) 7825. O. Yesiltas, M. Simsek, R. Sever and C. Tezcan, Physica Scripta, T67 (2003) 472; H. Taseli, J. Phys. A 31 (1998) 779; M. Znojil, Phys. Lett. A, 264 (1999) 108; G. Levai and M. Znojil, J. Phys. A, 35 (2002) 8793. C. M. Bender and S. Boettcher, J. Phys. A, 31 (1998) L273; M. Znojil, J. Phys. A, 33 (2000) 4203. A. Khare and U. Sukhatme, Phys. Lett. A, 324 (2004) 406. G. Faridfathi, R. Sever and M. Aktas, J. Math. Chem., 38 (2005) 533; M. Aktas, R. Sever, Mod. Phys. Lett. A, 19 (2004) 2871. G. Levai and M. Znojil, J. Phys. A, 33 (2000) 7165. Parthasarathi, D. Parashar and R. S. Kaushal, J.Phys. A, 37 (2004) 781. P. A. M. Dirac, Rev. Mod. Phys., 17 (1945) 195; P. A. M. Dirac, Proc. R. Soc. London, 113A (1927) 621. P. Jordan, Z. Phys., 38 (1926) 513. J. Schwinger, Quantum Electrodynamics (NY: Dover Pub.) (1958). A. D. Alhaidari,Phys. Rev. A 66, (2002) 042116. B. Roy, P. Roy, arXiv: quant-ph/0106028; Phys. Lett. A, 340 70 (2005). J. Yu, S. H. Dong, Phys. Lett. A, 325 (2004) 194. G. Chen, Z. D. Chen, Phys. Lett. A, 331 (2004) 312. J. Yu, S. H. Dong, G. H. Sun, Phys. Lett. A, 322 (2004) 290. K. Bencheikh, K. Berkane, S. Bouizane, J. Phys. A, 37 (2004) 10719. V. Milanovic, Z. Ikovic, J. Phys. A, 32 (1999) 7001. C. Quesne, V. M. Tkachuk, J. Phys. A, 37 (2004) 4267. Y. C. Ou, Z. Q. Cao, Q. H. Shen, J. Phys. A, 37 (2004) 4283. B. Bagchi, P. Gorain, C. Quesne, R. Roychoudhury, Mod. Phys. Lett. A, 19 (2004) 2765; Czech J. Phys., 54 (2004) 1019. B. Bagchi, A. Banerjee, C. Quesne, V. M. Tkachuk, J. Phys. A, 38 (2005) 2929. Omar Mustafa, S. Habib Mazharimousavi, J. Phys. A: Math. Gen., 39 (2006) 10537. OL. Serra, E. Lipparini, Europhys. Lett., 40 (1997) 667. F. Arias de Saavedra, J. Boronati, A. Polis, A. Fabrocini, Phys. Rev. B, 50 (1994) 4248. S. L. Dudarev, Philosophical Magazine, 83 (2003) 3577-97. M. Barranco, M. Pi, S.M. Gatica, E.S. Hernandez, J. Navarro, Phys. Rev. B, 56 (1997) 8997. G.T. Einevoll, P.C. Hemmer, J. Thomson, Phys. Rev. B, 42 (1990) 3485. P. Ring, P. Schuck,“The Nuclear Many-Body Problem”, (Springer-Verlag, New York, 1980). Phys. Rev. B, 42 (1990) 3485. R. Montemayer,Phys. Rev. A, 36 (1987) 1562. G. Junker,J. Phys. A, 23 (1990) L881. R. De, R. Dutt, U. Sukhatme,J. Phys. A, 25 (1992) L843. R. Dutt, A. Khare, Y. P. Varshni, J. Phys. A, 28 (1995) L107. A. D. Alhaidari,Phys. Rev. A, 66 (2002) 042116. S. H. Dong, M. Lozada-Cassou,Phys. Lett. A, 337 (2005) 313. C. S. Jia, Y. F. Diao, M. Li, Q. B. Yang, L. T. Sun, R. Y. Huang, J. Phys. A, 37 (2004) 11275. R. Dutt, A. Khare, Y. P. Varshni,J. Phys. A, 28 (1995) L107. L. Jiang, L. Z. Yi, C. S. Jia,Phys. Lett. A, 345 (2005) 279. [^1]: Corresponding author: sever@metu.edu.tr

Ł[[L]{}]{} [$\tilde{\phantom{a}}$]{} [**Bessel Beams**]{}\ Kirk T. McDonald\ [*Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544*]{}\ (June 17, 2000) Problem ======= Deduce the form of a cylindrically symmetric plane electromagnetic wave that propagates in vacuum. A scalar, azimuthally symmetric wave of frequency $\omega$ that propagates in the positive $z$ direction could be written as $$\psi({\bf r},t) = f(\rho) e^{i(k_z z - \omega t)}, \label{eq1}$$ where $\rho = \sqrt{x^2 + y^2}$. Then, the problem is to deduce the form of the radial function $f(\rho)$ and any relevant condition on the wave number $k_z$, and to relate that scalar wave function to a complete solution of Maxwell’s equations. The waveform (\[eq1\]) has both wave velocity and group velocity equal to $\omega / k_z$. Comment on the apparent superluminal character of the wave in case that $k_z < k = \omega / c$, where $c$ is the speed of light. Solution ======== As the desired solution for the radial wave function proves to be a Bessel function, the cylindrical plane waves have come to be called Bessel beams, following their introduction by Durnin [@Durnin1; @Durnin2]. The question of superluminal behavior of Bessel beams has recently been raised by Mugnai [@Mugnai]. Bessel beams are a realization of super-gain antennas [@Schelkunoff; @Bouwkamp; @Yaru] in the optical domain. A simple experiment to generate Bessel beams is described in [@McQueen]. Sections 2.1 and 2.2 present two methods of solution for Bessel beams that satisfy the Helmholtz wave equation. The issue of group and signal velocity for these waves is discussed in sec. 2.3. Forms of Bessel beams that satisfy Maxwell’s equations are given in sec. 2.4. Solution via the Wave Equation ------------------------------ On substituting the form (\[eq1\]) into the wave equation, $$\nabla^2 \psi = { 1 \over c^2} {\partial^2 \psi \over \partial t^2}, \label{eq2}$$ we obtain $${d^2 f \over d\rho^2} + {1 \over \rho} {d f \over d \rho} + (k^2 - k_z^2) f = 0. \label{eq3}$$ This is the differential equation for Bessel functions of order 0, so that $$f(\rho) = J_0(k_r \rho), \label{eq4}$$ where $$k_\rho^2 + k_z^2 = k^2. \label{eq5}$$ The form of eq. (\[eq5\]) suggests that we introduce a (real) parameter $\alpha$ such that $$k_\rho = k \sin \alpha, \qquad \mbox{and} \qquad k_z = k \cos\alpha. \label{eq6}$$ Then, the desired cylindrical plane wave has the form $$\psi({\bf r},t) = J_0(k \sin\alpha \, \rho) e^{i(k \cos\alpha \, z - \omega t)}, \label{eq7}$$ which is commonly called a Bessel beam. The physical significance of parameter $\alpha$, and that of the group velocity $$v_g = {d \omega \over d k_z} = {\omega \over k_z} = v_p = {c \over \cos\alpha} \label{eq8}$$ will be discussed in sec. 2.3. While eq. (\[eq7\]) is a solution of the Helmholtz wave equation (\[eq2\]), assigning $\psi({\bf r},t)$ to be a single component of an electric field, say $E_x$, does not provide a full solution to Maxwell’s equations. For example, if ${\bf E} = \psi \hat{\bf x}$, then $\nabla \cdot {\bf E} = \partial \psi / \partial x \neq 0$. Bessel beams that satisfy Maxwell’s equations are given in sec. 2.4. Solution via Scalar Diffraction Theory -------------------------------------- The Bessel beam (\[eq7\]) has large amplitude only for $\abs{\rho} \lsim 1/ k \sin\alpha$, and maintains the same radial profile over arbitrarily large propagation distance $z$. This behavior appears to contradict the usual lore that a beam of minimum transverse extent $a$ diffracts to fill a cone of angle $1/a$. Therefore, the Bessel beam (\[eq7\]) has been called “diffraction free” [@Durnin2]. Here, we show that the Bessel beam does obey the formal laws of diffraction, and can be deduced from scalar diffraction theory. According to that theory [@Jackson], a cylindrically symmetric wave $f(\rho)$ of frequency $\omega$ at the plane $z = 0$ propagates to point [**r**]{} with amplitude $$\psi({\bf r},t) = {k \over 2 \pi i} \int \int \rho' d\rho' d\phi f(\rho') {e^{i(k R - \omega t)} \over R}, \label{eq9}$$ where $R$ is the distance between the source and observation point. Defining the observation point to be $(\rho,0,z)$, we have $$R^2 =z^2 + \rho^2 + \rho^{'2} - 2 \rho \rho' \cos\phi, \label{eq10}$$ so that for large $z$, $$R \approx z + {\rho^2 + \rho^{'2} - 2 \rho \rho' \cos\phi \over 2 z}. \label{eq11}$$ In the present case, we desire the amplitude to have form (\[eq1\]). As usual, we approximate $R$ by $z$ in the denominator of eq. (\[eq9\]), while using approximation (\[eq11\]) in the exponential factor. This leads to the integral equation $$\begin{aligned} f(\rho) e^{i k_z z} & = & {k \over 2 \pi i} {e^{ik z} e^{i k \rho^2 / 2 z} \over z} \int_0^\infty \rho' d\rho' f(\rho') e^{i k \rho^{'2} / 2z} \int_0^{2 \pi} d\phi e^{-i k \rho \rho' \cos\phi / z} \nonumber \\ & = & {k \over i} {e^{ik z} e^{i k \rho^2 / 2 z} \over z} \int_0^\infty \rho' d\rho' f(\rho') J_0(k \rho \rho' / z) e^{i k \rho^{'2} / 2z}, \label{eq12}\end{aligned}$$ using a well-known integral representation of the Bessel function $J_0$. It is now plausible that the desired eigenfunction $f(\rho)$ is a Bessel function, say $J_0(k_\rho \rho)$, and on consulting a table of integrals of Bessel functions we find an appropriate relation [@Gradshteyn], $$\int_0^\infty \rho' d\rho' J_0(k_{\rho} \rho') J_0(k \rho \rho' / z) e^{i k \rho^{'2} / 2z} = {i z \over k} e^{-i k \rho^2 / 2 z} e^{- i k_\rho^2 z / 2 k} J_0(k_\rho \rho). \label{eq13}$$ Comparing this with eq. (\[eq12\]), we see that $f(\rho) = J_0(k_\rho \rho)$ is indeed an eigenfunction provided that $$k_z = k - {k_\rho^2 \over 2 k}. \label{eq14}$$ Thus, if we write $k_\rho = k \sin\alpha$, then for small $\alpha$, $$k_z \approx k (1 - \alpha^2 / 2) \approx k \cos\alpha, \label{eq15}$$ and the desired cylindrical wave again has form (\[eq7\]). Strictly speaking, the scalar diffraction theory reproduces the “exact” result (\[eq7\]) only for small $\alpha$. But the scalar diffraction theory is only an approximation, and we predict with confidence that an “exact” diffraction theory would lead to the form (\[eq7\]) for all values of parameter $\alpha$. That is, “diffraction-free” beams are predicted within diffraction theory. It remains that the theory of diffraction predicts that an infinite aperture is needed to produce a beam whose transverse profile is invariant with longitudinal distance. That a Bessel beam is no exception to this rule is reviewed in sec. 2.3. The results of this section were inspired by [@Jiang]. One of the first solutions for Gaussian laser beams was based on scalar diffraction theory cast as an eigenfunction problem [@Boyd]. Superluminal Behavior --------------------- In general, the group velocity (\[eq8\]) of a Bessel beam exceeds the speed of light. However, this apparently superluminal behavior cannot be used to transmit signals faster than lightspeed. An important step towards understanding this comes from the interpretation of parameter $\alpha$ as the angle with respect to the $z$ axis of the wave vectors of an infinite set of ordinary plane waves whose superposition yields the Bessel beam [@Eberly]. To see this, we invoke the integral representation of the Bessel function to write eq. (\[eq7\]) as $$\begin{aligned} \psi({\bf r},t) & = &J_0(k \sin\alpha \, \rho) e^{i(k \cos\alpha \, z - \omega t)} \nonumber \\ & = & {1 \over 2 \pi} \int_0^{2 \pi} d \phi e^{i(k \sin\alpha \, x \cos\phi + k \sin\alpha \, y \sin\phi + k \cos\alpha \, z - \omega t)} \label{eq16} \\ & = & {1 \over 2 \pi} \int_0^{2 \pi} d \phi e^{i({\bf q} \cdot {\bf r} - \omega t)}, \nonumber\end{aligned}$$ where the wave vector [**q**]{}, given by $${\bf q} = k (\sin\alpha \cos\phi, \sin\alpha \sin\phi, \cos\alpha), \label{eq17}$$ makes angle $\alpha$ to the $z$ axis as claimed. We now see that a Bessel beam is rather simple to produce in principle [@Durnin2]. Just superpose all possible plane waves with equal amplitude and a common phase that make angle $\alpha$ to the $z$ axis, According to this prescription, we expect the $z$ axis to be uniformly illuminated by the Bessel beam. If that beam is created at the plane $z = 0$, then any annulus of equal radial extent in that plane must project equal power into the beam. For large $\rho$ this is readily confirmed by noting that $J_0^2(k \sin\alpha\, \rho) \approx \cos^2(k \sin\alpha\, \rho + \delta)/ (k \sin\alpha\, \rho)$, so the integral of the power over an annulus of one radial period, $\Delta \rho = \pi / (k \sin\alpha)$, is independent of radius. Thus, from an energy perspective a Bessel beam is not confined to a finite region about the $z$ axis. If the beam is to propagate a distance $z$ from the plane $z = 0$, it must have radial extent of at least $ \rho = z \tan\alpha$ at $z = 0$. An arbitrarily large initial aperture, and arbitrarily large power, is required to generate a Bessel beam that retains its “diffraction-free” character over an arbitrarily large distance. Each of the plane waves that makes up the Bessel beam propagates with velocity $c$ along a ray that makes angle $\alpha$ to the $z$ axis. The intersection of the $z$ axis and a plane of constant phase of any of these wave moves forward with superluminal speed $c / \cos\alpha$, which is equal to the phase and group velocities (\[eq8\]). This superluminal behavior does not represent any violation of special relativity, but is an example of the “scissors paradox" that the point of contact of a pair of scissors could move faster than the speed of light while the tips of the blades are moving together at sublightspeed. A ray of sunlight that makes angle $\alpha$ to the surface of the Earth similarly leads to a superluminal velocity $c / \cos\alpha$ of the point of contact of a wave front with the Earth. However, we immediately see that a Bessel beam could not be used to send a signal from, say, the origin, $(0,0,0)$, to a point $(0,0,z)$ at a speed faster than light. A Bessel beam at $(0,0,z)$ is made of rays of plane waves that intersect the plane $z = 0$ at radius $\rho = z \tan\alpha$. Hence, to deliver a message from $(0,0,0)$ to $(0,0,z)$ via a Bessel beam, the information must first propagate from the origin out to at least radius $\rho = z \tan\alpha$ at $z = 0$ to set up the beam. Then, the rays must propagate distance $z/\cos\alpha$ to reach point $z$ with the message. The total distance traveled by the information is thus $z(1 + \sin\alpha)/\cos\alpha$, and the signal velocity $v_s$ is given by $$v_s \approx c {\cos\alpha \over 1 + \sin\alpha}, \label{eq18}$$ which is always less than $c$. The group velocity and signal velocity for a Bessel beam are very different. Rather than being a superluminal carrier of information at its group velocity $c / \cos\alpha$, a modulated Bessel beam could be used to deliver messages only at speeds well below that of light. Solution via the Vector Potential --------------------------------- To deduce all components of the electric and magnetic fields of a Bessel beam that satisfies Maxwell’s equation starting from a single scalar wave function, we follow the suggestion of Davis [@Davis] and seek solutions for a vector potential [**A**]{} that has only a single component. We work in the Lorentz gauge (and Gaussian units), so that the scalar potential $\Phi$ is related by $$\nabla \cdot {\bf A} + {1 \over c} {\partial \Phi \over \partial t} = 0. \label{e1}$$ The vector potential can therefore have a nonzero divergence, which permits solutions having only a single component. Of course, the electric and magnetic fields can be deduced from the potentials via $${\bf E} = - \nabla \Phi - {1 \over c} {\partial {\bf A} \over \partial t}, \label{e2}$$ and $${\bf B} = \nabla \times {\bf A}. \label{e3}$$ For this, the scalar potential must first be deduced from the vector potential using the Lorentz condition (\[e1\]). We consider waves of frequency $\omega$ and time dependence of the form $e^{-i \omega t}$, so that $\partial \Phi / \partial t = - i k \Phi$. Then, the Lorentz condition yields $$\Phi = - {i \over k} \nabla \cdot {\bf A}, \label{e4}$$ and the electric field is given by $${\bf E} = ik \left[ {\bf A} + {1 \over k^2} {\bf \nabla} ({\bf \nabla} \cdot {\bf A}) \right]. \label{e5}$$ Then, $\nabla \cdot {\bf E} = 0$ since $\nabla^2 (\nabla \cdot {\bf A}) + k^2 (\nabla \cdot {\bf A}) = 0$ for a vector potential [**A**]{} of frequency $\omega$ that satifies the wave equation (\[eq2\]), We already have a scalar solution (\[eq7\]) to the wave equation, which we now interpret as the only nonzero component, $A_j$, of the vector potential for a Bessel beam that propagates in the $+z$ direction, $$A_j({\bf r},t) = \psi({\bf r},t) \propto J_0(k \sin\alpha\, \rho) e^{i(k \cos\alpha\, z - \omega t)}. \label{e6}$$ We consider five choices for the meaning of index $j$, namely $x$, $y$, $z$, $\rho$, and $\phi$, which lead to five types of Bessel beams. Of these, only the case of $j = z$ corresponds to physical, azimuthally symmetric fields, and so perhaps should be called the Bessel beam. ### $j = x$ In this case, $$\nabla \cdot {\bf A} = {\partial \psi \over \partial x} = - {k \sin\alpha \, x \over \rho} J_1(k \sin\alpha\, \rho) e^{i(k \cos\alpha\, z - \omega t)}. \label{e7}$$ In calculating $\nabla(\nabla \cdot {\bf A})$ we use the identity $J_1' = (J_0 - J_2)/2$. Also, we divide [**E**]{} and [**B**]{} by the factor $ik$ to present the results in a simpler form. We find, $$\begin{aligned} E_x & = & \left\{ J_0(\varrho) - {\sin^2\alpha\ \over \rho^2} \left[ {y^2 J_1(\varrho) \over \varrho} - {x^2 \over 2} \left(J_0(\varrho) - J_2(\varrho) \right) \right] \right\} e^{i(k \cos\alpha\, z - \omega t)}, \nonumber \\ E_y & = & {\sin^2\alpha\, x y \over \rho^2} \left[ { J_1(\varrho) \over \varrho} - {1 \over 2} \left(J_0(\varrho) - J_2(\varrho) \right) \right] e^{i(k \cos\alpha\, z - \omega t)}, \label{e8} \\ E_z & = & - i \sin 2\alpha {x \over 2 \rho} J_1(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \nonumber\end{aligned}$$ where $$\varrho \equiv k \sin\alpha\, \rho, \label{e9}$$ and $$\begin{aligned} B_x & = & 0, \nonumber \\ B_y & = & \cos\alpha\, J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \label{e10} \\ B_z & = & - i \sin\alpha {x \over \rho} J_1(\varrho) e^{i(k \cos\alpha\, z - \omega t)}. \nonumber\end{aligned}$$ A Bessel beam that obeys Maxwell’s equations and has purely $x$ polarization of its electric field on the $z$ axis includes nonzero $y$ and $z$ polarization at points off that axis, and does not exhibit the azimuthal symmetry of the underlying vector potential. ### $j = y$ This case is very similar to that of $j = x$. ### $j = z$ In this case the electric and magnet fields retain azimuthal symmetry, so that it is convenient to display the $\rho$, $\phi$ and $z$ components of the fields. First, $$\nabla \cdot {\bf A} = {\partial \psi \over \partial z} = i k \cos\alpha \, J_0(k \sin\alpha\, \rho) e^{i(k \cos\alpha\, z - \omega t)}. \label{e11}$$ Then, we divide the electric and magnetic fields by $k \sin\alpha$ to find the relatively simple forms: $$\begin{aligned} E_\rho & = & \cos\alpha\, J_1(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \nonumber \\ E_\phi & = & 0, \label{e12} \\ E_z & = & i \sin\alpha\, J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \nonumber\end{aligned}$$ and $$\begin{aligned} B_\rho & = & 0, \nonumber \\ B_\phi & = & J_1(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \label{e13} \\ B_z & = & 0. \nonumber\end{aligned}$$ This Bessel beam is a transverse magnetic (TM) wave. The radial electric field $E_\rho$ vanishes on the $z$ axis (as it must if that axis is charge free), while the longitudinal electric field $E_z$ is maximal there. Cylindrically symmetric waves with radial electric polarization are often called axicon beams [@McLeod]. ### $j = \rho$ In this case, $$\nabla \cdot {\bf A} = {1 \over \rho} {\partial \rho \psi \over \partial \rho} = \left[ {J_0(k \sin\alpha\, \rho) \over \rho} - k \sin\alpha\, J_1(k \sin\alpha\, \rho) \right] e^{i(k \cos\alpha\, z - \omega t)}. \label{e14}$$ After dividing by $ik$, the electric and magnetic fields are $$\begin{aligned} E_\rho & = & \left\{ J_0(\varrho) - \sin^2\alpha \left[ {J_0(\varrho) \over \varrho^2} + {J_1(\varrho) \over \varrho} + {1 \over 2} (J_0(\varrho - J_2(\varrho)) \right] \right\} e^{i(k \cos\alpha\, z - \omega t)}, \nonumber \\ E_\phi & = & 0, \label{e15} \\ E_z & = & i \cos\alpha \sin\alpha \left[ {J_0(\varrho) \over \varrho} - J_1(\varrho) \right] e^{i(k \cos\alpha\, z - \omega t)}, \nonumber\end{aligned}$$ and $$\begin{aligned} B_\rho & = & 0, \nonumber \\ B_\phi & = & \cos\alpha\, J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \label{e16} \\ B_z & = & 0. \nonumber\end{aligned}$$ The radial electric field diverges as $1 / \rho^2$ for small $\rho$, so this case is unphysical. ### $j = \phi$ Here, $$\nabla \cdot {\bf A} = {1 \over \rho} {\partial \psi \over \partial \phi} = 0. \label{e17}$$ After dividing by $ik$, the electric and magnetic fields are $$\begin{aligned} E_\rho & = & 0, \nonumber \\ E_\phi & = & J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \label{e18} \\ E_z & = & 0, \nonumber\end{aligned}$$ and $$\begin{aligned} B_\rho & = & - \cos\alpha\, J_0(\varrho) e^{i(k \cos\alpha\, z - \omega t)}, \nonumber \\ B_\phi & = & 0, \label{e19} \\ B_z & = & - i \sin\alpha\ \left[ {J_0(\varrho) \over \varrho} - J_1(\varrho) \right] e^{i(k \cos\alpha\, z - \omega t)}. \nonumber\end{aligned}$$ These fields are unphysical due to the finite value of $E_\phi$ at $\rho = 0$, and the divergence of $B_z$ as $\rho \to 0$. [99]{} J. Durnin, [*Exact solutions for nondiffracting beams. I. The scalar theory*]{}, J. Opt. Soc. Am. [**A**]{} 4, 651-654 (1987). J. Durnin, J.J. Miceli, Jr. and J.H. Eberly, [*Diffraction-free beams*]{}, Phys. Rev. Lett. [**778**]{},1499-1501 (1987). D. Mugnai, A. Ranfagni and R. Ruggeri, [*Observation of Superluminal Behavior in Wave Propagation*]{}, Phys. Rev. Lett. [**84**]{}, 4830-4833 (2000). S.A. Schelkunoff, [*A methematical theory of linear arrays*]{}, Bell. Sys. Tech. J. [**22**]{}, 80-107 (1943). C.J. Bouwkamp and N.G. deBruijn, [*The problem of optimum antenna current distribution*]{}, Philips Res. Rep. [**1**]{}, 135-158 (1946). N. Yaru, [*A Note on Super-Gain Antenna Arrays*]{}, Proc. I.R.E. [**39**]{}, 1081-1085 (1951). C.A. McQueen, J. Arlt and K. Dholkia, [*An experiment to study a “nondiffracting" light beam*]{}, Am. J. Phys. [**67**]{}, 912-915 (1999). J.D. Jackson, [*Classical Electrodynamics*]{}, 3d ed.(Wiley, New York, 1999). I.S. Gradshteyn and I.M. Ryzhik, [*Table of Integrals, Series, and Products*]{}, 5th ed.(Academic Press, San Diego, 1994), integral 6.633.2. A. Zhiping, Q. Lu and Z. Liu, [*Propagation of apertured Bessel beams*]{}, Appl. Opt. [**34**]{}, 7183-7185 (1995). G.D. Boyd and J.P. Gordon, [*Confocal Multimode Resonator for Millimeter Through Optical Wavelength Masers*]{}, Bell Sys. Tech. J. [**40**]{}, 489-509 (1961). P.W. Milonni and J.H. Eberly, [*Lasers*]{} (Wiley Interscience, New York, 1988), sec. 14.14. L.W. Davis, [*Theory of electromagnetic beams*]{}, Phys. Rev. A [**19**]{}, 1177-1179 (1979). J.H. McLeod, [*The Axicon: A New Type of Optical Element*]{}, J. Opt. Soc. Am. [**44**]{}, 592-597 (1954).

--- author: - | [^1]\ Laboratoire de l’Accélérateur Linéaire, Univ. Paris-Sud 11 et IN2P3/CNRS, France\ E-mail: title: Top and EW Physics at the LHeC --- Introduction ============ Deep inelastic scattering (DIS) of a point-like lepton beam over a hadron has played a central role in establishing the quark-parton model and QCD starting with fixed target experiments in the late 1960s at SLAC. Later the Gargamelle neutrino-nucleon experiment at CERN has discovered weak neutral currents. HERA, operated at DESY from 1992 to 2007, was the only $ep$ collider of the world. It has extended the study of the proton structure and quark-gluon interaction dynamics up to a centre-of-mass energy ($\sqrt{s}$) of 320GeV corresponding to an extension by two orders of magnitude towards both higher negative four-momentum transfer squared $Q^2$ and lower Bjorken $x$ in comparison with the kinematic region covered by the fixed target experiments. The LHeC, if realised by adding to the LHC a separate 9km racetrack-shaped recirculating superconducting energy recovery linac providing a polarised electron (possibly also positron) beam of 60GeV, will be a new $ep$ collider of 1.3TeV, running in parallel with the high luminosity phase of the LHC. It has a rich and complementary physics programme to the LHC [@cdr; @1211.5102]. It would enable new precision studies of QCD in general and the precision determination of parton distributions functions (PDFs) in a largely extended kinematic region in particular. It has the potential to reveal new QCD dynamics in an unexplored low $x$ regime where the DGLAP evolution equations may no longer be valid as the latest QCD analysis of the newly combined inclusive neutral and charged current (NC and CC) cross sections at HERA may indicate [@herapdf2]. It would also provide additional and sometimes unique ways for studying top and electroweak (EW) physics as well as Higgs and physics beyond the Standard Model (BSM). This talk focuses on some of the selected topics on top and EW physics at the LHeC and the writeup is organised as follows. In Sec. \[sec:top\], expected limits on anomalous $Wtb$ couplings from the single top production are presented as an exemple. In Sec. \[sec:ew\], the expected precision determination of light quark couplings to the $Z$ boson and the scale dependence of the weak mixing angle $\sin^2\!\theta_W$ based either on the inclusive NC cross section measurements or on polarisation asymmetries of the NC interactions are shown, followed by a summary in Sec. \[sec:summary\]. Top physics {#sec:top} =========== The top quark is the heaviest particle in the SM, which is believed to be most sensitive to BSM physics. It has not been studied so far by any DIS experiments because of the kinematic limit or too small cross section. Therefore the LHeC will be the first DIS experiment capable to study the directly produced single top quark and top pairs in CC and NC interactions, respectively. In the five flavour scheme, the single top-quark production cross section of the $2\to 2$ $t$-channel process $e^-p\to \bar{t}\nu_e+X$ with $\bar{t}\to W^-b$ at $\sqrt{s}=1.3$TeV is predicted to be around 2pb for un polarised electron beam and increases by a factor of $1+P_e$ with $P_e$ being the degree of the longitudinal polarisation of the beam [@dgkm13]. This cross section value is comparable with that of the Tevatron and smaller by about two orders of magnitude than the LHC at 14TeV [@nk15]. The LHeC has however a much cleaner environment due to the absence of pile-up and underlying events. Therefore this process can be used for many precision measurements within the SM, such as the bottom-quark distribution of the proton, the CKM matrix element $V_{tb}$, the $t$-quark polarisation and the $W$ boson helicity. It can also be used to study deviations from the SM such as the anomalous couplings $Wtb$. In addition, the single top production in the NC protoproduction can be used to study top quark flavour changing neutral current couplings $tq\gamma$ with $q$ being a light quark [@cdr]. The top pair events are also produced at the LHeC in NC interactions. Even though the rate is lower than at the LHC, the potential for a better measurement of $tt\gamma$ than LHC is good [@bl13] as in the $t\bar{t}$ photoproduction at the LHeC, the highly energetic incoming photon couples only to the $t$ quark so that the cross section depends directly on the $tt\gamma$ vertex, whereas at the LHC the vertex is probed through $t\bar{t}\gamma$ production, where the outing going photon could come from other charged sources such as the top decays products. The DIS regime of $t\bar{t}$ production will also be able to probe the $ttZ$ coupling though with less sensitivity. A detailed study was performed in [@dgkm13] to evaluate the expected accuracy of measuring the anomalous $Wtb$ couplings at the LHeC based on the single anti-top quark production in $e^-p$ collisions in a model independent way by means of the following effective CP conserving Lagrangian [@dgkm13] $${\cal L}_{Wtb}=\frac{g}{\sqrt{2}}\left[W_\mu\bar{t}\gamma^\mu\left(V_{tb}f^L_1P_L+f^R_1P_R\right)b-\frac{1}{2m_W}W_{\mu\nu}\bar{t}\sigma^{\mu\nu}\left(f^L_2P_L+f^R_2P_R\right)b\right] + h.c.$$ where $f^L_1(\equiv 1+\Delta f^L_1)$ and $f^R_1$ are left- and right-handed vector couplings, $f^{L,R}_2$ are left- and right-handed tansor couplings, $W_{\mu\nu}=\partial_\mu W_\nu-\partial_\nu W_\mu$, $P_{L,R}=\frac{1}{2}(1\mp\gamma_5)$ are left- and right-handed projection operators, $\sigma^{\mu\nu}=i/2(\gamma^\mu\gamma^\nu-\gamma^\nu\gamma^\mu)$ and $g=2/\sin\theta_W$. In the SM, $f^L_1\equiv 1$ and $\Delta f^L_1=f^R_1=f^{L,R}_2\equiv 0$. Several analyses were performed using a simulated event sample corresponding to an integrated luminosity of 100fb$^{-1}$ for three different systematic uncertainties of 1%, 5% and 10%. One of them was based on a $\chi^2$ analysis using differential distributions of a few relevant kinematic variables in the leptonic and hadronic decay modes, respectively. Contours at 68% and 95% confidence level (CL) on two dimensional plane for any coupling combination were presented. One example is shown in Fig. \[fig:wtb\]. The corresponding results in comparison with other results from Tevatron, LHC and indirect one from $B$ decays are shown in Table \[tab:wtb\]. The conservative LHeC limits are thus competitive with or better than similar results from other determinations. ![Contours at (left) 68% and (right) 95% CL on the plane of $|V_{tb}|\Delta^L_1$ and $f^R_2$ for a systematic error of 1%, 5% and 10% on a sample with an integrated luminosity of 100fb$^{-1}$ (figures taken from Ref. [@dgkm13]).[]{data-label="fig:wtb"}](f1l_f2r68_had "fig:"){width=".495\textwidth"} ![Contours at (left) 68% and (right) 95% CL on the plane of $|V_{tb}|\Delta^L_1$ and $f^R_2$ for a systematic error of 1%, 5% and 10% on a sample with an integrated luminosity of 100fb$^{-1}$ (figures taken from Ref. [@dgkm13]).[]{data-label="fig:wtb"}](f1l_f2r95_had "fig:"){width=".495\textwidth"} Upper limit [(95% CL)]{} $|\Delta f^L_1|$ $|f^R_1|$ $|f^L_2|$ $|f^R_2|$ -------------------------- ------------------ --------------------- --------------------- ----------------- LHeC [@dgkm13] $0.005-0.03$ $0.01-0.1$ $0.01-0.1$ $0.01-0.1$ D0 [@d0:wtb] 0.548 0.324 0.347 LHC [@lhc:wtb] $0.03-0.06$ $0.22-0.34$ $0.06-008$ $0.06-0.08$ $B$ decays [@b:wtb] $[-0.13, 0.03]$ $[-0.0007, 0.0025]$ $[-0.0013, 0.0004]$ $[-0.15, 0.57]$ : Comparison of expected upper limits at 95% CL at the LHeC (100fb$^{-1}$, hadronic modes, $\delta_{\rm sy}=0.01-0.1$) [@dgkm13] with the actual limits from D0 (5.4fb$^{-1}$, $W$-helicity, single top) [@d0:wtb] and expected limits at the LHC (100fb$^{-1}$, $\gamma p\to WtX$) [@lhc:wtb] and $B$ decays (indirect) [@b:wtb].[]{data-label="tab:wtb"} EW physics {#sec:ew} ========== Inclusive NC and CC DIS interactions are two main processes which can be measured at the LHeC with high precision providing primary source not only for precision QCD studies but also for EW physics. Three examples are briefly presented in this section. The first example concerns a precision measurement of vector and axial-vector weak NC couplings of the $Z$ boson to light quarks $v_q$ and $a_q$. They were determined together with PDFs in a combined EW and QCD analysis of simulated inclusive NC and CC cross section data samples following Ref. [@h1ew]. This is possible since the double differential NC cross section $\frac{{\rm d}^2\sigma_{\rm NC}}{{\rm d}x{\rm d}Q^2}$ in e.g. $e^-p$ collisions may be expressed in terms of three structure functions as $\frac{2\pi\alpha^2}{xQ^2}\left[Y_+\tilde{F}_2+Y_-\tilde{F}_3-y^2\tilde{F}_L\right]$, where $\alpha$ is the electromagnetic fine structure constant and $Y_\pm=1\pm (1-y)^2$ with $y=Q^2/(xs)$ being the electron inelasticity. The generalised structure $\tilde{F}_2$ can be decomposed as $F_2+P_ea_e\kappa_ZF_2^{\gamma Z}+a^2_e\kappa^2_ZF^Z_2$ corresponding to $\gamma$ exchange, $\gamma Z$ interference and $Z$ exchange contributions. In this expression, $\kappa_Z^{-1}=\frac{2\sqrt{2}\pi\alpha}{G_FM^2_Z}\frac{Q^2+M^2_Z}{Q^2}$ and $a_e$ is the axial-vector coupling of the electron (due to the smallness of the vector coupling $v_e$, terms proportional to $v_e$ have been omitted). Similarly $x\tilde{F}_3=-a_e\kappa_ZxF_3^{\gamma Z}-P_ea^2_e\kappa^2_ZxF_3^{\gamma Z}$. These different structure functions can be further expressed in terms of PDFs $q, \bar{q}$ and the light quark couplings $v_q$ and $a_q$ as $\left[ F_2, F_2^{\gamma Z}, F_2^Z\right]=x\sum_q\left[e^2_q, 2e_qv_q, v^2_q+a^2_q\right]\{q+\bar{q}\}$ and $\left[xF_3^{\gamma Z}, xF_3^Z\right]=2x\sum_q\left[e_qa_q, v_aa_q\right]\{q-\bar{q}\}$, where $e_q$ being the electronic charge of quark $q$. The longitudinal structure function $\tilde{F}_L$ does not contribute at LO. The CC cross section is independent of these couplings but its inclusion in the fit helps to constrain the PDFs. In Ref. [@cdr], different scenarios were considered. The results of one of these, corresponding to $e^\pm$ beams of 50GeV with a longitudinal polarisation of 40% colliding with a proton beam of 7TeV for an integrated luminosity of 1fb$^{-1}$ per beam, are shown in Fig. \[fig:couplings\] in comparison with similar determinations from other experiments. The expected precision at the LHeC is indeed much better and any significant deviation from the SM expectations can thus be observed with this analysis. ![Determination of the vector and axial-vector weak neutral current couplings of the light quarks by LEP [@lep_couplings], D0 [@d0_couplings], H1 [@h1_couplings] and ZEUS [@zeus_couplings], compared with the simulated prospects for the LHeC [@cdr].[]{data-label="fig:couplings"}](couplings_u "fig:"){width=".495\textwidth"} ![Determination of the vector and axial-vector weak neutral current couplings of the light quarks by LEP [@lep_couplings], D0 [@d0_couplings], H1 [@h1_couplings] and ZEUS [@zeus_couplings], compared with the simulated prospects for the LHeC [@cdr].[]{data-label="fig:couplings"}](couplings_d "fig:"){width=".495\textwidth"} The second example is on the scale dependence of the weak mixing angle $\sin^2\!\theta_W$ obtained from a projected measurement of the polarisation asymmetry $A^-=\frac{\sigma^-_{\rm NC}(P_R)-\sigma^-_{\rm NC}(P_L)}{\sigma^-_{\rm NC}(P_R)+\sigma^-_{\rm NC}(P_L)}\simeq \frac{\kappa_Za_e(P_L-P_R)}{2}\frac{F_2^{\gamma Z}}{F_2}$ assuming a left-handed ($P_L$) or right-handed ($P_R$) polarisation of 80% and an integrated luminosity of 10fb$^{-1}$ per polarisation state [@cdr]. The results are compared in Fig. \[fig:wma\_cctot\] (left) with other determinations at different energies. The LHeC measurements are precise and cover a large energy range. It should be mentioned that the NC and CC cross section ratio is also sensitive to $\sin^2\!\theta_W$, provided that the PDFs related uncertainty is under control [@cdr]. ![Left: dependence of the weak mixing angle on the energy scale $\mu$ from [@cdr]. Right: energy dependence of the $\nu N$ cross section. The open points up to 50TeV correspond to the expected precision of the HERA measurements and the solid point corresponds to an expected measurement at the LHeC. The full line represents the predicted cross section including the $W$ propagator while the dashed line is a linear extrapolation from low energy measurements.[]{data-label="fig:wma_cctot"}](sintmu "fig:"){width=".575\textwidth"} ![Left: dependence of the weak mixing angle on the energy scale $\mu$ from [@cdr]. Right: energy dependence of the $\nu N$ cross section. The open points up to 50TeV correspond to the expected precision of the HERA measurements and the solid point corresponds to an expected measurement at the LHeC. The full line represents the predicted cross section including the $W$ propagator while the dashed line is a linear extrapolation from low energy measurements.[]{data-label="fig:wma_cctot"}](cctot_lhec "fig:"){width=".43\textwidth"} The third example is on a measurement of the CC total cross section (Fig. \[fig:wma\_cctot\] (right)). The LHeC measurement together with those from HERA illustrates in a spectacular way the impact of the propagator mass of the $W$ boson on the CC cross section. The dependence on the polarisation of the CC total cross section can further be used to set a stringent lower mass limit on a right-handed $W$ boson following Ref. [@h1cc]. Summary {#sec:summary} ======= A few selected examples of the LHeC measurements from top and EW sectors clearly demonstrate that the realisation of the LHeC can greatly enhance the physics programme and discovery potential of the LHC in a complementary manner. Indeed the LHeC is also considered as the next machine for studying the Higgs boson and a luminosity upgrade by a factor of 10 reaching $10^{34}\,{\rm cm}^{-2}s^{-1}$ is under study [@max], compared to the Conceptional Design Report [@cdr]. More studies are desirable from both theoretical and experimental communities, which will certainly reveal even larger potential of the LHeC. [99]{} J. L. Abelleira Fernandez [*et al.*]{} \[LHeC Study Group\], “A Large Hadron Electron Collider at CERN", J. Phys. G. [**39**]{} (2012) 075001, arXiv:1206.2913 \[physics.acc-ph\]. J. L. Abelleira Fernandez [*et al.*]{} \[LHeC Study Group Collaboration\], “On the Relation of the LHeC and the LHC", arXiv:1211.5102 \[hep-ex\]. F. D. Aaron [*et al.*]{} \[H1 and ZEUS Collaborations\], JHEP [**1001**]{} (2010) 109, arXiv:0911.0884 \[hep-ex\]. S. Dutta, A. Goyal, M. Kumar and B. Mellado, arXiv:1307.1688 \[hep-ph\]. N. Kidonakis, arXiv:1509.02528 \[hep-ph\]. A. O. Bouzas and F. Larios, Phys. Rev. D [**88**]{} (2013) 9, 094007 \[arXiv:1308.5634 \[hep-ph\]\]. V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Lett. B [**713**]{} (2012) 165 \[arXiv:1204.2332 \[hep-ex\]\]. B. Sahin and A. A. Billur, Phys. Rev. D [**86**]{} (2012) 074026 \[arXiv:1210.3235 \[hep-ph\]\]. B. Grzadkowski and M. Misiak, Phys. Rev. D [**78**]{} (2008) 077501 \[Phys. Rev. D [**84**]{} (2011) 059903\] \[arXiv:0802.1413 \[hep-ph\]\]. A. Aktas [*et al.*]{} \[H1 Collaboration\], Phys. Lett. B [**632**]{} (2006) 35 \[hep-ex/0507080\]. S. Schael [*et al.*]{} \[ALEPH and DELPHI and L3 and OPAL and SLD and LEP Electroweak Working Group and SLD Electroweak Group and SLD Heavy Flavour Group Collaborations\], Phys. Rept.  [**427**]{} (2006) 257 \[hep-ex/0509008\]. V. M. Abazov [*et al.*]{} \[D0 Collaboration\], Phys. Rev. D [**84**]{} (2011) 012007 \[arXiv:1104.4590 \[hep-ex\]\]. Z. Zhang (for the H1 Collaboration), Combined electroweak and QCD fits including NC and CC data with polarised electron beam at HERA-2, PoS DIS2010 (2010) 056. ZEUS Collaboration, ZEUS-prel-07-027. A. Aktas [*et al.*]{} \[H1 Collaboration\], Phys. Lett. B [**634**]{} (2006) 173 \[hep-ex/0512060\]. O. Bruening and M. Klein, Mod. Phys. Lett. A [**28**]{} (2013) 16, 1330011 \[arXiv:1305.2090 \[physics.acc-ph\]\]. [^1]: for the LHeC Study Group

--- abstract: 'The quantum many-body problem can be rephrased as a variational determination of the two-body reduced density matrix, subject to a set of $N$-representability constraints. The mathematical problem has the form of a semidefinite program. We adapt a standard primal-dual interior point algorithm in order to exploit the specific structure of the physical problem. In particular the matrix-vector product can be calculated very efficiently. We have applied the proposed algorithm to a pairing-type Hamiltonian and studied the computational aspects of the method. The standard $N$-representability conditions perform very well for this problem.' author: - Brecht Verstichel - Helen van Aggelen - Dimitri Van Neck - Patrick Bultinck - Stijn De Baerdemacker bibliography: - 'primal\_dual.bib' title: 'A primal-dual semidefinite programming algorithm tailored to the variational determination of the two-body density matrix' --- Introduction ============ It was realized in the 1950’s [@husimi; @lowdin] that the energy of a quantum many-body system can be expressed in terms of the two-body reduced density matrix (2DM), when only one- and two-body interactions are present. This insight led to the idea of variationally determining the 2DM by minimizing the energy, henceforth referred to as the v2DM method. Once the 2DM is known, all other physical properties that can be expressed as one- or two-body operators can be extracted. In this way the 2DM effectively replaces the wave function and we have “quantum mechanics without wave functions” [@coleman_book]. Early attempts, however, produced unrealistic results [@mayer] and it was soon realized [@tredgold] that non-trivial constraints are needed to ensure that the 2DM is derivable from a physical wave function. These constraints were called $N$-representability conditions by Coleman [@coleman], and Garrod and Percus [@garrod] derived two such conditions, the so-called $Q$ and $G$ conditions, which can be expressed as matrix-positivity constraints. With these constraints there were some attempts, some of which quite successful, to solve this problem numerically in the 1970s [@fusco; @garrod_comp; @rosina; @mihailovic]. However the method was soon abandoned because of the computational cost. Interest in the subject was renewed at the beginning of this century, when first Nakata [@nakata_first] and then Mazziotti [@mazziotti] realized that the v2DM problem can be formulated as a semidefinite program (SDP) for which general-purpose primal-dual SDP solvers can be used [@vandenberghe], and they calculated the ground-state properties of small atoms and molecules. Primal-dual interior point methods are the “Rolls Royce” of SDP algorithms, having several appealing features, but they require a lot of storage and are computationally expensive. These early calculations were therefore limited to small systems (minimal basis set). Mazziotti [@maz_prl] then developed an algorithm that transforms the SDP into a non-linear optimization program solved by a gradient-only method. This reduced the cost of the storage and the basic floating point operations, but at the cost of these nice convergence properties of the interior point methods. In this paper we adapt a standard primal-dual interior point algorithm [@sturm] to the specific case of v2DM, in an attempt to retain the nice convergence properties, while reducing the storage and computational cost. In Sec. \[v2DM\] we present an introduction to the theory of $N$-representability, v2DM and some mathematical properties of the constraints. In Sec. \[SDP\] we discuss the representation of the problem as a primal-dual semidefinite program, and introduce the method we use to solve it. Then we apply the algorithm to a BCS (Bardeen-Cooper-Shrieffer) [@BCS] or pairing-type Hamiltonian in Sec. \[app\] and present the physical results and computational aspects. A summary is provided in Sec. \[sum\]. \[v2DM\] Variational density matrix determination ================================================= When only two-body interactions are present, the Hamiltonian of a physical system can be written as: $$\hat{H} = \sum_{\alpha\gamma}t_{\alpha\gamma} a^\dagger_\alpha a_\gamma + \frac{1}{4}\sum_{\alpha\beta\gamma\delta}V_{\alpha\beta;\gamma\delta}a^\dagger_\alpha a^\dagger_\beta a_\delta a_\gamma~,$$ using second quantized notation where $a^\dagger_\alpha$ ($a_\alpha$) creates (annihilates) a fermion in a single-particle (sp) state $\alpha$ [@bijbel]. The expectation value of the energy in an arbitrary $N$-particle state ${|\Psi^N\rangle}$ can be expressed in terms of the 2DM only, $$E(\Gamma) = \mathrm{Tr}~\Gamma H^{(2)} = \sum_{\alpha<\beta;\gamma<\delta}\Gamma_{\alpha\beta;\gamma\delta}H^{(2)}_{\alpha\beta;\gamma\delta}~, \label{ener_func}$$ with the 2DM defined as: $$\Gamma_{\alpha\beta;\gamma\delta} = {\langle \Psi^N|}a^\dagger_\alpha a^\dagger_\beta a_\delta a_\gamma {|\Psi^N\rangle}~, \label{2DM}$$ and the reduced two-particle Hamiltonian, $$H^{(2)}_{\alpha\beta;\gamma\delta} = \frac{1}{N-1}\left(\delta_{\alpha\gamma}t_{\beta\delta} - \delta_{\alpha\delta}t_{\beta\gamma} - \delta_{\beta\gamma}t_{\alpha\delta} + \delta_{\beta\delta}t_{\alpha\gamma}\right) + V_{\alpha\beta;\gamma\delta}~.$$ The idea of v2DM is to determine the ground-state energy and other two- or one-body properties by minimizing the energy (\[ener\_func\]) using the 2DM as a variable. The 2DM is a much more compact object than the wave function because one keeps the dimension of two-particle (tp) space, no matter how many particles are involved. The problem is that there is no straightforward way to know whether an arbitrary matrix in tp-space $\Gamma$ is derivable from a physical wave function as in Eq. (\[2DM\]). Actually, it is sufficient that $\Gamma$ is derivable from an ensemble of $N$-particle wave functions, and this is called the $N$-representability problem [@coleman]. Some obvious necessary $N$-representability constraints are apparent from the definition (\[2DM\]): $$\begin{aligned} \text{trace condition}\qquad\mathrm{Tr}~\Gamma &=& \sum_{\alpha<\beta}\Gamma_{\alpha\beta;\alpha\beta}=\frac{N(N-1)}{2}~,\\ \text{antisymmetry}\qquad\Gamma_{\alpha\beta;\gamma\delta} &=& -\Gamma_{\beta\alpha;\gamma\delta} = -\Gamma_{\alpha\beta;\delta\gamma} = \Gamma_{\beta\alpha;\delta\gamma}~,\\ \text{Hermiticity}\qquad\Gamma_{\alpha\beta;\gamma\delta} &=& \Gamma_{\gamma\delta;\alpha\beta}~,\end{aligned}$$ but it turns out that there are many non-trivial constraints needed to ensure that a 2DM is physical. $N$-representability -------------------- The necessary and sufficient conditions for $N$-representability are formally known [@payers]. A tp-matrix is $N$-representable if and only if, for every two-body Hamiltonian $\hat{H}_\nu$, the following inequality is satisfied: $$\mathrm{Tr}~H^{(2)}_\nu \Gamma \geq E_0(H_\nu)~,$$ where $E_0(H_\nu)$ is the exact $N$-particle ground-state energy corresponding to the Hamiltonian. This is hardly a practical approach, as one needs to know the ground-state energy of every two-body Hamiltonian. Therefore one resorts to certain classes of Hamiltonians for which a lower bound to the ground-state energy is known. A Hamiltonian class that is used as necessary constraint is $$\label{stand_constr_tp} {\langle \Psi^N|}B^\dagger B{|\Psi^N\rangle} \geq 0~,$$ which leads to positivity conditions of linear matrix maps of the 2DM. If we want (\[stand\_constr\_tp\]) to be restricted to tp-space there are three possible forms of the operator $B^\dagger$, leading to three conditions on the density matrix: #### $B^\dagger = \sum_{\alpha\beta}p_{\alpha\beta}a^\dagger_\alpha a^\dagger_\beta$ leads to the trivial $\mathcal{P}$-condition: $$\mathcal{P}(\Gamma) = \Gamma \succeq 0~,$$ which imposes positive semidefiniteness on the 2DM. #### $B^\dagger = \sum_{\alpha\beta}q_{\alpha\beta}a_\alpha a_\beta$ leads to the $\mathcal{Q}$-condition: $$\mathcal{Q}(\Gamma) \succeq 0~,$$ where the linear matrix map $\mathcal{Q}$ is defined as $$\begin{aligned} \nonumber\mathcal{Q}(\Gamma)_{\alpha\beta;\gamma\delta} &=& {\langle \Psi^N|}a_\alpha a_\beta a^\dagger_\delta a^\dagger_\gamma{|\Psi^N\rangle}\\ \nonumber&=&\Gamma_{\alpha\beta;\gamma\delta} + (\delta_{\alpha\gamma}\delta_{\beta\delta} - \delta_{\alpha\delta}\delta_{\beta\delta})\frac{\bar{\bar{\Gamma}}}{N(N-1)}\\ &&- \delta_{\alpha\gamma}\rho_{\beta\delta} + \delta_{\alpha\delta}\rho_{\beta\gamma} + \delta_{\beta\gamma}\rho_{\alpha\delta} - \delta_{\beta\delta}\rho_{\alpha\gamma}~, \label{Q}\end{aligned}$$ with $$\rho_{\alpha\gamma} = \frac{1}{N-1}\bar{\Gamma}_{\alpha\gamma} = \frac{1}{N-1}\sum_{\beta}\Gamma_{\alpha\beta;\gamma\beta}~,\\$$ the one-body reduced density matrix (1DM), and with $$\bar{\bar{\Gamma}} = \sum_{\alpha\beta}\Gamma_{\alpha\beta;\alpha\beta}~,$$ the unrestricted trace of the 2DM. #### $B^\dagger = \sum_{\alpha\beta}g_{\alpha\beta}a^\dagger_\alpha a_\beta$ which leads to the $\mathcal{G}$-condition: $$\mathcal{G}(\Gamma) \succeq 0~,$$ with the linear matrix map $\mathcal{G}$ defined as $$\begin{aligned} \nonumber\mathcal{G}(\Gamma)_{\alpha\beta;\gamma\delta} &=& {\langle \Psi^N|}a^\dagger_\alpha a_\beta a^\dagger_\delta a_\gamma{|\Psi^N\rangle}\\ &=& \delta_{\beta\delta}\rho_{\alpha\gamma} - \Gamma_{\alpha\delta;\gamma\beta}~. \label{G_up}\end{aligned}$$ Another Hamiltonian class for which a lower bound to the ground-state energy is known gives rise to the so-called three-index conditions: $${\langle \Psi^N|}\left\{B^\dagger,B\right\}{|\Psi^N\rangle} \geq 0~. \label{three_index}$$ In this article we will use two conditions that come from Eq. (\[three\_index\]). #### $B^\dagger = \sum_{\alpha\beta\gamma}t_{\alpha\beta\gamma}a^\dagger_\alpha a^\dagger_\beta a^\dagger_\gamma$ leads to the $\mathcal{T}_1$-condition: $$\mathcal{T}_1(\Gamma) \succeq 0~,$$ with the linear matrix map $\mathcal{T}_1$ defined as $$\begin{aligned} \nonumber\mathcal{T}_1\left(\Gamma\right)_{\alpha\beta\gamma;\delta\epsilon\zeta} &=& {\langle \Psi^N|}a^\dagger_\alpha a^\dagger_\beta a^\dagger_\gamma a_\zeta a_\epsilon a_\delta + a_\alpha a_\beta a_\gamma a^\dagger_\zeta a^\dagger_\epsilon a^\dagger_\delta{|\Psi^N\rangle}\\ \nonumber&=&\left(\delta_{\gamma\zeta}\delta_{\beta\epsilon}\delta_{\alpha\delta} - \delta_{\gamma\epsilon}\delta_{\alpha\delta}\delta_{\beta\zeta} + \delta_{\alpha\zeta}\delta_{\gamma\epsilon}\delta_{\beta\delta} - \delta_{\gamma\zeta}\delta_{\alpha\epsilon}\delta_{\beta\delta} + \delta_{\beta\zeta}\delta_{\alpha\epsilon}\delta_{\gamma\delta} -\delta_{\alpha\zeta}\delta_{\beta\epsilon}\delta_{\gamma\delta}\right)\frac{\bar{\bar{\Gamma}}}{N(N - 1)}\\ \nonumber&& -\left(\delta_{\gamma\zeta}\delta_{\beta\epsilon} - \delta_{\beta\zeta}\delta_{\gamma\epsilon}\right)\rho_{\alpha\delta} + \left(\delta_{\gamma\zeta}\delta_{\alpha\epsilon} - \delta_{\alpha\zeta}\delta_{\gamma\epsilon}\right)\rho_{\beta\delta} - \left(\delta_{\beta\zeta}\delta_{\alpha\epsilon}- \delta_{\alpha\zeta}\delta_{\beta\epsilon}\right)\rho_{\gamma\delta}\\ \nonumber&& + \left(\delta_{\gamma\zeta}\delta_{\beta\delta} - \delta_{\beta\zeta}\delta_{\gamma\delta}\right)\rho_{\alpha\epsilon} - \left(\delta_{\gamma\zeta}\delta_{\alpha\delta} - \delta_{\alpha\zeta}\delta_{\gamma\delta}\right)\rho_{\epsilon\beta} + \left(\delta_{\beta\zeta}\delta_{\alpha\delta} - \delta_{\alpha\zeta}\delta_{\beta\delta}\right)\rho_{\gamma\epsilon}\\ \nonumber&& - \left(\delta_{\beta\delta}\delta_{\gamma\epsilon} - \delta_{\beta\epsilon}\delta_{\gamma\delta}\right)\rho_{\alpha\zeta} + \left(\delta_{\gamma\epsilon}\delta_{\alpha\delta} - \delta_{\alpha\epsilon}\delta_{\gamma\delta}\right)\rho_{\beta\zeta} - \left(\delta_{\beta\epsilon}\delta_{\alpha\delta} - \delta_{\alpha\epsilon}\delta_{\beta\delta}\right)\rho_{\gamma\zeta}\\ \nonumber&&+ \delta_{\gamma\zeta}\Gamma_{\alpha\beta;\delta\epsilon} - \delta_{\beta\zeta}\Gamma_{\alpha\gamma;\delta\epsilon} + \delta_{\alpha\zeta}\Gamma_{\beta\gamma;\delta\epsilon} - \delta_{\gamma\epsilon}\Gamma_{\alpha\beta;\delta\zeta} + \delta_{\beta\epsilon}\Gamma_{\alpha\gamma;\delta\zeta} - \delta_{\alpha\epsilon}\Gamma_{\beta\gamma;\delta\zeta}\\ && + \delta_{\gamma\delta}\Gamma_{\alpha\beta;\epsilon\zeta} - \delta_{\beta\delta}\Gamma_{\alpha\gamma;\epsilon\zeta} + \delta_{\alpha\delta}\Gamma_{\beta\gamma;\epsilon\zeta}~. \label{T1_up}\end{aligned}$$ #### $B^\dagger = \sum_{\alpha\beta\gamma}t_{\alpha\beta\gamma}a^\dagger_\alpha a^\dagger_\beta a_\gamma$ leads to the $\mathcal{T}_2$-condition $$\mathcal{T}_2(\Gamma) \succeq 0~,$$ with the linear matrix map $\mathcal{T}_2$ defined as $$\begin{aligned} \label{T2_up}\mathcal{T}_2(\Gamma)_{\alpha\beta\gamma;\delta\epsilon\zeta} &=& {\langle \Psi^N|}a^\dagger_\alpha a^\dagger_\beta a_\gamma a^\dagger_\zeta a_\epsilon a_\delta + a^\dagger_\gamma a_\beta a_\alpha a^\dagger_\delta a^\dagger_\epsilon a_\zeta{|\Psi^N\rangle}\\ \nonumber&=& \left(\delta_{\alpha\delta}\delta_{\beta\epsilon} - \delta_{\alpha\epsilon}\delta_{\beta\delta}\right)\rho_{\gamma\zeta} + \delta_{\gamma\zeta}\Gamma_{\alpha\beta;\delta\epsilon} - \delta_{\alpha\delta}\Gamma_{\gamma\epsilon;\zeta\beta} + \delta_{\beta\delta}\Gamma_{\gamma\epsilon;\zeta\alpha} + \delta_{\alpha\epsilon}\Gamma_{\gamma\delta;\zeta\beta} - \delta_{\beta\epsilon}\Gamma_{\gamma\delta;\zeta\alpha}~.\end{aligned}$$ The optimization problem that we have to solve can be summarized as: $$\min_{\Gamma} \mathrm{Tr}~\Gamma H^{(2)}~,$$ under the condition that $$\begin{aligned} \mathrm{Tr}~\Gamma &=& \frac{N(N-1)}{2}~,\\ \mathcal{L}(\Gamma) &\succeq& 0 \qquad \forall \mathcal{L} \in \{\mathcal{P,Q,G},\mathcal{T}_1,\mathcal{T}_2\}~.\end{aligned}$$ Hermitian adjoint maps ---------------------- For the following it is useful to introduce the Hermitian adjoints of matrix maps introduced in the previous section. The Hermitian adjoint maps are defined through: $$\label{gen_herm} \mathrm{Tr}~\mathcal{L}_i(\Gamma) A = \mathrm{Tr}~\mathcal{L}_i^\dagger(A)\Gamma~,$$ in which $A$ is a matrix of the same dimension as the image of the map $\mathcal{L}_i$ in question (*e.g.* a three-particle matrix for a $\mathcal{T}_1$ map, *etc.*), and the traces sum over the appropriate indices. The $\mathcal{P}$ and $\mathcal{Q}$ maps are Hermitian, so they are identical to their Hermitian adjoints. For the other maps however this is not the case. Using Eq. (\[gen\_herm\]) the Hermitian adjoint of the $\mathcal{G}$ map can be shown to have the form: $$\begin{aligned} \label{G_down} \mathcal{G}^\dagger\left(A\right)_{\alpha\beta;\gamma\delta} &=& \frac{1}{N-1}\left[\delta_{\beta\delta}\bar{A}_{\alpha\gamma} - \delta_{\alpha\delta}\bar{A}_{\beta\gamma} - \delta_{\beta\gamma}\bar{A}_{\alpha\delta} + \delta_{\alpha\gamma}\bar{A}_{\beta\delta}\right]\\ \nonumber&&\qquad\qquad - A_{\alpha\delta;\gamma\beta} + A_{\beta\delta;\gamma\alpha} + A_{\alpha\gamma;\delta\beta} - A_{\beta\gamma;\delta\alpha}~,\end{aligned}$$ in which a particle-hole matrix $A$ is mapped onto tp-matrix space and $$\bar{A}_{\alpha\gamma} = \sum_\lambda A_{\alpha\lambda;\gamma\lambda}~.$$ The $\mathcal{T}_1$-operator maps a tp-matrix onto a three-particle matrix, so its Hermitian adjoint has to map a three-particle matrix $A$ onto tp-space. Solving Eq. (\[gen\_herm\]) with $\mathcal{L} = \mathcal{T}_1$ one finds that: $$\begin{aligned} \label{T1_down} \mathcal{T}^\dagger_1\left(A\right)_{\alpha\beta;\gamma\delta} &=& \frac{2}{N(N-1)}\left(\delta_{\alpha\gamma}\delta_{\beta\delta} - \delta_{\alpha\delta}\delta_{\beta\gamma}\right)\mathrm{Tr}~A + \bar{A}_{\alpha\beta;\gamma\delta}\\ \nonumber&&-\frac{1}{2(N-1)}\left[\delta_{\beta\delta}\bar{\bar{A}}_{\alpha\gamma} - \delta_{\alpha\delta}\bar{\bar{A}}_{\beta\gamma} - \delta_{\beta\gamma}\bar{\bar{A}}_{\alpha\delta} + \delta_{\alpha\gamma}\bar{\bar{A}}_{\beta\delta}\right]~,\end{aligned}$$ with $$\begin{aligned} \bar{A}_{\alpha\beta;\gamma\delta} &=& \sum_\lambda A_{\alpha\beta\lambda;\gamma\delta\lambda}~,\\ \bar{\bar{A}}_{\alpha\gamma} &=& \sum_{\lambda\kappa} A_{\alpha\lambda\kappa;\gamma\lambda\kappa}~.\end{aligned}$$ In the same way one can derive for $\mathcal{L}=\mathcal{T}_2$ that $$\begin{aligned} \label{T2_down} \mathcal{T}^\dagger_2(A)_{\alpha\beta;\gamma\delta} &=& \frac{1}{2(N-1)}\left[\delta_{\beta\delta}\tilde{\tilde{A}}_{\alpha\gamma} - \delta_{\alpha\delta}\tilde{\tilde{A}}_{\beta\gamma} - \delta_{\beta\gamma}\tilde{\tilde{A}}_{\alpha\delta} + \delta_{\alpha\gamma}\tilde{\tilde{A}}_{\beta\delta}\right] + \bar{A}_{\alpha\beta;\gamma\delta}\\ \nonumber&&-\left[\tilde{A}_{\delta\alpha;\beta\gamma} - \tilde{A}_{\delta\beta;\alpha\gamma} - \tilde{A}_{\gamma\alpha;\beta\delta} + \tilde{A}_{\gamma\beta;\alpha\delta}\right]~,\end{aligned}$$ with this time $A$ a matrix on two-particle-one-hole space and $$\begin{aligned} \tilde{\tilde{A}}_{\alpha\gamma} &=& \sum_{\lambda\kappa}A_{\lambda\kappa\alpha;\lambda\kappa\gamma}~,\\ \bar{A}_{\alpha\beta;\gamma\delta} &=& \sum_{\lambda}A_{\alpha\beta\lambda;\gamma\delta\lambda}~,\\ \tilde{A}_{\alpha\beta;\gamma\delta} &=& \sum_{\lambda}A_{\lambda\alpha\beta;\lambda\gamma\delta}~.\end{aligned}$$ \[SDP\] Primal-dual semidefinite program ======================================== The variational method described in the previous section can be formulated as a primal-dual semidefinite program. A general 2DM, describing an $N$-particle system can be expanded in an arbitrary orthogonal basis $\{f^i\}$ of traceless matrix space as $$\Gamma = \frac{N(N - 1)}{M(M-1)} \mathbb{1}_{\text{tp}} + \sum_i \gamma_i f^i~,$$ with $M$ the dimension of single-particle (sp) space, and the unit matrix on tp space defined as $$\left(\mathbb{1}_{\text{tp}}\right)_{\alpha\beta;\gamma\delta} = \delta_{\alpha\gamma}\delta_{\beta\delta} - \delta_{\alpha\delta}\delta_{\beta\gamma}~.$$ The energy of the system can be written as a function of the $\gamma$’s as $$\mathrm{Tr}~\Gamma H^{(2)} = \frac{N(N - 1)}{M(M-1)}\mathrm{Tr}~H^{(2)} + \sum_i \gamma_i \mathrm{Tr}~H^{(2)}f^i~.$$ Because the necessary $N$-representability conditions can be written as linear homogeneous matrix maps of $\Gamma$, we can also write them as a function of the $\gamma$’s: $$\mathcal{L}\left({\Gamma}\right) = \frac{N(N - 1)}{M(M-1)}\mathcal{L}\left({\mathbb{1}_\text{tp}}\right) + \sum_{i} \gamma_i \mathcal{L}\left({f^i}\right) \succeq 0~.$$ If we now consider the direct sum of the linear spaces associated with the maps and define the block matrices: $$u^0 = \frac{N(N - 1)}{M(M-1)}\bigoplus_k \mathcal{L}_k\left(\mathbb{1}_\text{tp}\right) \qquad\text{and}\qquad u^i = \bigoplus_k \mathcal{L}_k\left(f^i\right)~,$$ then we can formulate v2DM as a standard dual semidefinite program [@vandenberghe]: $$\min_\gamma~\gamma^T h \qquad \text{on condition that} \qquad Z = u^0 + \sum_i \gamma_i u^i \succeq 0~, \label{primal}$$ in which $h^i = \mathrm{Tr}~H^{(2)}f^i$. The primal problem corresponding to (\[primal\]) optimizes the matrixvariable $X$, the problem being defined as: $$\max_X~ \left(-\mathrm{Tr}~Xu^0\right) \qquad \text{on condition that} \qquad \mathrm{Tr}~Xu^i = h^i \qquad \text{and} \qquad X\succeq 0~. \label{dual}$$ $X$ will be a block matrix because the $u$-matrices are block matrices. The primal-dual gap $\eta$ is defined as the difference between the primal and the dual cost function for a certain primal-dual point $(X,Z)$: $$\eta = \gamma^T h + \mathrm{Tr}~u^0 X = \sum_i \gamma_i \mathrm{Tr}~X u^i + \mathrm{Tr}~X u^0 = \mathrm{Tr}~X Z \geq 0~,$$ as $X$ and $Z$ are positive semidefinite matrices. We can see that the smallest value of $\eta$ will be reached when both the primal and the dual problem are optimal. It can be proven that if the primal and the dual problem are both strictly feasible, then the primal-dual gap vanishes at their solution [@vandenberghe]. This means that the primal-dual gap can be used as a convergence criterion for the algorithm. Even better, at any point during the optimization, the error on the current value is limited from above by the primal-dual gap. Note that in our previous implementation [@atomic], a dual-only algorithm was used. The properties of the present primal-dual method can lead to a serious reduction in computation time since we can stop the algorithm at a prescribed error estimate. Equations of motion ------------------- There are several known methods to solve a semidefinite program. In this paper a path-following interior point method is used. The central path is defined as the set of primal-dual points for which $$\label{cent_path} X Z = \frac{\eta}{n} \mathbb{1}_\text{sup}~,$$ with $n$ the total dimension of the $X$ and $Z$ matrices and $\mathbb{1}_\text{sup}$ the direct sum of the unity matrices on the different constraint spaces: $$\mathbb{1}_\text{sup} = \bigoplus_k \mathbb{1}_k~.$$ In the path-following algorithm [@sturm] we try to follow the central path, reducing the primal-dual gap along the way. Consider a primal-dual point $(X,Z)$ on the central path with primal-dual gap $\eta$. We want to know what is the primal-dual point on the central path with primal-dual gap scaled down with a factor $\nu$. Rephrasing, we are looking for the $(\Delta_X,\Delta_Z)$ that solve: $$(X + \Delta_X)(Z + \Delta_Z) = \frac{\nu\eta}{n}\mathbb{1}_\text{sup}~. \label{EOM}$$ There are several ways to symmetrize these equations. Using the method proposed by [@sturm], two equivalent equations (called the dual and the primal) are obtained, *i.e.* one has to solve the equations $$\begin{aligned} \label{P_eom}(\text{dual}) : \Delta_X + D^{-1}\Delta_Z D^{-1} &=& \frac{\nu\eta}{n} Z^{-1} - X~,\\ \label{D_eom}(\text{primal}) : \Delta_Z + D~\Delta_{X}~D &=& \frac{\nu\eta}{n} X^{-1} - Z~,\end{aligned}$$ and under the condition that: $$\label{eom_constr} \mathrm{Tr}~\Delta_Xu^i = 0 \qquad \text{and} \qquad \Delta_Z = \sum_i \left(\delta \gamma\right)_i u^i~,$$ and with $$D(X,Z) = X^{-\frac{1}{2}}\left(X^{\frac{1}{2}}ZX^{\frac{1}{2}}\right)^{\frac{1}{2}}X^{-\frac{1}{2}}~. \label{metric}$$ ### Solution to the dual equation In order to obtain the primal-dual direction $(\Delta_X,\Delta_Z)$ , the dual equation (\[P\_eom\]) is first projected onto the space spanned by the non-orthogonal basis $\{u^i\}$ (which we will call $\mathcal{U}$-space). With $B$ denoting the right-hand side of (\[P\_eom\]) and making use of Eq. (\[eom\_constr\]) we obtain: $$\label{primal_cg} \sum_j \underbrace{\left(\mathrm{Tr}~D^{-1} u^j D^{-1} u^i\right)}_{\mathcal{H}^D_{ij}} \Delta\gamma_j = \mathrm{Tr}~B u^i~,$$ which can be seen to be a symmetrical, positive-definite linear system and as such can be solved iteratively using the linear conjugate gradient method. This can be done without explicit construction of the dual Hessian matrix $\mathcal{H}^D$ or any reference to the non-orthogonal basis set $\{u^i\}$. This is because $\mathcal{H}^D$ can be seen as a map from traceless tp-matrix space onto itself, by using the Hermitian adjoints of the linear maps $\mathcal{L}$. Consider an arbitrary traceless tp-matrix: $$\epsilon = \sum_j \epsilon_j f^j~.$$ Using (\[gen\_herm\]) and the fact that the $\mathcal{L}$’s are linear and homogeneous we obtain that the image of $\epsilon$ under the dual Hessian map can be written as: $$\mathcal{H}^D\epsilon = \hat{P}_{\text{Tr}}\left[\sum_k\mathcal{L}^\dagger_k\left(D_k^{-1}\mathcal{L}_k\left(\epsilon\right) D_k^{-1}\right)\right]~,$$ in which the $D_k$ are the blocks of the $D$ matrix corresponding to the different constraints $\mathcal{L}_k$, and $\hat{P}_\text{Tr}$ stands for the projection operator onto traceless tp-matrix space: $$\hat{P}_{\text{Tr}}(A) = A - \frac{2\mathrm{Tr}~A}{M(M-1)}\mathbb{1}_\text{tp}~.$$ ### Solution to the primal equation The solution of the primal equation (\[D\_eom\]) is obtained in the same manner, by projecting this equation onto $\mathcal{C}$-space, the orthogonal complement of $\mathcal{U}$-space. With $B$ denoting the right-hand side of the equation (\[D\_eom\]) and making use of Eq. (\[eom\_constr\]) one gets: $$\label{dual_cg} \sum_j \underbrace{\left(\mathrm{Tr}~D~c^j~D~c^i\right)}_{\mathcal{H}^P_{ij}} \delta x_j = \mathrm{Tr}~B c^i~,$$ where we have used $$\Delta_X = \sum_i \delta x_i~c^i~.$$ This is again a symmetrical positive-definite system of linear equations that can be solved iteratively using the linear conjugate gradient method. As with the dual equation it can be solved without explicit construction of the Hessian matrix $\mathcal{H}^P$, or any reference to the basisset $\{c^i\}$, because $\mathcal{H}^P$ can be seen as a map from $\mathcal{C}$-space onto itself. For an arbitrary matrix in $\mathcal{C}$-space: $$\epsilon = \sum_i \epsilon_i c^i~,$$ the image of $\epsilon$ under the primal Hessian map is $$\mathcal{H}^P\epsilon = \hat{P}_{\mathcal{C}}\left[D\epsilon D\right]~.$$ in which $\hat{P}_\mathcal{C}$ is the projection onto $\mathcal{C}$-space. This projection can be executed quickly by using the inverse of the overlap matrix of the $\mathcal{U}$-space basis vectors. Suppose we have an arbitrary block matrix $A$ of the same dimension as $X$ and $Z$. First we project it onto the space spanned by the basis $\{u^0,u^i\} = \{u^\alpha\}$. The projected matrix $A'$ reads as: $$A' = \sum_{\alpha\beta} \mathrm{Tr}~\left[Au^\alpha\right] \left(\mathcal{S}^{-1}\right)_{\alpha\beta}u^\beta~,$$ where the overlap matrix $\mathcal{S}$ appears because of the non-orthogonality of the basis. Due to the special properties of the linear matrix maps $\mathcal{L}$ that determine the basis matrices $u^\alpha$, the inverse overlap matrix can also be considered as a map from tp space onto itself (see Appendix \[overlapmatrix\] and \[inverse\_overlapmatrix\] for the actual analytic expression of this map). The projected matrix $A'$ can now be written in block-matrix form as: $$A' = \bigoplus_l \mathcal{L}_l\left[\mathcal{S}^{-1}\left(\sum_k \mathcal{L}^\dagger_k\left(A_k\right)\right)\right]~.$$ To project $A$ onto $\mathcal{U}$-space we still have to remove the component along the $u^0$-matrix: $$\hat{P}_{\mathcal{U}}A = A' - \left(\frac{\mathrm{Tr}~u^0 A'}{\mathrm{Tr}~u^0u^0}\right)u^0~.$$ Since $\mathcal{C}$-space is the orthogonal complement of the $\mathcal{U}$-space, the desired projection of $A$ onto the $\mathcal{C}$-space is simply given by $$\hat{P}_{\mathcal{C}}A = A - \hat{P}_{\mathcal{U}}A~.$$ Outline of the algorithm ------------------------ In this section a short outline of the algorithm will be presented. The first step is to initialize the primal-dual variables, after which they are directed towards the central path. Then the actual minimization of the primal-dual gap takes place, which is done in a predictor-corrector loop. ### Initialization We need a feasible primal-dual starting point. An initial feasible dual point $Z^{(0)}$, *i.e.* a matrix that satisfies the inequality (\[primal\]), is easily found by setting $$Z^{(0)} = u^0~,$$ which corresponds to setting al the $\gamma_i$’s equal to zero. A feasible primal starting point will have to satisfy Eq. (\[dual\]). To construct such a point we take a completely random matrix $X$ and project it onto a matrix $X'$ for which $$\label{projham} \mathrm{Tr}~X'u^i = h^i~.$$ This is again achieved using the inverse overlap matrix of the $\{u^\alpha\}$ basis, $$X' = X - \underbrace{\sum_{\alpha\beta}\left(\mathrm{Tr}~Xu^\alpha - h^\alpha\right)\mathcal{S}^{-1}_{\alpha\beta}u^\beta}_{X^\perp}~.$$ The last term on the right-hand side can be computed as: $$X^\perp = \bigoplus_l \mathcal{L}_l\left[\mathcal{S}^{-1}\left(\sum_k \mathcal{L}^\dagger_k\left(X_k\right) - H^{(2)}\right)\right]~.$$ At this point, $X'$ satifies the equality constraint (\[projham\]), and one just has to add $u^{0}$, with a positive scaling factor that is large enough to ensure positive semidefiniteness: $$X^{(0)} = X' + \alpha u^{0}\succeq 0~.$$ ### \[centering\]Centering run Before the actual program can be started, a couple of centering steps have to be taken, which is done by solving the equations (\[P\_eom\]) and (\[D\_eom\]) with $\nu = 1$. The purpose is to go sufficiently near the central path, without bothering about the primal-dual gap. In a first step, Eq. (\[primal\_cg\]) which has the smallest dimension, is solved using the conjugate gradient method, and the dual solution $\Delta_Z$ is obtained. The primal solution $\Delta_X$ then follows from the dual equation (\[P\_eom\]) by substitution. For these initial centering steps, both linear systems are so well conditioned that hardly any iterations are needed for convergence. As a measure for the distance from the center we use the potential [@vandenberghe]: $$\Phi(X,Z) = -\ln\det X - \ln \det Z~,$$ which is minimal (for points with the same primal-dual gap $\eta = \mathrm{Tr}~XZ$) on the central path for which Eq. (\[cent\_path\]) is satisfied: $$\Phi(X^c,Z^c) = -n \ln{\frac{\eta}{n}}~.$$ When the potential difference (which is always positive): $$\begin{aligned} \nonumber\Psi(X,Z) &=& \Phi(X,Z) - \Phi(X^c,Z^c)\\ \label{logpot}&=& {n}\ln \mathrm{Tr}~XZ -{n}\ln{n}-\ln\det X - \ln \det Z~,\end{aligned}$$ is sufficiently small, the centering run is stopped. ### Predictor-corrector run In this part of the program the primal-dual gap is minimized by alternating predictor and corrector steps. A predictor step tries to reduce the primal-dual gap by solving the equations (\[P\_eom\]) and (\[D\_eom\]) with $\nu = 0$. This is done in exactly the same way as for the centering run, by first solving (\[primal\_cg\]) for $\Delta_Z$, then substituting into (\[P\_eom\]) to obtain an approximate primal step $\Delta_X$. The final primal step $\Delta_X$ is obtained by solving (\[dual\_cg\]) using the conjugate gradient method with the approximate $\Delta_X$ as a starting point. Note that when the primal-dual gap decreases, the condition number of the primal and dual Hessian matrices increases and more iterations are needed before convergence is reached. One can adjust the convergence criteria of the primal and dual conjugate gradient loops, in order to minimize the combined number of iterations. At this point we have a predictor direction $(\Delta_X,\Delta_Z)$. The logarithmic potential $\phi(\alpha) = \Psi(X + \alpha \Delta_X,Z + \alpha \Delta_Z)$ in the predictor direction (see Eq. (\[logpot\])) can be simply evaluated for any value of $\alpha$ by precomputing the eigenvalues $\lambda^X_i$ of $X^{-\frac{1}{2}}\Delta_X X^{-\frac{1}{2}}$ and $\lambda^Z_i$ of $Z^{-\frac{1}{2}}\Delta_Z Z^{-\frac{1}{2}}$. One then has $$\begin{aligned} \phi(\alpha) &=& \Psi(X,Z) + \ln\left[1 + \alpha (c_X + c_Z)\right] - \sum_i \ln (1 + \alpha \lambda^X_i) - \sum_i \ln(1 + \alpha \lambda^Z_i)~,\end{aligned}$$ where $$c_Z = \frac{1}{\eta}\mathrm{Tr}~X\Delta_Z \qquad\text{and}\qquad c_X = \frac{1}{\eta}\mathrm{Tr}~Z\Delta_X~,$$ With a standard bisection method one can now compute the stepsize $\alpha$ corresponding to the maximal deviation from the central path we want to allow. After the predictor step, a corrector step is taken, which is equivalent to the centering step described previously (see Sec. \[centering\]). The alternation of predictor and corrector steps continues until the primal-dual gap is smaller then the desired value. ![\[energy\] The ground-state energy as calculated by v2DM(PQG) and the Richardson-Gaudin equations (RG), together with the pair occupation in the groundstate by v2DM(PQG), as a function of the pairing interaction strength $g$.](double_plot.pdf) ![\[diff\] The difference between the ground-state energy calculated by v2DM with various constraints, and the exact solution, as a function of pairing strength $g$.](diff.pdf) \[app\] Application to the BCS Hamiltonian ========================================== \[bcsham\]The BCS Hamiltonian ----------------------------- The algorithm introduced in Sec. \[SDP\] is applied to the BCS Hamiltionian [@BCS]. The BCS Hamiltonian is an interesting system that models the competition between a single-particle operator and a schematic pairing interaction: $$\label{BCS_ham} \hat{H} = \sum_{i\sigma}\epsilon_i a^\dagger_{i\sigma}a_{i\sigma} - g \sum_{ij} a^\dagger_{i\uparrow}a^\dagger_{i\downarrow}a_{j\downarrow}a_{j\uparrow}~.$$ Here the single-particle levels are denoted with an index $i=1,\ldots,M$, and the up (down) spin as $\sigma = \uparrow(\downarrow)$. When the pairing strength $g$ is small compared to the single-particle level spacing, the energy is minimized by filling up the single-particle orbitals up to the fermi level. With increasing $g$ however, it becomes advantageous to form pairs, *i.e.* it is energetically favorable to maximize the ground-state occupation of the fermion pair state $\sum_i a^\dagger_{i\uparrow}a^\dagger_{i\downarrow}$. This problem is hard to solve using standard perturbative methods as these tend to break down when pairs are formed. An exact solution based on the Bethe-ansatz exists for this problem, however, and involves solving a system of non-linear equations [@richardson]. These equations are notoriously difficult to solve because, for certain critical values of $g$, the equations become singular. Several approaches have been suggested for solving these equations [@richardson2; @rombouts]. In this paper we follow the approach recently proposed by De Baerdemacker [@stijn]. The exact ground-state energies as a function of $g$ are compared to the v2DM results calculated within the present formalism. Results ------- We have studied the Hamiltonian Eq. (\[BCS\_ham\]) with $M=12$ doubly degenerate equidistant single-particle levels and $N=12$ fermions, and $g$ ranging from $0$ to $5$ in steps of $0.01$. v2DM calculations were performed with respectively $PQG$, $PQGT_1$ and $PQGT_1T_2$ constraints. The resulting ground-state energy is compared to the exact solution in Fig. \[energy\]. For all values of $g$ the agreement is already remarkably good at the $PQG$ level. To appreciate how the result improves when constraints are added the difference between the various v2DM results and the exact solution is plotted in Fig. \[diff\]. Note that the difference is always negative, since v2DM provides a variational lower bound. As one observes, all approximations describe exactly the non-interacting small-$g$ limit. When $g$ becomes larger, there is competition between different types of ground states and the performance of $PQG$ gets worse up to $g\approx1.4$. For larger $g$ the $PQG$ result becomes better again. In fact, we checked (by omitting the single-particle piece) that also the $g\rightarrow\infty$ limit becomes exact for $PQG$, which is a peculiarity of the schematic pairing force. The $PQGT_1$ results show that the $T_1$ constraint only becomes active around $g = 2.5$, and ensures faster convergence to the exact $g\rightarrow\infty$ limit. Somewhat surprisingly, adding the $T_2$ condition is sufficient for obtaining the exact solution at all values of $g$. Computational Performance ------------------------- Some of the computational aspects of the algorithm are worth pointing out. It is interesting to see *e.g.* that depending on the pairing interaction parameter $g$, the convergence properties of the algorithm change. In Fig. \[nr\_newton\_it\] the joint number of predictor and corrector steps needed for convergence, is plotted as a function of $g$. One observes a sharp peak at fairly small $g$, just when the perturbative regime is left and the structure of the ground state changes. For $g=0.25$, which is at the position of the peak in Fig. \[nr\_newton\_it\], we have plotted in Fig. \[nr\_iter\_bad\] the number of conjugate gradient iterations needed for convergence, of both the dual and the primal linear system, as a function of the primal-dual gap $\eta$. As expected, the number of iterations for the dual problem increases with decreasing primal-dual gap, as the linear system grows ill-conditioned. The primal conjugate gradient loop only becomes active for small values of $\eta$. This signals that the numerical stability becomes too small to generate a high quality approximation for $\Delta_X$ using the $\Delta_Z$ obtained in the dual conjugate gradient loop. Anyway, the needed number of primal iteration remains insignificant compared to the dual ones, for all values of $\eta$. The situation at $g=0.25$ is the worst case. For larger values of $g$, where the number of predictor-corrector steps is smaller and approximately constant (see Fig. \[nr\_newton\_it\]), the number of conjugate gradient iterations is also drastically reduced. A typical behaviour is plotted in Fig. \[nr\_iter\_good\] for $g=4$. ![\[nr\_newton\_it\] Number of predictor and corrector steps needed for convergence, as a function of the pairing strength $g$ in v2DM(PQG).](nr_newton_it.pdf) ![\[nr\_iter\_bad\] Number of primal and dual conjugate gradient iterations needed for convergence, as a function of the primal-dual gap $\eta$ for $g=0.25$ in v2DM(PQG).](nr_iter_bad.pdf) ![\[nr\_iter\_good\] Number of primal and dual conjugate gradient iterations needed for convergence, as a function of the primal-dual gap $\eta$ for $g=4$ in v2DM(PQG).](nr_iter_good.pdf) \[sum\]Summary and discussion ============================= Interacting quantum many-particle systems lie at the heart of most issues in condensed matter, molecular/atomic and nuclear physics. Their analysis may be rephrased as the problem of minimizing the energy, expressed as a linear function of a two-body density matrix, subject to the $N$-representability constraint that the 2DM can be derived from a physical $N$-particle system. By working solely with the 2DM, rather than with the $N$-particle wave function itself, the problem of the exponentially exploding dimension of $N$-particle Hilbert space with increasing $N$ is circumvented. The complexity of the problem is shifted, however, to the characterization of the $N$-particle representable 2DM’s. In practice, a limited set of necessary but not sufficient conditions for $N$-representability are imposed during the minimization, resulting in a strict lower bound to the energy, which converges to the exact energy when more and more $N$-representability conditions are imposed. Commonly used $N$-representability conditions impose the positive semidefiniteness of a set of linear matrix functionals of the 2DM. In this way the quantum many-body problem is converted into a well established field of optimization techniques called semidefinite programming. Standard packages for SDP, however, fail to take into account properties of the physical problem that can be exploited. Using specific mathematical properties of the constraints for the v2DM problem, we have adapted a standard primal-dual interior point method to be computationally cheaper, both in storage as in floating point operations. We make extensize use of the algebra of linear matrix maps to calculate efficiently some intermediate quantities. During the Newton minimization procedure, a new direction in 2DM space is found iteratively using the conjugate gradient algorithm, thereby exploiting the fact that the product of the Hessian with a 2DM is considerably cheaper for the physical problem at hand than in a general situation. As an example we have applied the algorithm to a BCS-type Hamiltonian. We found that the standard constraints work very well for this kind of problem. The computational performance of the method was analyzed, and it was shown that the convergence behaviour is dependent on the value of the pairing strength parameter. As in our previous algorithm [@atomic] the method slows down near the solution, because the matrices involved become ill conditioned. The present primal-dual algorithm allows to control this since the primal-dual gap provides an upper bound to the remaining error. Therefore the algorithm can be stopped when the required accuracy is reached, saving many unnecessary iterations. Acknowledgements ================ We gratefully acknowledge financial support from FWO-Flanders and the research council of Ghent University. We would like to thank Paul W. Ayers for his useful suggestions. B.V., H.V.A., P.B. and D.V.N. are Members of the QCMM alliance Ghent-Brussels. \[overlapmatrix\] Calculation of the overlap-matrix map ======================================================= The overlap matrix of the non-orthogonal basisset $\{u^\alpha\}$ is defined as: $$\mathcal{S}_{\alpha\beta} = \mathrm{Tr}~u^\alpha u^\beta~.$$ Using the Hermitian adjoints of the linear maps $\mathcal{L}$ we can rewrite this as: $$\mathcal{S}_{\alpha\beta} = \sum_k \mathrm{Tr}~\left[\mathcal{L}_k^\dagger\left(\mathcal{L}_k\left(f^\alpha\right)\right) f^\beta\right]~,$$ in which $\{f^\alpha\}$ *is* an orthogonal basis of tp-matrix space. This means that the overlap matrix can be seen as a linear map from tp-space onto itself, whose action onto a tp-matrix $\Gamma$ is: $$\mathcal{S}\left(\Gamma\right) = \sum_k \mathcal{L}_k^\dagger\left(\mathcal{L}_k\left(\Gamma\right)\right)~.$$ It turns out that this map can be written as a generalized $\mathcal{Q}$ map, which is defined as: $$\mathcal{Q}(a,b,c)\left(\Gamma\right)_{\alpha\beta;\gamma\delta} = a\Gamma_{\alpha\beta;\gamma\delta} + b\left(\delta_{\alpha\gamma}\delta_{\beta\delta} - \delta_{\alpha\delta}\delta_{\beta\gamma}\right)\bar{\bar{\Gamma}} - c\left(\delta_{\alpha\gamma}\bar{\Gamma}_{\beta\delta} - \delta_{\beta\gamma}\bar{\Gamma}_{\alpha\delta} - \delta_{\alpha\delta}\bar{\Gamma}_{\beta\gamma} + \delta_{\beta\delta}\bar{\Gamma}_{\alpha\gamma}\right)~. \label{Q_like}$$ This is like a $\mathcal{Q}$-map (\[Q\]) but with general coefficients $(a,b,c)$. The proof is somewhat tedious and proceeds by considering every $\mathcal{L}_k$ separately. $\mathcal{P}^2$ --------------- It is trivial to see that $\mathcal{P}^2(\Gamma) = \Gamma$ and that this is a generalized $\mathcal{Q}$ map with coefficients $$a = 1\qquad b = 0\qquad c = 0~.$$ $\mathcal{Q}^2$ --------------- To reexpress $\mathcal{Q}^2$ we first calculate the various pieces, $$\begin{aligned} \bar{\mathcal{Q}}(\Gamma)_{\alpha\gamma} &=& \left[\frac{M-N-1}{N(N-1)}\right]\delta_{\alpha\gamma}\bar{\bar{\Gamma}} - \left[\frac{M-N-1}{N-1}\right]\bar{\Gamma}_{\alpha\gamma}~,\\ \bar{\bar{\mathcal{Q}}}(\Gamma) &=& \left[\frac{(M-N)(M-N-1)}{N(N-1)}\right]\bar{\bar{\Gamma}}~.\end{aligned}$$ Substitute into Eq. (\[Q\]) leads once again to a generalized $\mathcal{Q}$ map with coefficients: $$a = 1\qquad b = \frac{4N^2 + 2N - 4NM + M^2 -M}{N^2(N-1)^2}\qquad c = \frac{2N-M}{(N-1)^2}~.$$ $\mathcal{G}^\dagger\mathcal{G}$ -------------------------------- With the same strategy one finds on the basis of Eq. (\[G\_down\]) and $$\bar{\mathcal{G}}(\Gamma)_{\alpha\gamma} = \frac{M-1}{N-1}\bar{\Gamma}_{\alpha\gamma}~,$$ that substituting into (\[G\_down\]) leads to another generalized $\mathcal{Q}$ map with coefficients: $$a = 4\qquad b = 0 \qquad c = \frac{2N - M - 2}{(N-1)^2}~.$$ $\mathcal{T}_1^\dagger \mathcal{T}_1$ ------------------------------------- The needed terms are now: $$\begin{aligned} \bar{\mathcal{T}}_1\left(\Gamma\right)_{\alpha\beta;\gamma\delta} &=& (M-4)\Gamma_{\alpha\beta;\gamma\delta} + \left[\frac{M-N-2}{N(N-1)}\right]~(\delta_{\alpha\gamma}\delta_{\beta\delta} - \delta_{\alpha\delta}\delta_{\beta\gamma})\bar{\bar{\Gamma}}~,\\ &&- \left[\frac{M-N-2}{N-1}\right]\hat{A}\left[\delta_{\alpha\gamma}\bar{\Gamma}_{\beta\delta} - \delta_{\beta\gamma}\bar{\Gamma}_{\alpha\delta}-\delta_{\alpha\delta}\bar{\Gamma}_{\beta\gamma} + \delta_{\beta\delta}\bar{\Gamma}_{\alpha\gamma}\right]~,\\ \bar{\bar{\mathcal{T}}}_1\left(\Gamma\right)_{\alpha\gamma} &=& \left[\frac{(M-N-2)(M-N-1)}{N(N-1)}\right]\delta_{\alpha\gamma}\bar{\bar{\Gamma}} - \left[\frac{(M-3)(M-2N)}{N-1}\right]~\bar{\Gamma}_{\alpha\gamma}~,\\ \bar{\bar{\bar{\mathcal{T}}}}_1\left(\Gamma\right) &=& \left[\frac{(M-2)(M(M-1) - 3N(M-N))}{N(N-1)}\right]\bar{\bar{\Gamma}}~,\end{aligned}$$ and substitution into Eq. (\[T1\_down\]) leads to the coefficients: $$\begin{aligned} a &=& M-4~,\\ b &=& \frac{M^3-6M^2N-3M^2+12MN^2+12MN+2M-18N^2-6N^3}{3N^2(N-1)^2}~,\\ c &=& -\frac{M^2 + 2N^2 - 4MN - M + 8N - 4}{2(N-1)^2}~.\end{aligned}$$ $\mathcal{T}_2^\dagger \mathcal{T}_2$ ------------------------------------- Finally, needed for the calculation of the last map are: $$\begin{aligned} \bar{\mathcal{T}}_2\left(\Gamma\right)_{\alpha\beta;\gamma\delta} &=& \frac{\bar{\bar{\Gamma}}}{N - 1}(\delta_{\alpha\gamma}\delta_{\beta\delta} - \delta_{\alpha\delta}\delta_{\beta\gamma}) + M~ \Gamma~,\\ &&- \left[\delta_{\alpha\gamma}\bar{\Gamma}_{\beta\delta} - \delta_{\beta\gamma}\bar{\Gamma}_{\alpha\delta} - \delta_{\alpha\delta}\bar{\Gamma}_{\beta\gamma} + \delta_{\beta\delta}\bar{\Gamma}_{\alpha\gamma}\right]~,\\ \tilde{\mathcal{T}}_2\left(\Gamma\right)_{\alpha\beta;\gamma\delta} &=& \frac{M - N}{N - 1}\bar{\Gamma}_{\beta\delta}\delta_{\alpha\gamma} + \delta_{\beta\delta}\bar{\Gamma}_{\alpha\gamma} - (M - 2)\Gamma_{\alpha\delta;\gamma\beta}~,\\ \tilde{\tilde{\mathcal{T}}}_2\left(\Gamma\right)_{\alpha\gamma}&=& \left[\frac{M(M - N) - (N - 1)(M - 2)}{N - 1}\right]\bar{\Gamma}_{\alpha\gamma} + \delta_{\alpha\gamma}\bar{\bar{\Gamma}}~,\end{aligned}$$ which, when substituted into Eq. (\[T2\_down\]) gives the following coefficients: $$a = 5M - 8\qquad b = \frac{2}{N - 1}\qquad c = \frac{2N^2 + (M - 2)(4N-3) - M^2}{2(N - 1)^2}~.$$ The overlap-matrix map is just the sum of the various terms obtained, and hence also a generalized $\mathcal{Q}$ map with rather complex coefficients. \[inverse\_overlapmatrix\]Inverse of generalized $\mathcal{Q}$ map ================================================================== The inverse of a generalized $\mathcal{Q}$ map can be shown to be another generalized $\mathcal{Q}$ map. Consider for brevity the notation: $$\mathcal{Q}(a,b,c)(\Gamma) = Q~,$$ then applying partial trace operations on Eq. (\[Q\_like\]) leads to: $$\begin{aligned} \bar{\bar{\Gamma}} &=& \frac{\bar{\bar{Q}}}{a + M(M - 1)b - 2(M - 1)c}~,\\ \bar{\Gamma}_{\alpha\gamma} &=& \frac{1}{a - c(M - 2)}\left[\bar{Q}_{\alpha\gamma} - \frac{b(M - 1) - c}{a + M(M - 1)b - 2(M - 1)c}\delta_{\alpha\gamma}\bar{\bar{Q}}\right]~.\end{aligned}$$ Upon substitution into Eq. (\[Q\_like\]) and solving for $\Gamma$ one obtains, $$\Gamma = \mathcal{Q}^{-1}(a,b,c)(Q) = \mathcal{Q}(a',b',c')(Q)~,$$ where $$\begin{aligned} a' &=& \frac{1}{a}~,\\ b' &=& \frac{ba + bcM -2c^2}{a\left[c(M - 2) - a\right]\left[a + bM(M - 1) - 2c(M - 1)\right]}~,\\ c' &=& \frac{c}{a\left[c(M - 2) - a\right]}~.\end{aligned}$$ These are important relations since they allow to evaluate the action of the inverse overlap matrix on a tp matrix as fast as a $\mathcal{Q}$ map. *i.e.* at a computational cost which is negligible compared to the other matrix manipulations.

--- abstract: 'Motivated by change point problems in time series and the detection of textured objects in images, we consider the problem of detecting a piece of a Gaussian Markov random field hidden in white Gaussian noise. We derive minimax lower bounds and propose near-optimal tests.' author: - 'Ery Arias-Castro[^1]' - 'Sébastien Bubeck[^2]' - 'Gábor Lugosi[^3]' - 'Nicolas Verzelen[^4]' bibliography: - 'ref.bib' title: Detecting Markov Random Fields Hidden in White Noise --- Introduction {#sec:intro} ============ Anomaly detection is important in a number of applications, including surveillance and environment monitoring systems using sensor networks, object tracking from video or satellite images, and tumor detection in medical imaging. The most common model is that of an object or signal of unusually high amplitude hidden in noise. In other words, one is interested in detecting the presence of an object in which the mean of the signal is different from that of the background. We refer to this as the *detection-of-means* problem. In many situations, anomaly manifests as unusual dependencies in the data. This *detection-of-correlations* problem is the one that we consider in this paper. Setting and hypothesis testing problem {#sec:setting} -------------------------------------- It is common to model dependencies by a Gaussian random field $\X = (X_i : i \in \cV)$, where $\cV \subset \cV_\infty$ is of size $|\cV| = n$, while $\cV_\infty$ is countably infinite. We focus on the important example of a $d$-dimensional integer lattice \[lattice\] = {1, …, m}\^d \_= \^d. We formalize the task of detection as the following hypothesis testing problem. One observes a realization of $\X = (X_i : i \in \cV)$, where the $X_i$’s are known to be standard normal. Under the null hypothesis $\cH_0$, the $X_i$’s are independent. Under the alternative hypothesis $\cH_1$, the $X_i$’s are correlated in one of the following ways. Let $\cC$ be a class of subsets of $\cV$. Each set $S\in \cC$ represents a possible anomalous subset of the components of $\X$. Specifically, when $S \in \cC$ is the anomalous subset of nodes, each $X_i$ with $i \notin S$ is still independent of all the other variables, while $(X_i : i \in S)$ coincides with $(Y_i : i \in S)$, where $\Y=(Y_i : i \in \cV_\infty)$ is a stationary Gaussian Markov random field. We emphasize that, in this formulation, the anomalous subset $S$ is only known to belong to $\cC$. We are thus addressing the problem of detecting a region of a Gaussian Markov random field against a background of white noise. This testing problem models important detection problems such as the detection of a piece of a time series in a signal and the detection of a textured object in an image, which we describe below. Before doing that, we further detail the model and set some foundational notation and terminology. Tests and minimax risk {#sec:tests} ---------------------- We denote the distribution of $\X$ under $\cH_0$ by $\PROB_0$. The distribution of the zero-mean stationary Gaussian Markov random field $\Y$ is determined by its covariance operator $\bGamma=(\bGamma_{i,j} : i,j\in \cV_\infty)$ defined by $\bGamma_{i,j}=\EXP[Y_i Y_j]$. We denote the distribution of $\X$ under $\cH_1$ by $\PROB_{S,\bGamma}$ when $S\in \cC$ is the anomalous set and $\bGamma$ is the covariance operator of the Gaussian Markov random field $\Y$. A *test* is a measurable function $f: \bbR^\cV \to \{0,1\}$. When $f(\X)=0$, the test accepts the null hypothesis and it rejects it otherwise. The probability of *type I* error of a test $f$ is $\PROB_0\{f(\X)=1\}$. When $S\in \cC$ is the anomalous set and $\Y$ has covariance operator $\bGamma$, the probability of *type II* error is $\PROB_{S,\bGamma}\{f(\X)=0\}$. In this paper we evaluate tests based on their *worst-case risks*. The risk of a test $f$ corresponding to a covariance operator $\bGamma$ and class of sets $\cC$ is defined as \[risk-known-f\] R\_[,]{}(f) = \_0{f()=1} + \_[S ]{} \_[S,]{}{f()=0} . Defining the risk this way is meaningful when the distribution of $\Y$ is known, meaning that $\bGamma$ is available to the statistician. In this case, the minimax risk is defined as \[risk-known\] R\^\*\_[,]{} = \_f R\_[,]{}(f) , where the infimum is over all tests $f$. When $\bGamma$ is only known to belong to some class of covariance operators $\mathfrak{G}$, it is more meaningful to define the risk of a test $f$ as \[risk-unknown-f\] R\_[,]{}(f) = \_0{f()=1} + \_ \_[S ]{} \_[S,]{}{f()=0} . The corresponding minimax risk is defined as \[risk-unknown\] R\_[,]{}\^\* = \_f R\_[,]{}(f) . In this paper we consider situations in which the covariance operator $\Gamma$ is known (i.e., the test $f$ is allowed to be constructed using this information) and other situations when $\Gamma$ is unknown but it is assumed to belong to a class $\mathfrak{G}$. When $\bGamma$ is known (resp. unknown), we say that a test $f$ *asymptotically separates the two hypotheses* if $R_{\cC,\bGamma}(f) \to 0$ (resp. $R_{\cC,\mathfrak{G}}(f) \to 0$), and we say that the hypotheses *merge asymptotically* if $R_{\cC,\bGamma}^* \to 1$ (resp. $R_{\cC,\mathfrak{G}}^* \to 1$), as $n = |\cV| \to \infty$. We note that, as long as $\bGamma \in \mathfrak{G}$, $R_{\cC,\bGamma}^* \le R_{\cC,\mathfrak{G}}^*$, and that $R_{\cC,\mathfrak{G}}^* \le 1$, since the test $f \equiv 1$ (which always rejects) has risk equal to $1$. At a high-level, our results are as follows. We characterize the minimax testing risk for both known ($R^*_{\cC,\bGamma}$) and unknown ($R_{\cC,\mathfrak{G}}^*$) covariances when the anomaly is a Gaussian Markov random field. More precisely, we give conditions on $\bGamma$ or $\mathfrak{G}$ enforcing the hypotheses to merge asymptotically so that detection problem is nearly impossible. Under nearly matching conditions, we exhibit tests that asymptotically separate the hypotheses. Our general results are illustrated in the following subsections. Example: detecting a piece of time series {#sec:intro-tseries} ----------------------------------------- As a first example of the general problem described above, consider the case of observing a time series $X_1,\ldots,X_n$. This corresponds to the setting of the lattice in dimension $d=1$. Under the null hypothesis, the $X_i$’s are i.i.d. standard normal random variables. We assume that the anomaly comes in the form of temporal correlations over an (unknown) interval $S = \{i+1, \dots, i+k\}$ of, say, known length $k<n$. Here, $i\in \{0,1\ldots,n-k\}$ is thus unknown. Specifically, when $S$ is the anomalous interval, $(X_{i+1}, \dots, X_{i+k}) \sim (Y_{i+1}, \dots, Y_{i+k})$, where $(Y_i: i \in \bbZ)$ is an autoregressive process of order $h$ (abbreviated $\ar_h$) with zero mean and unit variance, that is, \[ARp\] Y\_i = \_1 Y\_[i-1]{} + + \_h Y\_[i-h]{} + Z\_i, i , where $(Z_i: i \in \bbZ)$ are i.i.d. standard normal random variables, $\psi_1, \dots, \psi_h \in \bbR$ are the coefficients of the process—assumed to be stationary—and $\sigma>0$ is such that $\Var(Y_i) = 1$ for all $i$. Note that $\sigma$ is a function of $\psi_1, \dots, \psi_h$, so that the model has effectively $h$ parameters. It is well-known that the parameters $\psi_1,\ldots,\psi_h$ define a stationary process when the roots of the polynomial $z^p - \sum_{i=1}^p \psi_i z^{p-i}$ in the complex plane lie within the open unit circle. See [@MR1093459] for a standard reference on time series. In the simplest setting $h=1$ and the parameter space for $\psi$ is $(-1,1)$. Then, the hypothesis testing problem is to distinguish $$\cH_0: X_1, \dots, X_n \iid \cN(0,1),$$ versus $$\cH_1: \exists i \in \{0,1,\ldots,n-k\} \text{ such that }$$ $$X_1, \dots, X_i, X_{i+k+1}, \dots, X_n \iid \cN(0,1)$$ and $(X_{i+1}, \dots, X_{i+k})$ is independent of $X_1, \dots, X_i, X_{i+k+1}, \dots, X_n$ with $$%X_{i+1} \sim \cN(0,1), \quad X_{i+j+1} - \psi X_{i+j} \iid \cN(0,1-\psi^2), \quad \forall j \in \{1, \dots, k-1\}\ .$$ Typical realizations of the observed vector under the null and alternative hypotheses are illustrated in . ![Top: a realization of the observed time series under the null hypothesis (white noise). Bottom: a realization under the alternative with anomalous interval $S = \{201, \dots, 250\}$, assuming an $\ar_1$ covariance model with parameter $\psi = 0.9$.[]{data-label="fig:timeseries"}](tseries-H0 "fig:"){width="0.9\linewidth"}\ ![Top: a realization of the observed time series under the null hypothesis (white noise). Bottom: a realization under the alternative with anomalous interval $S = \{201, \dots, 250\}$, assuming an $\ar_1$ covariance model with parameter $\psi = 0.9$.[]{data-label="fig:timeseries"}](tseries-H1 "fig:"){width="0.9\linewidth"} Gaussian autoregressive processes and other correlation models are special cases of Gaussian Markov random fields, and therefore this setting is a special case of our general framework, with $\cC$ being the class of discrete intervals of length $k$. In the simplest case, the length of the anomalous interval is known beforehand. In more complex settings, it is unknown, in which case $\cC$ may be taken to be the class of all intervals within $\cV$ of length at least $k_{\rm min}$. This testing problem has been extensively studied in the slightly different context of change-point analysis, where under the null hypothesis $X_1, \dots, X_n$ are generated from an $\ar_h(\psi^0)$ process for some $\psi^0\in \mathbb{R}^h$, while under the alternative hypothesis there is an $i \in \cV$ such that $X_1, \dots, X_i$ and $X_{i+1}, \dots, X_n$ are generated from $\ar_h(\psi^0)$ and $\ar_h(\psi^1)$, with $\psi^0 \ne \psi^1$, respectively. The order $h$ is often given. In fact, instead of assuming autoregressive models, nonparametric models are often favored. See, for example, [@MR0269062; @MR1331669; @springerlink:10.1007/BF00970969; @MR1791905; @MR2597589; @MR2301477; @MR809433; @MR1200409] and many other references therein. These papers often suggest maximum likelihood tests whose limiting distributions are studied under the null and (sometimes fixed) alternative hypotheses. For example, in the special case of $h=1$, such a test would reject $\cH_0$ when $|\hat \psi|$ is large, where $\hat \psi$ is the maximum likelihood estimate for $\psi$. In particular, from [@MR809433], we can speculate that such a test can asymptotically separate the hypotheses in the simplest setting described above when $\psi k^\alpha \to \infty$ for some $\alpha < 1/2$ fixed. See also [@MR2301477; @MR2597589] for power analyses against fixed alternatives. Our general results imply the following in the special case when the anomaly comes in the form of an autoregressive process with unknown parameter $\psi \in \bbR^h$. We note that the order of the autoregressive model $h$ is allowed to grow with $n$ in this asymptotic result. \[cor:AR\] Assume $n, k \to \infty$, and that $h = o\big(\sqrt{k/\log(n)}\wedge k^{1/4}\big)$. Denote by $\mathfrak{F}(h,r)$ the class of covariance operators corresponding to $AR_h$ processes with valid parameter $\psi=(\psi_1,\ldots, \psi_h)$ satisfying $\|\psi\|_2^2\geq r^2$. Then $R_{\cC,\mathfrak{F}(h,r)}^* \to 1$ when \[AR1\] r\^2 C\_1 ((n/k)/k + /k) . Conversely, if $f$ denotes the pseudo-likelihood test of Section \[sec:fisher\], then $R_{\cC,\mathfrak{F}(h,r)}(f) \to 0$ when \[AR2\] r\^2 C\_2 ((n)/k + /k) . In both cases, $C_1$ and $C_2$ denote numerical constants. In the interesting setting where $k = n^\kappa$ for some $\kappa > 0$ fixed, the lower and upper bounds provided by match up to a multiplicative constant that depends only on $\kappa$. Despite an extensive literature on the topic, we are not aware of any other minimax optimality result for time series detection. Example: detecting a textured region {#sec:intro-images} ------------------------------------ In image processing, the detection of textured objects against a textured background is relevant in a number of applications, such as in the detection of local fabric defects in the textile industry by automated visual inspection [@4418522], the detection of a moving object in a textured background [@yilmaz2006object; @Kim2005172], the identification of tumors in medical imaging [@1229852; @breast-tumor], the detection of man-made objects in natural scenery [@10.1109/CVPR.2003.1211345], the detection of sites of interest in archeology [@litton] and of weeds in crops [@MR1959083]. In all these applications, the object is generally small compared to the size of the image. Common models for texture include Markov random fields [@4767341] and joint distributions over filter banks such as wavelet pyramids [@531803; @springerlink:10.1023/A:1026553619983]. We focus here on textures that are generated via Gaussian Markov random fields [@1164641; @springerlink:10.1023/A:1007925832420]. Our goal is to detect a textured object hidden in white noise. For this discussion, we place ourselves in the lattice setting in dimension $d=2$. Just like before, under $\cH_0$, the $(X_i : i\in \cV)$ are independent standard normal random variables. Under $\cH_1$, when the region $S \subset \cV$ is anomalous, the $(X_i : i\notin S)$ are still i.i.d. standard normal, while $(X_i : i \in S) \sim (Y_i : i \in S)$, where $(Y_i: i \in \bbZ^2)$ is such that for each $i\in \bbZ^2$, the conditional distribution of $Y_i$ given the rest of the variables $Y^{(-i)} := (Y_j : j \ne i)$ is normal with mean \[markov-lattice\] \_[(t\_1,t\_2)\^2{(0,0)}]{} \_[t\_1, t\_2]{} Y\_[i + (t\_1, t\_2)]{} and variance $\sigma_{\phi}^2$, where the $\phi_{t_1,t_2}$’s are the coefficients of the process and $\sigma_{\phi}$ is such that $\Var(Y_i) = 1$ for all $i$. The set of valid parameters $\phi$ is defined in Section \[sec:prelim\_gmrf\]. A simple sufficient condition is $\|\phi\|_1= \sum_{(t_1,t_2)\in [-h,h]^2\setminus \{(0,0)\}} |\phi_{t_1, t_2}|<1$. In this model, the *dependency neighborhood* of $i \in \bbZ^2$ is $i + [-h,h]^2 \cap \bbZ^2$. One of the simplest cases is when $h=1$ and $\phi_{t_1, t_2} = \phi$ when $(t_1,t_2) \in \{(\pm1,0), (0,\pm1)\}$ for some $\phi \in (-1/4, 1/4)$, and the anomalous region is a discrete square; see Figure \[fig:texture\] for a realization of the resulting process. This is a special case of our setting. While intervals are natural in the case of time series, squares are rather restrictive models of anomalous regions in images. We consider instead the “blob-like” regions (to be defined later) that include convex and star-shaped regions. ![Left: white noise, no anomalous region is present. Right: a squared anomalous region is present. In this example on the $50 \times 50$ grid, the anomalous region is a $15 \times 15$ square piece from a Gaussian Markov random field with neighborhood radius $h=1$ and coefficient vector $\phi_{t_1, t_2} = \phi := \frac14(1-10^{-4})$ when $(t_1,t_2) \in \{(\pm1,0), (0,\pm1)\}$, and zero otherwise.[]{data-label="fig:texture"}](texture-H0 "fig:"){width=".30\linewidth"}   ![Left: white noise, no anomalous region is present. Right: a squared anomalous region is present. In this example on the $50 \times 50$ grid, the anomalous region is a $15 \times 15$ square piece from a Gaussian Markov random field with neighborhood radius $h=1$ and coefficient vector $\phi_{t_1, t_2} = \phi := \frac14(1-10^{-4})$ when $(t_1,t_2) \in \{(\pm1,0), (0,\pm1)\}$, and zero otherwise.[]{data-label="fig:texture"}](texture-H1 "fig:"){width=".30\linewidth"}   A number of publications address the related problems of texture classification [@kervrann1995markov; @springerlink:10.1023/A:1007925832420; @varma05] and texture segmentation [@jain1991unsupervised; @buhmann98; @MR1966045; @sharon; @malik-belongie-al]. In fact, this literature is quite extensive. Only very few papers address the corresponding change-point problem [@shahrokni; @palenichka:158] and we do not know of any theoretical results in this literature. Our general results (in particular, Corollary \[cor:lower\_hypercube\]) imply the following. \[cor:image\] Assume $n, k \to \infty$, and that $h = o\big(\sqrt{k/\log(n)}\wedge k^{1/5}\big)$. Denote by $\mathfrak{G}(h,r)$ the class of covariance operators corresponding to stationary Gaussian Markov Random Fields with valid parameter (see Section \[sec:prelim\_gmrf\] for more details) $\phi=(\phi_{i,j})_{(i,j)\in \{-h,\ldots, h\}^2\setminus \{0\}}$ satisfying $\|\phi\|_2^2\geq r^2$. Then $R_{\cC,\mathfrak{G}(h,r)}^* \to 1$ when \[CAR1\] r\^2 C\_1 . Conversely, if $f$ denotes the pseudo-likelihood test of Section \[sec:fisher\], then $R_{\cC,\mathfrak{G}(h,r)}(f) \to 0$ when \[CAR2\] r\^2 C\_2  . In both cases, $C_1$ and $C_2$ denote positive numerical constants. Informally, the lower bound on the magnitude of the coefficient vector $\phi$, namely $r^2$, quantifies the extent to which the variables $Y_i$ are explained by the rest of variables $Y^{(-i)}$ as in . Although not in the literature on change-point or object detection, [@anandkumar2009detection] is the only other paper developing theory in a similar context. It considers a spatial model where points $\{x_i, i \in [N]\}$ are sampled uniformly at random in some bounded region and a nearest-neighbor graph is formed. On the resulting graph, variables are observed at the nodes. Under the (simple) null hypothesis, the variables are i.i.d. zero mean normal. Under the (simple) alternative, the variables arise from a Gaussian Markov random with covariance operator of the form $\Gamma_{i,j} \propto g(\|x_i - x_j\|)$, where $g$ is a known function. The paper analyzes the large-sample behavior of the likelihood ratio test. More related work ----------------- As we mentioned earlier, the detection-of-means setting is much more prevalent in the literature. When the anomaly has no a priori structure, the problem is that of multiple testing; see, for example, [@Ingster99; @baraud; @MR2065195] for papers testing the global null hypothesis. Much closer to what interests us here, the problem of detecting objects with various geometries or combinatorial properties has been extensively analyzed, for example, in some of our earlier work [@maze; @combin; @cluster] and elsewhere [@MR2604703; @morel]. We only cite a few publications that focus on theory. The applied literature is vast; see [@cluster] for some pointers. Despite its importance in practice, as illustrated by the examples and references given in Sections \[sec:intro-tseries\] and \[sec:intro-images\], the detection-of-correlations setting has received comparatively much less attention, at least from theoreticians. Here we find some of our own work [@correlation-detect; @multidim]. In the first of these papers, we consider a sequence $X_1, \dots, X_n$ of standard normal random variables. Under the null, they are independent. Under the alternative, there is a set $S$ in a class of interest $\cC$ where the variables are correlated. We consider the unstructured case where $\cC$ is the class of all sets of size $k$ (given) and also various structured cases, and in particular, that of intervals. This would appear to be the same as in the present lattice setting in dimension $d=1$, but the important difference is that that correlation operator $\bGamma$ is not constrained, and in particular no Markov random field structure is assumed. The second paper extends the setting to higher dimensions, thus testing whether some coordinates of a high-dimensional Gaussian vector are correlated or not. When the correlation structure in the anomaly is arbitrary, the setting overlaps with that of sparse principal component analysis [@berthet; @cai2013sparse]. The problem is also connected to covariance testing in high-dimensions; see, e.g., [@cai2013optimal]. We refer the reader to the above-mentioned papers for further references. Contribution and content ------------------------ The present paper thus extends previous work on the detection-of-means setting to the detection-of-correlations setting in the (structured) context of detecting signals/objects in time series/images. The paper also extends some of our own work on the detection-of-correlations to Markov random field models, which are typically much more appropriate in the context of detection in signals and images. The theory in the detection-of-correlations setting is more complicated than in the the detection-of-means setting, and in particular deriving exact minimax (first-order) results remains an open problem. Compared to our previous work on the detection-of-correlations setting, the Markovian assumption makes the problem significantly more complex as it requires handling Markov random fields which are conceptually more complex objects. As a result, the proof technique is by-and-large novel, at least in the detection literature. The rest of the paper is organized as follows. In we lay down some foundations on Gaussian Markov Random Fields, and in particular, their covariance operators, and we also derive a general minimax lower bound that is used several times in the paper. In the remainder of the paper, we consider detecting correlations in a finite-dimensional lattice , which includes the important special cases of time series and textures in images. We establish lower bounds, both when the covariance matrix is known (Section \[sec:known\]) or unknown (Section \[sec:unknown\]) and propose test procedures that are shown to achieve the lower bounds up to multiplicative constants. In Section \[sec:cubes\], we specialize our general results to specific classes of anomalous regions such as classes of cubes, and more generally, “blobs.” In we outline possible generalizations and further work. The proofs are gathered in . Preliminaries {#sec:prelim} ============= In this paper we derive upper and lower bounds for the minimax risk, both when $\bGamma$ is known as in and when it is unknown as in , the latter requiring a substantial amount of additional work. For the sake of exposition, we sketch here the general strategy for obtaining minimax lower bounds by adapting the general strategy initiated in [@ingster93a] to detection-of-correlation problems. This allows us to separate the technique used to derive minimax lower bounds from the technique required to handle Gaussian Markov random fields. Some background on Gaussian Markov random fields {#sec:prelim_gmrf} ------------------------------------------------ We elaborate on the setting described in Sections \[sec:setting\] and \[sec:tests\]. As the process $Y$ is indexed by $\mathbb{Z}^d$, note that all the indices $i$ of $\phi$ and $\bGamma$ are $d$-dimensional. Given a positive integer $h$, denote by $\bbN_h$ the integer lattice $\{-h, \dots, h\}^d \setminus\{0\}^d$ with $(2h+1)^d-1$ nodes. For any nonsingular covariance operator $\bGamma$ of a stationary Gaussian Markov random field over $\mathbb{Z}^d$ with unit variance and neighborhood $\bbN_h$, there exists a unique vector $\phi$ indexed by the nodes of $\bbN_h$ satisfying $\phi_i=\phi_{-i}$ such that, for all $i,j\in \bbZ^d$, \^[-1]{}\_[i,j]{}/\^[-1]{}\_[i,i]{}= - \_[i-j]{}& 1|i-j|\_h,\ 1 &i=j,\ 0 & \[eq:conditions\] where $\bGamma^{-1}$ denotes the inverse of the covariance operator $\bGamma$. Consequently, there exists a bijective map from the collection of invertible covariance operators of stationary Gaussian Markov random fields over $\mathbb{Z}^d$ with unit variance and neighborhood $\bbN_h$ to some subset $\Phi_h\subset \bbR^{\bbN_h}$. Given $\phi\in \Phi_h$, $\bGamma(\phi)$ denotes the unique covariance operator satisfying $\bGamma_{i,i}=1$ and . It is well known that $\Phi_h$ contains the set of vectors $\phi$ whose $\ell_1$-norm is smaller than one, that is, $$\{\phi \in \bbR^{\bbN_h} : \|\phi\|_1 < 1\} \subset \Phi_h\ ,$$ as the corresponding operator $\bGamma^{-1}(\phi)$ is diagonally dominant in that case. In fact, the parameter space $\Phi_h$ is characterized by the Fast Fourier Transform (FFT) as follows $$\Phi_h=\Big\{\phi:\quad 1+ \sum_{1\leq |i|_{\infty}\leq h } \phi_i \cos(\langle i,\omega\rangle) >0,\quad \forall \omega \in (-\pi,\pi]^{d}\Big\}\ ,$$ where and $i\in \mathbb{Z}^d$ and $\langle \cdot , \cdot\rangle$ denotes the scalar product in $\mathbb{R}^d$. The interested reader is referred to [@MR1344683 Sect.1.3] or [@MR2130347 Sect.2.6] for further details and discussions. For $\phi\in \Phi_h$, define $\sigma^{2}_{\phi}=1/\bGamma _{i,i}^{-1}(\phi)$. The correlated process $Y=(Y_i : i\in \bbZ^d)$ is centered Gaussian with covariance operator $\bGamma(\phi)$ is such that, for each $i\in \mathbb{Z}^d$, the conditional distribution of $Y_i$ given the rest of the variables $Y^{(-i)}$ is \[eq:conditional\_definition\] Y\_[i]{}|Y\^[(-i)]{}  \~ (\_[j\_h]{} \_[j]{}Y\_[i+j]{}, \^2\_)  . Define the *$h$-boundary* of $S$, denoted $\Delta_h(S)$, as the collection of vertices in $S$ whose distance to $\bbZ^d\setminus S$ is at most $h$. We also define the *$h$-interior* $S$ as $S^h = S\setminus \Delta_h(S)$. If $S\subset \cV$ is a finite set, we denote by $\bGamma_{S}$ the principal submatrix of the covariance operator $\bGamma$ indexed by $S$. If $\bGamma$ is nonsingular, each such submatrix is invertible. A general minimax lower bound ----------------------------- As is standard, an upper bound is obtained by exhibiting a test $f$ and then upper-bounding its risk—either or according to whether $\bGamma$ is known or unknown. In order to derive a lower bound for the minimax risk, we follow the standard argument of choosing a prior distribution on the class of alternatives and then lower-bounding the minimax risk with the resulting *average risk*. When $\bGamma$ is known, this leads us to select a prior on $\cC$, denoted by $\nu$, and consider \[risk-nu-gamma\] |[R]{}\_[,]{}(f) = \_0{f()=1} + \_[S]{} (S) \_[S,]{}{f()=0} |[R]{}\^\*\_[,]{} = \_f |[R]{}\_[,]{}(f) . The latter is the *Bayes risk* associated with $\nu$. By placing a prior on the class of alternative distributions, the alternative hypothesis becomes effectively simple (as opposed to composite). The advantage of this is that the optimal test may be determined explicitly. Indeed, the Neyman-Pearson fundamental lemma implies that the likelihood ratio test $f_{\nu,\bGamma}^*(x) = \IND{L_{\nu,\bGamma}(x) > 1}$, with $$L_{\nu,\bGamma} = \sum_{S\in \cC} \nu(S) \frac{{\rm d}\PROB_{S,\bGamma}}{{\rm d}\PROB_0}~,$$ minimizes the average risk. In most of the paper, $\nu$ will be chosen as the uniform distribution on the class $\cC$. In this because the sets in $\cC$ play almost the same role (although not exactly because of boundary effects). When $\bGamma$ is only known to belong to some class $\mathfrak{G}$ we also need to choose a prior on $\mathfrak{G}$, which we denote by $\pi$, leading to \[risk-nu-pi\] |[R]{}\_[,]{}(f) = \_0{f()=1} + \_[S]{} (S) \_[S,]{}{f()=0} ([d]{}) |[R]{}\^\*\_[,]{} = \_f |[R]{}\_[,]{}(f) . In this case, the likelihood ratio test becomes $f_{\nu,\pi}^*(x) = \IND{L_{\nu,\pi}(x) > 1}$, where $$L_{\nu,\pi} = \sum_{S\in \cC} \nu(S) \frac{{\rm d}\PROB_{S,\pi}}{{\rm d}\PROB_0}~, \quad \PROB_{S,\pi} = \int \PROB_{S,\bGamma} \pi({\rm d}\bGamma)~,$$ minimizes the average risk. In both cases, we then proceed to bound the second moment of the resulting likelihood ratio under the null. Indeed, in a general setting, if $L$ is the likelihood ratio for $\P_0$ versus $\P_1$ and $R$ denotes its risk, then [@TSH Problem 3.10] \[LR-risk-general\] R = 1 - 12 \_0 | L(X) - 1 | 1 - 12  , where the inequality follows by the Cauchy-Schwarz inequality. Working with the minimax risk (as we do here) allows us to bypass making an explicit choice of prior, although one such choice is eventually made when deriving a lower bound. Another advantage is that the minimax risk is monotone with respect to the class $\cC$ in the sense that if $\cC'\subset \cC$, then the minimax risk corresponding to $\cC'$ is at most as large as that corresponding to $\cC$. This monotonicity does not necessarily hold for the Bayes risk. See [@combin] for a discussion in the context of the detection-of-means problem. We now state a general minimax lower bound. (Recall that all the proofs are in .) Although the result is stated for a class $\cC$ of disjoint subsets, using the monotonicity of the minimax risk, the result can be used to derive lower bounds in more general settings. It is particularly useful in the context of detecting blob-like anomalous regions in the lattice. (The same general approach is also fruitful in the detection-of-means setting.) We emphasize that this result is quite straightforward given the work flow outlined above. The technical difficulties will come with its application to the context that interest us here, which will necessitate a good control of below. Recall the definition . \[prp:non\_overlap\_parametric\] Let $\{\bGamma(\phi): \phi\in \Phi\}$ be a class of nonsingular covariance operators and let $\cC$ be a class of disjoint subsets of $\cV$. Put the uniform prior $\nu$ on $\cC$ and let $\pi$ be a prior on $\Phi$. Then $$\bar{R}^*_{\nu,\pi} \ge 1 - \frac{1}{2|\cC| } \Big(\sum_{S\in \cC} V_S\Big)^{1/2}\ ,$$ where \[V\_S\] V\_S := \_  , and the expected value is with respect to $\phi_1,\phi_2$ drawn i.i.d. from the distribution $\pi$. Known covariance {#sec:known} ================ We start with the case where the covariance operator $\bGamma$ is known. Although this setting is of less practical importance, as this operator is rarely known in applications, we treat this case first for pedagogical reasons and also to contrast with the much more complex setting where the operator is unknown, treated later on. Lower bound {#sec:lower-known} ----------- Recall the definition of the minimax risk and the average risk . (Henceforth, to lighten the notation, we replace subscripts in $\bGamma(\phi)$ with subscripts in $\phi$.) For any prior $\nu$ on $\cC$, the minimax risk is at least as large as the $\nu$-average risk, $R^*_{\cC,\phi} \ge \bar{R}^*_{\nu,\phi}$, and the following corollary of Proposition \[prp:non\_overlap\_parametric\] provides a lower bound on the latter. \[cor:non\_overlap\_GMRF\] Let $\cC$ be a class of disjoint subsets of $\cV$ and fix $\phi\in \Phi_h$ satisfying $\|\phi\|_1<1/2$. Then, letting $\nu$ denote the uniform prior over $\cC$, we have \[eq:lower\_R\*\_GMRF1\] |[R]{}\^\*\_[,]{} 1 - \^[1/2]{} . In particular, the corollary implies that, for any fixed $a\in (0,1)$, $R^*_{\cC,\phi} \ge 1-a$ as soon as \[eq:lower\_R\*\_GMRF1-2\] \_[S]{} . Furthermore, the hypotheses merge asymptotically (i.e., $R^*_{\cC,\phi} \to 1$) when \[nonoverlap3\] (||) - \_[S ]{} |S|  . The condition $\|\phi\|_1<1/2$ in is technical and likely an artifice of our proof method. This condition arises from the term $\det^{-1/2}(2\bGamma_S(\phi)-\bI_S)$ in $V_S$ in . For this determinant to be positive, the smallest eigenvalue of $\bGamma_S(\phi)$ has to be larger than $1/2$, which in turn is enforced by $\|\phi\|_1<1/2$. In order to remove, or at least improve on this constraint, we would need to adopt a more subtle approach than applying the Cauchy-Schwarz inequality in . We did not pursue this as typically one is interested in situations where $\phi$ is small — see, for example, how the result is applied in . Upper bound: the generalized likelihood ratio test {#sec:glrt} -------------------------------------------------- When the covariance operator $\bGamma(\phi)$ is known, the generalized likelihood ratio test rejects the null hypothesis for large values of $$\max_{S \in \cC} \ X_S^\top (\bI_S - \bGamma_S^{-1}(\phi)) X_S~.$$ We use instead the statistic \[U\] U(X) = \_[S ]{}   , which is based on the centering and normalization the statistics $X_S^\top (\bI_S - \bGamma_S^{-1}(\phi)) X_S$ where $S\in \cC$. In the following result, we implicitly assume that $|\cC| \to \infty$, which is the most interesting case. \[prp:glrt\] Assume that $\phi\in \Phi_h$ satisfies $\|\phi\|_1 \le \eta < 1$ and that $|S^h|\geq |S|/2$. The test $f(x)=\IND{U(x) > 4}$ has risk $R_{\cC,\phi}(f) \le 2/|\cC|$ when \[glrt4\] \_2\^2 \_[S ]{} |S| C\_0 (||) , where $C_0 > 0$ only depends on the dimension $d$ of the lattice and $\eta$. Comparing with Condition , we see that condition matches (up to constants) the minimax lower bound, so that (at least when $\|\phi\|_1 < 1/2$) the normalized generalized likelihood ratio test based on is asymptotically minimax up to a multiplicative constant. The $\ell_1$-norm $\|\phi\|_1$ arises in the proof of Corollary \[cor:non\_overlap\_GMRF\] when bounding the largest eigenvalue of $\bGamma(\phi)$ (see Lemma \[lem:spectrum\_gamma\]). Unknown covariance {#sec:unknown} ================== We now consider the case where the covariance operator $\bGamma(\phi)$ of the anomalous Gaussian Markov random field is unknown. We therefore start by defining a class of covariance operators via a class of vectors $\phi$. Given a positive integer $h>0$ and some $r>0$, define \[eq:definition\_phi\_h\_r\] \_h(r) := {\_h, \_2 r} , and let \[eq:G\] (h,r) := {() : \_h(r)} , which is the class of covariance operators corresponding to stationary Gaussian Markov Random Fields with parameter in the class . Lower bound {#sec:lower-unknown} ----------- The theorem below establishes a lower bound for the risk following the approach outlined in , which is based on the choice of a suitable prior $\pi$ on $\Phi_h$, defined as follows. By symmetry of the elements of $\Phi_h$, one can fix a sublattice $\bbN'_h$ of size $|\bbN_h|/2$ such that any $\phi\in\Phi_h$ is uniquely defined (via symmetry) by its restriction to $\bbN'_h$. Choose the distribution $\pi$ such that $\phi\sim \pi$ is the unique extension to $\bbN_h$ of the random vector $ r|\bbN_h|^{-1/2}\xi$, where the coordinates of the random vector $\xi$—indexed by $\bbN'_h$—are i.i.d. Rademacher random variables (i.e., symmetric $\pm 1$-valued random variables). Note that, if $r|\bbN_h|<1$, $\pi$ is acceptable since it concentrates on the set $\{\phi\in \Phi_h,\ \|\phi\|_2 = r\} \subset \Phi_h(r)$. Recall the definition of the minimax risk and the average risk . As before, for any priors $\nu$ on $\cC$ and $\pi$ on $\Phi_h(r)$, the minimax risk is at least as large as the average risk with these priors, $R^*_{\cC,\mathfrak{G}(h,r)} \ge \bar{R}^*_{\nu,\pi}$, and the following (much more elaborate) corollary of Proposition \[prp:non\_overlap\_parametric\] provides a lower bound on the latter. \[thrm3:non\_overlap\_GMRF\] There exists a constant $C_0>0$ such that the following holds. Let $\cC$ be a class of disjoint subsets of $\cV$ and let $\nu$ denote the uniform prior over $\cC$. Let $a\in (0,1)$ and assume that the neighborhood size $|\bbN_h|$ satisfies \[eq:condition\_Nh\] |\_h|\_[S]{} . Then $ \bar{R}^*_{\nu,\pi} \ge 1-a $ as soon as \[eq:condition\_r2\_lower\] r\^2 \_[S ]{} |S| C\_0  . This bound is our main impossibility result. Its proof relies on a number auxiliary results for Gaussian Markov Random Fields (Section \[sec:proofs\_technique\]) that may useful for other problems of estimating Gaussian Markov Random Fields. Notice that the second term in is what appears in , which we saw arises in the case where the covariance is known. In light of this fact, we may interpret the first term in as the ‘price to pay’ for adapting to an unknown covariance operator in the class of covariance operators of Gaussian Markov random fields with dependency radius $h$. Upper bound: a Fisher-type test {#sec:fisher} ------------------------------- We introduce a test whose performance essentially matches the minimax lower bound of Theorem \[thrm3:non\_overlap\_GMRF\]. Comparatively, the construction and analysis of this test is much more involved than that of the generalized likelihood ratio test of . Let $F_i = (X_{i+v} : 1 \le |v|_{\infty} \le h)$, seen as a vector, and let $\bF_{S,h}$ be the matrix with row vectors $F_i, i \in S^h$. Also, let $X_{S,h} = (X_i : i \in S^h)$. Under the null hypothesis, each variable $X_i$ is independent of $F_i$, although $X_i$ is correlated with some $(F_j, j\neq i)$. Under the alternative hypothesis, there exists a subset $S$ and a vector $\phi\in \Phi_h$ such that \[eq:conditional\] X\_[S,h]{}= \_[S,h]{}+ \_[S,h]{} , where each component $\epsilon_i$ of $\epsilon_{S,h}$ is independent of the corresponding vector $F_{i}$, but the $\epsilon_i$’s are not necessarily independent. Equation is the so-called conditional autoregressive (CAR) representation of a Gaussian Markov random field [@MR1344683]. For Gaussian Markov random fields, the celebrated pseudo-likelihood method [@besag:1975] amounts to estimating $\phi$ by taking least-squares in . Returning to our testing problem, observe that the null hypothesis is true if and only if all the parameters of the conditional expectation of $X_{S,h}$ given $\bF_{S,h}$ are zero. In analogy with the analysis-of-variance approach for testing whether the coefficients of a linear regression model are all zero, we consider a Fisher-type statistic \[phi-stat\] T\^\*= \_[S]{}T\_S , T\_S :=  , where $\boldsymbol{\Pi}_{S,h} := \bF_{S,h}(\bF_{S,h}^\top \bF_{S,h})^{-1}\bF^\top _{S,h}$ is the orthogonal projection onto the column space of $\bF_{S,h}$. Since in the linear model the response vector $X_{S,h}$ is not independent of the design matrix $\bF_{S,h}$, the statistic $T_S$ does not follow an $F$-distribution. Nevertheless, we are able to control the deviations of $T^*$, both under null and alternative hypotheses, leading to the following performance bound. Recall the definition . \[thm:LS1\] There exist four positive constants $C_1,C_2,C_3,C_4$ depending only on $d$ such that the following holds. Assume that \[eq:condition\_LS\] |\_h|\^4|\_h|\^2 (||) C\_1 \_[S]{}|S\^h| . Fix $\alpha$ and $\beta$ in $(0,1)$ such that \[alpha\_beta\] ()()C\_2  . Then, under the null hypothesis, \[eq:upper\_TS\_H0\] ¶{ T\^\* |\_h|+ C\_3}  , while under the alternative, \[eq:upper\_TS\_H1\] ¶{ T\^\* |\_h| + C\_4 } 1 -  . In particular, if $\alpha_n,\beta_n\to 0$ are arbitrary positive sequences, then the test $f$ that rejects the null hypothesis if $$T^* \ge |\bbN_h| + C_3\left[\sqrt{|\bbN_h|(\log(|\cC|)+ 1+\log(\alpha_n^{-1}) )}+ \log(|\cC|)+ \log(\alpha_n^{-1})\right]$$ satisfies $R_{\cC,\mathfrak{G}(h,r)}(f) \to 0$ as soon as \[eq:power\_LS\] r\^2 &gt;  , where $C_0 > 0$ depends only on $d$. Comparing with the minimax lower bound established in Theorem \[thrm3:non\_overlap\_GMRF\], we see that this test is nearly optimal with respect to $h$, the size of the collection $|\cC|$, and the size $|S|$ of the anomalous region (under the alternative). Examples: cubes and blobs {#sec:cubes} ========================= In this section we specialize our general results proved in the previous subsections to classes of cubes, and more generally, blobs. Cubes ----- Consider the problem of detecting an anomalous cube-shaped region. Let $\ell \in \{1, \dots, m\}$ and assume that $m$ is an integer multiple of $\ell$ (for simplicity). Let $\cC$ denote the class of all discrete hypercubes of side length $\ell$, that is, sets of the form $S = \prod_{s=1}^d \{b_s,\ldots,b_s+\ell-1\}$, where $b_s\in \{1,\ldots,m+1-\ell\}$. Each such hypercube $S \in \cC$ contains $|S| = k := \ell^d$ nodes, and the class is of size $|\cC| = (m-1-\ell)^d \le n$. The lower bounds for the risk established in and Theorem \[thrm3:non\_overlap\_GMRF\] are not directly applicable here since these results require subsets of the class $\cC$ to be disjoint. However, they apply to any subclass $\cC' \subset \cC$ of disjoint subsets and, as mentioned in , any lower bound on the minimax risk over $\cC'$ applies to the minimax risk over $\cC$. A natural choice for $\cC'$ here is that of all cubes of the form $S = \prod_{s=1}^d \{a_s \ell +1, \dots, (a_s+1) \ell\}$, where $a_s \in \{0, \dots, m/\ell-1\}$. Note that $|\cC'|=(m/\ell)^d= n/k$. [**$h$ bounded.**]{} Consider first the case where the radius $h$ of the neighborhood is bounded. We may apply to get $$R^*_{\cC,\phi} \geq 1- \frac{k^{1/2}}{2n^{1/2}}\exp\left\{\frac{5k\|\phi\|_2^2}{1-2\|\phi\|_1}\right\}\ .$$ For a given $r>0$ satisfying $2|\bbN_h|r\leq 1$, we can choose a parameter $\phi$ constant over $\bbN_h$ such that $\|\phi\|_2=r$ and $\|\phi\|_1=r\sqrt{(2h+1)^d-1}$. Since $R^*_{\cC,\mathfrak{G}(h,r)}\geq R^*_{\cC,\phi}$, we thus have $R^*_{\cC,\mathfrak{G}(h,r)} \to 1$ when $n \to \infty$, if $(k,\phi) = (k(n), \phi(n))$ satisfies $\log(n)\ll k\ll n$ and $r^2 \le \log(n/k)/(11 k)$. Comparing with the performance of the Fisher test of , in this particular case, Condition is met, and letting $\alpha = \alpha(n) \to 0$ and $\beta = \beta(n) \to 0$ slowly, we conclude from that this test (denoted $f$) has risk $R_{\cC,\mathfrak{G}(h,r)}(f) \to 0$ when $r^2 \ge C_0 \log(n)/k$ for some constant $C_0$. Thus, in this setting, the Fisher test, without knowledge of $\phi$, achieves the correct detection rate as long as $k \le n^b$ for some fixed $b < 1$. [**$h$ unbounded.**]{} When $h$ is unbounded, we obtain a sharper bound by using Theorem \[thrm3:non\_overlap\_GMRF\] instead of . Specialized to the current setting, we derive the following. \[cor:lower\_hypercube\] There exist two positive constants $C_1$ and $C_2$ depending only on $d$ such that the following holds. Assume that the neighborhood size $h$ is small enough that \[eq:condition\_neigbhorhood\] |\_h|C\_1 . Then the minimax risk tends to one when $n \to \infty$ as soon as $(k,h,r)=(k(n),h(n),r(n))$ satisfies $n/k\rightarrow \infty$ and \[eq:powerless\_gmrf\] r\^2 C\_2  . Note that, in the case of a square neighborhood, $|\bbN_h| = (2h+1)^d -1$. Comparing with the performance of the Fisher test, in this particular case, Condition is equivalent to $|\bbN_h| \le C_0 \big(k^{1/4} \wedge \sqrt{k/\log(n)}\big)$ for some constant $C_0$. When $k$ is polynomial in $n$, this condition is stronger than Condition unless $d \le 5$. In any case, assuming $h$ is small enough that both and hold, and letting $\alpha = \alpha(n) \to 0$ and $\beta = \beta(n) \to 0$ slowly, we conclude from that the Fisher test has risk $R_{\cC,\mathfrak{G}(h,r)}$ tending to zero when $$r^2 \ge C_0 \left[\frac{\log(n)}{k}\bigvee\frac{\sqrt{|\bbN_{h}|\log(n)}}{k}\right],$$ for some large-enough constant $C_0 > 0$, matching the lower bound up to a multiplicative constant as long as $k \le n^b$ for some fixed $b < 1$. In conclusion, whether $h$ is fixed or unbounded but growing slowly enough, the Fisher test achieves a risk matching the lower bound up to a multiplicative constant. Blobs ----- So far, we only considered hypercubes, but our results generalize immediately to much larger classes of blob-like regions. Here, we follow the same strategy used in the detection-of-means setting, for example, in [@MGD; @cluster; @MR2589191]. Fix two positive integers $\ell_\circ \le \ell^\circ$ and let $\cC$ be a class of subsets $S$ such that there are hypercubes $S_\circ$ and $S^\circ$, of respective side lengths $\ell_\circ$ and $\ell^\circ$, such that $S_\circ \subset S \subset S^\circ$. Letting $\cC_\circ$ and $\cC^\circ$ denote the classes of hypercubes of side lengths $\ell_\circ$ and $\ell^\circ$, respectively, our lower bound for the worst-case risk associated with the class $\cC^\circ$ obtained from applies directly to $\cC$—although not completely obvious, this follows from our analysis—while scanning over $\cC_\circ$ in the Fisher test yields the performance stated above for the class of cubes. In particular, if $\ell_\circ/\ell^\circ$ remains bounded away from $0$, the problem of detecting a region in $\cC$ is of difficulty comparable to detecting a hypercube in $\cC_\circ$ or $\cC^\circ$. When the size of the anomalous region $k$ is unknown, meaning that the class $\cC$ of interest includes regions of different sizes, we can simply scan over dyadic hypercubes as done in the first step of the multiscale method of [@MGD]. This does not change the rate as there are less than $2n$ dyadic hypercubes. See also [@cluster]. We note that when $\ell_\circ/\ell^\circ = o(1)$, scanning over hypercubes may not be very powerful. For example, for “convex" sets, meaning when $$\cC = \Big\{S = K \cap \cV: K \subset \bbR^d \text{ convex}, |K \cap \cV| = k\Big\}~,$$ it is more appropriate to scan over ellipsoids due to John’s ellipsoid theorem [@MR0030135], which implies that for each convex set $K \subset \bbR^d$, there is an ellipsoid $E \subset K$ such that ${\rm vol}(E) \ge d^{-d} {\rm vol}(K)$. For the case where $d=2$ and the detection-of-means problem, [@MR2589191]—expanding on ideas proposed in [@MGD]—scan over parallelograms, which can be done faster than scanning over ellipses. Finally, we mention that what we said in this section may apply to other types of regular lattices, and also to lattice-like graphs such as typical realizations of a random geometric graph. See [@cluster; @MR2604703] for detailed treatments in the detection-of-means setting. Discussion {#sec:discussion} ========== We provided lower bounds and proposed near-optimal procedures for testing for the presence of a piece of a Gaussian Markov random field. These results constitute some of the first mathematical results for the problem of detecting a textured object in a noisy image. We leave open some questions and generalization of interest. [*More refined results.*]{} We leave behind the delicate and interesting problem of finding the exact detection rates, with tight multiplicative constants. This is particularly appealing for simple settings such as finding an interval of an autoregressive process, as described in . Our proof techniques, despite their complexity, are not sufficiently refined to get such sharp bounds. We already know that, in the detection-of-means setting, bounding the variance of the likelihood ratio does not yield the right constant. The variant which consists of bounding the first two moments of a carefully truncated likelihood ratio, possibly pioneered in [@Ingster99], is applicable here, but the calculations are quite complicated and we leave them for future research. [*Texture over texture.*]{} Throughout the paper we assumed that the background is Gaussian white noise. This is not essential, but makes the narrative and results more accessible. A more general, and also more realistic setting, would be that of detecting a region where the dependency structure is markedly different from the remainder of the image. This setting has been studied in the context of time series, for example, in some of the references given in . However, we are not aware of existing theoretical results in higher-dimensional settings such as in images. [*Other dependency structures.*]{} We focused on Markov random fields with limited neighborhood range (quantified by $h$ earlier in the paper). This is a natural first step, particularly since these are popular models for time series and textures. However, one could envision studying other dependency structures, such as short-range dependency, defined in [@MR2379935] as situations where the covariances are summable in the following sense $$\sup_{i\in \cV_\infty} \sum_{j \in \cV_\infty \setminus i} |\bGamma_{i,j}| < \infty\ .$$ Proofs {#sec:proofs} ====== Proof of --------- The Bayes risk is achieved by the likelihood ratio test $f_{\nu,\pi}^*(x)=\IND{L_{\nu,\pi}(x)>1}$ where $$L_{\nu,\pi}(x)= \frac{1}{|\cC|} \sum_{S\in \cC} L_S(x)~, \text{ with} \quad L_S(x) = \int \frac{ {\rm d}\PROB_{S,\bGamma(\phi)}(x) } {{\rm d} \PROB_0(x)} \pi({\rm d}\phi)~.$$ In our Gaussian model, \[L\_S\] L\_S(x) = \_ , where the expectation is taken with respect to the random draw of $\phi \sim \pi$. Then, by , \[LR-risk\] |[R]{}\^\*\_[,]{} = 1 - 12 \_0 | L\_[,]{}() - 1 | 1 - 12  . (Recall that $\EXP_0$ stands for expectation with respect to the standard normal random vector $\X$.) We proceed to bound the second moment of the likelihood ratio under the null hypothesis. Summing over $S,T \in \cC$, we have && \_0\[L\_[,]{}()\^2\]\ &=& 1[||\^2]{} \_[S,T]{} \_0\[L\_S() L\_T()\]\ &=& 1[||\^2]{} \_[S T]{} \_0\[ L\_S()\] \_0 \[L\_T()\] + 1[||\^2]{} \_[S]{} \_0 \[L\_S\^2()\]\ &=& + 1[||\^2]{} \_[S]{} \_0 \_\ & & 1+ 1[||\^2]{} \_[S]{}\_\ & = & 1 + \_[S]{} V\_S  , where in the second equality we used the fact that $S \ne T$ are disjoint, and therefore $L_S(\X)$ and $L_T(\X)$ are independent, and in the third we used the fact that $\E_0 [L_S(\X)] = 1$ for all $S\in \cC$. Deviation inequalities ---------------------- Here we collect a few more-or-less standard inequalities that we need in the proofs. We start with the following standard tail bounds for Gaussian quadratic forms. See, e.g., Example 2.12 and Exercise 2.9 in [@BoLuMa13]. \[lem:normal-quad\] Let $Z$ be a standard normal vector in $\bbR^d$ and let $\bR$ be a symmetric $d\times d$ matrix. Then $$\pr{Z^\top \bR Z - \tr(\bR) \geq 2\|\bR\|_F \sqrt{t} + 2 \|\bR\|t \, } \le e^{-t}, \quad \forall t \ge 0~.$$ Furthermore, if the matrix $\bR$ is positive semidefinite, then $$\pr{Z^\top \bR Z - \tr(\bR) \leq -2\|\bR\|_F \sqrt{t} \, } \le e^{-t}, \quad \forall t \ge 0.$$ \[lem:chaos\_4\] There exists a positive constant $C$ such that the following holds. For any Gaussian chaos $Z$ up to order $4$ and any $t>0$, $$\P\left\{|Z-\E[Z]|\geq C\Var^{1/2}(Z)t^2 \right\}\leq e^{-t}\ .$$ This deviation inequality is a consequence of the hypercontractivity of Gaussian chaos. More precisely, Theorem 3.2.10 and Corollary 3.2.6 in [@MR1666908] state that $$\E\exp\left[\left(\frac{Z-\E[Z]}{C\Var^{1/2}(Z)}\right)^{1/2}\right]\leq 2\ ,$$ where $C$ is a numerical constant. Then, we apply Markov inequality to prove the lemma. \[lem:chaos\] There exists a positive constant $C$ such that the following holds. Let $F$ be a compact set of symmetric $r\times r$ matrices and let $Y\sim \cN(0,\bI_r)$. For any $t>0$, the random variable $Z:= \sup_{\bR \in F} \tr\left[\bR YY^\top \right]$ satisfies \[eq:chaos\_concentration\] {Z (Z) + t} (-C( )), where $W := \sup_{\bR \in F} \tr(\bR YY^\top \bR)$ and $B := \sup_{\bR \in F}\|\bR\|$. A slight variation of this result where $Z$ is replaced by $\sup_{\bR \in F} \tr\left[\bR (YY^\top -\bI_r)\right]$ is proved in [@MR2662346] using the exponential Efron-Stein inequalities of [@MR2123200]. Their arguments straightforwardly adapt to Lemma \[lem:chaos\]. \[lem:concentration\_vp\_wishart\] Let $\bW$ be a standard Wishart matrix with parameters $(n,d)$ satisfying $n>d$. Then for any number $0<x<1$, $$\begin{aligned} \mathbb{P}\left\{\lambda^{\rm max}(\bW) \geq n\left(1+\sqrt{d/n}+\sqrt{2x/n}\right)^2 \right \} & \leq &e^{-x}\ ,\nonumber \\ %\label{majoration_wishart_sous_gaussienne} \mathbb{P}\left\{\lambda^{\rm min}(\bW) \leq n\left(1-\sqrt{d/n}-\sqrt{2x/n}\right)_+^2 \right\} & \leq &e^{-x}\ . \nonumber\end{aligned}$$ Auxiliary results for Gaussian Markov random fields on the lattice {#sec:proofs_technique} ------------------------------------------------------------------ He we gather some technical tools and proofs for Gaussian Markov random fields on the lattice. Recall the notation introduced in Section \[sec:prelim\_gmrf\]. \[lem:spectrum\_gamma\] For any positive integer $h$ and $\phi\in \Phi_{h}$ with $\|\phi\|_1 <1$, we have that if $\lambda$ is an eigenvalue of the covariance operator $\bGamma(\phi)$, then $$\frac{\sigma_{\phi}^2}{1+\|\phi\|_1} \le \lambda \le \frac{\sigma_{\phi}^2}{1-\|\phi\|_1}~.$$ Also, we have \[eq:lb\_sigma\]  1- \_1 \_\^2 1 . Recall that $\|\cdot\|$ denotes the $\ell^2 \to \ell^2$ operator norm. First note that by the definition of $\phi$, $\sigma_\phi^2 \bGamma^{-1}(\phi)- \bI = (\phi_{i-j})_{i,j \in \bbZ^d}$, and therefore \[diag\_dominant\] \_\^2 \^[-1]{}() - \_1 , where whe used the bound $\|\bA\|\leq \sup_{i\in \bbZ^d}\sum_{j\in \bbZ^d}|\bA_{ij}|$. This implies that the largest eigenvalue of $\bGamma(\phi)$ is bounded by $\sigma^2_{\phi}/(1-\|\phi\|_1)$ if $\|\phi\|_1<1$ and that the smallest eigenvalue of $\bGamma(\phi)$ is at least $\sigma^2_{\phi}/(1+\|\phi\|_1)$. Considering the conditional regression of $Y_i$ given $Y_{-i}$ mentioned above, that is, $$Y_i= -\sum_{1\leq |j|_{\infty}\leq h}\phi_j Y_{i+j}+\epsilon_i$$ (with $\epsilon_i$ being standard normal independent of the $Y_j$ for $j\neq i$) and taking the variance of both sides, we obtain $$1-\sigma^2_{\phi}= \Var\left[ \sum_{1<|j|_{\infty}\leq h}\phi_j Y_{i+j}\right] = \phi^\top \bGamma(\phi) \phi \le \|\bGamma(\phi)\| \|\phi\|_2^2\leq \frac{\|\phi\|_2^2}{1-\|\phi\|_1}\sigma^2_{\phi}~,$$ and therefore 1 - \^[2]{}\_\^[2]{}\_ . Rearranging this inequality and using the fact that $\|\phi\|_2^2 \le \|\phi\|_1^2 \le \|\phi\|_1$, we conclude that $\sigma^{2}_{\phi}\geq 1-\|\phi\|_1$. The remaining bound $\frac{\|\phi\|_2^2}{1+ \|\phi\|_1}\leq \frac{1 - \sigma^{2}_{\phi}}{\sigma^2_{\phi}}$ is obtained similarly. Recall that for any $v\in \mathbb{Z}^d$, $\gamma_v$ is the correlation between $Y_i$ and $Y_{i+v}$ and is therefore equal to $\bGamma_{i, i+v}$. This definition does not depend on the node $i$ since $\bGamma$ is the covariance of a stationary process. \[lem:conditional\_variance\] For any $h$ and any $\phi\in \Phi_{h}$, let $Y\sim \cN(0,\bGamma(\phi))$. As long as $\|\phi\|_1<1$, the $l_2$ norm of the correlations satisfies \[eq:upper\_norm\_correlation\] \_[v0]{}\_v\^2+ ()\^2 In order to compute $\|\gamma\|^2_2$, we use the spectral density of $Y$ defined by $$f(\omega_1,\ldots,\omega_d)=\frac{1}{(2\pi)^d}\sum_{(v_1,\ldots,v_d)\in \mathbb{Z}^d}\gamma_{v_1,\ldots v_d}\exp\left(\iota\sum_{i=1}^dv_i\omega_i\right)\ , \quad (\omega_1,\ldots,\omega_d)\in (-\pi,\pi]^d\ .$$ Following [@MR1344683 Sect.1.3] or [@MR2130347 Sect.2.6.5], we express the spectral density in terms of $\phi$ and $\sigma_{\phi}^2$: $$\frac{1}{f(\omega_1,\ldots, \omega_d)}= \frac{(2\pi)^d}{\sigma_{\phi}^2}\left[1- \sum_{v, 1\leq |v|_{\infty}\leq h\in \mathbb{Z}^d}\phi_{v}e^{\iota \langle v,\omega\rangle}\right]\ ,$$ where $\langle \cdot , \cdot\rangle$ denotes the scalar product in $\mathbb{R}^d$. As a consequence, $$|f(\omega_1,\ldots, \omega_d)|\leq \sigma_{\phi}^2[(2\pi)^d(1-\|\phi\|_1)]^{-1}~.$$ Relying on Parseval formula, we conclude \_[v0]{}\_v\^2&=& (2)\^d\_[\[-;\]\^d]{}\^2d\_1…d\_d\ && \_[\[-;\]\^d]{}| -1|\^2d\_1…d\_d\ && \_[\[-;\]\^d]{}| - 1 - \_[v, 1|v|\_h\^d]{} \_[v]{}e\^[v,]{}|\^2d\_1…d\_d\ &&\ & & ()\^2+\ & & ()\^2+  , where we used in the last line. \[lem:covariance\_residuals\] For any $h$ and any $\phi\in \Phi_{h}$, let $Y\sim \cN(0,\bGamma(\phi))$. Then for any $i\in \bbZ^d$, the random variable $\epsilon_i$ defined by the conditional regression $Y_i=\sum_{v\in \bbN_h}\phi_vY_{i+v} +\epsilon_i$ satisfies that 1. $\epsilon_i$ is independent of all $X_j\ , j\neq i$ and $\Cov(\epsilon_i,X_i)=\Var(\epsilon_i)=\sigma_{\phi}^2$. 2. For any $i\neq j$, $\Cov(\epsilon_i,\epsilon_j)=-\phi_{i-j}\sigma_{\phi}^2$ if $|i-j|_{\infty}\leq h$ and 0 otherwise. The first independence property is a classical consequence of the conditional regression representation for Gaussian random vectors, see, for example, [@MR1419991]. Since $\Var(\epsilon_i)$ is the conditional variance of $Y_i$ given $Y^{(-i)}$, it equals $[(\bGamma^{-1}(\phi))_{i,i}]^{-1}=\sigma_{\phi}^2$. Furthermore, $$\Cov(\epsilon_i,Y_i) = \Var(\epsilon_i)+ \sum_{v\in \bbN_h}\phi_j\Cov(\epsilon_i,Y_{i+v})= \Var(\epsilon_i)\ ,$$ by the independence of $\epsilon_i$ and $Y^{(-i)}$. Finally, consider any $i\neq j$, (\_i,\_j)= (\_i,Y\_j) - \_[v\_h]{}\_v(\_i,Y\_[j+v]{}) , where all the terms are equal to zero with the possible exception of $v=i-j$. The result follows. \[lem:control\_Gamma\_S\] As long as $\|\phi\|_1<1$, the following properties hold: 1. If $i\in S^h$ or if $j \in S^h$, then $(\bGamma_S^{-1}(\phi))_{i,j}= (\bGamma^{-1}(\phi))_{i,j}$. 2. If $i\in S^h$ and $j\in \Delta_h(S)$, then $1\leq (\bGamma_S^{-1}(\phi))_{j,j}\leq (\bGamma_S^{-1}(\phi))_{i,i}$. 3. If $i\in \Delta_h(S)$, then $\sum_{j\in S:j\neq i}(\bGamma_S^{-1}(\phi))_{i,j}^2\leq \frac{2\|\phi\|_2^2}{(1-\|\phi\|_1)^3}$. We prove each part in turn. [*Part 1.*]{} Consider $i \in S^h$ and any $j \in S$. By the Markov property, conditionally to $(Y_{i+k},\ 1\leq |k|_{\infty}\leq h)$, $Y_i$ is independent of all the remaining variables. Since all vertices $i+k$ with $1\leq |k|_{\infty}\leq h$ belong to $S$, the conditional distribution of $Y_{i}$ given $Y^{(-i)}$ is the same as the conditional distribution of $Y_{i}$ given $(Y_{j}, j\in S\setminus\{i\})$. This conditional distribution characterizes the $i$-th row of the inverse covariance matrix $\bGamma^{-1}_S$. Also, the conditional variance of $Y_i$ given $Y^{(-i)}$ is $[(\bGamma^{-1}(\phi))_{i,i}]^{-1}$ and the conditional variance of $Y_i$ given $Y_{S}$ is $[(\bGamma_S^{-1}(\phi))_{i,i}]^{-1}$. Furthermore, $-(\bGamma^{-1}(\phi))_{i,j}/(\bGamma^{-1}(\phi))_{i,i}$ is the $j$-th parameter of the condition regression of $Y_i$ given $Y^{(i)}$, and therefore we conclude that $(\bGamma^{-1}(\phi))_{i,i}= (\sigma^2_{\phi})^{-1}=(\bGamma_S^{-1}(\phi))_{i,i} $ and $(\bGamma^{-1}(\phi))_{i,j}/(\bGamma^{-1}(\phi))_{i,i}= -\phi_{i-j} = (\bGamma_S^{-1}(\phi))_{i,j}/(\bGamma_S^{-1}(\phi))_{i,i}$. [*Part 2.*]{} Consider any vertex $i\in S^h$ and $j\in \Delta_h(S)$. Since $1/\bGamma_S^{-1}(\phi))_{j,j}$ and $1/\bGamma_S^{-1}(\phi))_{j,j}$ are the conditional variances of $Y_i$ and $Y_j$ given $Y_{k}, k \in S\setminus \{j\}$ and $Y_{k}, k \in S\setminus \{i\}$, respectively, we have 1/\_S\^[-1]{}())\_[j,j]{} &=& (Y\_j | Y\_[k]{}: k S{j})\ && (Y\_j |Y\^[(-j)]{} )\ &= & (Y\_i |Y\^[(-i)]{} )\ & = & (Y\_i |Y\_k: kS{i})\ &= & 1/\_S\^[-1]{}())\_[i,i]{}  . [*Part 3.*]{} Consider $i\in \Delta_h(S)$. The vector $(\bGamma_S^{-1}(\phi))_{i,-i}\defeq \left(-(\bGamma_S^{-1}(\phi))_{i,j}/ (\bGamma_S^{-1}(\phi))_{i,i}\right)_{j\in S: j\neq i}$ is formed by the regression coefficients of $Y_{i}$ on $(Y_{j}, j\in S\setminus\{i\})$. Since the conditional variance of $Y_{i}$ given $(Y_{j}, j\in S\setminus\{i\})$ is at least $\sigma_{\phi}^2$ (by Parts 1 and 2), we get 1 - \_\^2 && 1 - (Y\_i|Y\_j:  j S{i})\ &= &({Y\_i|Y\_j, jS{i}})\ & = & (\_[jS{i}]{} - Y\_j )\ && \_\^4 (\_[jS{i}]{} -(\_S\^[-1]{}())\_[i,j]{} Y\_j)\ & = & \_\^4 (\_S\^[-1]{}())\_[i,-i]{}\^\_S() (\_S\^[-1]{}())\_[i,-i]{}\ && (\_S\^[-1]{}())\_[i,-i]{}\_2\^2 , where the equality in the second line above we use $\Var(Y_j)= 1$ and the law of total variance (i.e., $\Var(Y)= \EXP[Var(Y|\cB)]+ Var(E[Y|\cB])$) and in the last line we use that the smallest eigenvalue of $\bGamma(\phi)$ (and also of $\bGamma_S(\phi)$) is larger than $\sigma_{\phi}^2/(1+\|\phi\|_1)$ (Lemma \[lem:spectrum\_gamma\]). Rearranging this inequality and using the fact that $\|\phi\|_1 < 1$, we arrive at (\_S\^[-1]{}())\_[i,-i]{}\_2\^2 && (1 + \_1)2\ &&\ && \[lem:det\] For any $\phi_1, \phi_2\in \Phi_h$, define $$B_{\phi_1,\phi_2}:=\left(\frac{\det(\bGamma_{S}^{-1}(\phi_1))\det(\bGamma_{S}^{-1}(\phi_2))}{\det(\bGamma_{S}^{-1}(\phi_1)+ \bGamma_{S}^{-1}(\phi_2) -\bI_S)}\right)^{1/2}\ .$$ (Note that $V_S$ defined in Proposition \[prp:non\_overlap\_parametric\] equals the expected value of $B_{\phi_1,\phi_2}$ when $\phi_1$ and $\phi_2$ are drawn independently from the distribution $\pi$.) Assuming that $\|\phi_1\|_1\vee \|\phi_2\|_1<1/5$, we have $$\log B_{\phi_1,\phi_2} \leq \frac{1}{2}|S|\langle \phi_1,\phi_2\rangle+ 8 Q_S\ ,$$ where Q\_S &:=& |S|\_[s\_1,s\_2,s\_3=1]{}\^2|\_[j,k\_h]{}\_[s\_1,j]{}\_[s\_2,k]{}\_[s\_3,k-j]{}|+ 15|S| (\_1\_2\^3\_2\_2\^3) + |\_[h]{}(S)| (\_1\_2\^2\_2\_2\^2)\ &&+  28 |\_[2h]{}(S)|(|\_[2h]{}(S)|(|\_h|+1))\^[1/2]{} (\_1\_2\^3\_2\_2\^3)  . Since for any $\phi$, the spectrum of $\bGamma_S^{-1}(\phi)$ lies between the extrema of the spectrum of $\bGamma^{-1}(\phi)$, by Lemma \[lem:spectrum\_gamma\], we have -1 \^(\_S\^[-1]{}()-\_S) \^[max]{}(\_S\^[-1]{}()-\_S) -1 , where $\lambda^{\min}(\bA)$ and $\lambda^{\max}(\bA)$ denote the smallest and largest eigenvalues of a matrix $\bA$. Since $\sigma^2_{\phi}\leq \var{Y_i}=1$, the left-hand side is larger than $-\|\phi\|_1$, while relying on , we derive $$\frac{1+\|\phi\|_1}{\sigma^2_{\phi}}-1\leq (\|\phi\|_1+1)\left[1+\frac{\|\phi\|_2^2}{1-\|\phi\|_1}\right]-1 \leq \frac{2\|\phi\|_1}{1-\|\phi\|_1}\ .$$ Consequently, as long as $\|\phi\|_1< 1/5$, the spectrum of $\bGamma_S^{-1}(\phi)$ lies in $(\tfrac45,\tfrac32)$. This allows us to use the Taylor series of the logarithm, which for a matrix $\bA$ with spectrum in $(\tfrac12, 2)$, gives |(()) - +| || . Applying this expansion to $\bGamma_{S}^{-1}(\phi_1)$, $\bGamma_{S}^{-1}(\phi_2)$ and $\bGamma_{S}^{-1}(\phi_1)+ \bGamma_{S}^{-1}(\phi_2)-\bI_S$, 2B\_[\_1,\_2]{} && V\_1+ V\_2+ 8V\_3+8V\_4 ,\ V\_1&:=&  ,\ V\_2&:= & ||+||  ,\ V\_3&:= & ||  ,\ V\_4 &:= & ||  . #### Control of $V_1$. We use the fact that $$\tr\Big[\big(\bGamma_{S}^{-1}(\phi_1)-\bI_S\big)\big(\bGamma_{S}^{-1}(\phi_2)-\bI_S\big)\Big] = \sum_{i,j\in S}\big((\bGamma_{S}^{-1}(\phi_1))_{i,j}-\delta_{i,j}\big)\big((\bGamma_{S}^{-1}(\phi_2))_{i,j}-\delta_{i,j}\big)~.$$ To bound the right-hand side, first consider any node $i \in S^h$ in the $h$-interior of $S$. By the first part of Lemma \[lem:control\_Gamma\_S\], the $i$-th row of $\bGamma_S^{-1}(\phi)$ equals the restriction to $S$ of the $i$-th row of $\bGamma^{-1}(\phi)$. Using the definition of $\phi_1,\phi_2$, we therefore have $$\begin{aligned} \label{v1-bound1} \lefteqn{\sum_{j\in S}\big((\bGamma_{S}^{-1}(\phi_1))_{i,j}-\delta_{i,j}\big)\big((\bGamma_{S}^{-1}(\phi_2))_{i,j}-\delta_{i,j}\big)} &&\\ \notag &=& (\bGamma^{-1}(\phi_1))_{i,i} (\bGamma^{-1}(\phi_2))_{i,i} \langle \phi_1,\phi_2\rangle + ((\bGamma^{-1}(\phi_1))_{i,i}-1)((\bGamma^{-1}(\phi_2))_{i,i}-1) \\ \notag &= & \frac{\langle \phi_1,\phi_2\rangle }{\sigma_{\phi_1}^{2}\sigma_{\phi_2}^{2}}+ \frac{(1-\sigma_{\phi_1}^{2})(1-\sigma_{\phi_2}^{2})}{\sigma_{\phi_1}^{2}\sigma_{\phi_2}^{2}} \\ \notag &= & \langle \phi_1,\phi_2\rangle+ \langle \phi_1,\phi_2\rangle \frac{1- \sigma_{\phi_1}^{2}\sigma_{\phi_2}^{2}}{\sigma_{\phi_1}^{2}\sigma_{\phi_2}^{2}}+\frac{(1-\sigma_{\phi_1}^{2})(1-\sigma_{\phi_2}^{2})}{\sigma_{\phi_1}^{2}\sigma_{\phi_2}^{2}} \\ \notag &\leq &\langle \phi_1,\phi_2\rangle+ \frac{3}{2}\frac{\|\phi_1\|_2^4+ \|\phi_2\|_2^4}{(1-\|\phi_1\|_1)(1-|\phi_2\|_1)}\ ,\end{aligned}$$ using in the last line. Next, consider a node $i \in \Delta_h(S)$, near the boundary of $S$. Relying on Lemmas \[lem:spectrum\_gamma\] and \[lem:control\_Gamma\_S\], we get $$\begin{aligned} \sum_{j\in S}\big((\bGamma_{S}^{-1}(\phi_1))_{i,j}-\delta_{i,j}\big)^2&\leq& \frac{2\|\phi_1\|_2^2}{(1-\|\phi_1\|_1)^3}+ \big(1/\sigma_{\phi_1}^{2}-1\big)^2\nonumber\\ &\leq & \frac{2\|\phi_1\|_2^2}{(1-\|\phi_1\|_1)^3}+ \frac{\|\phi_1\|_2^4}{(1-\|\phi_1\|_1)^2}\leq \frac{3\|\phi_1\|_2^2}{(1-\|\phi_1\|_1)^3} \ ,\label{eq:upper_delta_h} \end{aligned}$$ since we assume that $\|\phi\|_1<1$. By the Cauchy-Schwarz inequality, \[v1-bound2\] \_[jS]{}((\_[S]{}\^[-1]{}(\_1))\_[i,j]{}-\_[i,j]{})((\_[S]{}\^[-1]{}(\_2))\_[i,j]{}-\_[i,j]{})3 . Summing over $i\in S^h$ and over $i \in \Delta_h(S)$, we get $$V_1 \le |S|\langle \phi_1,\phi_2\rangle+ \frac32 |S|\frac{\|\phi_1\|_2^4+ \|\phi_2\|_2^4}{(1-\|\phi_1\|_1)(1-\|\phi_2\|_1)}+ 3|\Delta_h(S)|\frac{\|\phi_1\|_2^2\vee \|\phi_2\|_2^2}{(1-\|\phi_1\|_1\vee \|\phi_2\|_1)^3}\ .$$ #### Control of $V_2$. We proceed similarly as in the previous step. Note that $$\begin{aligned} \tr\Big[\big(\bGamma_{S}^{-1}(\phi_1)-\bI_S\big)^3\Big] & = & \sum_{i,j,k\in S}\big((\bGamma_{S}^{-1}(\phi_1))_{i,j}-\delta_{i,j}\big)\big((\bGamma_{S}^{-1}(\phi_1))_{j,k}-\delta_{j,k}\big)\big((\bGamma_{S}^{-1}(\phi_1))_{k,i}-\delta_{k,i}\big) \\ &\le & \sum_{i\in S}\left|\sum_{j,k\in S}\big((\bGamma_{S}^{-1}(\phi_1))_{i,j}-\delta_{i,j}\big)\big((\bGamma_{S}^{-1}(\phi_1))_{j,k}-\delta_{j,k}\big)\big((\bGamma_{S}^{-1}(\phi_1))_{k,i}-\delta_{k,i}\big)\right| ~.\end{aligned}$$ First, consider a node $i$ in $S\setminus \Delta_{2h}(S)$. Here, we use $\Delta_{2h}(S)$ instead of $\Delta_{h}(S)$ so that we may replace $\bGamma_S^{-1}(\phi)$ below with $\bGamma^{-1}(\phi)$. We use again Lemma \[lem:control\_Gamma\_S\] to replace $(\bGamma_S^{-1}(\phi))_{i,j}$ by $(\bGamma^{-1}(\phi))_{i,j}$ in the sum\ && |\_[j,k\_h]{}| + 3 \_[j\_h]{}|\_[1,j]{}|\^2+\ && |\_[j,k\_h]{}| + 4 , using in the last line. Next, consider a node $i\in \Delta_{2h}(S)$. If $i\notin\Delta_h(S)$, then the support of $(\bGamma_{S}^{-1}(\phi_1))_{i,-i}$ is of size $|\bbN_h|$. If $i\in \Delta_{h}(S)$, then $\Delta_{2h}(S)\setminus\{i\}$ separates $\{i\}$ from $S\setminus \Delta_{2h}(S)$ in the dependency graph and the Global Markov property [@MR1419991] entails that $$Y_i\indep (Y_{k},\ k\in S\setminus \Delta_{2h}(S))|(Y_{k},\ k\in \Delta_{2h}(S)\setminus\{i\})\ ,$$ and therefore the support of $(\bGamma_{S}^{-1}(\phi_1))_{i,-i}$ is of size smaller than $|\Delta_{2h}(S)|$. Using the Cauchy-Schwarz inequality and , we get &&\ && \_[jS]{}|(\_[S]{}\^[-1]{}(\_1))\_[i,j]{}-\_[i,j]{}| (\_[S]{}\^[-1]{}(\_1))\_[i,.]{}-\_[i,.]{}\_2 (\_[S]{}\^[-1]{}(\_1))\_[j,.]{}-\_[j,.]{}\_2\ && (\_[S]{}\^[-1]{}(\_1))\_[i,.]{}-\_[i,.]{}\_2(\_[S]{}\^[-1]{}(\_1))\_[i,.]{}-\_[i,.]{}\_2 \_[jS]{}(\_[S]{}\^[-1]{}(\_1))\_[j,.]{}-\_[j,.]{}\_2\ && 3\^[3/2]{} . In conclusion, V\_2&& |S| + 8|S|\ & & + 11 |\_[2h]{}(S)|(|\_[2h]{}(S)|(|\_h|+1))\^[1/2]{} . #### Control of $V_3+V_4$. Arguing as above, we obtain V\_3+V\_4&& |S| + 8|S|\ &+& 11 |\_[2h]{}(S)|(|\_[2h]{}(S)|(|\_h|+1))\^[1/2]{} . Proof of Corollary \[cor:non\_overlap\_GMRF\] --------------------------------------------- As stated in Lemma \[lem:spectrum\_gamma\], all eigenvalues of the covariance operator $\bGamma^{-1}(\phi)$ lie in $(1-\|\phi\|_1,\tfrac{1+\|\phi\|_1}{1-\|\phi\|_1} )$. Since the spectrum of $\bGamma^{-1}_{S}(\phi)$ lies between the extrema of the spectrum of $\bGamma^{-1}(\phi)$, and using the assumption that $\|\phi\|_1<1/2$, this entails \[gamma-id-op\] \_S()-\_S&lt;1 , We now apply Proposition \[prp:non\_overlap\_parametric\] with the probability measure $\pi$ concentrating on $\phi$. In this case, $$V_S = \frac{\det(\bGamma_{S}^{-1}(\phi))}{ \det(2 \bGamma_{S}^{-1}(\phi) - \bI_S)^{1/2}} = \det(\bI_S - (\bI_S - \bGamma_S(\phi))^2)^{-1/2},$$ and we get |[R]{}\^\*\_[, ]{} && 1 - \^[1/2]{}\ && 1 - \^[1/2]{}\ && 1 - \^[1/2]{} , where $\|\cdot\|_F$ denotes the Frobenius norm. The second inequality above is obtained by applying the inequality $1/(1-\lambda)\leq e^{\lambda/(1-\lambda)}$ for $0\leq \lambda<1$ to the eigenvalues of $(\bGamma_S(\phi)-\bI_S)^2$, while the third inequality follows from and the fact that $\|\phi\|_1 < 1/2$. It remains to bound $\|\bGamma_S(\phi)-\bI_S\|_F^2$: \_S() -\_S\_F\^2&=& \_[(i, j S),  ij]{}\^2(Y\_i,Y\_j)\ && |S| \_[v0]{}\_v\^2\ && 20|S| \_2\^2 , where we used Lemma \[lem:conditional\_variance\], $\sigma_{\phi}^2\le 1$, and $\|\phi\|_2\leq \|\phi\|_1\leq 1/2$ in the last line. Proof of Theorem \[thrm3:non\_overlap\_GMRF\] --------------------------------------------- Recall the definition of the prior $\pi$ defined just before the statement of the theorem. Taking the numerical constant $C$ in sufficiently small and relying on condition , we have $\|\phi\|_1= r \sqrt{\bbN_h}<1/5$. Consequently, the support of $\pi$ is a subset of the parameter space $\Phi_h$ and we are in position to invoke Lemma \[lem:det\]. Let $\phi_1,\phi_2$ be drawn independently according to the distribution $\pi$ and denote by $\xi_1$ and $\xi_2$ the corresponding random vectors defined on $\bbN'_h$. By Lemma \[lem:det\], $$\log B_{\phi_1,\phi_2} \leq |S|r^2\bbN_h^{-1} \langle \xi_1,\xi_2 \rangle + 8 Q_S\ ,$$ where $$Q_S \leq 23|S| r^{3}\sqrt{|\bbN_h|} + |\Delta_{h}(S)|r^2+ 28 |\Delta_{2h}(S)|(|\Delta_{2h}(S)|\vee (|\bbN_h|+1))^{1/2} r^3\ .$$ Since $\langle \xi_1, \xi_2\rangle $ is distributed as the sum of $|\bbN_h|/2$ independent Rademacher random variables, we deduce that V\_S&& ()\^[|\_h|/2]{} ( 383(|S||\_[2h]{}(S)|\^[3/2]{}) r\^[3]{} + 8|\_[h]{}(S)|r\^2)\ && ( + 383(|S||\_[2h]{}(S)|\^[3/2]{}) + 8|\_[h]{}(S)|r\^2) , since $\cosh(x)\leq \exp(x)\wedge \exp(x^2/2)$ for any $x>0$. Combining this bound with Proposition \[prp:non\_overlap\_parametric\], we conclude that the Bayes risk $\bar{R}^*_{\nu,\pi}$ is bounded from below by \[eq:lower\_gmrf\_complex\] 1 - \_[S]{}( + 383(|S||\_[2h]{}(S)|\^[3/2]{}) r\^[3]{} + 8|\_[h]{}(S)|r\^2) . If the numerical constant $C$ in Condition is sufficiently small, then $\tfrac{|S|^2r^4}{4|\bbN_h|} \bigwedge \tfrac{|S|r^2}{2}\leq 0.5\log(|\cC|/a)$. Also, choosing $C_0$ small enough in condition , relying on condition and on $|\bbN_h|\geq 1$, we also have $$383\left(|S|\sqrt{|\bbN_h|+1}\vee|\Delta_{2h}(S)|^{3/2}\right) r^{3} + 8|\Delta_{h}(S)|r^2\leq 0.5 \log(|\cC|/a) \ .$$ Thus, we conclude that $\bar{R}^*_{\nu,\pi}\geq 1-a$. Proof of Corollary \[cor:lower\_hypercube\] ------------------------------------------- We deduce the result by closely following the proof of Theorem \[thrm3:non\_overlap\_GMRF\]. We first prove that $5r\sqrt{|\bbN_h|}\leq 1$ is satisfied for $n$ large enough. Starting from , we have, for $n$ large enough, 5r&& 5C\_2\^[1/2]{} ()\^[1/2]{}\ && 5C\_2\^[1/2]{} (C\_1 )\^[1/2]{} , where we used Condition in the second line. Taking $C_1$ and $C_2$ small enough, we only have to bound $|\bbN_{h}|^{3/2}\sqrt{\log(\tfrac nk)}/k$. We distinguish two cases. [Case 1]{}: $|\bbN_h|\leq \log(n/k)$. Since $|\bbN_h|\leq C_1k/\log\left(\frac{n}{k}\right)$, it follows that $|\bbN_{h}|^{3/2}\sqrt{\log(\tfrac nk)}/k\leq C_1$. [Case 2]{}: $|\bbN_h|\geq \log(n/k)$. Then the second part of Condition enforces $\log^{4/5}(n/k) \leq C_1 k^{2/5}$. Using again the second part of Condition yields C\_1\^[3/2]{} C\_1\^[3/2]{} . As $5r\sqrt{|\bbN_h|}\leq 1$, we can use the same prior $\pi$ as in the proof of Theorem \[thrm3:non\_overlap\_GMRF\] and arrive at the same lower bound on $R^*_\pi$. It remains to prove that this lower bound goes to one, namely that $$\frac{2|S|^2r^4}{|\bbN_h|} \bigwedge (|S|r^2)+ 765\left(|S|\sqrt{|\bbN_h|+1}\vee|\Delta_{2h}(S)|^{3/2}\right) r^{3} + 16|\Delta_{2h}(S)|r^2- \frac{1}{2}\log(n/k)\to -\infty\ ,$$ where $S$ is a hypercube of size $k$. Taking the constant $C_2$ small enough in leads to $\frac{2k^2r^4}{|\bbN_h|} \bigwedge (kr^2)\leq \log(n/k)/4$ for $n$ large enough. k r\^[3]{}C\_2\^[3/2]{}\^[1/2]{}C\_2\^[3/2]{}(C\_1\^[1/2]{}C\_1\^[5/4]{})(n/k) , where we used again the second part of Condition . Taking $C_1$ and $C_2$ small enough ensures that $765k r^{3}\sqrt{|\bbN_h|+1}\leq \log(n/k)/8$ for $n$ large enough. Finally, it suffices to control $|\Delta_{2h}(S)|^{3/2}r^3$ since $|\Delta_{2h}(S)|r^2\leq |\Delta_{2h}(S)|^{3/2}r^3\vee 1$. Observe that $$|\Delta_{2h}(S)|= \ell^{d}- (\ell-4h)^d= \ell^d \left[1- (1-4h/\ell)^d\right]\leq 4\ell^ddh/\ell\leq 4d|\bbN_h|^{1/d}k^{\frac{d-1}{d}} .$$ It then follows from Condition that (d|\_h|\^[1/d]{}k\^)\^[3/2]{}r\^3 && C\_2\^[3/2]{}()\ && C\_2\^[3/2]{}() where we used again in the second line. Choosing $C_1$ and $C_2$ small enough concludes the proof. Proof of --------- We leave $\phi$ implicit throughout. Define $$U'_S = X_S^\top (\bI_S - \bGamma_S^{-1}) X_S - \tr(\bI_S - \bGamma_S^{-1}).$$ Under the null, $X$ is standard normal, so applying the union bound and gives $$\pr{U > 4} \le \sum_{\cS \in \cC}\P\Big\{ U'_S > 4\|\bI_S - \bGamma_S^{-1}\|_F\sqrt{\log(|\cC|)} + 4\|\bI_S - \bGamma_S^{-1}\|\log(|\cC|)\Big\} \le |\cC|^{-1}~.$$ Under the alternative where $S \in \cC$ is anomalous, $X_S$ has covariance $\bGamma_S$, so that we have $X_S^\top (\bI_S - \bGamma_S^{-1}) X_S \sim Z^\top (\bGamma_S - \bI_S) Z$, where $Z$ is standard normal in dimension $|S|$. Since $\Var(Y_i)=1$, the diagonal elements of $\bGamma_S - \bI_S$ are all equal to zero. We apply to get that ¶||\^[-1]{} , In view of the definition of $U$, we have $\P[U>4]\geq 1- |\cC|^{-1}$ as soon as\[eq:lower\_RS\] 4 + 6 (||) . Therefore, it suffices to bound $\|\bGamma_S - \bI_S\|_F$, $\|\bGamma_S^{-1}-\bI_S\|_F$, $\|\bGamma-\bI\|$, $\|\bGamma^{-1}-\bI\|$ and $\tr[\bGamma_S^{-1}-\bI_S]$. In the sequel, the $C$ denotes a large enough positive constant depending only on $\eta$, whose value may vary from line to line. From Lemma \[lem:conditional\_variance\], we deduce that $$\|\bGamma_S - \bI_S\|^2_F\leq C |S| \|\phi\|_2^2 \ .$$ Lemma \[lem:spectrum\_gamma\] implies that $$\|\bGamma-\bI\|\vee \|\bGamma^{-1}-\bI\| \leq C \ .$$ We apply Lemma \[lem:control\_Gamma\_S\] to obtain \_S\^[-1]{}-\_S\_F\^2 && C |S|\_2\^2+ |S|(\_\^[-2]{} - 1)\^2\ && C |S|\_2\^2 , where we used Lemma \[lem:spectrum\_gamma\] in the second line. Finally, we use again Lemmas \[lem:control\_Gamma\_S\] and \[lem:spectrum\_gamma\] to obtain &=& |S\^h| + \_[j\_h(S)]{}(\^[-1]{}\_[S]{})\_[j,j]{} -1\ && |S\^h| C |S|\^2\_2 . Consequently, holds as soon as $|S|\|\phi\|_2^2\geq C \log(|\cC|)$. Proof of --------- We use $C, C', C''$ as generic positive constants, whose actual values may change with each appearance. [**Under the null hypothesis**]{}. First, we bound the $1-\alpha$ quantile of $T^*$ under the null hypothesis. Denote $Z_S:= \|\boldsymbol{\Pi}_{S,h}X_{S,h}\|_2^2$ so that $T_S= Z_S|S^h|[\|X_{S,h}\|_2^2-Z_S]^{-1}$. Since $Z_S$ is the squared norm of the projection of $X_{S,h}$ onto the column space of $\bF_{S,h}$, we can express $Z_S$ as a least-squares criterion: $$Z_S= \max_{\phi\in \mathbb{R}^{\bbN_h}} \|X_{S,h}\|_2^2- \sum_{i\in S^h}\Big(X_i - \sum_{j \in \bbN_h }\phi_jX_{i+j} \Big)^2 \ .$$ Given $\phi\in \mathbb{R}^{\bbN_h}$, define the matrix $\bB_{\phi,S}\in \mathbb{R}^{S\times S}$ such that for any $i\in S^h$, and any $j$, $(\bB_{\phi,S})_{i,i+j}=\phi_j$, and all the remaining entries of $\bB_{\phi,S}$ are zero. It then follows that \[eq:definition\_ZS\] Z\_S= \_[\^[\_h]{}]{}  , \_[,S]{}:= (\_S-\^\_[,S]{})(\_S-\_[,S]{})-\_S . Observe that $Z_S$ can be seen as the supremum of a Gaussian chaos of order 2. As the collection of matrices in the supremum of is not bounded, we cannot directly apply Lemma \[lem:chaos\]. Nevertheless, upon defining defining $\tilde{Z}_S:= \max_{\|\phi\|_1\leq 1} \tr\left[\bR_{\phi,S}X_{S}X_{S}^\top \right]$, we have for any $t>0$, \[eq:deviation\_ZS\] ¶\[Z\_St\]¶\[\_St\]+ ¶\[\_SZ\_S\] , and we can control the deviations of $\tilde{Z}_S$ using Lemma \[lem:chaos\]. Observe that for any $\phi$ with $\|\phi\|_1\leq 1$, $\|\bI_S-\bB_{\phi,S}\| \le 2$, so that $\|\bR_{\phi,S}\|\leq 3$. Choose $\widehat{\phi}_S$ among the $\phi$’s achieving the maximum in , and note that $\P[\tilde{Z}_S\neq Z_S] = \P[\|\widehat{\phi}_S\|_1> 1]$. We bound the right-hand side below. In view of , we also need to bound $\E[\tilde{Z}_S]$ and $\E\big[\sup_{\|\phi\|_1\leq 1}\tr(\bR_{\phi,S} X_{S}X_{S}^\top \bR_{\phi,S})\big]$ in order to control $\P[\tilde{Z}_S\geq t]$. [*Control of $\P[\|\widehat{\phi}_S\|_1> 1]$*]{}. When $\bF_{S,h}^\top \bF_{S,h}$ is invertible, $\widehat{\phi}_S= (\bF_{S,h}^\top \bF_{S,h})^{-1}\bF_{S,h}X_{S,h}$. By the Cauchy-Schwarz inequality, $$\begin{aligned} \P\big[\|\widehat{\phi}_S\|_1> 1\big] &\leq & \P\big[\|\widehat{\phi}_S\|_2 > |\bbN_h|^{-1/2}\big]\nonumber\\ &\leq & \P\left[\lambda^{\min}(\bF_{S,h}^\top \bF_{S,h})\leq \tfrac12 |S^h|\right]+ \P\left[\|\bF_{S,h}X_{S,h}\|_2\geq \frac{|S^h|}{2|\bbN_h|^{1/2}} \right]\nonumber\\ &\leq &\P\left[\lambda^{\min}(\bF_{S,h}^\top \bF_{S,h})\leq \tfrac12 |S^h|\right]+ \P\left[\|\bF_{S,h}X_{S,h}\|_\infty\geq \frac{|S^h|}{2|\bbN_h|} \right] \label{eq:upper_phi_1}\ .\end{aligned}$$ First, we control the smallest eigenvalue of $\bF_{S,h}^\top \bF_{S,h}$. Under the null hypothesis, the vectors $F_{i}$ follow the standard normal distribution, but $\bF_{S,h}^\top \bF_{S,h}$ is not a Wishart matrix since the vectors $F_i$ are correlated. However, $\bF_{S,h}^\top \bF_{S,h}$ decomposes as a sum of $|\bbN_{h}|+1$ (possibly dependent) standard Wishart matrices. Indeed, define \[S\_i\] S\_i= S\^h{i + (2h+1)u, u\^d}, i\_h {0} , and then $\bA_i=\sum_{j\in S_i} F_jF_j^\top $. The vectors $(F_{j},\ j\in S_i)$ are independent since the minimum $\ell_{\infty}$ distance between any two nodes in $S_i$ is at least $2h+1$, so that $\bA_i$ is standard Wishart. Denoting $n_i=|S_i|$, we are in position to apply , to get $$\P\left[\lambda^{\min}(\bA_i)\le n_i - 2\sqrt{|\bbN_h| n_i}- 2\sqrt{2x n_i}\right]\leq e^{-x}\ , \quad \forall x > 0\ .$$ Since the $\{S_i : i \in \bbN_h \cup \{0\}\}$ forms a partition of $S^h$, we have $\bF_{S,h}^\top \bF_{S,h}= \sum_{i \in \bbN_h \cup \{0\}} \bA_i$, and in particular, $\lambda_{\rm min}(\bF_{S,h}^\top \bF_{S,h}) \ge \sum_{i} \lambda_{\rm min}(\bA_i)$. Using this, the tail bound for $\lambda^{\min}(\bA_i)$ with $x \gets x +\log(|\bbN_h|+1)$, some simplifying algebra, and the union bound, we conclude that, for all $x > 0$, \[eq:control\_eigen\_FF\] ¶e\^[-x]{} , since $$\sum_{i \in \bbN_h \cup \{0\}} \Big(n_i - 2\sqrt{|\bbN_h| n_i}- 2\sqrt{2x n_i}\Big) \ge \sum_{i \in \bbN_h \cup \{0\}} n_i - 2(\sqrt{|\bbN_h|} + \sqrt{2x}) \sum_{i \in \bbN_h \cup \{0\}} \sqrt{n_i}\ ,$$ with $\sum_{i \in \bbN_h \cup \{0\}} n_i = |S^h|$, and $\sum_{i \in \bbN_h \cup \{0\}} \sqrt{n_i}\leq \sqrt{|S^h|(|\bbN_h|+1)}$, by the Cauchy-Schwarz inequality. Taking $x=C |S^h|/(|\bbN_h|+1)$ in the above inequality for a sufficiently small constant $C$ and relying on Condition , we get $$\P\left\{\lambda^{\min}(\bF_{S,h}^\top \bF_{S,h}^\top )\leq \tfrac12 |S^h|\right\} \le \exp\left(-C |S^h|/(|\bbN_h|+1)\right) \ .$$ We now turn to bounding $\|\bF_{S,h}X_{S,h}\|_{\infty}$. Each component of $\bF_{S,h}X_{S,h}$ is of the form $Q_v:=\sum_{i\in S^h}X_{i}X_{i+v}$ for some $v\in \bbN_h$. Note that $Q_v$ is a quadratic form of $|S|$ standard normal variables, and the corresponding symmetric matrix has zero trace, Frobenius norm equal to $\sqrt{|S^h|/2}$, and operator norm smaller than $1$ by diagonal dominance. Combining Lemma \[lem:normal-quad\] with a union bound, we get ¶{\_[S,h]{}X\_[S,h]{}\_+ 2 (x+|\_h|) } 2e\^[-x]{} , x&gt;0 . Taking $x=C|S^h|/|\bbN_h|^2$ in the above inequality for a sufficiently small constant $C$ and using once again Condition allows us to get the bound $$\P\left\{ \|\bF_{S,h}X_{S,h}\|_\infty\geq \frac{|S^h|}{2|\bbN_h|} \right\} \le \exp[-C |S^h|/|\bbN_h|^2]\ .$$ Plugging these bounds into , we conclude that \[eq:bound\_phi\_l1\] ¶{\_S\_1&gt; 1} 3(-C ) . [*Control of $\E[\tilde{Z}_S]$*]{}. Since $$\tilde{Z}_S\leq Z_S = \|\boldsymbol{\Pi}_{S,h}X_{S,h}\|_2^2\leq \|(\bF_{S,h}^\top \bF_{S,h})^{-1}\|\|\bF_{S,h}X_{S,h}\|_2^2 \le \|X_{S,h}\|_2^2\ ,$$ we have, for any $a > 0$, && a +\ && a +  , where we used the Cauchy-Schwarz inequality in the second line. Since, under the null, $X_S\sim \cN(0,\bI_S)$, it follows that $\E\left[\|\bF_{S,h}X_{S,h}\|_2^2\right]=|\bbN_h(S)||S^h|$ and $\E\left[\|X_{S,h}\|_2^4\right]= |S^h|(|S^h|+2)$. Gathering this, the deviation inequality with $x=C|S^h|/|\bbN_h|^2$ with a small constant $C>0$, and Condition , and choosing as threshold $a =(|S^h|(1-|\bbN_h|^{-1/2})^{-1}$, leads to $$\begin{aligned} \E[\tilde{Z}_S]&\leq & \frac{|\bbN_h|}{1-|\bbN_h|^{-1/2}}+ \sqrt{3}|S^h|\sqrt{\P\left\{\lambda^{\min}(\bF_{S,h}^\top \bF_{S,h})\le 1/a\right\}}\nonumber\\ &\leq & |\bbN_h|+ C' |\bbN_h|^{1/2}+ \sqrt{3}|S^h| \exp\left(-C\frac{|S^h|}{|\bbN_h|^2}\right)\nonumber\\ &\leq & |\bbN_h|+ C |\bbN_h|^{1/2}\label{eq:upper_EZ}\ .\end{aligned}$$ [*Control of $\E\big[\sup_{\|\phi\|_1\leq 1} \tr(\bR_{\phi,S} X_{S}X_{S}^\top \bR_{\phi,S})\big]$*]{}. As explained above, $\|\bR_{\phi,S}\|\leq 3$ and we are therefore able to bound this expectation in terms of $\E[\tilde{Z}_S]$ as follows: \[eq:upper\_ER2\] 3C |\_h| , where we used in the last inequality. Combining the decomposition with Lemma \[lem:chaos\] and , and , we obtain $$\P\left\{Z_S\geq |\bbN_h|+ C\left(|\bbN_h|^{1/2}+\sqrt{|\bbN_h|t}+ t\right)\right\}\leq e^{-t}+ 3\exp\left(-C' \frac{|S^h|}{|\bbN_h|^2}\right)\ , \quad \forall t>0\ .$$ Since $$T_S= \frac{|S^h|Z_S}{\|X_{S,h}\|_2^2-Z_S}, \text{ where $\|X_{S,h}\|_2^2$ follows a $\chi^2$ distribution with $|S^h|$ degrees of freedom}\ ,$$ from Lemma \[lem:normal-quad\], we derive $$\P\Big[\|X_{S,h}\|_2^2\geq |S^h|-2 \sqrt{|S^h|t}-2t\Big]\leq e^{-t}\ ,$$ for any $t>0$, and from these two deviation inequalities, we get, for all $t\leq C'' |S^h|$, ¶2e\^[-t]{}+ 3 . Finally, we take a union bound over all $S\in \cC$ and invoke again Condition to conclude that, for any $t\leq C'' |S^h|$, $$\P\left\{\max_{S\in \cC} T_S \geq |\bbN_h|+ C\left(\sqrt{|\bbN_h|(\log(|\cC|)+ 1+t )}+ \log(|\cC|)+ t\right) \right\}\leq 2e^{-t}+ 3|\cC|\exp\left(-C' \frac{|S^h|}{|\bbN_h|^2}\right) \ .$$ To conclude, we let $t=\log(1/(4\alpha))$ in the above inequality, and use the condition on $\alpha$ in the statement of the theorem together with Condition , to get the following control of $T^*$ under the null hypothesis: $$\P\left\{\max_{S\in \cC} T_S \geq |\bbN_h|+ C\left(\sqrt{|\bbN_h|(\log(|\cC|)+ 1+\log(\alpha^{-1}) )}+ \log(|\cC|)+ \log(\alpha^{-1})\right) \right\} \leq \alpha \ .$$ [**Under the alternative hypothesis**]{}. Next we study the behavior of the test statistic $T^*$ under the assumption that there exists some $S\in \cC$ such that $X_S=Y_S\sim \cN(0, \bGamma_S(\phi))$. Since $T^*\geq T_S$, it suffices to focus on this particular $T_S$. For any $i \in S^h$, recall that $Y_i= \phi^\top F_i+ \epsilon_i$ where $F_i = (Y_{i+v} : 1 \le |v|_{\infty} \le h)$ and $\epsilon_i$ is independent of $F_i$. Hence, $Z_S$ decomposes as Z\_S &= &\_[S,h]{}Y\_[S,h]{}\_2\^2\ &= &\_[S,h]{}+ \_[S,h]{}\_[S,h]{} \_2\^2\ &= & \_[S,h]{}\_2\^2+ 2 \^\_[S,h]{}\^\_[S,h]{}+ \_[S,h]{}\_[S,h]{}\_2\^2 = ([I]{}) + ([II]{}) +([III]{}) . To bound the numerator of $T_S$, we bound each of these three terms. (I) and (II) are simply quadratic functions of multivariate normal random vectors and we control their deviations using Lemma \[lem:normal-quad\]. In contrast, (III) is more intricate and we use an ad-hoc method. In order to structure the proof, we state four lemmas needed in our calculations. We provide proofs of the lemmas further down. \[lem:control\_(I)\] Under condition , there exists a numerical constant $C>0$ such that \[eq:deviation\_I\] ¶{([I]{})\_\^2 } 1 -(-C) . \[lem:control\_(II)\] For any $t>0$, $$\begin{aligned} \label{eq:deviation_II-a} \P\left\{ ({\rm II})\geq - 2\sigma_{\phi}\|\phi\|_2\sqrt{2 |S^h| \|\bGamma(\phi)\|(2+\|\phi\|_1)t}- 12\big[\|\bGamma(\phi)\|\vee (1+\|\phi\|_1)\sigma_{\phi}^2\big]t \right\} &\geq& 1 -e^{-t}\ ,\\ \label{eq:deviation_II-b} \P\left\{({\rm II}) \geq -2\sqrt{2} \sigma_{\phi}\sqrt{(|\bbN_h|+1)[\log(|\bbN_h|+1)+ t]} \|\bF_{S,h} \phi\|_2\right\}&\geq& 1-e^{-t}\ .\end{aligned}$$ Recall that $\gamma_j= (\bGamma(\phi))_{0,j}$ denotes the covariance between $Y_0$ and $Y_j$. \[lem:control\_(III)\] Denote by $\bGamma_{\bbN_h}(\phi)$ the covariance matrix of $(Y_{i},\ i\in \bbN_h)$. For any $t\leq |S^h|$, \[eq:deviation\_wisharts\] \^[max]{}(\_[\_h]{}()\^[-1/2]{}\_[\_h]{}()\^[-1/2]{})1+ 4()\^[-1]{}()(+(|\_h|)) with probability larger than $1-2e^{-t}$. Also, for any $t\ge 1$, \[eq:control\_bSigma\] |\_h|- C(|\_h|\_1+ \^[-1]{}()\_2\^2+ )t\^2 with probability larger than $1-2e^{-t}$. To bound the denominator of $T_S$, we start from the inequality $$\|Y_{S,h}\|_2^2 - \|\boldsymbol{\Pi}_{S,h}Y_{S,h}\|_2^2= \|\epsilon_{S,h}\|_2^2- \|\boldsymbol{\Pi}_{S,h}\epsilon_{S,h}\|_2^2 \leq \|\epsilon_{S,h}\|_2^2$$ and then use the following result. \[lem:control\_(IV)\] Under condition , we have \[eq:upper\_epsilon\] ¶{\_[S,h]{}\^2 \_\^2|S\^h|(1+ |\_h|\^[-1/2]{})}1-(-C )  . With these lemmas in hand, we divide the analysis into two cases depending on the value of $\|\phi\|_2^2$. For small $\|\phi\|^2_2$, the operator norm of the covariance operator $\bGamma(\phi)$ remains bounded, which simplifies some deviation inequalities. For large $\|\phi\|^2_2$, we are only able to get looser bounds which are nevertheless sufficient as in that case $\|\phi\|_2^2$ is far above the detection threshold. [**Case 1**]{}: $\|\phi\|_2^2\leq (4|\bbN_h|)^{-1}$. This implies that $\|\phi\|_1\leq 1/2$ and also that $\|\bGamma(\phi)\|\leq 2\sigma_{\phi}^2$ by Lemma \[lem:spectrum\_gamma\]. Combining and together with the inequality $2xy\leq x^2+y^2$, we derive that for any $t>0$, \[eq:lower\_case1\] C(|S\^h|\_2\^2- t) with probability larger than $1-e^{-t} -\exp\left(-C\frac{|S^h|}{(|\bbN_h|+1)}\right)$. Turning to the third term, we have $$\frac{({\rm III})}{\sigma_{\phi}^2} \geq \lambda^{\rm max}\left(\bGamma_{\bbN_h}(\phi)^{-1/2}\frac{\bF_{S,h}^\top \bF_{S,h}}{|S^h|}\bGamma_{\bbN_h}(\phi)^{-1/2}\right)^{-1}\frac{\|\bGamma_{\bbN_h}(\phi)^{-1/2}\bF_{S,h}^\top \epsilon_{S,h}\|_2^2}{\sigma_{\phi}^2 |S^h|}\ .$$ Let $a>0$ be a positive constant whose value we determine later. For any $t>0$, with probability larger than $1-4e^{-t}$, we have &&\ && |\_h|- C(|\_h|\^[3/2]{}\_2+ \_2\^2+ )t\^2 - 16(+(|\_h|))\ &&|\_h|- C(|\_h|\^[3/2]{}\_2+ )t\^2 - C’(1+ t\^4)\ && |\_h|- a |S\^h|\^2\_2 - C(a\^[-1]{}t\^4 + t\^2 + (1+ t\^4))\ && |\_h| - a |S\^h|\^2\_2 - C(1+ (a\^[-1]{}+1)t\^4) . Here in the first line, we used Lemma \[lem:control\_(III)\]. In the second line, we used the fact that $(1-y)/(1+x)\geq 1-x-y$ for all $x,y \ge 0$, $\|\phi\|_1\leq \sqrt{|\bbN_h|}\|\phi\|_2$ by the Cauchy-Schwarz inequality, and $\|\bGamma(\phi)\|\vee \|\bGamma^{-1}(\phi)\|\leq 2$. In the third line, we applied the inequality $\sum_{j\neq 0}\gamma_{j}^2\leq 4\|\phi\|_2^4+ 16\|\phi\|_2^2\leq 20$, which is a consequence of $\|\phi\|_1\leq 1/2$ and Lemma \[lem:conditional\_variance\]. The last line is a consequence of Condition . Then, we take $a=C/2$ with $C$ as in and apply Lemma \[lem:control\_(IV)\] to control the denominator of $T_S$. This leads to $$\P\left\{T_S \geq C |S^h|\|\phi\|_2^2 + |\bbN_h|- C' \sqrt{\bbN_h}(1\vee t^4)\right\}\geq 1 - 4e^{-t} -2e^{-C'' \frac{|S^h|}{|\bbN_h|^2}}\ .$$ Taking $t=\log(8/\beta)$ and letting $C_2$ be small enough in , we get $$\P\left\{T_S \geq C |S^h|\|\phi\|_2^2 + |\bbN_h|- C' \sqrt{\bbN_h}(1+\log^{4}(\beta^{-1}))\right\}\geq 1 - \beta\ ,$$ proving in Case 1. [**Case 2**]{}: $\|\phi\|_2^2\geq (4 |\bbN_h|)^{-1}$. This condition entails $$\frac{2\|\phi\|_2^2}{1+\|\phi\|_1}\geq \frac{\|\phi\|_2}{\sqrt{|\bbN_h|}}\ .$$ Since the term $({\rm III})$ is non-negative, we can start from the lower bound $Z_S\geq ({\rm I})+ ({\rm II})$. We derive from Lemma \[lem:control\_(I)\] and the above inequality that \[eq:lower\_I-bis\] ¶{([I]{})|S\^h| }1- (-C) . Taking $t=C |S^h|/|\bbN_h|^2$ in for a constant $C$ sufficiently small, and using Condition , we get that $({\rm II}) \ge - 3 \sqrt{C} \sigma_\phi \sqrt{|S^h|/|\bbN_h|} \sqrt{({\rm I})}$ with probability at least $1 - e^{-t}$. Also, $\|\phi\|_2^2 \ge (4|\bbN_h|)^{-1}$ implies that the right-hand side exceeds $- \frac12 ({\rm I})$ when the event in holds and $C$ is small enough. Hence, we get $$\P\left\{({\rm I}) + ({\rm II})\geq |S^h|\frac{\sigma_{\phi}^2\|\phi\|_2}{8\sqrt{|\bbN_h|}} \right\}\geq 1- 2 \exp\Big(-C\frac{|S^h|}{|\bbN_h|^2}\Big)\ .$$ Finally, we combine this bound with and the condition $\|\phi\|_2^2 \ge (4|\bbN_h|)^{-1}$, to get ¶{T\_S }1- 3 (-C) 1-, where we used the condition on $\beta$. In view of Condition , we have proved . This concludes the proof of . It remains to prove the auxiliary lemmas. ### Proof of Lemma \[lem:control\_(I)\] Recall the definition of $S_i$ in . Let $\bF_{S_i}$ denote the matrix with row vectors $F_j, j \in S_i$. We have $$({\rm I}) = \|\bF_{S,h}\phi\|^2_2 = \sum_{i\in \bbN_h\cup\{0\}} \|\bF_{S_i}\phi\|^2_2 \ .$$ For any $u\in \bbR^{S_i}$, = u\_2\^2\_2\^2 \^(()) , since the indices $(v+j : v\in\bbN_{h}, j\in S_i)$ are all distinct. Since $\lambda^{\min}(\bGamma(\phi))\geq \frac{\sigma_{\phi}^2}{1+\|\phi\|_1}$ by Lemma \[lem:spectrum\_gamma\], $\frac{1+\|\phi\|_1}{\sigma_\phi^2\|\phi\|_2^2}\|\phi^\top \bF_{S_i}\|_2^2$ is stochastically lower bounded by a $\chi^2$ distribution with $|S_i|$ degrees of freedom. By Lemma \[lem:normal-quad\] and the union bound, we have that for any $t>0$, ([I]{})&& \_\^2 \_[i\_h{0}]{} |S\_i| -2\ && \_[\^2 ]{} (|S\^h|- 2 ) , with probability larger than $1-e^{-t}$. Finally we set $t=\tfrac{|S^h|}{32(|\bbN_h|+1)}$ and use Condition to conclude. ### Proof of Lemma \[lem:control\_(II)\] We first prove . Denote by $\tilde{\bSigma}_{\phi,S}$ the covariance matrix of the random vector $(\epsilon^\top _{\phi,S},\phi^\top \bF_{S,h}^\top )$ of size $2|S^h|$. Let $\bR$ be the block matrix defined by $$\bR=\begin{pmatrix} 0 &\bI_{S^h} \\ \bI_{S^h} & 0 \end{pmatrix}.$$ Letting $Z$ be a standard Gaussian vector of size $2|S^h|$, we have $2\phi^\top \bF_{S,h}^\top \epsilon_{S,h}\sim Z^\top \tilde{\bSigma}^{1/2}_{\phi,S}\bR \tilde{\bSigma}^{1/2}_{\phi,S}Z \ .$ From Lemma \[lem:normal-quad\] we get that for all $t>0$, with probability at least $1-e^{-t}$, $$\begin{aligned} 2\phi^\top \bF_{S,h}^\top \epsilon_{S,h}& \le& \tr[\tilde{\bSigma}^{1/2}_{\phi,S}\bR\tilde{\bSigma}^{1/2}_{\phi,S}]- 2\|\tilde{\bSigma}^{1/2}_{\phi,S}\bR\tilde{\bSigma}^{1/2}_{\phi,S}\|_F\sqrt{t}- 2 \|\tilde{\bSigma}^{1/2}_{\phi,S}\bR\tilde{\bSigma}^{1/2}_{\phi,S}\|t\ , \notag \\ %&\geq& - 2\|\tilde{\bSigma}^{1/2}_{\phi,S}\bR\tilde{\bSigma}^{1/2}_{\phi,S}\|_F\sqrt{t}- 2 \|\tilde{\bSigma}_{\phi,S}\|\|\bR\|t\\ &\le & - 2\|\tilde{\bSigma}^{1/2}_{\phi,S}\bR\tilde{\bSigma}^{1/2}_{\phi,S}\|_F\sqrt{t} - 2 \|\tilde{\bSigma}_{\phi,S}\| t\ , \label{phi_F_eps}\end{aligned}$$ where we used the fact that $\tr[\tilde{\bSigma}^{1/2}_{\phi,S}\bR\tilde{\bSigma}^{1/2}_{\phi,S}] = \E[\phi^\top \bF_{S,h}^\top \epsilon_{S,h}]=0$ and that $\|\bR\| = 1$. In order to bound the Frobenius norm above, we start from the identity $$\|\tilde{\bSigma}^{1/2}_{\phi,S}\bR\tilde{\bSigma}^{1/2}_{\phi,S}\|^2_F = \Var[2\phi^\top \bF_{S,h}^\top \epsilon_{S,h}] = \E\left[(2 \phi^\top \bF_{S,h}^\top \epsilon_{S,h})^2\right] = 4 \sum_{i,j\in S^h}\E[\epsilon_i\epsilon_j (\phi^\top F_i)(\phi^\top F_j)] \ ,$$ with $\eps_i$ being the $i$th component of $\eps_{S,h}$. For $i=j$, the expectation of the right-hand side is $\sigma_{\phi}^2\E[(\phi^\top F_i)^2]$, while if the distance between $i$ and $j$ is larger than $h$, then $\epsilon_i$ and $(\epsilon_j,F_i,F_j)$ are independent and the expectation of the right-hand side is zero. If $1 \le |i-j|\leq h$, then we use Isserlis’ theorem, together with the fact that $\eps_i \perp F_i$, to obtain ||&=& |+ E\[\_j\^F\_i\]|\ & & \_\^2|\_[i-j]{}|+ \^2\_[j-i]{}\_\^2 . Putting all the terms together, we obtain \^[1/2]{}\_[,S]{}\^[1/2]{}\_[,S]{}\^2\_F && 4 |S\^h|\_\^2 { (1 +\_1) + \_2\^2 }\ && 4 \_\^2|S\^h|\_2\^2 ()(2+\_1) , using the fact that $\|\bGamma(\phi)\| \ge 1$. Turning to $\|\tilde{\bSigma}_{\phi,S}\|$, denote $\bGamma(\phi)^{\epsilon}$ the covariance of the process $(\epsilon_i,\ i\in \bbZ^d)$. By Lemma \[lem:covariance\_residuals\], $(\bGamma(\phi)^{\epsilon})_{i,j}=\left[- \phi_{i-j}+ \1_{i=j}\right]\sigma_{\phi}^2$, and it follows that $\|\bGamma(\phi)^{\epsilon}\|\leq (1+\|\phi\|_1)\sigma_{\phi}^2$. Then, for all vectors $u,v \in \bbR^{S^h}$, (\_[iS\^h]{} u\_i\^F\_i + \_[iS\^h]{} v\_i\_i)&= & (\_[iS\^h]{} u\_iY\_i + \_[iS\^h]{} (v\_i-u\_i)\_i)\ && 2 (\_[iS\^h]{} u\_iY\_i )+ 2(\_[iS\^h]{} (v\_i-u\_i)\_i)\ && 2u\_2\^2 () + 2 u-v\_2\^2 ()\^\ && 6(u\^2\_2+ v\_2\^2)  . Consequently, $\|\tilde{\bSigma}_{\phi,S}\|\leq 6[\|\bGamma(\phi)\|\vee \|\bGamma(\phi)^{\epsilon}\|]\leq 6[\|\bGamma(\phi)\|\vee (1+\|\phi\|_1)\sigma_{\phi}^2]$. We conclude that holds by virtue of the two bounds we obtained for the two terms in . Turning to , we decompose $({\rm II})$ into $2 \sum_{i\in \bbN_h\cup\{0\}}\phi^\top \bF_{S_i}^\top \epsilon_{S_i}$. For any $j_1\neq j_2\in S_i$, $|j_1-j_2|_{\infty}\geq 2h+1$ and therefore $\epsilon_{j_1}$ is independent of $(Y_{j_2+v},\ v\in\bbN_h\cup\{0\})$. Since $\epsilon_{j_2}$ and $F_{j_2} \phi$ are linear combinations of this collection, we conclude that $\epsilon_{j_1}\perp (\epsilon_{j_2}^\top , \phi^\top F_{j_2}^\top )$. Consequently, $\epsilon_{S_i}/\sigma_\phi$ follows a standard normal distribution and is independent of $\bF_{S_i} \phi$. By conditioning on $\bF_{S_i} \phi$ and applying a standard Gaussian concentration inequality, we get $$\P\Big\{|\phi^\top \bF_{S_i}^\top \epsilon_{S_i}| \leq \sigma_\phi\|\bF_{S_i} \phi\|_2\sqrt{2t} \Big\} \leq e^{-t}\ ,$$ for any $t>0$. We then take a union bound over all $i\in \bbN_h\cup\{0\}$. For any $t>0$, ([II]{}) && -2 \_ \_[i\_h{0}]{}\^\_[S\_i]{}\_2\ && -2 \_ \_[S,h]{}\_2 , with probability larger than $1-e^{-t}$. ### Proof of Lemma \[lem:control\_(III)\] [**Proof of .**]{} Fix $(v_1,v_2)\in \bbN_h$ and consider the random variable $$(\bF_{S,h}^\top \bF_{S,h})_{v_1,v_2} = \sum_{i\in S^h}Y_{i+v_1}Y_{i+v_2} = Y_{S}^\top \bR Y_S = V^\top \bGamma_S(\phi)^{1/2}\bR \bGamma^{1/2}_{S}(\phi)V\ ,$$ which constitutes a definition for the symmetric matrix $\bR$, and $V\sim \cN(0,\bI_S)$. Observe that $\|\bR\|_F^2=|S^h|$ and $\|\bR\|\leq 1$ as the $l_1$ norm of each row of $\bR$ is smaller than one. We derive from Lemma \[lem:normal-quad\], and the fact that $\|\bGamma_S(\phi)^{1/2}\bR \bGamma^{1/2}_{S}(\phi)\|_F^2\leq \|\bR\|_F^2\|\bGamma^{1/2}_{S}(\phi)\|^4\leq |S^h|\|\bGamma(\phi)\|^2$ and $\|\bGamma_S(\phi)^{1/2}\bR \bGamma^{1/2}_{S}(\phi)\|\leq \|\bR\|\|\bGamma_S(\phi)\|\leq \|\bGamma(\phi)\|$, that for any $t>0$, $$\P\left\{\big|(\bF_{S,h}^\top \bF_{S,h})_{v_1,v_2}- |S^h|\gamma_{v_1,v_2}\big|\leq 2\|\bGamma(\phi)\|\sqrt{|S^h|t}+ 2\|\bGamma(\phi)\|t \right\}\leq 2e^{-t}~.$$ Then we bound the $\ell_2$ operator norm of $|S^h|^{-1}\bF_{S,h}^\top \bF_{S,h}- \bGamma_{\bbN_h}(\phi)$ by its $\ell_1$ operator norm and combine the above deviation inequality with a union bound over all $(v_1,v_2)\in \bbN_h$. Thus, for any $t\leq |S^h|$, - \_[\_h]{}()&& \_[v\_1\_h]{}\_[v\_2\_h]{}|- \_[v\_1,v\_2]{}|\ && 2()( + )\ && 4()(+ (|\_h|))  , with probability larger than $1-2e^{-t}$. Hence, under this event, $$\lambda^{\rm max}\left(\bGamma_{\bbN_h}(\phi)^{-1/2}\frac{\bF_{S,h}^\top \bF_{S,h}}{|S^h|}\bGamma_{\bbN_h}(\phi)^{-1/2}\right)\leq 1+ 4\|\bGamma(\phi)\|\|\bGamma^{-1}(\phi)\|\frac{|\bbN_h|}{|S^h|^{1/2}}\left(\sqrt{t}+\log(|\bbN_h|)\right)\ ,$$ since $\|\bGamma_{\bbN_h}(\phi)^{-1}\|\leq \|\bGamma^{-1}(\phi)\|$. This concludes the proof of . [**Proof of .**]{} Turning to the second deviation bound, we use the following decomposition $$\|\bGamma_{\bbN_h}(\phi)^{-1/2}\bF_{S,h}^\top \epsilon_{S,h}\|_2^2= \sum_{i\in S^h} \epsilon_{i}^2\|\bGamma_{\bbN_h}(\phi)^{-1/2}F_{i}\|_2^2+ \sum_{(i,j),\ i\neq j } \epsilon_i\epsilon_j F_j^\top \bGamma_{\bbN_h}(\phi)^{-1}F_i =: A+ B\ ,$$ with $\eps_i$ being the $i$th entry of $\eps_{S,h}$. Since both $A$ and $B$ are Gaussian chaos variables of order 4, we apply Lemma \[lem:chaos\_4\] to control their deviations. For any $t>0$, \[eq:decomposition\_A+B\] ¶{A+B- C(\^[1/2]{}(A)+ \^[1/2]{}(B))t\^2 }2e\^[-t]{} , using the fact that $\Var^{1/2}(A+B) \le \Var^{1/2}(A)+ \Var^{1/2}(B)$. Thus, it suffices to compute the expectation and variance of $A$ and $B$. First, we have $\E[A]= |S^h||\bbN_h|\sigma_{\phi}^2$, by independence of $\eps_i$ and $F_i$, and from this we get (A)&=&\_[i,j S\^h]{}(- \^4\_|\_h|\^2 ) =: \_[i,j S\^h]{}A\_[i,j]{} . If $|i-j|_{\infty}\leq h$, we may use the Cauchy-Schwarz inequality to get |A\_[i,j]{}|= 3\_\^4|\_h|(|\_[h]{}|+2) , again by independence of $\epsilon_i$ and $\|\bGamma_{\bbN_h}(\phi)^{-1/2}F_{i}\|_2^2$. If $|i-j|_{\infty}> h$, then $\epsilon_i$ is independent of $(F_i,F_j,\epsilon_j)$ and $\epsilon_j$ is independent of $(F_i,F_j,\epsilon_i)$, so we get &=& - |\_h|\^2\ & = & \_[v\_1,v\_2,v\_3,v\_4\_h]{} (\_[\_h]{}()\^[-1]{})\_[v\_1,v\_2]{}(\_[\_h]{}()\^[-1]{})\_[v\_3,v\_4]{}\ & &- |\_h|\^2 \ & =& \_[v\_1,v\_2,v\_3,v\_4\_h]{} (\_[\_h]{}()\^[-1]{})\_[v\_1,v\_2]{}(\_[\_h]{}()\^[-1]{})\_[v\_3,v\_4]{}where we apply Isserlis’ theorem in the second line and use the definition of $\bGamma_{\bbN_h}(\phi)$ in the last line. By symmetry, we get && 2\_[\_h]{}()\^[-1]{}\_\^2\_[v\_1,v\_2,v\_3,v\_4\_h]{}|\_[i+v\_1-j-v\_3]{}\_[i+v2-j-v\_4]{}|\ && 2\_[\_h]{}()\^[-1]{}\^2|\_h| \_[v\_1,v\_2\_h]{}\^2\_[i-j+v\_1-v\_2]{}\ && 2\^[-1]{}()\^2 |\_h|\^2 \_[v\_[2h]{}]{}\^2\_[i-j+v]{} , using the Cauchy-Schwarz inequality in the second line. Here $\|\bA\|_{\infty}$ denotes the supremum norm of the entries of $\bA$. Then, summing over all $j$ lying at a distance larger than $h$ from $i$, \_[jS\^h, |j-i|\_&gt;h]{} && 2\^[-1]{}()\^2 |\_h|\^2 \_[jS\^h, |j-i|\_&gt;h]{} \_[v\_[2h]{}]{}\^2\_[i-j+v]{}\ && 2\^[d+1]{}\^[-1]{}()\^2 |\_h|\^3 \_[j\^d{0}]{}\_j\^[2]{} . Putting the terms together, we conclude that \[eq:var\_A\] (A)\_\^4|S\^h||\_h|\^3(6+ 2\^[d+1]{}\_[j\^d{0}]{}\_j\^[2]{}) . Next we bound the first two moments of $B$. Consider $(i,j)\in S^h$ such that $|i-j|_{\infty}>h$. Then $\E\big[\epsilon_i\epsilon_j F_j^\top \bGamma_{\bbN_h}(\phi)^{-1}F_i\big] = 0$ by independence of $\epsilon_i$ with the other variables in the expectation. Suppose now that $|i-j|_{\infty}\leq h$. By Isserlis’ theorem, and the independence of $\eps_i$ and $F_i$, as well as $\eps_j$ and $F_j$, and symmetry, to get &=& \_[\_h]{}()\^[-1]{}+\ && - \^4\_|\_[i-j]{}|\^2\_[\_h]{}()\^[-1]{} -\_\^2|\_[i-j]{}| |\_h| , using the Cauchy-Schwarz inequality and Lemma \[lem:covariance\_residuals\]. As a consequence, \[eq:E\_B\] -\_\^4|S\^h|\_2\^2\^[-1]{}() - \_\^2|S\^h||\_h|\_1 . Turning to the variance, we obtain $$\Var(B)\leq \E[B^2] = \sum_{i_1\neq i_2} \sum_{i_3\neq i_4 }\E[V_{i_1,i_2,i_3,i_4}]\ ,$$ where $$V_{i_1,i_2,i_3,i_4} := \epsilon_{i_1}\epsilon_{i_2}\epsilon_{i_3}\epsilon_{i_4} F_{i_1}^\top \bGamma_{\bbN_h}(\phi)^{-1}F_{i_2} F_{i_3}^\top \bGamma_{\bbN_h}(\phi)^{-1}F_{i_4}\ .$$ Fix $i_1$. If one index among $(i_1,i_2,i_3,i_4)$ lies at a distance larger than $h$ from the three others, then the expectation of $V_{i_1,i_2,i_3,i_4}$ is equal to zero. If one index lies within distance $h$ of $i_1$ and the two remaining indices lie within distance $3h$ of $i_1$, we use the Cauchy-Schwarz inequality to get $$\E[V_{i_1,i_2,i_3,i_4}] \le \E\Big[\prod_{k=1}^4 |\epsilon_{i_k}| (F_{i_k}^\top \bGamma_{\bbN_h}(\phi)^{-1}F_{i_k})^{1/2}\Big] \\ \leq \E[\epsilon_1^4(F_1^\top \bGamma_{\bbN_h}(\phi)^{-1}F_1)^2]= 3\sigma_{\phi}^4|\bbN_h|(|\bbN_h|+2)\ .$$ Finally, if say $|i_1-i_2|_{\infty}\leq h$ and $|i_3-i_4|_{\infty}\leq h$ and $|i_k - i_\ell| > h$ for $k = 1,2$ and $\ell = 3,4$, then we use again Isserlis’ theorem and simplify the terms to get &=&\ &+& \_[\_h]{}()\^[-1]{}\_[\_h]{}()\^[-1]{}\ && \_\^4|\_[i\_2-i\_1]{}\_[i\_4-i\_3]{}||\_h|(|\_h|+2)+ \^[-1]{}()\^2\_\^8 \_[i\_2-i\_1]{}\^2\_[i\_4-i\_3]{}\^2  , where we used again Lemma \[lem:covariance\_residuals\] to control the terms involving $\epsilon$’s and the Cauchy-Schwarz inequality to bound the term in $(F_{i_k}, k=1,\ldots,4)$. Putting all the terms together, we conclude that \[eq:var\_B\] (B)C \_\^4(|S\^h||\_h|\^5 + |S\^h|\^2\_1\^2|\_h|\^2+ |S\^h|\^2\_\^2\_2\^4 \^[-1]{}()\^2) , since $\sigma_{\phi}^2\leq \Var[Y_i]= 1$. Plugging in the bounds that we obtained for the moments of $A$ and $B$ in , we conclude the proof of . ### Proof of Lemma \[lem:control\_(IV)\] Recall the definition of $S_i$ in . We decompose $\|\epsilon_{S,h}\|_2^2= \sum_{i\in \bbN_h\cup\{0\}}\|\epsilon_{S_i}\|^2_2$ and note that $\|\epsilon_{S_i}\|^2_2 \sim \sigma_\phi^2 \chi^2_{|S_i|}$. Applying the second deviation bound of Lemma \[lem:normal-quad\] together with a union bound, we obtain that for any $t>0$, \_[S,h]{}\_2\^2 && \_\^2 \_[i\_h{0}]{} (|S\_i|+ 2+ 2 t+ 2(|\_h|+1) )\ && \_\^2(|S\^h| + 2+ 2|\_h|(t+ (|\_h|+1))) , with probability larger $1-e^{-t}$. Relying on Condition , we derive that ¶{\_[S,h]{}\^2\_\^2|S\^h|(1+ |\_h|\^[-1/2]{})} 1-(-C ) , for a numerical constant $C>0$ small enough. Proof of Corollary \[cor:AR\] ----------------------------- It is well known—see, e.g., [@MR1419991]—that any $\ar_h$ process is also a Gaussian Markov random field with neighborhood radius $h$ (and vice-versa). Denote $\tau_\psi^2$ the innovation variance of an $\ar_h(\psi)$ process. The bijection between the parameterizations $(\psi,\tau_\psi^2)$ and $(\phi,\sigma_{\phi}^2)$ is given by the following equations $$\begin{aligned} \phi_{-i}~=~\phi_i&=&\frac{\psi_{i}- \sum_{k=i+1}^h \psi_k \psi_{k-i}}{1+ \|\psi\|_2^2}~,\quad \quad \text{for $i=1,\ldots, h$ ,} \label{eq:phi_psi_ar} \\ \sigma^2_{\phi}&=& \frac{\tau_\psi^2}{1+\|\psi\|_2^2} \ .\label{eq:variance_sigma_ar}\end{aligned}$$ This correspondence is maintained below. [**Lower bound**]{}. In this proof, $C$ is a positive constant that may vary from line to line. It follows from the above equations that $$\|\phi\|_2^2 \leq C\frac{\|\psi\|_2^2 + h\|\psi\|_2^4}{1+\|\psi\|_2^2} \ .$$ Consider any $r\leq 1/h$. In that case, if $\|\phi\|_2\geq r$ then the inequality above implies that $\|\psi\|_2\geq Cr$, and as a consequence, $R^*_{\cC,\mathfrak{G}(h,r)}\leq R^*_{\cC,\mathfrak{F}(h,Cr)}$. Therefore, since and our condition on $h$ together imply that $r \le 1/h$ eventually, it suffices to prove that $R^*_{\cC,\mathfrak{G}(h,r)} \to 1$. For that, we apply Corollary  \[cor:lower\_hypercube\]. Condition there is satisfied eventually under our assumptions ( and our condition on $h$). Consequently, we have $R^*_{\cC,\mathfrak{G}(h,r)}\to 1$ as soon as holds, which is the case when holds. [**Upper bound**]{}. It follows from and the inequality $\tau_\psi^2\leq 1$ that $$1- \sigma^2_{\phi}\geq \frac{\|\psi\|_2^2}{1+\|\psi\|_2^2}\ .$$ Denoting $u_n:=\log(n)/k + \sqrt{h\log(n)}/k$, observe as above that $u_n\ll 1/h$ by our assumption on $h$. Assume that $\|\psi\|_2^2\geq r^2$ for some $r^2\geq u_n$. If $\|\phi\|_1\leq 1/2$, it follows from the inequality $1-\sigma^2_{\phi}\leq \|\phi\|_2^2/(1-\|\phi\|_1)\leq 2\|\phi\|^2_2$ (Lemma \[lem:spectrum\_gamma\]) that $\|\phi\|_2^2\geq r^2/4$. And if $\|\phi\|_1> 1/2$, then $\|\phi\|_2^2 \geq (8h)^{-1}$ by the Cauchy-Schwarz inequality. Thus, when $r^2 \le 1/h$, we have $\|\phi\|_2^2\geq r^2/8$, and this implies $$R_{\cC,\mathfrak{F}(h,r)}(f)\leq R_{\cC,\mathfrak{G}(h,r/\sqrt{8})}(f)~, \quad \text{for any test $f$.}$$ When $r^2\geq 1/h$, we simply use a monotonicity argument $$R_{\cC,\mathfrak{F}(h,r)}(f)\leq R_{\cC,\mathfrak{F}(h,h^{-1/2})}(f)\leq R_{\cC,\mathfrak{G}(h,1/\sqrt{8h})}(f)~, \quad \text{for any test $f$.}$$ The result then follows from Theorem \[thm:LS1\]. Acknowledgements {#acknowledgements .unnumbered} ---------------- This work was partially supported by the US National Science Foundation (DMS-1223137, DMS-1120888) and the French Agence Nationale de la Recherche (ANR 2011 BS01 010 01 projet Calibration). The third author was supported by the Spanish Ministry of Science and Technology grant MTM2012-37195. [^1]: Department of Mathematics, University of California, San Diego [^2]: Department of Operations Research and Financial Engineering, Princeton University [^3]: ICREA and Department of Economics, Universitat Pompeu Fabra [^4]: (corresponding author) INRA, UMR 729 MISTEA, F-34060 Montpellier, FRANCE

--- author: - 'M. Juvela' - 'K. Mattila' - 'D. Lemke' - 'U. Klaas' - 'C. Leinert' - 'Cs. Kiss' date: 'Received 1 January 2005 / Accepted 2 January 2005' title: ' Determination of the cosmic far-infrared background level with the ISOPHOT instrument [^1]' --- [The cosmic infrared background (CIRB) consists mainly of the integrated light of distant galaxies. In the far-infrared the current estimates of its surface brightness are based on the measurements of the COBE satellite. Independent confirmation of these results is still needed from other instruments.]{} [In this paper we derive estimates of the far-infrared CIRB using measurements made with the ISOPHOT instrument aboard the ISO satellite. The results are used to seek further confirmation of the CIRB levels that have been derived by various groups using the COBE data. ]{} [We study three regions of very low cirrus emission. The surface brightness observed with the ISOPHOT instrument at 90, 150, and 180$\mu$m is correlated with hydrogen 21cm line data from the Effelsberg radio telescope. Extrapolation to zero hydrogen column density gives an estimate for the sum of extragalactic signal plus zodiacal light. The zodiacal light is subtracted using ISOPHOT data at shorter wavelengths. Thus, the resulting estimate of the far-infrared CIRB is based on ISO measurements alone. ]{} [In the range 150 to 180$\mu$m, we obtain a CIRB value of 1.08$\pm$0.32$\pm$0.30MJysr$^{-1}$ quoting statistical and systematic errors separately. In the 90$\mu$m band, we obtain a 2-$\sigma$ upper limit of 2.3MJysr$^{-1}$. ]{} [ The estimates derived from ISOPHOT far-infrared maps are consistent with the earlier COBE results. ]{} Introduction ============ The extragalactic background light (EBL) consists of the integrated light of all galaxies along the line of sight with possible additional contributions from intergalactic gas and dust and hypothetical decaying relic particles. It plays an important role in cosmological studies because most of the gravitational and fusion energy released in the universe since the recombination epoch is expected to reside in the EBL. Measurements of the cosmic infrared background, CIRB, help to address some central, but still largely open astrophysical problems, including the early evolution of galaxies, and the entire star formation history of the universe. An important issue is the balance between the UV-optical-NIR and the far-infrared backgrounds; the fraction of optical radiation lost by dust obscuration re-appears as dust emission at longer wavelengths. The absolute level of the CIRB, the fluctuations in the CIRB surface brightness, and the resolved bright end of the distribution of galaxies contributing to the CIRB all provide strong constraints on the models of galaxy evolution through different epochs. For reviews, see Hauser & Dwek ([@Hauser2001]) and Lagache, Puget, & Dole ([@Lagache2005]). The full analysis of the data from the DIRBE (Hauser et al. [@Hauser1998]; Schlegel et al. [@schlegel]) and FIRAS (Fixen et al. [@fixen]) experiments indicated a CIRB at a surprisingly high level of $\sim$1 MJysr$^{-1}$ between 140 and 240 $\mu$m. Preliminary results had been obtained by Puget et al. ([@puget]). Lagache et al. ([@Lagache1999]) claimed the detection of a component of Galactic dust emission associated with warm ionised medium. The removal of this component led to a CIRB level of 0.7 MJysr$^{-1}$ at 140 $\mu$m. Because the FIR CIRB is important for cosmology these results need to be confirmed by independent measurements. The ISOPHOT instrument (Lemke at el. [@Lemke96]), flown on the cryogenic, actively cooled ISO satellite, provided the capabilities for this. The ISOPHOT observation technique was different from COBE: (1) with its relatively small f.o.v. ISOPHOT was capable of looking into the darkest spots between the cirrus clouds; (2) ISOPHOT had high sensitivity in the important FIR window at 120-200 $\mu$m; (3) with its good spatial and multi-wavelength FIR spectral sampling ISOPHOT gave an improved possibility of separating and eliminating the emission of Galactic cirrus. The primary goal of the ISOPHOT EBL project is the determination of the absolute level of the FIR CIRB. The other goals are the measurement of the spatial CIRB fluctuations and the detection of the bright end of the FIR point source distribution. The bright end of the galaxy population contributing to the FIR CIRB signal was analysed by Juvela et al. ([@Juvela00]). The method ========== We examine three regions of low cirrus emission that were mapped with the ISOPHOT at 90, 150, and 180$\mu$m. Because of the high sensitivity of the ISOPHOT FIR detectors, we can directly correlate HI with ISOPHOT measurements for each FIR band separately. In the case of DIRBE, the original analysis performed by the DIRBE team used 100$\mu$m as an ISM template and, therefore, the accuracy of the CIRB detections at 140$\mu$m and 240$\mu$m also depended on the systematic uncertainties of the 100$\mu$m data (Hauser et al. [@Hauser1998]; Arendt et al. [@Arendt1998]). The HI lines are optically thin and their intensity traces the amount of neutral hydrogen along the line-of-sight. The level of FIR emission associated with the ionised medium is still uncertain and we will consider the possible effects later in the analysis. As a first step, a relation between the HI line area and the FIR surface brightness is obtained. The relation depends on the gas-to-dust ratio, grain properties, and the radiation field illuminating the interstellar medium (ISM) along the line-of-sight. No significant variations have been observed in the gas-to-dust ratio apart from those associated with large scale metallicity variations. Similarly, because of the diffuse nature of the HI clouds, no small scale changes in the intrinsic dust properties or dust temperature are expected. Under these conditions the FIR signal should have a linear dependence on the HI column density. Because each field is considered individually, possible differences in the HI–FIR relation towards different regions can and will be taken into account. For each field, an extrapolation to zero HI intensity eliminates emission associated with the neutral ISM (for details, see Sect. \[Sect:cirrus\_HI\]). The remaining signal is equal to the sum of the zodiacal light (ZL) and the CIRB. These components are not removed because they are uncorrelated with the HI emission. Furthermore, the ZL has a smooth distribution and remains practically constant within each of the areas covered by individual ISOPHOT maps (see Ábrahám et al. [@Abraham_1997]). If the ZL level is known, the absolute value of the CIRB can be obtained. The ZL estimation is described in detail in Sect. \[sect:ZL\]. Observations {#sect:obs} ============ We study three low surface brightness fields that are labelled NGP, EBL22, and EBL26. The field NGP is located at the North Galactic Pole, the field EBL22 is similarly at a high ecliptic latitude, while the third one, EBL26, lies close to the ecliptic plane (see Table \[table:fields\]). EBL26 was selected as a field with high ZL level with the purpose of estimating the ZL contribution at the different wavelengths observed in this project. The observations of the hydrogen 21cm line were made with the Effelsberg radio telescope in May 2002. The telescope beam has a FWHM of 9 arcminutes. The areas mapped with the ISOPHOT instrument were covered with pointings at steps of FWHM/2. The stray radiation was removed with a program developed by P. Kalberla (see Kalberla [@Kalberla1982], Hartmann et al. [@Hartmann1996], Kalberla et al. [@Kalberla2005]). For details of the observations of the EBL fields and the associated data reduction, see Appendix \[app:obs\]. The principles of ISOPHOT data reduction and calibration of surface brightness measurements are explained in Appendix \[sect:techcal\]. ----------- ----------- ---------------- -------------------------------------------- Field $\lambda$ Offset Slope ($\mu$m) (MJysr$^{-1}$) ($10^{-3}$ MJysr$^{-1}$K$^{-1}$km$^{-1}$s) EBL22 90 5.53 (0.30) 35.15 (4.26) EBL22 150 3.56 (0.34) 38.91 (4.38) EBL22 180 3.10 (0.38) 27.95 (5.42) EBL26$^1$ 90 18.66 (1.22) 17.28 (6.57) EBL26$^1$ 150 6.38 (1.33) 27.42 (7.36) EBL26$^1$ 180 6.00 (1.30) 25.76 (7.41) NGP 90 6.31 (0.17) 24.60 (3.98) NGP 150 3.53 (0.23) 28.61 (5.58) NGP 180 3.20 (0.19) 31.91 (4.22) ----------- ----------- ---------------- -------------------------------------------- : Parameters of linear fits of FIR surface brightness versus the HI line area. The 1-$\sigma$ error estimates determined with the bootstrap method are given in parentheses. For NGP, the results correspond to a fit to the combined data of the northern and southern sub-fields (see Appendix \[sect:isophot\_reduction\]). $^1$Fit to data with $W(HI)<$200Kkms$^{-1}$ only. \[table:fit\] Analysis and results ==================== Subtraction of Galactic cirrus emission using HI data {#Sect:cirrus_HI} ----------------------------------------------------- The FIR surface brightness was correlated at each observed wavelength with the integrated line area of the HI spectra. At each observed HI position the average FIR signal was calculated using spatial weighting with a gaussian with FWHM equal to 9$\arcmin$. Only those pointings are used where the centre of the Effelsberg beam falls inside the FIR map. In addition to the observational uncertainties, each data point was weighted in direct proportion to the fraction of the HI FWHM beam that was covered by FIR observations. Therefore, the data close to FIR map boundaries get lower weight in the following analysis. The obtained correlations are shown in Fig. \[fig:HI\_corr\]. For FIR observations the plotted error bars are based on the statistical uncertainties reported by the PIA. The figures include linear fits that take into account the estimated uncertainties in both FIR and HI data. The slopes and zero points of the fit are given in Table \[table:fit\]. In field EBL26 there is a clear break in the relation above $W$(HI)=200Kkms$^{-1}$ that may indicate the presence of molecular gas. There is also one fairly bright galaxy that is located in the region of higher cirrus emission and may have affected the correlation. Therefore, in the field EBL26 the linear fitting was carried out using only data below $W(HI)=200$Kkms$^{-1}$. In the other fields the hydrogen column densities are in general smaller, $W(HI) \la 100$Kkms$^{-1}$, so that the fraction of molecular gas can be expected to be insignificant. The offsets thus obtained correspond to an extrapolation to zero HI column density. To the extent to which the remaining contributions of ionised and molecular gas can be ignored (see below), the values correspond to the sum of CIRB and the zodiacal light. Subtraction of the zodiacal light {#sect:ZL} --------------------------------- The zodiacal light (ZL) emission is assumed to have a pure black body spectrum. The colour temperature of the spectrum depends on the ecliptic coordinates of the source and the solar elongation at the time of the observations. Leinert et al. ([@Leinert2002]) have studied the variations of mid-infrared ZL spectra over the sky using a set of observations made with the ISOPHOT spectrometer. We use their results to fix the colour temperature of the ZL spectra. The absolute intensity of the ZL emission in the FIR is estimated with the help of shorter wavelength ISOPHOT observations made using the ISOPHOT P detector in the absolute photometry observing mode PHT-05 (Laureijs et al. [@Handbook]). Because the observations were made in regions of low cirrus emission, the mid-infrared signal is completely dominated by the ZL. The measurements were carried out close to the larger raster maps, in terms of both time and position. Therefore, they give a good estimate for the zodiacal light emission present in the raster maps. FIR absolute photometry measurements were made at the same time and at the same positions. These are used to make a correction for the contribution that the interstellar dust has, conversely, on the measured mid-infrared values. The complete list of observations is given in Table \[table:ZL\_observations\]. The derived ZL values obtained from the fits (ZL+cirrus) are listed in Table \[table:ZL\_estimates\]. The values are given at the nominal wavelengths assuming a spectrum $\nu I_{\nu}=$constant. The uncertainties were estimated based on the quality of the fits (see Appendix \[sect:zlfit\]). In fields EBL26 and NGP, because error estimate of each of the two measurements is itself uncertain, we conservatively take the average of the two error estimates as the uncertainty of the mean. --------------- -------------- --------- --------- ---------- ---------- ----- -- Field $T_{\rm ZL}$ $I$(25) $I$(90) $I$(150) $I$(180) rms (K) error EBL22 280 40.51 7.69 3.03 2.19 13% EBL26\_ZL1 270 106.05 21.03 8.34 6.02 21% EBL26\_ZL2 270 100.50 19.84 7.87 5.68 7% EBL26 (aver.) 270 100.98 19.94 7.91 5.71 14% NGP\_ZL1 260 28.65 5.95 2.38 1.72 5% NGP\_ZL2 260 29.15 6.06 2.42 1.75 18% NGP (aver.) 260 28.69 5.96 2.38 1.72 12% --------------- -------------- --------- --------- ---------- ---------- ----- -- : The estimated zodiacal light emission. The columns are: (1) name of the EBL field (see Appendix, Table \[table:ZL\_observations\]), (2) temperature of the zodiacal light spectrum (Leinert et al. [@Leinert2002]), (3)-(6) estimated intensity of the zodiacal light at 25$\mu$m, 90$\mu$m, 150$\mu$m, and 180$\mu$m, and (7) relative uncertainty of the ZL value calculated on the basis of the difference of the fitted ZL model and the observations in the range 7.3–90$\mu$m. The zodiacal light estimates are given at the nominal wavelengths assuming a spectrum $\nu I_{\nu}$=constant. For EBL26 and NGP, two separate positions were observed (see Fig. \[fig:allsky\] and Table \[table:ZL\_observations\]). \[table:ZL\_estimates\] Estimated CIRB levels and their uncertainties {#sect:estimates} --------------------------------------------- Table \[table:EBL\_estimates\] lists the CIRB levels that are estimated based on the linear fits between FIR and HI data (Table \[table:fit\]) and the zodiacal light values of Table \[table:ZL\_estimates\]. The uncertainties are obtained by adding in quadrature the estimated errors of the offsets from Table \[table:fit\], the errors of the zodiacal light values from Table \[table:ZL\_estimates\], and the error resulting from the dark current subtraction (see Appendix \[sect:calibration\]), $$\sigma_{\rm tot}^2 = \sigma_{\rm offset}^2 + \sigma_{\rm ZL}^2 + \sigma_{\rm DC}^2.$$ The uncertainty due to the dark current is estimated to be $\sigma_{\rm DC}=$0.25-0.30MJysr$^{-1}$ and it is likely to be the main factor affecting the uncertainty of the zero point of the FIR observations (see Appendix \[sect:calibration\]). The results obtained for the three individual fields can be combined, deriving our final estimates for the CIRB and its uncertainty. In the case of the field EBL26 the values are relatively unprecise because of the high ZL level. This uncertainty is reflected in the error estimates. Combining the results we get average values $-0.54\pm$0.65MJysr$^{-1}$, 0.83$\pm$0.41MJysr$^{-1}$, 1.26$\pm$0.37MJysr$^{-1}$, at 90$\mu$m, 150$\mu$m, and 180$\mu$m, respectively, as given in the last line of Table \[table:EBL\_estimates\]. The 90$\mu$m values are very low, because in both the EBL22 and EBL26 fields negative values are obtained. In the case of EBL26 the negative value is not surprising, because the expected CIRB level is only a small fraction of the zodiacal light which itself has a considerable statistical uncertainty. Therefore, the result is sensitive also to any systematic errors of the ZL estimates. Apart from the results at 90$\mu$m, the variation between fields is only slightly larger than expected on the basis of the quoted error estimates. At 150$\mu$m a negative value is obtained for EBL26 which, nevertheless, is less than 2-$\sigma$ below the highest values. At 90$\mu$m we can derive for EBL only an upper limit. The 150$\mu$m and 180$\mu$m bands are close to each other and the CIRB values should be very similar. Therefore, based on the three fields and the two frequency bands, we can calculate, as a weighted average, an estimate for the CIRB in the range 150–180$\mu$m. The result is 1.08$\pm$0.32MJysr$^{-1}$. The result would not change significantly (less than $1-\sigma$) even if either EBL22 or EBL26 were omitted from the analysis. ----------- ---------------- ---------------- ---------------- Field $I$(90$\mu$m) $I$(150$\mu$m) $I$(180$\mu$m) (MJysr$^{-1}$) (MJysr$^{-1}$) (MJysr$^{-1}$) EBL22 -2.16 (1.04) 0.53 (0.52) 0.91 (0.47) EBL26$^1$ -1.28 (3.05) -1.53 (1.73) 0.29 (1.53) NGP 0.35 (0.74) 1.15 (0.37) 1.48 (0.28) average -0.54 (0.65) 0.83 (0.41) 1.26 (0.37) ----------- ---------------- ---------------- ---------------- : Estimated level of the CIRB for the individual fields. The values correspond to the difference between the offsets listed in Table \[table:fit\] and the ZL estimates of Table \[table:ZL\_estimates\]. No colour correction was applied. The 1-$\sigma$ estimated statistical errors are given in parentheses (see text). \ $^1$Fit to data with $W(HI)<$200Kkms$^{-1}$ only. \[table:EBL\_estimates\] The reliability of the CIRB values {#sect:reliability} ---------------------------------- In addition to the statistical uncertainties, the results are affected by systematic errors. The CIRB estimates are not affected by the HI antenna temperature scale. However, the presence of unsubtracted stray radiation could affect the HI zero point of the HI data and, thus, lower the CIRB values. We cannot directly estimate the presence of residual stray radiation in the HI data. However, in Sect. \[sect:HImeas\] we compare some of our HI spectra with data from the Leiden/Dwingeloo survey (Hartmann & Burton [@Hartmann1997]; Kalberla et al. [@Kalberla2005]) and we find that the residual stray radiation is likely to be less than 4Kkms$^{-1}$ which, assuming a slope of 29$\times 10^{-3}$MJysr$^{-1}$ (Kkms$^{-1}$)$^{-1}$ (see Table \[table:fit\]), corresponds to 0.12MJysr$^{-1}$. In this case HI stray radiation would not be a major source of error. The relative calibration of the ISOPHOT FIR cameras and the P-detectors directly affects the estimated FIR ZL levels and is probably the most important source of systematic errors. According to the ISOPHOT Handbook (Laureijs et al. [@Handbook]) the absolute accuracy of the C100 and C200 cameras and the P-detectors is typically of the order of 20%. If there were a difference of 10% in the [*relative*]{} calibration of the MIR and FIR bands, this would cause a similar percentual error to the ZL estimates. The fact that we obtained negative CIRB values at 90$\mu$m, especially when the absolute level of the ZL is high, suggests that the FIR ZL levels may have been overestimated. The effect of an error of 10% would range from $\sim$2MJysr$^{-1}$ at 90$\mu$m in the field EBL26 to $\sim$0.2MJysr$^{-1}$ at 150–180$\mu$m in the field NGP. Taking into account the relative weighting of the three fields, a 10% error in ZL corresponds to an uncertainty of 0.3MJysr$^{-1}$ in the 150-180$\mu$m CIRB estimate. Assuming a systematic uncertainty of this magnitude, the CIRB estimate can be written as 1.08$\pm$0.32$\pm$0.30MJysr$^{-1}$ where the first error estimate refers to statistical and the second to systematic uncertainties. At 90$\mu$m the negative value obtained for EBL26 carries very little weight and an additional 10% systematic uncertainty in the ZL would correspond to an additional uncertainty of 2MJysr$^{-1}$. In EBL22 the CIRB values was -2.16$\pm$1.04 and a 10% systematic error in the ZL values would correspond to 0.77MJysr$^{-1}$. The CIRB value is $\sim$2-$\sigma$ below zero and suggests that the ZL values may contain a systematic error of $\sim$10–20% that has reduced the obtained CIRB values. In the field NGP the 90$\mu$m CIRB estimate was 0.35$\pm$0.74MJysr$^{-1}$. Assuming a 10% systematic uncertainty in the ZL and adding the error estimates in quadrature, the CIRB estimate becomes 0.35$\pm$0.95MJy$sr^{-1}$ and we obtain a 2-$\sigma$ upper limit of 2.3MJysr$^{-1}$. Discussion ========== Dust emission associated with ionised gas ----------------------------------------- Our analysis was based on the correlation of HI emission and FIR intensity. So far we have omitted the possible effect that dust mixed with ionised gas might have. The ionised component can affect the results only as far as it is uncorrelated with the HI emission. Lagache et al. ([@Lagache2000]) decomposed the DIRBE FIR intensity into components correlated with the neutral and the ionised medium. The column density of ionised hydrogen, $N(H^{+})$, was estimated based on the $H_{\alpha}$ line. They found that the infrared emissivity of dust associated with the ionised medium would be very similar to the emissivity of dust within the neutral medium. However, Odegard et al. ([@Odegard2007]) recently re-examined this issue, obtaining significantly lower emissivity values for the ionised medium. The derived 2-$\sigma$ upper limit for the 100$\mu$m emissivity per hydrogen ion was typically only 40% of the emissivity in the neutral atomic medium. We use the all-sky H$\alpha$ map produced by Finkbeiner ([@Finkbeiner2003]) to examine the possible contribution of the ionised medium to the FIR emission. The resolution of the $H_{\alpha}$ data is 6$\arcmin$ for fields EBL22 and EBL26 and one degree at the location of the field NGP. The average $H_{\alpha}$ emission in the EBL22, EBL26, and NGP fields is $\sim$0.7R, $\sim$0.5R, and $\sim$0.6R, respectively. The $H_{\alpha}$ background contains small scale structure that may be caused by faint point sources, mainly stars. Therefore, the quoted $H_{\alpha}$ levels are not caused by the diffuse ISM only. For example, in NGP the $H_{\alpha}$ image is dominated by an unresolved ($\sim$ one degree) emission peak at the centre of the field, the nature of which remains unknown. Apart from this, the $H_{\alpha}$ background does not show any significant gradients or correlation with the FIR emission. Therefore, we consider only the effect on the average FIR signal. Using the Lagache et al. ([@Lagache2000]) conversion factors an $H_{\alpha}$ signal of 0.6R would correspond to $\sim$0.5MJysr$^{-1}$ in the FIR. Therefore, the CIRB values could be overestimated by a similar amount. However, adopting the Odegard et al. ([@Odegard2007]) 1-$\sigma$ upper limits, the contribution from the ionised medium should remain below 0.1MJy/sr. Furthermore, in our analysis we have correlated the FIR emission only with HI while in the quoted studies the FIR signal was correlated simultaneously with both HI and H$^{+}$. Therefore, since HI and H$^{+}$ are themselves correlated, the correction to be applied to our results should be correspondingly smaller. Therefore, we believe that the possible effects due to the presence of an ionised medium are small compared with the other uncertainties given above. Dust emission associated with molecular gas ------------------------------------------- If molecular gas is present, the HI lines will underestimate the total column density of gas and, because the fraction of molecular gas increases with column density, the relation between FIR emission and the HI intensity becomes steeper. Our fields have low column density and, therefore, the fraction of molecular gas should be low. Hydrogen molecules cannot survive in clouds with visual extinction below $A_{\rm V}\sim 0.1^{\rm m}$ and, consequently, no molecular gas should exist in clouds with column densities below $N(H)\sim 2\times 10^{20}$cm$^{-2}$. Arendt et al. ([@Arendt1998]) detected a steepening in the FIR vs. HI relation which, however, in different regions took place at different column densities. The effect could start already around $N(H)\sim2\times 10^{20}$cm$^{-2}$ which corresponds to an HI line area of $W(HI)\sim100\,$Kkms$^{-1}$. Kiss et al. ([@Kiss2003]) observed a change in the spatial power spectra of FIR surface brightness around $N(H)\sim 10^{21}$cm$^{-2}$. This was similarly interpreted as a sign of the transition between atomic and molecular phases. In the EBL fields, molecular emission could be significant only in the EBL26 region, where the slope between HI and FIR data appears to change at $W(HI) \sim 200\,$Kkms$^{-1}$ (see Fig. \[fig:HI\_corr\]). Below this limit there is a good, linear correlation between the FIR surface brightness and HI line area which also shows that toward those positions the fraction of molecular gas is low. In the EBL estimation only data below $W(HI)=200\,$Kkms$^{-1}$ were used. Comparison with earlier results ------------------------------- The earlier CIRB results in the FIR range are all based on measurements of the COBE satellite. We have derived our CIRB estimates using the ISOPHOT measurements, without relying on the COBE data even in the determination of the ZL levels. Therefore, our result is the first completely independent CIRB estimate after the COBE detections. Table \[table:comparison\] in Appendix \[sect:dirbe\_ebl\] lists the existing CIRB estimates in the FIR wavelength range. In the range 150–180$\mu$m our value is consistent with the DIRBE results at 140$\mu$m. According to Kiss et al. ([@Kiss2006]) the COBE/DIRBE and ISOPHOT FIR surface brightness values agree to within $\sim$15% and, therefore, the differences in the surface brightness scales are likely to be smaller than our statistical uncertainty. In the ZL subtraction, the relative calibration of the ISOPHOT-P detector and the FIR cameras could introduce a systematic error that has a magnitude comparable to that of the statistical uncertainty. The low, even negative CIRB estimates obtained at 90$\mu$m suggest that this systematic error causes the ZL estimates to be $\sim$10% too large. Taking into account our statistical and systematic uncertainties at 150–180$\mu$m, we cannot exclude even the highest DIRBE estimates close to 1.5MJy$^{-1}$. At 90$\mu$m our 2-$\sigma$ upper limit of 2.3MJysr$^{-1}$ is consistent with the existing DIRBE results. Based on the above values, the galaxies resolved with ISO FIR observations account for some 5% of the total CIRB (e.g., Juvela et al. [@Juvela00]; Héraudeau et al. [@Heraudeau04]; Lagache & Dole [@lagache_dole_01]; Kawara et al. [@Kawara04]). A stacking analysis of Spitzer measurements (Dole et al. [@Dole2006]) showed that galaxies detected at 24$\mu$m contribute some 0.7MJysr$^{-1}$ to the 160$\mu$m sky surface brightness. Therefore, the results from galaxy counts and measurements of the absolute level of CIRB are converging, and probably more than half of the sources responsible for the CIRB have already been identified. Conclusions =========== For the ISOPHOT EBL project far-infrared raster maps were obtained in selected low-cirrus regions. We have analysed these observations and, by correlating the FIR surface brightness with HI line areas measured with the Effelsberg radio telescope, we derive estimates for the cosmic infrared background in the wavelength range 90–180$\mu$m. We determined the level of ZL emission using shorter wavelength ISOPHOT observations, without relying on a model of the spatial distribution of the ZL emission on the sky. Therefore, our results are independent of the existing COBE results. [Based on this study we conclude the following:]{} - At 90$\mu$m we derived a 2-$\sigma$ upper limit of 2.3MJysr$^{-1}$ for the CIRB. - In the range 150–180$\mu$m we obtained a CIRB value of 1.08$\pm$0.32$\pm$0.30MJysr$^{-1}$, where we quote separately the estimated statistical and systematic uncertainties. - The accuracy of the results was determined mostly by the accuracy of the zodiacal light estimates and the dark signal subtraction. - Assuming the latest emissivity values of dust associated with the ionised medium, the uncertainty related to the presence of ionised medium was small compared with the other sources of uncertainty. We thank the anonymous referee for valuable comments. This work was supported by the Academy of Finland grants no. 115056, 107701, 124620, and 119641. ISOPHOT and the Data Centre at MPIA, Heidelberg, were funded by the DLR and the Max-Planck-Gesellschaft. We thank P. Kalberla for his help in the planning of the HI measurements and for performing the stray radiation correction of these observations. Ábrahám P., Leinert Ch., Lemke D., 1997, A&A 328, 702 Acosta-Pulido, J.A., Gabriel, C., and Castañeda, O., 2000 Experimental Astronomy 10, 333 Arendt R.G., Odegard N., Weiland J.L. et al. 1998, ApJ 508, 74 del Burgo, C., Héraudeau, P., and Ábrahám, P., 2002 in: Proceedings of the Symposium “Exploiting the ISO Data Archive – Infrared Astronomy in the Internet Age”, ESA SP-5111, Sigüenza, p. 339 Dwek E. 1998, ApJ. 501, 643 Dole H., Lagache G., Puget J.-L., et al. 2006, A&A 451, 417 Finkbeiner D.P., Davis M., Schlegel D.J. 2000, ApJ 544, 81 Finkbeiner D.P.A. 2003, ApJS 146, 407 Fixen, D.J. 1997, ApJ, 490, 482 Hartmann D., Burton W.B. 1997, Atlas of Galactic Neutral Hydrogen, Cambridge University Press, Hartmann D., Kalberla P.M.W., Burton W.B., Mebold, U. 1996, A&AS 119, 115 Hauser, M.G., et al. 1998, ApJ 508, 25 Hauser M.G., Dwek E. 2001, ARA&A 39, 249 Héraudeau P., Oliver S., del Burgo C., et al. 2004, MNRAS 354, 924 Juvela M., Mattila L., Lemke D. 2000, A&A 360, 813 Kalberla P.M.W., Mebold U., Reif K. 1982, A&A 106,190 Kalberla P.M.W., Burton W.B., Hartmann D., et al. 2005, A&A, 440, 775 Kawara K., Matsuhara H., Okuda H., et al. 2004, A&A 413, 843 Kelsall et al. 1998, ApJ, 508, 44 Kessler M.F., Müller T.G., Leech, K. et al., 2003, The ISO Handbook I, Mission & Satellite Overview, SAI/2000-035/Dc, Noordwijk, The Netherlands Kiss C., Ábrahám P., Klaas U., Juvela M., Lemke D. 2001, A&A 379, 1161 Kiss C., Ábrahám P., Klaas U., et al. 2003, A&A 399, 177 Kiss Cs., Ábrahám P., Laureijs R.J., Moór A., Birkmann S.M. 2006, MNRAS 373, 1213 Klaas, U., Ábrahám, P., Acosta-Pulido, J.A., et al., 2001 in: Proceedings of the Conference “The Calibration Legacy of the ISO Mission”, ISO Data Centre, Villafranca del Castillo, Madrid, p. 19 Lagache G., Abergel A., Boulanger F., Desert F.X., Puget, J.-L. 1999, A&A 344, 322 Lagache G., Dole H. 2001. A&A 372, 702 Lagache G., Haffner L.M., Reynolds R.J., Tufte S.L. 2000, A&A 354, 247 Lagache G., Puget J.-L. 2000, A&A 355, 17 Lagache G., Puget J.-L., Dole H. 2005, ARA&A 43, 727 Laureijs R.J., Klaas, U., Richards, P.J., & Schulz B., Ábrahám P., 2003, The ISO Handbook, Volume IV: PHT – The Imaging Photo-Polarimeter, ESA SP-1262 (http://isowww.estec.esa.nl/users/handbook/) Leinert Ch., Ábráham P., Acosta-Pulido J., Lemke D., Siebenmorgen R., 2002, A&A 393, 1073 Lemke D., Klaas U., Abolins J., et al., 1996, A&A 315, L64 Lemke D., Kranz Th., Klaas U., et al., 2001, Proceedings of the Conference “The Calibration Legacy of the ISO Mission”, ESA SP-481, p. 219 Li A., Draine B. 2001, ApJ 554, 778 Odegard N., Arendt R.G., Dwek E., et al. 2007, ApJ 667, 110 Puget J.-L., Abergel A., Bernard J.-P., Boulanger F., Burton W.B., et al. 1996, A&A 308, L5 Schlegel D.J., Finkbeiner D.P., Davis M. 1998, ApJ 500, 525 Stark A.A., Gammie C.F., Wilson R.W. et al. 1992, ApJS 79, 77 The principles of surface brightness observations with ISOPHOT: data reduction and calibration {#sect:techcal} ============================================================================================== The most detailed description of the ISOPHOT instrument, its observing modes (so called Astronomical Observation Templates, AOTs) and the corresponding data analysis and calibration steps is given in the ISOPHOT Handbook (Laureijs et al. [@Handbook]). In the following we describe recent calibration techniques which are beyond the scope of the Handbook and which are essential for the determination of the EBL surface brightness. Absolute surface brightness calibration of ISOPHOT observations {#sect:abssurfacebrightnesscal} --------------------------------------------------------------- ISOPHOT was absolutely calibrated against a flux grid of celestial point source standards consisting of stars, asteroids and planets, thus covering a fair fraction of the entire dynamic flux range from $\approx$100mJy up to about 1000Jy. Each detector aperture/pixel was individually calibrated against these standards. Therefore, the basic ISOPHOT calibration is in Jypixel$^{-1}$. In order to derive proper surface brightness values in MJysr$^{-1}$, the solid angles of each detector aperture/pixel must be accurately known $$B_{\rm \lambda} = \frac{f_{\rm psf}^{aper}(0,0)}{ \Omega^{aper}_{\rm eff}} \cdot F_{\rm \lambda},$$ with $B_{\rm \lambda}$ being the surface brightness, $\Omega^{aper}_{\rm eff}$ the effective solid angle of the pixel/aperture, $F_{\rm \lambda}$ the total flux of a celestial standard and $f_{\rm psf}^{aper}(0,0)$ the fraction of the Point Spread Function contained in the pixel/aperture (i.e. the convolution of the PSF with the aperture response) when being centred at position (0,0). Hence, $F_{\rm \lambda} \cdot f_{\rm psf}^{aper}(0,0)$ is the flux per pixel. ISOPHOT’s effective solid angles have been determined by 2D-scanning of a point source over the pixel/aperture in fine steps dx’ and dy’ and measuring the resulting intensity at each measurement point $(x'_{\rm i},y'_{\rm j})$, the footprint, taking into account a non-flat aperture/pixel response: $$\Omega^{aper}_{\rm eff} = \sum_{i}^{A}\sum_{j}\,f_{\rm psf}^{aper}(x'_{\rm i},y'_{\rm j})\,dx'\,dy'$$ If the peak of the point source was located outside the aperture by $\sim$1/2 of the aperture size, the S/N of the resulting intensity dropped so much that at this border the summation was complemented by a model of the broad band telescope PSF and adding up the corresponding PSF fractions $f_{\rm psf}^{aper}(x'_{\rm i},y'_{\rm j})$ out to $\pm$10arcmin assuming a flat response, but taking into account a cut by ISO’s pyramidal central mirror feeding the 4 instrument beams. An example of such a synthetic footprint is shown in Fig. \[fi:c100\_60umsyntheticfootprint\]. The values of the solid angles used in PIA V11.3 are listed in Tables \[tab:c100omega\] and \[tab:c200omega\]. [lcccc]{} filter & $\lambda_{\rm c}$ &    i = 1 $\ldots$ 9\ & ($\mu$m) &\ C\_50 & 65 & 0.3737 & 0.4005 & 0.3929\ & & 0.4167 & 0.3963 & 0.4120\ & & 0.4064 & 0.3690 & 0.3943\ C\_60 & 60 & 0.3570 & 0.3925 & 0.3816\ & & 0.4087 & 0.3897 & 0.4044\ & & 0.3980 & 0.3608 & 0.3827\ C\_70 & 80 & 0.4252 & 0.4253 & 0.4276\ & & 0.4418 & 0.4177 & 0.4367\ & & 0.4326 & 0.3940 & 0.4299\ C\_100 & 100 & 0.4855 & 0.4517 & 0.4666\ & & 0.4679 & 0.4396 & 0.4632\ & & 0.4603 & 0.4194 & 0.4701\ C\_105 & 105 & 0.5006 & 0.4588 & 0.4764\ & & 0.4751 & 0.4454 & 0.4698\ & & 0.4678 & 0.4260 & 0.4804\ C\_90 & 90 & 0.4577 & 0.4403 & 0.4492\ & & 0.4568 & 0.4304 & 0.4519\ & & 0.4484 & 0.4086 & 0.4520\ [lccccc]{} filter & $\lambda_{\rm c}$ &    i = 1 $\ldots$ 4\ & ($\mu$m) &\ C\_160 & 170 & 1.782 & 1.940 & 1.895 & 1.781\ C\_200 & 200 & 1.810 & 1.996 & 1.988 & 1.878\ C\_180 & 180 & 1.792 & 1.960 & 1.927 & 1.815\ C\_135 & 150 & 1.759 & 1.898 & 1.825 & 1.711\ C\_120 & 120 & 1.722 & 1.843 & 1.708 & 1.587\ ![Synthetic (outer part, i.e. green and blue coloured areas, modelled) footprints (convolution of the ISO telescope PSF with the pixel aperture response) of the 3 $\times$ 3 pixels of ISOPHOT’s C100 array for the 60$\mu$m broad band filter. The solid angles of each pixel are obtained by integration over the footprint area.[]{data-label="fi:c100_60umsyntheticfootprint"}](mjuvela_04.eps){width="8.5cm"} It should be noted that an absolute surface brightness calibration is more accurate than an absolute calibration of a compact source of similar brightness, since no background subtraction has to be performed, which introduces an additional uncertainty. The accuracies quoted in the ISOPHOT Handbook (Laureijs et al. [@Handbook]), Table 9.1 for extended emission take COBE/DIRBE photometry as the reference. By not referring to COBE/DIRBE photometry, the absolute surface brightness calibration for ISOPHOT’s C100 and C200 array is as good as that for bright compact sources, i.e. better than 15%. New calibration products and strategies for PIA V11.3 ----------------------------------------------------- For the very sensitive analysis needed for the EBL determination and, in particular, an absolute surface brightness calibration that is as accurate as possible, a number of calibration upgrades and new calibration features have been developed and implemented in PIA V11.3. For the ones which are not described in the ISOPHOT Handbook (Laureijs et al. [@Handbook]), we provide a description and examples for the C100 detector in the following. An overview of the individual calibration steps associated with different instrument components is shown in Fig. \[fi:phtcalibrationschema\]. By application of all these steps, instrumental artefacts are minimized, the resulting detector signals are homogenized and a high calibration reproducibility and accuracy is achieved. ![Scheme of the ISOPHOT calibration steps associated with the different instrument components. The meaning of the abbreviations is the following: BSL = Bypassing Sky Light correction, DS = detector Dark Signal, RL = Ramp Linearisation, TC = signal Transient Correction, and RIC = Reset Interval Correction.[]{data-label="fi:phtcalibrationschema"}](mjuvela_05.eps){width="8.5cm"} ### Detector responsivity calibration The absolute photometric calibration of an individual measurement is performed via a transfer calibration using the internal calibration sources. This measures the actual responsivity of the detector and provides the absolute signal-to-flux conversion. It is a separate measurement of each observation mode by deflecting the chopper mirror to the field of view of the internal calibrator (Fine Calibration Source, FCS). The illumination level of the internal calibrator was not fixed but adjusted as much as possible to the expected brightness level of the sky. This was achieved by selecting an appropriate heating power for the internal source. There exists a calibration relation between this heating power and the optical power received by each detector pixel which is established from measurements on celestial standards. {width="5cm"} {height="5cm"} {height="5cm"} {height="5cm"} {width="5cm"} Therefore, for reliable and accurate transfer calibrations, the following requirements are put on the FCS: - High reproducibility. This was better than 1%, since the monitoring of the flux of faint standards was reproducible within a few percent, and this uncertainty was dominated by the signal noise (Klaas et al. [@Klaas01]). - A very detailed characterization of the illuminated power depending on the heating power applied to the source. This is illustrated in Fig. \[fi:fcscalschema\]. It involves the following steps: - For each C100 and C200 array filter all measurements of celestial standards done in raster map mode were evaluated such that for each pixel the background signal was properly subtracted and the resulting source signal was associated with the celestial standard flux. The ratio of the source signal and the simultaneously obtained FCS signal gave the illumination power by the FCS for the selected heating power. The discrete results were fitted and the reliable lower and upper heating power limits covered by measurements were identified (Fig. \[fi:fcscalschema\] upper left). The heating power ranges were not identical or equally large for each filter (Fig. \[fi:fcscalschema\] upper right). In general they were shifted to smaller heating power values for longer wavelengths. - For fine discrete steps in heating power the inband powers were read from the relations and were converted to monochromatic surface brightnesses by applying the bandpass conversions derived from the relative system response profiles, see ISOPHOT Handbook (Laureijs et al. [@Handbook]), section A.2, and the solid angles of Tables \[tab:c100omega\] and \[tab:c200omega\]. These fluxes were fitted with a modified BB curve after appropriate colour correction (Fig. \[fi:fcscalschema\] middle left). If for a certain filter the selected heating power was outside the reliable limits, the value of this filter was excluded from the fit. The fit gave the temperature of the FCS for the selected heating power. An additional constraint was that the temperature had to be the same for the fit curves of all pixels. C100 and C200 filter values had to be fitted independently because of the different detector areas and hence illumination factors, however, the fits were checked for consistent temperatures, because the illuminating FCS was the same for both detectors. - This was achieved for the heating power range from 0.07 up to 6.5mW adopting an emissivity of the source $\propto \lambda^{1.25}$ yielding the temperature vs. heating power relation as shown in Fig. \[fi:fcscalschema\] middle right. - By applying this FCS temperature model and the established illumination factors for each pixel it was possible to establish homogeneous calibration curves of the internal reference source, thus polishing out measurement outliers affecting the initial empirical curves. The multi-filter approach connecting all curves and not treating them individually enabled a large extension and a common range for all filters: compare Fig. \[fi:fcscalschema\] lower centre with Fig. \[fi:fcscalschema\] upper right. ### Bypassing sky light correction of FCS signal As a safety design against single point failures, ISOPHOT was not equipped with any cold shutter to suppress straylight when performing internal calibration measurements. Therefore, when deflecting the chopper onto the illuminated internal calibration (FCS) sources, some fraction of the power received on the detector did not come from the FCS but from sky light bypassing along non nominal light paths. Since this depends on the sky brightness it is subtracted in the transfer calibration measurements on celestial standards and hence has to be subtracted for any FCS measurement in order to get a reproducible zero point. This was achieved by performing a number of measurements on the switched-off, i.e. cold FCS, so that only the bypassing sky light contribution was measured. The result for one C100 array pixel is shown in Fig. \[fi:c100pix4bypassskylightcorr\] which demonstrates a linear dependence of the bypassing sky light contribution to the FCS signal on the sky background. This correction was established for all C100 and C200 array pixels. The bypassing sky light contribution contains the detector dark signal contribution, cf. Sect. \[sect:darksignal\]. ![Bypassing sky light contribution to the FCS signal depending on the sky background.[]{data-label="fi:c100pix4bypassskylightcorr"}](mjuvela_11.eps){width="8.5cm"} ### Effective pixel/aperture solid angles These are described in the previous section \[sect:abssurfacebrightnesscal\] and their values are compiled in Tables \[tab:c100omega\] and \[tab:c200omega\]. ### Filter profiles The bandpass system responses and the conversion factors from inband power to a monochromatic flux, as well as colour correction factors are described in the ISOPHOT Handbook (Laureijs et al. [@Handbook]). ### Detector dark signal {#sect:darksignal} The detector dark signals were re-analyzed as described in del Burgo ([@delBurgo02]). In this analysis special care was given to exclude those dark measurements suffering from memory effects by preceding bright illuminations, thus not representing the true dark level. An example of the new results is shown in Fig. \[fi:c100pix5darksignal\] for the central pixel 5 of the C100 array. A slight orbital dependence is visible with an increase of the dark signal towards the beginning and the end of the observational window. It can also be noticed that there is a scatter of the dark signals at the same orbit position and there are occasional large outliers. These are not due to signal determination uncertainties, but are real variations due to space weather effects on different revolutions over the ISO mission. ![Orbit dependent dark signal determination for the central pixel 5 of ISOPHOT’s C100 array. Dots represent individual measurements obtained during the entire ISO mission, filled and open signals identify a different reset interval in the integration of the dark signal. The solid line is the fit to the measurements providing the so-called default dark level. The dotted line is the default dark level of an older calibration version used before 2001.[]{data-label="fi:c100pix5darksignal"}](mjuvela_12.eps){width="8.5cm"} ### Ramp linearisation This was performed as described in the ISOPHOT Handbook (Laureijs et al. [@Handbook]). For ISOPHOT’s far-infrared detectors two types of effects cause non-linearities of the integration ramps: - De-biasing effects of the photoconductors operated with low bias caused by feed-back from the integration capacitor. - Non-linearities generated in the cold read-out electronics. ### Signal dependence on reset interval correction Despite the ramp linearisation step, signals obtained under constant illumination, but with different reset intervals show a systematic difference, see Fig. \[fi:c100resetintervalcorr\] upper panel. In order to have a consistent signal handling of measurements with different reset interval settings applied - to optimize the dynamic range of the signal - all signals were converted as if they were taken with a 1/4s reset interval. The correction relations were established from special calibration measurements applying the full suite of reset intervals under constant illumination and this for different illumination levels. In this way signal corrections were established for all reset intervals in the range 1/32s to 8s (Fig. \[fi:c100resetintervalcorr\] middle and lower panel). While previously, as still described in the ISOPHOT Handbook (Laureijs et al. [@Handbook]) a linear correlation with offset was used, a re-analysis (del Burgo et al. [@delBurgo02]) yielded non-linear relations as shown in Fig. \[fi:c100resetintervalcorr\]. This latter analysis also found a bi-modal behaviour for C100 array pixels, such that the pixels on the main diagonal, \#1, 5 and 9, behaved differently from the rest of the pixels. For the C200 array all pixels behaved in the same way. ![Correction of the signal dependence on the selected reset interval. [*Upper panel:*]{} Demonstration of the effect, showing the resulting signal versus the selected reset interval over the range from 1/32s up to 8s (reset intervals were commanded in powers of 2) under constant illumination. [*Middle panel:*]{} Solid line: Correction relation for a reset interval of 8s w.r.t. the reference reset interval of 1/4s for all C100 array pixels, except the ones on the main diagonal. Dotted line: old linear correlation used before the re-analysis. [*Lower panel:*]{} Solid line: Correction relation for a reset interval of 8s w.r.t. the reference reset interval of 1/4s for all C100 array pixels on the main diagonal (pixels \#1, 5, and 9). Dotted line: old linear correlation used before the re-analysis (same as for middle panel).[]{data-label="fi:c100resetintervalcorr"}](mjuvela_13.eps "fig:"){width="8.5cm"} ![Correction of the signal dependence on the selected reset interval. [*Upper panel:*]{} Demonstration of the effect, showing the resulting signal versus the selected reset interval over the range from 1/32s up to 8s (reset intervals were commanded in powers of 2) under constant illumination. [*Middle panel:*]{} Solid line: Correction relation for a reset interval of 8s w.r.t. the reference reset interval of 1/4s for all C100 array pixels, except the ones on the main diagonal. Dotted line: old linear correlation used before the re-analysis. [*Lower panel:*]{} Solid line: Correction relation for a reset interval of 8s w.r.t. the reference reset interval of 1/4s for all C100 array pixels on the main diagonal (pixels \#1, 5, and 9). Dotted line: old linear correlation used before the re-analysis (same as for middle panel).[]{data-label="fi:c100resetintervalcorr"}](mjuvela_14.eps "fig:"){width="8.5cm"} ![Correction of the signal dependence on the selected reset interval. [*Upper panel:*]{} Demonstration of the effect, showing the resulting signal versus the selected reset interval over the range from 1/32s up to 8s (reset intervals were commanded in powers of 2) under constant illumination. [*Middle panel:*]{} Solid line: Correction relation for a reset interval of 8s w.r.t. the reference reset interval of 1/4s for all C100 array pixels, except the ones on the main diagonal. Dotted line: old linear correlation used before the re-analysis. [*Lower panel:*]{} Solid line: Correction relation for a reset interval of 8s w.r.t. the reference reset interval of 1/4s for all C100 array pixels on the main diagonal (pixels \#1, 5, and 9). Dotted line: old linear correlation used before the re-analysis (same as for middle panel).[]{data-label="fi:c100resetintervalcorr"}](mjuvela_15.eps "fig:"){width="8.5cm"} ### Signal transient correction The ISOPHOT detectors were photoconductors operated under low background conditions provided by a cryogenically cooled spacecraft. Under these conditions they showed the behaviour that the output signal was not instantaneously adjusted to a flux change but rather, following an initial jump by a certain fraction of the flux step, the signal adjusted with some time constant to the final level, see e.g. Acosta et al. ([@Acosta-Pulido00]). In particular the ISOPHOT C100 detector showed significant transient behaviour. This time constant depended on the detector material (doping of the semi-conductor and its contacts), the flux step, the direction of the flux step (dark to bright versus bright to dark) and the illumination history. Attempts had been made to model this behaviour (Acosta et al. [@Acosta-Pulido00]), but no unique description could be found for the FIR detectors. To overcome this effect at least partly the method of transient recognition was implemented in the ISOPHOT analysis software as described in the ISOPHOT Handbook (Laureijs et al. [@Handbook]) using the most stable part of the measurement for signal determination. Finally, another approach was to use a data base of long measurements with signals stabilising and to determine the deviation from the end level for shorter intermediate times (del Burgo et al. [@delBurgo02]), see Fig. \[fi:c100transientcorr\] for an illustration. A measurement time of 128s was used as reference, because - Most calibration measurements in staring mode were performed with this basic measurement time. - In most cases the signals stabilised within this time. For the C200 array pixels the signal transient effect is considerably smaller and faster and therefore it is sufficient to apply the transient recognition as described in the ISOPHOT Handbook (Laureijs et al. [@Handbook]). ![Empirical signal transient correction for ISOPHOT’s C100 array. The left column shows the signal loss for integration times of 4, 8, 16, 32, and 64s (commendable integration times of ISOPHOT detectors) with regard to the reference time of 128s. The red line is a fit through the measured points over the covered signal range and is used as the correction relation. The right column shows the residuals after applying this correction.[]{data-label="fi:c100transientcorr"}](mjuvela_16.eps){width="8.5cm"} Observations and data reduction for the EBL fields {#app:obs} ================================================== ISOPHOT observations -------------------- The following tables give details of the ISOPHOT observations used in the paper. Table \[table:tdt\] lists the raster maps and absolute photometry measurements that were made at 90, 150, and 180$\mu$m. Correspondingly, Table \[table:ZL\_observations\] lists observations used for the determination of the zodiacal light levels. These include both mid-infrared measurements carried out with the ISOPHOT-P detector and longer wavelength absolute photometry measurements carried out with the C100 and C200 cameras. --------- ----------- ------------ ------------- -------------- ---------- -- Field / $\lambda$ RA DEC Size TDT AOT ($\mu$m) (J2000) (J2000) number EBL22 PHT22 90 2 26 34.4 -25 53 49 32$\times$3 82101111 PHT22 150 $-''-$ $-''-$ $-''-$ 81901910 PHT22 180 $-''-$ $-''-$ $-''-$ 81901509 PHT22 90 2 32 47.9 -25 54 6.0 32$\times$1 82101115 PHT22 180 $-''-$ $-''-$ $\times$132 81901513 PHT22 150 $-''-$ $-''-$ $\times$1 32 81901914 PHT25 150 2 26 34.4 -25 53 52.8 1 81901927 PHT25 180 2 26 34.4 -25 53 52.8 1 81901529 PHT25 90 2 26 34.4 -25 53 52.9 1 82101126 EBL26 PHT22 90 1 18 14.5 1 56 39 32$\times$4 58600319 PHT22 150 $-''-$ $-''-$ $-''-$ 58600418 PHT22 180 $-''-$ $-''-$ $-''-$ 58600517 PHT25 150 1 17 39.3 2 3 9.8 1 60302067 PHT25 180 $-''-$ $-''-$ 1 60302275 PHT25 90 1 19 30.2 1 23 59.2 1 58600350 PHT25 90 $-''-$ $-''-$ 1 58600342 PHT25 150 1 19 30.2 1 23 59.2 1 58600443 PHT25 150 $-''-$ $-''-$ 1 58600451 PHT25 180 $-''-$ $-''-$ 1 58600545 PHT25 180 $-''-$ $-''-$ 1 58600553 NGP(N) PHT22 90 13 43 53.8 40 11 42 32$\times$4 55800211 PHT22 150 $-''-$ $-''-$ $-''-$ 56200110 PHT22 180 $-''-$ $-''-$ $-''-$ 56300109 PHT22 180 13 42 32.2 40 29 12 15$\times$15 21301803 PHT25 90 13 43 16.3 40 28 46.5 1 55800226 PHT25 90 13 43 16.3 40 28 46.5 1 55800234 PHT25 150 13 43 16.3 40 28 46.6 1 56200127 PHT25 150 13 43 16.3 40 28 46.5 1 56200135 PHT25 180 13 43 16.3 40 28 46.7 1 56300129 PHT25 180 13 43 16.3 40 28 46.5 1 56300137 NGP(S) PHT22 90 13 49 44.5 39 07 37 32$\times$4 55800215 PHT22 150 $-''-$ $-''-$ $-''-$ 56200114 PHT22 180 $-''-$ $-''-$ $-''-$ 56300113 --------- ----------- ------------ ------------- -------------- ---------- -- : List of ISOPHOT observations of EBL fields carried out in the PHT-22 and PHT-25 observation modes. \[table:tdt\] ---------------- ------------ ----------- ----------- ------- ------------ ------------ Field RA DEC $\lambda$ AOT TDT number $\Delta t$ (J2000) (J2000) ($\mu$m) (days) EBL22 2 26 34.4 -25 53 53 25 PHT05 82101132 +2 90 PHT25 82101126 +2 150 PHT25 81901927 180 PHT25 81901529 EBL26\_ZL1$^1$ 1 20 3.0 1 32 30 7.3 PHT05 58600239 25 PHT05 58600240 60 PHT25 58600341 170 PHT25 58600444 EBL26\_ZL2 1 19 30.2 1 23 59 3.6 PHT05 58600646 7.3 PHT05 58600647 25 PHT05 58600648 60 PHT25 58600349 90 PHT25 58600342 90 PHT25 58600350 150 PHT25 58600443 150 PHT25 58600451 170 PHT25 58600452 180 PHT25 58600545 180 PHT25 58600553 NGP\_ZL1 13 43 16.3 40 28 47 7.3 PHT05 55800331 -4 25 PHT05 55800332 -4 60 PHT25 55800233 -4 90 PHT25 55800226 -4 90 PHT25 55800234 -4 150 PHT25 56200127 150 PHT25 56200135 170 PHT25 56200136 180 PHT25 56300129 +1 180 PHT25 56300137 +1 NGP\_ZL2$^2$ 13 43 32.4 40 33 35 7.3 PHT05 55800123 -4 25 PHT05 55800124 -4 170 PHT25 56200128 ---------------- ------------ ----------- ----------- ------- ------------ ------------ $^1$The position is outside the raster maps.\ $^2$The position is outside maps other than the 180$\mu$m map consisting of 15$\times$15 rasters (TDT number 21301803). \[table:ZL\_observations\] Each field was mapped in the PHT22 staring raster map mode (ISOPHOT Handbook, Laureijs et al. [@Handbook]) using filters C\_90, C\_135, and C\_180. The corresponding reference wavelengths of the filters are 90$\mu$m, 150$\mu$m, and 180$\mu$m. The 90$\mu$m observations were made with the C100 detector consisting of 3$\times$3 pixels, with 43.5$\arcsec \times$43.5$\arcsec$ each. The longer wavelength observations were made with the C200 detector which has $2\times 2$ detector pixels, with 89.4$\arcsec \times$89.4$\arcsec$ each. The same raster maps were used in Juvela et al. ([@Juvela00]). Table \[table:fields\] lists the coordinates and the sizes of the maps. Additionally, we make use of PHT25 absolute photometry measurements (see ISOPHOT Handbook, Laureijs et al. [@Handbook]) made at the same three wavelengths. Two positions in NGP, two positions in EBL26, and one position in EBL22 were observed in this mode. Field Ra (2000) Dec (2000) $l$ $b$ $\lambda$. $\beta$ Rasters Area ($\square\degr$) Notes -------- ------------ ------------ ------- ------- ------------ --------- -------------- ----------------------- ---------------------------------------- EBL22 02 26 34.5 -25 53 43 215.0 -68.7 23.8 -37.9 32$\times$3 0.19 02 32 48.0 -25 54 06 215.6 -67.3 25.5 -38.5 31$\times$1 0.07 1D scan into a region of higher cirrus EBL26 01 18 14.5 01 56 40 136.5 -60.2 18.8 -5.9 32$\times$4 0.27 NGP(N) 13 43 53.0 40 11 35 86.5 73.0 184.2 46.5 32$\times$4 0.27 13 42 32.2 40 29 12 87.9 73.0 183.6 46.6 15$\times$15 0.56 180$\mu$m only NGP(S) 13 49 43.7 39 07 30 81.3 72.9 186.4 46.1 32$\times$4 0.27 \[table:fields\] Reduction of EBL field observations {#sect:isophot_reduction} ----------------------------------- The ISOPHOT data were processed with PIA (PHT Interactive Analysis) program version 11.3. For details of the analysis steps, see the ISOPHOT Handbook (Laureijs et al. [@Handbook]) and Appendix, section \[sect:techcal\]. For C100 a method of signal transient correction was introduced in PIA 11.3. This procedure was used for all C100 measurements. Nevertheless, some of the internal calibrator (FCS) measurements show residual drifts. In those cases we applied transient recognition which removes the initial, unstabilised part of the measurements. The flux density calibration was made using the internal calibrator measurements (FCS1) performed immediately before and after each map for actual detector response assessment. The calibration was applied using the average response of the two FCS measurements. The reduced data contained a few artifacts. These include short time scale detector drifts at the beginning of some C100 observations, temporary signal variations caused by cosmic ray glitches, and occasional drifting of some detector pixels that may also be connected with cosmic ray hits. The time ordered data were examined by eye. For rasters and detector pixels affected by clear anomalies (glitches or drifting) the corresponding PIA error estimates were scaled upwards, typically by a factor of a few. For each detector pixel the signal values were scaled so that their average value over a map became equal to the overall average over all detector pixels. The scaling takes into account the already manually adjusted error estimates. The flat fielding would actually not be necessary, because FIR fluxes are compared only with observed HI 21cm lines and, therefore, averaged over areas that are large compared with the size of the ISOPHOT rasters. Long term detector response drifts are not taken out by a simple averaging of the FCS measurements, nor is an initial non-linear drift corrected for by linear interpolation between the two FCS measurements. Both could introduce an artificial gradient in the time ordered data and, because of the systematic scan pattern, also in the maps. The maps were compared with IRAS data in order to see if there were any gradients uncorrelated with the IRAS 100$\mu$m signal. The only significant difference was found in the C200 observations of the southern NGP field. The gradient was removed while keeping the average surface brightness unchanged. The correction has little effect on the subsequent analysis. Apart from the EBL22 field, all maps contain four detector scans that run alternatively in opposite directions along the longer map dimension. When data are correlated with the lower resolution HI observations, the subsequent scan legs tend to cancel out any long term drifts. The raster map observations themselves do not contain any direct measurement of the dark current. In such cases one usually relies on the orbit dependent “default” dark current estimates included in the PIA. However, absolute photometry PHT-25 measurements were carried out within a couple of hours before or after each raster map. The data reduction was carried out also using the dark current and cold FCS values obtained from those measurements. In the subsequent analysis, we use maps that are averages of those obtained using default dark current values and those obtained using PHT-25 dark current measurements. When absolute photometry points were inside the mapped area they were compared with the surface brightness of the raster maps. The maps were re-scaled so that the final surface brightness corresponds to the average of the original FCS calibrated maps and the values given by the absolute photometry measurements. This causes systematic lowering of the surface brightness values of the original maps. For EBL26, NGP(N), and NGP(S) the change is typically $\sim$4%, for both C100 and C200 observations. In the case of EBL22 the correction is larger, some 20%, for the C200 detector. In the region NGP there are separate northern and southern fields that overlap by a few arc minutes. The maps, each containing 32$\times$4 raster points, were fitted together using the overlapping area, where the final map is at a level equal to the average of the northern and the southern maps. The resulting change in the surface brightness levels of individual maps was $\sim$5% or less. In the north there is yet another 15$\times$15 raster map that was observed only at 180$\mu$m. Because that measurement includes only very short FCS measurements, it was scaled to fit the already combined long 180$\mu$m map. This required scaling of the surface brightness values by a factor of 1.05. The main maps of the field EBL22 cover an area of low cirrus emission. There are additional one-directional scans that extend to a region of higher surface brightness in the west. In the absence of scans in the opposite direction, it is not possible to directly determine the presence of detector response drifts. However, these observations were reduced using the average of the responsivities given by the two FCS measurements and the error bars reflect also the difference in the responsivity before and after the measurement. Using the overlapping area, the $32\times 1$ raster strips were scaled to the same level with the $32\times 3$ raster maps. The scalings applied were 0.97, 1.02, and 0.84 at 90$\mu$m, 150$\mu$m, and 180$\mu$m, respectively. The final FIR errorbars show the uncertainty for the weighted means over the Effelsberg beam. The noise of each HI spectrum was estimated separately using the velocity channels outside the line. The uncertainty of the line area was calculated assuming the same, uncorrelated noise for the integrated velocity interval. This might underestimate the total uncertainty, because it ignores the uncertainties in the stray radiation subtraction that do not affect the signal in the line wings. However, for a small field the stray radiation causes a constant systematic error rather than statistical uncertainty and does not affect the weighting of the observations when the linear fit is made. For selected positions there exist mid-infrared observations made with the ISOPHOT P-detectors as well as further absolute photometry measurements with the C100 and C200 cameras (see Appendix, Table \[table:ZL\_observations\]). These observations were performed for the purpose of estimating the zodiacal light. The data reduction of P-detector data is similar to that of the C100 and C200 cameras, except that also signal linearisation is included. HI measurements {#sect:HImeas} --------------- The observations of the hydrogen 21cm line were made with the Effelsberg radio telescope in May 2002. The observed positions, $~580$ in number, are indicated in Fig. \[fig:HI\_positions\]. The integration times were 30s in EBL22 and 62s in EBL26. In the field NGP the observations were done with 62s integrations except for the northern part where the integration time was 94s. The average noise estimated from the velocity intervals outside the HI line is 0.15K per channel of 1km/s. This corresponds to a typical uncertainty of 1.7Kkms$^{-1}$ in the integrated line area. For calibration purposes and for precise subtraction of the stray radiation, regular observations of the standard region S7 were made. The stray radiation subtraction is crucial because it affects the zero point of the estimated HI column densities. The observed fields, NGP in particular, have some of the lowest line-of-sight column densities over the whole sky. Under these conditions the stray radiation received by the telescope side lobes becomes a significant fraction of the total signal. The stray radiation was removed with a program developed by P. Kalberla (Kalberla et al. [@Kalberla2005]; see Sect. \[sect:obs\]). In Fig. \[fig:HI\_comparison\] we compare our data with spectra from the Leiden/Dwingeloo survey (Hartmann & Burton [@Hartmann1997]; Kalberla et al. [@Kalberla2005]). For this comparison, in order to match the resolution of the Leiden/Dwingeloo survey, the Effelsberg data were convolved with a gaussian with FWHM equal to 36$\arcmin$. The HI profiles agree very well. Part of the differences may be caused by the fact that our HI maps do not cover the whole area of the 36$\arcmin$ beams. Nevertheless, the figure shows that the observations and the stray radiation subtraction (see Sect. [sect:obs]{}) are consistent with the Kalberla ([@Kalberla2005]) results. Calibration accuracy {#sect:calibration} ==================== The error estimates listed in Table \[table:fit\] are based on the statistical uncertainties in the fits between FIR and HI data. The scatter of data points around the fitted lines is usually larger than their estimated uncertainty. This could be a sign of underestimated measurement uncertainties but is more likely caused by true scatter in the relation. If the formal uncertainties of the line parameters were estimated based on the error estimates of the individual points, the uncertainties could be severely underestimated. Therefore, instead of relying only on the measurement uncertainties, the uncertainty of the fit parameters was estimated separately with the bootstrap method so that they reflect the true scatter of observed points. The error estimates corresponding to a 67% confidence interval are given in Table \[table:fit\]. These uncertainties do not include estimates for the systematic errors introduced by the independent calibration of each map or the absolute accuracy of the overall ISOPHOT calibration. There are both multiplicative and additive sources of uncertainty. The former include, for example, uncertainties in the internal calibration source (FCS) measurements (e.g., detector drifts) that alter the estimated detector response. The uncertainties that affect the zero point of the intensity scale are more critical, because the CIRB is small compared with the observed signal and can be recovered only as the residual after the subtraction of the ZL. Table \[table:uncertainties\] lists an assessment of uncertainty that, using data in Table \[table:fit\], have been converted into uncertainty of the FIR flux at zero HI column density. The quoted values are half of the difference of two values obtained in two independent ways. Thereby the quoted values are also $\sim$1-$\sigma$ estimates for the uncertainty of the average of the two values. In Table \[table:uncertainties\] column 4 has been obtained by comparing the fine calibration source measurements performed before and after each map. The numbers indicate the statistical uncertainty of the detector response measurements. The FCS measurements are generally very consistent, particularly in the case of the C200 detector. On the other hand, the effect of the drift affecting the first FCS measurement of the one-dimensional strip map of EBL22 is clearly visible at 90$\mu$m. The dark signal subtraction is the most important correction affecting the zero point of the FIR intensity. Close to each of the raster map observations, we have one or two absolute photometry observations which include dark signal measurements of their own. In PIA, the default dark current calibration is based on a larger set ($\sim$70) of dark current measurements for which the orbit trend has been determined. Therefore, the PIA default dark current calibration is less affected by the noise of individual measurements but may not take into account short time scale variations in the detector dark current on a specific orbit. The maps were reduced using the default dark current values and the actually measured dark current values. In Table \[table:uncertainties\] column 5 shows the associated uncertainty in the FIR signal at zero HI column density. The observed uncertainty in the dark current values is comparable with the variation observed in the systematic analysis of a large sample of ISOPHOT observations (del Burgo et al. [@delBurgo02]; see also Fig. \[fi:c100pix5darksignal\]). When absolute photometry measurements existed within mapped areas, those were used to re-scale the surface brightness values of the maps (see Sect. \[sect:isophot\_reduction\]). The difference in the absolute photometry and mapping measurements is used to derive the values in column 6 of Table \[table:uncertainties\]. The final column reflects the difference in the surface brightness in areas where two independently calibrated maps overlap. The numbers in columns 6 and 7 include, of course, dark current and FCS uncertainties as one of their components. For the C100 observations at 90$\mu$m the uncertainty is close to 1MJysr$^{-1}$, i.e., comparable with the expected EBL signal. On the other hand, for the C200 detector the uncertainty of an individual map is $\sim$0.3MJysr$^{-1}$. Most of this is caused by the uncertainty in the dark current values. ------------------------------- ----------- -------------- --------------- -------------- ---------------- ---------------- Field $\lambda$ $<S>$ $\Delta$(FCS) $\Delta$(DC) $\Delta$(Abs.) $\Delta$(Join) ($\mu$m) MJysr$^{-1}$ MJysr$^{-1}$ MJysr$^{-1}$ MJysr$^{-1}$ MJysr$^{-1}$ EBL22 90 9.0 0.12 0.43 -1.15 - EBL22 150 5.7 -0.06 1.44 -0.64 - EBL22 180 4.5 -0.38 0.48 -0.28 - EBL22$^1$ 90 6.5 -1.09 0.22 - -0.17 EBL22$^1$ 150 3.6 0.02 0.30 - 0.07 EBL22$^1$ 180 4.5 0.04 1.00 - -0.50 EBL26 90 20.6 -0.09 -0.80 -1.66 - EBL26 150 4.3 -0.05 -0.28 -0.09 - EBL26 180 3.7 0.06 -0.20 0.11 - NGP(N) 90 7.8 0.49 -0.19 -0.50 0.58 NGP(N) 150 5.1 0.06 -0.30 -0.35 0.19 NGP(N) 180 4.8 0.22 -0.22 -0.26 0.39 NGP(S) 90 6.9 -0.32 0.26 - -0.49 NGP(S) 150 4.7 0.04 -0.30 - -0.18 NGP(S) 180 4.7 0.05 -0.31 - -0.31 [$^1$The narrow strip map.]{} ------------------------------- ----------- -------------- --------------- -------------- ---------------- ---------------- \[table:uncertainties\] Straylight radiation -------------------- Straylight may be another instrumental artefact affecting the zero level of the FIR surface brightness. By design and operation ISO’s viewing direction stayed by several tens of degrees away from the brightest FIR emitters in the sky, the Sun, the Earth and the Moon (Kessler et al. [@Kessler2003]). A dedicated straylight program was executed verifying by deep “differential” integrations that the uniform straylight level due to these sources was below ISOPHOT’s detection limit, even under the most unfavourable pointing conditions close to the visibility constraints (Lemke et al. [@Lemke2001]). Specular straylight by the second brightest class of objects, the giant planets Jupiter and Saturn, was observed when pointing to within 15$\arcmin$ to 1$\degr$ of the planet, expressing itself as finger-like stripes or faint ghost rings (Kessler et al. [@Kessler2003], Lemke et al. [@Lemke2001]). The NGP and EBL 22 fields are far away from the ecliptic and can thus not suffer from this type of straylight. For EBL 26 we checked the positions of the planets Mars, Jupiter, Saturn, Uranus and Neptune at the time of the observations, 1997-06-26 and 1997-07-11, respectively. Mars, Jupiter, Uranus, Neptune were all far off. Saturn was at a distance of 3.25 degrees, which is still more than a factor of 3 off of any known straylight-critical distance. Determination of the ZL levels {#sect:zlfit} ============================== The ZL level was estimated by fitting ZL and cirrus templates to ISOPHOT observations in the wavelength range from 7.3$\mu$m to 200$\mu$m (see Sect. \[sect:ZL\]). Figure \[fig:zlfit\] shows the results of these fits. In the field EBL22 we had observations of one position and in the fields EBL26 and NGP of two positions (see Table \[table:ZL\_observations\]). For the latter two fields, the figures show the fit to data combined from the two positions. Table \[table:ZL\_observations\] lists the time difference between the listed observations and the observations of the raster maps. In the case of NGP these are relative to the 150$\mu$m observations. The 90$\mu$m maps were observed four days before and the 180$\mu$m one day after the 150$\mu$m maps. According to the Kelsall et al. ([@Kelsall98]) ZL model the four day difference causes only $\sim$1.5% change in the expected ZL. The combined NGP map is almost 1.5 degrees long. In the Kelsall model the difference in the centre positions of the southern and northern parts corresponds to about 1% difference in the ZL. Therefore, we use only one zodiacal estimate value for both NGP(N) and NGP(S) and for all observations made during the five day interval. In the fields EBL26 and NGP, MIR observations exist for two separate positions (see Fig. \[fig:allsky\]). In both fields, the measurements at these two positions are close to each other, both in time and position. Therefore, their ZL values should be identical and also the cirrus levels should be very similar. Comparison of the fits performed using these independent sets of measurements gives the first indication of the statistical uncertainty of the ZL values. In both fields, the ZL values obtained for the two positions agree within 10%. The observations are fitted as a sum of ZL and cirrus components. The ZL template is a black body curve at the temperature obtained from Leinert et al. ([@Leinert2002]). The cirrus template is based on the model by Li & Draine ([@Li2001]). Using the ISOPHOT filter profiles we calculate for both radiation components, ZL and cirrus, and for each filter the in-band power values that can be directly compared with the observed values. In the fit we have only two free parameters, the intensity of the ZL component and the intensity of the cirrus component. The ZL estimates should be based mainly on data between 10$\mu$m and 60$\mu$m where the ZL is clearly the dominant component. Therefore, in the fit, the weight of the data points in this wavelength range is increased by a factor of two. The level of the cirrus component is determined mostly by the longer wavelength data. In reality, the component corresponds to the sum of the cirrus and CIRB signals. As long as the component is small in the MIR, the ZL estimates are almost independent of the exact shape of this template. We confirmed this by replacing the Li & Draine ([@Li2001]) cirrus template by a pure CIRB template, using the model curve from Dole et al. ([@Dole2006]; Fig. 13). The resulting change in the ZL estimates was less than one per cent. The actual statistical errors of the ZL values are estimated using the standard deviation of the relative errors when observations are compared with the fitted ZL curve. The last column of Table \[table:ZL\_estimates\] lists the corresponding error of the mean, calculated using data points between 7.3$\mu$m and 90$\mu$m. In the case of fields NGP and EBL26, the error estimates are calculated from the fits where we have combined the data from the two measured positions within each field. In all three fields, the obtained relative uncertainties are $\sim$10%. In the fields EBL26 and NGP the uncertainties are also consistent with the difference of the ZL values obtained for the two individual positions. The ZL fits are shown in Fig. \[fig:zlfit\]. In this paper we have used original ISOPHOT observations without applying colour corrections. Therefore, in the fitting procedure also the ZL and cirrus templates were converted to corresponding values using the ISOPHOT filter profiles. However, for Fig. \[fig:zlfit\] we have performed colour corrections. The templates are plotted by connecting the values at the nominal wavelengths by a straight line. The template spectra used in the ZL fitting are colour corrected using their respective spectral shapes. In the figure, the colour correction of the observed surface brightness values is done assuming the blackbody ZL spectrum below 90$\mu$m, and a modified black body cirrus spectrum, $B_{\nu}(T=18\,{\rm K}) \nu^2$, at 90$\mu$m and longer wavelengths. The plots include DIRBE values from the DIRBE weekly maps. These correspond to the DIRBE pixel closest to the centre of the corresponding ISOPHOT map. Linear interpolation was performed between the weeks in order to accurately match the solar elongation of the ISOPHOT observations. In addition to the DIRBE value that corresponds directly to the ISOPHOT observations (solid squares) we plot the DIRBE value for the same solar aspect angle and opposite sign of the solar elongation. Assuming that the zodiacal dust cloud is symmetric along the ecliptic, the two values should be identical. The predictions of the ZL model of Kelsall et al. ([@Kelsall98]) are also plotted. The DIRBE values are colour corrected. As in the case of ISOPHOT data, colour correction of the observations assumes a blackbody ZL spectrum at and below 60$\mu$m, and a modified black body cirrus spectrum, $B_{\nu}(T=18\,{\rm K}) \nu^2$, at the longer wavelengths, 100, 140, and 240$\mu$m. There is a clear difference in the ISOPHOT and DIRBE surface brightness scales. The DIRBE values are consistently lower by some 20–30%, in the MIR range. In the FIR bands the extended cirrus structures combined with the much larger pixel size and noise in the DIRBE pixels precludes direct comparison. The determination of the CIRB values is not directly affected by a possible calibration difference between DIRBE and ISOPHOT because, in this paper, we use exclusively ISOPHOT measurements. Systematic uncertainties affecting all ISOPHOT bands have only little impact on the derived CIRB values. The relative calibration accuracy between the FIR cameras and the ISOPHOT-P photometer is more important, because the zodiacal light estimates are based on the latter. When the absolute level of the zodiacal light was estimated we calculated the scatter between the SED model and the observations at different wavelengths (see Table \[table:ZL\_estimates\]). The scatter was typically $\sim$10–20%. The importance of this error source depends, of course, on the absolute level of the ZL emission. The field EBL26 is located near the ecliptic plane and at 90$\mu$m the observed signal and the ZL are both of the order of 20MJysr$^{-1}$. Therefore, a relative uncertainty of 10% would already correspond to about twice the expected level of the CIRB. For EBL22 and especially for NGP the zodiacal light level is much lower so that more meaningful limits can be derived for the CIRB also at 90$\mu$m. The quoted ZL error estimates reflect the uncertainty in the determined ZL level in the mid-infrared. If there were a systematic difference in the calibration of the mid- and FIR-bands, the ZL estimates could be wrong by the corresponding amount. Generally the relative calibration accuracy is considered to be within 15%. This uncertainty would not necessarily be reflected in the quality of the ZL spectrum fits, because a systematic calibration error could have been partly compensated by a change in the intensity of the cirrus component. The ZL spectrum was assumed to be a pure black body with the temperature given by Leinert et al. ([@Leinert2002]). As far as the mid-infrared points are concerned, a wrong temperature would, at some level, be reflected also in our error estimate. However, if the ZL spectrum deviated from the assumed shape only in the FIR this could again be masked by a change in the fitted cirrus component without a corresponding increase in the rms value. Therefore, we must explicitly assume that the same ZL temperature is applicable both at mid-infrared and far-infrared wavelengths. However, because a 5K change in the ZL temperature corresponds to only $\sim$2% relative change in the ratio of 150$\mu$m and 25$\mu$m intensities, this source of uncertainty is unimportant compared with the uncertainty in the relative calibrations of the different detectors. Comparison with DIRBE EBL estimates {#sect:dirbe_ebl} =================================== This present study represents the first determination of the absolute level of the FIR EBL that is independent of measurements of the COBE DIRBE instrument. In Table \[table:comparison\] we list FIR EBL estimates given in seven publications based on the DIRBE measurements. Included are also our 2-$\sigma$ upper limit at 90$\mu$m and the EBL estimate for the range 150-180$\mu$m. ----------- ----------------- ------------------------ ------------- $\lambda$ $I_{\nu}$ Reference Instrument ($\mu$m) (MJysr$^{-1}$) 90 $<$2.3 this paper ISO/ISOPHOT 150/180 1.08 (0.32)$^2$ this paper ISO/ISOPHOT 100 $<1.1^1$ Hauser et al. 1998 COBE/DIRBE 0.73 (0.20)$^1$ 100 0.37 (0.10) Dwek et al. 1998 COBE/DIRBE 100 0.78 (0.20) Lagache et al. 2000 COBE/DIRBE 100 0.83 (0.27) Finkbeiner et al. 2000 COBE/DIRBE 140 1.49 (0.33) Schlegel et al. 1998 COBE/DIRBE 140 1.17 (0.33) Hauser et al. 1998 COBE/DIRBE 140 0.70 (0.28) Hauser et al. 1998 COBE/FIRAS 140 0.70 (0.28) Lagache et al. 1999 COBE/DIRBE 140 1.12 (0.56) Lagache et al. 2000 COBE/DIRBE 140 1.17 (0.37) Odegard et al. 2007 COBE/DIRBE 240 1.36 (0.16) Schlegel et al. 1998 COBE/DIRBE 240 1.12 (0.24) Hauser et al. 1998 COBE/DIRBE 240 1.04 (0.16) Hauser et al. 1998 COBE/FIRAS 240 0.88 (0.16) Lagache et al. 1999 COBE/DIRBE 240 0.88 (0.56) Lagache et al. 2000 COBE/DIRBE 240 1.04 (0.24) Odegard et al. 2007 COBE/DIRBE ----------- ----------------- ------------------------ ------------- : Comparison of existing CIRB estimates in the FIR range. The error estimates quoted by the authors are shown in parenthesis. In our case, we include only the statistical uncertainty. $^1$ Hauser et al. did not claim detection at 100$\mu$m, because the CIRB signal failed test for isotropy.\ $^2$Only the statistical error is quoted. \[table:comparison\] [^1]: Based on observations with the Infrared Space Observatory ISO. ISO is an ESA project with instruments funded by ESA member states (especially the PI countries France, Germany, The Netherlands, and the UK) and with participation of ISAS and NASA.

--- abstract: 'We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The argument is based on the construction of a continuous function which is not the Jacobian of a biLipschitz map.' author: - | Dmitri Burago [^1]\ Bruce Kleiner[^2] bibliography: - 'refs.bib' title: Separated nets in Euclidean space and Jacobians of biLipschitz maps --- Introduction ============ A subset $X$ of a metric space $Z$ is a [*separated net*]{} if there are constants $a,\,b>0$ such that $d(x,x')>a$ for every pair $x,\,x'\in X$, and $d(z,X)<b$ for every $z\in Z$. Every metric space contains separated nets: they may be constructed by finding maximal subsets with the property that all pairs of points are separated by some distance $a>0$. It follows easily from the definitions that two spaces are quasi-isometric if and only if they contain biLipschitz equivalent separated nets. One may ask if the choice of these nets matters, or, in other words, whether any two separated nets in a given space are biLipschitz equivalent. To the best of our knowledge, this problem was first posed by Gromov [@asyinv p.23]. The answer is known to be yes for separated nets in non-amenable spaces (under mild assumptions about local geometry), see [@Gromov; @McMullen; @Whyte]; more constructive proofs in the case of trees or hyperbolic groups can be found in [@Papas; @Bogop]. In this paper, we prove the following theorem: \[T1\] There exists a separated net in the Euclidean plane which is not biLipschitz equivalent to the integer lattice. The proof of Theorem \[T1\] is based on the following result: \[T2\] Let $I:=[0,1]$. Given $c>0$, there is a continuous function $\rho:I^2{\rightarrow}[1,1+c]$, such that there is no biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ with $$Jac(f):=Det(Df)=\rho \quad a.e.$$ 1\. Although we formulate and prove these theorems in the 2-dimensional case, the same proofs work with minor modifications in higher dimensional Euclidean spaces as well. We only consider the 2-dimensional case here to avoid cumbersome notation. 2\. Theorem \[T2\] also works for Lipschitz homeomorphisms; we do not use the lower Lipschitz bound on $f$. 3\. After the first version of this paper had been written, Curt McMullen informed us that he also had a proof of Theorems \[T1\] and \[T2\]. See [@McMullen] for a discussion of the the linear analog of Theorem \[T2\], and the Hölder analogs of the mapping problems in Theorems \[T1\] and \[T2\]. The problem of prescribing Jacobians of homeomorpisms has been studied by several authors. In [@DacMos] Dacorogna and Moser proved that every ${\alpha}$-Holder continuous function is locally the Jacobian of a $C^{1,{\alpha}}$ homeomorphism, and they then raised the question of whether any continuous function is (locally) the Jacobian of a $C^1$ diffeomorphism. [@RivYe; @Ye] consider the prescribed Jacobian problem in other regularity classes, including the cases when the Jacobian is in $L^{\infty}$ or in a Sobolev space. Overview of the proofs [*Theorem \[T2\] implies Theorem \[T1\].*]{} Let $\rho:I^2{\rightarrow}{\mathbb R}$ be measurable with $0<\inf\rho\leq \sup\rho <\infty$. We will indicate why $\rho$ would be the Jacobian of a biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ if all separated nets in ${\mathbb E}^2$ were biLipschitz equivalent. Take a disjoint collection of squares $S_i\subset{\mathbb E}^2$ with side lengths $l_i$ tending to infinity, and “transplant” $\rho$ to each $S_i$ using appropriate similarities ${\alpha}_i:I^2{\rightarrow}S_i$, i.e. set $\rho_i\defeq \rho\circ{\alpha}_i^{-1}$. Then construct a separated net $L\subset {\mathbb E}^2$ so that the “local average density” of $L$ in each square $S_i$ approximates $\rho_i^{-1}$. If $g:L{\rightarrow}{\mathbb Z}^2$ is a biLipschitz homeomorphism, consider “pullbacks” of $g{\mbox{\Large \(|\)\normalsize}}_{S_i}$ to $I^2$, i.e. pre and post-compose $g{\mbox{\Large \(|\)\normalsize}}_{S_i}$ with suitable similarities so as to get a sequence of uniformly biLipschitz maps $g_i:I^2\supset Z_i{\rightarrow}{\mathbb E}^2$. Then extract a convergent subsequence of the $g_i$’s via the Arzela-Ascoli theorem, and obtain a limit map $f:I^2{\rightarrow}{\mathbb E}^2$ with Jacobian $\rho$. [*Theorem \[T2\].*]{} The observation underlying our construction is that if the Jacobian of $f:I^2{\rightarrow}{\mathbb E}^2$ oscillates in a rectangular neighborhood $U$ of a segment $\ol{xy}\subset I^2$, then $f$ will be forced to stretch for one of two reasons: either it maps $\ol{xy}$ to a curve which is far from a geodesic between its endpoints, or it maps $\ol{xy}$ close to the segment $\ol{f(x)f(y)}$ but it sends $U$ to a neighborhood of $\ol{f(x)f(y)}$ with wiggly boundary in order to have the correct Jacobian. By arranging that $Jac(f)$ oscillates in neighborhoods of a hierarchy of smaller and smaller segments we can force $f$ to stretch more and more at smaller and smaller scales, eventually contradicting the Lipschitz condition on $f$. We now give a more detailed sketch of the proof. We first observe that it is enough to construct, for every $L>1,\,\bar c>0$, a continuous function $\rho_{L,\bar c}:I^2{\rightarrow}[1,1+\bar c]$ such that $\rho_{L,\bar c}$ is not the Jacobian of an L-biLipschitz map $I^2{\rightarrow}{\mathbb E}^2$. Given such a family of functions, we can build a new continuous function $\rho:I^2{\rightarrow}[1,1+c]$ which is not the Jacobian of any biLipschitz map $I^2{\rightarrow}{\mathbb E}^2$ as follows. Take a sequence of disjoint squares $S_k\subset I^2$ which converge to some $p\in I^2$, and let $\rho:I^2{\rightarrow}[1,1+c]$ be any continuous function such that $\rho{\mbox{\Large \(|\)\normalsize}}_{S_k}=\rho_{k,\min(c,\frac{1}{k})}\circ{\alpha}_k$ where ${\alpha}_k:S_k{\rightarrow}I^2$ is a similarity. Also, note that to construct $\rho_{L,\bar c}$, we really only need to construct a measurable function with the same property: if $\rho^k_{L,\bar c}$ is a sequence of smoothings of a measurable function $\rho_{L,\bar c}$ which converge to $\rho_{L,\bar c}$ in $L^1$, then any sequence of $L$-biLipschitz maps $\phi_k:I^2{\rightarrow}{\mathbb E}^2$ with $Jac(\phi_k)=\rho^k_{L,\bar c}$ will subconverge to a biLipschitz map $\phi:I^2{\rightarrow}{\mathbb E}^2$ with $Jac(\phi)=\rho_{L,\bar c}$. We now fix $L>1,\,c>0$, and explain how to construct $\rho_{L,c}$. Let $R$ be the rectangle $[0,1]\times [0,\frac{1}{N}]\subset {\mathbb E}^2$, where $N\gg 1$ is chosen suitably depending on $L$ and $c$, and let $S_i=[\frac{i-1}{N},\frac{i}{N}]\times[0,\frac{1}{N}]$ be the $i^{th}$ square in $R$. Define a “checkerboard” function $\rho_1:I^2{\rightarrow}[1,1+c]$ by letting $\rho_1$ be $1+c$ on the squares $S_i$ with $i$ even and $1$ elsewhere. Now subdivide $R$ into $M^2N$ squares using $M$ evenly spaced horizontal lines and $MN$ evenly spaced vertical lines. We call a pair of points [*marked*]{} if they are the endpoints of a horizontal edge in the resulting grid. The key step in the proof (Lemma \[MLE\]) is to show that any biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ with Jacobian $\rho_1$ must stretch apart a marked pair quantitatively more than it stretches apart the pair $(0,0),\,(1,0)$; more precisely, there is a $k>0$ (depending on $L,\, c$) so that $\frac{d(f(p),f(q))}{d(p,q)}>(1+k)d(f(0,0),f(1,0))$ for some marked pair $p,\,q$. If this weren’t true, then we would have an $L$-biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ which stretches apart all marked pairs by a factor of at most $(1+\eps)d(f(0,0),f(0,1))$, where $\eps\ll 1$. This would mean that $f$ maps horizontal lines in $R$ to “almost taut curves”. Using triangle inequalities one checks that this forces $f$ to map most marked pairs $p,\,q$ so that vector $f(q)-f(p)$ is $\approx d(p,q)(f(1,0)-f(0,0))$; this in turn implies that for some $1\leq i\leq N$, [*all*]{} marked pairs $p,\,q$ in the adjacent squares $S_i,\,S_{i+1}$ are mapped by $f$ so that $f(q)-f(p)\approx d(p,q) (f(1,0)-f(0,0))$. Estimates then show that $f(S_i)$ and $f(S_{i+1})$ have nearly the same area, which contradicts the assumption that $Jac(f)=\rho_1$, because $\rho_1$ is $1$ on one of the squares and $1+c$ on the other. Our next step is to modify $\rho_1$ in a neighborhood of the grid in $R$: we use thin rectangles (whose thickness will depend on $L,\, c$) containing the horizontal edges of our grid, and define $\rho_2:I^2{\rightarrow}[1,1+c]$ by letting $\rho_2$ be a “checkerboard” function in each of these rectangles and $\rho_1$ elsewhere. Arguing as in the previous paragraph, we will conclude that some suitably chosen pair of points in one of these new rectangles will be stretched apart by a factor $>d(f(0,0),f(1,0))(1+k)^2$ under the map $f$. Repeating this construction at smaller and smaller scales, we eventually obtain a function which can’t be the Jacobian of an $L$-biLipschitz map. The paper is organized as follows. In Section 2 we prove that Theorem \[T2\] implies Theorem \[T1\]. Section 3 is devoted to the proof of Theorem \[T2\]. Reduction of Theorem \[T1\] to Theorem \[T2\] ============================================= Recall that every biLipschitz map is differentiable a.e., and the area of the image of a set is equal to the integral of the Jacobian over this set. We formulate our reduction as the following Lemma: Let $\rho:I^2{\rightarrow}[1,1+c]$ be a measurable function which is not the Jacobian of any biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$ with $$Jac(f):=det(Df)=\rho \quad a.e.$$ Then there is a separated net in ${\mathbb E}^2$ which is not biLipschitz homeomorphic to ${\mathbb Z}^2$. In what follows, the phrase “subdivide the square $S$ into subsquares will mean that $S$ is to be subdivided into squares using evenly spaced lines parallel to the sides of $S$. Let ${\cal S}=\{S_k\}_{k=1}^\infty$ be a disjoint collection of square regions in ${\mathbb E}^2$ so that each $S_k$ has integer vertices, sides parallel to the coordinate axes, and the side length $l_k$ of $S_k$ tends to $\infty$ with $k$. Choose a sequence $m_k\in(1,\infty)$ with $\lim_{k{\rightarrow}\infty}m_k=\infty$ and $\lim_{k{\rightarrow}\infty}\frac{m_k}{l_k}=0$. Let $\phi_k:I^2{\rightarrow}S_k$ be the unique affine homeomorphism with scalar linear part, and define $\rho_k:S_k{\rightarrow}[1,1+c]$ by $\rho_k=(\frac{1}{\rho})\circ \phi_k^{-1}$. Subdivide $S_k$ into $m_k^2$ subsquares of side length $\frac{l_k}{m_k}$. Call this collection ${\cal T}_k=\{T_{ki}\}_{i=1}^{m_k^2}$. For each $i$ in $\{1,\ldots,m_k^2\}$, subdivide $T_{ki}$ into $n_{ki}^2$ subsquares $U_{kij}$ where $n_{ki}$ is the integer part of $\sqrt{\int_{T_{ki}}\rho_kd{\cal L}}$. Now construct a separated net $X\subset{\mathbb E}^2$ by placing one point at the center of each integer square not contained in $\cup S_k$, and one point at the center of each square $U_{kij}$. We now prove the lemma by contradiction. Suppose $g:X{\rightarrow}{\mathbb Z}^2$ is an $L$-biLipschitz homeomorphism. Let $X_k=\phi_k^{-1}(X)\subset I^2$, and define $f_k:X_k{\rightarrow}{\mathbb E}^2$ by $$f_k(x)=\frac{1}{l_k}(g\circ\phi_k(x)-g\circ\phi_k(\star_k))$$ where $\star_k$ is some basepoint in $X_k$. Then $f_k$ is an $L$-biLipschitz map from $X_k$ to a subset of ${\mathbb E}^2$, and the $f_k$’s are uniformly bounded. By the proof of the Arzela-Ascoli theorem we may find a subsequence of the $f_k$’s which “converges uniformly” to some biLipschitz map $f:I^2{\rightarrow}{\mathbb E}^2$. By the construction of $X$, the counting measure on $X_k$ (normalized by the factor $\frac{1}{l_k^2}$) converges weakly to $\frac{1}{\rho}$ times Lebesgue measure, while the (normalized) counting measure on $f_k(X_k)$ converges weakly to Lebesgue measure. It follows that $f_*((\frac{1}{\rho}){\cal L})={\cal L}{\mbox{\Large \(|\)\normalsize}}_{f(I^2)}$, i.e. $Jac(f)=\rho$. 501em $\square$=0 Construction of a continuous function which is not a Jacobian of a biLipschitz map ================================================================================== The purpose of this section is to prove Theorem \[T2\]. As explained in the introduction, Theorem \[T2\] follows from \[ML\] For any given $L$ and $c>1$, there exists a continuous function $\rho:S=I^2{\rightarrow}[1,1+c]$, such that there is no $L$-biLipschitz homeomorphism $f:I^2{\rightarrow}{\mathbb E}^2$ with $$Jac(f)=\rho \quad a.e.$$ From now on, we fix two constants $L$ and $c$ and proceed to construct of a continuous function $\rho: I^2 \rightarrow [1,1+c]$ which is not a Jacobian of an $L$-biLipschitz map. By default, all functions which we will describe, take values between $1$ and $1+c$. We say that two points $x,\,y\in I^2$ are $A$-stretched (under a map $f:I^2 \rightarrow {\mathbb E}^2$) if $d(f(x),f(y)) \geq Ad(x,y)$. For $N\in{\mathbb N}$, $R_N$ be the rectangle $[0,1]\times [0,\frac{1}{N}]$ and define a “checkerboard” function $\rho_N:R_N \rightarrow [1,1+c]$ by $\rho_N(x,y)=1$ if $[Nx]$ is even and $1+c$ otherwise. It will be convenient to introduce the squares $S_i=[\frac{i-1}{N},\frac{i}{N}] \times [0, \frac{1}{N}]$, $i=1, \dots,N$; $\rho_N$ is constant on the interior of each $S_i$. The cornerstone of our construction is the following lemma: \[MLE\] There are $k>0,\,M,\,\mu$, and $N_0$ such that if $N\geq N_0$, $\eps\leq \frac{\mu}{N^2}$ then the following holds: if the pair of points $(0,0)$ and $(1,0)$ is $A$-stretched under an $L$-biLipschitz map $f:R_N \rightarrow {\mathbb E}^2$ whose Jacobian differs from $\rho_N$ on a set of area no bigger than $\epsilon$ , then at least one pair of points of the form $((\frac{p}{NM}, \frac{s}{NM}), (\frac{q}{NM}, \frac{s}{NM}))$ is $(1+k)A$-stretched (where $p$ and $q$ are integers between $0$ and $NM$ and $s$ is an integer between $0$ and $M$). We will need constants $k,\,l,\,m,\,\eps\in(0,\infty)$ and $M,\,N\in {\mathbb N}$, which will be chosen at the end of the argument. We will assume that $N>10$ and $c,\,l<1$. Let $f:R_N{\rightarrow}{\mathbb E}^2$ be an $L$-biLipschitz map such that $Jac(f)=\rho_N$ off a set of measure $\eps$. Without loss of generality we assume that $f(x)=(0,0)$ and $f(y)=(z,0)$, $z \geq A$. We will use the notation $x_{pq}^i:=(\frac{p+M(i-1)}{NM}, \frac{q}{NM})$, where $i$ is an integer between 1 and $N$, and $p$ and $q$ are integers between $0$ and $M$. We call these points [*marked*]{}. Note that the marked points in $S_i$ are precisely the vertices of the subdivision of $S_i$ into $M^2$ subsquares. The index $i$ gives the number of the square $S_i$, and $p$ and $q$ are “coordinates” of $x_{pq}^i$ within the square $S_i$. We will prove Lemma \[MLE\] by contradiction: we assume that all pairs of the form $x_{pq}^i, x_{sq}^j$ are no more than $(1+k)A$-stretched. If $x^i_{pq}\in S_i$ is a marked point, we say that $x^{i+1}_{pq}\in S_{i+1}$ is the [*marked point corresponding to $x^i_{pq}$*]{}; corresponding points is obtained by adding the vector $(\frac{1}{N},0)$, where $\frac{1}{N}$ is the side length of the square $S_i$. We are going to consider vectors between the images of marked points in $S_i$ and the images of corresponding marked points in the neighbor square $S_{i+1}$. We denote these vectors by $W_{pq}^i:=f(x_{pq}^{i+1})-f(x_{pq}^i)$. We will see that most of the $W_{pq}^i$’s have to be extremely close to the vector $W:=(A/N,0)$, and, in particular, we will find a square $S_i$ where $W_{pq}^i$ is extremely close to $W$ for [*all*]{} $0\leq p,q\leq M$. This will mean that the areas of $f(S_i)$ and $f(S_{i+1})$ are very close, since $f(S_{i+1})$ is very close to a translate of $f(S_i)$. On the other hand, except for a set of measure $\eps$, the Jacobian of $f$ is $1$ in one of the square $S_i,\,S_{i+1}$ and $1+c$ in the other. This allows us to estimate the difference of the areas of their images from below and get a contradiction. If $l\in (0,1)$, we say that a vector $W_{pq}^i=f(x_{pq}^{i+1})-f(x_{pq}^i)$, (or the marked point $x_{pq}^i$), is [*regular*]{} if the length of its projection to the x-axis is greater than $\frac{(1-l)A}{N}$. We say that a square $S_i$ is [*regular*]{} if all marked points $x_{pq}^i$ in this square are regular. There exist $k_1=k_1(l)>0$, $N_1=N_1(l)$, such that if $k \leq k_1,\,N\geq N_1$, there is a regular square. Reasoning by contradiction, we assume that all squares are irregular. By the pigeon-hole principle, there is a value of $ s $ (between 0 and $M$) such that there are at least $\frac{N}{2M+2}$ irregular vectors $W_{p_js}^{ i_j}$, $j=1,2,\dots J \geq \frac{N}{2M+2}$, where $i_j$ is an increasing sequence with a fixed parity. This means that we look for $l$-irregular vectors between marked points in the same row $s$ and only in squares $S_i$’s which have all indices $i$’s even or all odd. The latter assumption guarantees that the segments $[x_{p_js}^{ i_j},x_{p_js}^{ i_j+1}]$ do not overlap. We look at the polygon with marked vertices $$(0,0),x_{0s}^0= (0, s/MN),x_{p_1s}^{ i_1}, x_{p_1s}^{ i_1+1},x_{p_2s}^{ i_2},x_{p_2s}^{ i_2+1} \dots ,x_{p_Js}^{ i_J}, x_{p_Js}^{ i_J+1}, x_{Ms}^N=(1, s/MN), (1,0)$$ The image of this polygon under $f$ connects $(0,0)$ and $(z,0)$ and, therefore, the length of its projection onto the x-axis is at least $z \geq A$. On the other hand, estimating this projection separately for the images of $l$-irregular segments $[x_{p_js}^{ i_j},x_{p_js}^{ i_j+1}]$, the “horizontal” segments $[x_{p_js}^{ i_j+1},x_{p_{j+1}s}^{ i_{j+1}}]$ and the two “vertical” segments $[(0,0), x_{0s}^0]$ and $[x_{Ms}^N, (1,0)]$ , one gets that the lengths of this projection is no bigger than $$\label{projlength} ( \frac{N}{2M+2}) (\frac{(1-l)A}{N}) + (\frac{(1+k)A}{N})(N-\frac{N}{2M+2})+2\frac{L}{N}.$$ The first term in (\[projlength\]) bounds the total length of projections of images of irregular segments by the definition of irregular segments and total number of them. The second summand is maximum stretch factor $(1+k)A$ between marked points times the total length of remaining horizontal segments. The third summand estimates the lengths of images of segments $[(0,0), x_{0s}^0]$ and $[x_{Ms}^N, (1,0)]$ just by multiplying their lengths by our fixed bound $L$ on the Lipschitz constant. Recalling that this projection is at least $z$, which in its turn is no less than $A$, we get $$( \frac{N}{2M+2}) (\frac{(1-l)A}{N}) + (\frac{(1+k)A}{N})(N-\frac{N}{2M+2})+2\frac{L}{N}\geq A.$$ One easily checks that this is impossible when $k$ is sufficiently small and $N$ is sufficiently large. This contradiction proves Claim 1. 501em $\square$=0 Let $W=(\frac{A}{N}, 0)$. Given any $m>0$, there is an $l_0=l_0(m)>0$ such that if $l \leq l_0$ and $k \leq l$, then $|W-W_{pq}^i| \leq \frac{m}{N}$ for every regular vector $W_{pq}^i$. Consider a regular vector $W_{pq}^i=(X,Y)$. Since $W_{pq}^i$ is regular, $X \geq \frac{(1-l)A}{N}$. On the other hand, $X^2+Y^2 \leq \frac{(1+k)^2A^2}{N^2}$ and $X \leq \frac{(1+k)A}{N}$. Thus the difference of the $x$-coordinates of $W_{pq}^i$ and $W$ is bounded by $\frac{(l+k)A}{N} <\frac{2lA}{N}$. Substituting the smallest possible value $\frac{(1-l)A}{N}$ for $X$ into $X^2+Y^2 \leq \frac{(1+k)^2A^2}{N^2}$, we get $Y^2 \leq \frac{2(l+k)A^2}{N^2} \leq \frac{4lA^2}{N^2}$ . This implies that $$\label{w'sclose} N|W-W_{pq}^i| \leq 2A\sqrt{l^2+l}\leq 2L\sqrt{l^2+l}.$$ The right-hand side of (\[w’sclose\]) tends to zero with $l$, so Claim 2 follows. 501em $\square$=0 There are $m_0>0$, $M_0$ such that if $m<m_0$ and $M>M_0$, then the following holds: if for some $1\leq i\leq N$ and every $p,\,q$ we have $|W-W_{pq}^i|\leq\frac{m}{N}$, then $$\label{areasclose} |Area(f(S_{i+1}))-Area(f(S_i))|<\frac{c}{2N^2}.$$ We assume that $i$ is even and therefore $\rho$ takes the value $1$ on $S_i$ and $1+c$ on $S_{i+1}$; the other case is analogous. We let $Q:=f(S_i)$ and $R=f(S_{i+1})$. $Q$ is bounded by a curve (which is the image of the boundary of $S_i$). Consider the result $\tilde R :=Q+W$ of translating $Q$ by the vector $W=(A/N, 0)$. The area of $\tilde R$ is equal to the area of $Q$. The images of the marked points on the boundary of $S_i$ form an $\frac{L}{NM}$-net on the boundary of $Q$, and the images of marked points on the boundary of $S_{i+1}$ form an $\frac{L}{NM}$-net on $R$. By assumption the difference between $W$ and each vector $W_{pq}^i$ joining the image of a marked point on the boundary of $S_i$ and the image of the corresponding point on the boundary of $S_{i+1}$ is less than $\frac{m}{N}$. We conclude that the boundary of $\tilde R$ lies within the $\frac{m}{N}+\frac{2L}{MN}$-neighborhood of the boundary of $R$. Since $f$ is $L$-Lipschitz, the length of the boundary of $R$ is $\leq 4L/N$. Using a standard estimate for the area of a neighborhood of a curve, we obtain: $$|Area(R)-Area(Q)|=|Area(R)-Area(\tilde R)| \leq \frac{2L}{N}(\frac{m}{N}+\frac{2L}{MN})+\pi(\frac{m}{N}+\frac{2L}{MN})^2.$$ Therefore (\[areasclose\]) holds if $m$ is sufficiently small and $M$ is sufficiently large. 501em $\square$=0 Now assume $m<m_0$, $M>M_0$, $l\leq l_0(m)$, $k\leq\min(l,k_1(l))$, $N\geq N_1(l)$, and $\eps\leq\frac{c}{8N^2L^2}$. Combining claims 1, 2, and 3, we find a square $S_i$ so that (\[areasclose\]) holds. On the other hand, since $Jac(f)$ coincides with $\rho$ off a set of measure $\eps$, $Area(f(S_i))\leq 1/N^2+\eps L^2$ and $Area(f(S_{i+1})\geq (1+c)(1/N^2-\epsilon)$. Using the assumption that $\eps\leq\frac{c}{8N^2L^2}$ we get $$Area(R)-Area(Q)\geq \frac{c}{2N^2},$$ contradicting (\[areasclose\]). This contradiction proves Lemma \[MLE\]. 501em $\square$=0 We will use an inductive construction based on Lemma \[MLE\]. Rather than dealing with an explicit construction of pairs of points as in Lemma \[MLE\], it is more convenient to us to use the following lemma, which is an obvious corollary of Lemma \[MLE\]. (To deduce this lemma from Lemma \[MLE\], just note that all properties of interest persist if we scale our coordinate system.) \[IL\] There exists a constant $k>0$ such that, given any segment $\ol{xy}\subset I^2$ and any neighborhood $\ol{xy}\subset U\subset I^2$, there is a measurable function $\rho:U \rightarrow [1,1+c]$, $\eps>0$ and a finite collection of non-intersecting segments $\ol{l_kr_k}\subset U$ with the following property: if the pair $x,\,y$ is $A$-stretched by an $L$-biLipschitz map $f:U \rightarrow {\mathbb E}^2$ whose Jacobian differs from $\rho$ on a set of area $<\epsilon$ , then for some $k$ the pair $l_k,\,r_k$ is $(1+k)A$-stretched by $f$. The function $\rho$ may be chosen to have finite image. We will prove Lemma \[ML\] by induction, using the following statement. (It is actually even slightly stronger than Lemma \[ML\] since it not only guarantees non-existence of $L$-biLipschitz maps with a certain Jacobian, but also gives a finite collection of points, such that at least one distance between them is distorted more than by factor $L$.) \[FF\] For each integer $i$ there is a measurable function $\rho_i: I^2 \rightarrow [1, 1+c]$ , a finite collection ${\cal S}_i$ of non-intersecting segments $\ol{l_kr_k}\subset I^2$, and $\epsilon_i>0$ with the following property: For every $L$-biLipschitz map $f:I^2 \rightarrow {\mathbb E}^2$ whose Jacobian differs from $\rho_i$ on a set of area $<\epsilon_i$ , at least one segment from ${\cal S}_i$ will have its endpoints $\frac{(1+k)^i}{L}$-stretched by $f$. The case $i=0$ is obvious. Assume inductively that there are $\rho_{i-1},\,\eps_{i-1}$, and a disjoint collection of segments ${\cal S}_{i-1}=\{\ol{l_kr_k}\}$ which satisfy the conditions of the lemma. Let $\{U_k\}$ be a disjoint collection of open sets with $U_k\supset\ol{l_kr_k}$ and with total area $<\frac{\eps_{i-1}}{2}$. For each $k$ apply Lemma \[IL\] to $U_k$ to get a function $\hat\rho_k:U_k{\rightarrow}[1,1+c],\,\hat\eps_k>0$, and a disjoint collection $\hat{\cal S}_k$ of segments. Now let $\rho_i:I^2{\rightarrow}[1,1+c]$ be the function which equals $\hat\rho_k$ on each $U_k$ and equals $\rho_{i-1}$ on the complement of $\cup U_k$; let ${\cal S}_i=\cup \hat{\cal S}_k$, and $\eps_i=\min\hat\eps_k$. The required properties follow immediately. 501em $\square$=0 Lemma \[ML\] and (Theorem \[T2\]) follows from Lemma \[FF\]. [^1]: Supported by a Sloan Foundation Fellowship and NSF grant DMS-95-05175. [^2]: Supported by a Sloan Foundation Fellowship and NSF grants DMS-95-05175 and DMS-96-26911.

--- abstract: 'In this paper, the performance of the selection combining (SC) scheme over Fisher-Snedecor $\mathcal{F}$ fading channels with independent and non-identically distributed (i.n.i.d.) branches is analysed. The probability density function (PDF) and the moment generating function (MGF) of the maximum i.n.i.d. Fisher-Snedecor $\mathcal{F}$ variates are derived first in terms of the multivariate Fox’s $H$-function that has been efficiently implemented in the technical literature by various software codes. Based on this, the average bit error probability (ABEP) and the average channel capacity (ACC) of SC diversity with i.n.i.d. receivers are investigated. Moreover, we analyse the performance of the energy detection that are widely employed to perform the spectrum sensing in cognitive radio networks via deriving the average detection probability (ADP) and the average area under the receiver operating characteristics curve (AUC). To validate our analysis, the numerical results are affirmed by the Monte Carlo simulations.' author: - 'Hussien Al-Hmood,  and H. S. Al-Raweshidy,  [^1][^2] [^3]' title: 'Selection Combining Scheme over Non-identically Distributed Fisher-Snedecor $\mathcal{F}$ Fading Channels' --- [Author 1 : Bare Demo of IEEEtran.cls for Journals]{} Selection combining, Fisher-Snedecor $\mathcal{F}$ fading, average bit error probability, average channel capacity, energy detection. Introduction ============ mitigate the impacts of the multipath fading and shadowing on the performance of wireless communications systems, diversity reception techniques have been used in the open technical literature. Selection combining (SC) approach has been considered as an efficient diversity scheme to improve the signal-to-noise-ratio (SNR) at the receiver side. This is because it’s a non-coherent combining technique where the branch with a high SNR is selected among many branches \[1\]. The statistical properties, namely, the probability density function (PDF), the cumulative distribution function (CDF), and the moment generating function (MGF), of the maximum of random variables (RVs) of the fading channels are widely employed to study the SC diversity \[2\]-\[5\]. In this context, the SC receivers over independent and non-identically distributed (i.n.i.d.) generalized $K_G$ fading channels was investigated in \[2\]. The authors in \[3\] studied the average bit error probability (ABEP) of SC technique with i.n.i.d. branches over $\kappa-\mu$ shadowed fading channels. In \[4\], the PDF, the CDF, and the MGF of the maximum of $\eta-\mu$/gamma RVs were derived and used in the analysis of average channel capacity (ACC) of wireless communications systems. Based on the results of \[4\], the behaviour of energy detection (ED) that is one of the most utilised spectrum sensing methods was analysed in \[5\] by providing unified expressions for the average detection probability (ADP) and the average area under the receiver operating characteristics (ROC) curve (AUC). More recently, the Fisher-Snedecor $\mathcal{F}$ fading channel has been proposed as a composite of Nakagami-$m$/inverse Nakagami-$m$ distributions to model device-to-device (D2D) fading channels at 5.8 GHz in both indoor and outdoor environments \[6\]. In contrast to the generalised-$K$ fading channel, the statistics of the Fisher-Snedecor $\mathcal{F}$ fading channel are expressed in simple analytic functions. Furthermore, it includes Nakagami-$m$, Rayleigh, and one-sided Gaussian as special cases. In addition, the Fisher-Snedecor $\mathcal{F}$ fading channel can be utilised for both line-of-sight (LoS) and non-LoS (NLoS) communications scenarios with better fitting to the empirical measurements than the generalised-$K$ ($K_G$) fading model. The authors in \[7\] derived the basic statistics of the sum of i.n.i.d. Fisher-Snedecor $\mathcal{F}$ RVs with applications to maximal ratio combining (MRC) receivers. The ADP and the average AUC of ED with square law selection (SLS) branches over arbitrarily distributed Fisher-Snedecor $\mathcal{F}$ fading channels were given in \[8\]. The product of multiple Fisher-Snedecor $\mathcal{F}$ RVs, namely, cascaded fading model, was addressed in \[9\]. To the best authors’ knowledge, the statistical characteristics of the maximum of i.n.i.d. Fisher-Snedecor $\mathcal{F}$ variates have not been yet reported in the open literature. Motivated by this and based on the above observations, this paper derives exact analytic closed-form mathematically tractable of the PDF and the MGF of the maximum of i.n.i.d. Fisher-Snedecor $\mathcal{F}$ RVs. To this end, the performance of SC scheme is analysed by deriving the ABEP, the ACC, the ADP and the average AUC of ED in terms of the multivariate Fox’s $H$-function. The PDF and MGF of the Maximum I.N.I.D. Fisher-Snedecor $\mathcal{F}$ Variates ============================================================================== The CDF of the received instantaneous SNR, $\gamma$, at $i$th branch of a SC receiver over Fisher-Snedecor $\mathcal{F}$ fading channel is expressed as \[6, eq. (11)\] \[eqn\_1\] $$\begin{aligned} F_{\gamma_i}(\gamma)&=\frac{\Xi_i^{m_i} \gamma^{m_i}}{m_i B(m_i,m_{s_i})} {_2F_1(m_i+m_{s_i},m_i;1+m_i;-\Xi_i \gamma)}\end{aligned}$$ where $\Xi_i=\frac{m_i}{m_{s_i} \bar{\gamma}_i}$, for $i=1,\cdots,L$, $m_i$, $m_{s_i}$, $L$, and $\bar{\gamma}_i$ stand for the multipath index, the shadowing parameter, the number of diversity branches, and the average SNR, respectively, $B(.,.)$ is the beta function \[10, eq. (8.380.1)\] and $_2F_1(.,.;.;.)$ is the Gauss hypergeometric function \[10, eq. (9.14.1)\]. Recalling the identity \[11, eq. (1.132)\] and performing some mathematical simplifications with the aid of \[10, eq. (8.384.1)\] and \[10, eq. (8.331.1)\], (1) can be equivalently rewritten as \[eqn\_2\] $$\begin{aligned} F_{\gamma_i}(\gamma)=&\frac{\Xi_i^{m_i} \gamma^{m_i} }{\Gamma(m_i) \Gamma(m_{s_i})} \nonumber\\ &\times H^{1,2}_{2,2} \bigg[ \Xi_i \gamma \bigg\vert \begin{matrix} (1-m_i-m_{s_i},1), (1-m_i,1)\\ (0,1),(-m_i,1)\\ \end{matrix} \bigg]\end{aligned}$$ where $\Gamma(.)$ is the gamma function and $H^{m,n}_{p,q}[.]$ is the univariate Fox’s $H$-function defined in \[11, eq. (1.2)\]. \[Proposition\_1\] Let all RVs, $\gamma_i$ $\forall \in \{i,\cdots,L \}$, follow i.n.i.d. Fisher-Snedecor $\mathcal{F}$ distribution. Thus, the PDF of $\gamma=\text{max}\{\gamma_1,\cdots,\gamma_L\}$ is given as \[eq\_3\] $$\begin{aligned} &f_{\gamma}(\gamma)=\bigg(\prod_{i=1}^{L}\frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\bigg)\nonumber\\ &\times \gamma^{\Omega-1}H^{0,1:[1,2]_{i=1:L}}_{1,1:[2,2]_{i=1:L}} \bigg[ \Xi_1 \gamma,\cdots,\Xi_L \gamma\bigg\vert \begin{matrix} (-\Omega;\{1\}_{i=1:L})\\ (1-\Omega;\{1\}_{i=1:L})\\ \end{matrix}\bigg\vert \nonumber\\ &\begin{matrix} [(1-m_i-m_{s_i},1), (1-m_i,1)]_{i=1:L}\\ [(0,1),(-m_i,1)]_{i=1:L}\\ \end{matrix} \bigg]\end{aligned}$$ where $\Omega=\sum_{i=1}^L m_i$ and $H^{m,n:m_1,n_1;\cdots;m_L,n_L}_{p,q:p_1,q_1;\cdots;p_L,q_L}[.]$ is the multivariate Fox’s $H$-function \[11, eq. (A.1)\]. An efficient MATLAB code that is readily implemented by \[12\] to compute the multivariate Fox’s $H$-function is used in this work. This because this function is not yet available as a built-in in the popular software packages such as MATLAB and MATHEMATICA. \[Proposition\_1\] The CDF of the maximum i.n.i.d. variates can be computed by \[1\] \[eqn\_4\] $$\begin{aligned} F_{\gamma}(\gamma)=\prod_{i=1}^L F_{\gamma_i}(\gamma)\end{aligned}$$ Substituting (2) into (4), yielding \[eqn\_5\] $$\begin{aligned} F_{\gamma}(\gamma)&=\prod_{i=1}^L \frac{\Xi_i^{m_i} \gamma^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})} \nonumber\\ &\times H^{1,2}_{2,2} \bigg[ \Xi_i \gamma \bigg\vert \begin{matrix} (1-m_i-m_{s_i},1), (1-m_i,1)\\ (0,1),(-m_i,1)\\ \end{matrix} \bigg]\end{aligned}$$ After using the definition of the single variable Fox’s $H$-function \[11, eq. (1.2)\], (5) can be expressed in multiple Barnes-type closed contours as \[eqn\_6\] $$\begin{aligned} &F_{\gamma}(\gamma)=\Bigg(\prod_{i=1}^L \frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\Bigg) \frac{1}{(2 \pi j)^L} \int_{\mathbb{U}_1} \cdots \int_{\mathbb{U}_L} \nonumber\\ & \bigg\{\prod_{i=1}^L \frac{\Gamma(u_i) \Gamma(m_i+m_{s_i}-u_i) \Gamma(m_i-u_i)}{\Gamma(1+m_i-u_i)}\bigg\} \Xi^{-u_1}_1 \cdots \Xi^{-u_L}_L \nonumber\\ & \gamma^{\sum_{i=1}^{L} m_i-u_i} du_1 \cdots du_L\end{aligned}$$ where $j=\sqrt{-1}$ and $\mathbb{U}_i$ is the $i$th suitable contours in the $u$-plane from $\sigma_i-j\infty$ to $\sigma_i+j\infty$ with $\sigma_i$ is a constant value. Differentiating (6) with respect to $\gamma$ to obtain $f_\gamma(\gamma)$, i.e. $f_\gamma(\gamma)=dF_\gamma(\gamma)/d\gamma$ and then employing the identity $\Gamma(1+x)=x\Gamma(x)$ \[10, eq. (8.331.1)\]. Thus, this yields \[eqn\_7\] $$\begin{aligned} &f_{\gamma}(\gamma)=\Bigg(\prod_{i=1}^L \frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\Bigg) \frac{1}{(2 \pi j)^L} \int_{\mathbb{U}_1} \cdots \int_{\mathbb{U}_L} \nonumber\\ & \bigg\{\prod_{i=1}^L \frac{\Gamma(u_i) \Gamma(m_i+m_{s_i}-u_i) \Gamma(m_i-u_i)}{\Gamma(1+m_i-u_i)}\bigg\} \Xi^{-u_1}_1 \cdots \Xi^{-u_L}_L\nonumber\\ & \frac{\Gamma(1+\sum_{i=1}^{L} m_i-u_i)}{\Gamma(\sum_{i=1}^{L} m_i-u_i)} \gamma^{\sum_{i=1}^{L} m_i-u_i-1} du_1 \cdots du_L\end{aligned}$$ With the help of \[11, eq. (A.1)\], (7) can be written in exact closed-form expression as in (3), which completes the proof. \[Proposition\_2\] The MGF of $\gamma=\text{max}\{\gamma_1,\cdots,\gamma_L\}$, $\mathcal{M}_\gamma(s)$, is given as \[eq\_8\] $$\begin{aligned} &\mathcal{M}_\gamma(s)=\bigg(\prod_{i=1}^{L}\frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\bigg)\frac{1}{s^\Omega}\nonumber\\ &\times H^{0,1:[1,2]_{i=1:L}}_{1,0:[2,2]_{i=1:L}} \bigg[ \frac{\Xi_1}{s},\cdots,\frac{\Xi_L}{s} \gamma\bigg\vert \begin{matrix} (-\Omega;\{1\}_{i=1:L})\\ -\\ \end{matrix}\bigg\vert \nonumber\\ &\begin{matrix} [1-m_i-m_{s_i}, 1-m_i]_{i=1:L}\\ [0,-m_i]_{i=1:L}\\ \end{matrix} \bigg]\end{aligned}$$ The MGF can be calculated by plugging (6) into $\mathcal{M}_\gamma(s)=s \mathcal{L} \{F_\gamma(\gamma);-s\}$ where $\mathcal{L}\{.\}$ denotes the Laplace transform. Hence, we have \[eqn\_9\] $$\begin{aligned} &\mathcal{M}_\gamma(s)=\Bigg(\prod_{i=1}^L \frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\Bigg) \frac{1}{(2 \pi j)^L} \int_{\mathbb{U}_1} \cdots \int_{\mathbb{U}_L} \nonumber\\ & \bigg\{\prod_{i=1}^L \frac{\Gamma(u_i) \Gamma(m_i+m_{s_i}-u_i) \Gamma(m_i-u_i)}{\Gamma(1+m_i-u_i)}\bigg\} \Xi^{-u_1}_1 \cdots \Xi^{-u_L}_L \nonumber\\ & s\mathcal{L}\{\gamma^{\sum_{i=1}^{L} m_i-u_i};-s\} du_1 \cdots du_L\end{aligned}$$ The Laplace transform in (8) is recoded in \[10, eq. (3.381.4)\]; thus, $\mathcal{M}_\gamma(s)$ can be derived as \[eqn\_10\] $$\begin{aligned} &\mathcal{M}_\gamma(s)=\Bigg(\prod_{i=1}^L \frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\Bigg) \frac{1}{(2 \pi j)^L} \int_{\mathbb{U}_1} \cdots \int_{\mathbb{U}_L} \nonumber\\ & \bigg\{\prod_{i=1}^L \frac{\Gamma(u_i) \Gamma(m_i+m_{s_i}-u_i) \Gamma(m_i-u_i)}{\Gamma(1+m_i-u_i)}\bigg\} \Xi^{-u_1}_1 \cdots \Xi^{-u_L}_L \nonumber\\ & \frac{\Gamma(1+\sum_{i=1}^{L} m_i-u_i)}{s^{\sum_{i=1}^{L} m_i-u_i}} du_1 \cdots du_L\end{aligned}$$ Again, with the aid of \[11, eq. (A.1)\], (8) is deduced and the proof is accomplished. Performance of SC over Non-Identically Distributed Fisher-Snedecor $\mathcal{F}$ Fading Channels ================================================================================================ Due to the space limitations, the following unified framework can be utilised \[eqn\_11\] $$\begin{aligned} &\mathcal{P}=\int_0^\infty \mathcal{P}(\gamma) f_\gamma(\gamma) d\gamma\end{aligned}$$ where $\mathcal{P}$ and $\mathcal{P}(\gamma)$ are the average and the conditional of the performance metric, respectively. Substituting (7) into (11), we have \[eqn\_12\] $$\begin{aligned} &\mathcal{P}=\Bigg(\prod_{i=1}^L \frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\Bigg) \nonumber\\ & \frac{1}{(2 \pi j)^L} \int_{\mathbb{U}_1} \cdots \int_{\mathbb{U}_L} \frac{\Gamma(1+\Omega-\sum_{i=1}^{L}u_i)}{\Gamma(\Omega-\sum_{i=1}^{L}u_i)} \nonumber\\ & \bigg\{\prod_{i=1}^L \frac{\Gamma(u_i) \Gamma(m_i+m_{s_i}-u_i) \Gamma(m_i-u_i)}{\Gamma(1+m_i-u_i)}\bigg\} \Xi^{-u_1}_1 \cdots \Xi^{-u_L}_L \nonumber\\ & \underbrace{\int_0^\infty \gamma^{\Omega-\sum_{i=1}^{L}u_i-1} \mathcal{P}(\gamma) d\gamma}_{\mathcal{I}} du_1 \cdots du_L\end{aligned}$$ Average Bit Error Probability ----------------------------- The ABEP can be evaluated by \[1\] \[eqn\_13\] $$P_e=\int_0^\infty Q(\sqrt{2 \rho \gamma}) f_\gamma(\gamma) d\gamma$$ where $Q(.)$ is the Gaussian $Q$-function presented in \[1, eq. (4.1)\] and $\rho$ represents the modulation parameter. For example, $\rho = 1$ for binary phase shift keying (BPSK), while $\rho = 0.5$ for binary frequency shift keying (BFSK). Inserting (7) in (13) and invoking the identity \[13, eq. (13)\], $\mathcal{I}$ of (12) is obtained as \[eqn\_14\] $$\begin{aligned} \mathcal{I}&=\frac{1}{2 \sqrt{\pi}}\int_0^\infty \gamma^{\Omega-\sum_{i=1}^{L} u_i-1} H^{2,0}_{1,2} \bigg[ \rho \gamma \bigg\vert \begin{matrix} (1,1)\\ (0,1),(0.5,1)\\ \end{matrix} \bigg] d\gamma \nonumber\\ &\stackrel{(a)}{=} \rho^{-\Omega+\sum_{i=1}^{L}u_i} \frac{\Gamma(\Omega-\sum_{i=1}^{L}u_i) \Gamma(0.5+\Omega-\sum_{i=1}^{L}u_i)}{\Gamma(1+\Omega-\sum_{i=1}^{L}u_i)}\end{aligned}$$ where $(a)$ follows \[11, eq. (2.8)\]. Next, plugging (14) in (12), performing some mathematical straightforward simplifications and using \[11, eq. (A.1)\], $P_e$ is obtained as \[eq\_15\] $$\begin{aligned} &P_e=\bigg(\prod_{i=1}^{L}\frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\bigg)\frac{1}{2\sqrt{\pi} \rho^\Omega}\nonumber\\ &\times H^{0,1:[1,2]_{i=1:L}}_{1,0:[2,2]_{i=1:L}} \bigg[ \frac{\Xi_1}{\rho},\cdots,\frac{\Xi_L}{\rho}\bigg\vert \begin{matrix} (0.5-\Omega;\{1\}_{i=1:L})\\ -\\ \end{matrix}\bigg\vert \nonumber\\ &\begin{matrix} [(1-m_i-m_{s_i},1), (1-m_i,1)]_{i=1:L}\\ [(0,1),(-m_i,1)]_{i=1:L}\\ \end{matrix} \bigg]\end{aligned}$$ Average Channel Capacity ------------------------ According to Shannon theory, the ACC, $\bar{C}$, can be computed by \[eqn\_16\] $$\bar{C}=\frac{B}{\text{ln}2}\int_0^\infty \text{ln}(1+\gamma) f_\gamma(\gamma) d\gamma$$ where $B$ is the bandwidth of the channel. Inserting (7) in (16), $\mathcal{I}$ of (12) for $\bar{C}$ becomes \[eqn\_17\] $$\begin{aligned} \mathcal{I}&=\frac{B}{\text{ln}2} \int_0^\infty \gamma^{\Omega-\sum_{i=1}^L u_i-1} \text{ln}(1+\gamma)d\gamma \nonumber\\ &\stackrel{(b)}{=} \frac{B}{\text{ln}2} \frac{ \Gamma(1-\Omega+\sum_{i=1}^L u_i) [\Gamma(\Omega-\sum_{i=1}^L u_i)]^2}{\Gamma(1+\Omega-\sum_{i=1}^L u_i)}\end{aligned}$$ where $(b)$ follows after employing \[10, eq. (4.293.10)\] and making use of the properties \[10, eq. (8.334.3)\] and \[10, eq. (8.331.1)\]. Now, substituting $(b)$ of (17) into (12) and doing some algebraic manipulations, $\bar{C}$ is yielded as follows \[eq\_18\] $$\begin{aligned} &\bar{C}=\bigg(\prod_{i=1}^{L}\frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\bigg)\frac{B}{\text{ln}2} H^{1,1:[1,2]_{i=1:L}}_{1,1:[2,2]_{i=1:L}} \bigg[\Xi_1,\cdots,\Xi_L\bigg\vert \nonumber\\ & \begin{matrix} (1-\Omega;\{1\}_{i=1:L})\\ (1-\Omega;\{1\}_{i=1:L})\\ \end{matrix}\bigg\vert \begin{matrix} [(1-m_i-m_{s_i},1), (1-m_i,1)]_{i=1:L}\\ [(0,1),(-m_i,1)]_{i=1:L}\\ \end{matrix} \bigg]\end{aligned}$$ It can be noted that (18) reduces to \[15, eq. (18)\] for $L=1$. ED with SC over Fisher-Snedecor $\mathcal{F}$ fading conditions --------------------------------------------------------------- ### Average Detection Probability The ADP can be evaluated by \[9, eq. (9)/eq. (4)\] \[eqn\_19\] $$\begin{aligned} \bar{P}_d=\int_0^\infty Q_u(\sqrt{2 \gamma}, \sqrt{\lambda}) f_\gamma(\gamma) d\gamma\end{aligned}$$ where $\lambda$ is the threshold value, $u=TW$ stands for the time-bandwidth product and $Q_u(., .)$ is the generalized Marcum $Q$-function. \[eq\_22\] $$\begin{aligned} \bar{P}_d=&1-\pi\bigg(\frac{\lambda}{2}\bigg)^u \bigg(\prod_{i=1}^{L}\frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\bigg) H^{0,3:1,0;1,0;[1,2]_{i=1:L}}_{3,2:0,1;1,3;[2,2]_{i=1:L}} \bigg[ \frac{\lambda}{2},\frac{\lambda}{2},\Xi_1,\cdots,\Xi_L \bigg\vert\begin{matrix} (1-u;1,1,\{0\}_{i=1:L}), (1-\Omega;0,1,\{1\}_{i=1:L})\\ (-u;1,1,\{0\}_{i=1:L})\\ \end{matrix}\bigg\vert \nonumber\\ &\begin{matrix} (-\Omega;0,0,\{1\}_{i=1:L})\\ (1-\Omega;0,0,\{1\}_{i=1:L})\\ \end{matrix}\bigg\vert \begin{matrix} -\\ (0,1)\\ \end{matrix} \bigg\vert \begin{matrix} (0.5,1)\\ (0,1),(1-u,1),(0.5,1)\\ \end{matrix} \bigg\vert \begin{matrix} [(1-m_i-m_{s_i},1), (1-m_i,1)]_{i=1:L}\\ [(0,1),(-m_i,1)]_{i=1:L}\\ \end{matrix} \bigg]\end{aligned}$$ \[eq\_26\] $$\begin{aligned} &\bar{A}=1-\bigg(\prod_{i=1}^{L}\frac{\Xi_i^{m_i}}{\Gamma(m_i) \Gamma(m_{s_i})}\bigg)\sum_{k=0}^{u-1}\sum_{l=0}^{k}{{k+u-1}\choose{k-l}}\frac{1}{2^{k+\Omega+u}l!}H^{0,2:[1,2]_{i=1:L}}_{2,1:[2,2]_{i=1:L}} \bigg[ 2\Xi_1,\cdots,2\Xi_L \bigg\vert \nonumber\\ & \hspace{4 cm} \begin{matrix} (-\Omega,\{1\}_{i=1:L}),(1-l-\Omega,\{1\}_{i=1:L})\\ (1-\Omega,\{1\}_{i=1:L})\\ \end{matrix} \bigg\vert \begin{matrix} [(1-m_i-m_{s_i},1), (1-m_i,1)]_{i=1:L}\\ [(0,1),(-m_1,1)]_{i=1:L}\\ \end{matrix} \bigg]\end{aligned}$$ It can be observed that the generalized Marcum $Q$-function can be expressed as \[eqn\_20\] $$\begin{aligned} &Q_u(\sqrt{2 \gamma}, \sqrt{\lambda})\nonumber\\ & \stackrel{(c_1)}{=} 1-\frac{e^{-\gamma}}{2^{\frac{u+1}{2}} \gamma^{\frac{u-1}{2}}} \int_0^\lambda x^{\frac{u-1}{2}} e^{-\frac{x}{2}} I_{u-1}(\sqrt{2\gamma x}) dx \nonumber\\ &\stackrel{(c_2)}{=} 1- \frac{ \pi e^{-\gamma}}{2^u } \int_0^\lambda x^{u-1} H^{1,0}_{0,1} \bigg[ \frac{x}{2} \bigg\vert \begin{matrix} -\\ (0,1)\\ \end{matrix} \bigg] \nonumber\\ & \times H^{1,0}_{1,3} \bigg[ \frac{\gamma x}{2} \bigg\vert \begin{matrix} (0.5,1)\\ (0,1),(1-u,1),(0.5,1)\\ \end{matrix}\bigg] dx \nonumber\\ &\stackrel{(c_3)}{=}1-\pi\bigg(\frac{\lambda}{2}\bigg)^u e^{-\gamma} \frac{1}{(2 \pi j)^2}\int_{\mathbb{R}_1} \int_{\mathbb{R}_2} \frac{\Gamma(u-r_1-r_2)}{\Gamma(1+u-r_1-r_2)}\nonumber\\ &\frac{\Gamma(r_1) \Gamma(r_2)}{\Gamma(0.5+r_2) \Gamma(u-r_2) \Gamma(0.5-r_2)} \bigg(\frac{\lambda}{2}\bigg)^{-r_1} \bigg(\frac{\lambda \gamma}{2}\bigg)^{-r_2} dr_1 dr_2\end{aligned}$$ where $(c_1)$ and $(c_2)$ arise after employing \[1, eq. (4.60)\] and then respectively utilising the properties \[11, eq. (1.39)\] and \[14, eq. (03.02.26.0067.01)\] for the exponential function and $I_{a}(.)$, which represents the modified Bessel function of the first kind and $a$th-order. Using the definition of the univariate Fox’s $H$-function \[11, eq. (1.2)\] and solving the integral of $(c_2)$, then $(c_3)$ follows in terms of the contour integral form where $\mathbb{R}_1$ and $\mathbb{R}_2$ are the suitable closed contours in the complex $r$-plane. Now, plugging (7) and (20) into (19) and using the fact that $\int_0^\infty f_\gamma(\gamma)d\gamma \triangleq1$, $\mathcal{I}$ of (12) is deduced as follows \[eqn\_21\] $$\begin{aligned} &\mathcal{I}=1-\pi\bigg(\frac{\lambda}{2}\bigg)^u e^{-\gamma} \frac{1}{(2 \pi j)^2}\int_{\mathbb{R}_1} \int_{\mathbb{R}_2} \frac{\Gamma(u-r_1-r_2)}{\Gamma(1+u-r_1-r_2)}\nonumber\\ &\frac{\Gamma(r_1) \Gamma(r_2)}{\Gamma(0.5+r_2) \Gamma(u-r_2) \Gamma(0.5-r_2)} \bigg(\frac{\lambda}{2}\bigg)^{-r_1} \bigg(\frac{\lambda \gamma}{2}\bigg)^{-r_2} \nonumber\\ & \int_0^\infty \gamma^{\Omega-\sum_{i=1}^{L}u_i-r_2-1} e^{-\gamma} dr_1 dr_2\end{aligned}$$ Recalling \[10, eq. (3.381.4)\] for the inner integral of (21), substituting the result into (12) and making employ of \[11, eq. (A.1)\], then $\bar{P}_d$ is obtained as shown on the top of this page. In contrast to \[9, eq. (14)\] and \[16, eq. (14)\] that are derived for no diversity scenario in terms of the infinite series, (22) for $L=1$ can be obtained in exact closed-from computationally tractable expression in terms of a single variable Fox’s $H$-function. ### Average AUC The average AUC is a single figure of merit that can be used in the analysis of performance of the ED when the plotting of the ADP versus the probability of false alarm, namely, ROC, doesn’t provide a clear insight into the behaviour of the system. The average AUC, $\bar{A}$, can be calculated by \[9, eq. (36)\] \[eqn\_23\] $$\begin{aligned} &\bar{A}=\int_0^\infty A(\gamma) f_\gamma(\gamma)d\gamma\end{aligned}$$ where $A(\gamma)$ is the AUC at the instantaneous SNR. The $A(\gamma)$ is given as \[9, eq. (35)\] \[eqn\_24\] $$\begin{aligned} &A(\gamma)=1-\sum_{k=0}^{u-1}\sum_{l=0}^{k}{{k+u-1}\choose{k-l}}\frac{1}{2^{k+l+u}l!}\gamma^l e^{-\frac{l}{2}}\end{aligned}$$ where ${{b}\choose{a}}$ denotes the binomial coefficient. Substituting (24) and (7) into (23) and invoking $\int_0^\infty f_\gamma(\gamma)d\gamma \triangleq1$, we have $\mathcal{I}$ of (12) as \[eqn\_25\] $$\begin{aligned} &\mathcal{I}=1-\sum_{k=0}^{u-1}\sum_{l=0}^{k}{{k+u-1}\choose{k-l}}\frac{1}{2^{k+l+u}l!} \nonumber\\ & \times \int_0^\infty \gamma^{l+\Omega-\sum_{i=1}^{L}u_i-r_2-1} e^{-\frac{l}{2}} d\gamma\end{aligned}$$ Utilising \[10, eq. (3.381.4)\] to evaluate the integral of (25) and plugging the result in (12), we have a closed-form expression of $\bar{A}$ as given on the top of this page. {width="3.5" height="2.5"} Analytical and Simulation Results ================================= In this section, to validate our derived PDF and MGF of the maximum of i.n.i.d. Fisher-Snedecor $\mathcal{F}$ variates, the ABEP, the ACC, the ADP, and the average AUC of SC diversity are analysed. The Monte Carlo simulations that are obtained via generating $10^7$ realizations for each RV are compared with the analytical results. In all figures, the multivariate Fox’s $H$-function has been evaluated by the MATLAB code that was implemented by \[12\]. Additionally, the solid lines corresponds to the simulations results whereas the markers represents the numerical results. Three scenarios of the shadowing impact, which are heavy, moderate, and light shadowing are studied by using $m_s = 0.5$, $m_s = 5$ and $m_s = 50$, respectively. {width="3.5" height="2.5"} {width="3.5" height="2.5"} {width="3.5" height="2.5"} Figs. 1, 2, and 4 illustrate the ABEP for BPSK, the normalised ACC, and the complementary AUC ($1-\bar{A}$) with single receiver, dual, and triple SC branches over i.n.i.d. Fisher-Snedecor $\mathcal{F}$ fading channels versus the average SNR per branch, $\bar{\gamma}$, respectively, for different scenarios of the fading parameters. In the same context, Fig. 3 explains the complementary ROC, which plots the average probability of missed-detection ($1-\bar{P}_d$) versus the probability of false alarm $P_f(\lambda)=\Gamma(u,\lambda/2)/\Gamma(u)$ for $u=3$ and $\bar{\gamma} = 15$ dB[^4]. As anticipated, the performance of the communication systems becomes better when the SC diversity is employed and monotonically improves with the increasing in the number of diversity branches. The reason has been widely presented in the literature, which is the received average SNR of SC scheme is higher than the no-diversity and its increases when $L=3$ is used rather than $L=2$. For comparison purpose, the scenario $m=[3.5, 4.5, 5.5]$ and $m_s =50$ that was studied in \[7, Fig. 3\], has been utilised here. As expected, the MRC diversity provides less ABEP than the SC branches but with high implementation complexity. In all provided figures, the perfect matching between the numerical results and their Monte Carlo simulation counterparts can be observed, which confirms the validation of our derived expressions. Conclusions =========== In this paper, the PDF and the MGF of the maximum of not necessarily identically distributed Fisher-Snedecor $\mathcal{F}$ RVs were derived in terms of the multivariate Fox’s $H$-function that has been widely used and implemented in the literature. These statistics were then employed to analyse the performance of SC diversity with non-identically distributed branches. To be specific, the ABEP, the ACC, the ADP, and the AUC of ED technique were obtained in exact mathematically tractable closed-form expressions. Comparisons of our results with previous works that were achieved by using a single receiver and MRC scheme as well as the numerical and simulation results for different scenarios have been carried out via using the same simulation parameters. [1]{} M. K. Simon and M.-S. Alouini, *Digital Communications over Fading Channels*. New York: Wiley, 2005. P. S. Bithas, P. T. Mathiopoulos, and S. A. Kotsopoulos, $``$Diversity reception over generalized-K (KG) fading channels,$"$ *IEEE Trans. Wireless Commun.*, vol. 6, no. 12, pp. 4238-4243, Dec. 2007. J. Paris, $``$Statistical characterization of $\kappa-\mu$ shadowed fading,$"$ *IEEE Trans. Veh. Technol.*, vol. 63, no. 2, pp. 518-526, Feb. 2014. H. Al-Hmood, and H. S. Al-Raweshidy, $``$On the sum and the maximum of non-identically distributed composite $\eta-\mu$/gamma variates using a mixture gamma distribution with applications to diversity receivers,$"$ *IEEE Trans. Veh. Technol.*, vol. 65, no. 12, pp. 10048-10052, Dec. 2016. H. Al-Hmood, *Performance Analysis of Energy Detector over Generalised Wireless Channels in Cognitive Radio*. PhD Thesis, Brunel University London, 2015. S. K. Yoo, S. L. Cotton, P. C. Sofotasios, M. Matthaiou, M. Valkama, and G. K. Karagiannidis, $``$The Fisher-Snedecor $\mathcal{F}$ distribution: A simple and accurate composite fading model,$"$ *IEEE Commun. Lett.*, vol. 21, no. 7, pp. 1661-1664, March 2017. O. S. Badarneh, D. B. Da Costa, P. C. Sofotasios, S. Muhaidat, and S. L. Cotton, $``$On the sum of Fisher-Snedecor $\mathcal{F}$ variates and its application to maximal-ratio combining,$"$ *IEEE Commun. Lett.*, vol. 7, no. 6, pp. 966-969, Dec. 2018. O. S. Badarneh, S. Muhaidat, P. C. Sofotasios, S. L. Cotton, K. Rabie and D. B. da Costa, $``$The N$*$Fisher-Snedecor $\mathcal{F}$ cascaded fading model,$"$ in *Proc. IEEE WiMob*, Oct. 2018, pp. 1-7. S. K. Yoo *et al.*, $``$Entropy and energy detection-based spectrum sensing over $\mathcal{F}$ composite fading channels,$"$ *IEEE Trans. Commun.*, 2019, pp. 1-1. I. S. Gradshteyn, and I. M. Ryzhik, *Table of Integrals, Series and Products*, 7th edition. Academic Press Inc., 2007. A. M. Mathai, R. K. Saxena, and H. J. Haubold, *The H-Function: Theory and Applications.* Springer, 2009. H. Chergui, M. Benjillali, and M.-S. Alouini, (2018) $``$Rician $K$-factor-based analysis of XLOS service probability in 5G outdoor ultra-dense networks,$"$ \[Online\]. Available: https://arxiv.org/abs/1804.08101 O. S. Badarneh and F. S. Almehmadi, $``$Performance analysis of $L$-branch maximal ratio combining over generalised $\eta-\mu$ fading channels with imperfect channel estimation,$"$ *IET Commun.*, vol. 10, no. 10, pp. 1175-1182, July 2016. $``$The Wolfram Functions Website.$"$ (Last accessed April 2019). S. K. Yoo *et al.*, $``$A comprehensive analysis of the achievable channel capacity in $\mathcal{F}$ composite fading channels,$"$ *IEEE Access*, vol. 7, pp. 34078-34094, March 2019. H. Al-Hmood, $``$Performance of cognitive radio systems over shadowed with integer and Fisher-Snedecor $\mathcal{F}$ fading channels,$"$ in *Proc. IEEE IICETA,* May 2018, pp. 130-135. [^1]: Manuscript received May 6, 2019; xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx xxxxx. [^2]: Hussien Al-Hmood is with the Department of Electronic and Computer Engineering, College of Engineering, Design and Physical Sciences, Brunel University London, UB8 3PH, U.K., e-mails: hussien.al-hmood@brunel.ac.uk, h.a.al-hmood@ieee.org. [^3]: H. S. Al-Raweshidy is with the Department of Electronic and Computer Engineering, College of Engineering, Design and Physical Sciences, Brunel University London, UB8 3PH, U.K., e-mail: hamed.al-raweshidy@brunel.ac.uk. [^4]: Here, $\Gamma(.,.)$ represents the upper incomplete gamma function \[10, eq. (8.350.2)\].

--- abstract: 'We establish the $L_p$-solvability for time fractional parabolic equations when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend a recent result in [@MR3581300] to a large extent.' address: - 'Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA' - 'Department of Mathematics, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul, 02841, Republic of Korea' author: - Hongjie Dong - Doyoon Kim title: '$L_p$-estimates for time fractional parabolic equations with coefficients measurable in time' --- [^1] [^2] Introduction ============ In this paper, we consider time fractional parabolic equations with a non-local type time derivative term of the form $$\label{eq0525_01} - \partial_t^\alpha u + a^{ij}(t,x) D_{ij} u + b^i(t,x) D_i u + c(t,x) u = f(t,x)$$ in $(0,T) \times \bR^d$, where $\partial_t^\alpha u$ is the Caputo fractional derivative of order $\alpha \in (0,1)$: $$\partial_t^\alpha u(t,x) = \frac{1}{\Gamma(1-\alpha)} \frac{d}{dt} \int_0^t (t-s)^{-\alpha} \left[ u(s,x) - u(0,x) \right] \, ds.$$ See Sections \[sec2\] and \[Sec3\] for a precise definition and properties of $\partial_t^\alpha u$. Our main result is that, for a given $f \in L_p\left((0,T) \times \bR^d \right)$, there exists a unique solution $u$ to the equation in $(0,T) \times \bR^d$ with the estimate $$\||\partial_t^\alpha u|+|u|+| Du|+|D^2u|\|_{L_p\left((0,T) \times \bR^d \right)} \le N \|f\|_{L_p\left((0,T) \times \bR^d \right)}.$$ The assumptions on the coefficients $a^{ij}$, $b^i$, and $c$ are as follows. The leading coefficients $a^{ij}=a^{ij}(t,x)$ satisfy the uniform ellipticity condition and have no regularity in the time variable. Dealing with such coefficients in the setting of $L_p$ spaces is the main focus of this paper. As functions of $x$, locally the coefficients $a^{ij}$ have small (bounded) mean oscillations (small BMO). See Assumption \[assump2.2\]. The lower-order coefficients $b^i$ and $c$ are assumed to be only bounded and measurable. If the fractional (or non-local) time derivative $\partial_t^\alpha u$ is replaced with the local time derivative $u_t$, the equation becomes the usual second-order non-divergence form parabolic equation $$\label{eq0525_02} -u_t + a^{ij} D_{ij} u + b^i D_i u + c u = f.$$ As is well known, there is a great amount of literature on the regularity and solvability for equations as in in various function spaces. Among them, we only refer the reader to the papers [@MR2304157; @MR2352490; @MR2771670], which contain corresponding results of this paper to parabolic equations as in . More precisely, in these papers, the unique solvability results are proved in Sobolev spaces for elliptic and parabolic equations/systems. In particular, for the parabolic case, the leading coefficients are assumed to satisfy the same conditions as mentioned above. This class of coefficients was first introduced by Krylov in [@MR2304157] for parabolic equations in Sobolev spaces. In [@MR2352490], the results in [@MR2304157] were generalized to the mixed Sobolev norm setting, and in [@MR2771670] to higher-order elliptic and parabolic systems. Thus, one can say that the unique solvability of solutions in Sobolev spaces to parabolic equations as in is well established when coefficients are merely measurable in the time variable. On the other hand, it is well known that the $L_p$-solvability of elliptic and parabolic equations requires the leading coefficients to have some regularity conditions in the spatial variables. See, for instance, the paper [@MR3488249], where the author shows the impossibility of finding solutions in $L_p$ spaces to one spatial dimensional parabolic equations if $p \notin (3/2,3)$ and the leading coefficient are merely measurable in $(t,x)$. In view of mathematical interests and applications, it is a natural and interesting question to explore whether the corresponding $L_p$-solvability results hold for equations as in for the same class of coefficients as in [@MR2304157; @MR2352490; @MR2771670]. In a recent paper [@MR3581300] the authors proved the unique solvability of solutions in mixed $L_{p,q}$ spaces to the time fractional parabolic equation under the stronger assumption that the leading coefficients are piecewise continuous in time and uniformly continuous in the spatial variables. Hence, the results in this paper can be regarded as a generalization of the results in [@MR3581300] to a large extent, so that one can have the same class of coefficients as in [@MR2304157; @MR2352490; @MR2771670] for the time non-local equation in $L_p$ spaces. We note that in [@MR3581300] the authors discussed the case $\alpha \in (0,2)$, whereas in this paper we only discuss the parabolic regime $\alpha \in (0,1)$. It is also worth noting that, for parabolic equations as in , it is possible to consider more general classes of coefficients than those in [@MR2304157; @MR2352490; @MR2771670]. Regarding this, see [@DK15], where the classes of coefficients under consideration include those $a^{ij}(t,x)$ measurable both in one spatial direction and in time except, for instance, $a^{11}(t,x)$, which is measurable either in time or in the spatial direction. Besides [@MR3581300], there are a number of papers about parabolic equations with a non-local type time derivative term. For divergence type time fractional parabolic equations in the Hilbert space setting, see [@MR2538276], where the time fractional derivative is a generalized version of the Caputo fractional derivative. One can find De Giorgi-Nash-Moser type Hölder estimates for time fractional parabolic equations in [@MR3038123], and for parabolic equations with fractional operators in both $t$ and $x$ in [@MR3488533]. For other related papers and further information about time fractional parabolic equations and their applications, we refer to [@MR3581300] and the references therein. As a standard scheme in $L_p$-theory, to establish the main results of this paper, we prove a priori estimates for solutions to . In [@MR3581300] a representation formula for a solution to the time fractional heat operator $-\partial_t^\alpha u + \Delta u$ is used, from which the $L_p$-estimate is derived for the operator. Then for uniformly continuous coefficients, a perturbation argument takes places to derive the main results of the paper. Our proof is completely different. Since $a^{ij}$ are measurable in time, it is impossible to treat the equation via a perturbation argument from the time fractional heat equation. Thus, instead of considering a representation formula for equations with coefficients measurable in time, which does not seem to be available, we start with the $L_2$-estimate and solvability, which can be obtained from integration by parts. We then exploit a level set argument originally due to Caffarelli and Peral [@MR1486629] as well as a “crawling of ink spots” lemma, which was originally due to Safonov and Krylov [@MR579490; @MR563790]. The main difficulty arises in the key step where one needs to estimate local $L_\infty$ estimates of the Hessian of solutions to locally homogeneous equations. Starting from the $L_2$-estimate and applying the Sobolev type embedding results proved in Appendix, we are only able to show that such Hessian are in $L_{p_1}$ for some $p_1>2$, instead of $L_\infty$. Nevertheless, this allows us to obtain the $L_p$ estimate and solvability for any $p\in [2,p_1)$ and $a^{ij}=a^{ij}(t)$ by using a modified level set type argument. Then we repeat this procedure and iteratively increase the exponent $p$ for any $p\in [2,\infty)$. In the case when $p\in (1,2)$, we apply a duality argument. For equations with the leading coefficients being measurable in $t$ and locally having small mean oscillations in $x$, we apply a perturbation argument (see, for instance, [@MR2304157]). This is done by incorporating the small mean oscillations of the coefficients into local mean oscillation estimates of solutions having compact support in the spatial variables. Then, the standard partition of unity argument completes the proof. In forthcoming work, we will generalize our results for time fraction parabolic equations with more general coefficients considered, for example, in [@DK15]. We will also consider solutions in Sobolev spaces with mixed norms as in [@MR3581300] as well as equations in domains. The remainder of the paper is organized as follows. In the next section, we introduce some notation and state the main results of the paper. In Section \[Sec3\], we define function spaces for fractional time derivatives and show some of their properties. In Section \[sec4\], we prove the $L_2$ estimate and solvability for equations with coefficients depending only on $t$, and then derive certain local estimates, which will be used later in the iteration argument. We give the estimates of level sets of Hessian in Section \[sec5\] and complete the proofs of the main theorems in Section \[sec6\]. In Appendix, we establish several Sobolev type embedding theorems involving time fractional derivatives and prove a “crawling of ink spots” lemma adapted to our setting. Notation and main results {#sec2} ========================= We first introduce some notation used through the paper. For $\alpha \in (0,1)$, denote $$I^\alpha \varphi(t) = I_0^\alpha \varphi(t) = \frac{1}{\Gamma(\alpha)} \int_0^t (t-s)^{\alpha - 1} \varphi(s) \, ds$$ for $\varphi \in L_1(\bR^+)$, where $$\Gamma(\alpha) = \int_0^\infty t^{\alpha - 1} e^{-t} \, dt.$$ In [@MR1544927] $I^\alpha \varphi$ is called $\alpha$-th integral of $\varphi$ with origin $0$. For $0 < \alpha < 1$ and sufficiently smooth function $\varphi(t)$, we set $$D_t^\alpha \varphi(t) = \frac{d}{dt} I^{1-\alpha} \varphi(t) = \frac{1}{\Gamma(1-\alpha)} \frac{d}{dt} \int_0^t (t-s)^{-\alpha} \varphi(s) \, ds,$$ and $$\begin{aligned} \partial_t^\alpha \varphi(t) &= \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} \varphi'(s) \, ds\\ &= \frac{1}{\Gamma(1-\alpha)} \frac{d}{dt} \int_0^t (t-s)^{-\alpha} \left[ \varphi(s) - \varphi(0) \right] \, ds.\end{aligned}$$ Note that if $\varphi(0) = 0$, then $$\partial_t(I^{1-\alpha} \varphi) = \partial_t^\alpha \varphi.$$ Let $\cD$ be a subset (not necessarily open) of $\bR^k$, $k \in \{1,2, \ldots\}$. By $\varphi \in C_0^\infty(\cD)$, we mean that $\varphi$ is infinitely differentiable in $\cD$ and is supported in the intersection of $\cD$ and a bounded open subset in $\bR^d$. In particular, $\varphi$ may not be zero on the boundary of $\cD$, unless $\cD$ is an open subset of $\bR^k$. For $\alpha \in (0,1)$, we denote $$Q_{R_1,R_2}(t,x) = (t-R_1^{2/\alpha}, t) \times B_{R_2}(x) \quad \text{and} \quad Q_R(t,x)=Q_{R,R}(t,x).$$ We often write $B_R$ and $Q_R$ instead of $B_R(0)$ and $Q_R(0,0)$, respectively. In this paper, we assume that there exists $\delta \in (0,1)$ such that $$a^{ij}(t,x)\xi_j \xi_j \geq \delta |\xi|^2,\quad |a^{ij}| \leq \delta^{-1}$$ for any $\xi \in \bR^d$ and $(t,x) \in \bR \times \bR^d$. Our first main result is for equations with coefficients $a^{ij}$ depending only on the time variable without any regularity assumptions. \[thm0412\_1\] Let $\alpha \in (0,1)$, $T \in (0,\infty)$, $a^{ij} = a^{ij}(t)$, and $p \in (1,\infty)$. Suppose that $u \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ satisfies $$\label{eq0411_03} -\partial_t^\alpha u + a^{ij} D_{ij} u = f$$ in $\bR^d_T := (0,T) \times \bR^d$. Then there exists $N = N(d,\delta,\alpha,p)$ such that $$\label{eq0411_04} \|\partial_t^\alpha u\|_{L_p(\bR^d_T)} + \|D^2 u\|_{L_p(\bR^d_T)} \leq N \|f\|_{L_p(\bR^d_T)}.$$ Moreover, for $f \in L_p(\bR^d_T)$, there exists a unique $u \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ satisfying and . We refer the reader to Section \[Sec3\] for the definitions of function spaces including $\bH_{p,0}^{\alpha,2}(\bR^d_T)$. We also consider more general operators with lower-order terms and with coefficients depending on both $t$ and $x$. In this case, we impose the following VMO$_x$ condition on the leading coefficients. \[assump2.2\] There is a constant $R_0\in (0,1]$ such that for each parabolic cylinder $Q_r(t_0,x_0)$ with $r\le R_0$ and $(t_0,x_0)\in \bR^{d+1}$, we have $$\sup_{i,j}\dashint_{Q_r(t_0,x_0)}|a^{ij}-\bar a^{ij}(t)|\,dx\,dt\le \gamma_0,$$ where $\bar a^{ij}(t)$ is the average of $a^{ij}(t,\cdot)$ in $B_r(x_0)$. \[rem2.3\] From the above assumption, we have that for any $x_0\in \bR^d$ and $a, b \in \bR$ such that $b - a > R_0^{2/\alpha}$, there exists $\bar{a}^{ij}(t)$ satisfying the ellipticity condition and $$\dashint_{\!a}^{\,\,\,b} \dashint_{B_{R_0}(x_0)} |a^{ij} - \bar{a}^{ij}(t)| \, dx \, dt \leq 2 \gamma_0.$$ Indeed, find $k \in \{1,2, \ldots\}$ such that $$b - (k+1) R_0^{2/\alpha} \leq a < b - k R_0^{2/\alpha},\quad \text{i.e.,}\,\, \frac{1}{k+1} \leq \frac{R_0^{2/\alpha}}{b-a} < \frac{1}{k},$$ and set $\bar a^{ij}(t)$ to be the average of $a^{ij}(t,\cdot)$ in $B_{R_0}(x_0)$. We then see that $$\begin{aligned} &\dashint_{\!a}^{\,\,\,b} \dashint_{B_{R_0}(x_0)} |a^{ij} - \bar{a}^{ij}(t)| \, dx \, dt = \frac{1}{b-a} \int_a^b \dashint_{B_{R_0}(x_0)} |a^{ij} - \bar{a}^{ij}(t)| \, dx \, dt\\ &\leq \frac{R_0^{2/\alpha}}{b-a} \sum_{j=0}^k \dashint_{\!b-(j+1)R_0^{2/\alpha}}^{\,\,\,b-j R_0^{2/\alpha}} \dashint_{B_{R_0}(x_0)} |a^{ij} - \bar{a}^{ij}(t)| \, dx \, dt\\ &\leq \frac{R_0^{2/\alpha}}{b-a} (k+1) \gamma_0 \leq \frac{k+1}{k} \gamma_0 \leq 2 \gamma_0.\end{aligned}$$ We also assume that the lower-order coefficients $b^i$ and $c$ satisfy $$|b^i|\le \delta^{-1},\quad |c|\le \delta^{-1}.$$ \[main\_thm\] Let $\alpha \in (0,1)$, $T \in (0,\infty)$, and $p \in (1,\infty)$. There exists $\gamma_0\in (0,1)$ depending only on $d$, $\delta$, $\alpha$, and $p$, such that, under Assumption \[assump2.2\] ($\gamma_0$), the following hold. Suppose that $u \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ satisfies $$\label{eq0411_03c} -\partial_t^\alpha u + a^{ij} D_{ij} u+b^i D_i u+cu = f$$ in $\bR^d_T$. Then there exists $N = N(d,\delta,\alpha,p,R_0,T)$ such that $$\label{eq0411_04c} \|u\|_{\bH_p^{\alpha,2}(\bR^d_T)} \leq N \|f\|_{L_{p}(\bR^d_T)}.$$ Moreover, for $f \in L_{p}(\bR^d_T)$, there exists a unique $u \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ satisfying and . Function spaces {#Sec3} =============== Let $\Omega$ be a domain (open and connected, but not necessarily bounded) in $\bR^d$. For $T > 0$, we denote $$\Omega_T = (0,T) \times \Omega \subset \bR \times \bR^d.$$ Thus, if $\Omega = \bR^d$, we write $\bR^d_T = (0,T) \times \bR^d$. For $S>-\infty$ and $\alpha\in (0,1)$, let $I_S^{1-\alpha} u$ be the $(1-\alpha)$-th integral of $u$ with origin $S$: $$I_S^{1-\alpha} u = \frac{1}{\Gamma(1-\alpha)}\int_S^t (t-s)^{-\alpha} u(s, x) \, ds.$$ Throughout the paper, $I_0^{1-\alpha}$ is denoted by $I^{1-\alpha}$. For $1 \le p \le \infty$, $\alpha \in (0,1)$, $T > 0$, and $k \in \{1,2,\ldots\}$, we set $$\widetilde{\bH}_p^{\alpha,k}(\Omega_T) = \left\{ u \in L_p(\Omega_T): D_t^\alpha u, \, D^\beta_x u \in L_p(\Omega_T), \, 0 \leq |\beta| \leq k \right\}$$ with the norm $$\|u\|_{\widetilde{\bH}_p^{\alpha,k}(\Omega_T)} = \|D_t^\alpha u\|_{L_p(\Omega_T)} + \sum_{0 \leq |\beta| \leq k}\|D_x^\beta u\|_{L_p(\Omega_T)},$$ where by $D_t^\alpha u$ or $\partial_t(I^{1-\alpha}u) (= \partial_t (I_0^{1-\alpha}u))$ we mean that there exists $g \in L_p(\Omega_T)$ such that $$\label{eq0122_01} \int_0^T\int_\Omega g(t,x) \varphi(t,x) \, dx \, dt = - \int_0^T\int_\Omega I^{1-\alpha}u(t,x) \partial_t \varphi(t,x) \, dx \, dt$$ for all $\varphi \in C_0^\infty(\Omega_T)$. If we have a domain $(S,T) \times \Omega$ in place of $\Omega_T$, where $-\infty < S < T < \infty$, we write $\widetilde{\bH}_p^{\alpha,k}\left((S,T) \times \Omega\right)$. In this case $$D_t^\alpha u(t,x) = \partial_t I_S^{1-\alpha} u(t,x).$$ Now we set $$\bH_p^{\alpha,k}(\Omega_T) = \left\{ u \in \widetilde{\bH}_p^{\alpha,k}(\Omega_T): \text{\eqref{eq0122_01} is satisfied for all}\,\, \varphi \in C_0^\infty\left([0,T) \times \Omega\right)\right\}$$ with the same norm as for $\widetilde{\bH}_p^{\alpha,k}(\Omega_T)$. Similarly, we define $\bH_p^{\alpha,k}((S,T)\times\Omega)$. If holds for all functions $\varphi \in C_0^\infty\left([0,T) \times \Omega \right)$, then one can regard that $I^{1-\alpha}u(t)|_{t=0} = 0$ in the trace sense with respect to the time variable. In Lemma \[lem0123\_1\] below, we show that, if $\alpha \leq 1 - 1/p$, then $\bH_p^{\alpha,k}(\Omega_T)=\widetilde{\bH}_p^{\alpha,k}(\Omega_T)$. \[lem0123\_1\] Let $p \in [1,\infty]$, $\alpha \in (0,1)$, $k \in \{1,2,\ldots\}$ and $$\alpha \le 1 - 1/p.$$ Then, for $u \in \widetilde{\bH}_p^{\alpha,k}(\Omega_T)$, the equality holds for all $\varphi \in C_0^\infty\left([0,T) \times \Omega\right)$. Let $\eta_k(t)$ be an infinitely differentiable function such that $0 \leq \eta_k(t) \leq 1$, $\eta_k(t) = 0$ for $t \leq 0$, $\eta_k(t) = 1$ for $t \geq 1/k$, and $|\partial_t \eta_k(t)| \leq 2k$. Then $$\begin{aligned} &\int_0^T I^{1-\alpha} u(t,x) \partial_t (\varphi(t) \eta_k(t)) \, dt\\ &= \int_0^T I^{1-\alpha}u(t,x) \partial_t \varphi(t) \eta_k(t) \, dt + \int_0^T I^{1-\alpha} u(t,x) \varphi(t) \partial_t \eta_k(t) \, dt.\end{aligned}$$ To prove the desired equality, we only need to show that $$\int_0^T \int_\Omega I^{1-\alpha}u(t,x) \varphi(t) \partial_t \eta_k(t)\, dx \, dt \to 0$$ as $k \to \infty$. Note that $$\int_0^T I^{1-\alpha}u(t,x) \varphi(t) \partial_t \eta_k(t) \, dt = \int_0^{1/k} I^{1-\alpha}u(t,x) \varphi(t) \partial_t \eta_k(t) \, dt =:J_k(x).$$ Then, by Lemma \[lem1018\_01\] with $1-\alpha$ in place of $\alpha$, for any $q \in [1,\infty]$ satisfying $$1- \alpha - 1/p > -1/q,$$ we have $$\begin{aligned} |J_k(x)| &\leq N k \int_0^{1/k} I^{1-\alpha} |u(t,x)| \, dt \leq N k^{1/q} \left( \int_{0}^{1/k} \left|I^{1-\alpha}|u(t,x)|\right|^q \, dt \right)^{1/q}\\ &\leq N k^{\alpha - 1 + 1/p}\|u(\cdot,x)\|_{L_p(0,1/k)} \to 0\end{aligned}$$ as $k \to \infty$, provided that $\alpha \le 1 - 1/p$. The lemma is proved. We now prove that every function in $\bH_p^{\alpha,k} (\Omega_T)$ can be approximated by infinitely differentiable functions up to the boundary with respect to the time variable. \[prop0120\_1\] Let $p \in [1,\infty)$, $\alpha \in (0,1)$, and $k \in \{1,2,\ldots\}$. Then functions in $C^\infty\left([0,T] \times \Omega\right)$ vanishing for large $|x|$ are dense in $\bH_p^{\alpha,k}(\Omega_T)$. We prove only the case when $\Omega = \bR^d$. More precisely, we show that $C^\infty_0\left([0,T] \times \bR^d\right)$ is dense in $\bH_p^{\alpha,k}(\bR^d_T)$. The proof of the case when $\Omega = \bR^d_+$ is similar. For a general $\Omega$, the claim is proved using a partition of unity with respect to the spatial variables. See, for instance, [@MR0164252]. Let $u \in \bH_p^{\alpha,k}(\bR^d_T)$. Let $\eta(t,x)$ be an infinitely differentiable function defined in $\bR^{d+1}$ satisfying $\eta \ge 0$, $$\eta(t,x) = 0 \quad \text{outside} \,\,(0,1)\times B_1,\quad \int_{\bR^{d+1}} \eta \, dx \, dt = 1.$$ Set $$\eta_\varepsilon(t,x) = \frac{1}{\varepsilon^{d+2/\alpha}} \eta(t/\varepsilon^{2/\alpha}, x/\varepsilon)$$ and $$u^{(\varepsilon)}(t,x) = \int_\bR \int_{\bR^d} \eta_{\varepsilon}(t-s,x-y) u(s,y) I_{0 < s < T} \, dy \, ds.$$ Then it follows easily that $u^{(\varepsilon)}(t,x) \in C^\infty(\bR^{d+1})$ and, for $(t,x) \in (0,T) \times \bR^d$ and $0 \leq |\beta| \leq k$, $$\label{eq0120_01} D^\beta_x u^{(\varepsilon)}(t,x) = \int_\bR \int_{\bR^d} \eta_{\varepsilon}(t-s,x-y) D^\beta_x u(s,y) I_{0<s<T} \, dy \, ds.$$ Moreover, for $(t,x) \in (0,T) \times \bR^d$, $$\label{eq0120_02} D_t^\alpha u^{(\varepsilon)}(t,x) = \int_{\bR}\int_{\bR^d} \eta_{\varepsilon}(t-s,x-y) D^\alpha_t u(s,y) I_{0<s<T} \, dy \, ds.$$ To see , we first check that $$\label{eq0124_01} I^{1-\alpha} u^{(\varepsilon)}(t,x) = (I^{1-\alpha} u)^{(\varepsilon)}(t,x).$$ Indeed, $$\begin{aligned} &\Gamma(1-\alpha) I^{1-\alpha} u^{(\varepsilon)}(t,x)\\ &= \int_0^t (t-s)^{-\alpha}\int_0^T \int_{\bR^d} \eta_\varepsilon(s-r,x-y) u(r,y) \, dy \, dr \, ds\\ &= \int_{\bR^d} \int_0^T \int_0^t (t-s)^{-\alpha} \eta_\varepsilon(s-r,x-y) u(r,y) \, ds \, dr \, dy\\ &= \int_{\bR^d} \int_0^t \int_r^t (t-s)^{-\alpha} \eta_\varepsilon(s-r,x-y) u(r,y) \, ds \, dr \, dy,\end{aligned}$$ where we used the fact that $\eta(t,x) = 0$ if $t \leq 0$. Then by the change of variable $\rho = t-s+r$ in the integration with respect to $s$, we have $$\begin{aligned} &\Gamma(1-\alpha) I^{1-\alpha} u^{(\varepsilon)}(t,x) = \int_{\bR^d} \int_0^t \int_r^t (\rho-r)^{-\alpha} \eta_\varepsilon(t-\rho,x-y) u(r,y) \, d\rho \, dr \, dy\\ &= \int_{\bR^d} \int_0^t \eta_\varepsilon(t-\rho,x-y) \int_0^\rho (\rho-r)^{-\alpha} u(r,y) \, dr \, d\rho \, dy\\ &= \int_{\bR^d} \int_0^T \eta_\varepsilon(t-\rho,x-y) \int_0^\rho (\rho-r)^{-\alpha} u(r,y) \, dr \, d\rho \, dy\\ &= \Gamma(1-\alpha) (I^{1-\alpha}u)^{(\varepsilon)}(t,x).\end{aligned}$$ Hence, the inequality is proved. Now observe that $$\begin{aligned} &\int_{\bR}\int_{\bR^d} \eta_{\varepsilon}(t-s,x-y) D^\alpha_t u(s,y) I_{0<s<T} \, dy \, ds\\ &= \int_0^T \int_{\bR^d} \eta_\varepsilon(t-s,x-y) \partial_s I^{1-\alpha}u(s,y) \, dy \, ds\\ &= \int_0^T \int_{\bR^d} (\partial_t \eta_\varepsilon) (t-s,x-y) I^{1-\alpha}u(s,y) \, dy \, ds\\ &= \partial_t \left[\int_0^T \int_{\bR^d} \eta_\varepsilon(t-s,x-y) I^{1-\alpha} u(s,y) \, dy \, ds\right]\\ &= \partial_t (I^{1-\alpha} u)^{(\varepsilon)}(t,x) = \partial_t I^{1-\alpha} u^{(\varepsilon)}(t,x) = D_t^\alpha u^{(\varepsilon)}(t,x),\end{aligned}$$ where in the second equality we used the fact that $u$ satisfies for all $\varphi \in C_0^\infty\left([0,T) \times \bR^d\right)$ and, by the choice of $\eta$, $\eta_\varepsilon(t-T,x-y) = 0$. From the equalities and , we see that $$\|u^{(\varepsilon)} - u\|_{\bH_p^{\alpha,k}(\bR^d_T)} \to 0$$ as $\varepsilon \to 0$. Finally, we take a smooth cutoff function $\zeta\in C_0^\infty(\bR^d)$ such that $\operatorname{supp} \zeta \subset B_2$ and $\zeta=1$ in $B_1$, and denote $\zeta_\varepsilon(x)=\zeta(x/\varepsilon)$. Then by the uniform bound of $ \|u^{(\varepsilon)}\|_{\bH_p^{\alpha,k}(\bR^d_T)}$, it is easily seen that $$\|u^{(\varepsilon)} - u^{(\varepsilon)}\zeta_\varepsilon\|_{\bH_p^{\alpha,k}(\bR^d_T)} \to 0$$ as $\varepsilon \to 0$. The lemma is proved. \[rem0606\_1\] If the boundary of $\Omega$ is sufficiently smooth, for instance $\Omega$ is a Lipschitz domain, then $C^\infty\big([0,T] \times \overline{\Omega} \big)$ is dense in $\bH_p^{\alpha,k}(\Omega_T)$. Lemma \[lem0123\_1\] shows that $\bH_p^{\alpha,k}(\Omega_T) = \widetilde{\bH}_p^{\alpha,k}(\Omega_T)$ whenever $\alpha \leq 1 - 1/p$, $p \in [1,\infty]$. Hence, by Proposition \[prop0120\_1\], it follows that functions in $C^\infty\left([0,T] \times \Omega\right)$ vanishing for large $|x|$ are dense in $\widetilde{\bH}_p^{\alpha,k}(\Omega_T)$, provided that $\alpha \leq 1 - 1/p$, $p \in [1,\infty)$, $\alpha \in (0,1)$, and $k \in \{1,2,\dots\}$. However, in the case $\alpha > 1 - 1/p$, we have $$\bH_p^{\alpha,k}(\Omega_T) \subsetneq \widetilde{\bH}_p^{\alpha,k}(\Omega_T).$$ To see this, let $$u(t)=t^{\alpha-1},$$ where $\alpha \in (1-1/p,1)$ and $p \in [1,\infty)$. Then $u \in L_p(0,T)$ and $$I^{1-\alpha} u(t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} s^{\alpha-1} \, ds = \Gamma(\alpha),$$ which is a nonzero constant, so that $$\partial_t I^{1-\alpha} u=0.$$ Thus, $$u, \, D_t^\alpha u \in L_p(0,T).$$ However, clearly the integration by parts formula does not hold for $\varphi \in C_0^\infty[0,T)$. The above example also shows that, even though we have $$u, \, D_t^\alpha u\in L_p((0,T))$$ for $\alpha > 1-1/p$, it is not likely to gain better integrability or regularity (up to the boundary) of $u$, as apposed to the usual Sobolev embedding results. To deal with solutions with the zero initial condition, we define $\bH_{p,0}^{\alpha,k}((S,T)\times\Omega)$ to be functions in $\bH_p^{\alpha,k}((S,T)\times\Omega)$ each of which is approximated by a sequence $\{u_n(t,x)\} \subset C^\infty\left([S,T]\times \Omega\right)$ such that $u_n$ vanishes for large $|x|$ and $u_n(S,x) = 0$. For $u \in \bH_{p,0}^{\alpha,k}((S,T)\times\Omega)$ and for any approximation sequences $\{u_n\}$ such that $u_n \to u$ in $\bH_p^{\alpha,k} ((S,T)\times\Omega)$ with $u_n \in C^\infty\left([S,T] \times \Omega\right)$ and $u_n(S,x) = 0$, we have $$\partial_t^\alpha u_n = D_t^\alpha u_n.$$ Thus, when, for instance, $S=0$, for $u \in \bH_{p,0}^{\alpha,k}(\Omega_T)$, we define $$\partial_t^\alpha u := D_t^\alpha u = \frac{1}{\Gamma(1-\alpha)} \partial_t \int_0^t (t-s)^{-\alpha} u(s,x) \, ds.$$ \[lem0206\_1\] Let $p \in [1,\infty)$, $\alpha \in (0,1)$, $k \in \{1,2,\ldots\}$, $-\infty < S < t_0 < T < \infty$, and $u \in \bH_{p,0}^{\alpha,k}\left( (t_0,T) \times \Omega \right)$. If $u$ is extended to be zero for $t \leq t_0$, denoted by $\bar{u}$, then $\bar{u} \in \bH_{p,0}^{\alpha,k}\left((S,T) \times \Omega\right)$. Without loss of generality, we assume $t_0 = 0$ so that $$- \infty < S < 0 < T < \infty.$$ For $u \in \bH_{p,0}^{\alpha,k}(\Omega_T)$, let $\{u_n\}$ be an approximating sequence of $u$ such that $u_n \in \bH_{p,0}^{\alpha,k}(\Omega_T) \cap C^\infty\left([0,T] \times \Omega\right)$, $u_n$ vanishes for large $|x|$, and $u_n(0,x) = 0$. Extend $u_n$ to be zero for $t \leq 0$, denoted by $\bar{u}_n$. It is readily seen that, for $0 \leq |\beta| \leq k$, $$D_x^\beta \bar{u}_n = \left\{ \begin{aligned} D_x^\beta u_n, \quad 0 \leq t \leq T, \\ 0, \quad S \leq t < 0, \end{aligned} \right.$$ $$D^\beta_x \bar{u}_n \in L_p\left((S,T) \times \Omega \right).$$ Now we check that $$\label{eq0120_03} D_t^\alpha \bar{u}_n = \partial_t I_S^{1-\alpha} \bar{u}_n = \left\{ \begin{aligned} \partial_t I_0^{1-\alpha} u_n, \quad 0 \leq t \leq T, \\ 0, \quad S \leq t < 0, \end{aligned} \right.$$ $$D_t^\alpha \bar{u}_n \in L_p\left((S,T) \times \Omega\right).$$ To see this, note that $I_S^{1-\alpha}\bar{u}_n(t,x) = 0$ for $S \leq t < 0$. For $0 \leq t \leq T$, we have $$\begin{aligned} &I_S^{1-\alpha} \bar{u}_n = \frac{1}{\Gamma(1-\alpha)} \int_S^t (t-s)^{-\alpha} \bar{u}_n(s,x) \, dy\\ &= \frac{1}{\Gamma(1-\alpha)} \int_0^t (t-s)^{-\alpha} u_n(s,x) \, dy = I_0^{1-\alpha} u_n(t,x).\end{aligned}$$ We now observe that, for $\varphi \in C_0^\infty\left( (S,T) \times \Omega\right)$, $$\begin{aligned} &\int_S^T \int_\Omega I_S^{1-\alpha} \bar{u}_n(t,x) \varphi_t(t,x) \, dx \, dt = \int_0^T \int_\Omega I_0^{1-\alpha} u_n (t,x) \varphi_t(t,x) \, dx \, dt\\ &= - \int_0^T \int_\Omega \partial_t I_0^{1-\alpha} u_n(t,x) \varphi(t,x) \, dx \, dt,\end{aligned}$$ where we used the fact that $I_0^{1-\alpha}u_n(0,x) = 0$. This proves Since $\{\bar{u}_n\}$ is Cauchy in $\bH_p^{\alpha,k}\left((S,T) \times \Omega\right)$ and $\bar{u}_n \to \bar{u}$ in $L_p\left((S,T) \times \Omega\right)$, we see that $\bar{u} \in \bH_p^{\alpha,k}\left((S,T) \times \Omega\right)$. Moreover, since $\bar{u}_n(S,x) = 0$, $\bar{u} \in \bH_{p,0}^{\alpha,k}\left((S,T) \times \Omega\right)$. In fact, $\bar{u}_n$’s are not necessarily in $C^\infty\left( [S,T] \times \Omega \right)$, but by using mollifications from $\bar{u}_n$ one can easily obtain $v_n \in C^\infty\left( [S,T] \times \Omega \right)$ vanishing for large $|x|$ such that $v_n(S,x) = 0$ and $$v_n \to \bar{u} \quad \text{in} \quad \bH_p^{\alpha,k}\left((S,T) \times \Omega\right).$$ The lemma is proved. \[lem0207\_1\] Let $p \in [1,\infty)$, $\alpha \in (0,1)$, $k \in \{1,2,\ldots\}$, $-\infty < S < t_0 < T < \infty$, and $v \in \bH_p^{\alpha,k}\left((S,T) \times \Omega \right)$. Then, for any infinitely differentiable function $\eta$ defined on $\bR$ such that $\eta(t)=0$ for $t \leq t_0$ and $$|\eta'(t)| \le M, \quad t \in \bR,$$ the function $\eta v$ belongs to $\bH_{p,0}^{\alpha,k}\left( (t_0,T) \times \Omega \right)$ and $$\label{eq9.45} \partial_t^\alpha (\eta v)(t,x) = \partial_t I_{t_0}^{1-\alpha} (\eta v)(t,x) = \eta(t) \partial_t I_S^{1-\alpha} v (t,x) - g(t,x),$$ for $(t,x) \in (t_0,T) \times \Omega$, where $$\label{eq0207_04} g(t,x) = \frac{\alpha}{\Gamma(1-\alpha)} \int_S^t (t-s)^{-\alpha-1} \left(\eta(s) - \eta(t)\right) v(s,x) \, ds$$ satisfies $$\label{eq0207_01} \|g\|_{L_p\left((t_0,T) \times \Omega\right)} \le N(\alpha, p, M, T, S) \|v\|_{L_p\left( (S,T) \times \Omega\right)}.$$ As in Lemma \[lem0206\_1\], we assume that $t_0 = 0$. First we check . Note that since $|\eta'(t)| \leq M$, we have $$\begin{aligned} &\left| \int_S^t (t-s)^{-\alpha-1} \left(\eta(t) - \eta(s) \right) v(s,x) \, ds \right|\\ &\leq M \int_S^t (t-s)^{-\alpha}|v(s,x)| \, ds = M \Gamma(1-\alpha) I^{1-\alpha}_S |v(t,x)|\end{aligned}$$ for $(t,x) \in \Omega_T$. Hence, the inequality follows from Lemma \[lem1018\_01\] with $1-\alpha$ in place of $\alpha$ (also see Remark \[rem0120\_1\]). Since $v \in \bH_p^{\alpha,2}\left((S,T) \times \Omega \right)$, there exists a sequence $\{v_n\} \subset \bH_p^{\alpha,2}\left((S,T) \times \Omega \right) \cap C^\infty\left([S,T] \times \Omega \right)$ such that $v_n$ vanishes for large $|x|$ and $$\|\partial_t I_S^{1-\alpha} (v_n - v) \|_{L_p\left((S,T) \times \Omega \right)} + \sum_{0 \leq |\beta| \leq 2} \|D_x^\beta(v_n - v)\|_{L_p\left((S,T) \times \Omega\right)} \to 0$$ as $n \to \infty$. Let $$g_n(t,x)= \eta(t) \partial_t I_S^{1-\alpha} v_n (t,x) - \partial_t I_0^{1-\alpha} (\eta v_n) (t,x).$$ Then $$\begin{aligned} &- \Gamma(1-\alpha) g_n(t,x)\\ &= \partial_t \int_0^t (t-s)^{-\alpha} \eta(s) v_n(s,x) \, ds - \eta(t) \partial_t \int_S^t (t-s)^{-\alpha} v_n(s,x) \, ds\\ &= \frac{\partial}{\partial t}\left[\int_0^t (t-s)^{-\alpha} \eta(s) v_n(s,x) \, ds - \eta(t) \int_S^t (t-s)^{-\alpha} v_n(s,x) \, ds \right]\\ &\qquad + \eta'(t) \int_S^t (t-s)^{-\alpha} v_n(s,x) \, ds\\ &= \frac{\partial}{\partial t} \left[ \int_S^t (t-s)^{-\alpha} \left(\eta(s) - \eta(t)\right) v_n(s,x) \, ds \right] + \eta'(t) \int_S^t (t-s)^{-\alpha} v_n(s,x) \, ds\\ &= -\alpha \int_S^t (t-s)^{-\alpha-1} \left(\eta(s) - \eta(t)\right) v_n(s,x) \, ds.\end{aligned}$$ Hence, $$g_n(t,x) = \frac{\alpha}{\Gamma(1-\alpha)} \int_S^t (t-s)^{-\alpha-1} \left( \eta(s) - \eta(t) \right) v_n(s,x) \, ds$$ for $(t,x) \in \Omega_T$. Clearly, $$\eta(t) \partial_t I_S^{1-\alpha} v_n(t,x) \to \eta(t) \partial_t I_S^{1-\alpha} v(t,x)$$ in $L_p(\Omega_T)$. From the estimate for $g$ with $v_n - v$ in place of $v$, it follows that $$\left\| g_n - g \right\|_{L_p(\Omega_T)} \to 0$$ as $n \to \infty$. That is, $$\partial_t I_0^{1-\alpha}(\eta v_n)(t,x) \to \eta(t) \partial_t I_S^{1-\alpha} v(t,x) - g(t,x)$$ in $L_p(\Omega_T)$. This together with $I_0^{1-\alpha} (\eta v_n) \to I_0^{1-\alpha} (\eta v)$ in $L_p(\Omega_T)$ implies and $\partial_t I_0^{1-\alpha}(\eta v_n)(t,x) \to \partial_t I_0^{1-\alpha}(\eta v)(t,x)$ in $L_p(\Omega_T)$. Obviously, $D_x^{\beta} (\eta v_n) \to D_x^\beta (\eta v)$ in $L_p(\Omega_T)$ for $0 \leq |\beta| \leq k$. Then from the fact that $\eta v_n \in C_0^\infty\left([0,T] \times \Omega \right)$ vanishing for large $|x|$ with $(\eta v_n)(0,x) = 0$, we conclude that $\eta v \in \bH_{p,0}^{\alpha,k}(\Omega_T)$. Auxiliary results {#sec4} ================= Throughout this section, we assume that $a^{ij}$ are measurable functions of only $t \in \bR$. That is, $a^{ij} = a^{ij}(t)$. \[prop0720\_1\] Theorem \[thm0412\_1\] holds when $p=2$. A version of this result for divergence type equations can be found in [@MR2538276]. Roughly speaking, the results in this proposition can be obtained by taking the spatial derivatives of the equation in [@MR2538276]. For the reader’s convenience, we present here a detailed proof. By the results from [@MR3581300] and the method of continuity, we only prove the a priori estimate . Moreover, since infinitely differentiable functions with compact support in $x$ and with the zero initial condition are dense in $\bH_{2,0}^{\alpha,2}(\bR^d_T)$, it suffices to prove for $u$ in $C_0^\infty\left([0,T] \times \bR^d\right)$ satisfying $u(0,x) =0$ and . Multiplying both sides of by $\Delta u$ and then integrating on $(0,T) \times \bR^d$, we have $$\label{eq0125_02} - \int_{\bR^d_T} \partial_t^\alpha u \Delta u \, dx \, dt + \int_{\bR^d_T} a^{ij}(t) D_{ij} u \Delta u \, dx \, dt = \int_{\bR^d_T} f \Delta u \, dx \, dt.$$ By integration by parts and the ellipticity condition, it follows that $$\begin{aligned} &\int_{\bR^d_T} a^{ij}(t) D_{ij} u \Delta u \, dx \, dt = \int_{\bR^d_T} \sum_{k=1}^d\sum_{i,j=1}^d a^{ij}(t) D_{ij} u D_k^2 u \, dx \, dt\\ &= \int_{\bR^d_T} \sum_{k=1}^d \sum_{i,j=1}^d a^{ij}(t) D_{ki} D_{kj}u \, dx \, dt \geq \delta \int_{\bR^d_T} \sum_{i,k = 1}^d |D_{ki}u|^2 \, dx \, dt.\end{aligned}$$ The term on the right-hand side of is taken care of by Young’s inequality. Moreover, the estimate for the term $\partial_t^\alpha u$ follows from that of $D^2u$ and the equation. Thus, to obtain we only need to see that the first integral in is non-negative. To do this, by setting $\nabla u = v$, we have $$- \int_{\bR^d_T} \partial_t^\alpha u \, \Delta u \, dx \, dt = \int_{\bR^d_T} \partial_t^\alpha v \cdot v \, dx \, dt.$$ We claim that, for each $(t,x) \in \bR^d_T$, $$\label{eq0904_01} \partial_t^\alpha v(t,x) \cdot v(t,x) \ge \frac{1}{2} \partial_t^\alpha |v|^2(t,x) .$$ To see this, for fixed $t\in (0,T)$ and $x\in \bR^d$, let $$F_1(s)=\frac 1 2 |v(s,x)|^2,\quad F_2(s)=v(s,x)\cdot v(t,x),$$ and $$F(s)=\frac 1 2 (|v(s,x)|^2-|v(t,x)|^2)-(v(s,x)-v(t,x))\cdot v(t,x).$$ Because $$F(s)=\frac 1 2|v(s,x)-v(t,x)|^2\ge 0$$ on $[0,T]$ with the equality at $s=t$, integration by parts clearly yields that $$\int_0^t (t-s)^{-\alpha}(F_1'(s)-F_2'(s))\,ds =\int_0^t (t-s)^{-\alpha}F'(s)\,ds\le 0,$$ which together with the definition of $\partial_t^\alpha$ implies . Therefore, because $F_1(0)=0$ we have $$\begin{aligned} &2 \Gamma(1-\alpha)\int_0^T\partial_t^\alpha v(t,x) \cdot v(t,x)\,dt\\ &\geq \int_0^T \frac{\partial}{\partial t}\left[ \int_0^t (t-s)^{-\alpha} |v(s,x)|^2 \, ds \right] \, dt = \left[\int_0^t (t-s)^{-\alpha} |v(s,x)|^2 \, ds\right]_{t=0}^{t=T}\\ &= \int_0^T (T-s)^{-\alpha} |v(s,x)|^2 \, ds \geq 0,\end{aligned}$$ where we used the fact that $v(s,x)$ is bounded on $[0,T] \times \bR^d$ so that $$\begin{aligned} &\int_0^t (t-s)^{-\alpha} |v(s,x)|^2 \, ds = \int_0^1 (t - tr)^{-\alpha} |v(tr,x)|^2 t \, dr\\ &= t^{1-\alpha} \int_0^1 (1-r)^{-\alpha} |v(tr,x)|^2 \, dr \to 0\end{aligned}$$ as $t \to 0$. \[lem0731\_1\] Let $p\in (1,\infty)$, $\alpha \in (0,1)$, $T \in (0,\infty)$, and $0 < r < R < \infty$. If Theorem \[thm0412\_1\] holds with this $p$ and $v \in \bH_{p,0}^{\alpha,2}\left((0, T) \times B_R\right)$ satisfies $$-\partial_t^\alpha v + a^{ij}(t) D_{ij} v = f$$ in $(0,T) \times B_R$, then $$\begin{aligned} &\| \partial_t^\alpha v \|_{L_p\left((0,T) \times B_r\right)} + \| D^2 v\|_{L_p\left((0,T) \times B_r\right)} \\ &\le \frac{N}{(R-r)^2} \|v\|_{L_p\left((0,T) \times B_R\right)} + N \|f\|_{L_p\left((0,T) \times B_R\right)},\end{aligned}$$ where $N = N(d,\delta,\alpha,p)$. Set $$r_0 = r, \quad r_k = r+(R-r)\sum_{j=1}^k \frac{1}{2^j}, \quad k = 1, 2, \ldots.$$ Let $\zeta_k = \zeta_k(x)$ be an infinitely differentiable function defined on $\bR^d$ such that $$\zeta_k = 1 \quad \text{on} \quad B_{r_k}, \quad \zeta = 0 \quad \text{outside} \quad \bR^d \setminus B_{r_{k+1}},$$ and $$|D_x \zeta_k(x)| \le \frac{2^{k+2}}{R-r}, \quad |D^2_x \zeta_k(x)| \le \frac{2^{2k+4}}{(R-r)^2}.$$ Then $v \zeta_k$ belongs to $\bH_{p,0}^{\alpha,2}(\bR^d_T)$ and satisfies $$-\partial_t^\alpha (v \zeta_k) + a^{ij}D_{ij} (v \zeta_k) = 2 a^{ij} D_i v D_j \zeta_k + a^{ij} v D_{ij} \zeta_k + f\zeta_k$$ in $(0,T) \times \bR^d$. By Theorem \[thm0412\_1\], it follows that $$\begin{aligned} \label{eq0728_01} &\|D^2 (v\zeta_k)\|_{L_p(\bR^d_T)}\nonumber\\ &\le \frac{N2^k}{R-r} \|Dv\|_{L_p\left((0,T)\times B_{r_{k+1}}\right)} + \frac{N2^{2k}}{(R-r)^2} \|v\|_{L_p\left((0,T)\times B_{r_{k+1}}\right)} + N \|f\|_{L_p\left((0,T)\times B_R\right)}\nonumber \\ &\le \frac{N 2^k}{R-r} \|D(v\zeta_{k+1})\|_{L_p(\bR^d_T)} + \frac{N 2^{2k}}{(R-r)^2} \|v\|_{L_p\left((0,T)\times B_R\right)} + N \|f\|_{L_p\left((0,T)\times B_R\right)}, \end{aligned}$$ where $N = N(d,\delta,\alpha,p)$. By an interpolation inequality with respect to the spatial variables, $$\begin{aligned} &\frac{2^k}{R-r}\|D(v\zeta_{k+1})\|_{L_p(\bR^d_T)}\\ &\le \varepsilon \|D^2(v \zeta_{k+1})\|_{L_p(\bR^d_T)} + N \varepsilon^{-1} \frac{2^{2k}}{(R-r)^2} \|v \zeta_{k+1}\|_{L_p(\bR^d_T)}\end{aligned}$$ for any $\varepsilon \in (0,1)$, where $N=N(d,p)$. Combining this inequality with , we obtain that $$\begin{aligned} &\|D^2 (v \zeta_k) \|_{L_p(\bR^d_T)}\\ &\le \varepsilon \|D^2 (v \zeta_{k+1}) \|_{L_p(\bR^d_T)} + N \varepsilon^{-1} \frac{4^k}{(R-r)^2}\|v\|_{L_p\left((0,T) \times B_R\right)} + N \|f\|_{L_p\left((0,T)\times B_R\right)},\end{aligned}$$ where $N = N(d,\delta,\alpha,p)$. By multiplying both sides of the above inequality by $\varepsilon^k$ and making summation with respect to $k = 0, 1, \ldots$, we see that $$\begin{aligned} &\|D^2(v\zeta_0)\|_{L_p(\bR^d_T)} + \sum_{k=1}^\infty \varepsilon^k \|D^2(v\zeta_k)\|_{L_p(\bR^d_T)}\\ &\le \sum_{k=1}^\infty \varepsilon^k \|D^2(v\zeta_k)\|_{L_p(\bR^d_T)} + N \frac{\varepsilon^{-1}}{(R-r)^2} \|v\|_{L_p\left((0,T) \times B_R\right)} \sum_{k=0}^\infty (4\varepsilon)^k\\ &+ N \|f\|_{L_p\left((0,T)\times B_R\right)} \sum_{k=0}^\infty \varepsilon^k,\end{aligned}$$ where the convergence of the summations are guaranteed if $\varepsilon = 1/8$. We then obtain the desired inequality in the lemma after we remove the same terms from both sides of the above inequality and use the fact that $\zeta_0 = 1$ on $B_r$. \[lem0211\_1\] Let $p \in [1,\infty)$, $\alpha \in (0,1)$, $0 < T < \infty$, and $0 < r < R < \infty$. If $v \in \bH_{p,0}^{\alpha,2}\left((0, T) \times B_R\right)$, then, for $\varepsilon \in (0, R-r)$, $$D_x v^{(\varepsilon)}(t,x) \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_r\right),$$ where $v^{(\varepsilon)}$ is a mollification of $v$ with respect to the spatial variables, that is, $$v^{(\varepsilon)}(t,x) = \int_{B_R} \phi_\varepsilon(x-y) v(t,y) \, dy, \quad \phi_\varepsilon(x) = \varepsilon^{-d} \phi(x/\varepsilon),$$ and $\phi\in C_0^\infty(B_1)$ is a smooth function with unit integral. Since $v \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_R\right)$, there exists a sequence $\{v_n\} \subset C^\infty\big([0,T] \times B_R\big)$ such that $v_n(0,x) = 0$ and $$\left\| v_n - v \right\|_{\bH_p^{\alpha,2}\left((0,T) \times B_R \right)} \to 0$$ as $n \to \infty$. Then, $D_x v_n^{(\varepsilon)} \in C^\infty\left([0,T]\times B_r\right)$ and $D_x v_n^{(\varepsilon)}(0,x) = 0$. For $(t,x) \in (0,T) \times B_r$, we have $$D_x^k D_x v^{(\varepsilon)}(t,x) = \int_{B_R} (D_x \phi_\varepsilon)(x-y) D_x^k v(t,y) \, dy, \quad k = 0,1,2,$$ $$D_t^\alpha D_x v^{(\varepsilon)}(t,x) = \int_{B_R} (D_x \phi_\varepsilon)(x-y) \partial_t^\alpha v(t,y) \, dy.$$ We also have the same expressions for $v_n$ in place of $v$. Hence, we see that $$\left\| D_x v_n^{(\varepsilon)} - D_x v^{(\varepsilon)} \right\|_{\bH_p^{\alpha,2}\left((0,T) \times B_r \right)} \to 0$$ as $n \to \infty$. This shows that $D_x v^{(\varepsilon)} \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_r\right)$. If $v \in \bH_{p,0}^{\alpha,2}\left((S,T) \times \bR^d\right)$ is a solution to a homogenous equation, one can improve its regularity as follows. \[lem0731\_2\] Let $p\in (1,\infty)$, $\alpha \in (0,1)$, $-\infty< S < t_0 < T < \infty$, and $0 < r < R < \infty$. Suppose that Theorem \[thm0412\_1\] holds with this $p$ and $v \in \bH_{p,0}^{\alpha,2}\left((S, T) \times B_R\right)$ satisfies $$-\partial_t^\alpha v + a^{ij}(t) D_{ij} v = f$$ in $(S,T) \times B_R$, where $f(t,x) = 0$ on $(t_0,T) \times B_R$ and, as we recall, $$\partial_t^\alpha v (t,x) = \frac{\partial}{\partial t} I_S^{1-\alpha} v(t,x) = \frac{1}{\Gamma(1-\alpha)} \frac{\partial}{\partial t} \int_S^t (t-s)^{-\alpha} v(s,x) \, ds.$$ Then, for any infinitely differentiable function $\eta$ defined on $\bR$ such that $\eta(t)=0$ for $t \leq t_0$, the function $D^2 (\eta v) = D^2_x (\eta v)$ belongs to $\bH_{p,0}^{\alpha,2}\left( (t_0,T) \times B_r \right)$ and satisfies $$- \partial_t^\alpha (D^2(\eta v)) + a^{ij}(t) D_{ij} (D^2(\eta v)) = \cG$$ in $(t_0,T) \times B_r$, where $\partial_t^\alpha = \partial_t I_{t_0}^{1-\alpha}$ and $\cG$ is defined by $$\cG(t,x) = \frac{\alpha}{\Gamma(1-\alpha)} \int_S^t (t-s)^{-\alpha-1}\left(\eta(t) - \eta(s)\right) D^2 v(s,x) \, ds.$$ Moreover, $$\label{eq0208_02} \|D^4 (\eta v)\|_{L_p\left((0,T) \times B_r\right)} \le \frac{N}{(R-r)^2} \|D^2 v\|_{L_p\left((0,T) \times B_R\right)} + N \|\cG\|_{L_p\left((0,T) \times B_R\right)},$$ where $N = N(d,\delta,\alpha,p)$. Without loss of generality we assume $t_0 = 0$ so that $$- \infty < S < 0 < T < \infty.$$ By Lemma \[lem0207\_1\] and the fact that $f(t,x) = 0$ on $(0,T) \times B_R$, we have that $\eta v$ belongs to $\bH_{p,0}^{\alpha,2}\left((0,T) \times B_R\right)$ and satisfies $$- \partial_t^\alpha (\eta v) + a^{ij}(t) D_{ij}(\eta v) = g$$ in $(0,T) \times B_R$, where $g \in L_p\left((0,T) \times B_R\right)$ is from . Find $r_i$, $i=1,2,3$, such that $r = r_1 < r_2 < r_3 < R$. Set $w = \eta v$ and consider $w^{(\varepsilon)}$, $\varepsilon \in (0,R-r_3)$, from Lemma \[lem0211\_1\], which is a mollification of $w$ with respect to the spatial variables. Since $w \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_R\right)$, by Lemma \[lem0211\_1\], $D_x w^{(\varepsilon)}$ belongs to $\bH_{p,0}^{\alpha,2}\left((0,T) \times B_{r_3}\right)$ and satisfies $$- \partial_t^\alpha (D_x w^{(\varepsilon)} ) + a^{ij}(t) D_{ij} (D_x w^{(\varepsilon)} w) = D_x g^{(\varepsilon)}$$ in $(0,T) \times B_{r_3}$, where $$D_x g^{(\varepsilon)}(t,x) = \frac{\alpha}{\Gamma(1-\alpha)} \int_S^t (t-s)^{-\alpha-1} \left(\eta(s) - \eta(t)\right) D_x v^{(\varepsilon)}(s,x) \, ds.$$ It then follows from Lemma \[lem0731\_1\] that $$\begin{gathered} \label{eq0209_01} \| \partial_t^\alpha (D_x w^{(\varepsilon)})\|_{L_p\left((0,T) \times B_{r_2}\right)} + \| D^2 (D_x w^{(\varepsilon)})\|_{L_p\left((0,T) \times B_{r_2}\right)} \\ \le \frac{N}{(r_3-r_2)^2} \|D_x w^{(\varepsilon)}\|_{L_p\left((0,T) \times B_{r_3}\right)} + N \|D_x g^{(\varepsilon)}\|_{L_p\left((0,T) \times B_{r_3}\right)},\end{gathered}$$ where $N = N(d,\delta,p)$. Note that $$\label{eq0211_01} \|D_x w^{(\varepsilon)} - D_x w\|_{L_p\left((0,T) \times B_{r_3}\right)} \to 0, \quad \|D_x g^{(\varepsilon)} - \cG_0 \|_{L_p\left((0,T) \times B_{r_3}\right)} \to 0,$$ where $\cG_0$ is defined as $\cG$ with $Dv$ in place of $D^2 v$. In particular, the latter convergence in is guaranteed by and the properties of mollifications. Recall that $D_x w^{(\varepsilon)} \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_{r_3}\right)$. Then, from and , we conclude that $D_x w$ belongs to $\bH_{p,0}^{\alpha,2}\left((0,T) \times B_{r_2}\right)$ and satisfies $$- \partial_t^\alpha (D_x w ) + a^{ij}(t) D_{ij} (D_x w) = \cG_0$$ in $(0,T) \times B_{r_2}$. We now repeat the above argument with $Dw$, $r_1$, and $r_2$ in place of $w$, $r_2$, and $r_3$, respectively, along with the observation that the limits in hold with $Dw$ in place of $w$. In particular, the estimate with $Dw$ in place of $w$ implies . The lemma is proved. Level set arguments {#sec5} =================== Recall that $Q_{R_1,R_2}(t,x) = (t-R_1^{2/\alpha}, t) \times B_{R_2}(x)$ and $Q_R(t,x)=Q_{R,R}(t,x)$. For $(t_0,x_0) \in \bR \times \bR^d$ and a function $g$ defined on $(-\infty,T) \times \bR^d$, we set $$\label{eq0406_03b} \cM g(t_0,x_0) = \sup_{Q_{R}(t,x) \ni (t_0,x_0)} \dashint_{Q_{R}(t,x)}|g(s,y)| I_{(-\infty,T) \times \bR^d} \, dy \, ds$$ and $$\label{eq0406_03} \cS\cM g(t_0,x_0) = \sup_{Q_{R_1,R_2}(t,x) \ni (t_0,x_0)} \dashint_{Q_{R_1,R_2}(t,x)}|g(s,y)| I_{(-\infty,T) \times \bR^d} \, dy \, ds.$$ The first one is called the (parabolic) maximal function of $g$, and second one is called the strong (parabolic) maximal function of $g$. \[prop0406\_1\] Let $p\in (1,\infty)$, $\alpha \in (0,1)$, $T \in (0,\infty)$, and $a^{ij} = a^{ij}(t)$. Assume that Theorem \[thm0412\_1\] holds with this $p$ and $u \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ satisfies $$-\partial_t^\alpha u + a^{ij} D_{ij} u = f$$ in $(0,T) \times \bR^d$. Then there exists $$p_1 = p_1(d, \alpha,p)\in (p,\infty]$$ satisfying $$\label{eq0411_05} p_1 > p + \min\left\{\frac{2\alpha}{\alpha d + 2 - 2\alpha}, \alpha, \frac{2}{d} \right\}$$ and the following. For $(t_0,x_0) \in [0,T] \times \bR^d$ and $R \in (0,\infty)$, there exist $$w \in \bH_{p,0}^{\alpha,2}((t_0-R^{2/\alpha}, t_0)\times \bR^d), \quad v \in \bH_{p,0}^{\alpha,2}((S,t_0) \times \bR^d),$$ where $S = \min\{0, t_0 - R^{2/\alpha}\}$, such that $u = w + v$ in $Q_R(t_0,x_0)$, $$\label{eq8.13} \left( |D^2w|^p \right)_{Q_R(t_0,x_0)}^{1/p} \le N \left( |f|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p},$$ and $$\begin{gathered} \label{eq0411_01} \left( |D^2v|^{p_1} \right)_{Q_{R/2}(t_0,x_0)}^{1/p_1} \leq N \left( |f|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p} \\ + N \sum_{k=0}^\infty 2^{-k\alpha} \left( \dashint_{\!t_0 - (2^{k+1}+1)R^{2/\alpha}}^{\,\,\,t_0} \dashint_{B_R(x_0)} |D^2u(s,y)|^p \, dy \, ds \right)^{1/p},\end{gathered}$$ where $N=N(d,\delta, \alpha,p)$. Here we understand that $u$ and $f$ are extended to be zero whenever $t < 0$ and $$\left( |D^2v|^{p_1} \right)_{Q_{R/2}(t_0,x_0)}^{1/p_1} = \|D^2v\|_{L_\infty(Q_{R/2}(t_0,x_0))},$$ provided that $p_1 = \infty$. We extend $u$ and $f$ to be zero, again denoted by $u$ and $f$, on $(-\infty,0) \times \bR^d$. Thanks to translation, it suffices to prove the desired inequalities when $x_0 = 0$. Moreover, we assume that $R = 1$. Indeed, for $R > 0$, we set $$\tilde{u}(t,x) = R^{-2}u(R^{2/\alpha}t, R x), \quad \tilde{a}^{ij} = a^{ij}(R^{2/\alpha}t), \quad \tilde{f}(t,x) = f(R^{2/\alpha}t, Rx).$$ Then $$- \partial_t^\alpha \tilde{u} + \tilde{a}^{ij}(t) D_{ij} \tilde{u} = \tilde{f}$$ in $(0,R^{-2/\alpha} T) \times \bR^d$. We then apply the result for $R=1$ to this equation on $$(\tilde{t}_0-1,\tilde{t}_0) \times B_1, \quad \tilde{t}_0 = R^{-2/\alpha} t_0$$ and return to $u$. For $R=1$ and $t_0 \in (0,\infty)$, set $\zeta = \zeta(t,x)$ to be an infinitely differentiable function defined on $\bR^{d+1}$ such that $$\zeta = 1 \quad \text{on} \quad (t_0-1, t_0) \times B_1,$$ and $$\zeta = 0 \quad \text{on} \quad \bR^{d+1} \setminus (t_0-2^{2/\alpha}, t_0+2^{2/\alpha}) \times B_2.$$ Using Theorem \[thm0412\_1\], find a solution $w\in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ to the problem $$\left\{ \begin{aligned} -\partial_t^\alpha w + a^{ij}(t) D_{ij} w &= \zeta(t,x) f(t,x)\quad \text{in} \,\, (t_0-1, t_0) \times \bR^d, \\ w(t_0 - 1,x) &= 0 \quad \text{on} \quad \bR^d. \end{aligned} \right.$$ where we recall that $$\partial_t^\alpha w = \frac{1}{\Gamma(1-\alpha)} \partial_t \int_{t_0-1}^t (t-s)^{-\alpha} w(s,x) \, ds.$$ Again extend $w$ to be zero on $(-\infty,t_0-1) \times \bR^d$. From Theorem \[thm0412\_1\] it follows that $$\label{eq1214_01} \|\partial_t^\alpha w\|_{L_p\left(Q_r(t_0,0)\right)} + \|D^2 w \|_{L_p\left(Q_r(t_0,0)\right)} \le N \|f\|_{L_p\left(Q_2(t_0,0)\right)}$$ for any $r > 0$. Set $v = u - w$ so that $$v = \left\{ \begin{aligned} u-w, &\quad t \in (t_0 - 1,t_0), \\ u, &\quad t \in (-\infty, t_0 - 1], \end{aligned} \right.$$ where we note that it is possible to have $t_0 - 1 < 0$. Then by Lemma \[lem0206\_1\], $v$ belongs to $\bH_{p,0}^{\alpha,2}\left((S,t_0) \times \bR^d\right)$ for $S := \min \{0, t_0 -1\}$ and satisfies $$\partial_t^\alpha w = \partial_t I_{t_0-1}^{1-\alpha} w = \partial_t I_S^{1-\alpha} w, \quad \partial_t^\alpha u = \partial_t I_0^{1-\alpha} u = \partial_t I_S^{1-\alpha} u,$$ and $$-\partial_t^\alpha v + a^{ij} D_{ij} v = h$$ in $(S,t_0)\times \bR^d$, where $$h(t,x) = \left\{ \begin{aligned} \left( 1 -\zeta(t,x)\right) f(t,x) \quad &\text{in} \,\, (t_0 - 1, t_0) \times \bR^d, \\ f(t,x) \quad &\text{in} \,\, (S, t_0 - 1) \times \bR^d. \end{aligned} \right.$$ In particular, we note that $h = 0$ in $(t_0 - 1,t_0) \times B_1$. Find an infinitely differentiable function $\eta$ defined on $\bR$ such that $$\eta = \left\{ \begin{aligned} 1 \quad &\text{if} \quad t \in (t_0-(1/2)^{2/\alpha},t_0), \\ 0 \quad &\text{if} \quad t \in \bR \setminus (t_0-1,t_0+1), \end{aligned} \right.$$ and $$\left|\frac{\eta(t)-\eta(s)}{t-s}\right| \le N(\alpha).$$ By Lemma \[lem0731\_2\], $D^2(\eta v)$ belongs to $\bH_{p,0}^{\alpha,2}\left((t_0-1,t_0)\times B_{3/4}\right)$ and satisfies $$-\partial_t^\alpha \left( D^2(\eta v) \right) + a^{ij} D_{ij} D^2(\eta v) = \cG$$ in $(t_0-1,t_0)\times B_{3/4}$, where $$\cG(t,x) = \frac{\alpha}{\Gamma(1-\alpha)} \int_S^t (t-s)^{-\alpha-1}\left(\eta(t) - \eta(s)\right) D^2 v(s,x) \, ds.$$ If $p \leq 1/\alpha$, take $p_1$ satisfying $$p_1 \in \left(p, \frac{1/\alpha + d/2}{1/(\alpha p) + d/(2 p) -1}\right) \quad \text{if} \quad p \leq d/2,$$ $$p_1 \in \left(p, p(\alpha p + 1)\right) \quad \text{if} \quad p > d/2.$$ If $p > 1/\alpha$, take $p_1$ satisfying $$p_1 \in \left(p, p + 2p^2/d\right) \quad \text{if} \quad p \leq d/2,$$ $$p_1 \in (p, 2p) \quad \text{if} \quad p > d/2, \quad p \leq d/2 + 1/\alpha,$$ $$p_1 = \infty \quad \text{if} \quad p > d/2 + 1/\alpha.$$ Note that $p_1$ satisfies and the increment $\min \{2\alpha/(\alpha d + 2 - 2\alpha), \alpha, 2/d\}$ is independent of $p$. By Lemma \[lem0731\_2\] and the embedding results in Appendix (Corollary \[cor1211\_1\], Theorem \[thm1207\_2\], Corollary \[cor0225\_1\], Theorem \[thm0214\_1\], and Theorem \[thm5.18\]), we have $$\begin{aligned} \label{eq0715_01} &\|D^2 v\|_{L_{p_1}\left(Q_{1/2}(t_0,0)\right)} \le \|D^2(\eta v)\|_{L_{p_1}\left((t_0-1,t_0)\times B_{1/2}\right)}\nonumber \\ &\le N\| |D^2(\eta v)| + |D^4(\eta v)| + |D_t^\alpha D^2 (\eta v)| \|_{L_p\left((t_0-1,t_0)\times B_{3/4}\right)}\nonumber \\ &\le N \||D^2(\eta v)| + |\cG|\|_{L_p\left((t_0-1,t_0)\times B_1\right)} \le N \||D^2 v| + |\cG|\|_{L_p\left((t_0-1,t_0)\times B_1\right)}\nonumber \\ &\leq N \| |D^2 u| + |D^2 w| + |\cG| \|_{L_p\left((t_0-1,t_0)\times B_1\right)},\end{aligned}$$ where $N = N(d, \delta, \alpha,p,p_1)$ and we used the fact that $$D_t^\alpha D^2(\eta v) = a^{ij} D_{ij} D^2 (\eta v) - \cG$$ in $(t_0-1,t_0) \times B_{3/4}$. Since $D^2 v = 0$ for $t \leq S$, we write $$\begin{aligned} &\frac{\Gamma(1-\alpha)}{\alpha} \cG(t,x) = \int_{-\infty}^t (t-s)^{-\alpha-1} \left( \eta(s) - \eta(t) \right) D^2 v(s,x) \, ds\\ &= \int_{t-1}^t (t-s)^{-\alpha-1}\left( \eta(s) - \eta(t) \right) D^2 v(s,x) \, ds\\ &\quad + \int_{-\infty}^{t-1} (t-s)^{-\alpha-1}\left( \eta(s) - \eta(t) \right) D^2 v(s,x) \, ds := I_1(t,x)+I_2(t,x),\end{aligned}$$ where $$\begin{aligned} |I_1(t,x)| \le N \int_{t-1}^t |t-s|^{-\alpha} |D^2 v(s,x)|\,ds= N \int_0^1 |s|^{-\alpha} |D^2 v(t-s,x)|\, ds,\end{aligned}$$ From this we have $$\begin{gathered} \label{eq0715_02} \|I_1\|_{L_p\left((t_0-1,t_0) \times B_1\right)} \le N \|D^2 v\|_{L_p\left((t_0-2,t_0)\times B_1\right)} \\ = N \|D^2 v\|_{L_p\left((t_0-1,t_0)\times B_1\right)} + N \|D^2 u\|_{L_p\left((t_0-2,t_0-1)\times B_1\right)}.\end{gathered}$$ To estimate $I_2$, we see that $\eta(s) = 0$ for any $s \in (-\infty, t-1)$ with $t \in (t_0-1,t_0)$. Thus we have $$I_2(t,x) = -\eta(t) \int_{-\infty}^{t-1} (t-s)^{-\alpha-1} D^2 v(s,x) \, ds.$$ Then, $$\begin{aligned} |I_2(t,x)| &\le \int_{-\infty}^{t-1} |t-s|^{-\alpha-1} |D^2 v(s,x)| \, ds\\ &= \sum_{k=0}^\infty \int_{t-2^{k+1}}^{t-2^k} |t-s|^{-\alpha-1} |D^2 v(s,x)|\,ds\\ &\le \sum_{k=0}^\infty \int_{t-2^{k+1}}^{t-2^k } 2^{-k(\alpha+1)} |D^2 v(s,x)|\,ds.\end{aligned}$$ From this we have $$\|I_2\|_{L_p\left((t_0-1,t_0) \times B_1\right)}\le \sum_{k=0}^\infty 2^{-k(\alpha+1)} \left\| \int_{t-2^{k+1}}^{t-2^k} |D^2 v(s,x)| \, ds \right\|_{L_p\left((t_0-1,t_0) \times B_1\right)}.$$ Since $t_0 - 1 < t < t_0$, $$\int_{t-2^{k+1}}^{t-2^k} |D^2 v(s,x)|\, ds \leq \int_{t_0-(2^{k+1}+1)}^{t_0-2^k} |D^2 v(s,x)|\, ds.$$ Hence, by the Minkowski inequality, $$\begin{aligned} &\left\| \int_{t-2^{k+1}}^{t-2^k } |D^2 v(s,x)| \, ds \right\|_{L_p\left(Q_1(t_0,0)\right)}\\ &\le \int_{t_0 - (2^{k+1}+1)}^{t_0 - 2^k} \left( \int_{B_1} |D^2 v(s,x)|^p \, dx \right)^{1/p} \, ds\\ &\le 2^{k+2}\left( \dashint_{\!t_0 - (2^{k+1}+1)}^{\,\,\,t_0} \dashint_{B_1} |D^2 v(s,x)|^p \, dx \, ds \right)^{1/p}.\end{aligned}$$ It then follows that $$\begin{aligned} &\|I_2\|_{L_p\left(Q_1(t_0,0)\right)}\\ &\le \sum_{k=0}^\infty 2^{-k\alpha+2}\left( \dashint_{\!t_0 - (2^{k+1}+1)}^{\,\,\,t_0} \dashint_{B_1} |D^2 v(s,x)|^p \, dx \, ds \right)^{1/p}\\ &\le \sum_{k=0}^\infty 2^{-k\alpha+2} \left( \dashint_{\!t_0 - (2^{k+1}+1)}^{\,\,\,t_0} \dashint_{B_1} |D^2 u(s,x)|^p \, dx \, ds \right)^{1/p}\\ &\quad + \sum_{k=0}^\infty 2^{-k\alpha+2}\left( \dashint_{\!t_0 - (2^{k+1}+1)}^{\,\,\,t_0} \dashint_{B_1} |D^2 w(s,x)|^p \, dx \, ds \right)^{1/p},\end{aligned}$$ where $$\sum_{k=0}^\infty 2^{-k\alpha+2}\left( \dashint_{\!t_0 - (2^{k+1}+1)}^{\,\,\,t_0} \dashint_{B_1} |D^2 w(s,x)|^p \, dx \, ds \right)^{1/p} \leq N(\alpha) \left( |D^2 w|^p \right)^{1/p}_{Q_1(t_0,0)}.$$ Combining the above inequalities, , and , we get $$\begin{aligned} &\|D^2 v\|_{L_{p_1}\left(Q_{1/2}(t_0,0)\right)} \leq N \left( |D^2 w|^p \right)_{Q_1(t_0,0)}^{1/p}\\ &\quad + N \sum_{k=0}^\infty 2^{-k\alpha} \left( \dashint_{\!t_0 - (2^{k+1}+1)}^{\,\,\,t_0} \dashint_{B_1(x_0)} |D^2u(s,y)|^p \, dy \, ds \right)^{1/p}.\end{aligned}$$ We then use with $r=1$ to obtain with $R = 1$. The proposition is proved. Let $\gamma\in (0,1)$, and let $p\in (1,\infty)$ and $p_1=p_1(d,\alpha,p)$ be from the above proposition. Denote $$\label{eq0406_04} \cA(s) = \left\{ (t,x) \in (-\infty,T) \times \bR^d: |D^2 u(t,x)| > s \right\}$$ and $$\begin{gathered} \label{eq0406_05} \cB(s) = \big\{ (t,x) \in (-\infty,T) \times \bR^d: \\ \gamma^{-1/p}\left( \cM |f|^p (t,x) \right)^{1/p} + \gamma^{-1/p_1}\left( \cS\cM |D^2 u|^p(t,x)\right)^{1/p} > s \big\},\end{gathered}$$ where, to well define $\cM$ and $\cS\cM$ (recall the definitions in and ), we extend a given function to be zero for $t \leq S$ if the function is defined on $(S,T) \times \bR^d$. Set $$\label{eq0606_01} \cC_R(t,x) = (t-R^{2/\alpha},t+R^{2/\alpha}) \times B_R(x),\quad \hat \cC_R(t,x)=\cC_R(t,x)\cap \{t\le T\}.$$ \[lem0409\_1\] Let $p\in (1,\infty)$, $\alpha \in (0,1)$, $T \in (0,\infty)$, $a^{ij} = a^{ij}(t)$, $R \in (0,\infty)$, and $\gamma \in (0,1)$. Assume that Theorem \[thm0412\_1\] holds with this $p$ and $u \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ satisfies $$-\partial_t^\alpha u + a^{ij} D_{ij} u = f$$ in $(0,T) \times \bR^d$. Then, there exists a constant $\kappa = \kappa(d,\delta,\alpha,p) > 1$ such that the following hold: for $(t_0,x_0) \in (-\infty,T] \times \bR^d$ and $s>0$, if $$\label{eq0406_02} |\cC_{R/4}(t_0,x_0) \cap \cA(\kappa s)| \geq \gamma |\cC_{R/4}(t_0,x_0)|,$$ then we have $$\hat\cC_{R/4}(t_0,x_0) \subset \cB(s).$$ By dividing the equation by $s$, we may assume that $s = 1$. We only consider $(t_0,x_0) \in (-\infty,T] \times \bR^d$ such that $t_0 + (R/4)^{2/\alpha} \geq 0$, because otherwise, $$\cC_{R/4}(t_0,x_0) \cap \cA(\kappa) \subset \left\{ (t,x) \in (-\infty,0] \times \bR^d: |D^2 u(t,x)| > s \right\} = \emptyset$$ as $u(t,x)$ is extended to be zero for $t < 0$. Suppose that there is a point $(s,y) \in \hat\cC_{R/4}(t_0,x_0)$ such that $$\label{eq0406_01} \gamma^{-1/p}\left( \cM |f|^p (s,y) \right)^{1/p} + \gamma^{-1/p_1} \left( \cS\cM |D^2 u|^p(s,y)\right)^{1/p} \leq 1.$$ Set $$t_1 := \min \{ t_0 + (R/4)^{2/\alpha}, T\} \quad \text{and} \quad x_1 := x_0.$$ Then $(t_1,x_1) \in [0,T] \times \bR^d$ and by Proposition \[prop0406\_1\] there exist $p_1 = p_1(d,\alpha,p) \in (p,\infty]$ and $w \in \cH_{p,0}^{\alpha,2}\left((t_1-R^{2/\alpha}, t_1) \times \bR^d\right)$, $v \in \cH_{p,0}^{\alpha,2}\left((S, t_1) \times \bR^d\right)$, where $S=\min\{0,t_1-R^{2/\alpha}\}$, such that $u = w + v$ in $Q_R(t_1,x_1)$, $$\label{eq0409_01} \left(|D^2 w|^p\right)^{1/p}_{Q_R(t_1,x_1)} \leq N \left( |f|^p \right)_{Q_{2R}(t_1,x_1)}^{1/p},$$ and $$\begin{gathered} \label{eq0409_02} \left( |D^2v|^{p_1} \right)_{Q_{R/2}(t_1,x_1)}^{1/p_1} \le N \left( |f|^p \right)_{Q_{2R}(t_1,x_1)}^{1/p} \\ + N \sum_{k=0}^\infty 2^{-\kappa \alpha} \left(\dashint_{\! t_1 - (2^{k+1}+1)R^{2/\alpha}}^{\,\,\,t_1} \dashint_{B_R(x_1)} |D^2u(\ell,z)|^p \, dz \, d \ell \right)^{1/p},\end{gathered}$$ where $N=N(d,\delta, \alpha,p)$. Since $t_0 \leq T$, we have $$(s,y) \in \hat\cC_{R/4}(t_0,x_0) \subset Q_{R/2}(t_1,x_1) \subset Q_{2R}(t_1,x_1),$$ $$(s,y) \in \hat\cC_{R/4}(t_0,x_0) \subset (t_1- (2^{k+1}+1)R^{2/\alpha}, t_1) \times B_R(x_1)$$ for all $k = 0,1,\ldots$. From these set inclusions, in particular, we observe that $$\dashint_{\! t_1 - (2^{k+1}+1)R^{2/\alpha}}^{\,\,\,t_1} \dashint_{B_R(x_1)} |D^2u(\ell,z)|^p \, dz \, d \ell \leq \cS\cM |D^2u|^p(s,y)$$ for all $k=0,1,2,\ldots$. Thus the inequality along with and implies that $$\left( |D^2v|^{p_1} \right)_{Q_{R/2}(t_1,x_1)}^{1/p_1} \leq N \gamma^{1/p} + N \gamma^{1/p_1} \leq N_0 \gamma^{1/p_1},$$ $$\left(|D^2 w|^p\right)^{1/p}_{Q_R(t_1,x_1)} \leq N_1 \gamma^{1/p},$$ where $N_0$ and $N_1$ depend only on $d$, $\delta$, $\alpha$, and $p$. Note that, for a sufficiently large $K_1$, $$\begin{aligned} &|\cC_{R/4}(t_0,x_0) \cap \cA(\kappa)| = |\{(t,x) \in \cC_{R/4}(t_0,x_0), t \in (-\infty,T): |D^2u(t,x)| > \kappa\}|\\ &\leq \left|\{ (t,x) \in Q_{R/2}(t_1,x_1): |D^2 u(t,x)| > \kappa\}\right|\\ &\leq \left|\{(t,x) \in Q_{R/2}(t_1,x_1): |D^2 w(t,x)| > \kappa - K_1 \}\right|\\ &\quad + \left|\{(t,x) \in Q_{R/2}(t_1,x_1): |D^2 v(t,x)| > K_1 \}\right|\\ &\leq (\kappa-K_1)^{-p} \int_{Q_{R/2}(t_1,x_1)} |D^2 w|^p \, dx \, dt + K_1^{-p_1} \int_{Q_{R/2}(t_1,x_1)} |D^2v|^{p_1} \, dx \, dt\\ &\leq \frac{N_1^p \gamma |Q_R|}{(\kappa - K_1)^p} + \frac{N_0^{p_1}\gamma|Q_{R/2}|}{K_1^{p_1}}I_{p_1 \neq \infty}\\ &\leq N(d,\alpha) |\cC_{R/4}| \left(\frac{N_1^p \gamma }{(\kappa - K_1)^p} + \gamma \left(\frac{N_0}{K_1}\right)^p I_{p_1 \neq \infty} \right) < \gamma |\cC_{R/4}(t_0,x_0)|,\end{aligned}$$ provided that we choose a sufficiently large $K_1(\ge N_0)$ depending only on $d$, $\delta$, $\alpha$, and $p$, so that $$N(d,\alpha) (N_0/K_1)^p < 1/2,$$ and then choose a $\kappa$ depending only on $d$, $\delta$, $\alpha$, and $p$, so that $$N(d,\alpha) N_1^p/(\kappa-K_1)^p < 1/2.$$ Considering , we get a contradiction. The lemma is proved. $L_p$-estimates {#sec6} =============== Now we are ready to give the proof of Theorem \[thm0412\_1\]. We first consider the case when $p\in [2,\infty)$ by using an iterative argument to successively increase the exponent $p$. When $p=2$, the theorem follows from Proposition \[prop0720\_1\]. Now suppose that the theorem is proved for some $p_0\in [2,\infty)$. Let $p_1=p_1(d,\alpha,p_0)$ be from Proposition \[prop0406\_1\], and $p\in (p_0,p_1)$. As in the proof of Proposition \[prop0720\_1\] we assume $u \in C_0^\infty\left([0,T] \times \bR^d\right)$ with $u(0,x) = 0$ and prove the a priori estimate . Note that $$\label{eq8.47} \|D^2u\|_{L_p(\bR^d_T)}^p = p \int_0^\infty |\cA(s)| s^{p-1} \, ds = p \kappa^p \int_0^\infty |\cA(\kappa s)| s^{p-1} \, ds.$$ By Lemmas \[lem0409\_1\] and \[lem0409\_2\] it follows that $$\label{eq8.46} |\cA(\kappa s)| \leq N(d,\alpha) \gamma|\cB(s)|$$ for all $s \in (0,\infty)$. Hence, by the Hardy-Littlewood maximal function theorem, $$\begin{aligned} &\|D^2u\|_{L_p(\bR^d_T)}^p \leq N p \kappa^p \gamma \int_0^\infty |\cB(s)| s^{p-1} \, ds\\ &\le N\gamma \int_0^\infty\left|\left\{ (t,x) \in (-\infty,T) \times \bR^d:\gamma^{-\frac 1{ p_1}}\left( \cS\cM |D^2 u|^{p_0}(t,x)\right)^{\frac 1 {p_0}} > s/2 \right\}\right| s^{p-1} \, ds\\ &\quad + N\gamma \int_0^\infty\left|\left\{ (t,x) \in (-\infty,T) \times \bR^d:\gamma^{-\frac 1 {p_0}}\left( \cM |f|^{p_0} (t,x) \right)^{\frac 1 {p_0}} > s/2 \right\}\right| s^{p-1} \, ds\\ &\leq N \gamma^{1-p/p_1} \|D^2u\|^p_{L_p(\bR^d_T)} + N \gamma^{1-p/p_0} \|f\|^p_{L_p(\bR^d_T)},\end{aligned}$$ where $N = N(d,\delta,\alpha,p)$. Now choose $\gamma \in (0,1)$ so that $$N \gamma^{1-p/p_1} < 1/2,$$ which is possible because $p\in (p_0,p_1)$. Then we have $$\|D^2u\|_{L_p(\bR^d_T)} \leq N \|f\|_{L_p(\bR^d_T)}.$$ From this and the equation, we arrive at for $p \in (p_0,p_1)$. We repeat this procedure. Recall , which shows that each time the increment from $p_0$ to $p_1$ can be made bigger than a positive number depending only on $d$ and $\alpha$. Thus in finite steps, we get a $p_0$ which is larger than $d/2 + 1/\alpha$, so that $p_1=p_1(d,\alpha,p_0)=\infty$. Therefore, the theorem is proved for any $p\in [2,\infty)$. For $p \in (1,2)$, we use a duality argument. We only prove the a priori estimate . Without loss of generality, assume that $u \in C_0^\infty\left([0,T] \times \bR^d\right)$ with $u(0,x) = 0$ satisfies $$-\partial_t^\alpha u + a^{ij} D_{ij} u = f$$ in $(0,T) \times \bR^d$. Let $\phi \in L_q(\bR^d_T)$, where $1/p+1/q=1$. Then $$\phi(-t,x) \in L_q\left((-T,0) \times \bR^d\right).$$ Find $w \in \bH_{q,0}^{\alpha,2}\left((-T,0) \times \bR^d\right)$ satisfying $$- \partial_t^\alpha w + a^{ij}(-t) D_{ij} w = \phi(-t,x)$$ in $(-T,0) \times \bR^d$ with the estimate $$\|D^2w\|_{L_q\left((-T,0) \times \bR^d\right)} \leq N \|\phi(-t,x)\|_{L_q\left((-T,0) \times \bR^d\right)} = N \|\phi\|_{L_q(\bR^d_T)},$$ where $$\partial_t^\alpha w = \partial_t I_{-T}^{1-\alpha} w.$$ Considering $w_k \in C_0^\infty\left([-T,0] \times \bR^d\right)$ with $w_k(-T,0) = 0$ such that $w_k \to w$ in $\bH_{q,0}^{\alpha,2}\left((-T,0) \times \bR^d \right)$, we observe that $$\begin{aligned} &\int_0^T \int_{\bR^d} \phi D^2 u \, dx \, dt = \int_{-T}^0 \int_{\bR^d} \phi(-t,x) D^2 u(-t,x) \, dx \, dt\\ &= \int_{-T}^0 \int_{\bR^d} \left(-\partial_t^\alpha w + a^{ij}(-t) D_{ij} w \right) D^2 u(-t,x) \, dx \, dt\\ &= \int_0^T \int_{\bR^d} \left( -\partial_t^\alpha u(t,x) + a^{ij}(t)D_{ij} u(t,x) \right) D^2 w(-t,x) \, dx \, dt\\ &= \int_0^T \int_{\bR^d} f(t,x) D^2 w(-t,x) \, dx \, dt \leq N\|f\|_{L_p(\bR^d_T)} \|\phi\|_{L_q(\bR^d_T)}.\end{aligned}$$ It then follows that $$\|D^2u\|_{L_p(\bR^d_T)} \leq N \|f\|_{L_p(\bR^d_T)},$$ from which and the equation, we finally obtain . To prove Theorem \[main\_thm\], we extend Proposition \[prop0406\_1\] to the case when $a^{ij}=a^{ij}(t,x)$ satisfying Assumption \[assump2.2\]. \[prop0515\_1\] Let $p\in (1,\infty)$, $\alpha,\gamma_0 \in (0,1)$, $T \in (0,\infty)$, $\mu\in (1,\infty)$, $\nu=\mu/(\mu-1)$, and $a^{ij} = a^{ij}(t,x)$ satisfying Assumption \[assump2.2\] ($\gamma_0$). Assume that $u \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ vanishes for $x\notin B_{R_0}(x_1)$ for some $x_1\in \bR^d$, and satisfies in $(0,T) \times \bR^d$. Then there exists $$p_1 = p_1(d, \alpha,p)\in (p,\infty]$$ satisfying and the following. For $(t_0,x_0) \in [0,T] \times \bR^d$ and $R \in (0,\infty)$, there exist $$w \in \bH_{p,0}^{\alpha,2}((t_0-R^{2/\alpha}, t_0)\times \bR^d), \quad v \in \bH_{p,0}^{\alpha,2}((S,t_0) \times \bR^d),$$ where $S = \min\{0, t_0 - R^{2/\alpha}\}$, such that $u = w + v$ in $Q_R(t_0,x_0)$, $$\left( |D^2w|^p \right)_{Q_R(t_0,x_0)}^{1/p} \le N \left( |f|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p}+N\gamma_0^{1/(p\nu)}\left( |D^2 u|^{p\mu} \right)_{Q_{2R}(t_0,x_0)}^{1/(p\mu)},$$ and $$\begin{gathered} \left( |D^2v|^{p_1} \right)_{Q_{R/2}(t_0,x_0)}^{1/p_1} \leq N \left( |f|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p}+N\gamma_0^{1/(p\nu)}\left( |D^2 u|^{p\mu} \right)_{Q_{2R}(t_0,x_0)}^{1/(p\mu)} \\ + N \sum_{k=0}^\infty 2^{-k\alpha + 2} \left( \dashint_{\!t_0 - (2^{k+1}+1)R^{2/\alpha}}^{\,\,\,t_0} \dashint_{B_R(x_0)} |D^2u(s,y)|^p \, dy \, ds \right)^{1/p},\end{gathered}$$ where $N=N(d,\delta, \alpha,p,\mu)$. Denote $$Q:=\left\{ \begin{array}{ll} Q_{2R}(t_0,x_0) & \hbox{when $2R\le R_0$;} \\ (t_0 - (2R_0)^{2/\alpha}, t_0)\times B_{R_0}(x_1) & \hbox{otherwise.} \end{array} \right.$$ Note that in both cases $|Q|\le |Q_{2R}(t_0,x_0)|$. Thus, by Assumption \[assump2.2\] and Remark \[rem2.3\], we can find $\bar a^{ij}=\bar a^{ij}(t)$ such that $$\label{eq8.09} \sup_{i,j}\dashint_{Q_{2R}(t_0,x_0)}|a^{ij}-\bar a^{ij}(t)|1_Q\,dx\,dt \le \sup_{i,j}\dashint_{Q}|a^{ij}-\bar a^{ij}(t)|\,dx\,dt\le 2\gamma_0,$$ where $1_Q$ is the indicator function of $Q$. We then rewrite into $$-\partial_t^\alpha u + \bar a^{ij}(t)D_{ij} u = \tilde f :=f+(\bar a^{ij}(t)-a^{ij})D_{ij} u.$$ Now that Theorem \[thm0412\_1\] holds for this equation with the same $p$, it follows from Proposition \[prop0406\_1\] that there exist $$w, \, v \in \bH_p^{\alpha,2}((t_0-R^{2/\alpha}, t_0)\times \bR^d)$$ such that $u = w + v$ in $Q_R(t_0,x_0)$, and – hold with $\tilde f$ in place of $f$. To conclude the proof, it remains to notice that by Hölder’s inequality and , $$\begin{aligned} &\left( |\tilde f|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p} \le \left( |f|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p}+ \left( |(\bar a^{ij}(t)-a^{ij})D_{ij} u|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p}\\ &\le \left( |f|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p}+ N\left( |(\bar a-a)1_{Q}|^{p\nu} \right)_{Q_{2R}(t_0,x_0)}^{1/(p\nu)} \left( |D^2 u|^{p\mu} \right)_{Q_{2R}(t_0,x_0)}^{1/(p\mu)}\\ &\le N \left( |f|^p \right)_{Q_{2R}(t_0,x_0)}^{1/p}+N\gamma_0^{1/(p\nu)}\left( |D^2 u|^{p\mu} \right)_{Q_{2R}(t_0,x_0)}^{1/(p\mu)}.\end{aligned}$$ Now we define $\cA(s)$ as in , but instead of we define $$\begin{gathered} \cB(s) = \big\{ (t,x) \in (-\infty,T) \times \bR^d: \gamma^{-1/p}\left( \cM |f|^p (t,x) \right)^{1/p} \\ +\gamma^{-1/p}\gamma_0^{1/(p\nu)}\left( \cM |D^2 u|^{p\mu} (t,x) \right)^{1/(p\mu)} + \gamma^{-1/p_1}\left( \cS\cM |D^2 u|^p(t,x)\right)^{1/p} > s \big\}.\end{gathered}$$ By following the proof of Lemma \[lem0409\_1\] with minor modifications, from Proposition \[prop0515\_1\], we get the following lemma. \[lem6.3\] Let $p\in (1,\infty)$, $\alpha,\gamma_0,\gamma \in (0,1)$, $T \in (0,\infty)$, $R \in (0,\infty)$, $\mu\in (1,\infty)$, $\nu=\mu/(\mu-1)$, and $a^{ij} = a^{ij}(t,x)$ satisfying Assumption \[assump2.2\] ($\gamma_0$). Assume that $u \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$ vanishes for $x\notin B_{R_0}(x_1)$ for some $x_1\in \bR^d$, and satisfies in $(0,T) \times \bR^d$. Then, there exists a constant $\kappa = \kappa(d,\delta,\alpha,p,\mu) > 1$ such that the following hold: for $(t_0,x_0) \in (-\infty,T] \times \bR^d$ and $s>0$, if $$|\cC_{R/4}(t_0,x_0) \cap \cA(\kappa s)| \geq \gamma |\cC_{R/4}(t_0,x_0)|,$$ then we have $$\hat\cC_{R/4}(t_0,x_0) \subset \cB(s).$$ Finally, we give the proof of Theorem \[main\_thm\]. As before we may assume that $u \in C_0^\infty\left([0,T] \times \bR^d\right)$ with $u(0,x) = 0$ and prove the a priori estimate . We divide the proof into three steps. [*Step 1.*]{} We assume that $u$ vanishes for $x\notin B_{R_0}(x_1)$ for some $x_1\in \bR^d$, and $b\equiv c\equiv 0$. We take $p_0\in (1,p)$ and $\mu\in (1,\infty)$ depending only on $p$ such that $p_0<p_0\mu<p<p_1$, where $p_1=p_1(d,\alpha,p_0)$ is taken from Proposition \[prop0515\_1\]. By Lemmas \[lem6.3\] and \[lem0409\_2\], we have , which together with and the Hardy-Littlewood maximal function theorem implies that [$$\begin{aligned} &\|D^2u\|_{L_p(\bR^d_T)}^p \leq N p \kappa^p \gamma \int_0^\infty |\cB(s)| s^{p-1} \, ds\\ &\le N\gamma \int_0^\infty\left|\left\{ (t,x) \in (-\infty,T) \times \bR^d:\gamma^{-\frac 1 {p_1}}\left( \cS\cM |D^2 u|^{p_0}(t,x)\right)^{\frac 1 {p_0}} > s/3 \right\}\right| s^{p-1} \, ds\\ &\quad+ N\gamma \int_0^\infty\left|\left\{ (t,x) \in (-\infty,T) \times \bR^d:\gamma^{-\frac 1 {p_0}}\left( \cM |f|^{p_0} (t,x) \right)^{\frac 1 {p_0}} > s/3 \right\}\right| s^{p-1} \, ds\\ &\quad+ N\gamma \int_0^\infty\left|\left\{ (t,x) \in (-\infty,T) \times \bR^d:\gamma^{-\frac 1 {p_0}}\gamma_0^{\frac 1 {p_0\nu}} \left( \cM |D^2 u|^{p_0\mu} (t,x) \right)^{\frac 1 {p_0\mu}} > s/3 \right\}\right| s^{p-1} \, ds\\ &\leq N (\gamma^{1-p/p_1}+\gamma^{1-p/p_0}\gamma_0^{p/(p_0\nu)}) \|D^2u\|^p_{L_p(\bR^d_T)} + N \gamma^{1-p/p_0} \|f\|^p_{L_p(\bR^d_T)},\end{aligned}$$]{} where $N = N(d,\delta,\alpha,p)$. Now choose $\gamma \in (0,1)$ sufficiently small and then $\gamma_0$ sufficiently small, depending only on $d$, $\delta$, $\alpha$, and $p$, so that $$N (\gamma^{1-p/p_1}+\gamma^{1-p/p_0}\gamma_0^{p/(p_0\nu)}) < 1/2.$$ Then we have $$\|D^2u\|_{L_p(\bR^d_T)} \leq N(d,\delta,\alpha,p) \|f\|_{L_p(\bR^d_T)}.$$ From this and the equation, we arrive at . [*Step 2.*]{} In this step, we show that under the assumptions of the theorem with $\gamma_0$ being the constant from the previous step, we have $$\label{eq8.59} \|\partial_t^\alpha u\|_{L_p(\bR^d_T)} + \|D^2 u\|_{L_p(\bR^d_T)} \leq N \|f\|_{L_p(\bR^d_T)}+N_1\|u\|_{L_p(\bR^d_T)},$$ where $N=N(d,\delta,\alpha,p)$ and $N_1=N_1(d,\delta,\alpha,p,R_0)$. By moving the lower-order terms to the right-hand side of the equation, and using interpolation inequalities, without loss of generality, we may assume that $b\equiv c\equiv 0$. Now follows a standard partition of unity argument with respect to $x$ and interpolation inequalities. [*Step 3.*]{} In this step, we show how to get rid of the second term on the right-hand side of and conclude the proof of . By and Lemma \[lem1110\_1\], we can find $q\in (p,\infty)$, depending on $\alpha$ and $p$, such that for any $T'\in (0,T]$, $$\begin{aligned} \|u\|_{L_p\left(\bR^d; L_q(0,T')\right)} &\le N(\alpha,p,T)\|\partial^t_\alpha u\|_{L_p((0,T');L_p(\bR^d))}\\ &\le N\|f\|_{L_p(\bR^d_{T'})}+N_1\|u\|_{L_p(\bR^d_{T'})},\end{aligned}$$ where $N=N(d,\delta,\alpha,p,T)$ and $N_1=N_1(d,\delta,\alpha,p,T,R_0)$. Next we take a sufficiently large integer $m=m(d,\delta,\alpha,p,T,R_0)$ such that $N_1(T/m)^{1/p-1/q}\le 1/2$. Then for any $j=0,2,\ldots,m-1$, by Hölder’s inequality and the above inequality with $T'=(j+1)T/m$, we have $$\begin{aligned} &\|u\|_{L_p((jT/m,(j+1)T/m);L_p(\bR^d))} \le (T/m)^{1/p-1/q}\|u\|_{L_p\left(\bR^d;L_q(jT/m,(j+1)T/m)\right)}\\ &\le N\|f\|_{L_p(\bR^d_T)}+\frac 1 2\|u\|_{L_p((0,(j+1)T/m);L_p(\bR^d))}.\end{aligned}$$ This implies that $$\|u\|_{L_p((jT/m,(j+1)T/m);L_p(\bR^d))} \le N\|f\|_{L_p(\bR^d_T)}+\|u\|_{L_p((0,jT/m);L_p(\bR^d))}.$$ By an induction on $j$, we obtain $$\|u\|_{L_p(\bR^d_T)}\le N\|f\|_{L_p(\bR^d_T)},$$ which together with yields . The theorem is proved. Acknowledgment {#acknowledgment .unnumbered} ============== The authors would like to thank Nicolai V. Krylov for telling us a simple proof of , and the referee for helpful comments. The authors also thank Kyeong-hun Kim for bringing our attention to the problems discussed in this paper. Sobolev embeddings for $\bH_{p,0}^{\alpha,2}$ and a “crawling of ink spots” lemma ================================================================================= In the proof of Lemma \[lem0123\_1\] as well as in several places of this paper, we use the following properties of the operator $I^\alpha$. In the sequel, let $T\in (0,\infty)$ be a constant. \[lem1115\_1\] Let $p \in (1,\infty)$, $q \in (1,\infty)$, and $\alpha \in (0, 1/p)$ satisfy $$q > p, \quad \alpha - 1/p = - 1/q.$$ Then we have $$\|I^\alpha \psi\|_{L_q(0,T)} \le N(\alpha,p)\|\psi\|_{L_p(0,T)}$$ for $\psi \in L_p(0,T)$. See [@MR1544927 Theorem 4]. \[lem1018\_01\] Let $\alpha \in (0,1)$, $\psi \in L_p(0,T)$, and $p \in [1,\infty]$ and $q \in [1,\infty]$ satisfy $$\alpha - 1/p > - 1/q.$$ Then we have $$\|I^\alpha \psi\|_{L_q(0,T)} \le N(\alpha,p,q)T^{\alpha-1/p+1/q} \|\psi\|_{L_p(0,T)}.$$ First, consider $p=1$. In this case, $q \in [1,1/(1-\alpha))$. Then $$\begin{aligned} &\Gamma(\alpha)|I^\alpha \psi(t)| \le \int_0^t (t-s)^{\alpha-1} |\psi(s)| \, ds = \int_0^t (t-s)^{\alpha-1} |\psi(s)|^{\frac{1}{q}} |\psi(s)|^{\frac{q-1}{q}} \, ds\\ &\le \left(\int_0^t (t-s)^{(\alpha-1)q}|\psi(s)| \, ds\right)^{\frac{1}{q}} \left( \int_0^t |\psi(s)| \, ds \right)^{\frac{q-1}{q}}.\end{aligned}$$ Thus, $$\begin{aligned} &\| I^{\alpha} \psi(t)\|_{L_q(0,T)} \leq N(\alpha) \|\psi\|_{L_1(0,T)}^{1-\frac{1}{q}} \left(\int_0^T \int_0^t (t-s)^{(\alpha-1)q}|\psi(s)| \, ds \, dt\right)^{\frac{1}{q}}\\ &\leq N(\alpha,q) T^{\alpha -1 + 1/q} \|\psi\|_{L_1(0,T)},\end{aligned}$$ where we used the condition that $(\alpha-1)q > -1$. If $p \in (1,\infty]$, $q \in [1,\infty)$, and $\alpha - 1/p > -1/q$, then one can find $p_1 \in (1,p]$, $q_1 \in [q,\infty)$ such that $\alpha - 1/p_1 = -1/q_1$. The result then follows from Lemma \[lem1115\_1\] and Hölder’s inequality. Finally, if $$p \in (1,\infty], \quad q = \infty, \quad \alpha - 1/p > 0,$$ then $$\begin{aligned} &\Gamma(\alpha)|I^\alpha \psi(t)| \le \int_0^t (t-s)^{\alpha-1} |\psi(s)| \, ds\\ &\le \left(\int_0^t (t-s)^{(\alpha-1)\frac{p}{p-1}} \, ds\right)^{\frac{p-1}{p}} \left( \int_0^t |\psi(s)|^p \, ds\right)^{\frac{1}{p}}\\ &\le T^{\frac{\alpha p - 1}{p}} \left(\frac{p-1}{\alpha p - 1}\right)^{1-1/p} \|\psi\|_{L_p(0,T)},\end{aligned}$$ where we again use the condition that $(\alpha-1)p/(p-1) > -1$. The lemma is proved. \[rem0120\_1\] From Lemma \[lem1018\_01\], if $u = u(t,x) \in L_p(\Omega_T)$, $1\leq p \leq \infty$, then $I^\alpha u \in L_p(\Omega_T)$ and $$\|I^\alpha u \|_{L_p(\Omega_T)} \leq N(\alpha, p) T^\alpha \|u\|_{L_p(\Omega_T)}.$$ \[lem1115\_3\] Let $\psi \in C^1([0,T])$ and $\psi(0) = 0$. Then $$I^\alpha D_t^\alpha \psi = I^\alpha \partial_t^\alpha \psi = \psi.$$ Since $\psi(0) = 0$, we have $$D_t^\alpha \psi(t) = \partial_t^\alpha \psi(t) = \frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha} \psi'(s) \, ds.$$ Then $$\begin{aligned} &I^\alpha D_t^\alpha \psi (t) = \frac{1}{\Gamma(\alpha)}\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{\alpha-1} \int_0^s (s-r)^{-\alpha}\psi'(r) \, dr \, ds\\ &=\frac{1}{\Gamma(\alpha)}\frac{1}{\Gamma(1-\alpha)} \int_0^t \psi'(r) \int_r^t (t-s)^{\alpha-1}(s-r)^{-\alpha} \, ds \, dr = \int_0^t \psi'(r) \, dr = \psi(t).\end{aligned}$$ \[lem1115\_2\] Let $p, q \in (1,\infty)$ and $\alpha \in (0,1/p)$ satisfy $$\alpha - 1/p = - 1/q.$$ Then $$\|\psi\|_{L_q(0,T)} \le N(\alpha,p)\|\partial_t^\alpha \psi\|_{L_p(0,T)}$$ for $\psi \in C^1([0,T])$ such that $\psi(0) = 0$. Using Lemmas \[lem1115\_3\] and \[lem1115\_1\], we obtain that $$\|\psi\|_{L_q(0,T)} = \|I^\alpha \partial_t^\alpha \psi\|_{L_q(0,T)} \le N(\alpha,p)\|\partial_t^\alpha \psi\|_{L_p(0,T)}.$$ \[lem1110\_1\] Let $\psi \in C^1([0,T])$ such that $\psi(0) = 0$. Then $$\|\psi\|_{L_q(0,T)} \le N(\alpha,p,q)T^{\alpha-1/p+1/q}\|\partial_t^\alpha \psi\|_{L_p(0,T)},$$ where $p \in [1,\infty]$, $q \in [1,\infty]$, and $$\alpha - 1/p > - 1/q.$$ We have $$\|\psi\|_{L_q(0,T)} = \|I^\alpha \partial_t^\alpha \psi\|_{L_q(0,T)} \le N(\alpha,p,q)T^{\alpha-1/p+1/q} \|\partial_t^\alpha \psi\|_{L_p(0,T)},$$ where the second inequality is due to Lemma \[lem1018\_01\]. \[lem1115\_4\] Let $p, q, r \in (1,\infty)$ and $\alpha \in (0,1/p)$. Let $\psi \in C^1([0,T])$ such that $\psi(0) = 0$. Then $$\|\psi\|_{L_q(0,T)} \le N(\alpha,p,\theta) \|\partial_t^\alpha \psi\|_{L_p(0,T)}^{\theta} \|\psi\|_{L_r(0,T)}^{1-\theta}$$ for all $\theta \in [0,1]$ satisfying $$\label{eq1115_02} 1/q = \left( 1/p - \alpha \right) \theta + (1-\theta)/r.$$ By Lemma \[lem1115\_2\], we can clearly assume that $\theta \in (0,1)$. Under the conditions $\alpha < 1/p$ and , we see that $$\frac{(1-\theta)q}{r} < 1.$$ Note that by Hölder’s inequality, $$\|\psi\|_{L_q(0,T)} \leq \left(\int_0^T |\psi|^r \, dt \right)^{\frac{(1-\theta)}{r}}\left(\int_0^T |\psi|^{\theta q A'} \, dt \right)^{\frac{1}{qA'}},$$ where $A'$ satisfies $$\frac{(1-\theta)q}{r} + \frac{1}{A'}=1.$$ Hence, by Lemma \[lem1115\_2\] and the fact that $$\alpha < 1/p, \quad \theta q A' > 1, \quad \alpha - \frac{1}{p} = - \frac{1}{\theta q A'},$$ it follows $$\|\psi\|_{L_q(0,T)} \leq \|\psi\|_{L_r(0,T)}^{(1-\theta)} \| \psi\|_{L_{\theta q A'}(0,T)}^{\theta} \leq N(\alpha, p)^{\theta} \|\psi\|_{L_r(0,T)}^{(1-\theta)} \|\partial_t^\alpha \psi\|_{L_p(0,T)}^{\theta}.$$ The lemma is proved. \[thm1204\_1\] Let $\alpha \in (0, 1)$ and $p, q \in (1,\infty)$ satisfy $$p < \min\{1/\alpha, d/2\}, \quad p < q < q^* := \frac{1/\alpha + d/2}{1/(\alpha p) + d/(2p) - 1}.$$ Then $$\label{eq0213_01} \|\psi\|_{L_q(\bR^d_T)} \le N \|D_x^2 \psi\|_{L_p(\bR^d_T)}^{\theta} \|\partial_t^\alpha \psi \|_{L_p(\bR^d_T)}^{\tau(1-\theta)} \|\psi\|_{L_p(\bR^d_T)}^{(1-\tau)(1-\theta)}$$ for $\psi \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$, where $$\theta = \frac{d}{2}\left(\frac{1}{p} - \frac{1}{q}\right) \in (0,1), \quad \tau = \frac{2}{\alpha d} \frac{\theta}{1-\theta} \in (0,1),$$ and $N = N(d,\alpha,p,q)$, but independent of $T$. If $q = q^*$, then $$\label{eq0213_02} \|\psi\|_{L_q(\bR^d_T)} \le N \|D_x^2 \psi\|_{L_p(\bR^d_T)}^{\alpha d/(2+\alpha d)} \|\partial_t^\alpha \psi \|_{L_p(\bR^d_T)}^{2/(2+\alpha d)}.$$ By the definition of $\bH_{p,0}^{\alpha,2}(\bR^d_T)$, we may assume that $\psi \in C_0^\infty\left([0,T] \times \bR^d \right)$ and $\psi(0,x) = 0$. By the Sobolev embedding in $x$, we have $$\label{eq11.55} \|\psi\|_{L_p((0,T);L_{pd/(d-2p)}(\bR^d))} \le N\|D_x^2 \psi\|_{L_p(\bR^d_T)}.$$ Similarly, by Lemma \[lem1115\_4\] with $\theta = 1$, we have $$\|\psi\|_{L_p(\bR^d;L_{p/(1-\alpha p)}((0,T)))} \le N\|\partial_t^\alpha \psi\|_{L_p(\bR^d_T)},$$ which together with the Minkowski inequality implies that $$\label{eq11.56} \|\psi\|_{L_{p/(1-\alpha p)}((0,T);L_p(\bR^d))} = \left\| \int_{\bR^d} |\psi(\cdot,x)|^p \, dx \right\|_{L_{\frac{1}{1-\alpha p}}(0,T)}^{\frac{1}{p}} \le N\|\partial_t^\alpha \psi\|_{L_p(\bR^d_T)}.$$ By , , and Hölder’s inequality, we immediately get . Finally, follows from and Hölder’s inequality. From Theorem \[thm1204\_1\] the following corollary follows easily. \[cor1211\_1\] Let $\alpha \in (0, 1)$ and $p, q \in (1,\infty)$ satisfy $$p < \min\{1/\alpha,d/2\}, \quad p<q \leq q^* := \frac{1/\alpha + d/2}{1/(\alpha p) + d/(2p) - 1}.$$ Then we have $$\|\psi\|_{L_q\left((0,T) \times B_1\right)} \le N \|\psi\|_{\bH_p^{\alpha,2}\left((0,T) \times B_1\right)}$$ for any $\psi \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_1\right)$, where $N = N(d,\alpha,p,q)$, but independent of $T$. If $p \leq d/2$ and $p \leq 1/\alpha$, then the same estimate holds for $q\in [1,q^*)$ with $N$ depending also on $T$. If $p < d/2$ and $p < 1/\alpha$, the result follows easily from Theorem \[thm1204\_1\] with an extension of $\psi$ to a function in $\bH_{p,0}^{\alpha,2}(\bR^d_T)$. If $p = d/2$ or $p = 1/\alpha$, then find $\varepsilon > 0$ such that $$q < \frac{1/\alpha + d/2}{1/(\alpha (p-\varepsilon)) + d/(2(p-\varepsilon)) - 1} < \frac{1/\alpha + d/2}{1/(\alpha p) + d/(2p) - 1}.$$ Then $$\|\psi\|_{L_q\left((0,T) \times B_1\right)} \leq N \|\psi\|_{\bH_{p-\varepsilon}^{\alpha,2}\left((0,T) \times B_1\right)} \leq N \|\psi\|_{\bH_p^{\alpha,2}\left((0,T) \times B_1\right)}.$$ The corollary is proved. \[thm1207\_2\] Let $\alpha \in (0, 1)$ and $p, q \in (1,\infty)$ satisfy $$\frac{d}{2} < p < \frac{1}{\alpha}, \quad p < q \leq p(\alpha p + 1).$$ Then, for $\psi \in \bH_{p,0}^{\alpha, 2}\left((0,T) \times B_1\right)$, we have $$\label{eq0213_04} \|\psi\|_{L_q((0,T)\times B_1)} \le N \left( \sum_{0 \leq |\beta| \leq 2} \|D_x^\beta \psi\|_{L_p((0,T)\times B_1)} \right)^{1-\theta} \|\partial_t^\alpha\psi\|_{L_p((0,T)\times B_1)}^\theta,$$ where $N = N(d,\alpha,p,q)$, but independent of $T$, and $$\theta = \frac{1}{\alpha} \left( \frac{1}{p} - \frac{1}{q} \right) \in (0,1).$$ If $d/2 < p \leq 1/\alpha$, then the same estimate holds for $q$ satisfying $$1 \leq q < p(\alpha p + 1)$$ with $N$ depending also on $T$. As above, we assume that $\psi \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_1\right) \cap C^\infty\big([0,T] \times B_1\big)$ and $\psi(0,x) = 0$. Since $p>d/2$, by the Sobolev embedding in $x$, we have $$\label{eq12.01} \|\psi\|_{L_p((0,T);L_{\infty}(B_1))} \le N\left( \sum_{0 \leq |\beta| \leq 2} \|D_x^\beta \psi\|_{L_p((0,T)\times B_1)} \right).$$ Similarly, by Lemma \[lem1115\_4\] with $\theta = 1$ and the Minkowski inequality, we have $$\begin{gathered} \label{eq11.56b} \|\psi\|_{L_{p/(1-\alpha p)}((0,T);L_p(B_1))} = \left\| \int_{B_1} |\psi(\cdot,x)|^p \, dx \right\|_{L_{1/(1-\alpha p)}((0,T))}^{1/p} \\ \leq \|\psi\|_{L_p(B_1; L_{p/(1-\alpha p)}((0,T)))} \le N\|\partial_t^\alpha \psi\|_{L_p((0,T)\times B_1)}.\end{gathered}$$ By , , and Hölder’s inequality, we immediately get with $q=p(\alpha p+1)$ and $\theta=1/(\alpha p+1)$. The general case then follows from Hölder’s inequality. Let $\alpha \in (0,1)$ and $p, q \in (1,\infty)$ such that $$\frac{1}{\alpha} < p < \frac{d}{2}, \quad p < q \leq p + \frac{2p^2}{d}.$$ Then, for $\psi \in \bH_{p,0}^{\alpha,2}(\bR^d_T)$, $$\|\psi\|_{L_q(\bR^d_T)} \le N T^{\alpha\left(1-\frac{p}{q}\right)-\frac{1}{p}+\frac{1}{q}} \|\partial_t^\alpha \psi\|_{L_p(\bR^d_T)}^{1-p/q} \|D_x^2 \psi\|_{L_p(\bR^d_T)}^{\theta p/q} \|\psi\|_{L_p(\bR^d_T)}^{(1-\theta)p/q},$$ where $N=N(d,\alpha,p,q)$ and $ \theta = d(q-p)/(2p^2) \in (0,1]$. As above, we assume that $\psi \in C_0^\infty\left([0,T] \times \bR^d \right)$ and $\psi(0,x) = 0$. Since $\alpha>1/p$, by Lemma \[lem1110\_1\] and the Minkowski inequality, we have $$\label{eq12.14} \|\psi\|_{L_{\infty}((0,T);L_p(\bR^d_T))} \le \|\psi\|_{L_p(\bR^d;L_{\infty}((0,T)))}\le NT^{\alpha-1/p}\|\partial_t^\alpha \psi\|_{L_p(\bR^d_T)}.$$ By the Sobolev embedding in $x$, we have $$\label{eq12.19} \|\psi\|_{L_p((0,T);L_{dp/(d-2p)}(\bR^{d}))}\le N\|D_x^2 \psi\|_{L_p(\bR^d_T)}.$$ By , , and Hölder’s inequality, we get the desired estimate with $q=p+2p^2/d$. The general case then follows from Hölder’s inequality. By extending $\psi \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_1 \right)$ to a function in $\bH_{p,0}^{\alpha,2}(\bR^d_T)$ and using the above theorem, we get \[Embedding with $\alpha$-time derivative and $2$-spatial derivatives with $1/\alpha < p < d/2$\] \[cor0225\_1\] Let $\alpha \in (0,1)$ and $p, q \in (1,\infty)$ such that $$\frac{1}{\alpha} < p < \frac{d}{2}, \quad p < q \leq p + \frac{2p^2}{d}.$$ Then, for $\psi \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_1\right)$, $$\|\psi\|_{L_q\left((0,T) \times B_1\right)} \le N T^{\alpha\left(1-\frac{p}{q}\right)-\frac{1}{p}+\frac{1}{q}} \|\psi\|_{\bH_p^{\alpha,2}\left((0,T) \times B_1\right)},$$ where $N=N(d,\alpha,p,q)$. If $1/\alpha < p \leq d/2$, the same estimate holds for $q$ satisfying $$1 \leq q < p + 2p^2/d$$ with $N$ depending also on $T$. \[thm0214\_1\] Let $\alpha \in (0,1)$ and $p, q \in (1,\infty)$ such that $$\max\{1/\alpha ,d/2\} < p \leq d/2 + 1/\alpha, \quad p < q \le 2p.$$ Then, for $\psi \in \bH_{p,0}^{\alpha,2}\left((0,T) \times B_1\right)$, $$\begin{aligned} &\|\psi\|_{L_q\left((0,T) \times B_1\right)} \\ &\le N T^{\frac{\alpha p}{q} - \frac{1}{p} + \frac{1}{q}} \left( \sum_{0 \leq |\beta|\leq 2}\|D_x^\beta \psi\|_{L_p\left((0,T) \times B_1\right)}\right)^{1-\theta} \|\partial_t^\alpha \psi\|_{L_p\left((0,T) \times B_1\right)}^\theta,\end{aligned}$$ where $N=N(d,\alpha,p,q)$ and $ \theta = p/q \in (0,1)$. Again we assume that $\psi \in \bH_{p,0}^{\alpha,2}\left((0,T)\times B_1\right) \cap C^\infty\left([0,T] \times B_1 \right)$ and $\psi(0,x) = 0$. We set $q':=p^2/(2p-q)\in (p,\infty]$. Since $\alpha - 1/p > 0$, from Lemma \[lem1110\_1\] and the Minkowski inequality, $$\|\psi\|_{L_{q'}((0,T);L_p(B_1))} \le \|\psi\|_{L_p(B_1;L_{q'}((0,T)))} \le N T^{\alpha - 1/p + 1/q'} \|\partial_t^\alpha \psi\|_{L_p((0,T)\times B_1)},$$ where $N = N(\alpha,p,q')$. This, , and Hölder’s inequality yield the desired inequality. \[lem0217\_2\] Let $p \in (1, \infty]$, $\alpha > 1/p$, and $\psi \in C^1([0,T])$ with $\psi(0) = 0$. Then $$|\psi(t_2) - \psi(t_1)| \leq N(\alpha,p) (t_2 - t_1)^{\alpha-1/p} \|\partial_t^\alpha\psi\|_{L_p(0,T)}$$ for $0 \leq t_1 < t_2 \leq T$. Note that $$\psi(t_2) - \psi(t_1) = (I^\alpha \partial_t^\alpha \psi)(t_2) - (I^\alpha \partial_t^\alpha \psi)(t_1).$$ Set $\partial_t^\alpha \psi(t) = \Psi(t)$. Then $$\begin{aligned} &\Gamma(\alpha) \left(\psi(t_2) - \psi(t_1)\right) = \int_0^{t_2} (t_2-s)^{\alpha-1} \Psi(s) \, ds - \int_0^{t_1} (t_1 - s)^{\alpha-1}\Psi(s) \, ds\\ &= \int_0^{t_1} (t_2-s)^{\alpha-1} \Psi(s) \, ds - \int_0^{t_1} (t_1 - s)^{\alpha-1}\Psi(s) \, ds + \int_{t_1}^{t_2} (t_2-s)^{\alpha-1} \Psi(s) \, ds\\ &:= J_1 + J_2 + J_3.\end{aligned}$$ Note that $$\begin{aligned} &J_1 + J_2 = \int_0^{t_1} \left( (t_2-s)^{\alpha-1} - (t_1-s)^{\alpha-1} \right) \Psi(s) \, ds\\ &\leq \left(\int_0^{t_1} \left| (t_2-s)^{\alpha-1} - (t_1-s)^{\alpha-1} \right|^{p/(p-1)} \, ds\right)^{(p-1)/p} \left(\int_0^{t_1} |\Psi(s)|^p \, ds\right)^{1/p},\end{aligned}$$ where $$\begin{aligned} &\int_0^{t_1} \left| (t_2-s)^{\alpha-1} - (t_1-s)^{\alpha-1} \right|^{p/(p-1)} \, ds\\ &= \int_0^{t_1} \left[ (t_1-s)^{\alpha-1} - (t_2-s)^{\alpha-1} \right]^{p/(p-1)} \, ds =: K_1.\end{aligned}$$ If $2 t_1 \leq t_2$, since $(\alpha-1)\frac{p}{p-1}>-1$, it follows that $$K_1 \leq \int_0^{t_1} (t_1-s)^{(\alpha-1)\frac{p}{p-1}} \, ds \leq N(\alpha, p)t_1^{(\alpha-1)\frac{p}{p-1}+1} \leq N(\alpha,p)(t_2 - t_1)^{(\alpha-1)\frac{p}{p-1}+1}.$$ If $2t_1 > t_2$, that is, $2t_1 - t_2 >0$, then $$\begin{aligned} &K_1 = \int_0^{2t_1-t_2} \left[ (t_1-s)^{\alpha-1} - (t_2-s)^{\alpha-1} \right]^{p/(p-1)} \, ds\\ &\qquad + \int_{2t_1-t_2}^{t_1} \left[ (t_1-s)^{\alpha-1} - (t_2-s)^{\alpha-1} \right]^{p/(p-1)} \, ds\\ &\leq (1-\alpha)^{\frac{p}{p-1}} (t_2-t_1)^{\frac{p}{p-1}}\int_0^{2t_1-t_2} (t_1-s)^{(\alpha-2)\frac{p}{p-1}} \, ds + \int_{2t_1-t_2}^{t_1} (t_1-s)^{(\alpha-1)\frac{p}{p-1}} \, ds\\ &= N (t_2-t_1)^{\frac{p}{p-1}} \left[ (t_2-t_1)^{(\alpha-2)\frac{p}{p-1} + 1} - t_1^{(\alpha-2)\frac{p}{p-1} + 1}\right] + N (t_2-t_1)^{(\alpha-1)\frac{p}{p-1} + 1}\\ &\leq N(\alpha,p) (t_2-t_1)^{(\alpha-1)\frac{p}{p-1} + 1},\end{aligned}$$ where $N = N(\alpha,p)$ and we used the fact that $$(\alpha-2)\frac{p}{p-1} + 1 < 0.$$ Hence, $$J_1 + J_2 \leq N(\alpha,p) (t_2-t_1)^{\alpha - 1/p} \|\Psi\|_{L_p(0,T)}.$$ For the term $I_3$, we see that $$J_3 \leq \left(\int_{t_1}^{t_2} (t_2-s)^{(\alpha-1)\frac{p}{p-1}} \,ds\right)^{\frac{p-1}{p}} \|\Psi\|_{L_p(0,T)} \leq N(\alpha,p) (t_2-t_1)^{\alpha-1/p}\|\Psi\|_{L_p(0,T)}.$$ Therefore, $$\begin{aligned} &|\psi(t_2,x)-\psi(t_1,x)| \leq N(\alpha,p) (t_2-t_1)^{\alpha-1/p}\|\Psi\|_{L_p(0,T)}\\ &= N(\alpha,p) (t_2-t_1)^{\alpha-1/p}\|\partial_t^\alpha \psi\|_{L_p(0,T)}.\end{aligned}$$ Recall that $$Q_R(t,x) =Q_{R,R}(t,x) = (t-R^{2/\alpha}, t) \times B_R(x).$$ For the Hölder semi-norm, we denote $$[u]_{C^{\sigma_1, \sigma_2}(\cD)} = \sup_{\substack{(t,x),(s,y) \in \cD \\ (t,x) \neq (s,y)}}\frac{|u(t,x) - u(s,y)|}{|t-s|^{\sigma_1} + |x-y|^{\sigma_2}},$$ where $\cD \subset \bR \times \bR^d$. \[lem0225\_1\] Let $\alpha \in (0,1)$ and $p \in (1,\infty)$ such that $$\sigma := 2-(d+2/\alpha)/p \in (0,1).$$ Assume that $\psi \in C^\infty\big(\overline{(0,1) \times B_1}\big)$ and $\psi(0,x) = 0$. For any $\varepsilon \in (0,1/2)$ and $$(t,x), (t,y) \in (0,1) \times B_1,$$ we have $$\begin{gathered} \label{eq0226_01} |\psi(t,x)-\psi(t,y)| \leq (2^{\sigma-1} + 3 \varepsilon^\sigma) |x-y|^\sigma K \\ + N \varepsilon^{-2/(\alpha p) - d/p + 1/p} |x-y|^\sigma \|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)},\end{gathered}$$ provided that either $B_h(x) \subset B_1$ or $B_h(y) \subset B_1$, $h := |x-y|$, where $N=N(d,\alpha,p)$ and $$\label{eq0225_02} K = \sup_{\substack{(t,x),(s,y) \in (0,1)\times B_1 \\ (t,x) \neq (s,y)}}\frac{|\psi(t,x) - \psi(s,y)|}{|t-s|^{\sigma \alpha/2} + |x-y|^\sigma}.$$ Without loss of generality, we assume that $B_h(x) \subset B_1$. Due to an appropriate orthogonal transformation, we assume that $$x=(x_1,x'), \quad y = (x_1-h,x').$$ Since $B_h(x) \subset B_1$, we have $$(x_1+h,x') \in \overline{B_1}.$$ For any $\varepsilon \in (0,1/2)$, set $$\rho = \varepsilon h.$$ We write $$\begin{aligned} &\psi(t,x_1,x')-\psi(t,x_1-h,x') = \frac{1}{2}\left[ \psi(t,x_1 + h,x') - \psi(t,x_1-h,x') \right]\\ &\qquad - \frac{1}{2} \left[ \psi(t,x_1+h,x') - 2 \psi(t,x_1,x') + \psi(t,x_1-h,x') \right].\end{aligned}$$ Thus, $$\begin{gathered} \label{eq0225_03} |\psi(t,x)-\psi(t,y)| \leq \frac{1}{2} (2h)^\sigma K \\ + \frac{1}{2} \left|\psi(t,x_1+h,x') - 2 \psi(t,x_1,x') + \psi(t,x_1-h,x')\right|.\end{gathered}$$ To estimate the last term in the above inequalities, we observe that $$\begin{aligned} \label{eq0225_04} &\left|\psi(t,x_1+h,x') - 2 \psi(t,x_1,x') + \psi(t,x_1-h,x')\right|\nonumber \\ &\leq |\psi(t,x_1+h-\rho,x')-2\psi(t,x_1,x')+\psi(t,x_1-h+\rho,x')|\nonumber \\ &\quad + |\psi(t,x_1+h,x') - \psi(t,x_1+h-\rho,x')| + |\psi(t,x_1-h,x') - \psi(t,x_1-h+\rho,x')|\nonumber \\ &\leq 2 K \rho^\sigma + |\psi(t,x_1+h-\rho,x')-2\psi(t,x_1,x')+\psi(t,x_1-h+\rho,x')|\nonumber \\ &:= 2K \rho^\sigma + J.\end{aligned}$$ We consider $$\big((t-\rho^{2/\alpha}, t+\rho^{2/\alpha}) \cap (0,1)\big) \times B_\rho'(x') \subset \bR \times \bR^{d-1},$$ where $$B_\rho'(x') := \{y' \in \bR^{d-1}: |y'-x'|< \rho\}.$$ We see that $(x_1-h+\rho,z'), (x_1,z'), (x_1+h-\rho,z') \in B_1$ if $z \in B_\rho'(x')$. Moreover, $$[x_1-h + \rho, x_1+h-\rho] \times B_\rho'(x') \subset B_1$$ because if $$(z_1,z') \in [x_1-h + \rho, x_1+h-\rho] \times B_\rho'(x'),$$ then $$|x_1-z_1| \leq h-\rho, \quad |x'-z'| < \rho,$$ and $$\begin{aligned} |(z_1,z')| &\le |(z_1,z') - (x_1,x')| + |x| \leq |x_1-z_1|+|x'-z'| + |x|\\ &< h-\rho + \rho + |x| \leq 1,\end{aligned}$$ where, for the last inequality, we used the assumption that $B_h(x) \subset B_1$. For $$(s,z') \in \big((t-\rho^{2/\alpha}, t+\rho^{2/\alpha}) \cap (0,1)\big) \times B_\rho'(x'),$$ we write $$\begin{aligned} \label{eq0225_05} J &\leq \left|\psi(s,x_1 + h - \rho,z') - 2 \psi(s,x_1,z') + \psi(s,x_1 - h+\rho, z')\right|\nonumber \\ &\quad + \left|\psi(t,x_1 +h-\rho,x') - \psi(s,x_1 + h-\rho,z')\right|\nonumber \\ &\quad + 2 \left|\psi(s,x_1,z') - \psi(t,x_1,x')\right|\nonumber \\ &\quad + \left|\psi(t,x_1 - h + \rho, x') - \psi(s,x_1 - h + \rho, z')\right|\nonumber \\ &\leq 4 K \rho^\sigma + \left|\psi(s,x_1 + h-\rho,z') - 2 \psi(s,x_1,z') + \psi(s,x_1 - h+\rho, z')\right|,\end{aligned}$$ where $$\begin{aligned} &\psi(s,x_1 + h-\rho,z') - 2 \psi(s,x_1,z') + \psi(s,x_1 - h+\rho, z')\\ &= \int_{x_1}^{x_1+h-\rho} \int_{2x_1-r}^r D_1^2 \psi(s,z_1,z') \, d z_1 \, dr.\end{aligned}$$ Hence, from this along with , , and , we obtain that $$\begin{gathered} \label{eq0225_01} |\psi(t,x)-\psi(t,y)| \leq \frac{1}{2} (2h)^\sigma K + K \rho^\sigma + 2 K \rho^\sigma \\ + \frac{1}{2} \int_{x_1}^{x_1+h-\rho}\int_{2x_1-r}^r |D_1^2\psi(s,z_1,z')|\, dz_1 \, dr\end{gathered}$$ for any $(s,z')$ satisfying $$\big((t-\rho^{2/\alpha}, t+\rho^{2/\alpha}) \cap (0,1)\big) \times B_\rho'(x') =: \cD.$$ By taking the average of both sides of over the domain $\cD$ with respect to $(s,z')$ along with Hölder’s inequality (note that $h-\rho > h/2$), we finally arrive at . \[lem0225\_2\] Under the assumptions of Lemma \[lem0225\_1\], for any $\varepsilon$ satisfying $$\label{eq0225_06} 0 < \varepsilon < (1-2^{\sigma-1})^{1/\sigma},$$ we have $$K_1 \leq \frac{2^{1+\sigma}}{1-2^{\sigma-1}-\varepsilon^\sigma} M + \frac{2\varepsilon^\sigma}{1-2^{\sigma-1}-\varepsilon^\sigma}K + N \frac{\varepsilon^{-2/(\alpha p) - d/p + 1/p}}{1-2^{\sigma-1} - \varepsilon^\sigma}\|D^2_x \psi\|_{L_p\left((0,1) \times B_1 \right)},$$ where $N = N(d,\alpha,p)$, $$M = \sup_{(t,x) \in (0,1) \times B_1} |\psi(t,x)|,$$ $$K_1 = \sup_{\substack{(t,x),(t,y) \in (0,1)\times B_1 \\ (t,x) \neq (t,y), y = \theta x, \theta \in \bR}}\frac{|\psi(t,x) - \psi(t,y)|}{|x-y|^\sigma},$$ and $K$ is defined as in . The quantity $K_1$ is the Hölder semi-norm of $\psi$ when $x$ and $y$ are on the same line passing through the origin. Thanks to an appropriate transformation, to estimate $K_1$, it is enough to estimate $$\frac{|\psi(t,x_1,0) - \psi(t,y_1,0)|}{|x_1-y_1|^\sigma}$$ for $x_1, y_1 \in (-1,1)$. For $x_1, y_1 \in (-1,1)$ such that $h:=|x_1 - y_1| \geq 1/2$, we see that $$\label{eq0224_06} \frac{|\psi(t,x_1,0)-\psi(t,y_1,0)|}{|x_1-y_1|^\sigma} \leq 2^{1+\sigma} M.$$ When $h < 1/2$, either $2x_1-y_1$ or $2y_1-x_1$ is in $(-1,1)$. Without loss of generality we assume that $$y_1 = x_1 - h,\quad x_1+h\in (-1,1).$$ Set $$\rho = \varepsilon h,$$ where $\varepsilon$ is a number satisfying . Since $$\begin{aligned} &\psi(t,x_1,0)-\psi(t,x_1-h,0) = \frac{1}{2}\left[ \psi(t,x_1 + h,0) - \psi(t,x_1-h,0) \right]\\ &\quad - \frac{1}{2} \left[ \psi(t,x_1+h,0) - 2 \psi(t,x_1,0) + \psi(t,x_1-h,0) \right],\end{aligned}$$ we have $$\begin{gathered} \label{eq0224_02} |\psi(t,x_1,0)-\psi(t,y_1,0)| \leq \frac{1}{2} (2h)^\sigma K_1 \\ + \frac{1}{2} \left|\psi(t,x_1+h,0) - 2 \psi(t,x_1,0) + \psi(t,x_1-h,0)\right|.\end{gathered}$$ To estimate the last term in the above inequalities, we observe that $$\begin{aligned} \label{eq0224_03} &\left|\psi(t,x_1+h,0) - 2 \psi(t,x_1,0) + \psi(t,x_1-h,0)\right|\nonumber \\ &\leq |\psi(t,x_1+h-\rho,0)-2\psi(t,x_1,0)+\psi(t,x_1-h+\rho,0)|\nonumber \\ &\quad + |\psi(t,x_1+h,0) - \psi(t,x_1+h-\rho,0)| + |\psi(t,x_1-h,0) - \psi(t,x_1-h+\rho,0)|\nonumber \\ &\leq 2 K_1 \rho^\sigma + |\psi(t,x_1+h-\rho,0)-2\psi(t,x_1,0)+\psi(t,x_1-h+\rho,0)|.\end{aligned}$$ We note that $(x_1-h+\rho,z'), (x_1,z'), (x_1+h-\rho,z') \in B_1$. Moreover, $$[x_1-h + \rho, x_1+h-\rho] \times B_\rho'(0) \subset B_1,$$ where $B_\rho'(0) = \{y' \in \bR^{d-1}: |y'|< \rho\}$. To estimate the last term in , for $$(s,z') \in \big((t-\rho^{2/\alpha}, t+\rho^{2/\alpha}) \cap (0,1)\big) \times B_\rho'(0),$$ we write $$\begin{aligned} \label{eq0224_04} &|\psi(t,x_1+h-\rho,0)-2\psi(t,x_1,0)+\psi(t,x_1-h+\rho,0)|\nonumber \\ &\leq \left|\psi(s,x_1 + h - \rho,z') - 2 \psi(s,x_1,z') + \psi(s,x_1 - h+\rho, z')\right|\nonumber \\ &\qquad+ \left|\psi(t,x_1 +h-\rho,0) - \psi(s,x_1 + h-\rho,z')\right|+ 2 \left|\psi(s,x_1,z') - \psi(t,x_1,0)\right|\nonumber \\ &\qquad + \left|\psi(t,x_1 - h + \rho, 0) - \psi(s,x_1 - h + \rho, z')\right|\nonumber \\ &\leq 4 K \rho^\sigma + \left|\psi(s,x_1 + h-\rho,z') - 2 \psi(s,x_1,z') + \psi(s,x_1 - h+\rho, z')\right|.\end{aligned}$$ Note that $$\begin{aligned} &\psi(s,x_1 + h-\rho,z') - 2 \psi(s,x_1,z') + \psi(s,x_1 - h+\rho, z')\\ &= \int_{x_1}^{x_1+h-\rho} \int_{2x_1-r}^r D_1^2 u(s,z_1,z') \, d z_1 \, dr.\end{aligned}$$ Hence, from this along with , , and , we obtain that $$\begin{gathered} \label{eq0224_05} |\psi(t,x_1,0)-\psi(t,y_1,0)| \leq \frac{1}{2} (2h)^\sigma K_1 + K_1 \rho^\sigma + 2 K \rho^\sigma \\ + \frac{1}{2} \int_{x_1}^{x_1+h-\rho}\int_{2x_1-r}^r |D_1^2u(s,z_1,z')|\, dz_1 \, dr\end{gathered}$$ for any $(s,z')$ satisfying $$\big((t-\rho^{2/\alpha}, t+\rho^{2/\alpha}) \cap (0,1)\big) \times B_\rho'(0) =: \cD.$$ By taking the average of both sides of over the domain $\cD$ with respect to $(s,z')$ along with Hölder’s inequality (note that $h-\rho > h/2$), we arrive at $$\begin{aligned} &|\psi(t,x_1,0)-\psi(t,y_1,0)| \leq (2^{\sigma-1} h^\sigma + \varepsilon^\sigma h^\sigma) K_1 + 2 \varepsilon^\sigma h^\sigma K\\ &\qquad+ N \varepsilon^{-2/(\alpha p) - d/p + 1/p} h^\sigma \|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}\end{aligned}$$ whenever $h = |x_1-h_1| < 1/2$. From this and , we conclude that $$K_1 \leq 2^{1+\sigma}M + (2^{\sigma-1} + \varepsilon^\sigma) K_1 + 2 \varepsilon^\sigma K + N \varepsilon^{-2/(\alpha p) - d/p + 1/p} \|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}$$ for any $\varepsilon$ satisfying , where $N = N(d,\alpha,p)$. This shows that $$\begin{aligned} K_1 &\leq \frac{2^{1+\sigma}}{1-2^{\sigma-1}-\varepsilon^\sigma} M + \frac{2\varepsilon^\sigma}{1-2^{\sigma-1}-\varepsilon^\sigma}K\\ &\quad + N \frac{\varepsilon^{-2/(\alpha p) - d/p + 1/p}}{1-2^{\sigma-1} - \varepsilon^\sigma}\|D^2_x \psi\|_{L_p\left((0,1) \times B_1 \right)}.\end{aligned}$$ The lemma is proved. \[thm5.18\] Let $\alpha \in (0,1)$ and $p \in (1,\infty)$ such that $$\sigma := 2-(d+2/\alpha)/p \in (0,1).$$ Then, for $\bH_{p,0}^{\alpha,2}\left((0,1) \times B_1\right)$, we have $$\label{eq0224_01} [\psi]_{C^{\sigma \alpha/2, \sigma}\left((0,1) \times B_1\right)} \leq N(d,\alpha,p) \|\psi\|_{\bH_p^{\alpha,2}\left((0,1) \times B_1\right)}.$$ By the definition of $\bH_{p,0}^{\alpha,2}\left((0,1) \times B_1\right)$ and Remark \[rem0606\_1\], we may assume that $\psi \in C^\infty_0 \big(\overline{(0,1)\times B_1}\big)$ and $\psi(0,x) = 0$. To prove , we take $(t_1,x), (t_2,y) \in (0,1) \times B_1$, $(t_1,x) \neq (t_2,y)$, and set $$\rho = \varepsilon \left( |t_1-t_2|^{\alpha/2} + |x-y| \right),$$ where $\varepsilon \in (0,1)$ is to be specified below. We write $$|\psi(t_1,x) - \psi(t_2,y)| \leq |\psi(t_1,x) - \psi(t_2,x)| + |\psi(t_2,x) - \psi(t_2,y)| := J_1 + J_2.$$ To estimate $J_1$, for $z \in B_\rho(x) \cap B_1$, we have $$\begin{aligned} J_1 &\leq |\psi(t_1,x)-\psi(t_1,z)| + |\psi(t_1,z) - \psi(t_2,z)| + |\psi(t_2,z) -\psi(t_2,x)|\\ &\leq 2 K \rho^\sigma + |\psi(t_1,z) - \psi(t_2,z)|,\end{aligned}$$ where by Lemma \[lem0217\_2\] we see that $$|\psi(t_1,z) - \psi(t_2,z)| \le N(\alpha,p)|t_1-t_2|^{\alpha-1/p} \|\partial_t^\alpha \psi(\cdot,z)\|_{L_p(0,1)}.$$ Then by taking the average of $J_1$ over $B_\rho(x) \cap B_1$ with respect to $z$ along with Hölder’s inequality (note that $|B_\rho(x) \cap B_1| \ge N(d) |B_\rho(x)|$), we get $$\begin{aligned} J_1 &\leq 2K \rho^\sigma + N |t_1-t_2|^{\alpha-1/p} \rho^{-d/p} \| \partial_t^\alpha \psi\|_{L_p\left((0,1) \times B_1\right)}\\ &\leq 2 K \rho^\sigma + N \varepsilon^{-2+2/(\alpha p)} \rho^\sigma \| \partial_t^\alpha \psi\|_{L_p\left((0,1) \times B_1\right)},\end{aligned}$$ where $N = N(d,\alpha,p)$. We now estimate $J_2$. First, recall the definitions of $M$, $K$, and $K_1$ from Lemmas \[lem0225\_1\] and \[lem0225\_2\]. If $|x-y| \ge 1/8$, we have $$\frac{|\psi(t_2,x)-\psi(t_2,y)|}{|x-y|^\sigma} \leq 2 \cdot 8^\sigma M.$$ Assume that $|x-y| =: h < 1/8$. If $B_h(x) \subset B_1$ or $B_h(y) \subset B_1$, by Lemma \[lem0225\_1\] we have $$\frac{|\psi(t,x)-\psi(t,y)|}{|x-y|^\sigma} \leq (2^{\sigma-1} + 3 \varepsilon_1^\sigma) K + N \varepsilon_1^{-2/(\alpha p) - d/p + 1/p} \|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}$$ for any $\varepsilon_1 \in (0,1/2)$. Now we consider the case that $x, y \in B_1$, $h:=|x-y| < 1/8$, and $$B_h(x) \not\subset B_1 \quad \text{and} \quad B_h(y) \not\subset B_1.$$ Without loss of generality, we assume that $|y| \ge |x|$. Then we see that $$|y| \ge 7/8, \quad |x| \geq 7/8, \quad |y| - h > 0.$$ Set $$\tilde{y} = \frac{|y|-h}{|y|}y, \quad \tilde{x} = \frac{|y|-h}{|y|}x.$$ Then $$|y - \tilde{y}| = h, \quad |x - \tilde{x}| = h |x|/|y| \leq h,$$ $$|\tilde{x}-\tilde{y}| = (1-h/|y|)h =: \tilde{h} < h.$$ Moreover, $$B_{\tilde{h}}(\tilde{y}) \subset B_1$$ because, for any $z \in B_{\tilde{h}}(\tilde{y})$, $$|z| \leq |z-\tilde{y}| + |\tilde{y}| < \tilde{h} + |y| - h < 1.$$ We observe that $$\begin{aligned} &|h|^{-\sigma} J_2 = \frac{|\psi(t_2,x) - \psi(t_2,y)|}{|h|^\sigma}\\ &\leq \frac{|\psi(t_2,x) - \psi(t_2,\tilde{x})|}{|h|^\sigma} + \frac{|\psi(t_2,\tilde{x}) - \psi(t_2,\tilde{y})|}{|h|^\sigma} + \frac{|\psi(t_2,\tilde{y}) - \psi(t_2,y)|}{|h|^\sigma}\\ &\leq \frac{|\psi(t_2,x) - \psi(t_2,\tilde{x})|}{|x-\tilde{x}|^\sigma} + \frac{|\psi(t_2,\tilde{x}) - \psi(t_2,\tilde{y})|}{|\tilde{h}|^\sigma} + \frac{|\psi(t_2,\tilde{y}) - \psi(t_2,y)|}{|h|^\sigma}\\ &=: J_{2,1} + J_{2,2} + J_{2,3},\end{aligned}$$ where we note that $x$ and $\tilde{x}$ are on the same line passing through the origin, so do $y$ and $\tilde{y}$. Thus, by Lemma \[lem0225\_2\], $$\begin{aligned} &J_{2,1} + J_{2,3}\\ &\leq \frac{2^{2+\sigma}}{1-2^{\sigma-1}-\varepsilon_2^\sigma} M + \frac{4\varepsilon_2^\sigma}{1-2^{\sigma-1}-\varepsilon_2^\sigma}K + N \frac{\varepsilon_2^{-2/(\alpha p) - d/p + 1/p}}{1-2^{\sigma-1} - \varepsilon_2^\sigma}\|D^2_x \psi\|_{L_p\left((0,1) \times B_1 \right)}\end{aligned}$$ for any $\varepsilon_2$ satisfying . For $J_{2,2}$, since $B_{\tilde{h}}(\tilde{y}) \subset B_1$, by Lemma \[lem0225\_1\], we obtain that $$J_{2,2} \leq (2^{\sigma-1} + 3 \varepsilon_3^\sigma) K + N \varepsilon_3^{-2/(\alpha p) - d/p + 1/p} \|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}$$ for any $\varepsilon_3 \in (0,1/2)$. We collect the estimates for $J_1$ and $J_2$ along with those for $J_{2,1}$, $J_{2,2}$, and $J_{2,3}$ as follows. Set $$J = \frac{|\psi(t_1,x) - \psi(t_2,y)|}{|t_1-t_2|^{\sigma \alpha/2} + |x-y|^\sigma}.$$ If $|x-y| \geq 1/8$, then $$J \leq 2K \varepsilon^\sigma + N \varepsilon^{-d/p} \|\partial_t^\alpha \psi\|_{L_p\left((0,1) \times B_1\right)} + 2 \cdot 8^\sigma M$$ for $\varepsilon \in (0,1)$. If $h := |x-y| < 1/8$ and $B_h(x) \subset B_1$ or $B_h(y) \subset B_1$, then $$\begin{aligned} &J \leq 2K \varepsilon^\sigma + N \varepsilon^{-d/p} \|\partial_t^\alpha \psi\|_{L_p\left((0,1) \times B_1\right)}\\ &\quad + (2^{\sigma-1} + 3 \varepsilon_1^\sigma) K + N \varepsilon_1^{-2/(\alpha p) - d/p + 1/p} \|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}\end{aligned}$$ for $\varepsilon \in (0,1)$ and $\varepsilon_1 \in (0,1/2)$. If $h=|x-y| < 1/8$, and $B_h(x) \not\subset B_1$ and $B_h(y) \not\subset B_1$, then $$\begin{aligned} J &\leq 2K \varepsilon^\sigma + N \varepsilon^{-d/p} \|\partial_t^\alpha \psi\|_{L_p\left((0,1) \times B_1\right)}\\ &\,\, + \frac{2^{2+\sigma}}{1-2^{\sigma-1}-\varepsilon_2^\sigma} M + \frac{4\varepsilon_2^\sigma}{1-2^{\sigma-1}-\varepsilon_2^\sigma}K + N \frac{\varepsilon_2^{-2/(\alpha p) - d/p + 1/p}}{1-2^{\sigma-1} - \varepsilon_2^\sigma}\|D^2_x \psi\|_{L_p\left((0,1) \times B_1 \right)}\\ &\,\,+ (2^{\sigma-1} + 3 \varepsilon_3^\sigma) K + N \varepsilon_3^{-2/(\alpha p) - d/p + 1/p} \|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}\end{aligned}$$ for $\varepsilon \in (0,1)$, $\varepsilon_2$ satisfying , and $\varepsilon_3 \in (0,1/2)$. The above three inequalities show that $$\begin{aligned} J &\leq 2K \varepsilon^\sigma + N \varepsilon^{-d/p} \|\partial_t^\alpha \psi\|_{L_p\left((0,1) \times B_1\right)} + 2 \cdot 8^\sigma M\\ &\,\, + 2^{\sigma-1}K + 3 (\varepsilon_1^\sigma + \varepsilon_3^\sigma)K\\ &\,\, + N (\varepsilon_1^{-2/(\alpha p) - d/p + 1/p} + \varepsilon_3^{-2/(\alpha p) - d/p + 1/p})\|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}\\ &\,\, + \frac{2^{2+\sigma}}{1-2^{\sigma-1}-\varepsilon_2^\sigma} M + \frac{4\varepsilon_2^\sigma}{1-2^{\sigma-1}-\varepsilon_2^\sigma}K + N \frac{\varepsilon_2^{-2/(\alpha p) - d/p + 1/p}}{1-2^{\sigma-1} - \varepsilon_2^\sigma}\|D^2_x \psi\|_{L_p\left((0,1) \times B_1 \right)}\end{aligned}$$ for any $\varepsilon \in (0,1)$, $\varepsilon_1, \varepsilon_3 \in (0,1/2)$, and $\varepsilon_2$ satisfying , where it is crucial that there is only one term $2^{\sigma-1}K$ on the right-hand side of the inequality. Since $(t_1,x)$ and $(t_2,y)$ are arbitrary points in $(0,1) \times B_1$, we see that $$\begin{aligned} K &\leq 2K \varepsilon^\sigma + N \varepsilon^{-d/p} \|\partial_t^\alpha \psi\|_{L_p\left((0,1) \times B_1\right)} + 2 \cdot 8^\sigma M\\ &\,\, + 2^{\sigma-1}K + 3 (\varepsilon_1^\sigma + \varepsilon_3^\sigma)K\\ &\,\, + N (\varepsilon_1^{-2/(\alpha p) - d/p + 1/p} + \varepsilon_3^{-2/(\alpha p) - d/p + 1/p})\|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}\\ &\,\, + \frac{2^{2+\sigma}}{1-2^{\sigma-1}-\varepsilon_2^\sigma} M + \frac{4\varepsilon_2^\sigma}{1-2^{\sigma-1}-\varepsilon_2^\sigma}K + N \frac{\varepsilon_2^{-2/(\alpha p) - d/p + 1/p}}{1-2^{\sigma-1} - \varepsilon_2^\sigma}\|D^2_x \psi\|_{L_p\left((0,1) \times B_1 \right)}.\end{aligned}$$ Upon using the fact that $2^{\sigma-1} < 1$, we fix $\varepsilon$, $\varepsilon_1$, $\varepsilon_2$, and $\varepsilon_3$ small enough depending on $d$, $\alpha$, and $p$ so that $$1-2^{\sigma-1} - 2\varepsilon^\sigma - 3(\varepsilon_1^\sigma+\varepsilon_3^\sigma) - \frac{4 \varepsilon_2^\sigma}{1-2^{\sigma-1}-\varepsilon_2^\sigma} > 0.$$ Then $$K \leq N M + N \|\partial_t^\alpha \psi\|_{L_p\left((0,1) \times B_1\right)} + N \|D_x^2\psi \|_{L_p\left((0,1) \times B_1 \right)}.$$ Finally, we observe that, by interpolation inequalities, for any $\varepsilon_4 > 0$, $$\label{eq2.36} M \leq \varepsilon_4 K + N(d,\alpha,p,\varepsilon_4) \|\psi\|_{L_p\left((0,1) \times B_1\right)}.$$ The theorem is proved. Let $\alpha \in (0,1)$ and $p \in (1,\infty)$ such that $$\sigma := 2-(d+2/\alpha)/p \in (0,1).$$ Then, for $\psi \in \bH_{p,0}^{\alpha,2}\left((0,1) \times \bR^d\right)$, we have $$[\psi]_{C^{\sigma \alpha/2, \sigma}\left((0,1) \times \bR^d\right)} \leq N(d,\alpha,p) \|\psi\|_{\bH_p^{\alpha,2}\left((0,1) \times \bR^d\right)}.$$ As in the proof Theorem \[thm1204\_1\], we assume that $\psi \in C^\infty_0 \left([0,1] \times \bR^d\right)$ and $\psi(0,x) = 0$. Consider $$\frac{|\psi(t,x) - \psi(s,y)|}{|t-s|^{\sigma \alpha/2} + |x-y|^\sigma}$$ for two different points $(t,x),(s,y) \in (0,1)\times \bR^d$. If $|x-y|<1$, then we apply Theorem \[thm5.18\] with a shift of the coordinates. If $|x-y|>1$, then the above quantity is bounded by $2\|\psi\|_{L_\infty((0,1)\times \bR^d)}$, and it suffices to use the interpolation inequality . The following is a version of the “crawling of ink spots” lemma to be used in the proofs of the main results of this paper. Note that the underlying set $(-\infty,T) \times \bR^d$ is unbounded. Recall the definitions of $\cC_R(t,x)$ and $\hat{\cC}_R(t,x)$ in . \[lem0409\_2\] Let $\gamma \in (0,1)$ and $|E| < \infty$. Suppose that $E \subset F \subset (-\infty,T) \times \bR^d$ and, for any $(t,x) \in (-\infty,T] \times \bR^d$ and for all $R \in (0,\infty)$ with $$\left| \cC_R(t,x) \cap E \right| \ge \gamma |\cC_R(t,x)|,$$ we have $$\hat\cC_R(t,x) \subset F.$$ Then $$\label{eq0409_03} |E| \leq N(d,\alpha) \gamma |F|.$$ For every $(t,x) \in E$, define $$\varphi_{(t,x)}(r) := \frac{|E \cap \cC_r(t,x)|}{|\cC_r(t,x)|} \leq \frac{|E|}{|\cC_r(t,x)|} \to 0$$ as $r \to \infty$. On the other hand, by the Lebesgue differentiation theorem, for almost every $(t,x) \in E$, $$\lim_{r \to 0} \varphi_{(t,x)}(r) = 1.$$ Moreover, $\varphi_{(t,x)}(r)$ is continuous on $(0,\infty)$. Since $\gamma \in (0,1)$, for almost every $(t,x) \in E$, there exits $r \in (0,\infty)$ such that $$\varphi_{(t,x)}(r) = \gamma.$$ Then we set $$R = R(t,x) = \sup\{ r \in (0,\infty): \varphi_{(t,x)}(r) = \gamma\},$$ where we understand that $\inf \emptyset = \infty$. Then $0 < R(t,x) \leq \infty$. Define $$\Gamma_1 = \{\cC = \cC_{R(t,x)}(t,x): (t,x) \in E, \,\, R(t,x) < \infty\}$$ and $$R^*_1 = \sup \{R(t,x): \cC_{R(t,x)}(t,x) \in \Gamma_1\}.$$ Note that $$E \setminus N \subset \bigcup_{\cC_{R(t,x)}(t,x) \in \Gamma_1} \cC_{R(t,x)}(t,x),$$ where $N$ is a null set. If $R^*_1 = \infty$, then $\Gamma_1$ contains a sequence of $\cC_{R_k}(t_k,x_k) := \cC_{R(t_k,x_k)}(t_k,x_k)$ with $$|\cC_{R_k}(t_k,x_k)| \to \infty$$ as $k \to \infty$. In this case, choose $k_1 \in \bN$ such that $$|\cC_{R_{k_1}}(t_{k_1},x_{k_1})| \geq 2 \gamma^{-1}|E|.$$ Since $$|\cC_{R_{k_1}}(t_{k_1},x_{k_1}) \cap E | = \gamma |\cC_{R_{k_1}}(t_{k_1},x_{k_1})|,$$ by the assumption in the lemma, we have $$\hat\cC_{R_{k_1}}(t_{k_1},x_{k_1}) \subset F.$$ It then follows that $$|E| \leq \frac{\gamma}{2} |\cC_{R_{k_1}}(t_{k_1},x_{k_1})| \leq \gamma |\hat\cC_{R_{k_1}}(t_{k_1},x_{k_1})| \leq \gamma |F|.$$ Hence, we obtain . If $R^*_1 < \infty$, we find a countable sub-collection $\Gamma_0$ of $\Gamma_1$ as follows. Choose $\cC_{R_1}(t_1,x_1):=\cC_{R(t_1,x_1)}(t_1,x_1)$ from $\Gamma_1$ such that $R_1 > R^*_1/2$. Now spit $\Gamma_1 = \Gamma_2 \cup \Gamma_2'$, where $\Gamma_2$ consists of those $\cC_{R(t,x)}(t,x)$ disjoint from $\cC_{R_1}(t_1,x_1)$, and $\Gamma_2'$ of those which intersect $\cC_{R_1}(t_1,x_1)$. Now we note that $$\cC_{R(t,x)}(t,x) \subset \cC_{5R_1}(t_1,x_1),$$ whenever $\cC_{R(t,x)}(t,x) \in \Gamma_2'$. Now assume that $\cC_{R_k}(t_k,x_k)$ and $\Gamma_{k+1}$ are chosen. If $\Gamma_{k+1}$ is empty, the process ends. If not, we choose $\cC_{R_{k+1}}(t_{k+1}, x_{k+1})$ from $\Gamma_{k+1}$ such that $$R_{k+1} > \frac{1}{2} R^*_{k+1}, \quad R^*_{k+1}:= \sup_{\cC_{R(t,x)}(t,x) \in \Gamma_{k+1}} R(t,x).$$ Then split $\Gamma_{k+1} = \Gamma_{k+2} \cup \Gamma_{k+2}'$, where $\Gamma_{k+2}$ consists of those $\cC_{R(t,x)}(t,x)$ disjoint from $\cC_{R_{k+1}}(t_{k+1}, x_{k+1})$, and $\Gamma_{k+2}'$ of those which intersect $\cC_{R_{k+1}}(t_{k+1}, x_{k+1})$. Now we set $$\Gamma_0 = \{\cC_{R_k}(t_k,x_k) : k = 1, 2, \ldots \}.$$ Clearly, we have $\cC_{R_k}(t_k,x_k) \cap \cC_{R_j}(t_j,x_j) = \emptyset$ if $k \neq j$. Now we prove when $R^*_1 < \infty$. First, consider the case that $\Gamma_0$ contains only finitely many elements or $\Gamma_0$ has infinitely many elements with $R_k^* \searrow 0$. Then $$\label{eq0409_05} \Gamma_1 = \bigcup_{k=2}^\infty \Gamma_k'.$$ In particular, when $R_k^* \searrow 0$, if there exits $\cC_{R(t,x)}(t,x) \in \Gamma_1$ such that $$\cC_{R(t,x)}(t,x) \not\in \bigcup_{k=2}^\infty \Gamma_k',$$ then $\cC_{R(t,x)}(t,x) \in \Gamma_k$ for all $k=1,2,\ldots$. This means that $R(t,x) = 0$, which is a contradiction because $R(t,x) > 0$. From and the fact that $$\cC_{R(t,x)}(t,x) \subset \cC_{5 R_k(t_k,x_k)}$$ for any $\cC_{R(t,x)}(t,x) \in \Gamma_{k+1}'$ and $k = 1,2,\ldots$, we have $$\label{eq0409_04} E \setminus N \subset \bigcup_{\cC_{R(t,x)}(t,x) \in \Gamma_1} \cC_{R(t,x)}(t,x) \subset \bigcup_{k=1}^\infty \cC_{5R_k}(t_k,x_k),$$ where $N$ is a null set. Note that, for each $k = 1,2,\ldots$, $$|\cC_{R_k}(t_k,x_k) \cap E| = \gamma|\cC_{R_k}(t_k,x_k)|,$$ $$|\cC_{5R_k}(t_k,x_k) \cap E| < \gamma|\cC_{5R_k}(t_k,x_k)|.$$ Hence, by the assumption, $$\hat\cC_{R_k}(t_k,x_k) \subset F,$$ and by the disjointness of $\cC_{R_k}(t_k,x_k)$ and we have $$\begin{aligned} &|E| \leq | \bigcup_{k=1}^\infty E \cap \cC_{5R_k}(t_k,x_k)| \leq \sum_{k=1}^\infty |E \cap \cC_{5R_k}(t_k,x_k)|\\ &\leq \gamma \sum_{k=1}^\infty |\cC_{5R_k}(t_k,x_k)| = 5^{d + 2/\alpha} \gamma \sum_{k=1}^\infty |\cC_{R_k}(t_k,x_k)| = 5^{d + 2/\alpha} \gamma \left|\bigcup_{k=1}^\infty \cC_{R_k}(t_k,x_k)\right|\\ &\leq 5^{d+2/\alpha} 2 \gamma \left| \bigcup_{k=1}^\infty \hat\cC_{R_k}(t_k,x_k)\right| \leq N(d,\alpha) \gamma |F|.\end{aligned}$$ Thus we obtain . Now assume that there exists a number $\varepsilon_0 > 0$ such that $R_k^* \ge \varepsilon_0$ for all $k = 1,2, \ldots$. This means that $$\left|\bigcup_{k=1}^M \cC_{R_k}(t_k,x_k)\right| = \sum_{k=1}^M |\cC_{R_k}(t_k,x_k)| \to \infty$$ as $M \to \infty$. Then we find $M$ such that $$\left|\bigcup_{k=1}^M \cC_{R_k}(t_k,x_k)\right| \geq 2 \gamma^{-1}|E|.$$ Since $\hat\cC_{R_k}(t_k,x_k) \subset F$, we have $$|E| \leq \frac{\gamma}{2} \sum_{k=1}^M |\cC_{R_k}(t_k,x_k)| \leq \gamma \left| \bigcup_{k=1}^M \hat\cC_{R_k}(t_k,x_k) \right| \leq \gamma |F|.$$ Thus we again arrive at . [10]{} Mark Allen, Luis Caffarelli, and Alexis Vasseur. A parabolic problem with a fractional time derivative. , 221(2):603–630, 2016. L. A. Caffarelli and I. Peral. On [$W^{1,p}$]{} estimates for elliptic equations in divergence form. , 51(1):1–21, 1998. Hongjie Dong and Doyoon Kim. On [$L_p$]{}-estimates for elliptic and parabolic equations with [$A_p$]{} weights. 370(7):5081–5130, 2018. Hongjie Dong and Doyoon Kim. On the [$L_p$]{}-solvability of higher order parabolic and elliptic systems with [BMO]{} coefficients. , 199(3):889–941, 2011. G. H. Hardy and J. E. Littlewood. Some properties of fractional integrals. [I]{}. , 27(1):565–606, 1928. Ildoo Kim, Kyeong-Hun Kim, and Sungbin Lim. An [$L_q(L_p)$]{}-theory for the time fractional evolution equations with variable coefficients. , 306:123–176, 2017. N. V. Krylov. Parabolic and elliptic equations with [VMO]{} coefficients. , 32(1-3):453–475, 2007. N. V. Krylov. Parabolic equations with [VMO]{} coefficients in [S]{}obolev spaces with mixed norms. , 250(2):521–558, 2007. N. V. Krylov. On parabolic equations in one space dimension. , 41(4):644–664, 2016. N. V. Krylov and M. V. Safonov. A property of the solutions of parabolic equations with measurable coefficients. , 44(1):161–175, 239, 1980. Norman G. Meyers and James Serrin. . , 51:1055–1056, 1964. M. V. Safonov. Harnack’s inequality for elliptic equations and [H]{}ölder property of their solutions. , 96:272–287, 312, 1980. Boundary value problems of mathematical physics and related questions in the theory of functions, 12. Rico Zacher. Weak solutions of abstract evolutionary integro-differential equations in [H]{}ilbert spaces. , 52(1):1–18, 2009. Rico Zacher. A [D]{}e [G]{}iorgi–[N]{}ash type theorem for time fractional diffusion equations. , 356(1):99–146, 2013. [^1]: H. Dong was partially supported by the NSF under agreement DMS-1600593. [^2]: D. Kim was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B03934369).

--- abstract: 'A starting point in the investigation of intersecting systems of subsets of a finite set is the elementary observation that the size of a family of pairwise intersecting subsets of a finite set $[n]=\{1,\ldots,n\}$, denoted by $2^{[n]}$, is at most $2^{n-1}$, with one of the extremal structures being family comprised of all subsets of $[n]$ containing a fixed element, called as a *star*. A longstanding conjecture of Chvátal aims to generalize this simple observation for all *downsets* of $2^{[n]}$. In this note, we prove this conjecture for all downsets where every subset contains at most $3$ elements.' author: - | Eva Czabarka[^1]\ Glenn Hurlbert[^2] [^3]\ Vikram Kamat [^4] title: 'Chvátal’s conjecture for downsets of small rank' --- Introduction ============ Let $[n]=\{1,\ldots,n\}$ and let $2^{[n]}$ (resp. $\binom{[n]}{k}$) denote the family of all subsets (resp. $r$-sized subsets) of $[n]$. A set system containing sets of size $r$ ($r\geq 1$) is called $r$-*uniform*. Additionally, let $\binom{[n]}{\leq r}$ be the family of all subsets of size at most $r$, for any $1\leq r\leq n$. For a family of subsets $\cF\sse 2^{[n]}$, call $\cF$ a *downset* if $A\in \cF$ and $B\sse A$ implies $B\in \cF$. Denote by $\cF^r$ those sets of $\cF$ having size $r$. A family $\cF\sse 2^{[n]}$ is called *intersecting* if $A\cap B\neq \mt$ for every $A,B\in \cF$. For any $\cF\sse 2^{[n]}$, let $\cF_x=\{A\in \cF:x\in A\}$, called the $\cF$-star centered at $x$. Call any $\cG\sse \cF_x$ a partial $\cF$-star centered at $x$, and call $x$ a [*center*]{} of such a family. As a family may have more than one center, we call the set of all centers of $\cG$ the [*head*]{} of $\cG$ — it equals the intersection of all the sets of $\cG$.\ A starting point in the study of intersecting set systems states that any intersecting set system on $[n]$ can contain at most $2^{n-1}$ subsets, as for any pair $(A,[n]\setminus A)$, where $A\sse [n]$, at most one can be in the intersecting family (see [@Ande]). It is clear that the *star* is one of the structures that attains this maximum size. The seminal Erdős–Ko–Rado theorem [@ErKoRa] proves a similar, more non-trivial result for *uniform* set systems. \[ekr\] [@ErKoRa] Let $r\leq n/2$ and let $\cF\sse \binom{[n]}{r}$ be intersecting. Then $|\cF|\leq \binom{n-1}{r-1}$. Furthermore, if $r<n/2$, then equality holds if and only if $\cF=\binom{n}{r}_x$, for some $x\in [n]$. In this note, we consider a famous longstanding conjecture of Chvátal (see [@Chva]), which deals with the “Erdős–Ko–Rado” property of downsets. Before we state the conjecture, we formulate the following definitions. For $\cF\sse 2^{[n]}$ we set $\i(\cF)$ to be the size of the largest intersecting subfamily of $\cF$ and $\s(\cF)=\emph{max}_{x\in [n]}|\cF_x|$. A set system $\cF\sse 2^{[n]}$ is *EKR* if $\i(\cF)=\s(\cF)$. Moreover, $\cF$ is *strictly* EKR if all of the largest intersecting subfamilies of $\cF$ are $\cF$-stars. \[chvatal\] [@Chva] If $\cH\sse 2^{[n]}$ is a downset, then $\cH$ is EKR. There have been a handful of results confirming this conjecture. For example, the trivial case $\cH=2^{[n]}$ is mentioned in [@Ande], and Theorem \[ekr\] implies the case for which $\cH=\binom{[n]}{\le k}$. Schonheim [@Scho] solved the case for which the maximal elements of $\cH$ share a common element, while Chvátal [@Chva] handled the case for which the maximal sets of $\cH$ can be partitioned into two sunflowers (see definition below), each with core size 1. In [@Chva] is also found the case for compressed $\cH$; Snevily [@Snev] strengthened this to $\cH$ being merely compressed with respect to some element (which also implies [@Scho]). Miklos [@Mikl] (and later Wang [@Wang]) verified the conjecture for $\cH$ satisfying $\i(\cH)\ge |\cH|/2$, and Stein [@Stei] verified it for those $\cH$ having $m$ maximal sets, every $m-1$ of which form a sunflower. Most recently, Borg [@Borg] solved a weighted generalization of [@Snev].\ In this paper, we prove Conjecture \[chvatal\] for $\cH\sse \binom{[n]}{\leq 3}$. We also prove a slightly weaker result, one that makes an additional assumption on the size of the maximum intersecting family in $\cH$. The advantage of this assumption is that the proof becomes significantly simpler, and the technique, which employs the famous Sunflower Lemma of Erdős and Rado, could potentially be extended for downsets containing larger subsets. Main Results {#main-results .unnumbered} ------------ We verify Conjecture \[chvatal\] for all downsets consisting of sets of size at most $3$. \[completechvatal\] Let $\cH\subseteq \binom{[n]}{\le 3}$ be a downset. Then $\cH$ is EKR. Moreover $\cH$ is strictly EKR, unless one of the following holds. 1. \[case:1\] There is a subset $K\in\binom{[n]}{4}$ such that - $\binom{K}{3}\subseteq \cH$, - for all $H\in\cH $, $H\subseteq K$ or $K\cap H=\emptyset$, and - the largest star in $\cH$ has size $7$. 2. \[case:2\] There are subsets $K\in\binom{[n]}{3}$ and (possibly empty) $M\sse [n]\setminus K$, and a subfamily $\cZ=\binom{K}{2}\cup\{Z\in\binom{K\cup M}{3}\mid |Z\cap K|=2 \}\sse\cH$ such that either - $K\notin\cH$ and the largest star in $\cH$ has size $|\cZ|=3|M|+3$, or - $K\in\cH$ and the largest star in $\cH$ has size $|\cZ|+1=3|M|+4$. We also prove the following weaker result, which is significantly stronger than the result of [@Mikl] for subfamilies of $\binom{[n]}{\leq 3}$. \[bigchvatal\] Let $\cH\sse \binom{[n]}{\leq 3}$ be a downset, and let $\cI\sse \cH$ be a maximum intersecting family. If $|\cI|\geq 31$, then $\cI$ is a star. Hence $\cH$ is EKR when $\i(\cH)\ge 31$. Of course, some intersecting family (in particular, some star) will be so large if $|\cH|>15n$ or $|\cH^3|>10n$, for example.\ Our proofs use the notion of *Sunflowers*, including the famous *Sunflower Lemma* of Erdős and Rado [@ErdRad], as well as a variant by H[å]{}stad, et al [@HaJuPu]. We state both the Sunflower Lemma and the variant below, after the following definitions. A set $S$ is a *covering set* for a set system $\cF$ if $S\cap F\neq \mt$ for every $F\in \cF$. The covering number of $\cF$, denoted by $\tau(\cF)$, is the size of the smallest covering set of $\cF$. \[sunflower\] A *sunflower* with $k$ petals and core $C$ is a set system $\{S_1,\ldots,S_k\}$ such that for any $i\neq j$, $S_i\cap S_j=C$. The sets $S_i\setminus C$ are the petals of the sunflower, and must be non-empty. If $k=1$ then we may choose $C$ to be any proper subset of $S_1$. For a set system $\cF$ and set $Y$, let $\cF_Y=\{F\setminus Y:F\in \cF,Y\sse F\}$. \[flower\] A $k$-flower with core $C$ is a set system $\cF$ with $\tau(\cF_C)\geq k$. \[sflemma\] [@ErdRad] If a family of sets $\cF$ is $r$-uniform and $|\cF|> r!(k-1)^r$ sets, then it contains a sunflower with $k$ petals. We will use the following variant of Theorem \[sflemma\]. \[kflemma\] [@HaJuPu] If $\cF$ is $r$-uniform and $|\cF|>(k-1)^r$, then $\cF$ contains a $k$-flower. Proof of Theorem \[completechvatal\] ==================================== Let $\cI$ be an intersecting subfamily of $\cH$ of maximum size. Our goal is to show that either $\cI$ must be a star or otherwise that $\cH$ contains a star of size equal to that of $\cI$, and to characterize the cases for which the latter happens.\ If $\cH$ does not contain a set of size $3$ then $\cI$ is a star unless $|\cI|=3$ and $\cI=\binom{K}{2}$ for some $K=\{x,y,z\}$. But then $\{\{x\},\{x,y\},\{x,z\}\}\subseteq \cH_x$, and so $|\cI|=|\cH_x|$, which is case (\[case:2\]) of the theorem with $M=\emptyset$.\ Thus we may assume that $\cH$ contains a set of size $3$ and, consequently, also contains a star of size $4$. Therefore $|\cI|\ge 4$. If $\cIo\not=\emptyset$ or $|\cIt|\ge 4$ then $\cI$ is a star and we are done; so we will assume that $\cIo=\emptyset$ and $|\cIt|\le 3$ (thus $\cIr\ne\emptyset$). Our proof splits into cases, based on $|\cIt|$.\ We first introduce some notation that we make use of below. Without loss of generality $\bigcup_{I\in\cI^2}I=[m]$ for some $m\le 4$. For $\emptyset\not=J\subset [m]$ define $\oJ=[m]\setminus J$, $\cA(J)=\{I\in\cIr\mid I\cap [m]=J\}$, and $C(J)=(\bigcup_{A\in\cA(J)}A)\setminus J$. In practice, we relax the notation somewhat to write $\cA(2,3)$ instead of $\cA(\{2,3\})$, and $C(\otwo)$ instead of $C(\overline{\{2\}})$, for example. Note that, when $m=3$, $|C(\oi)|=|\cA(\oi)|$ and $\cIr\setminus\bigcup_{i\in [3]}\cA(\oi)\subseteq\{[3]\}$.\ $|\cIt|=3$ ---------- ### $\cIt$ is a star We may assume that $\cIt=\{\{1,2\},\{1,3\},\{1,4\}\}$. If $\cIr=\cIro$, then $\cI$ is a star. Otherwise, we must have $\cIr\setminus\cIro=\{\{2,3,4\}\}$. Therefore $(\cI\setminus\{\{2,3,4\}\})\cup\{\{1\}\}\cup\{\{1,j\}\mid j\in I\in\cIro\}$ is a star subfamily of $\cH$ that has size at least $|\cI|$ and, in fact, is larger unless $I\subseteq [4]$ for every $I\in\cIro$. Therefore we must have that $|\cIro|\le 3$.\ If $|\cIro|<3$ then, without loss of generality, $\cIro\subseteq\{\{1,2,3\},\{1,2,4\}\}$, and then $(\cI\setminus\{\{1,3\},\{1,4\}\})\cup\{\{2\},\{2,3\},\{2,4\}\}$ is a larger intersecting subfamily of $\cH$, a contradiction. So we are left with the case in which $|\cIro|=3$ and, consequently, $|\cI|=7$ and $\cH\supseteq\binom{[4]}{\le 3}$.\ If there is an $H\in\cH$ such that both $H\cap [4]\not=\emptyset$ and $H\setminus [4]\not=\emptyset$ then, by taking $h\in H\cap [4]$, we have that $\binom{[4]}{\le 3}_h\cup\{H\}$ is a star in $\cH$ of size $8>7$, a contradiction. Hence there is no such $H$, which is case (\[case:1\]) of the theorem.\ ### $\cIt$ is a triangle We may assume that $\cIt= \{\{1,2\},\{1,3\},\{2,3\}\}$.\ Relabel, if necessary, so that $0\le|C(\oone)|\le |C(\otwo)|\le |C(\othree)|$. Then $(\cI\setminus(\cA(\oone)\cup\{\{2,3\}\}))\cup\{\{1,s\}\mid s\in C(\othree)\}\cup\{\{1\}\}$ is a star subfamily of $\cH$ of size $|\cI|+|cA(\othree)|-|\cA(\oone)|\ge|\cI|$, and so $\cH$ is EKR, and strictly so unless $|C(\oone)|=|C(\otwo)|=|C(\othree)|$, which we now assume.\ If not all the sets $C(\oi)$ are the same then, without loss of generality say $C(\oone)\not= C(\otwo)$, and so $|C(\oone)\cup C(\otwo)|>|C(\othree)|$. Then $(\cI\setminus(\cA(\othree)\cup\{\{1,2\}\}))\cup\{\{3,s\}\mid s\in C(\oone)\cup C(\otwo)\}\cup\{\{3\}\}$ is a star subfamily of $\cH$ of size $|\cI|+|A(\oone)\cup A(\otwo)|-|\cA(\othree)|>|\cI|$, a contradiction.\ Finally, if $C(\oone)=C(\otwo)=C(\othree)$ then $|\cH_1|=|\cI|$, so $\cH$ is EKR, but not strictly so, giving us case (\[case:2\]) of the theorem.\ $|\cIt|=2$ ---------- We may assume that $\cIt=\{\{1,2\},\{1,3\}\}$. For each $I\in\cIr$ we must have $1\in I$ or $\{2,3\}\subset I$. If $I\in\cIr\setminus(\cA(1)\cup\cA(2,3))$, then $1\in I$ and $\{2,3\}\cap I\ne\emptyset$.\ If $\cA(2,3)=\emptyset$, then $\cI$ is a star, so we assume that $\cA(2,3)\ne\emptyset$. It must be that $\cA(1)\ne\emptyset$, since otherwise $\cI\cup\{\{2,3\}\}$ would be a larger intersecting subfamily of $\cH$, a contradiction.\ Fix an $A\in\cA(1)$; then for each $k\in C(2,3)$ we must have that $k\in A$. Thus $|C(2,3)|\le|A\setminus\{1\}|=2\le |C(1)|$. Hence we have that $(\cI\setminus\cA(2,3))\cup\{\{1\}\}\cup\{\{1,i\}\mid i\in C(1)\}$ is a star of size $|\cI|+|C(1)|-|\cA(2,3)|+1>|\cI|$, a contradiction.\ $|\cIt|=1$ ---------- We may assume that $\cIt=\{\{1,2\}\}$. Without loss of generality, both of $\cA(1), \cA(2)$ are nonempty (otherwise $\cI$ is a star and we are done). If, for some $i\in\{1,2\}$, we have that $|\cA(i)|\le |\cA(1,2)|$ then $(\cI\cup\{A\setminus \{i\}\mid A\in\cA(1,2)\}\cup\{\{1,2\}\setminus\{i\}\})\setminus\cA(i)$ is a star-subfamily of $\cH$ of size larger than $\cI$, which is a contradiction. Thus we know that $|\cA(1,2)|<\min(|\cA(1)|,|\cA(2)|)$.\ If $|\cA(1)|=|\cA(2)|=1$, then $\cA(1,2)=\emptyset$, and $(\cI\cup\{\{1,j\}:j\in C(1)\}\cup\{\{1\}\})\setminus\cA(2)$ is a star subfamily of size larger that $\cI$, a contradiction., so we may assume without loss of generality that $|\cA(1)|\ge 2$.\ For $i\in\{1,2\}$, set $\cA^\pr(i)=\{A\setminus\{i\}\mid A\in \cA(i)\}$; then $|\cA^\pr(i)|=|\cA(i)|$. Clearly $\cA^\pr(1)$ and $\cA^\pr(2)$ cross-intersect. If, for some $i\in\{1,2\}$, $\cA^\pr(i)$ is an intersecting family then $\cI\cup\cA^\pr(i)\setminus\cA(1,2)$ is an intersecting subfamily of $\cH$ that is larger that $\cI$, a contradiction, so we have that neither $\cA^\pr(1)$ nor $\cA^\pr(2)$ is intersecting. Since $\cA^\pr(1)$ is not intersecting and $|\cA^\pr(1)|\ge 2$, we may assume (by relabeling, if necessary) that $\{\{3,4\},\{5,6\}\}\subseteq\cA^\pr(1)$. Because $\cA^\pr(2)$ cross-intersects $\cA^\pr(1)$ we have $\cA^\pr(2)\subseteq\{\{3,5\},\{3,6\},\{4,5\},\{4,6\}\}$. In particular, $|\cA(2)|=|\cA^\pr(2)|\le 4$ and, for each $x\in\{3,4,5,6\}$, $\{1,x\}$ is a subset of some set in $\cA(1)$. But then $(\cI\setminus\cA(2)) \cup \{\{1,x\}\mid x\in\{3,4,5,6\}\}\cup\{\{1\}\}$ is an intersecting subfamily of $\cH$ that is larger than $\cI$, a contradiction.\ $|\cIt|=0$ ---------- Here $\cIt=\emptyset$ and $\cI$ is an intersecting family of $3$-sets such that no 2-subset of $[n]$ is contained in every element of $\cI$ (otherwise that $2$-subset could be added to $\cI$).\ Let $\cS$ be the largest star in $\cI$ (clearly $|\cS|\ge 2$), and let $D$ be the head of $\cS$. If $\cS=\cI$ then we are done, so define $\cR=\cI\setminus\cS$ and assume that $\cR\ne\emptyset$. In particular, for every $R\in\cR$ we must have that $R\cap D=\emptyset$; otherwise $R$ could be added to $\cS$ to create a larger star. If $|D|\ge 2$ then for any $d\in D$ we have that $\cI\cup\{S\setminus\{d\}\mid S\in\cS\}$ is a larger intersecting subfamily of $\cH$ than $\cI$, a contradiction. Therefore $|D|=1$ and, without loss of generality, $D=\{1\}$.\ Let $\cF$ be the largest sunflower in $\cS$ with core $\{1\}$. Since any $R\in\cR$ must intersect every $F\in\cF$, we must have that $|\cF|\le 3$. If $|\cF|=1$, then $\{S\setminus\{1\}\mid S\in\cS\}$ forms an intersecting family, and from the fact that $|\cS|\ge 2$ and $D=\{1\}$, we have that $\cS=\{\{1,a,b\},\{1,a,c\},\{1,b,c\}\}$ for three different numbers $a,b,c$. Moreover, we must have $|R\cap\{a,b,c\}|\ge 2$ for every $R\in\cR$. This means that $\cI\cup\{\{a,b\}\}$ is a larger intersecting subfamily of $\cH$ than $\cI$, a contradiction. Therefore $2\le |\cF|\le 3$.\ Let $X=\left(\bigcup_{F\in\cF}F\right)$; then $|X|=2|\cF|+1$. Denote $X^*=X\setminus\{1\}$. Define $Y=\left(\bigcup_{S\in\cS}S\right)\setminus X$ and set $\cS(Y)=\{S\in\cS\mid S\cap Y\ne\emptyset\}$. Then we must have that, for all $y\in Y$, there is an $S\in\cS(Y)$ such that $\{1,y\}\subseteq S$ and, for all $x\in X$ (including $x=1$), there is an $F\in\cF$ such that $\{1,x\}\subseteq F$. If $|X\cup Y|=|X|+|Y|=2|\cF|+|Y|+1>|\cR|$, then $\cS\cup\{\{1,k\}\mid k\in X\cup Y\}$ is a star subfamily of $\cH$ of size larger than $\cI$, a contradiction. So in the rest we assume that $|\cR|\ge 2|\cF|+|Y|+1$.\ ### $|\cF|=3$ Without loss of generality, $\cF=\{\{1,2,3\},\{1,4,5\},\{1,6,7\}\}$. Set $\cE$ to be the family of $3$-element subsets of $X^*$ that intersect each of $\{2,3\},\{4,5\},\{6,7\}$. Then $|\cE|=8$ and $\cR\subseteq\cE$. However, if $R\in\cR$ then $X^*\setminus R\in\cE\setminus\cR$, and so $|\cR|\le 4< 7\le 2|\cF|+|Y|+1$, a contradiction.\ ### $|\cF|=2$ Without loss of generality, $\cF=\{\{1,2,3\},\{1,4,5\}\}$. We have $|\cR|\ge|Y|+5$.\ Define $\cS^*=\cS\setminus(\cF\cup\cS(Y))$. Clearly, $\sum_{x\in X^*}|\cS^*_x|=2|\cS^*|$, and $\cS^*\subseteq\{\{1,i,j\}\in\cS\mid i\in\{2,3\},j\in\{4,5\}\}$. Denote $\cR^*=\{R\in\cR\mid R\subseteq X^*\}$. Since $|\cR^*|\le 4<|Y|+5$, we know that $\cR\setminus\cR^*\ne\emptyset$.\ For each $x\in X^*$ we set $\hat{x}$ to be the integer and $C_x$ to be the $2$-set such that $\{\{x,\hat{x}\},C_x\}=\{\{2,3\},\{4,5\}\}$ (so, in particular, $C_x=C_{\hat{x}}$). Also define $Y_x=\{y\in Y\mid \{1,x,y\}\in\cS\}$. For $i\in\{2,3\}$ and $j\in\{4,5\}$ let $\cR(i,j)=\{R\in\cR\mid R\cap X^*=\{i,j\}\}$ and let $R(i,j)=\{y\mid \{i,j,y\}\in\cR(i,j)\}$. Note that $R(i,j)\sse Y$. The following properties are easy to see.\ 1. \[prop:partition\] The collection $\{\cR(i,j)\mid i\in\{2,3\},j\in\{4,5\}\}$ partitions $\cR\setminus\cR^*$; in particular, at least one of these sets is nonempty. 2. \[prop:sstar\] If $\{1,\hat{i},\hat{j}\}\in\cS^*$ then $\cR(i,j)=\emptyset$. (Since no element of $\cR(i,j)$ intersects $\{1,\hat{i},\hat{j}\}$.) 3. \[prop:rrstar\] $|\cS^*|\le 3$. (This follows from \[prop:partition\] and \[prop:sstar\].) 4. \[prop:qij\] If $\min(|R(i,j)|,|R(\hat{i},\hat{j})|)\ge 1$ then $R(i,j)=R(\hat{i},\hat{j})$ with $|R(i,j)|=1$. Therefore if $\min(|\cR(i,j)|,$ $|\cR(\hat{i},\hat{j})|)\ge 1$ then $|\cR(i,j)|=|\cR(\hat{i},\hat{j})|)=1$. (Since elements of $\cR(i,j)$ and $\cR(\hat{i},\hat{j}) $ can intersect in at most one element.) 5. \[prop:bi\] If $X^*\setminus\{x\}\in\cR^*$ then $Y_x=\emptyset$. (Since $X^*\setminus\{x\}$ does not intersect sets of the form $\{1,x,y\}$ for $y\in Y$.) 6. \[prop:ysmall\] If $y\in Y_x$ then, for $j\in C_x$, we have $\cR(\hat{x},j)\subseteq\{\{\hat{x},j,y\}\}$. (Since $\{1,x,y\}\in\cS$.) 7. \[prop:ylarge\] If $|Y_x|\ge 2$ then $\bigcup_{j\in C_x} \cR(\hat{x},j)=\emptyset$. (This follows from \[prop:ysmall\].) 8. \[prop:ymin\] Since $\cR^*\ne\cR$, we have $\min(|Y_x|,|Y_{\hat{x}}|)\le 1$ for every $x\in X^*$. (This follows from \[prop:partition\] and \[prop:ylarge\].) 9. \[prop:last\] If $|\cS^*|=3$ then $\cS_x^*\ne\emptyset$ for all $x\in X^*$. If, for some $x\in X^*$, we have $|Y_x|\ge 2$, and $|Y_{\hat{x}}|= 1$, then (from \[prop:partition\], \[prop:ysmall\], and \[prop:ylarge\]) $|\cR\setminus\cR^*|\le 2$ and (from \[prop:bi\]) $|\cR^*|\le 2$. But this means that $|\cR|\le 4< 5+|Y|$, a contradiction. Therefore we know that if $|Y_x|\ge 2$ then $Y_{\hat{x}}=\emptyset$.\ If, for some $x\in X^*$, we have $|Y_x|=|Y_{\hat{x}}|=1$ (we may assume by relabeling, if necessary, that $x=2$, so $\hat{x}=3$), then (from \[prop:partition\] and \[prop:ysmall\]) $|\cR\setminus\cR^*|\le 4$ and (from \[prop:bi\]) $|\cR^*|\le 2$. Therefore, from $Y\ne\emptyset$, we get $|\cR|\le 6\le 5+|Y|$, therefore $|\cR|=5+|Y|$ and $|Y|=1$. Without loss of generality $Y_2=Y_3=Y=\{6\}$. Also, $|\cR^*|=2$, and $\cR^*=\{\{2,3,5\},\{2,3,4\}\}$ and (from \[prop:bi\]) $Y_4=Y_5=\emptyset$; consequently $\cS(Y)=\{\{1,2,6\},\{1,3,6\}\}$. Moreover (using \[prop:ysmall\]), from $|\cR\setminus\cR^*|=4$ we get that, for each $i\in\{2,3\}$ and $j\in\{4,5\}$, we have $\cR(i,j)=\{\{i,j,6\}\}$. Thus (from \[prop:sstar\]) $\cS^*=\emptyset$. But this yields $|\cI_2|=5>4=|\cS|$, a contradiction.\ Therefore we can now assume, for all $x\in X^*$, that $\min(|Y_x|,|Y_{\hat{x}}|)=0$. Set $L=\{x\in X^*\mid Y_x\ne\emptyset\}$. Then we have that $|L|\le 1$ or $L=\{i,j\}$ for some $i\in\{2,3\}$ and $j\in\{4,5\}$. For each $x\in X^*$ we have that $$\Bigl(|\cF_x|+|Y_x|+|\cS_x^*|\Bigr) + \Bigl(\sum_{j\in C_x}|\cR(x,j)|+|\cR^*_x|\Bigr)\ \le\ |\cI_x|\ \le\ |\cS|\ , \label{eq:individual}$$ where we have counted the sets containing 1 before those not containing 1. Of course, $|\cF_x|=1$ and $|S^*_x|\le 2$. By summing over $X^*$, we obtain $$4+\sum_{x\in X^*} |Y_x|+2|\cS^*|+2\left(\sum_{i\in\{2,3\}}\sum_{j\in\{4,5\}} |{\cal R}(i,j)|\right)+3|{\cal R}^*|\le 4|\cS|\ ,$$ which simplifies to $$4+\sum_{x\in X^*} |Y_x|+2|\cS^*|+2|\cR|+|\cR^*|\le 4|\cS|\ .$$ In particular, $$|\cR|\le 2|\cS|-2-\frac{1}{2}\sum_{x\in X^*}|Y_x|-|\cS^*|-\frac{1}{2}|\cR^*|\ . \label{eq:general}$$ Now we consider the following three subcases, based on the size of $L$.\ [**Case**]{} $L=\emptyset$\ Then each $Y_x=Y=\emptyset$, $\cS(Y)=\emptyset$, $|\cS|=|\cS^*|+2$, and $|\cR|\ge 5+|Y|=5$. Using \[prop:rrstar\], equation (\[eq:general\]) becomes $$|\cR|\le 2+|\cS^*|-\frac{1}{2}|\cR^*|\le 2+3-0=5.$$ Thus $|\cR|=5$, $|\cS^*|=3$, $\cR^*=\emptyset$, $|\cS|=5$, and all inequalities hold with equality in inequality (\[eq:individual\]).\ We may assume, by relabeling, if necessary, that $\cS^*=\{\{1,2,4\},\{1,2,5\},\{1,3,4\}\}$. Then inequality (\[eq:individual\]) implies that, for $i\in\{2,4\}$, we have $\sum_{j\in C_i}|\cR(i,j)|=2$, and for $i\in\{3,5\}$ we have $\sum_{j\in C_i}|\cR(i,j)|=3$. This means that $|\cR(3,5)|-|\cR(2,4)|=1$ and, therefore, $|R(3,5)|>|R(2,4)|$. Hence (using \[prop:qij\]), we must have that $\cR(2,4)=\emptyset$ and $|\cR(3,5)|=1$. This means that $|\cR(2,5)|=|\cR(3,4)|=2$, which is a contradiction by \[prop:qij\].\ [**Case**]{} $|L|=1$\ Here we may assume that $L=\{2\}$. Then $Y=Y_2$, $|\cS|=2+|\cS^*|+|Y|$, $\{3,4,5\}\notin\cR^*$ (from \[prop:bi\], and from \[prop:ysmall\]) $|\cR(3,j)|\le 1$ for each $j\in \{4,5\}$. Using \[prop:rrstar\], equation (\[eq:general\]) becomes $$|\cR|\le 2+|\cS^*|+\frac{3}{2}|Y|-\frac{1}{2}|\cR^*|\le 5+\frac{3}{2}|Y|\ .$$\ 1. $|Y|=|Y_2|=1$\ Since $5+|Y|=6\le |\cR|\le 6+\frac{1}{2}-\frac{1}{2}|\cR^*|$, we get $|\cR|=6$, $|\cS^*|=3$, and $|\cR^*|\le 1$. From \[prop:sstar\] we know that there are $i\in\{2,3\}$ and $j\in\{4,5\}$ such that, if $\{\ell,m\}\ne\{i,j\}$ then $\cR(\ell,m)=\emptyset$. Since $\cR\setminus\cR^*=\cR(i,j)$, this implies that $|\cR(i,j)|\ge 5$. Moreover, since $|\cR(3,k)|\le 1$ for each $k\in\{4,5\}$, we have $i=2$. Then, using inequality (\[eq:individual\]), we obtain $7\le 1+|Y_2|+|\cS_2^*|+\sum_{k\in C_2}|\cR(2,k)|+|\cR^*_2|\le |\cS|=6$, a contradiction.\ 2. $|Y|=|Y_2|\ge 2$\ Here we have $\cR(3,4)=\cR(3,5)=\emptyset$, and so (since $\{3,4,5\}\notin\cR^*$) we get that $\sum_{j\in C_2}|\cR(2,j)|+|\cR^*_2|=|\cR|$. Using inequality (\[eq:individual\]) with $x=2$ yields $1+|Y|+|\cS_2^*|+|\cR|\le 2+|\cS^*|+|Y|$. In other words, $|\cR|\le 1+|\cS^*|-|\cS_2^*|<5$, a contradiction. [**Case**]{} $|L|=2$\ We may assume that $L=\{2,4\}$. Then $|\cS|=|\cS^*|+2+|Y_2|+|Y_4|$ and $Y=Y_2\cup Y_4$. From (\[prop:bi\]) we have $\cR^*\subseteq\{ \{2,3,4\},\{2,4,5\}\}$. We need only consider cases for which $|\cR|\ge 5+|Y|$.\ 1. $|Y_2|,|Y_4|\ge 2$\ From (\[prop:ylarge\]) we know that $\cR\setminus\cR^*=\cR(2,4)$. Using inequality (\[eq:individual\]) with $\{i,j\}=\{2,4\}$ and the fact that $5+|Y|\le|\cR|$, we get $|Y_i|+|S_i^*|+6+|Y|\le 1+|Y_i|+|\cS_i^*|+|\cR(2,4)|+|\cR^*|\le |\cS|=|\cS^*|+2+|Y_i|+|Y_j|$, which gives, for each $j\in\{2,4\}$, that $|Y|\le |\cS^*|-4+|Y_j|< |Y_j|$, a contradiction. 2. $|Y_2|=1$ and $|Y_4|\ge 2$ (the case $|Y_4|=1$ and $|Y_2|\ge 2$ is handled symmetrically)\ Without loss of generality, $Y_2=\{6\}$. From (\[prop:ylarge\]) we have $\cR(2,5)=\cR(3,5)=\emptyset$, and from (\[prop:ysmall\]) we know that $\cR(3,4)\subseteq\{\{3,4,6\}\}$ and $\cR\setminus\cR^*=\cR(3,4)\cup\cR(2,4)$. Thus, $\cR_4=\cR$. Also, $\cS^*\subseteq\{\{1,2,4\},\{1,2,5\},\{1,3,4\},\{1,3,5\}\}$. Set ${\cal P}=\cS^*\setminus\cS^*_4$. Since $\cR(2,4)\cup\cR(3,4)=\cR\setminus\cR^*\ne\emptyset$, we get from (\[prop:sstar\]) that $|{\cal P}|\le 1$. Thus, $\cI_4=\cI\setminus\left(\{\{1,2,6\},\{1,2,3\}\}\cup{\cal P}\right)$, so $|\cI|\le |\cI_4|+3$. Therefore the family $\cI_4\cup\{\{4\},\{1,4\}\}\cup\{\{4,y\}:y\in Y_4\}\}$ is a star subfamily of $\cH$ of size $|\cI_4|+2+|Y_4|\ge |\cI_4|+4> |\cI|$, a contradiction.\ 3. $|Y_2|=|Y_4|=1$ (so $1\le|Y|\le 2$)\ In this case $|\cS|=4+|\cS^*|$ and $\cR\subseteq\{\{2,3,4\},\{2,4,5\}\}$\ 1. $Y_2=Y_4=Y$\ Here we have $|Y|=1$ so, without loss of generality, $Y=\{6\}$. Then from \[prop:ysmall\] we learn that $\cR(2,5)\subseteq\{\{2,5,6\}\}$, $\cR(3,5)\subseteq\{\{3,5,6\}\}$ and $\cR(3,4)\subseteq\{\{3,4,6\}\}$. Therefore $4=6-2\le |\cR|-|\cR^*| = |\cR\setminus\cR^*|\le 3+|\cR(2,4)|$, and so $|\cR(2,4)|\ge 1$. By \[prop:sstar\] we know that $\{1,3,5\}\not\in\cS^*$, and thus $\cS^*\sse \cS^*_2\cup \{\{1,3,4\}\}$.\ 1. $|\cR(2,4)|>1$\ By \[prop:qij\] this implies that $\cR(3,5)=\emptyset$ and, hence, for $i\in\{2,4\}$, that $\sum_{k\in C_{i}}|\cR(i,k)|\ge|\cR\setminus\cR^*|-1$. Using equation (\[eq:individual\]) we get that $\sum_{k\in C_2}|\cR(2,k)|+|\cR^*|+2+|\cS_2^*|\le 4+|\cS^*|$. Since $7+|\cS_2^*|\le|\cR|+1+|\cS^*_2|\le \sum_{k\in C_2}|\cR(2,k)| +|\cR^*| +2 +|\cS_2^*|$, we obtain $3+|\cS^*_2|\le |\cS^*|\le 3$. This implies that $S_2^*=\emptyset$ and $|S^*|=3$, contradicting \[prop:last\].\ 2. $|\cR(2,4)|=1$\ Now we have (from \[prop:qij\]) that $|\cR(3,5)|\le 1$, so we know that $|\cR(i,j)|\le 1$ for every $i\in\{2,3\},j\in\{4,5\}$. Then $|\cR\setminus\cR^*|\ge 4$ implies that $|\cR(i,j)|=1$, $|\cR\setminus\cR^*|=4$, and $\cR^*=\{X^\pr_3,X^\pr_5\}$. Using equation (\[eq:individual\]) with $x\in\{2,4\}$, we get that $7+|\cS_x^*|\le 4+|\cS^*|$, and so $3\le |\cS^*|-|\cS_x^*|\le 3$, a contradiction.\ 2. $Y_2\ne Y_4$\ Here we have $|Y|=2$ and, without loss of generality, $Y_2=\{6\}$ and $Y_4=\{7\}$. From \[prop:ysmall\] we get that $\cR(3,j)\subseteq\{\{3,j,6\}\}$ for each $j\in \{4,5\}$ and $\cR(i,5)\subseteq\{\{i,5,7\}\}$ for each $i\in \{2,3\}$. This implies, in particular, that $\cR(3,5)=\emptyset$. Thus, for $i\in\{2,4\}$, we have $\sum_{k\in C_{i}}|\cR(i,k)|\ge|\cR\setminus\cR^*|-1$. In particular, $$7\le |\cR| \le \sum_{k\in C_2}|\cR(2,k)| + |\cR^*| +1 \le \sum_{k\in C_2}|\cR(2,k)| + |\cR^*_2| +2\ .$$ Using inequality (\[eq:individual\]) with $x=2$ we get that $$2 +|\cS_2^*| + \sum_{k\in C_2}|\cR(2,k)| + |\cR^*_2|\le |\cS^*|+4\ .$$ Together, these imply that $3+|\cS_2^*|\le |\cS^*|\le 3$, and so $|S^*|=3$ and $|S^*_2|=0$, which contradicts \[prop:last\].\ This completes the proof. Proof of Theorem \[bigchvatal\] =============================== We now proceed to a proof of Theorem \[bigchvatal\].\ \ Let $\cI_i=\cI\cap \binom{[n]}{i}$, for $i=1,2,3$. We can assume $\cI_1=\mt$, since otherwise, $\cI$ is a star. Similarly, we can assume $|\cI_2|\leq 3$. Thus, we have $|\cI_3|\geq 28$. Since $28=(4-1)^3+1$, we can use Theorem \[kflemma\] to conclude that $\cI_3$ contains a $4$-flower. Let $k\geq 4$ be maximum such that $\cS$ is a $k$-flower in $\cI_3$, and let $C$ be the core of $\cS$. As $\cI_3$ is $3$-uniform and intersecting, every subfamily $\cG\sse \cI$ has $\tau(\cG)\leq 3$, which implies that $C\neq \mt$. Suppose first that $C=\{a\}$, and suppose $\cI$ is not a star centered at $a$. Let $A\in \cI$ be such that $a\notin A$. Consider the family $\cS_C$. As $\tau(\cS_C)\geq 4$, there exists some $S_1\in \cS_C$ such that $A\cap S_1=\mt$. Consequently, if $S^\pr=S_1\cup \{a\}$, then $S^\pr\in \cI$ and $A\cap S^\pr=\mt$, a contradiction. As a result, we may assume that $C=\{a,b\}$. This implies that $\cS_C$ is a family of singletons. Consequently, $\cS$ is a sunflower with at least $4$ petals.[^5] Additionally, for every $A\in \cI_3$, $A\cap \{a,b\}\neq\mt$.\ Let $\cA=\{A\in \cI_3:A\cap C=\{a\}\}$, and let $\cB=\{B\in \cI_3:B\cap C=\{b\}\}$. We have $|\cI_3|=|S|+|\cA|+|\cB|$. Let $\cAp=\{A-\{a\}:A\in \cA\}$, and $\cBp=\{B-\{b\}:B\in \cB\}$. If $\cAp=\mt$ or $\cBp=\mt$, we can conclude that $\cI_3$, and hence, $\cI$ is a star (centered at either $a$ or $b$), so suppose both are non-empty. Since $\cI$ is intersecting, $\cAp$ and $\cBp$ are cross-intersecting families, i.e. for any $A\in \cAp$ and $B\in \cBp$, $A\cap B\neq \mt.$ Let $V(\cAp)$ and $V(\cBp)$ be the vertex sets of $\cAp$ and $\cBp$ respectively, and let $n(\cX)=|V(\cX)|$ for $\cX\in \{\cAp,\cBp\}$. We first prove the following claims. \[clm1\] If both $\cAp$ and $\cBp$ are intersecting, or $|\cAp|\geq 2$ and $|\cBp|\geq 2$, then, $|\cX|\leq 2+ n(\cX)$ for each $\cX\in \{\cAp,\cBp\}$. If $\cAp$ is intersecting, it is either a triangle, or a star. In either case, the bound follows trivially. A similar argument works for $\cBp$, so suppose, without loss of generality that $\cAp$ has two disjoint edges, say $\{xy,x^\pr y^\pr\}$. $\cBp\sse \{xy^\pr,y^\pr y,yx^\pr,x^\pr x\}$, giving the required bound for $\cBp$. Now, if $\cB$ has two disjoint edges, we can use a similar argument for $\cAp$, so suppose $\cBp$ is intersecting. Without loss of generality, suppose $\cBp= \{xy^\pr,y^\pr y\}$. Then $\cAp\sse \{xy, x^\pr y^\pr\}\cup \{A\in \binom{[n]}{2}:y^\pr\in A\}$, giving the bound $|n(\cAp)|\geq |\cAp|$. This completes the proof of the claim. \[clm2\] If $\cAp$ has a pair of disjoint edges, and $|\cBp|=1$, then $|\cAp|\leq n(\cAp)+(|S|+1)$. Let $\{xy,x^\pr y^\pr\}$ be a pair of disjoint edges in $\cAp$, and, wlog, let $\cBp=\{xx^\pr\}$. Let $\cAp_x=\{A\in \cAp:x\in A\}$, and let $\cAp_{x^\pr}=\{A\in \cAp:x^\pr\in A\}$. Let $X=\{v\in [n]:v\neq x^\pr, xv\in \cA_x\}$, $X^*=\{v\in [n]:v\neq x, x^\pr v\in \cA_{x^\pr}\}$ and $R=X\cap X^*$. Now, $|\cAp|\leq 2|R|+|X\setminus R|+|X^*\setminus R|+1$, and $n(\cAp)=2+|R|+|X\setminus R|+|X^*\setminus R|$. So, $n(\cAp)-|\cAp|\geq -(|R|+1)$. Since $|R|\leq |S|$ (otherwise, $R$ would be a bigger sunflower with core $\{a,x\}$ (or $\{a,x^\pr\}$), contradicting the choice of $S$), we have $n(\cAp)-|\cAp|\geq -(|S|+1)$.\ In the next claim, we give lower bounds on the sizes of $\cH_a$ and $\cH_b$.  \ - $|\cH_a|\geq 1+ (|S|+n(\cAp)+1)+(|S|+|\cAp|).$ - $|\cH_b|\geq 1+ (|S|+n(\cBp)+1)+(|S|+|\cBp|).$ We will only give the proof for $\cH_a$, as the proof for $\cH_b$ follows identically. We know that $|\cH_a|=\sum_{i=1}^3|\cH_a^i|$, where $\cH_a^i=\cH_a \cap \binom{[n]}{i}$ for $i\in \{1,2,3\}$. It is trivial to note that $|\cH_a^1|=1$. Now, consider $\cH_a^2$. First, $\{a,b\}\in \cH_a^2$. Also, for every $\{a,b,s\}\in S$, $\{a,s\}\in \cH_a^2$, as $\cH$ is a downset. Similarly, for every $s\in n(\cAp)$, there exists a $t\in n(\cAp)$ such that $\{a,s,t\}\in \cI_3$, and hence, $\{a,s\}\in \cH_a^2$. Thus, $|\cH_a^2|\geq |S|+n(\cAp)+1$. Also, it is not hard to see that $|\cH_a^3| \geq |S|+|\cAp|$. This completes the proof of the claim.\ We will now prove that either $\cH_a$ or $\cH_b$ is bigger than $\cI$, which will complete the proof of the theorem. It will be sufficient to prove the following claim. $|\cH_a|+|\cH_b|> 2(|\cI_3|+3).$ We will consider two cases, depending on whether or not the hypothesis of Claim \[clm1\] is true. Suppose the hypothesis of Claim \[clm1\] holds, so we have $n(\cX)-|\cX|\geq -2$, for $\cX\in \{\cAp,\cBp\}$. Thus, since $|S|>3$, we have $$\begin{aligned} |\cH_a|+|\cH_b| & \geq & 4+4|S|+|\cAp|+|\cBp|+n(\cAp)+n(\cBp) \\ & = & (2|S|+2|\cAp|+2|\cBp|+6)+(n(\cAp)-|\cAp|)+(n(\cBp)-|\cBp|)+2|S|-2 \\ & \geq & 2(|\cI_3|+3)+(2|S|-6) \\ & > & 2(|\cI_3|+3). \end{aligned}$$ Now, assume the hypothesis of Claim \[clm1\] is false, so, without loss of generality, suppose $\cAp$ has a pair of disjoint edges, and $|\cBp|=1$. Clearly, $n(\cBp)-|\cBp|=1$ and we can use Claim \[clm2\] to conclude that $n(\cAp)-|\cAp|\geq -(|S|+1)$. Thus, we have $$\begin{aligned} |\cH_a|+|\cH_b| & \geq & 4+4|S|+|\cAp|+|\cBp|+n(\cAp)+n(\cBp) \\ & \geq & (2|S|+2|\cAp|+2|\cBp|+6)-(|S|+1)+1+2|S|-2 \\ & \geq & 2(|\cI_3|+3)+|S|-2 \\ & > & 2(|\cI_3|+3). \end{aligned}$$\ \ This proves the theorem. Acknowledgements ================ The second and third authors would like to thank Dhruv Mubayi for productive discussions on approaches to proving Theorem \[bigchvatal\]. [10]{} I. Anderson, [*Combinatorics of Finite Sets*]{}, Oxford University Press, London, 1987. P. Borg, On Chvátal’s conjecture and a conjecture on families of signed sets, [*European J. Math.*]{} [**32**]{} (2011), no. 1, 140–145. V. Chvátal, Intersecting families of edges in hypergraphs having hereditary property, in: C. Berge, D.K. Ray-Chaudhuri (Eds.), [*Hypergraph Seminar, Lecture Notes in Mathematics*]{}, Vol. 411, Springer, Berlin, 1974, 61–66. P. Erdős, C. Ko, and R. Rado, Intersection theorems for systems of finite sets, [*Quart. J. Math. Oxford*]{} [**12**]{} (1961), 313–320. P. Erdős and R. Rado, Intersection theorems for systems of sets, [*J. London Math. Soc.*]{} [**35**]{} (1960), pp 85–90. J. H[å]{}stad, S. Jukna, and P. Pudlák, Top-down lower bounds for depth-three circuits. Comput. Complexity 5 (1995), no. 2, 99–112. D. Miklos, Great intersecting families of edges in hereditary hypergraphs, [*Discrete Math.*]{} [**48**]{} (1984), 95–99. J. Schonheim, Hereditary systems and Chvátal’s conjecture, [*Proc. of the Fifth British Combinatorial Conference*]{}, Aberdeen, 1975, 537–539. H. Snevily, A new result on Chvátal’s conjecture, [*J. Combin. Theory Ser. A*]{} [**61**]{} (1992), no. 1, 137–141. P. Stein, On Chvátal’s conjecture related to hereditary systems, [*Discrete Math.*]{} [**43**]{} (1983), 97–105. Y. Wang, Notes on Chvátal’s conjecture, [*Discrete Math.*]{} [**247**]{} (2002), no. 1–3, 255–259. [^1]: Department of Mathematics, University of S. Carolina, `czabarka@math.sc.edu` [^2]: Department of Mathematics and Applied Mathematics, Virginia Commonwealth University [^3]: `ghurlbert@vcu.edu`. Research partially supported by Simons Foundation Grant \#246436. [^4]: `vkamat@vcu.edu` [^5]: Note that every sunflower with $k$ petals is a $k$-flower, but the converse is not always true.

--- author: - 'S. Afach , C. A. Baker , G. Ban , G. Bison , K. Bodek , Z. Chowdhuri , M. Daum , M. Fertl [^1], B. Franke [^2], P. Geltenbort , K. Green , M. G. D. van der Grinten , Z. Grujic , P. G. Harris , W. Heil , V. Hélaine [^3], R. Henneck , M. Horras [^4], P. Iaydjiev [^5], S. N. Ivanov [^6], M. Kasprzak , Y. Kermaïdic , K. Kirch , P. Knowles [^7], H.-C. Koch , S. Komposch , A. Kozela , J. Krempel , B. Lauss , T. Lefort , Y. Lemière , A. Mtchedlishvili , O. Naviliat-Cuncic [^8], J. M. Pendlebury , F. M. Piegsa , G. Pignol , P. N. Prashant , G. Quéméner , D. Rebreyend , D. Ries , S. Roccia , P. Schmidt-Wellenburg , N. Severijns , A. Weis , E. Wursten , G. Wyszynski , J. Zejma , J. Zenner , G. Zsigmond' date: 'Received: date / Revised version: date' title: 'Measurement of a false electric dipole moment signal from $^{199}$Hg atoms exposed to an inhomogeneous magnetic field' --- =1 Introduction {#sec:intro} ============ Recent investigations characterizing frequency shifts for spins contained in vessels permeated with magnetic and electric fields $B$, $E$ have been motivated principally by the search for electric dipole moments (EDMs) of simple non-degenerate systems (neutron, atoms, molecules) and the potential discovery of new sources of CP violation [@ram2013]. Such experiments look for shifts, proportional to an applied electric field, of the Larmor precession frequency of stored particles. Any additional such shift is therefore a potential source of systematic errors. Among the few magnetic-field related spurious shifts, one is of particular concern: due to the motional magnetic field ${\mathbf E \times \mathbf v/{c^2} }$, a shift arises that is proportional to the electric-field strength and therefore mimics an EDM signal. Interestingly enough, ${\mathbf E \times \mathbf v/{c^2} }$ effects were already the main limiting factor for the early neutron beam experiments [@ramsey1982]. Then, with the advent of the storable ultra-cold neutrons (UCN), it was erroneously assumed for many years that this false EDM signal would vanish, based on the argument that the velocity of trapped particles averages to zero. The first correct and comprehensive calculation of this effect was given in Ref. [@pendlebury2004], in the context of an EDM experiment with stored particles. For completeness, it should be mentioned that Stark interference effects, such as the one reported for $^{199}{\rm Hg}$ in Ref. [@loftus2011], are also known to produce false EDM signals for atoms. The effect discussed in the present article is of a different nature, and to make the distinction we will refer to it as the motional false EDM. Our collaboration is conducting a program to search for the neutron EDM [@baker2011], using the new ultracold neutron (UCN) source [@Lauss2014] at the Paul Scherrer Institute (PSI). We are currently working with an upgraded version of the spectrometer [@baker2013] that was used to establish the best nEDM limit, $$\left| d_{{\rm n}}\right| < 2.9 \times 10^{-26}\, e\,\text{cm} \, (90\% \, \text{C.L.}),$$ at the Laue Langevin Institute (ILL) [@baker2006]. One distinct feature of this device is a mercury co-magnetometer [@green1998] using a spin-polarized vapor of $^{199}$Hg atoms that precess in the same volume as the neutrons. The nEDM analysis is then based on the ratio of the Larmor precession frequencies, $R=f_{{\rm n}}/f_{{{\rm Hg}}}$, which to first order is free of magnetic field fluctuations. However, both neutrons and mercury atoms are subject to a frequency shift that is proportional to the electric field, due to the unavoidable presence of magnetic-field gradients. As will be shown, the motional false neutron EDM, $d_{{\rm n}}^{\rm false}$, is negligible, at least at the current level of sensitivity. In contrast, the mercury-induced false nEDM $$d_{{\rm n}}^{{\rm false}, {{\rm Hg}}} = \frac{\gamma_{{\rm n}}}{\gamma_{{{\rm Hg}}}} d_{{{\rm Hg}}}^{\rm false} \approx 3.8\, d_{{{\rm Hg}}}^{\rm false},$$ where $d_{{{\rm Hg}}}^{\rm false}$ is the motional mercury false EDM and $\gamma_{{\rm n}}$, $\gamma_{{{\rm Hg}}}$ are the gyromagnetic ratios of the neutron and $^{199}$Hg respectively, is a major systematic effect that must be precisely controlled. One of the main improvements accomplished recently within the experiment is the installation of an array of cesium magnetometers that surrounds the precession chamber. This new device has made it possible to measure the magnetic field distribution, and thus to calculate the vertical gradient in the trap, which underlies the false EDM discussed here. In this article, we report on the first direct measurement of a motional false EDM signal for stored mercury atoms. A comparison to theoretical expectations is also presented. Theory of frequency shifts induced by magnetic field gradients: a brief reminder {#sec:theory} ================================================================================ Particles with a magnetic moment exposed to a magnetic field, ${\bf B_\textnormal{0}} = B_0 {\bf \hat{z}}$, precess at the Larmor frequency $f_{{\rm L}} = \gamma \, B_0 / 2 \pi$ where $\gamma$ is the gyromagnetic ratio. Because of experimentally unavoidable magnetic field gradients, the Larmor frequency of a particle moving through this field will be subject to a shift, known as the Ramsey-Bloch-Siegert (RBS) shift [@ramsey1955]. If an electric field ${\bf E}$ (parallel or anti-parallel to ${\bf B_\textnormal{0}}$) is applied – as is the case in experiments searching for EDMs – the moving particle will experience an additional motional magnetic field ${\mathbf B_v = \mathbf E \times \mathbf v/{c^2} }$. It is the interplay between this field and the magnetic field gradients that lies at the origin of a frequency shift proportional to the electric field strength, thus inducing a false EDM. As mentioned above, the first detailed calculation of such false EDMs for stored particles was given in Ref. [@pendlebury2004] in the context of the RAL-Sussex-ILL neutron EDM experiment [@baker2006]. The authors derived expressions for the two limiting cases: non adiabatic and adiabatic, corresponding to $2\pi f_{\rm L} \tau \gg 1$ and $2\pi f_{\rm L} \tau \ll 1$ respectively, where $\tau$ is the typical time particles take to cross the trap. Both regimes are of interest, since $^{199}{\rm Hg}$ atoms fall into the first category whereas UCNs fall into the second. More general results, valid for a broad range of frequencies, were obtained only for cylindrical symmetry and specular reflections. The expressions of the frequency shifts for the two limiting regimes are : $$\begin{aligned} \delta f_\textrm{L} &= \frac{\gamma^2 D^2}{32 \pi \, c^2} \frac{\partial B_0}{\partial z} E & \quad \textrm{(non adiabatic)} \label{eq_deltaOmegaNonAdiabatic}\\ \delta f_\textrm{L} &= \frac{v_{xy}^2}{4\pi\, B_0^2\, c^2} \frac{\partial B_0}{\partial z} E & \quad \textrm{(adiabatic),} \label{eq_deltaOmegaAdiabatic}\end{aligned}$$ where $\gamma$ is the gyromagnetic ratio, $D$ is the diameter of the trap, $c$ is the velocity of light and $v_{xy}$ is the particle velocity transverse to $B_0$. Note the absence of the gyromagnetic ratio in Eq. (\[eq\_deltaOmegaAdiabatic\]). Indeed, in the adiabatic case, the frequency shift can be interpreted as originating from a phase of purely geometric nature, or Berry’s phase [@ber1984; @commins1991], and is therefore independent of the coupling strength to the magnetic field. These results were then complemented and extended using the general theory of relaxation (Redfield theory) [@lamoreaux2005; @pignol2012], and then by solving the Schrödinger equation directly [@steyerl2014]. In Ref. [@pignol2012], an expression valid for arbitrary field distributions or trap shapes was obtained in the non-adiabatic limit : $$\begin{aligned} \delta f_\textrm{L} &= \frac{\gamma^2}{2 \pi c^2} \left\langle x B_x\, + yB_y \right\rangle E \quad \textrm{(non adiabatic),} \label{eq_deltaOmegaNonAdiabaticGeneralized}\end{aligned}$$ where the brackets refer to the average over the storage volume. For a cylindrical uniform gradient and a trap with cylindrical symmetry, Eq. (\[eq\_deltaOmegaNonAdiabaticGeneralized\]) reduces to Eq. (\[eq\_deltaOmegaNonAdiabatic\]). Using the relationship between the frequency shift and the false EDM, $${d}^{\rm false} = \frac{h}{2E} \delta f_{\rm L} (E)$$ where $h$ is Planck’s constant, together with Eqs. (\[eq\_deltaOmegaNonAdiabatic\]) and (\[eq\_deltaOmegaAdiabatic\]), one can now readily calculate the magnitude of the false EDMs for the mercury and for the neutron (both direct and mercury induced). Given our experimental conditions (see section \[sec:setup\]) and assuming a neutron velocity of 3 m/s, one obtains: $$\begin{aligned} &d_{{\rm n}}^{\rm false} = \frac{\partial B_0}{\partial z} \, 1.490 \times 10^{-29}\, e \, \text{cm}/\text{(pT/cm)} \\ & \nonumber\\ &d_{{{\rm Hg}}}^{\rm false} = \frac{\partial B_0}{\partial z} \, 1.148 \times 10^{-27}\, e \, \text{cm}/ \text{(pT/cm)} \label{falseHgEDMTheory}\\ & \nonumber \\ &d_{{\rm n}}^{\rm false, {{\rm Hg}}} = \frac{\partial B_0}{\partial z} \, 4.418 \times 10^{-27}\, e \, \text{cm}/\text{(pT/cm).}\end{aligned}$$ Considering a typical value of 10 pT/cm for the vertical ($z$ direction) gradient in our setup, we can conclude on the one hand that the direct false neutron EDM is negligible, at least at the current level of sensitivity. On the other hand, the mercury-induced false neutron EDM is a major systematic error that must be properly taken into account. Experimental apparatus {#sec:setup} ====================== The experimental study was performed with the nEDM spectrometer installed at the PSI UCN source. This room-temperature apparatus uses the Ramsey method of separated oscillatory fields [@green1998; @ramsey1950] to search for a shift, proportional to the strength of an applied electric field, in the neutron Larmor precession frequency. Under normal operation, polarized UCNs are stored in a $\sim20$ liter chamber (internal diameter [*D*]{} = 47 cm, height [*H*]{} = 12 cm), composed of a hollow polystyrene cylinder (coated with deuterated polystyrene) [@bodek2008; @kuzniak2008] and two disk-shaped aluminum electrodes coated with diamond-like carbon (Fig. \[fig:oILL\]). A cos$\theta$ coil produces a highly homogeneous magnetic field, $B_0 \approx 1 \, \mu\text{T}$, in the vertical direction while the two electrodes – the top one being connected to a high voltage (HV) source and the bottom one to ground potential – generate a strong electric field ($E \approx 10\, \text{kV/cm}$), either parallel or anti-parallel to ${\bf B_\textnormal{0} }$. In addition, a set of trim coils permits an optimization of the magnetic field uniformity at the $10^{-3}$ level. ![Schematic view of the precession chamber of the nEDM@PSI experiment.[]{data-label="fig:oILL"}](fig1){width="\linewidth"} The key to such experiments relies on the ability to control the magnetic field both in terms of stability and homogeneity. To this end, we use two highly sensitive and complementary atomic magnetometers based on mercury ($^{199}{\rm Hg}$) and cesium ($^{133}$Cs) atoms, respectively. Mercury is used in a co-magnetometer mode: polarized mercury atoms precess in the same volume as the neutrons, hence probing approximately the same space- and time-averaged magnetic field. Cesium is used in a set of external magnetometers surrounding the storage chamber. The former is an ideal tool to correct for field drifts, while the latter gives access to the spatial field distribution. The mercury co-magnetometer --------------------------- To date, $^{199}{\rm Hg}$ is the only atomic element that has been used as a co-magnetometer for a neutron EDM experiment. Thanks to its nuclear polarization, it benefits from long wall collision relaxation times, and polarization lifetimes larger than 100 s can be achieved. Moreover, it is one of the rare elements in which nuclear spin polarization can be created and monitored by optical means. It is worth noting that the best absolute EDM limit comes from an experiment using $^{199}{\rm Hg}$[@griffith2009][^9]: $$\left| d({\rm ^{199}Hg}) \right| < 3.1 \times 10^{-29}\, e\,\text{cm} \, (95\% \, {\rm CL}).$$ In our experiment, a vapor of mercury atoms is spin-polarized by optical pumping in a polarization chamber located underneath the precession chamber (Fig. \[fig:oILL\]). The operation of the co-magnetometer is synchronous with the nEDM measurement, and follows cycles about 300 s long. During neutron counting and filling, mercury atoms are continuously injected and optically pumped in the polarization chamber. Once the precession chamber is filled with UCNs, we let the vapor diffuse into the precession chamber where, after the application of a $\pi / 2$ pulse, the atoms freely precess around ${\bf B_\textnormal{0}}$ at a frequency of about 8 Hz. The interaction of the precessing atoms with a circularly polarized resonant probe beam produces a light-intensity modulation whose analysis yields the Larmor frequency of the atoms. One of the major drawbacks of the $^{199}{\rm Hg}$ co-magnetometer is its sensitivity to high voltage. As illustrated in Fig. \[fig:tauHg\], which displays the transverse polarization relaxation time T$_2$ versus the cycle number, sudden T$_2$ drops are systematically observed after each HV polarity reversal. The corresponding reduction of the signal amplitude directly affects the precision of the magnetometer. Fortunately, optimal performance can be recovered via discharge cleaning in an oxygen atmosphere. On average, the precision of the mercury co-magnetometer is of the order of 100 fT, equivalent to a magnetometric precision at the 0.1 ppm level per cycle. ![Transverse relaxation time T$_2$ of $^{199}\text{Hg}$ atoms (green points) together with the high voltage value (blue line) versus cycle number. Sudden drops of T$_2$ are observed after each polarity reversal.[]{data-label="fig:tauHg"}](fig2){width="\linewidth"} The array of cesium magnetometers {#sec:CsM} --------------------------------- An array of 16 cesium magnetometers (CsM) [@Knowles2009] allows measurement of the magnetic field distribution in the region of interest and, in particular, it gives us knowledge of the vertical gradient $\partial B_0 / \partial z$. Six HV-compatible (i.e. fully optically coupled) magnetometers were placed on top of the precession chamber, and ten standard ones below (Fig. \[fig:oILL\] and \[fig:HV-CsM\]). These laser-pumped magnetometers use a vapor of $^{133}$Cs atoms (gyromagnetic ratio $\gamma = 2 \pi \times 3.5\, \text{kHz}/\mu \text{T}$) and are operated in a phase-stabilized mode. They have a high statistical sensitivity ($\sim$ 100 fT for 40 s long measurements); however, they suffer from inaccurracies of their absolute field readings, with offsets that can be as high as 100 pT. They are therefore precise but not accurate. Finally, it is important to note that these magnetometers – like the mercury co-magnetometer – are scalar: they measure the magnitude of the magnetic field at the center of the bulb containing the cesium vapor. ![Picture of the six HV-compatible Cs magnetometers installed on the top HV electrode in Al enclosures. Optical fibers are also visible.[]{data-label="fig:HV-CsM"}](fig3){width="\linewidth"} Measurement and data analysis {#sec:analysis} ============================= A preliminary measurement with a limited number of CsM was performed in 2011, and led to a first result [@marlonThesis]. The present analysis is based on a dedicated data-taking period of 2 weeks’ duration in December 2013, where eight different gradient settings were explored: four with the magnetic field pointing upwards ($B_0^{\uparrow}$), and four downwards ($B_0^{\downarrow}$). Two trim coils were used to set a vertical gradient in addition to the $B_0$ field generated by the main coil. For each field configuration, about 500 cycles were recorded with a basic HV polarity pattern $(+\,-\,-\,+)$ and polarity changes every 20 cycles. The voltage was set to 120 kV, i.e. as high as possible to maximize the frequency shift while preserving a smooth operation (limited number of electrical breakdowns). As discussed above, the frequent polarity reversals induced a significant degradation of the mercury magnetometer’s sensitivity. Consequently, we decided to limit the free precession time to 40 s, a good compromise between sensitivity and the number of cycles. The mercury frequency was extracted using our standard “two windows” method [@chibane1995]. It consists of fitting the signal phase at either end of the signal, using data in two 15 s windows at the beginning and end of the time series. This method optimally takes into account possible frequency drifts during the precession time. During data taking, the mercury frequency uncertainty varied in the range 1-2 $\mu \text{Hz}$. Outputs from all 16 CsM were continuously recorded at a rate of 1 Hz, and a mean value of the magnetic field was calculated for time periods having an exact overlap with the mercury precession. We further made the approximation $$B_{\rm CsM} = \sqrt[2]{B_z^2 + B_T^2} \approx B_z(\vec{r}_{\rm CsM}),$$ where $B_T$ is a small transverse component. From several 3D mapping campaigns during which all coils (main and trim) were mapped, we know that this approximation is valid at the 10$^{-4}$ level. Gradient extraction {#subsec:gzExtrac} ------------------- We extracted the vertical gradient by fitting a harmonic polynomial expansion of the magnetic field to the CsM array data. The choice of harmonic polynomials ensures that the resulting expressions satisfy Maxwell’s equations. Due to the limited number of magnetometers the expansion was limited to the next-to-linear order (NLO), which involves 9 parameters: $$\begin{aligned} B_z(x,y,z) =\, &b_0 + g_x \, x + g_y \, y + g_z \, z + \nonumber \\ &g_{xx} (x^2 - z^2) + g_{yy} (y^2 - z^2)+ \nonumber \\ &g_{xy} xy + g_{xz} xz + g_{yz} yz. \label{eqn:fit_NLO} \end{aligned}$$ From expression (\[eqn:fit\_NLO\]), one can easily calculate the volume average of $B_z$ and of its vertical gradient, assuming a trap with cylindrical symmetry: $$\begin{aligned} B_0 \equiv \left\langle B_z \right\rangle &= b_0 +(g_{xx} + g_{yy})\left(\frac{D^2}{16}-\frac{H^2}{6}\right) \\ \left\langle \frac{\partial B_z}{\partial z}\right\rangle &= g_z. \label{eq:B_0}\end{aligned}$$ Let us now turn to the delicate task of estimating gradient uncertainties. The two main sources that have to be taken into account are the error on the magnetic field, and the extraction procedure. To assess their respective effects, extensive studies have been carried out using a toy model to generate known field distributions and check the extracted parameters [@victorThesis]. It was found that the errors coming both from the magnetometer offsets and from the expansion truncation never exceed 5 pT/cm. In addition, we used a technique known as the jackknife method to get an error directly from the data. It involves performing a series of $\chi^2$ minimizations (unweighted in our case) by removing one out of the 16 magnetometers at a time. The dispersion of the extracted parameters provides an estimate of the error. For the different field configurations, we systematically obtained errors in the range $10 \pm 5 \, \text{pT/cm}$, consistent with the model outcome. These jackknife errors were used subsequently in the analysis. Frequency shift measurement {#subsec:freqShift} --------------------------- A sample of a raw data time series $f_{{\rm Hg}}$ against cycle number is displayed in Fig. \[fig:fHg\_raw\], together with the corresponding high-voltage values. Despite the large point-to-point fluctuations and a slow linear drift, one can clearly observe a small but systematic correlation of the frequency shift with the electric field polarity. To correct for the slow magnetic field drift, we sliced the data relative to the electric field polarity and analyzed data sets corresponding to the $(+\,-\,-\,+)$ HV pattern. By doing so, any linear drift is exactly cancelled and higher orders are attenuated. For a data slice $(+\,-\,-\,+)$, corresponding to 40 cycles, the extracted frequency shift and its uncertainty are given by $$\delta f_{{{\rm Hg}}} = \left\langle f_{{{\rm Hg}}}^+ \right\rangle - \left\langle f_{{{\rm Hg}}}^- \right\rangle$$ and $$\Delta\delta f_{{{\rm Hg}}} = \sqrt{\Delta\left\langle f_{{{\rm Hg}}}^+ \right\rangle^2 + \Delta\left\langle f_{{{\rm Hg}}}^- \right\rangle^2},$$ where $\left\langle f_{{{\rm Hg}}}^{+(-)} \right\rangle$ and $\Delta\langle f_{{{\rm Hg}}}^{+(-)} \rangle$ stand for the mean frequency and its uncertainty as derived from the frequency distribution for the given HV polarity ($+$ or $-$). Finally, a weighted mean over the whole set of data slices was performed to estimate the electric-field induced frequency shift $\delta f_{\rm L} (E)$ for a given vertical gradient. ![Mercury frequency versus cycle number. The blue line shows the value of the applied high voltage.[]{data-label="fig:fHg_raw"}](fig4){width="\linewidth"} Results and discussion {#results} ====================== The final result is displayed in Fig. \[fig:dFalseVSgz\]. The motional false mercury EDM is plotted against the extracted vertical gradient $g_z$. The solid lines (red for $B_0^{\uparrow}$, blue for $B_0^{\downarrow}$) correspond to a global linear fit with a single free parameter, namely the slope $a$ ($\chi^2/\nu = 2.1/7$). ![Motional false mercury EDM versus the vertical gradient $g_z$ for $B_0^{\uparrow}$ (red up triangles) and $B_0^{\downarrow}$ (blue down triangles). The solid lines correspond to a linear fit, and the dashed line to the theory discussed in section \[sec:theory\]. The horizontal error bars are smaller than the symbol size. []{data-label="fig:dFalseVSgz"}](fig5){width="\linewidth"} We can now compare the measured slope to its theoretical expectation from Eq. 6: $$\begin{aligned} \left| a_{\rm exp} \right| &= 1.122(35) \times 10^{-27} \, e \,\text{cm}/(\text{pT/cm}),\\ \mathrm{and} \left| a_{\rm th} \right| &= 1.148 \times 10^{-27} \, e \,\text{cm}/(\text{pT/cm}).\end{aligned}$$ The agreement at the 1$\sigma$ level makes us confident that our magnetic gradient extraction procedure is reliable. This encouraging result is nonetheless not sufficient to directly control the mercury-induced false neutron EDM at the required level of sensitivity. Indeed, for an error of 10 pT/cm on the vertical gradient, Eq. (\[eq\_deltaOmegaNonAdiabatic\]) translates to a systematic error of $4.4 \times 10^{-26} e \,\text{cm}$ on the neutron EDM, which is already larger than the current limit. There is fortunately a way to circumvent this issue. In their last nEDM paper [@baker2006], the authors describe an analysis technique that enables one to find experimentally the working point with no vertical gradient and therefore no motional false EDM. This method, based on a tiny center of mass offset between the cold neutrons and the warmer mercury atoms, nevertheless induces some additional systematic errors. These errors were carefully assessed and found subdominant with a final result statistically limited. Whereas the use of the Hg comagnetometer is essential and does not limit our nEDM sensitivity for the time being, with a foreseen sensitivity of a few $10^{-27} e \,\text{cm}$ in the coming years, new magnetometry solutions will be needed in the future. We pursue an intensive R&D program on magnetometry using $^{133}$Cs but also $^3$He atoms [@koch2015]. In particular, efforts towards improving the absolute accuracy of the Cs magnetometers are currently underway [@grujic2015] as well as the implementation of Cs vector magnetometers [@afach2015]. In parallel, we have started design and construction of a next generation nEDM spectrometer [@baker2011] which, among other improvements, will benefit from a much better magnetic field control (passive and active). Advanced magnetometry and improved magnetic shielding will be combined with co-magnetometry or could even allow operation with only external magnetometers. Any possible mercury-induced false motional nEDM will therefore be much further suppressed or completely avoided. Conclusions {#sec:conclusions} =========== We have performed a measurement of a frequency shift proportional to the electric field strength for stored $^{199}$Hg atoms[^10], using a spectrometer devoted to the search for the neutron electric dipole moment at PSI. This shift, which we call the motional false EDM, originates from the combination of vertical magnetic field gradients with the motional magnetic field and could be measured for the first time thanks to the unique combination of a mercury co-magnetometer and an array of external cesium magnetometers. The agreement with a prediction based on the general Redfield theory of relaxation provides additional confidence in the validity of our gradient-extraction procedure as well as in our capability to measure and control the vertical gradient. The same method was used in a recent measurement of the neutron magnetic moment [@afach2014]. Acknowledgements {#acknowledgements .unnumbered} ================ We are grateful to the PSI staff (the accelerator operating team and the BSQ group) for providing excellent running conditions, and we acknowledge the outstanding support of M. Meier and F. Burri. Support by the Swiss National Science Foundation Projects 200020-144473 (PSI), 200021-126562 (PSI), 200020-149211 (ETH) and 200020-140421 (Fribourg) is gratefully acknowledged. The LPC Caen and the LPSC acknowledge the support of the French Agence Nationale de la Recherche (ANR) under reference ANR-09–BLAN-0046. The Polish partners acknowledge The National Science Centre, Poland, for the grant No. UMO-2012/04/M/ST2/00556. This work was partly supported by the Fund for Scientific Research Flanders (FWO), and Project GOA/2010/10 of the KU Leuven. The original apparatus was funded by grants from the UK’s PPARC (now STFC). J. Engel, M. J. Ramsey-Musolf and U. van Kolck, Prog. in Particle and Nuclear Phys. **71**,(2013)21-74. N. F. Ramsey, Rep. Prog. Phys., Vol. 45 (1982) 95. J. M. Pendlebury [*et al.*]{}, Phys. Rev. A **70** (2004) 032102. T. H. Loftus [*et al.*]{}, Phys. Rev. Letters [**106**]{} (2011) 253002. C.A. Baker [*et al.*]{}, Physics Procedia [**17**]{} (2011) 159. B. Lauss, Physics Procedia [**51**]{} (2014) 98. C.A. Baker [*et al.*]{}, Nucl. Instruments and Methods A **736** (2014) 184-203. C. Baker [*et al.*]{}, Phys. Rev. Letters [**97**]{} (2006) 131801. K. Green [*et al*]{}, Nucl. Instrum. Methods A **404** (1998) 381. N. F. Ramsey, Phys. Rev. **100** (1955) 1191. M. V. Berry, Proc. R. Soc. London A [**392**]{} (1984) 45-47. E. D. Commins, Am. J. Phys. **59** (1991) 12. S. K. Lamoreaux and R. Golub, Phys. Rev. A **71** (2005) 032104. G. Pignol and S. Roccia, Phys. Rev. A **85** (2012) 042105. A. Steyerl [*et al.*]{}, Phys. Rev. A **89** (2014) 052129. N. F. Ramsey, Phys. Rev. **76** (1949) 996; **78** (1950) 695. K. Bodek [*et al.*]{}, Nucl. Instruments and Methods A **597** (2008) 222-226. M. Kuźniak, PhD thesis, Jagiellonian University (2008). P. Knowles [*et al.*]{}, Nucl. Instrum. Methods A [**611**]{} (2009) 606. M. Horras, PhD thesis, ETH Zürich (2012). Y. Chibane, J.-M. Pendlebury and K. Green, Meas. Sci. Technol. **6** (1995) 11671. W. C. Griffith, M. D. Swallows, T. H. Loftus, [*et al.*]{}, Phys. Rev. Lett. [**102**]{} (2009) 101601. V. Helaine, PhD thesis, University of Caen Basse-Normandie (2014). H.-C. Koch [*et al.*]{}, submitted to Eur. Phys. J. D, arxiv.org/abs/1502.06366. Z. D. Grujić [*et al.*]{}, Eur. Phys. J. D [**69**]{} (2015). S. Afach [*et al.*]{}, submitted to Optics Express, arxiv.org/abs/1507.08523. S. Afach [*et al.*]{}, Phys Lett B [**739**]{} (2014) 128. [^1]: Now at University of Washington, Seattle WA, USA. [^2]: Now at Max-Planck-Institute of Quantum Optics, Garching, Germany. [^3]: Now at LPSC, Grenoble, France. [^4]: Present address: Hauptstrasse 60, CH-4455 Zunzgen, Switzerland. [^5]: On leave from INRNE, Sofia, Bulgaria. [^6]: On leave from PNPI, St. Petersburg, Russia. [^7]: Present address: Rilkeplatz 8/9, A-1040 Vienna, Austria. [^8]: Now at Michigan State University, East-Lansing, USA. [^9]: One may wonder why the effect discussed in the present article was not observed in that experiment. They actually use spectroscopy cells filled with 475 Torr of CO buffer gas acting as a UV quencher. Consequently, mercury atoms move in the diffusive regime where the motional false EDM essentially vanishes – in contrast to the ballistic regime of our mercury co-magnetometer. [^10]: It should be noted that, strictly speaking, we have only observed that frequency shifts were E-odd (measurements were done at different gradients but at a single HV value). However, the absence of physical justification for higher-order odd terms together with the excellent agreement with theory led us to disregard this possibility.

--- author: - | Thierry Mignon\ [*ENS Lyon, UMR CNRS 5669*]{}\ [*46 Allée d’Italie, 69364 Lyon Cedex, France*]{}\ [*e-mail : tmignon@@umpa.ens-lyon.fr*]{} title: An asymptotic existence theorem for plane curves with prescribed singularities --- Introduction {#introduction .unnumbered} ============ Let $d,m_1,\ldots,m_r$ be $(r+1)$ positive integers. Denote by $V(d;m_1,\ldots,m_r)$ the variety of irreducible (complex) plane curves of degree $d$ having exactly $r$ ordinary singularities of multiplicities $m_1,\ldots,m_r$. In most cases, it is still an open problem to know whether this variety is empty or not. In this paper, we will concentrate on the case where the $r$ singularities can be taken in a general position. Precisely, let $(P_1,\ldots,P_r)$ be a general $r$-tuple of point in $({\Bbb P}^2)^r$. Denote by $E$ the linear system of plane curves of degree $d$ passing through the points $P_i$ $(1\leq i\leq r)$ with multiplicity at least $m_i$. The expected dimension of $E$ is $\max (-1 ; d(d+3)/2-\sum m_i(m_i+1)/2)$. \[theorem\] Given a positive integer $m$, there exists an integer ${\mathbf{d}}'(m)$ such that, if $m_i\leq m$ for $1\leq i\leq r$ and $d\geq {\mathbf{d}}'(m)$, then : The system $E$ has the expected dimension $e$ and, if $e\geq 0$, then a general curve in $E$ is irreducible, smooth away from the $P_i$, and has an ordinary singularity of multiplicity $m_i$ at each point $P_i$. As a consequence, $V(d;m_1,\ldots,m_r)$ is not empty. The importance of this result comes from the fact that it is still valid when the expected dimension is small (which happens when the number $r$ of points is high) ; even say, when $e$ is zero. In this case, the curve is isolated in $E$, and Bertini’s theorem can not be used. Recent existence results have been proved by Greuel, Lossen and Shustin in the case of ordinary singularities, ([@gls.blowup], section 3.3) ; or even for general singularities [@gls.plane]. But in all these statements the dimension of the system $E$ must be, at least, quadratic in the degree $d$. Notice, however, that the method of [@gls.plane] together with the vanishing result of Alexander-Hirschowitz cited below (see [@al-hi.asymptotic]) would easily give theorem \[theorem\] as soon as $e\geq d+1$ (see also section \[sec.highdim\] for such considerations). As for zero dimensional systems, a previous theorem had been proved by the author [@mig.mono] for $m_i\leq 3$ and $d\geq 317$. An explicit value for ${\mathbf{d}}'(m)$ has been computed by the author : one may take ${\mathbf{d}}'(m)=2((m+2)38)^{2^{m-1}}$. According to a theorem of Alexander and Hirschowitz [@al-hi.asymptotic], it is already known that there exists a bound ${\mathbf{d}}(m)$ for the degree, above which $E$ has the expected dimension. In theorem \[theorem\] ${\mathbf{d}}'(m)$ is slightly greater than ${\mathbf{d}}(m)$ and is expressed in terms of it. It is now possible to follow the proof of [@al-hi.asymptotic] and give an explicit bound for ${\mathbf{d}}(m)$ (let us recall that [@al-hi.asymptotic] holds for any projective variety, since our bound only holds for ${\Bbb P}^2$). With this approach, it seems that the doubly exponential growth for the explicit value of ${\mathbf{d}}(m)$ is unavoidable. However this bound is far from being sharp. In fact, according to a conjecture of Hirschowitz, if the $m_i$ are in decreasing order and if $d$ is greater than $m_1+m_2+m_3$, then the system $E$ should have the expected dimension and contain an irreducible and smooth curve away from the $P_i$ (except in the well-known case $(d;m_1,\ldots,m_r)\neq (3n;n,\ldots,n)$ with $r=9$). Thus, the conjectural bound for ${\mathbf{d}}'(m)$ is $3m+1$. Due to its length, the computation of this explicit value is not described here. The author places it at the reader’s disposal. Theorem \[theorem\] is also interesting in view of recent results on the varieties $V(d;m_1,\ldots,m_r)$. Recall that the first variety of this type, $V=V(d;2,\ldots,2)$ was studied by Severi [@sev.anhangf]. He proved that $V$ is not empty and smooth if and only if $r\leq (d-1)(d-2)/2$. If in addition $r \leq d(d+3)/6$ we also know that the nodes can be taken in generic position except in the case $d=6$, $r=9$, $m_1=\cdots=m_9=2$ (case of an isolated double cubic) (see [@arb-cor.footnote] and [@treg.nodes]). In 1985, Harris [@ha.onseveri] completed this work, proving that $V(d;2,\ldots,2)$ is always irreducible. The questions of irreducibility and smoothness of general varieties of curves with prescribed singularities have been treated in many papers. Let us mention recent results for general singularities [@sh.equisingular] or for nodal curves on general surfaces in ${\Bbb P}^3$ [@ch-ci.sevvar]. However in the case considered here, i.e. plane curves with ordinary singularities, A. Bruno announced that, $V(d;m_1,\ldots,m_r)$ is irreducible, smooth and has the expected codimension assuming that it is not empty and that the singularities can be taken in generic position (conference in Toledo, September 98). This is exactly what is proved in theorem \[theorem\]. ***Strategy of the proof*** The proof of theorem \[theorem\] is based on a lemma proved by the author in [@mig.horgeo] (see also [@bossini] for a first –not differential– approach of this lemma). This result, which we called “Geometric Horace Lemma”, is inspired by the Horace method of Hirschowitz (see, for example, [@al-hi.asymptotic]). But, while the usual Horace method can only be used to compute the *dimension* of linear systems like $E$, the geometrical lemma also yields conclusions about the *irreducibility* and *smoothness* of the curves in $E$. The principle of the Geometric Horace Lemma is the following : Let us choose an irreducible and smooth plane curve $C$. Let us specialize some of the $r$ points on $C$. Denote by $y=(Q_1,\ldots,Q_r)$ this special point of $({\Bbb P}^2)^r$ and by $x$ the generic point of $({\Bbb P}^2)^r$. Two linear systems may be considered : $E_x=E$ when the points are in generic position and $E_y$ when they are in special position. The specialization from $x$ to $y$ is done in such a way, that $C$ is a base component of the system $E_y$. Thus a curve in $E_y$ is the union of $C$ and of a *residual* curve. Under some assumptions, detailed in \[horgeo\], if the generic residual curve is geometrically irreducible, smooth, and has ordinary singularities, then the general curve in $E_x$ also satisfies these properties. An important point must be mentioned : if we do not specialize enough points on $C$, then $C$ is not a base component of $E_y$ and the method fails. But, if we specialize too many points, then the dimension of the linear system grows : $\dim E_y > \dim E_x$. This phenomenon is controlled with the help of differential conditions. It means that we have to consider some sub-systems of curves bound to pass through infinitely near points. Here is the main point of the proof : by specializing too many points on the curve $C$, it is possible to make the dimension of $E$ grow considerably ; i.e. grow as high as the degree $d$. Then, assuming that some vanishing property holds true, the residual system is base point free, and Bertini’s theorem can be used. As a consequence, a general residual curve is smooth, irreducible, and the intersection variety described above is irreducible. To make all this strategy work, we still have to check the vanishing property referred to above. Roughly speaking, it means that the residual system has the expected dimension. To prove this, we make use of the following vanishing result of Alexander and Hirschowitz [@al-hi.asymptotic] : Given an integer $m$, there exists an integer ${\mathbf{a}}(m)$, and for $a\geq {\mathbf{a}}(m)$, there exists another bound $d_0(a,m)$ such that, if $C$ is the generic curve of degree $a$, if $d\geq d_0(a,m)$ and if the points $P_i$ are either generic in ${\Bbb P}^2$ or generic on $C$ (not too many of them) then the system $E$ has the expected dimension. In view of this result, the last choices are made : As for the curve $C$, we choose the generic curve of degree ${\mathbf{a}}(m)$ ; and we only consider systems of curves of degree higher than ${\mathbf{d}}(m)=d_0({\mathbf{a}}(m),m)$ (in fact, the final value ${\mathbf{d}}'(m)$ is greater than ${\mathbf{d}}(m)$, as appears in theorem \[theorem2\]). ***Contents*** The article is organized as follows : In the first part, notations and definitions are set. In particular, we describe the universal variety which parameterizes the curves we are studying. In the second and third sections, we restate respectively the Geometric Horace Lemma, and the vanishing theorem of Alexander and Hirschowitz. These are the two main tools in the proof of theorem \[theorem\]. The fourth section is devoted to the study of linear systems of “high” dimension, (precisely, a dimension greater than the degree $d$). In particular, when the $r$ points are in a good position, so that the vanishing lemma can be used, we show that theorem \[theorem\] is true for these systems. In the last section , theorem \[theorem\] is proved. Curves on rational surfaces {#sec.prelim} =========================== In the introduction, the situation has been described on the plane. Actually, most of the proofs will be done on the plane blown-up along the $r$ points $P_1,\ldots,P_r$. In this section, we shall describe the family of rational surfaces obtained by blowing up a family of $r$ disjoint sections in ${\Bbb P}^2$, and the families of curves on these surfaces. We shall also set up most of the notations used in the article. Families of rational surfaces and relative divisors --------------------------------------------------- Let $r$ be a positive integer, and $X\subset ({\Bbb P}^2)^r$ be the open subset of $r$-tuples of distinct points. The morphism ${\Bbb P}^2_X={\Bbb P}^2\times X{\longrightarrow}X$ is naturally endowed with $r$ sections : $$\begin{array}{cccc} \gamma_i : & X & {\longrightarrow}& {\Bbb P}^2\times X\\ &(P_1,\ldots,P_r)& {\longrightarrow}& (P_i,(P_1,\ldots,P_r)) \end{array}$$ Let $\Gamma_i$ be the image of $\gamma_i$ ; $\Gamma=\cup_{i=1}^r \Gamma_i$ is a nonsingular variety of ${\Bbb P}^2_X$. Blowing up ${\Bbb P}^2_X$ along $\Gamma$ produces a family of rational surfaces, parameterized by $X$ : $S_X{\stackrel{b}{\longrightarrow}}{\Bbb P}^2_X{\longrightarrow}X$. Let $\pi$ denote the composed morphism $S_X{\longrightarrow}X$. At any point $x=(P_1,\ldots,P_r)$ of $X$, the fiber of $\pi$ will be denoted by $S_x$. This surface $S_x$ is simply the projective plane blown up along the $r$ points $P_1,\ldots,P_r$. Let us keep in mind that a relative effective Cartier divisor of $S_X$ on $X$ is simply an ideal sheaf ${\mathcal{I}}_D$ on $S_X$, locally principal, and not a zero-divisor in any fiber of $\pi$ (see [@gro.pic]). These ideal sheaves are flat on $X$. **Examples** : 1 Consider a line $L$ on ${\Bbb P}^2$, $L\times X\subset {\Bbb P}^2\times X$ the trivial family of lines above $X$, and $H_X=b^{-1}(L\times X)$ the total transform of $L\times X$ in $S_X$. The ideal sheaf ${\mathcal{I}}_{H_X}$ is a relative effective Cartier divisor of $S_X$ on $X$. 2 Consider now $E_{i,X}$, the exceptional divisor obtained by blowing up the irreducible smooth variety $\Gamma_i$ ; the ideal sheaf ${\mathcal{I}}_{E_{i,X}}$ is also a relative effective Cartier divisor of $S_X$ on $X$. Intersection pairing and linear systems --------------------------------------- For any $x\in X$, the Picard group of the surface $S_x$ is endowed with the usual base : $[H_x],[-E_{1,x}],\ldots,[-E_{r,x}]$. The relative Cartier divisors being flat on $X$, these bases satisfy the following property : Let $D$ be a relative Cartier divisor of $S_X{\longrightarrow}X$. For any point $x\in X$, let $[D_x]\in { \Bbb Z}^{r+1}$ be the class of the sheaf ${\mathcal{O}}_{S_x}(D)$ expressed in the base defined above ; then $[D_x]$ does not depend on $x$. The canonical divisor of ${\ensuremath{\mathrm{Pic\ }}}S_x$ is ${\ensuremath{\underline{\omega}}}_x=(-3;(-1)^r)$ (the notation $(-1)^r$ means that the integer $(-1)$ is repeated $r$ times ; this convention will be kept in the sequel). The intersection pairing of ${\ensuremath{\mathrm{Pic\ }}}S_x$ is as follows : $(d;m_1,\ldots,m_r).(c;n_1,\ldots,n_r)=dc-m_1n_1-\ldots -m_rn_r$. Let ${\ensuremath{\underline{d}}}=(d;m_1,\ldots,m_r)\in {\ensuremath{\mathrm{Pic\ }}}S_x$. The sheaf ${\mathcal{O}}(dH_x-\sum_{i=1}^r m_iE_{i,x})$ will be denoted by ${\mathcal{O}}({\ensuremath{\underline{d}}})$, and the complete linear system of ${\mathcal{O}}({\ensuremath{\underline{d}}})$ (i.e. the projective space ${\Bbb P}(H^0(S_x,{\mathcal{O}}({\ensuremath{\underline{d}}})))$ will be denoted by ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$. The Riemann-Roch theorem for surfaces allows us to compute the Euler-Poincaré characteristic of ${\mathcal{O}}({\ensuremath{\underline{d}}})$ : $\chi({\ensuremath{\underline{d}}})=\frac{(d+1)(d+2)}{2}- \sum_{i=1}^r\frac{m_i(m_i+1)}{2}.$ One may also compute the arithmetical genus of the eventual sections of ${\mathcal{O}}({\ensuremath{\underline{d}}})$ : $g({\ensuremath{\underline{d}}})=\frac{(d-1)(d-2)}{2}- \sum_{i=1}^r\frac{m_i(m_i-1)}{2}.$ Suppose that $d\geq -2$. By Serre Duality Theorem, $h^2(S_x,{\mathcal{O}}({\ensuremath{\underline{d}}}))=0$. The *expected dimension* for ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ is then : $\max(\chi({\ensuremath{\underline{d}}})-1,-1)$. (An empty system is supposed to have dimension $-1$). A system ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ will be said to be *regular* if it has the expected dimension. More generally, a sheaf ${\mathcal{F}}$ on $S_x$ will be said to be regular if $h^0(S_x,{\mathcal{F}})= \max(\chi({\mathcal{F}}),0)$ and $h^1(S_x,{\mathcal{F}})= \max(-\chi({\mathcal{F}}),0)$. Universal family of divisors {#family} ---------------------------- Let ${\ensuremath{\underline{d}}}=(d;m_1,\ldots,m_r)$ be an $r$-tuple of integers, and let us define $m'_i=\max(m_i,0)$ and ${\ensuremath{\underline{d}}}'=(d;m'_1,\ldots,m'_r)$. Let $\Gamma_i$ be the image of the $i^{th}$ natural section $\gamma_i$ defined above, and let $Z\subset {\Bbb P}^2_X$ be the scheme defined by the ideal ${\mathcal{I}}_{\Gamma_1}^{m'_1}\ldots {\mathcal{I}}_{\Gamma_r}^{m'_r}$. The scheme $Z$ is a flat family above $X$ whose fibers $Z_x$ are unions of $r$ fat points of multiplicities $m'_1,\ldots,m'_r$ (see section \[sec.vanish\] for a definition of fat points). Consider $x=(P_1,\ldots,P_r)\in X$. The linear system $|{\mathcal{I}}_{Z_x}(d)|$ is the system of plane curves of degree $d$ passing through each point $P_i$ with multiplicity at least $m'_i$. One can easily see that this system is isomorphic to ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$. Consider now the linear system ${\Bbb P}(H^0({\Bbb P}^2,{\mathcal{O}}(d))){\stackrel{\sim}{\longrightarrow}}{\Bbb P}^{d(d+3)/2}$ of plane curves of degree $d$. One may define (here, only under a “set-theoretic” point of view, but it is endowed with a natural scheme structure) a subscheme $F$ of ${\Bbb P}^{d(d+3)/2}\times X$ in the following way : $\begin{array}{rcll} F& = & \{(D,(P_1,\ldots,P_r))\ | & \mbox{$D$ passes through each point} \\ & & &\mbox{$P_i$ with multiplicity at least $m'_i$}\}\\ \end{array}$ This scheme $F$ parameterizes a canonical family of curves ${\mathcal{D}}'\subset {\Bbb P}^2\times F$ : given $x=(D,(P_1,\ldots,P_r))$, the fiber ${\mathcal{D}}'_x$ simply is the curve $D$. As above, it is possible to blow-up the variety ${\Bbb P}^2_F={\Bbb P}^2\times F$ along the $r$ disjoint natural sections. Let $b_F:S_F{\longrightarrow}{\Bbb P}^2_F$ be this blowing-up. By assumption, the divisor ${\mathcal{D}}=b_F^{-1}({\mathcal{D}}')-\Sigma_{i=1}^r m_iE_i$ is effective : it is a relative effective Cartier divisor of the family $S_F{\longrightarrow}F$. Moreover, ${\mathcal{D}}$ is a universal divisor : Let ${\ensuremath{\underline{d}}}\in { \Bbb Z}^{r+1}$. The functor ${\mathbf{F}}_{{\ensuremath{\underline{d}}}}$ from $\mathbf{Schemes/X}$ to $\mathbf{Sets}$ such that ${\mathbf{F}}_{{\ensuremath{\underline{d}}}}(Y)=\{D\subset S_Y, \mbox{relative effective divisor of class ${\ensuremath{\underline{d}}}$ on $Y$}\}$ is represented by the couple $(F,{\mathcal{D}})$. (This proposition is detailed in [@mig.these] ; see also [@gro.pic] and [@no.planecur]). Let $p:F{\longrightarrow}X$ be the natural projection from $F\subset {\Bbb P}^{d(d+3)/2} \times X$ to $X$. The fiber of $p$ over a point $x\in X$ is nothing but the linear system ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$. Let $x'$ be the generic point of this fiber. By definition of the universal divisor ${\mathcal{D}}$, the curve ${\mathcal{D}}_{x'}$ is the generic curve of ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$. It will be denoted by ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$. Suppose now that a point $y\in X$ is a specialization of $x$. We will say that the curve ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ *specializes to the curve* ${\mathfrak{D}}_y({\ensuremath{\underline{d}}})$ if the generic point of $p^{-1}(y)={\mathcal{L}}_y({\ensuremath{\underline{d}}})$ is a specialization of $x'$. This notion is of special importance if one expects to find properties of ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ from those of ${\mathfrak{D}}_y({\ensuremath{\underline{d}}})$. In particular, if ${\mathfrak{D}}_y({\ensuremath{\underline{d}}})$ is geometrically irreducible, smooth, or has ordinary singularities, then the same holds for ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$. If the dimension of ${\mathcal{L}}_y({\ensuremath{\underline{d}}}) $ equals the dimension of ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$, one can easily see that ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ always specializes to ${\mathfrak{D}}_y({\ensuremath{\underline{d}}})$. If the dimension grows after the specialization, extra conditions are needed. In fact, it is sufficient to prove that the strata of the cohomological stratification (associated to the sheaf ${\mathcal{O}}_{S_X}({\ensuremath{\underline{d}}})$) have sufficiently big codimension. This can be done with the help of differential methods (see [@mig.horgeo], and the lemma \[horgeo\]). The Geometric Horace Lemma {#sec.horgeo} ========================== In this section, the Geometric Horace Lemma is restated and commented on. This lemma was proved by the author in [@mig.horgeo]. Let us first give some notations and conventions : Let $y=(Q_1,\ldots,Q_r)$ be a point of $X$ (the notation $(Q_1,\ldots,Q_r)$ is slightly incorrect, since $y$ is generally not a closed point). Let $G$ be a closed integral subscheme of ${\Bbb P}^2$. We will say that the $a$ points $Q_1,\ldots, Q_a$ $(0\leq a\leq r)$ are *generic and independent* on $G$ if $y$ is the generic point of a subvariety $Y\subset X$ such that $Y=G^a\times V$, where $V$ is an irreducible subscheme of $({\Bbb P}^2)^{r-a}$. Suppose now that the $i$-th point $Q_i$ is a nonsingular point of a plane curve $C$. On the rational surface $S_y$, the intersection point of the exceptional divisor $E_i$ and the strict transform ${\ensuremath{\widetilde{C}}}$ will be denoted by $Q_i^C$ (and its ideal sheaf, ${\mathcal{I}}_{Q_i^C}$). Let ${\mathcal{O}}({\ensuremath{\underline{d}}}=(d;m_1,\ldots,m_r))$ be an invertible sheaf on $S_x$. Global sections of the sheaf ${\mathcal{I}}_{Q_i^C}({\ensuremath{\underline{d}}})$ can be seen as plane curves of degree $d$ having multiplicity at least $m_j$ at each point $Q_j$ and, if the multiplicity at $Q_i$ is exactly $m_i$ having a branch tangent to $C$ at this point. \[horgeo\] Let ${\ensuremath{\underline{d}}}=(d;m_1,\ldots,m_r)$ be an $r$-tuple of positive integers, and $x=(P_1,\ldots,P_r)$, $y=(Q_1,\ldots,Q_r)$ be two points of $X$ such that $x$ specializes to $y$. Let $C$ be a plane curve, and ${\ensuremath{\widetilde{C}}}:={\ensuremath{\widetilde{C}}}_y$ its strict transform on $S_y$. Assume that ${\ensuremath{\widetilde{C}}}$ is geometrically irreducible and smooth, of class ${\ensuremath{\underline{c}}}\in {\ensuremath{\mathrm{Pic\ }}}S_y$ and of genus $g({\ensuremath{\underline{c}}})$. **Dimension and specialization** Suppose that : $-{\alpha}:=\chi({\mathcal{O}}_{{\ensuremath{\widetilde{C}}}}({\ensuremath{\underline{d}}}))={\ensuremath{\underline{d}}}.{\ensuremath{\underline{c}}}+1-g({\ensuremath{\underline{c}}})\leq 0$  ; At the point $y$, $g({\ensuremath{\underline{c}}})$ points are generic and independent on $C$. If ${\alpha}\geq 1$, there exist ${\alpha}+1$ integers $i_1,\ldots,i_{{\alpha}+1}$ such that :\ $P_{i_1},\ldots,P_{i_{{\alpha}+1}}$ are generic and independent in the plane, and\ $Q_{i_1},\ldots,Q_{i_{{\alpha}+1}}$ are generic and independent on $C$. If ${\alpha}=0$, $H^0(S,{\mathcal{O}}({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}))$ has the expected dimension : $\chi({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})=\chi({\ensuremath{\underline{d}}})$.\ If ${\alpha}\geq 1$, $H^0(S,{\mathcal{I}}_{Q_{i_1}^C\cup\cdots\cup Q_{i_{{\alpha}+1}}^C}({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}))$ has the expected dimension : $\chi({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})-{\alpha}-1 = \chi({\ensuremath{\underline{d}}})-1$, Then ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ is regular, and, if $\chi({\ensuremath{\underline{d}}})>0$, ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ specializes to ${\mathfrak{D}}_y({\ensuremath{\underline{d}}})$. **Irreducibility** Suppose, moreover, that $\chi({\ensuremath{\underline{d}}})>0$ and : the system ${\mathcal{L}}_x({\ensuremath{\underline{c}}})$ is empty. ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ is geometrically irreducible. Then ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ is geometrically irreducible. **Smoothness** Suppose finally that : If ${\alpha}=0$, $y$ is normal and the closure of $x$ ; ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ meets ${\ensuremath{\widetilde{C}}}$ transversally ; ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})\cap {\ensuremath{\widetilde{C}}}$ is irreducible (not: geometrically irreducible) ; ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ is a smooth curve, then ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ is smooth. The system ${\mathcal{L}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$, and the curve ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ are respectively called *residual system* and *residual curve*. In fact, ${\mathfrak{D}}_y({\ensuremath{\underline{d}}})$ is the union of ${\ensuremath{\widetilde{C}}}$ and ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$. The curve ${\ensuremath{\widetilde{C}}}$ being well-known, this lemma allows one to get information on ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$, from the properties of ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ and the relation between ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ and ${\ensuremath{\widetilde{C}}}$. Condition 9 is certainly best described in an example : assume that $\dim {\mathcal{L}}_y({\ensuremath{\underline{d}}})=0$, and denote by $Y$ the adherence of $y$ in $X$. There is only one curve in the system ${\mathcal{L}}_z({\ensuremath{\underline{d}}})$ for every closed point $z$ in an open subset of $Y$. Then the intersection of ${\ensuremath{\widetilde{C}}}$ and the residual curve in ${\mathcal{L}}_z({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ makes, as $z$ varies, a covering of degree $({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}).{\ensuremath{\underline{c}}}$ over the open subset of $Y$. The condition 9 means that this “intersection variety” is irreducible ; or in other words, that the monodromy group of this covering acts transitively on the intersection points of ${\ensuremath{\widetilde{C}}}$ and the residual curve ${\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$. A case of special interest is the case where the number ${\alpha}$ of 1 is positive. Roughly speaking, this situation arises when one specializes too many points on the curve $C$. Let us suppose that $\chi({\ensuremath{\underline{d}}})>0$. Considering the exact sequence $$\begin{aligned} \label{exact} 0{\longrightarrow}{\mathcal{O}}_{S_y}({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}){\longrightarrow}{\mathcal{O}}_{S_y}({\ensuremath{\underline{d}}}){\longrightarrow}{\mathcal{O}}_{{\ensuremath{\widetilde{C}}}}({\ensuremath{\underline{d}}}){\longrightarrow}0,\end{aligned}$$ we find that $\chi({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})=\chi({\ensuremath{\underline{d}}})+{\alpha}$. Since ${\mathcal{I}}_{Q_{i_1}^C\cup\cdots\cup Q_{i_{{\alpha}+1}}^C}({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}) $ is regular, ${\mathcal{L}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ also is regular and $\dim {\mathcal{L}}_y({\ensuremath{\underline{d}}})=\dim {\mathcal{L}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})=\dim {\mathcal{L}}_x({\ensuremath{\underline{d}}})+{\alpha}$. Thus, the dimension has grown by ${\alpha}$. \[ordinary\] [As regards to the ordinary singularities of the plane projection of ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$, the Geometric Horace Lemma yields no conclusion. But, if ${\mathfrak{D}}_y({\ensuremath{\underline{d}}})={\mathfrak{D}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})\cup {\ensuremath{\widetilde{C}}}$ meets the exceptional divisor $E_i$ in $m_i$ distinct points, ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ also possesses this property. Hence, its projection has only ordinary singularities of the expected multiplicity.]{.nodecor} [This is in particular the case if $C$ has ordinary singularities and ${\mathcal{L}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ is base point free (Bertini’s theorem applied to $E_i$ and the restricted system ${\mathcal{L}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})_{|E_i}$).]{.nodecor} An asymptotic vanishing theorem {#sec.vanish} =============================== In order to use the Geometric Horace Lemma, we have to check that some linear system is regular (condition 4 of \[horgeo\]). This will be done with the help of an asymptotic vanishing theorem of Alexander and Hirschowitz. In this section, we restate this result and the adequate definitions. As a corollary, we write down precisely the vanishing lemma used in the proof of theorem \[theorem\]. Here, opposed to the other sections, all the work is done on the plane, without blowing it up. It is a natural choice when dealing with the *dimension* of a linear system, without consideration of the smoothness of its sections. Let us first recall some definitions : As usual, a *fat point* of support $P\in {\Bbb P}^2$ is a subscheme $P^m$ of ${\Bbb P}^2$ defined by the ideal ${\mathcal{I}}_P^m$ ; the integer $m$ is called the *multiplicity* of $P$. If $Z$ is a zero dimensional subscheme of ${\Bbb P}^2$, the *degree* of $Z$, denoted by $\deg Z$, is the length of the ring ${\mathcal{O}}_Z$. As an example, $\deg P^m=m(m+1)/2$. Let $P\in {\Bbb P}^2$ be a nonsingular point of a plane curve $C$, and $i,m$ be two integers such that $0\leq i\leq m-1$. The *$i$-th residue point* supported by $P$, of multiplicity $m$, with respect to $C$, is the scheme defined by the ideal ${\mathcal{I}}_P^{m-1}\cap({\mathcal{I}}_C^i+{\mathcal{I}}_P^m)$. It is denoted by $D_{C}^i(P^m)$ or $D^i(P^m)$ if no confusion can arise. A residue of type $D^{m-1}(P^m)$ is called a *simple residue* ([@al-hi.asymptotic], 2.2.). Condition 4 of lemma \[horgeo\] is easily expressed with the help of residue points : For example, if $P_1,\ldots,P_r$ are points of ${\Bbb P}^2$, such that $P_1$ is a nonsingular point of a curve $C$, and if ${\ensuremath{\underline{d}}}=(d;m_1,\ldots,m_r)$ is an $(r+1)$-tuple of positive integers, we consider the zero dimensional scheme $Z=D^1(P_1^{m_1+1})\cup P_{2}^{m_{2}} \cup\cdots\cup P_r^{m_r}$. Then $|{\mathcal{I}}_{P_1^C}({\ensuremath{\underline{d}}})|$ is isomorphic to $|{\mathcal{I}}_Z(d)|$. What we still have to show is that $h^1({\Bbb P}^2,{\mathcal{I}}_Z(d))$ or $h^0({\Bbb P}^2,{\mathcal{I}}_Z(d))$ is zero. The vanishing result of Alexander and Hirschowitz deals with some special types of systems $|{\mathcal{I}}_Z(d)|$ defined below : \[defcand\] Let $d,m$ and $a$ be three positive integers ; denote by $C$ the generic plane curve of degree $a$. An *$(m,a)$-configuration* is a zero dimensional scheme $Z={\ensuremath{\mathrm{Const}}}(Z)\cup{\ensuremath{\mathrm{Free}}}(Z)$, where : ${\ensuremath{\mathrm{Free}}}(Z)$ is the *free* part of $Z$ ; it is a union of fat points, of generic and independent support in ${\Bbb P}^2$. ${\ensuremath{\mathrm{Const}}}(Z)$ is the *constrained* part of $Z$ ; it is a union of fat points or simple residue points, of generic and independent support in $C$. All these points are supposed to have a multiplicity less than or equal to $m$. A *$(d,m,a)$-candidate* is an $(m,a)$-configuration $Z$ such that $\chi({\mathcal{I}}_Z(d))\leq 0$ and $h^0(C,{\mathcal{O}}_{C}(d))\geq \deg(Z\cap C)$. A $(d,m,a)$-candidate such that $H^0({\Bbb P}^2,{\mathcal{I}}_Z(d))=0$ is said to be *winning*. \[remark\] [If $Z$ is an $(m,a)$-configuration such that $\chi({\mathcal{I}}_Z(d))>0$ and $h^0(C,{\mathcal{O}}_{C}(d))\geq \deg(Z\cap C)$, one also says that $Z$ is a $(d,m,a)$-candidate. But in this case, $Z$ is winning means that $h^1$ vanishes, whereas $h^0$ is positive.]{.nodecor} The crucial vanishing result is the following : \[theoalhi\] Given a positive integer $m$, there exists an integer ${\mathbf{a}}(m)$ and, for each $a\geq {\mathbf{a}}(m)$, an integer $d_0(a,m)$ such that : if $a\geq {\mathbf{a}}(m)$ and $d\geq d_0(a,m)$ then any $(d,m,a)$- candidate is winning. Unfortunately, this vanishing result involves *simple residues*, of type $D^{m-1}$, whereas the needed condition involves residues of type $D^1$. With our “asymptotical” point of view, this is essentially a technical problem. But to solve it, a little more Horace method is needed : Let $Z$ be a closed subscheme of ${\Bbb P}^2$, and $C$ be an irreducible and reduced plane curve. The *trace* of $Z$, denoted $Z\cap C$, is the scheme defined by the ideal ${\mathcal{I}}_Z+{\mathcal{I}}_C$. The residue of $Z$ with respect to $C$, denoted $Z'$, is the scheme defined by the conductor ideal $({\mathcal{I}}_Z:{\mathcal{I}}_C)$. \[hordiff\] Let $C$ be an irreducible and smooth plane curve of degree $a$ and genus $g=(a-1)(a-2)/2$, and $d$ be an integer greater than $a$. Let $Z=Z_0\cup P_1^{m_1}\cup\cdots \cup P_\beta ^{m_\beta }$ be a zero dimensional subscheme of ${\Bbb P}^2$ such that $P_1,\ldots,P_\beta $ are generic and independent points of ${\Bbb P}^2$. Denote also by $Q_1,\ldots,Q_\beta $, $\beta $ generic and independent points of $C$. Suppose that $\chi({\mathcal{I}}_Z(d))\leq 0$. If : $i)$ $H^1({\Bbb P}^2,{\mathcal{I}}_{Q_1^{m_1}\cup\cdots\cup Q_\beta ^{m_\beta }}(d-a))=0$ ; $ii)$ $\beta = da+1-g-\deg(Z\cap C)$ ; $iii)$ ${\mathcal{I}}_{(Z_0\cap C)\cup Q_1 \cup\cdots\cup Q_\beta }(d)$ is a regular invertible sheaf of $C$ ; $iv)$ $H^0({\Bbb P}^2,{\mathcal{I}}_{Z'_0\cup D^{m_1-1}(Q_1^{m_1})\cup\cdots\cup D^{m_\beta -1}(Q_{\beta}^{m_\beta })}(d-a))=0$ ; then $H^0({\Bbb P}^2,{\mathcal{I}}_Z(d))=0$, as expected. \[vanish\] Let $m,a$ be two positive integers, and $C$ be the generic plane curve of degree $a$. Denote by $x=(O_1,\ldots,O_t,P_1,\ldots,P_r)$ the generic point of $C^t\times ({\Bbb P}^2)^r$, and consider an integer ${\alpha}$ such that $0\leq {\alpha}\leq t$. Let ${\ensuremath{\underline{d}}}=(d;n_1,\ldots,n_t,m_1,\ldots,m_r)\in {\ensuremath{\mathrm{Pic\ }}}S_x$ such that $n_i\leq m-1,\ m_i\leq m$, and suppose that $\chi({\ensuremath{\underline{d}}})-{\alpha}= 0$. If $i)$ $a\geq \max({\mathbf{a}}(m)\ ,\ 4m)$ ; $ii)$ $d\geq \max(d_0(a,m)+a\ ,\ 2am)$ ; $iii)$ $da+1-g-n_1-\ldots-n_t-{\alpha}\geq 0$ ; then ${\mathcal{I}}_{O_1^C\cup\ldots\cup O_{{\alpha}}^C}({\ensuremath{\underline{d}}})$ is regular. [<span style="font-variant:small-caps;">Proof :  </span>]{}Let $Y_0=D^1(O_1^{n_1+1})\cup\cdots\cup D^1(O_{\alpha}^{n_{\alpha}+1})\cup O_{{\alpha}+1}^{n_{{\alpha}+1}} \cup\cdots\cup O_t^{n_t}$, and $Y=Y_0\cup P_1^{m_1}\cup\cdots\cup P_r^{m_r}$. Clearly, $\chi({\mathcal{I}}_Y(d))=\chi({\ensuremath{\underline{d}}})-{\alpha}=0$, therefore ${\mathcal{I}}_{O_1^C\cup\ldots\cup O_{{\alpha}}^C}({\ensuremath{\underline{d}}})$ is regular if and only if $H^0({\Bbb P}^2,{\mathcal{I}}_Y(d))=0$. $\bullet$ Let us first prove that there exists a non negative integer $s$ such that : $$\begin{aligned} \label{ajust} \beta&:=& da+1-g-{\alpha}-\sum_{i=1}^tn_i-\sum_{j=1}^sm_j\in [0;m-1]\\ \label{derivable} s+\beta&\leq r\end{aligned}$$ Since $0<m_i\leq m$, it will be enough to show that : $$\begin{array}{ll} \ &\!\! da+1-g-{\alpha}-\sum_{i=1}^t n_i-\sum_{j=1}^{r-m+1}m_j\leq m-1\\ \Longleftarrow_{(m_j\leq m)}&\!\! da+1-g-{\alpha}-\sum_{i=1}^t n_i-\sum_{j=1}^{r} m_j + (m-1)m\leq m-1\\ \Longleftarrow_{(m_j,n_i\leq m)}&\!\! da+1-g-{\alpha}\\ &\mbox{\qquad}-\frac{2}{m+1}\left(\sum \frac{n_i(n_i+1)}2 +\sum\frac{m_i(m_i+1)}2\right) +(m-1)^2 \leq 0\\ {\Longleftrightarrow}&\!\! da+1-g-{\alpha}-\frac{2}{m+1}\left(\frac{(d+1)(d+2)}2-\chi({\ensuremath{\underline{d}}})\right) + (m-1)^2 \leq 0\\ {\Longleftrightarrow}_{(\chi({\ensuremath{\underline{d}}})= {\alpha})} &\!\! da+1-g-\frac{(d+1)(d+2)}{(m+1)}-{\alpha}\left(1-\frac{2}{m+1}\right) + (m-1)^2 \leq 0\\ \Longleftarrow_{(m>0)} &\!\! da+1-\frac{(a-1)(a-2)}{2} -\frac{(d+1)(d+2)}{(m+1)} + (m-1)^2 \leq 0\\ \Longleftarrow_{\scriptscriptstyle{(\!d+1\geq a(\!m+1\!)\!)}} &\!\! 1-\frac{(a-1)(a-2)}{2} -2a + (m-1)^2 \leq 0\\ \Longleftarrow_{(a\geq 4 m)} &\!\! 1- \frac{(4 m-1)(4 m-2)}{2} -8 m + (m-1)^2 \leq 0\\ {\Longleftrightarrow}&\!\! -7m^2-4m+1\leq 0, \mbox{\ \ which is true.} \end{array}$$ $\bullet$ Consider $Q_1,\ldots,Q_{s+\beta}$, $(s+\beta)$ generic and independent points on $C$. Denote by $Z_0$ and $Z$ the schemes $$\begin{aligned} Z_0&:=& Y_0\cup Q_1^{m_1}\cup\cdots\cup Q_s^{m_s}\cup P_{s+\beta+1}^{m_{s+\beta+1}}\cup \cdots\cup P_r^{m_r}\\ Z &:=& Z_0\cup P_{s+1}^{m_{s+1}}\cup \cdots\cup P_{s+\beta}^{m_{s+\beta}}.\end{aligned}$$ By the Semicontinuity Theorem, if $H^0({\Bbb P}^2,{\mathcal{I}}_Z(d))=0$, then $H^0({\Bbb P}^2,{\mathcal{I}}_Y(d))=0$ as expected. To prove that $H^0({\mathcal{I}}_Z(d))$ is equal to zero, we make use of proposition \[hordiff\]. Let us specialize the points $P_{s+1},\ldots,P_{s+\beta}$ to the points $Q_{s+1},\ldots,Q_{s+\beta}$. The following relation,which bound the number of generic points on $C$, will be useful : $$\label{bigts} t+s+\beta \geq 2a^2-\frac{a^2}{2m}$$ This inequality comes from (\[ajust\]), which yields $\sum_{i=1}^t n_i + \sum_{j=1}^{s}m_j +\beta \geq da+1-g$. Since $n_i,m_j\leq m$ and $d\geq 2am$ one gets $m(t+s+\beta)\geq 2a^2m-a^2/2 +3a/2$. Let us check conditions $i$ to $iv$ of \[hordiff\] : $i)$ $H^1({\Bbb P}^2,{\mathcal{I}}_{Q_{s+1}^{m_{s+1}}\cup\cdots\cup Q_{s+\beta}^{m_{s+\beta}}}(d-a))=0$ by the lemma \[xu\] below. $ii)$ By definition of $Z$, $\deg Z\cap C= {\alpha}+\sum_{i=1}^t n_i+\sum_{j=1}^s m_j$. So that $\beta=da+1-g-\deg Z\cap C$. $iii)$ The divisor of $C$ defined by the ideal ${\mathcal{J}}={\mathcal{I}}_{(Z_0\cap C)\cup Q_{s+1} \cup\cdots\cup Q_{s+\beta}}$ is supported on the $t+s+\beta$ points $O_1\ldots,O_t,Q_1,\ldots,Q_{s+\beta}$ which are generic and independent on $C$. Hence, if $t+s+\beta\geq g$, ${\mathcal{J}}(d)$ is a nonspecial invertible sheaf. But, $t+s+\beta \geq 2a^2-a^2/(2m)$ (\[bigts\]) and then $t+s+\beta \geq a^2/2\geq g$. $iv)$ Let $T=Z'_0\cup D^{m_{s+1}-1}(Q_{s+1}^{m_{s+1}}) \cup\cdots\cup D^{m_{s+\beta}-1}(Q_{s+\beta}^{m_{s+\beta}})$. Since $Z'_0$ is a union of fat points (the $D^1$ have disappeared), $T$ is an $(m,a)$-configuration. Let us prove that it is a $(d-a,m,a)$-candidate. From the definition of $\beta$, one easily sees that $\chi({\mathcal{I}}_T(d-a))=\chi({\mathcal{I}}_Z(d))=0$. The second condition is : $h^0(C,{\mathcal{O}}_C(d-a))-\deg (T\cap C) \geq 0$. If $s+1\leq j\leq s+\beta$, then $\deg (D^{m_j-1}(Q_j^{m_j}))$ equals $m_j$ if $m_j> 1$ and $0$ if $m_j=1$. So the inequality can be checked as follows : $$\begin{array}{ll} & h^0(C,{\mathcal{O}}_C(d-a))-\deg (T\cap C)\\ \geq & (d-a)a+1-g-\sum_{i=1}^t (n_i\!-\!1)-\! \sum_{j=1}^s(m_j\!-\!1)\!-\sum_{j=s+1}^{s+\beta}m_j \\ =_{(\ref{ajust})}& -a^2+t+s+{\alpha}+\beta - \sum_{j=s+1}^{s+\beta}m_j \\ \geq_{(m_j\leq m)}& -a^2+t+s+{\alpha}+\beta -m(m-1)\\ \geq_{\mathrm{(\ref{bigts})}}& -a^2+(2a^2-a^2/2m) -m(m-1)\\ \geq_{(a\geq 4 m)}& 15m^2 -7m \geq 0. \end{array}$$ Thus $T$ is a $(d-a,m,a)$-candidate. By assumption, $a\geq {\mathbf{a}}(m)$, and $d-a\geq d_0(a,m)$ ; hence, by Proposition \[theoalhi\], $T$ is winning and $H^0({\mathcal{I}}_T(d-a))=0$. It is now allowed to apply proposition \[hordiff\] ; it gives $H^0({\mathcal{I}}_Z(d))=0$. [$\square$]{} \[xu\] Under the assumptions of corollary \[vanish\], consider $Q_1,\ldots,Q_\beta$, $\beta $ generic points of $C$, and $m_1,\ldots,m_\beta$, $\beta $ integers bounded by $m$. If $\beta \leq (m-1)$, then $H^1({\Bbb P}^2,{\mathcal{I}}_{Q_1^{m_1}\cup\cdots\cup Q_\beta^{m_\beta}}(d-a))=0$. [<span style="font-variant:small-caps;">Proof :  </span>]{}The only case we need to consider is $\beta=m-1$ and $m_1=\ldots=m_\beta=m$. By Xu’s theorem ([@xu.ample], theorem 3), $H^1({\Bbb P}^2,{\mathcal{I}}_{Q_1^{m}\cup\cdots\cup Q_\beta^{m}}(d-a))=0$, as soon as $d-a\geq 3m$ and $(d-a+3)^2>(10/9)\sum^{\beta}_{i=1} (m_i+1)^2=(10/9)(m-1)(m+1)^2$. By assumption, $d-a\geq 2am-a\geq am$ and $a\geq \sqrt 2 m$, hence $(d-a+3)^2\geq a^2m^2 \geq 2m^4 \geq (10/9)(m-1)(m+1)^2$ for any positive $m$. If $a\geq 3$ then $d-a \geq 3m$. If $a=2$ then $m$ necessary equals $1$, and the lemma is clearly true. [$\square$]{} Systems of high dimension {#sec.highdim} ========================= Given a system ${\mathcal{L}}({\ensuremath{\underline{d}}})$ of a sufficiently high dimension, Theorem \[theorem\] may be proved with the help of Bertini’s theorem. The main point consists in showing that ${\mathcal{L}}({\ensuremath{\underline{d}}})$ is base point free ; this is done here with the vanishing theorem of the preceding section and a kind of Castelnuovo-Mumford’s argument. \[highdim\] Let $m$ be a positive integer, $a\geq {\mathbf{a}}(m)$ (see prop. \[theoalhi\]) and $C$ the generic curve of degree $a$. Let $x=(P_1,\ldots,P_r)$ be the generic point of $C^s\times ({\Bbb P}^2)^{r-s}$, $(0\leq s\leq r)$ and ${\ensuremath{\underline{d}}}=(d;m_1,\ldots,m_r)\in {\ensuremath{\mathrm{Pic\ }}}S_x$ be such that $m_i\leq m$ ($1\leq i\leq r$). The class of ${\ensuremath{\widetilde{C}}}$ in ${\ensuremath{\mathrm{Pic\ }}}S_x$ is denoted by ${\ensuremath{\underline{c}}}$, and its genus $g$. Suppose that $d\geq d_0(a,m)+1$ (see \[theoalhi\]), $\chi({\ensuremath{\underline{d}}})\geq d+1$, and ${\ensuremath{\underline{d}}}.{\ensuremath{\underline{c}}}+1-g \geq a$. Then, ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ is base point free, ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ is geometrically irreducible and smooth, ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ meets ${\ensuremath{\widetilde{C}}}$ transversally, and ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})\cap {\ensuremath{\widetilde{C}}}$ is irreducible. [<span style="font-variant:small-caps;">Proof :  </span>]{}Let us first prove that ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ is base point free : the characteristic $\chi({\ensuremath{\underline{d}}})$ is greater than $1$, so we just have to show that, given a point $Q\in S_x$, $h^1(S_x, {\mathcal{I}}_Q({\ensuremath{\underline{d}}}))=0$. Whatever the position of $Q$ is (even on an exceptional divisor), there exists a line $L$ on ${\Bbb P}^2$ such that $Q$ belongs to the strict transform ${\ensuremath{\widetilde{L}}}$ of $L$. Let ${\ensuremath{\underline{l}}}\in {\ensuremath{\mathrm{Pic\ }}}S_x$ be the class of ${\ensuremath{\widetilde{L}}}$ ; one may write ${\ensuremath{\underline{l}}}=(1;{\varepsilon}_1,\ldots,{\varepsilon}_r)$, where ${\varepsilon}_i=1$ if $P_i\in L$ and $0$ otherwise. Consider the exact sequence : $$H^1(S_x,{\mathcal{O}}({\ensuremath{\underline{d}}}-{\ensuremath{\underline{l}}})){\longrightarrow}H^1(S_x,{\mathcal{I}}_Q({\ensuremath{\underline{d}}})){\longrightarrow}H^1({\ensuremath{\widetilde{L}}},{\mathcal{I}}_{Q,{\ensuremath{\widetilde{L}}}}({\ensuremath{\underline{d}}}))$$ The scheme $Z=P_1^{m_1-{\varepsilon}_1}\cup\cdots\cup P_r^{m_r-{\varepsilon}_r}$ is clearly an $(m,a)$-configuration. Since $({\ensuremath{\underline{d}}}-{\ensuremath{\underline{l}}}).{\ensuremath{\underline{c}}}+1-g\geq 0$, $h^0(C,{\mathcal{O}}(d-1))\geq (d-1)a+1-g\geq \deg (Z\cap C)$, hence $Z$ is a $(d-1,m,a)$-candidate (in the extended sense of remark \[remark\]). Moreover, $(d-1)\geq d_0(a,m)$ ; Proposition \[theoalhi\] shows that ${\mathcal{L}}_x({\ensuremath{\underline{d}}}-{\ensuremath{\underline{l}}})$ is a regular system. But $\chi({\ensuremath{\underline{d}}}-{\ensuremath{\underline{l}}})\geq\chi({\ensuremath{\underline{d}}})-(d+1)\geq 0$, therefore $h^1({\ensuremath{\underline{d}}}-{\ensuremath{\underline{l}}})=0$. Moreover, since ${\ensuremath{\widetilde{L}}}$ is a rational curve, $|{\mathcal{I}}_{Q,{\ensuremath{\widetilde{L}}}}({\ensuremath{\underline{d}}})|$ is a regular system of degree ${\ensuremath{\underline{d}}}.{\ensuremath{\underline{l}}}-1$. If ${\ensuremath{\underline{d}}}.{\ensuremath{\underline{l}}}\geq 0$, then $h^1({\mathcal{I}}_{Q,{\ensuremath{\widetilde{L}}}}({\ensuremath{\underline{d}}}))=0$, as expected. Otherwise, the exact sequence $ 0{\longrightarrow}{\mathcal{O}}_{S_x}({\ensuremath{\underline{d}}}-{\ensuremath{\underline{l}}}){\longrightarrow}{\mathcal{O}}_{S_x}({\ensuremath{\underline{d}}}){\longrightarrow}{\mathcal{O}}_{{\ensuremath{\widetilde{L}}}}({\ensuremath{\underline{d}}}){\longrightarrow}0$ shows that ${\ensuremath{\widetilde{L}}}$ is a base component of ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$. Consider another line $L'$ of ${\Bbb P}^2$ containing none of the $r$ points $P_1,\ldots,P_r$. The preceding argument is true, with $L'$ in place of $L$, showing that the point $Q'={\ensuremath{\widetilde{L}}}\cap{\ensuremath{\widetilde{L}}}'$ can not be a base point of ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$. This is a contradiction. Therefore, $h^1({\mathcal{O}}_{S_x}({\ensuremath{\underline{d}}}-{\ensuremath{\underline{l}}}))=h^1({\mathcal{I}}_{Q,{\ensuremath{\widetilde{L}}}}({\ensuremath{\underline{d}}}))=0$, and the first exact sequence yields $h^1({\mathcal{I}}_Q({\ensuremath{\underline{d}}}))=0$. $\bullet$ Thus ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ is base point free. Bertini’s theorem shows that ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ is a smooth curve. Suppose it is not geometrically irreducible, and denote by $D_1,\ldots,D_l \ (l\geq 2)$ its geometrically irreducible components (over a bigger base field) . Let ${\ensuremath{\underline{d}}}_i\in {\ensuremath{\mathrm{Pic\ }}}S_x$ be the class of $D_i$ $(1\leq i\leq l)$. Consider two integers $1\leq i\neq j \leq l$. The curve ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ being smooth, $D_i$ does not intersect $D_j$ and ${\ensuremath{\underline{d}}}_i.{\ensuremath{\underline{d}}}_j=0$. Let $P$ be a point of $D_i$. The dimension of ${\mathcal{L}}_x({\ensuremath{\underline{d}}}_j)$ being positive (this system is base point free), there exists a curve $D'_j\in {\mathcal{L}}_x({\ensuremath{\underline{d}}}_j)$ containing $P$. But $D'_j.D_i=0$, so $D_i$ is a component of $D'_j$, and ${\mathcal{L}}_x({\ensuremath{\underline{d}}}_j-{\ensuremath{\underline{d}}}_i)$ is effective. By the same argument ${\mathcal{L}}_x({\ensuremath{\underline{d}}}_i-{\ensuremath{\underline{d}}}_j)$ is effective too, hence ${\ensuremath{\underline{d}}}_i={\ensuremath{\underline{d}}}_j$ for every $i\neq j$. Thus ${\ensuremath{\underline{d}}}=l{\ensuremath{\underline{d}}}_1$ and ${\ensuremath{\underline{d}}}^2=0$. The equality $\chi({\ensuremath{\underline{d}}})+g({\ensuremath{\underline{d}}})={\ensuremath{\underline{d}}}^2+2$ yields $g({\ensuremath{\underline{d}}})\leq 1-d$. Moreover, one can easily see that $g(l{\ensuremath{\underline{d}}}_1)=lg({\ensuremath{\underline{d}}}_1)-(l-1)$. Therefore, $lg({\ensuremath{\underline{d}}}_1)\leq l-d$ and (since $g({\ensuremath{\underline{d}}}_1)\geq 0$), $l\geq d$. The only possibility is $l=d$. Then $D_1$ is the strict transform of a line such that $D_1^2=0$. One may suppose that ${\ensuremath{\underline{d}}}_1=(1;1)$, and ${\ensuremath{\underline{d}}}=(d;d)$. This situation never happens since, by assumption $d\geq d_0(a,m)>m$. $\bullet$ We still have to prove that ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ meets ${\ensuremath{\widetilde{C}}}$ transversally and that ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})\cap {\ensuremath{\widetilde{C}}}$ is irreducible. The first point comes from Bertini’s theorem, applied to the curve ${\ensuremath{\widetilde{C}}}$ and the restricted (base point free) linear system ${\mathcal{L}}_x({\ensuremath{\underline{d}}})_{|{\ensuremath{\widetilde{C}}}}$. As for the second point, consider ${\mathcal{G}}\subset S_x\times {\mathcal{L}}({\ensuremath{\underline{d}}})$, the universal divisor over ${\mathcal{L}}({\ensuremath{\underline{d}}})$. Let $I={\mathcal{G}}\cap ({\ensuremath{\widetilde{C}}}\times {\mathcal{L}}({\ensuremath{\underline{d}}}))$ be the intersection variety. Since ${\mathcal{L}}({\ensuremath{\underline{d}}})$ is base point free, the fibers of the natural projection $I{\longrightarrow}{\ensuremath{\widetilde{C}}}$ are projective spaces of constant dimension. Therefore, $I$ is irreducible and ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})\cap {\ensuremath{\widetilde{C}}}$, which is the generic fiber of $I{\longrightarrow}{\mathcal{L}}({\ensuremath{\underline{d}}})$ also. [$\square$]{} Proof of Theorem 1 {#sec.proof} ================== In this section we gather the preceding results to prove the announced theorem. Actually, the work is not made directly on the projective plane but, rather, on the plane blown up at the $r$ points. However, the theorem proved below clearly implies the statement of the introduction. \[theorem2\] Let $m$ be a positive integer, $x$ the generic point of $({\Bbb P}^2)^r$ and ${\ensuremath{\underline{d}}}=(d;m_1,\ldots,m_r)\in {\ensuremath{\mathrm{Pic\ }}}S_x$ such that $0<m_i\leq m$ ($1\leq i\leq r$). With the notations of \[theoalhi\], let $a={\mathbf{a}}'(m)=\max ({\mathbf{a}}(m), 4 m) ; $ ${\mathbf{d}}'(m)= \max(d_0(a,m)+2a, a(2m+1)).$ If $d\geq {\mathbf{d}}'(m)$ then ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ is regular, and if $\dim {\mathcal{L}}_x({\ensuremath{\underline{d}}})\geq 0$, the generic curve ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ is geometrically irreducible, smooth, and meets each exceptional divisor $E_i$ in $m_i$ distinct points ($1\leq i\leq r$). As a consequence, the image of ${\mathfrak{D}}_x({\ensuremath{\underline{d}}})$ on the plane has an ordinary singularity of the prescribed multiplicity $m_i$ at every $P_i$. [<span style="font-variant:small-caps;">Proof :  </span>]{}Let $C$ be the generic curve of degree $a$ and genus $g$. If $\chi({\ensuremath{\underline{d}}})\leq 0$, then the scheme $Z=P_1^{m_1}\cup\cdots\cup P_r^{m_r}$ is a $(d,m,a)$-candidate with an empty constraint part. By Proposition \[theoalhi\], $Z$ is winning, ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ is empty and the theorem is true. If $\chi({\ensuremath{\underline{d}}})>1$ one may consider $\chi({\ensuremath{\underline{d}}})-1$ more general points $P_{r+1},\ldots,P_{r+\chi({\ensuremath{\underline{d}}})-1}$ of ${\Bbb P}^2$ and study the sub-system of curves in ${\mathcal{L}}_x({\ensuremath{\underline{d}}})$ passing through these supplementary points with multiplicity at least $1$. It is equivalent to study the curves in ${\mathcal{L}}(d;m_1,\ldots,m_r)$ or in ${\mathcal{L}}(d;m_1,\ldots,m_r,1^{\chi({\ensuremath{\underline{d}}})-1})$. As a consequence, we can make the assumption that $\chi({\ensuremath{\underline{d}}})=1$. $\bullet$ There exists a positive integer $s\leq r$ such that : $$\begin{aligned} \label{ajuste} -{\alpha}&= & da-m_1-\cdots -m_s +1 -g \in [-d+a-m,-d+a-1]\\ \label{bigs} s&\geq& (2da-a^2)/(2m)\end{aligned}$$ The second inequality follows from the first one : (\[ajuste\]) together with $m_i\leq m$ gives $da -ms +1-g\leq -d+a-1$ hence (since $g\leq a^2/2$), $ms\geq (2da-a^2)/2$. As for (\[ajuste\]), since $0<m_i\leq m$, it is sufficient to show that $da-\sum_{i=1}^r m_i +1-g\leq -d+a-1$. This is a consequence of the assumption $\chi({\ensuremath{\underline{d}}})=1$ : $$\begin{array}{ll} & da-(m_1+\cdots +m_r)+1-g\leq -d+a-1\\ \Longleftarrow_{(m_i\leq m,g>0)}& da-\frac{2}{m+1} \left(\frac{m_1(m_1+1)}2 +\cdots +\frac{m_r(m_r+1)}2\right)\leq -d+a-1\\ {\Longleftrightarrow}& da-\frac{2}{m+1}\left(\frac{(d+1)(d+2)}2-\chi({\ensuremath{\underline{d}}})\right)\leq -d+a-1\\ {\Longleftrightarrow}_{(\chi({\ensuremath{\underline{d}}})=1)} & d(a+1)-\frac{d(d+3)}{(m+1)}-a+1\leq 0 \end{array}$$ which is true when $d\geq a(2m+1)$. $\bullet$ Let $x=(P_1,\ldots,P_r)$ denote the generic point of $({\Bbb P}^2)^r$, and $y=(Q_1,\ldots,$ $Q_s,P_{s+1},\ldots,P_r)$ the generic point of $C^s\times ({\Bbb P}^2)^{r-s}$. The $r$-tuple $y$ is a specialization of $x$. The class of ${\ensuremath{\widetilde{C}}}_y$ is ${\ensuremath{\underline{c}}}=(a;1^s,0^{r-s})$. In order to apply the Geometric Horace Lemma we are going to check the points ${\ensuremath{1^\circ\!\textit )}}$ to ${\ensuremath{10^\circ\!\textit )}}$ of lemma \[horgeo\]. Condition 1 is nothing but the relation (\[ajuste\]) above. The assumption $d\geq a(2m+1)$ and (\[bigs\]) yield $s\geq a^2\geq g$, hence 2 is true ; moreover (\[bigs\]) and $a\geq 4 m$ also give $s\geq 4 d -2a\geq d-a+m+1$, hence 3 is true. As for the regularity of ${\mathcal{I}}_{Q_1^C\cup\cdots\cup Q_{{\alpha}+1}^C} ({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}))$, we make use of the corollary \[vanish\]. Since $\chi({\ensuremath{\underline{d}}})=1$, the exact sequence \[exact\] (section \[sec.horgeo\]) and \[ajuste\] yield $\chi({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})-({\alpha}+1)=0$. The choice of ${\mathbf{a}}'(m)$ and ${\mathbf{d}}'(m)$ gives $a\geq \max({\mathbf{a}}(m),4 m)$ and $d-a\geq \max(d_0(a,m)+a, 2am)$. Therefore, the only remaining condition of \[vanish\] is $$\begin{array}{ll} & ({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}).{\ensuremath{\underline{c}}} - {\alpha}-1 +1-g \geq 0\\ {\Longleftrightarrow}& (d-a).a-\sum_{i=1}^s(m_i-1)+1-g-({\alpha}+1)\geq 0\\ {\Longleftrightarrow}_{(\ref{ajuste})}& -2{\alpha}-a^2+s-1\geq 0 \\ \Longleftarrow_{(\ref{bigs})}& -2{\alpha}- a^2 +(2da-a^2)/(2m)-1 \geq 0\\ \Longleftarrow_{({\alpha}\leq d-a+m)}& d\bigl(\frac a m - 2\bigr) -a^2 \bigl( 1 +\frac 1 {2m}\bigr) +2a - 2m -1 \geq 0\\ \Longleftarrow_{(d \geq a(2m+1))}& (2a^2 +\frac {a^2} m -4am-2a)-a^2 - \frac{a^2}{2m}+2a-2m-1 \geq 0\\ {\Longleftrightarrow}& a^2 - 4am +\frac{a^2}{2m} -2m -1 \geq 0 \\ \Longleftarrow_{(a\geq 4m)}& 8m -2m -1 \geq 0 \end{array}$$ which is true since $m\geq 1$. $\bullet$ We are now left with the “irreducibility” and “smoothness” part of lemma \[horgeo\]. The sheaf ${\mathcal{O}}({\ensuremath{\underline{c}}})$ is effective if and only if $s\leq a(a+3)/2$, which is not the case by (\[bigs\]) ; so ${\ensuremath{5^\circ\!\textit )}}$ is true. The $7$-th point is empty since ${\alpha}\geq d-a+1>0$. Now, the residual system ${\mathcal{L}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ has a “high” dimension : Precisely, the exact sequence \[exact\] shows that $\dim {\mathcal{L}}_x({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})={\alpha}\in [d-a+1,d-a+m]$. Thus, the remaining condition of the Horace lemma can be proved with the proposition \[highdim\]. By assumption $d-a\geq d_0(a,m)+1$. It is then sufficient to prove that $({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}).{\ensuremath{\underline{c}}}+1-g\geq a$. But the preceding computation shows that $({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}}).{\ensuremath{\underline{c}}}+1-g\geq {\alpha}+ 1\geq d-a+2 \geq _{(d\geq 3a)} a$. $\bullet$ Let us turn now to the question of ordinary singularities . Recalling the remark \[ordinary\], we only have to check that the system ${\mathcal{L}}_y({\ensuremath{\underline{d}}}-{\ensuremath{\underline{c}}})$ is base point free, which is the case by Proposition \[highdim\]. [$\square$]{} [99]{} J. Alexander et A. Hirschowitz, An asymptotic vanishing theorem for generic unions of multiple points, *preprint alg-geom/9703037*, to appear at *Invent. Math.*. E. Arbarello and M. Cornalba, Footnotes to a paper of Benjamino Segre, *Math. Ann. 256* (1981), 341–362. S. Bossini, *Classification des courbes sur les surfaces rationnelles génériques*, Thèse, Université de Nice-Sophia Antipolis (1994). L. Chiantini and C. Ciliberto, On the Severi varieties of surfaces in ${\Bbb P}^3$, *J. Algebraic Geometry* (1999), 67–83. G. M. Greuel, C. Lossen, E. Shustin, Plane curves of minimal degree with prescribed singularities, *Invent. Math. 133* (1998), 539–580. G. M. Greuel, C. Lossen, E. Shustin, Geometry of families of nodal curves on the blown-up projective plane, *Trans. Amer. Math. Soc., Vol. 350, No. 1* (1998), 251–274. A. Grothendieck, Les schémas de Picard : théorèmes d’existence, technique de descente et théorèmes d’existence en géométrie algébrique, *Séminaire Bourbaki, exposé 232* (1961-1962). J. Harris, On the Severi problem, *Invent. Math. 84* (1986), 445–461. A. Hirschowitz, Une conjecture pour la cohomologie des diviseurs sur les surfaces rationnelles génériques, *J. Reine Angew. Math. 397* (1989), 208–213. T. Mignon, *Systèmes linéaires de courbes planes*, Thèse, Université de Nice (1997). T. Mignon, Courbes lisses sur les surfaces rationnelles génériques : un lemme d’Horace géométrique, *Prépublications de l’ENS Lyon n$^\circ$ 245* (1999). T. Mignon, Intersection des courbes planes et construction de courbes à singularités ordinaires, *Prépublications de l’ENS Lyon n$^\circ$ 250*, to appear at J. of Pure and Applied Algebra (2000). A. Nobile, On families of singular plane projective curves, *Annali di Matematica pura ed applicata (IV) CXXXVIII* (1984), 341–378. F. Severi, Vorlesungen " uber Algebraische Geometrie, Anhang F, Teubner, Leipzig (1921). E. Shustin, Geometry of equisingular families of plane algebraic curves, *J. Algebraic Geometry* **5** (1996), 209–234. R. Treger, Plane curves with nodes, *Can. J. Math. **41**, no. 2* (1989), 193–212. G. Xu, Ample line bundles on smooth surfaces, *J. Reine Angew. Math. 107* (1995), 99–209.

--- abstract: 'Multichannel processing is widely used for speech enhancement but several limitations appear when trying to deploy these solutions in the real world. Distributed sensor arrays that consider several devices with a few microphones is a viable solution which allows for exploiting the multiple devices equipped with microphones that we are using in our everyday life. In this context, we propose to extend the distributed adaptive node-specific signal estimation approach to a neural network framework. At each node, a local filtering is performed to send one signal to the other nodes where a mask is estimated by a neural network in order to compute a global multichannel Wiener filter. In an array of two nodes, we show that this additional signal can be leveraged to predict the masks and leads to better speech enhancement performance than when the mask estimation relies only on the local signals.' bibliography: - 'strings.bib' - 'refs.bib' title: 'DNN-based distributed multichannel mask estimation for speech enhancement in microphone arrays' --- Speech enhancement, microphone arrays, distributed processing. Introduction {#sec:intro} ============ Almost all voice-based applications such as mobile communications, hearing aids or human to machine interfaces require a clean version of speech for an optimal use. Single-channel speech enhancement can substantially improve the speech intelligibility and speech recognition of a noisy mixture [@Gerkmann2012; @Weninger2014]. However improvement with a single-channel filter is limited by the distortions introduced during the filtering operation. The distortion can be reduced in multichannel processing which exploits spatial information [@Frost1972; @Vincent2018]. The [@Doclo2002] for example yields the optimal filter in the sense and can be extended to a where the noise reduction is balanced by the speech distortion [@Doclo2007]. Up to a certain point, the effectiveness of these algorithms increases with the number of microphones. More microphones can allow for a wider coverage of the acoustic scene and a more accurate estimation of the statistics of the source signals. In large rooms, or even in flats, this implies the need of huge microphone arrays, which, if they are constrained, can become prohibitively expensive and lacks flexibility. However, in our daily life, with the omnipresence of computers, telephones and tablets, we are surrounded by an increased number of embedded microphones. They can be viewed as unconstrained ad hoc microphone arrays which are promising but also challenging [@Bertrand2015]. A algorithm [@Bertrand2010], where the nodes exchange a single linear combination of their local signals, was proposed for a fully connected microphone array. It was shown to converge to the centralized [@Bertrand2010a]. The constraint of a fully connected array can be lifted with randomized gossiping-based algorithms, where beamformer coefficients are computed in a distributed fashion [@Zeng2015]. Message passing [@Heusdens2012] or diffusion-based [@Oconnor2014] algorithms can increase the rather slow convergence rate of these solutions. Another way to exploit the broad covering of the acoustic field by ad hoc microphone arrays is to gather the microphones into clusters dominated by a single common source which can be estimated more efficiently [@Gergen2018]. All these algorithms require the knowledge of either the or the speech activity to compute the filters and are sensitive to signal mismatches [@Vorobyov2003] or detection errors [@Doclo2007]. Deep learning-based approaches have been proposed to estimate accurately these quantities through the prediction of a mask [@Narayanan2013; @Heymann2016; @Perotin2018b] or of the spectrum of the desired signals [@Nugraha2016]. Although often used in a multichannel context, most of these solutions use single-channel data as input of their . Multichannel information was first taken into account through spatial features [@Jiang2014], but can also be exploited using the magnitude and phase of several microphones as the input of a [@Adavanne2018; @Chakrabarty2019]. This yields better results than single-channel prediction but combining all the sensor signals is not scalable and seems suboptimal because of the redundancy of the data. Coping with the redundancy, Perotin et al. [@Perotin2018a] combined a single estimate of the source signals with the input mixture and used the resulting tensor to train a . In this paper, we consider a fully connected microphone array with synchronized sensors. This allows for using the -based algorithm which was reported to achieve good speech enhancement performance [@Bertrand2010a]. Following the results shown by Perotin et al. [@Perotin2018a], we take advantage of the paradigm [@Bertrand2010a] by combining at each node one local signal with the estimations of the target signal sent by the other nodes. This uses the multichannel context for the mask estimation but avoids the redundancy brought by the signals of a same node. Additionally, this scheme takes advantage of the internal filter operated in and reduces the costs in terms of bandwidth and computational power compared to a network combining all the sensor signals. The paper is organised as follows. The problem formulation and are described in Section \[sec:problem\_formulation\]. In Section \[sec:tf\_estimation\] we present our solution to estimate the masks. The experimental setup is described in Section \[sec:setup\] and results are discussed in Section \[sec:results\] before we conclude the paper. Problem formulation {#sec:problem_formulation} =================== Signal model {#subsec:signal_model} ------------ We consider an additive noise model expressed in the domain as $y(f, t) = s(f, t) + n(f, t)$ where $y(f, t)$ is the recorded mixture at frequency index $f$ and time frame index $t$. The speech target signal is denoted $s$ and the noise signal $n$. For the sake of conciseness, we will drop the time and frequency indexes $f$ and $t$. The signals are captured by $M$ microphones and stacked into a vector $\mathbf{y}~=~[y_{1}, ..., y_{M}]^T$. In the following, regular lowercase letters denote scalars; bold lowercase letters indicate vectors and bold uppercase letters indicate matrices. Multichannel Wiener filter {#subsec:danse} -------------------------- The operates in a fully connected microphone array. It aims at estimating the speech component $s_{i}$ of a reference signal at microphone $i$. Without loss of generality, we take the reference microphone as $i=1$ in the remainder of the paper. The $\mathbf{w}$ minimises the cost function expressed as follows: $$\label{eq:mse_cost} J(\mathbf{w}) = \mathbb{E}\{|s_{1} - \mathbf{w}^H\mathbf{y}|^2\}.$$ $\mathbb{E}\{\cdot\}$ is the expectation operator and $\cdot^H$ denotes the Hermitian transpose. The solution to (\[eq:mse\_cost\]) is given by $$\label{eq:mwf_w} \mathbf{\hat{w}} = \mathbf{R}_{yy}^{-1}\mathbf{R}_{ys}\mathbf{e}_1\,,$$ with $\mathbf{R}_{yy} = \mathbb{E}\{\mathbf{y}\mathbf{y}^H\}$, $\mathbf{R}_{ys} = \mathbb{E}\{\mathbf{y}\mathbf{s}^H\}$ and $\mathbf{e}_1 = [1\; 0 \cdots 0]^T$. Under the assumption that speech and noise are uncorrelated and that the noise is locally stationary, $\mathbf{R}_{ys} = \mathbf{R}_{ss} = \mathbb{E}\{\mathbf{s}\mathbf{s}^H\} = \mathbf{R}_{yy} - \mathbf{R}_{nn}$ where $\mathbf{R}_{nn} = \mathbb{E}\{\mathbf{n}\mathbf{n}^H\}$. Computing these matrices requires the knowledge of noise-only periods and speech-plus-noise periods. This is typically obtained with a [@Doclo2007; @Bertrand2010a]. The provides a trade-off between the noise reduction and the speech distortion [@Doclo2007]. The filter parameters minimise the cost function $$\label{eq:cost_sdw} J(\mathbf{w}) = \mathbb{E}\{|s_{1} - \mathbf{w}^H\mathbf{s}|^2\} + \mu \mathbb{E}\{|\mathbf{w}^H\mathbf{n}|^2\}\,,$$ with $\mu$ the trade-off parameter. The solution to (\[eq:cost\_sdw\]) is given by $$\label{eq:sdw_w} \mathbf{\hat{w}} = \big(\mathbf{R}_{ss} + \mu\mathbf{R}_{nn}\big)^{-1}\mathbf{R}_{ss}\mathbf{e}_1.$$ If the desired signal comes from a single source, the speech covariance matrix is theoretically of rank 1. Under this assumption, Serizel et al. [@Serizel2014] proposed a rank-1 approximation of $\mathbf{R}_{ss}$ based on a , delivering a filter that is more robust in low SNR scenarios and provides a stronger noise reduction. In this section, we briefly describe the algorithm under the assumption that a single target source is present. We consider $M$ microphones spread over $K$ nodes, each node $k$ containing $M_k$ microphones. The signals of one node $k$ are stacked in $\mathbf{y}_k~=~[y_{k,1}, ..., y_{k,M_k}]^T$. As can be seen in (\[eq:mwf\_w\]), the array wide should be computed from all signals of the array, which can result in high bandwidth and computational costs. In , only a single compressed signal $z_j$ is sent from node $j$ to the other nodes. So a node $k$ has $M_k + K -1$ signals, stacked in $\tilde{\mathbf{y}}_k = \left[ \mathbf{y}_k^T,~\mathbf{z}_{-k}^T\right]^T$, where $\mathbf{z}_{-k}$ is a column vector gathering the compressed signals coming from the other nodes ${j\ne k}$. Replacing $\mathbf{y}$ by $\tilde{\mathbf{y}}_k$ and solving (\[eq:cost\_sdw\]) yields the solution to the : $$\label{eq:danse_w} \mathbf{\tilde{w}}_k = \big(\mathbf{R}_{ss,k} + \mu\mathbf{R}_{nn,k}\big)^{-1}\mathbf{R}_{ss,k}\mathbf{e}_1\,,$$ where $\mathbf{\tilde{w}}_k$, the filter at node $k$, can be decomposed into two filters as $\mathbf{\tilde{w}}_k~=~\left[ \mathbf{w}_{kk}^T,~\mathbf{g}_{k-k}^T\right]$. The first filter $\mathbf{w}_{kk}^T$ is applied on the local signals and $\mathbf{g}_{k-k}$ is applied on the compressed signals sent from the other nodes. The covariance matrices $\mathbf{R}_{ss,k}$ and $\mathbf{R}_{nn,k}$ are computed from the speech and noise components of $\mathbf{\tilde{y}}_k$. The compressed signal $z_{k}$ is computed as $z_{k} = \mathbf{w}_{kk}^H\mathbf{y}_k$. Bertrand and Moonen proved that this solution converges to the solution with $\mu = 1$, while dividing the bandwidth load by a factor $M_k$ at each node [@Bertrand2010a]. In this paper, we will focus on the batch-mode algorithm where the speech and noise statistics are computed based on the whole signal in order to focus on the interactions between the mask estimated by the and the filters. Deep neural network based distributed multichannel Wiener filter {#sec:tf_estimation} ================================================================ Heymann et al. predicted masks out of a single signal of the microphone array [@Heymann2016]. Perotin et al. [@Perotin2018a] or Chakrabarty and Habets [@Chakrabarty2019] included several other signals to improve the speech recognition or speech enhancement performance. We propose to extend these scenarios to the multi-node context of . In , at node $k$, a single is used to estimate the source and noise statistics required for both filters $\mathbf{w}_{kk}$ and $\mathbf{w}_{k}$. The first part of our contribution is to replace the by a mask predicted by a . Besides, since the compressed signals ${z}_{k}$ are sent from one node to the others, we also examine the option of exploiting this extra source of information by using it for the mask prediction. The schematic principle of is depicted in Figure \[fig:danse\]. As it can be seen, an initialisation phase is required to compute the initial signal $z_k$. We propose to do this with a first neural network. The second stage of is represented in the greyed box in Figure \[fig:danse\] and expended in Figure \[fig:solution\]. Our second contribution is highlighted with the red arrow. It is to exploit the presence of $\mathbf{z}_{-k}$ at one node to better predict the masks with the . Several iterations are necessary for the filter $\mathbf{w}_{kk}$ to converge to the solution (\[eq:sdw\_w\]). In , iterations are done at every time step. As we developed an offline batch-mode algorithm, we stopped the processing after the first iteration. To analyse the effectiveness of combining $\mathbf{z}_{-k}$ with a reference signal to predict the mask, we compare our solution with a single-channel prediction, where the masks required for both initialisation and iteration stages are predicted by a single-channel model seeing only the local signal $y_{k,1}$. We compare two different architectures for each of these schemes. The first architecture is a bidirectional introduced by Heymann et al. [@Heymann2016]. When additional inputs are used with a , they are stacked over the frequency axis [@Perotin2018a]. Although this might deliver improved performance compared to the single-channel version, stacking it over the frequency axis is not efficient as many connections are used to represent relations between bins that might not be related. That is why we propose a architecture which is more appropriate to process multichannel data. At each node, the compressed signals $\mathbf{z}_{-k}$ and the local reference signal $y_{k,1}$ are considered as separate convolutional channels. During the training, in order to take into account the spectral shape of the speech, we weight the loss between the predicted mask $\hat{\mathbf{m}}$ and the ground truth mask $\mathbf{m}$ by the frame of the input $\mathbf{y}$, corresponding to the predicted frame. Both models are thus trained to minimise the cost function $$\mathcal{L}(\mathbf{m}, \hat{\mathbf{m}}) = E\{|(\mathbf{m} - \hat{\mathbf{m}})\cdot\mathbf{y}|^2\},$$ where $E\{\cdot\}$ represents the empirical mean. Lastly, since the filter $\mathbf{w}_{kk}$ is also applied on $\mathbf{z}_{-k}$, we use the of the covariance matrices to compute the of equation (\[eq:sdw\_w\]). Contrary to equation (\[eq:mwf\_w\]), this does not explicitly take the first microphone as a reference. It also assigns higher importance to the compressed signals, which is desirable since they are pre-filtered with potentially higher than the local signals. ![Block diagram of principle. Bold arrows represent vectors, simple ones represent scalars.[]{data-label="fig:danse"}](danse_block.pdf){width="\linewidth"} ![Expansion of the iterated step in Figure \[fig:danse\]. Red parts are the modifications proposed to . Bold arrows represent multichannel signals.[]{data-label="fig:solution"}](solution.pdf){width="\linewidth"} Experimental setup {#sec:setup} ================== Dataset {#subsec:data} ------- Training as well as test data was generated by convolving clean speech and noise signals with simulated , and then by mixing the convolved signals at a specific . The anechoic speech material was taken from the clean subset of LibriSpeech [@Panayotov2015]. The were obtained with the Matlab toolbox *Roomsimove*[^1] simulating shoebox-like rooms. In the training set, the length of the room was drawn uniformly as $l \in [3, 8]$m, the width as $w \in [3, 5]$m, the height as $h \in [2, 3]$m. Two nodes of four microphones each recorded the acoustic scene. The distance between the nodes was set to $1$m, the microphones being $10$cm away from the node centre. Each node was at least 1m away from the closest wall. One source of noise and one of speech were placed at $2.5$ m from the array centre. Both sources had an angular distance $\alpha \in [25, 90]^{\circ}$ relative to the array centre. The microphones as well as the sources were at the constant height of $1.5$ m. The was drawn uniformly between $-5$dB and $+15$dB. The noise was white noise modulated in the spectral domain by the long term spectrum of speech. We generated $10,000$ files of 10 seconds each, corresponding to about 25 hours of training material. The test configuration was the same as the training configuration but with restricted values for some parameters. The length of the room was randomly selected among $l \in \llbracket3, 8\rrbracket$m, the width among $w \in \llbracket3, 5\rrbracket$m, and the height was set to $h = 2.5$m. The angular distance $\alpha$ between the sources was randomly selected in $\alpha = \{25, 45, 90\}^{\circ}$. The noise was a random part of the third CHiME challenge dataset [@chime] in the cafeteria or pedestrian environment. We generated $1,000$ files representing about 2 hours of test material. Setup {#subsec:setup} ----- All the data was sampled at 16 kHz. The was computed with an FFT-length of 512 samples (32 ms), 50% overlap and a Hanning window. Our model was composed of three convolutional layers with 32, 64 and 64 filters respectively. They all had $3\times3$ kernels, with stride $1\times1$ and ReLU activation functions. Each convolutional layer was followed by a batch normalization over the frequency axis and a maximum pooling layer of size $4\times1$ (along the frequency axis). The recurrent part of the network was a layer with 256 gated recurrent units, and the last layer was a fully connected layer with a sigmoid activation function. The input data of both CRNN and RNN networks was made of sequences of 21 frames and the mask corresponding to the middle frame was predicted. We trained them with the optimizer [@rmsprop]. Results {#sec:results} ======= We evaluate the speech enhancement performance based on the , and [@Vincent2006] computed with the *mir\_eval*[^2] toolbox. The performance reported corresponds to the mean over the $1,000$ test samples of the objective measures computed at the node with the best input . We also report the 95% confidence interval. The filter does not explicitly take one sensor signal as the reference signal to minimise the cost function, but a projection of the input signals into the space spanned by the common eigenvectors of the covariance matrices. Because of that, the objective measures computed with respect to the convolved signals did not give results that were coherent with perceptual listening tests performed internally on random samples. Indeed, differences between the enhanced signal and the reference signal are interpreted as artefacts whereas they are due to the decomposition of the input signals into the eigenvalue space of the covariance matrices. Therefore, we compute the objective measures using the dry (source) signals as reference signals. This decreases the because the reverberation is then considered as an artefact but the comparison between methods correlates more with the perceptual listening tests. We present the objective metrics for the different approaches in Table \[tab:res\_oim\]. In this table, single node filters are referred to as MWF (upper part of the table) and distributed filters as DANSE (lower part of the table). For each filter, the architecture used to obtain the masks is indicated between parenthesis. RNN refers to Heymann’s architecture and CRNN to the network introduced in Section \[subsec:setup\]. The subscript of the network architecture indicates the channels considered at the input. The results obtained with the single-channel models are denoted with “SC”. When the compressed signals $\mathbf{z}_{-k}$ were used as additional input to the to predict the mask of the second filtering stage, models are denoted with “MC”. Additionally, we report the number of trainable parameters of each model in Table \[tab:params\]. Oracle performance {#subsec:oracle_se} ------------------ The gives information about the speech-plus-noise and noise-only periods in a wide-band manner only, whereas a mask gives spectral information that enables a finer estimation of the speech and noise covariance matrices. This additional information is translated into an improvement of the speech enhancement performance with both types of filters (MWF and DANSE). In the following section, we analyse whether this conclusion still holds when the masks are predicted by a neural network. Performance with predicted masks {#subsec:mc_mp} -------------------------------- (dB) SAR SIR SDR ------------------------------ --------------- --------------- -------------- MWF (oracle ) 2.4$\pm$0.3 24.7$\pm$0.3 2.3$\pm$0.3 MWF (oracle mask) 4.0$\pm$0.3 26.7$\pm$0.3 3.9$\pm$0.3 MWF (RNN) 3.4$\pm$0.3 25.1$\pm$0.4 3.3$\pm$0.3 MWF (CRNN) 3.3$\pm$0.3 25.1$\pm$0.4 3.2$\pm$0.3 DANSE (oracle VAD) 2.6$\pm$ 0.3 25.2$\pm$ 0.3 2.6$\pm$ 0.3 DANSE (oracle mask) 4.8$\pm$ 0.3 27.6$\pm$ 0.3 4.8$\pm$ 0.3 DANSE (RNN$_{\mathrm{SC}}$) $4.0 \pm 0.3$ 26.0$\pm$0.4 $4.0\pm0.3$ DANSE (CRNN$_{\mathrm{SC}}$) $4.0\pm0.3$ $26.0\pm0.4$ $4.0\pm0.3$ DANSE (RNN$_{\mathrm{MC}}$) $4.1\pm 0.3$ $26.1\pm0.4$ $4.0\pm0.3$ DANSE (CRNN$_{\mathrm{MC}}$) 4.7$\pm$0.3 27.4$\pm$0.3 4.6$\pm$0.3 : Speech enhancement results in dB with oracle activity detectors and predicted ones.[]{data-label="tab:res_oim"} First, replacing the oracle by masks brings significant improvement in terms of all objective measures. This confirms the idea that masks are better activity detectors than , even oracle ones. Second, the objective measures corresponding to the output signals of DANSE filters are always better than those of the MWF filters. This confirms the benefit of using the algorithm. Although these differences are not high, increasing the number of nodes and the distance between them might enhance the utility of the distributed method. From the results in Table \[tab:res\_oim\], there is no clear advantage of using a over using a in the single channel case. Indeed, the objective measures of RNN$_{\mathrm{SC}}$ and CRNN$_{\mathrm{SC}}$ match in all points. In the multichannel case, the performance of the -based approach does not increase. This tends to confirm that the is not able to efficiently exploit multichannel information. Since the delivered good results in the single-channel scenario, this leads to the conclusion that stacking multichannel input on the frequency axis is not appropriate. In addition, as shown in Table \[tab:params\], the number of parameters of the almost doubles when a second signal is used, whereas it barely increases for the . This is due to the convolutional layers of the which can process multichannel data much more efficiently than recurrent layers. The solution can exploit the multichannel inputs efficiently and the performance increases for all metrics. The biggest improvement is obtained for the . Indeed, one of the main difficulties for the models is to predict noise-only regions, because of people talking in the noise CHiME database. Since the compressed signals are pre-filtered, they contain less noise and they are less ambiguous. This makes it easier for the model to recognize noise-only regions, without degrading its predictions of speech-plus-noise regions. Model Number of parameters ---------------------- ---------------------- RNN$_{\mathrm{SC}}$ $1,717,773 $ CRNN$_{\mathrm{SC}}$ $911,109$ RNN$_{\mathrm{MC}}$ $2,244,109$ CRNN$_{\mathrm{MC}}$ $911,397$ : Number of trainable parameters of the neural networks.[]{data-label="tab:params"} Conclusion and future work {#sec:conclusion} ========================== We introduced an efficient way of estimating masks in a multi-node context. We developed multichannel models combining an estimation of the target signals sent by the other nodes with a local sensor. This proved to better predict masks, which led to higher speech enhancement performance that outperformed the results obtained with an oracle . A was compared to a and the could exploit much better the multichannel information. In addition, the architecture is limited by its number of parameters, especially if the number of nodes had to increase. In such scenarios, the difference between single-channel and multichannel models performance might be even more important but this still has to be explored. To attain performance closer to the oracle ones, several options are possible. First, the rather simple architectures that were used could be replaced by state-of-the art architectures. Besides, given the increase in performance when the target estimation is given, it would also be interesting to additionally give the noise estimation at the input of the models. \[sec:refs\] [^1]: homepages.loria.fr/evincent/software/Roomsimove\_1.4.zip [^2]: https://github.com/craffel/mir\_eval/

--- abstract: 'We approach the non-perturbative regime in finite temperature QCD within a formulation in Polyakov gauge. The construction is based on a complete gauge fixing. Correlation functions are then computed from Wilsonian renormalisation group flows. First results for the confinement-deconfinement phase transition for $SU(2)$ are presented. Within a simple approximation we obtain a second order phase transition within the Ising universality class. The critical temperature is computed as $T_c \simeq 305$ MeV.' author: - Florian Marhauser - 'Jan M. Pawlowski' title: Confinement in Polyakov Gauge --- Introduction {#sec:intro} ============ One of the remaining problems in low energy QCD is the quantitative field theoretical description of the confinement-deconfinement phase transition. Apart from its genuine importance for a first principle understanding of the confining physics in QCD it also is a key input for the evaluation of the QCD phase diagram. In the past decade much progress has been achieved both in continuum studies as well as with lattice computations for our understanding of the low energy sector of QCD, for reviews see e.g.[@Svetitsky:1985ye; @Alkofer:2000wg; @Litim:1998nf; @Schaefer:2006sr; @Fischer:2008uz]. For an analytical description of the low energy sector, topological degrees of freedom are likely to play an important role for the confinement-deconfinement phase transition as well as for chiral symmetry breaking, see e.g. [@Schafer:1996wv]. The latter has been very successfully described within instanton models, whereas the confining properties of the theory are harder to incorporate within semi-classical descriptions. Indeed, tracking down those topological degrees of freedoms relevant for confinement in the physical vacuum has its intricacies as the physical vacuum is more likely to contain a rather dense packing of topological configurations, making their detection difficult. Moreover, models of confinement are rather based on topological defects instead of stable topological objects, the construction of which out of these defects is plagued by non-localities. Still, these defects are manifest in the Polyakov loop, the order parameter in pure Yang-Mills theory [@Polyakov:1978vu], and can be extracted by an appropriate gauge fixing, see e.g.[@Reinhardt:1997rm; @Ford:1998bt]. Gauge fixing is also mandatory in most continuum formulations of QCD for removing the redundant gauge degrees of freedom. This is mostly seen as a liability of such an approach, as a formulation of QCD in gauge-variant variables complicates the access to gauge invariant observables. However, gauge fixing is nothing but a reparameterisation of the path integral and can be used for even facilitating the computation of at least a subset of observables. Indeed, this point of view has been exploited much in the discussion of confinement mechanisms based on topological defects. More recently it also has become clear that these are not competing physics mechanisms but rather different facets of the same global physics picture which still awaits a fully gauge invariant description, see e.g.[@Greensite:2004mh]. Despite this final step we have learned much from the combined investigations which together built a nearly complete mosaic. The effective potential of the Polyakov loop has also been used as an input for effective field theories that give some access to the QCD phase diagram [@Pisarski:2000eq]. At finite temperature and vanishing density, these models have led to impressive results in particular for thermodynamical quantities. At finite chemical potential, the back-reaction of the matter sector to the gauge sector is difficult to quantify in these models, and the chiral and confinement-deconfinement phase transitions are sensitive to the details of this back-reaction. This also holds true for the question of a quarkyonic phase with confinement and chiral symmetry at finite density [@McLerran:2007qj]. For an extension of these models one has to resort to a field-theoretical description of the gauge sector which allows to systematically study the impact of a finite chemical potential on the dynamics of the gauge field, [@Braun:2008pi]. In summary, the evaluation of Green functions of the Polyakov loop allows for a direct access to the physics in the strongly coupled sector of QCD, and in particular the confinement-deconfinement phase transition. In the present work we initiate a non-perturbative study of QCD in Polyakov gauge. In this gauge the Polyakov loop takes a particularly simple form and is directly related to the temporal component of the gauge field. After integrating-out the spatial components of the gauge field, and formulated with Polyakov loop variables, the gauge field sector of QCD resembles a scalar model. The dynamics of low energy Yang-Mills theory is then incorporated by evaluating Wilsonian flows for the effective action [@Wetterich:1993yh; @Litim:1998nf; @Schaefer:2006sr; @Berges:2000ew; @Bagnuls:2000ae; @Pawlowski:2005xe]. We derive the flow equation for QCD in Polyakov gauge, and solve it for the full effective potential of the Polyakov loop. Due to the formulation in Polyakov gauge a simple truncation already suffices to encode the physics of the confinement-deconfinement phase transition. The results include the temperature dependence of the Polyakov loop, and the critical temperature. We also compare the present approach to lattice studies [@Fingberg:1992ju], and to a recent continuum computation in Landau gauge [@Braun:2007bx]. QCD in Polyakov gauge {#sec:QCDinPol} ===================== In QCD with static quarks the expectation value of a static quark $\langle q(\vec x)\rangle$ serves as an order parameter for confinement. It is proportional to the free energy $F_q$ of such a state, $\langle q(\vec x)\rangle\sim \exp(-\beta F_q)$, where $\beta =1/T$ is the inverse temperature. Hence in the confining phase at low temperature, the expectation value vanishes, whereas at high temperatures in the deconfined phase, it is non-zero. The Polyakov loop variable, [@Polyakov:1978vu], is the creation operator for a static quark, $$\label{eq:Polloop} L(\vec x)=\frac{1}{{N_{\text{c}}}} {\mathrm{tr}}\, \CP(\vec x)\,,$$ where the trace in is done in the fundamental representation, and the Polyakov loop operator is a Wegner-Wilson loop in time direction, $$\label{eq:Polop} {\cal P}(\vec x) =\CP\, \exp \left( {\mathrm{i}}g \int_0^\beta dx_0\, {A}_0(x_0,\vec x) \right)\,.$$ Here ${\cal P}$ stands for path ordering. We conclude that $\langle q(\vec x)\rangle \simeq \langle L(\vec x)\rangle$, and thus the negative logarithm of the Polyakov loop expectation value relates to the free energy of a static fundamental color source. Moreover, $\langle L\rangle$ measures whether center symmetry is realised by the ensemble under consideration, see e.g.[@Polyakov:1978vu; @Svetitsky:1985ye; @Schafer:1996wv; @Reinhardt:1997rm; @Ford:1998bt; @Greensite:2004mh]. More specifically we consider gauge transformations $U(x_0,x)$ with $U(0,\vec x) U^{-1}(\beta, \vec x) =Z$, where $Z$ is a center element. In $SU(2)$ the center is $Z_2$, whereas in physical QCD with $SU(3)$ it is $Z_3$. Under such center transformations the Polyakov loop operator ${\cal P}(\vec x)$ in is multiplied with a center element $Z$, $$\label{eq:centertrafo} {\cal P}(\vec x)\to Z\,{\cal P}(\vec x)\,,$$ and so does the Polyakov loop, $L(\vec x)\to Z\, L(\vec x)$. Hence, a center-symmetric confining (disordered) ground state ensures $\langle L\rangle=0$, whereas deconfinement $\langle L\rangle\neq 0$ signals the ordered phase and center-symmetry breaking, $$\begin{aligned} \nonumber T<T_c: &\qquad \langle L(\vec x)\rangle = 0\,,\quad F_q=\infty\,, \\ T>T_c: &\qquad \langle L(\vec x)\rangle \neq 0\,, \quad F_q<\infty\,. \label{eq:orderdisorder}\end{aligned}$$ The expectation value of the Polyakov loop can be deduced from the equations of motion of its effective potential $V_L[\langle L\rangle]$. We shall argue, that the computation of the latter greatly simplifies within an appropriate choice of gauge. Indeed, gauge fixing is nothing but the choice of a specific parameterisation of the path integral, and a conveniently chosen parameterisation can simplify the task of computing physical observables. In the present case our choice of gauge is guided by the demand of a particularly simple representation of the Polyakov loop variable . A gauge ensuring time-independent $A_0$ leads to both, a trivial integration in and renders the path ordering irrelevant. Having done this one can still rotate the Polyakov loop operator ${\cal P}(\vec x)$, , into the Cartan subgroup. The above properties are achieved for time-independent gauge field configurations in the Cartan subalgebra, i.e. $A_0(t_0,\vec x)=A_0^c(\vec x)$. For $SU(2)$ this reads $$\begin{aligned} \label{eq:A0} A_0(t_0,\vec x)=A_0(\vec x)\, \0{\sigma_3}{2}\end{aligned}$$ and entails a particularly simple relation between $A_0$ and $L$, $$\label{eq:polsu(2)} L(\vec x) = \cos\, \012 g \beta A_0(\vec x) \,,$$ Note that this simple relation is not valid on the level of expectation values of $L$ and $A_0$, in $SU(2)$ we have $\langle L\rangle \neq \cos\, \012 g \beta \langle A_0\rangle$. However, in the present work we consider an approach that gives direct access to the effective potential $V_{\rm eff}[ \langle A_0\rangle]$ for the gauge field, as distinguished to those for the Polyakov loop, $U_{\rm eff}[\langle L\rangle]$ [^1]. Here, we argue that $ L[\langle A_0\rangle ]$ also serves as an order parameter: to that end we show that the order parameter $ \langle L[ A_0 ]\rangle $ is bounded from above by $ L[\langle A_0\rangle ]$. It follows that $ L[\langle A_0\rangle ]$ is non-vanishing in the center-broken phase. Furthermore we show that in the center-symmetric phase with vanishing order parameter, $\langle L[ {A_0}] \rangle=0$, also the observable $L[\langle {A_0} \rangle]$ vanishes. For the sake of simplicity we restrict the explicit argument to $SU(2)$, but it straightforwardly extends to general $SU(N)$. First we note that we can use for expressing the expectation value of $A_0$ in terms of $L$, $$\label{eq:A_0fromL} \012 g \beta \langle A_0\rangle =\langle \arccos L\rangle\,.$$ We emphasise that the rhs of defines an observable as it is the expectation value of an gauge invariant object. This observable happens to agree with $\langle A_0\rangle $ in Polyakov gauge. It follows from the Jensen inequality that the expectation value of the Polyakov loop, the order parameter for confinement, is bounded from above by $L[\langle {A_0} \rangle]$, see [@Braun:2007bx] $$\begin{aligned} \label{eq:jensen} L[\langle {A_0} \rangle]\geq \langle L[A_0]\rangle\,. \end{aligned}$$ for gauge fields $g\beta \langle A_0\rangle/2 \in [0,\pi/2]$. Note that it is sufficient to consider the above interval due to periodicity and center symmetry of the potential. This means we restrict the Polyakov loop expectation value to $\langle L\rangle \geq 0$. Negative values for $\langle L\rangle$ are then obtained by center transformations, $L\to \pm L$. is easily proven for $SU(2)$ from as $\cos(x)$ is concave for $x\in [0,\pi/2]$. Thus, for $\langle L\rangle >0$ it necessarily also follows that $g\beta \langle A_0\rangle/2 <\pi/2$. In turn we can show that $g\beta \langle A_0\rangle/2=\pi/2$, if the Polyakov loop variable $\langle L[A_0]\rangle$ vanishes. This then entails that $L[\langle {A_0} \rangle]=0$. To that end we expand $L$ about its mean value $\langle L\rangle $, that is $L=\langle L\rangle +\delta L$. Inserting this expansion into we arrive at $$\label{eq:expandA_0} \012 g \beta \langle A_0\rangle =\arccos \langle L\rangle-\0{1}{\sqrt{1- \langle L\rangle ^2} } \langle \delta L\rangle + O\left(\langle \delta L^2\rangle\right)\,.$$ In the center-symmetric phase $\langle L\rangle =0$, c.f. . Under center transformations $L$ transforms according to (\[eq:centertrafo\]) $L\to Z\, L$ with $Z=\pm 1$ and hence $ \delta L\to Z\, \delta L$. It follows that $\langle \delta L^{2n+1}\rangle = Z\langle \delta L^{2n+1}\rangle =0$, and all odd powers in vanish. The even powers vanish since $\arccos$ is an odd function and hence has vanishing even Taylor coefficients $\arccos^{(2n)}(0)$. Thus, in the center-symmetric phase we have $$\label{eq:expandA_0centersym} \012 g \beta \langle A_0\rangle =\arccos \langle L\rangle=\0{\pi}{2}\,.$$ In summary we have shown $$\begin{aligned} \nonumber T<T_c: &\di\quad L[\langle {A_0} \rangle]=0 \quad \Leftrightarrow \quad\012 g \beta \langle A_0(\vec x)\rangle = \0\pi2\,, \\ T>T_c: &\di\quad L[\langle {A_0} \rangle]\neq 0 \quad \Leftrightarrow \quad \012 g \beta \langle A_0(\vec x)\rangle <\0\pi2\,. \label{eq:A0orderdisorder}\end{aligned}$$ We conclude that $\langle A_0\rangle $ in Polyakov gauge serves as an order parameter for the confinement-deconfinement (order-disorder) phase transition, as does $L[\langle A_0\rangle]$. Thus, we only have to compute the effective potential $V_{\rm eff}[\langle A_0\rangle]$ in order to extract the critical temperature, and e.g. critical exponents. This potential is more easily accessed than that for the Polyakov loop. It is here were the specific gauge comes to our aid as it allows the direct physical interpretation of a component of the gauge field. This property has been already exploited in the literature, where it has been shown that $\langle{A}_0\rangle$ in Polyakov gauge is sensitive to topological defects related to the confinement mechanism [@Reinhardt:1997rm; @Ford:1998bt]. Quantisation {#sec:quant} ============ We proceed by discussing the generating functional of Polyakov gauge Yang-Mills theory. For its derivation we use the Faddeev-Popov method. Specifying to $SU(2)$, the Polyakov gauge is implemented by the gauge fixing conditions $$\label{eq:Pol1} \partial_0 {\mathrm{tr}}\,\sigma_3 A_0=0\,, \qquad {\mathrm{tr}}\,(\sigma_1\pm i\sigma_2) A_0 = 0\,,$$ where the $\sigma_i$ are the Pauli matrices. However, the gauge fixing is not complete. It is unchanged under time-independent gauge transformations in the Cartan sub-group. These remaining gauge degrees of freedom are completely fixed by the following conditions, $$\begin{aligned} \nonumber & \partial_1 \int dx_0 \, {\mathrm{tr}}\, \sigma_3 A_1 =0\,,\qquad \partial_2 \int dx_0 dx_1 \, {\mathrm{tr}}\, \sigma_3 A_2 = 0\,,&\\ & \partial_3 \int dx_0 dx_1 dx_2\, {\mathrm{tr}}\,\sigma_3 A_3 = 0\,.& \label{eq:Pol2}\end{aligned}$$ The gauge fixings are integral conditions and are the weaker the more integrals are involved. Basically they eliminate the corresponding zero modes. This can be seen directly upon putting the theory into a box with periodic boundary conditions, $T^4$, see e.g.[@Ford:1998bt]. The gauge fixing conditions , lead to the Faddeev-Popov determinant $$\begin{aligned} \Delta_{FP}[A] = (2 T)^2 \left[ \prod_{x} \sin^2 \left( \frac{g A_0^3 (\vec x)}{ 2 T } \right) \right]\,, \label{eq:FPdet}\end{aligned}$$ the computation of which is detailed in appendix \[app:FPdet\]. The integration over the longitudinal gauge fields precisely cancels the Faddeev-Popov determinant in the static approximation $\partial_i A_0^c=0$, see Appendix \[app:FPdet\]. Finally we are left with the action $$\begin{aligned} S_{\rm eff}[A] &\simeq & -\frac{1}{2} \beta \int d^3x\, A_0 \vec \partial{\,}^2 A_0\\\nonumber &&\hspace{-1cm}-\frac{1}{2} \int_T d^4x\, A^a_{\bot,i} \left[(D_0^2)^{ab} + \vec \partial^2\delta^{ab} \right] A_{\bot,i}^a +O(A_{\bot,i}^3)\end{aligned}$$ with $D_0^{ab} = \partial_0 \delta^{ab} + A^3_0 g f^{a3b}$ and transversal spatial gauge fields, $\partial_i A_{\bot,i}=0$. The generating functional of Yang-Mills theory in Polyakov gauge then reads $$\begin{aligned} \nonumber \hspace{-.5cm}Z[J]&=&\int dA_0\,dA_{\bot,i} \,\exp\Bigl\{-S_{\rm eff}[A]\\ & &\hspace{.3cm} +\int d^3 x\,J_0 A_0+\int_T d^4 x \, J_{\bot,i} A_{\bot,i} \Bigr\}\,. \label{eq:Zpol}\end{aligned}$$ In we have normalised the temporal component $J_0$ of the current with a factor $\beta$. The classical action $S_{\rm eff}$ is inherently non-local as is contains one-loop terms, the Faddeev-Popov determinant as well as the integration over the longitudinal gauge fields. Instead of computing $Z[J]$ in we shall compute the effective action $\Gamma$ within a functional renormalisation group approach [@Wetterich:1993yh; @Litim:1998nf; @Schaefer:2006sr; @Berges:2000ew; @Bagnuls:2000ae; @Pawlowski:2005xe]. To that end we introduce an infrared cut-off for the transversal spatial gauge fields and in the temporal gauge fields by modifying the action, $S\to S_{\rm eff}+\Delta S_k[A_0]+\Delta S_{\bot,k}[\vec A_{\bot}]$, with infrared scale $k$, and cut-off terms $$\begin{aligned} \nonumber \Delta S_k[A_0]&=&\012 \beta \int d^3 x \, A_0\, R_{0,k}\, A_0\\ \Delta S_{k,\bot}[\vec A_{\bot}]&=& \int_T d^4 x\, A^a_{\bot, i}\, R_{\bot,k}\, A^a_{\bot, i}\,. \label{eq:Cutoff} \end{aligned}$$ The regulators $R_k$ in are chosen to be momentum-dependent and required to provide masses at low momenta and to vanish at large momenta. For $k\to 0$ they vanish identically. They can be written as one single regulator $R_{A,{\mu\nu}}$, which is a block-diagonal matrix in field space with entries $R_{A,{00}}=R_{0,k}$ and $R_{A,{ij}}=R_{\bot,k} \Pi_{\bot,ij}$, where the transversal projector is defined by $$\begin{aligned} \label{eq:transverse} \Pi_{\bot,ij} = \delta_{ij} -\frac{p_i p_j}{\vec p^2}\,.\end{aligned}$$ The above structure is induced by the fact the $A_{\bot,i}$ are transversal, and hence $R_{\bot,k}$ only couples to the transversal degrees of freedom. The flow of the cut-off dependent effective action $\Gamma_k$ is then given by Wetterich’s equation [@Wetterich:1993yh; @Berges:2000ew; @Bagnuls:2000ae] for Yang-Mills theory [@Litim:1998nf; @Pawlowski:2005xe] in Polyakov gauge, $$\begin{aligned} \nonumber \hspace{-.5cm} \partial_t \Gamma_{k}& = & \frac{\beta}{2} \int \0{d^3 p}{(2 \pi)^3} \left(\frac{1}{\Gamma_k^{(2)} + R_A}\right)_{00}\partial_t R_{0,k}\\ & & + \frac{T}{2} \sum_{n\in \Z} \int \0{d^3 p}{(2 \pi)^3} \left(\frac{1}{\Gamma_k^{(2)} + R_A}\right)_{ii}\partial_t R_{\bot, k}\,, \label{eq:flow}\end{aligned}$$ where $t$ is the RG time $t = \ln (k / \Lambda)$, and $\Lambda$ is some reference scale. Approximation scheme {#sec:approx} ==================== together with an initial effective action at some initial ultraviolet scale $k=\Lambda_{\rm UV}$ provides a definition of the full effective action at vanishing cut-off scale $k=0$ via the integrated flow. For the solution of we have to resort to approximations to the full effective action. In gauge theories such an approximation also requires the control of gauge invariance, see e.g.  [@Pawlowski:2005xe]. Here we shall argue that in Polyakov gauge a rather simple approximation to the full effective action already suffices to describe the confinement-deconfinement phase transition, and, in particular, to estimate the critical temperature. We compute the flow of the effective action $\Gamma[A_0,\vec A_{\bot}]$ in the following truncation $$\begin{aligned} \nonumber \hspace{-.4cm}\Gamma_k[A_0,\vec A_{\bot}] &\!=&\! \beta \int d^3x\, \left( -\0{Z_{0}}{2} A_0 \vec \partial{\,}^2 A_0+V_{k}[A_0]\right) \\ &&\hspace{-.8cm}-\frac{1}{2} \int_T d^4x\, Z_{i} \vec A_{\bot}^a \left[(D_0^2)^{ab} + \vec \partial{\,}^2\delta^{ab} \right] \vec A_{\bot}^a\,, \label{eq:effact}\end{aligned}$$ with $k$-dependent wave function renormalisations $Z_0,Z_i$. The effective action relates to the order parameter $\langle L(\vec x)\rangle$ as well as its two point correlation $\langle L(\vec x) L^\dagger (\vec y) \rangle$ via the effective potential $V_{\rm eff}[A_0]=V_k[A_0]$ as explained in section \[sec:QCDinPol\]. The expectation value $\langle L(\vec x)\rangle$, or $L[\langle A_0\rangle]$, is used to determine the phase transition temperature $T_c$ as well as critical exponents. The temperature-dependence of the Polyakov loop two-point function relates to the string tension. In the confining phase, for $T<T_c$, and large separations $|\vec x-\vec y|\to\infty$, the two-point function falls off like $$\label{eq:string} \lim_{|\vec x-\vec y|\to\infty} \langle L(\vec x) L^\dagger (\vec y) \rangle_{c} \simeq \exp \left\{-\beta\, \sigma |\vec x-\vec y|\right\}\,.$$ Here, $\langle\cdots \rangle_c$ stands for the connected part of the related correlation function, i.e. $ \langle L(\vec x) L^\dagger (\vec y) \rangle_{c} = \langle L(\vec x) L^\dagger (\vec y) \rangle-\langle L(\vec x)\rangle \langle L(\vec y)\rangle$. In turn, its Fourier transform shows the momentum dependence $$\label{eq:stringmomentum} \lim_{|p|\to 0} \langle L(0) L^\dagger (p) \rangle_c \simeq \lim_{|p|\to 0} \0{1}{\pi^2} \0{\beta\sigma} { ((\beta\sigma)^2+p^2)^2}=\0{1}{\pi^2} \0{1} { (\beta\sigma)^3} \,.$$ We conclude that the Polyakov loop variable has a massive propagator. This directly relates to a massive propagator of $A_0$ in Polyakov gauge. The approximation scheme is fully set by specifying the regulators $R_{0,k}$ and $R_{\bot,k}$. Naively one would identify the cut-off parameter $k$ in the regulators with the physical cut-off scale $k_{\rm phys}$. For general regulators this is not possible and one deals with two distinct physical cut-off scales, $k_{0,\rm phys}$ and $k_{\bot,\rm phys}$ related to $R_{0,k}$ and $R_{\bot,k}$ respectively, for a detailed discussion see [@Pawlowski:2005xe]. However, within the approximation it is crucial to have a unique effective cut-off scale $k_{\rm phys}=k_{0,\rm phys}=k_{\bot,\rm phys}$, as different physical cut-off scales $k_{0,\rm phys}\neq k_{\bot,\rm phys}$ necessarily introduce a momentum transfer into the flow which carries part of the physics. This momentum transfer is only fully captured with a non-local approximation to the effective action. In turn, a local approximation such as requires $k_{0,\rm phys}=k_{\bot,\rm phys}$. In other word, a local approximation works best if the momentum transfer in the flow is minimised. More details about such a scale matching and its connection to optimisation [@Litim:2000ci; @Pawlowski:2005xe] can be found in [@Pawlowski:2005xe]. Note in this context that in the present case we also have to deal with the subtlety that $A_0$ only depends on spatial coordinates whereas $\vec A_\bot$ is space-time dependent. However, the requirement of minimal momentum transfer in the flow is a simple criterion which is technically accessible. More specifically we restrict ourselves to regulators [@Litim:2006ag] $$R_{A,00} = Z_0 R_{{\rm opt},k}(\vec p^2)\,,\quad R_{A,ij} = Z_i \Pi_{\bot,ij}(\vec p)R_{\rm opt,k_\bot }(\vec p^2)\,, \label{eq:cutoffs}$$ where [@Litim:2000ci] $$\begin{aligned} R_{{\rm opt},k}(\vec p^2)=(k^2-\vec p^2)\theta(k^2-\vec p^2)\,. \label{eq:opt}\end{aligned}$$ The detailed scale-matching argument is deferred to Appendix \[app:match\], and results in a relation $k_{\bot}=k_\bot(k)$ depicted in Fig. \[fig:kbotk\] in the appendix. It is left to determine the effective cut-off scale $k_{\rm phys}$. This cut-off scale can be determined from the numerical comparison of the flows of appropriate observables: one computes the flow with the three-dimensional regulator $R_{{\rm opt},k_\bot }(\vec p^2)$ in , as well as with the four-dimensional regulator $R_{{\rm opt},k_{\rm phys}}(p^2)$. Then the respective physical scales are identified, i.e. $k_{\bot,\rm phys}(k_\bot)=k_{\rm phys}$. The results for this matching procedure are depicted in Fig. \[fig:kskphys\] in Appendix \[app:match\]. Another estimate comes from the flow related to the three-dimensional $A_0$-fluctuations, where we can directly identity $k_{\rm phys}=k$. We use the above choices as limiting cases for an estimate of the systematic error in our computation. Our explicit results are obtained for the best choice that works in all physics constraints. Flow {#sec:flow} ==== We are now in the position to integrate the flow equation . To begin with, we can immediately integrate out the spatial gauge fields $\vec A_\bot$ for $Z_i=1$, that is the second line in . This part of the flow only carries an explicit dependence on the cut-off $k$, details of the calculation can be found in Appendix \[app:intoutAI\]. It results in a non-trivial effective potential $V_{\bot,k}[A_0]$ that approaches the Weiss potential [@Weiss:1980rj] in the limit $k/T \to 0$, and falls off like $\exp(-\beta k_\bot (k)) \cos (g \beta A_0) $ in the UV limit $k/T\to \infty$, see Fig. \[fig:VPreWeiss3D\]. In terms of the effective action, after the integration over $\vec A_\bot$, we are led to an effective action of $A_0$, $$\label{eq:truncated_eff_actioncopy} \Gamma_{k}[A_0] = \beta \int d^3x \left(\frac{ Z_0}{2} (\vec \partial A_0)^2 + \Delta V_{k}[A_0] + V_{\bot,k} [A_0] \right)\,.$$ follows from with $\Gamma_k[A_0]=\Gamma_k[A_0,\vec A_\bot =0]$, and $$\label{eq:Vk} V_k[A_0]= \Delta V_{k}[A_0] + V_{\bot,k} [A_0]\,.$$ The full effective potential is given by $V_{\mathrm{eff}}[A_0] = \Delta V_{k=0}[A_0] + V_{\bot,k=0}[A_0]$. We are left with the task to determine $\Delta V_k$, which is the part of the effective potential induced by $A_0$-fluctuations. In Polyakov gauge, these fluctuations carry the confining properties of the Polyakov loop variable, whereas the spatial fluctuations generate a deconfining effective potential for $A_0$, see Appendix \[app:intoutAI\]. We emphasise that this structure is not present for spatial confinement which is necessarily also driven by the spatial fluctuations, and solely depends on these fluctuations in the high temperature limit. We hope to report on this matter in the near future. Here we proceed with the integration of the flow for the potential $\Delta V_{k}$. To that end we reformulate the flow as a flow for $\Delta V_{k}$ with the external input $V_{\bot,k}$, see . The flow equation for $\Delta V_{k}$ reads $$\label{eq:deltaV_FRGeq} \beta\, \partial_{t} {\Delta V}_{k} = \frac{1}{2} \int \0{d^3 p}{(2 \pi)^3} \frac{ \partial_{t} R_{0,k}}{Z_0\vec p{\,}^2 + \partial^2_{A_0}( \Delta V_{k} + V_{\bot,k}) + R_{0,k}} \,.$$ With the specific regulator $R_k$ in we can perform the momentum integration analytically. We also introduce the scalar field $\varphi = g \beta A_0$, and arrive at $$\begin{aligned} \label{eq:preflowV} \beta \partial_k \Delta V_k = \frac{2}{3 (2 \pi)^2} \frac{(1+\eta_0/5) k^2 }{1+\frac{ g_{k}^2 \beta^2}{ k^2 } \partial^2_{\varphi} ( V_{\bot,k} + \Delta V_k)}\,, \end{aligned}$$ where the coupling $g_k^2$ has to run with the effective cut-off scale $k_{\rm phys}$, and is estimated by an appropriate choice of the running coupling $\alpha_s$, $$\label{eq:runningg} g_k^2=\frac{g^2}{Z_0}\,,\qquad {\rm with}\qquad g_k^2=4 \pi \alpha_s(k_{\rm phys}^2)\,,$$ see also Appendix \[app:intoutAI\]. This also entails that the anomalous dimension $\eta_0$ is linked to the running coupling by $$\label{eq:eta0} \eta_0=-\partial_t \log \alpha_s(k_{\rm phys}^2)\,.$$ At its root is an equation for the dimensionless effective potential $\hat V = \beta^4 V_k$ in terms of $\hat V_\bot= \beta^4 V_{\bot,k}$ and $\hat \Delta V=\beta^4 \Delta V_k$. The infrared RG-scale $k$ naturally turns into the modified RG-scale $\hat k = k \beta$, that is all scales are measured in temperature units. Then the flow equation is of the form $$\begin{aligned} \label{eq:inter} \partial_{\hat k} \Delta \hat{ V} = \frac{2 }{3 (2 \pi)^2} \frac{ (1+\eta_0/5) \hat k^2 }{1+\frac{ g_{k}^2 }{ \hat k^2 } \partial^2_\varphi ( \hat{V}_{\bot} + \Delta \hat{ V})}\,.\end{aligned}$$ The potential $\hat V$ and hence $\hat\Delta V$ has a field-independent contribution which is related to the pressure. For the present purpose it is irrelevant and we can conveniently normalise the flow such that it vanishes at fields where $\partial_\varphi^2(\hat{V}_{\bot} + \Delta \hat{ V})=0$. This is achieved by subtracting $2(1+\eta_0/5)\, \hat k^2 / (3 (2\pi)^2)$ in and we are left with $$\begin{aligned} \label{eq:finalRGeq} \partial_{\hat k} \Delta \hat{ V} = -\frac{ 1 }{ 6 \pi^2} \left(1+\frac{\eta_0}{5}\right) \frac{ \ g_{k}^2 \ \partial^2_\varphi \, ( \hat{V}_{\bot} + \Delta \hat{ V}) }{1 +\frac{ g_{k}^2 }{ \hat k^2 } \partial^2_\varphi \, (\hat{V}_{\bot} + \Delta \hat{ V})}\,, \end{aligned}$$ where we have kept the notation $\partial_{\hat k}\Delta \hat V$ for $\partial_{\hat k}\Delta \hat V-2(1+\eta_0/5)\, \hat k^2 / (3 (2\pi)^2)$. In this form it is evident, that the flow vanishes for fields where $\partial_\varphi^2(\hat{V}_{\bot} + \Delta \hat{ V})=0$, i.e. once a region of the potential becomes convex, this part is frozen, unless the external input $\hat V_{\bot}$ triggers the flow again. We close this section with a discussion of the qualitative features of . It resembles the flow equation of a real scalar field theory, and due to $V_\bot$, the flow is initialised in the broken phase. It relies on two external inputs, $V_\bot$ and $\alpha_s$. The first input, $\hat V_\bot$, is computed in a perturbative approximation to the spatial gluon sector, and its computation is deferred to Appendix \[app:intoutAI\]. It is shown in Fig. \[fig:VPreWeiss3D\] for various values of the RG time $\hat k$, and approaches the perturbative Weiss potential [@Weiss:1980rj] for vanishing cutoff $\hat k=0$. ![$\hat V_{\bot}$ for different values of $\hat k$[]{data-label="fig:VPreWeiss3D"}](PreWeiss.eps "fig:"){width="8cm"}\ We have argued that within Polyakov gauge this approximation should capture the qualitative feature of its contribution to the Polyakov loop potential. We emphasise again that this is not so for the question of spatial confinement, and the related potential of the spatial Wilson loops. The second input is $\alpha_s= g_k^2/(4 \pi)$, the running gauge coupling. It runs with the physical cut-off scale $k_{\rm phys}$ derived in Appendix \[app:match\], $\alpha_s= \alpha_s(k_{\rm phys}^2)$. In the present work we model $\alpha_s$ with a temperature-dependent coupling that runs into a three-dimensional fixed point $\alpha_{*,3d}k_{\rm phys}/T$ for low cut-off scales $k_{\rm phys}/T\ll 1$. This choice carries some uncertainty as the running coupling in Yang-Mills theory is not universal beyond two loop order. Here we have chosen the Landau gauge couplings $\alpha_{{\rm Landau},4d} (k_{\rm phys}^2)$ at cut-off scales $k_{\rm phys}/T\gg 1$, see [@Alkofer:2000wg; @jan; @Fischer:2008uz; @von; @Smekal:1997is; @Bonnet:2001uh; @Lerche:2002ep]. The corresponding three-dimensional fixed point $\alpha_{*,3d}= 1.12$ is obtained from [@Lerche:2002ep]. A specific choice for such a running coupling is given in Fig. \[fig:alpha\]. ![$\alpha_s$ for temperatures $T=0,150,300,600$ MeV[]{data-label="fig:alpha"}](coupling.eps "fig:"){width="8cm"}\ The normalisation of the momentum scale has been done by the comparison of continuum Landau gauge propagators to their lattice analogues. Fixing the lattice string tension to $\sqrt{\sigma}=440$ MeV, we are led to the above momentum scales. For a comparison with the Landau gauge results obtained in [@Braun:2007bx] we have also computed the temperature-dependence of the Polyakov loop by using $\alpha_{{\rm Landau},4d}$ for all cut-off scales. Indeed, this over-estimates the strength of $\alpha_s$, as can be seen from Fig. \[fig:alpha\], However, qualitatively this does not make a difference: for infrared scales far below the temperature scale, $\hat k\to 0$, the flow switches off for fields $\varphi$ with $\partial^2_\varphi(\hat{V}_{\bot} + \Delta \hat{ V})\geq 0$, that is for the convex part of the potential. This happens both for $g_k^2\to {\rm const}$, and for $g_k^2(\hat k^2\to 0) \sim \hat k$. In other words, the minimum of the potential freezes out in this regime. For the non-convex part of the potential, $\partial^2_\varphi(\hat{V}_{\bot} + \Delta \hat{ V})<0$, the flow does not tend to zero but simply flattens the potential, thus arranging for convexity of the effective potential $V_{\rm eff}=V_{k=0}$. The uncertainty in $g^2_k$ is taken into account by evaluating the limiting cases. Together with the error estimate on the physical cutoff scale $k_{\rm phys}$ in Appendix \[app:match\] this leads to an estimate for the systematic error of the results presented below. This error includes that related to our specific choice of the running coupling. For example, a viable alternative choice to Fig. \[fig:alpha\] is provided by the background field coupling derived in [@Braun:2005uj] which is covered by the above error estimate. Integration {#sec:integration} =========== The numerical solution of is done on a suitably chosen grid or parameterisation of $\Delta \hat V$ and its derivatives. As $\hat V$, $\hat V_\bot$ and $\Delta \hat V$ are periodic, one is tempted to solve in a Fourier decomposition, see e.g. [@Braun:2005cn]. However, as can be seen already at the example of the perturbative Weiss potential $V_W = V_{\bot,0}$, , this periodicity is deceiving. The Weiss potential is polynomial of order four in $\tilde \varphi=\varphi\mod 2 \pi$, its periodicity comes from the periodic $\tilde \varphi(\varphi)$, [@Weiss:1980rj]. Consequently the third derivative $\partial_\varphi^3 V_W$ jumps at $\varphi =2 \pi n$ with $n\in \Z$. Moreover, $\partial_\varphi^3 V_W[\varphi\to 0_+]=-\partial_\varphi^3 V_W[\varphi\to 0_-]\neq 0$. A periodic expansion of $V_W$, e.g. in trigonometric functions cannot capture this property at finite order. This does not only destabilise the parameterisation, but also fails to capture important physics: the flow of the position of the minima is proportional to $\partial_\varphi^3 \hat V$. This follows from $\partial_t \hat V[\varphi_{\rm min,k}]=0$. Expanding this identity leads to $$\label{eq:phimin} \partial_t \varphi_{min,k} = -\left.\0{\partial_t \hat V'[\varphi]}{\hat V''[\varphi]} \right|_{\varphi=\varphi_{\min,k}}\,,$$ where $\hat V'=\partial_\varphi \hat V$ and $\hat V''=\partial_\varphi^2 \hat V$. The flow $\partial_t \hat V'[\varphi]$ is proportional to $\partial_\varphi^3 \hat V$, which e.g. can be seen by taking the $\varphi$-derivative of . Hence, as a Fourier-decomposition enforces $\partial_\varphi^3 \hat V=0$ at any finite order, the minimum does not flow in such an approximation, and the theory always remains in the deconfined phase. Note also that the resulting effective potential at $\hat k=0$ for smooth periodic potentials and flows vanishes identically as it has to be convex. In the present case this is not so, as the potential is rather polynomial (in $\tilde \varphi$) and convexity does not enforce a vanishing effective potential. In turn, a standard polynomial expansion about the minimum $\rho_{\min,k}$ already captures the flow towards the confining phase. Here, however, we use a grid evaluation of the flow of $\Delta \hat V$ with $\varphi\in [0\,,\,2 \pi]$ while taking special care of the boundary conditions at $\varphi=0,2 \pi$: we have extrapolated the second derivative to $\varphi=0$ and $\varphi = 2 \pi$. It suffices to use a first order extrapolation, and we have explicitly checked that the resulting flow is insensitive to the precision of the extrapolation. An alternative procedure is an expansion in terms of Chebyshev polynomials that also works quite well and is also a very fast and efficient way of integrating the flow. A comparison between the results obtained on a grid and with Chebyshev polynomials shows that both parameterisations agree nicely and the corresponding flows deviate from each other only for small values of $k$. This is due to an expected failure of the standard Chebyshev-expansion for those small $\hat k$ where the position of the minimum is almost settled and the potential flattens out in the regions that are not convex. This is better resolved with a grid than with polynomials. On a grid implementation we see the potential becoming convex as $\hat k \to 0$. Results {#sec:results} ======= In Fig. \[fig:Veff\] we show the full effective potential for temperatures ranging from $T=500$ MeV in the deconfined phase to $T=250$ MeV in the confined phase. The expectation value $\langle \varphi\rangle$ in the center-broken deconfined phase is given by the transition point between decreasing part of the potential for small $\varphi$ and the flat region in the middle of the plot. It can also be explicitly computed from . In the center-symmetric confined phase it is just given by the minimum at $\varphi=\pi$.\ ![Full effective potential $\hat V_{\rm eff}$, normalised to 0 at $\varphi =0$[]{data-label="fig:Veff"}](VeffforT.eps "fig:"){width="8cm"}\ The temperature-dependence of the order parameter $L[\langle A_0\rangle ] = \cos(\langle \varphi / 2 \rangle)$ is shown in Fig. \[fig:LofT\], and we observe a second order phase transition from the confined to the deconfined phase at a critical temperature $$\label{eq:Tc} T_c = 305^{+ 40}_{-55}\, {\rm MeV},\qquad \quad {T_c}/{\sqrt{\sigma}} =0.69^{+.04}_{-.12}\,,$$ with the string tension $\sqrt{\sigma}=440$ MeV. The corresponding value on the lattice is ${T_c}/{\sqrt{\sigma}}=.709$, [@Fingberg:1992ju], and agrees within the errors with our result. The estimate of the systematic error in is dominated by that of the uncertainty of the identification of $k_{\rm phys}$, see Appendix \[app:match\]. We would also like to comment on the difference of the temperature-dependence of $L[\langle A_0\rangle ]$ depicted in Fig. \[fig:LofT\] and that of the Polyakov loop $\langle L[A_0]\rangle$. It has been shown in section \[sec:QCDinPol\] that in the confined phase they both vanish and both are non-zero in the deconfined phase. However, the Jensen inequality entails that the present observable $L[\langle A_0\rangle ]$ takes bigger values than the Polyakov loop $\langle L[A_0]\rangle$, which is in agreement with lattice results. ![Temperature dependence of the Polyakov loop $L[\langle A_0\rangle]=\cos(\langle \varphi \rangle / 2)$ in $SU(2)$[]{data-label="fig:LofT"}](ptrans.eps "fig:"){width="8cm"}\ The critical physics should not depend on this issue. Here we compute the critical exponent $\nu$, a quantity well-studied in the $O(1)$ model which is in the same universality class as $SU(2)$ Yang-Mills theory. Moreover, in Polyakov gauge the effective action $\Gamma[A_0]$ after integrating-out the spatial gauge field is close to that of an $O(1)$-model. Studies using the FRG in local potential approximation with an optimised cut-off for the $O(1)$ model yield $\nu = 0.65$, see [@Litim:2001hk]. The critical exponent is related to the screening mass of temporal gauge field by $$m^2(T) \propto |T-T_c|^{2 \nu}, \label{eq:critexp}$$ where $m^2 = V''(\varphi_{min,0}) / 2$. We have computed the temperature-dependence of the screening mass in the confined phase near the phase transition, and extracted the critical exponent $\nu$ from a linear fit to the data. This is shown in Fig. \[fig:critexp\]. The fit yields the anticipated value of $$\nu = 0.65^{+0.02}_{-0.01}\,,$$ for the critical exponent $\nu$. The critical exponent $\beta$ agrees within the errors with the Ising exponent $\beta=0.33$. ![Critical exponent $\nu$ from $m^2=V''(\varphi_{min,0}) / 2$[]{data-label="fig:critexp"}](critexpnu.eps "fig:"){width="8cm"}\ Finally we would like to compare the results obtained here with the results of [@Braun:2007bx]. There, the effective potential $V_{\rm eff}[A_0]$ was computed from the flow [@jan; @Fischer:2008uz] of Landau gauge propagators [@Lerche:2002ep; @von; @Smekal:1997is; @Bonnet:2001uh] within a background field approach in Landau-DeWitt gauge. In this gauge the confining properties of the theory are encoded in the non-trivial momentum dependence of the gluon and ghost propagators. Indeed, in [@Braun:2007bx] the effective potential $V_k$ was computed solely from this momentum dependence and was not fed back into the flow. In $SU(2)$ Landau gauge Yang-Mills this is expected to be a good approximation with the exception of temperatures close to the phase transition, see [@Braun:2007bx]. The back-reaction of the effective potential is particularly important for the critical physics, and the value of the critical temperature [@BGMP]. For the comparison we have computed the present flow with the zero-temperature running coupling in Fig. \[fig:alpha\] for all temperatures. This mimics the approximation used in [@Braun:2007bx], which implicitly relies on the zero-temperature running coupling $\alpha_s$. We also remark that the quantity $L[\langle A_0\rangle]$ in general is gauge-dependent, and only the critical temperature derived from it is not. However, in Landau-DeWitt gauge with backgrounds $A_0$ in Polyakov gauge temporal fluctuations about this background include those in Polyakov gauge. For this reason we might expect a rather quantitative agreement for the quantity $L[\langle A_0\rangle]$ in both approaches. The results for the temperature dependence of the Polyakov loop are depicted in Fig. \[fig:compare\]. ![Comparison of $L[\langle A_0\rangle]$ computed in Polyakov gauge and in Landau-DeWitt gauge from [@Braun:2007bx].[]{data-label="fig:compare"}](LGcompare.eps "fig:"){width="8cm"}\ The coincidence between the two gauges is very remarkable, particularly since the mechanisms driving confinement are quite different in the different approaches, as are the approximations used in both cases. This provides further support for the respective results. It also sustains the argument concerning the lack of gauge dependence made above. The quantitative deviations in the vicinity of the phase transition are due to the truncation used in [@Braun:2007bx], that cannot encode the correct critical physics yet, as has been already discussed there. Summary and outlook {#sec:summ} =================== In the present work we have put forward a formulation of QCD in Polyakov gauge. We have argued that this gauge is specifically well-adapted for the investigation of the confinement-deconfinement phase transition as the order parameter, the Polyakov loop expectation value $\langle L[A_0]\rangle $, has a simple representation in terms of the temporal gauge field. Moreover, we have shown that $L[\langle A_0\rangle]$ also serves as an order parameter. In summary this allows us to access the phase transition within a simple approximation to the full effective action. The computation was done for the gauge group $SU(2)$, where we observe a second order phase transition at a critical temperature of $T_c=305^{+40}_{-55}$ MeV, as well as the Ising critical exponents $\nu$ and $\beta$ to the precision achieved within our approximation. The temperature-dependence of the order parameter $L[\langle A_0\rangle]$ agrees well with a recent computation in Landau gauge [@Braun:2007bx]. This is very remarkable: firstly the latter computation is technically different as in Landau gauge the full momentum-dependence of the propagators is needed to cover confinement. Secondly the order parameter $L[\langle A_0\rangle]$ is gauge dependent, only the critical temperature is not. In the present analysis we used several external inputs which we plan to remove in future work. First of all we proceed with computing the running coupling within Polyakov gauge, that is the momentum-dependence of the temporal gauge field. As it is one of the advantages of the computation in Polyakov gauge that the momentum dependence of Green functions is rather mild we expect only minor deviations from the computations shown here. As argued in the present work, the momentum-dependence of the $A_0$-propagator also gives access to the string tension. For a description of spatial confinement one has to treat the spatial components of the gauge field beyond the present perturbative approximation. Moreover, the present analysis is extended to $SU(3)$, which is conceptually straightforward but technically more challenging. For the matter sector one can revert to the plethora of results with the present renormalisation group methods, ranging from results in effective theories to that in QCD-based approaches, see e.g.[@Berges:2000ew; @Pawlowski:1996ch; @Braun:2005uj; @Braun:2008pi; @Schaefer:2006sr].\ [*Acknowledgements –*]{} We thank J. Braun, H. Gies, F. Lamprecht, D. F. Litim, A. Maas and B.-J. Schaefer for discussions. We thank O. Jahn for discussions and collaboration at an early state of this project. FM acknowledges financial support from the state of Baden-Württemberg and the Heidelberg Graduate School of Fundamental Physics. Faddeev-Popov determinant {#app:FPdet} ========================= From the gauge fixing functionals and we can compute the Faddeev-Popov determinant given by $$\begin{aligned} \Delta_{FP}[A] = \mathrm{det}\left[\frac{ \delta F^a(A^\omega) }{ \delta \omega^b } \right]\,,\end{aligned}$$ where $A^\omega$ is the gauge transformed gauge field $A$. For infinitesimal gauge transformations it is given by $$\begin{aligned} A_\mu^\omega &=& A_\mu - (\partial_\mu \sigma^a + i g A_\mu^b [\sigma^a, \sigma^b ]) \omega^a\,.\end{aligned}$$ In the following we use the representation $\omega^a \sigma^a = \omega^+ \sigma^- + \omega^- \sigma^+ + \omega^3 \sigma^3 $, and the related derivatives w.r.t. $\omega^{\pm},\omega^3$. The matrix elements related to $\omega$-derivatives of $F^+$ read $$\begin{aligned} \nonumber \frac{\delta F^+ (A^\omega)}{ \delta \omega^+ } &=& - {\mathrm{Tr}}\,\sigma^+ \left( \partial_0 \sigma^- + i A_0^3 [\sigma^-, \sigma^3] \right) \,,\\\nonumber \frac{\delta F^+ (A^\omega)}{ \delta \omega^- } &=&0\,,\\ \frac{\delta F^+ (A^\omega)}{ \delta \omega^3 } &=& -{\mathrm{Tr}}\,\sigma^+ \left( \partial_0 \sigma^3 + i A_0^+ [\sigma^3, \sigma^-] \right)\,. \hspace{.5cm} \label{eq:coef+}\end{aligned}$$ Analogously we get for the $\omega$-derivatives of $F^-$ $$\begin{aligned} \nonumber \frac{\delta F^- (A^\omega)}{ \delta \omega^+ } &=&0\,,\\\nonumber \frac{\delta F^- (A^\omega)}{ \delta \omega^- } &=& -{\mathrm{Tr}}\,\sigma^- \left( \partial_0 \sigma^+ + i A_0^3 [\sigma^+, \sigma^3] \right)\,,\\ \frac{\delta F^- (A^\omega)}{ \delta \omega^3 } &=& -{\mathrm{Tr}}\,\sigma^- \left( \partial_0 \sigma^3 + i A_0^- [\sigma^3, \sigma^+] \right)\,. \label{eq:coef-}\end{aligned}$$ The $\omega$-derivatives of $F^3$ yield long expressions, and we only display the parts proportional to $\partial_0 {\mathrm{Tr}}\,\sigma^3 A_0$, where we have abbreviated additional terms proportional to the spatial gauge fields by dots, $$\begin{aligned} \nonumber \frac{\delta F^3 (A^\omega)}{ \delta \omega^+ } &=& -i \partial_0 A_0^- \, {\mathrm{Tr}}\, \sigma^3 [\sigma^-, \sigma^+] + \cdots\,, \\\nonumber \frac{\delta F^3 (A^\omega)}{ \delta \omega^- } &=& - i \partial_0 A_0^+ \,{\mathrm{Tr}}\, \sigma^3 [\sigma^+, \sigma^-] + \cdots\,, \\ \frac{\delta F^3 (A^\omega)}{ \delta \omega^3 } &=&- 2 \partial_0^2 + \cdots \,. \label{eq:coef3}\end{aligned}$$ Evaluating the traces ,, we can compute the Faddeev-Popov determinant. Again we only concentrate on the terms dependent on $A_0$, and use the gauge fixing condition $A_0^+ = A_0^- = 0$ for eliminating some of the off-diagonal elements, $$\begin{aligned} \Delta_{FP}[A] &=& - \mathrm{det}\left[ \left( \begin{array}{ccc} \partial_0 +ig A_0^3 & 0 & 0\\ 0 & \partial_0 - ig A_0^3 & 0\\ -4 ig \int dx_0 \partial_1 A_1^- + \cdots & 4 i g \int dx_0 \partial_1 A_1^+ + \cdots & 1/2( \partial_0^2 + \int dx_0 \partial_1^2 + \cdots) \end{array} \right) \right] \nonumber\\ &=& - \mathrm{det}[(\partial_0 + ig A_0^3) (\partial_0 -ig A_0^3) \frac{1}{2} \left( \partial_0^2 + \int dx_0 \partial_1^2 + \int dx_0 dx_1 \partial_2^2+ \int dx_0 dx_1 dx_2 \partial_3^2 \right)]\end{aligned}$$ Using the third gauge fixing condition, $\partial_{0} A_{0}^3 = 0$, we can write the Faddeev-Popov determinant as $$\begin{aligned} \nonumber \Delta_{FP}[A] &=& \frac{1}{2}\mathrm{det} \left[ \left( \partial_0^2 + \left(g A_0^3\right)^2\right)\right] \mathrm{det}[\left( \partial_0^2 + \cdots \right)]\,. \label{eq:FPdet0} \end{aligned}$$ We note that the second determinant in is independent of the gauge fields and hence can be absorbed in the normalisation of the path integral. The first determinant is evaluated in frequency space, we get $$\prod_{\vec x} \left( (g A_0^3 (\vec x)) \prod_{n=1}^{n = \infty} \left((2 \pi T n)^2 - (g A_0^3 (\vec x))^2\right) \right)^2\,. \label{eq:FPdet1}$$ Multiplying the determinant with a further constant normalisation $$\begin{aligned} \mathcal{N} = \left( \prod_{n=1}^{n = \infty} (2 \pi T n)^2 \right)^{-2}\,,\end{aligned}$$ we arrive at $$\mathcal{N} \mathrm{det} \left[ G_{A_0}\right] = \prod_{x} (g A_0^3 (x))^2 \prod\limits_{n=1}^{n = \infty} \left(1 - \left( \frac{g A_0^3 (x)}{ 2 \pi n T} \right)^2 \right)\,. \label{eq:FPdet2}$$ is just a product representation of the sine-function, $\sin(x) = x \prod_{n=1}^{n = \infty} \left(1 - \frac{x^2}{ (\pi n)^2 }\right)$, and the final result for the Faddeev-Popov determinant is $$\begin{aligned} \Delta_{FP}[A] = \mathcal{N}' (2 T)^2 \left[ \prod_{x} \sin^2 \left( \frac{g A_0^3 (x)}{ 2 T } \right) \right],\end{aligned}$$ where $\mathcal{N}' $ is a further normalisation constant. Integrating out spatial gluons {#app:intoutAI} ============================== After integrating out the longitudinal gauge fields the action $S_{\rm eff}= \frac{1}{4}\int_T d^4x F_{\perp, \mu \nu}^a F_{\perp,\mu \nu}^a$ reads $$S_{\rm eff}= - \frac{1}{2} \beta\int d^3 x\, Z_0 A_0 \vec \partial^2 A_0 - \frac{1}{2} \int_T d^4x\, A_i^a \left[ (\partial_0^2 + \vec \partial^2) \delta_{ij} - \partial_i \partial_j + 2 g f^{a3b} (A_0 \partial_0 +g^2 A_0^2 (\delta^{ab} - \delta^{a3}\delta^{b3})\delta_{ij}\right] A_j^b +O(A_i^3)$$ Writing $A_0^3 = \varphi/(g \beta ) + a_0$, where $\varphi$ is a constant and $a_0$ the fluctuating field, this expression is given to second order in the fluctuating fields by $$\begin{aligned} S_{YM} &\approx& \frac{1}{2} \int d\tau d^3x\, \left\{ Z_0 (\vec \partial a_0)^2 -2 \varphi f^{a3c} (\partial_0 A_i^a) A_i^c+ \right. \nonumber\\ & & \hspace{1.9cm} \varphi^2 (\delta^{ab} - \delta^{a3}\delta^{b3})A_i^a A_i^b - \nonumber \\ & & \hspace{1.9cm} \left. A_i^a \left( (\partial_0^2 + \vec \partial^2) \delta_{ij} - \partial_i \partial_j \right)A_j^a \right\} \nonumber \\ &=& \frac{1}{2} \int d\tau d^3x \ \left\{ (\vec \partial a_0)^2 - A_i^a ( \vec \partial^2 - \partial_i \partial_j) A_j^a - \right. \nonumber\\ & & \hspace{1.9cm}\left. A_i^a D_0^{ac}D_0^{cb} A_i^b \right\}, \end{aligned}$$ where we have defined $$\begin{aligned} D_0^{ab} = \partial_0 \delta^{ab} + A^3_0 g f^{a3b} .\end{aligned}$$ In the present work we neglect back-reactions of the $A_0$ potential on the transversal gauge fields. Assuming an expansion around $A_i^a = 0$, $\Gamma^{(2)}$ is block-diagonal, like the regulators, cf. eq. (\[eq:cutoffs\]), and we can decompose the flow equation (\[eq:flow\]) into a sum of two contributions, schematically written as $$\begin{aligned} \label{eq:FRG_before_intout} \partial_t{\Gamma}_{k} &=& \frac{1}{2}\mathrm{Tr} \left(\frac{1}{\Gamma_{k}^{(2)} + R_A} \right)_{00} \partial_t R_k + \nonumber \\ & &\hspace{2cm}{\mathrm{Tr}}\, \partial_t \left[\ln (S_{YM}^{(2)} + R_{A}) \right]_{ii}. \end{aligned}$$ The first term on the rhs encodes the quantum fluctuations of $A_0$, the second line encodes those of the transversal spatial components of the gauge field. In the present truncation the second line is a total derivative w.r.t. $t$, and does not receive contributions from the first term. Therefore we can evaluate the flow of the second contribution, and use its output $V_{\bot,k}(A_0)$ as an input for the remaining flow. The computation is done for the regulators . As explained below in section \[sec:approx\], the cut-off parameters $k$, and $k_\bot $ in $R_k$ for the fluctuations of $A_0$ and $R_{k,\bot}$ for the fluctuations of $\vec A_\bot$ respectively satisfy a non-trivial relation $k_\bot =k_\bot (k)$ for coinciding physical infrared cut-offs $k_0$ for $A_0$ and $k_\bot $ for $\vec A_\bot$. The computation is similar to those done in one loop perturbation theory in $SU(2)$ by Weiss [@Weiss:1980rj], the only difference being the infrared cut-off. We infer from the second line in that $$\begin{aligned} \label{eq:preVbot} V_{\bot,k} &=& V_{\bot,\Lambda_{\rm UV}} +\left. \012 {\mathrm{Tr}}\, \left[\ln (S_{YM}^{(2)} + R_{A})\right]_{ii} \right|^k_{\Lambda_{\rm UV}} \\\nonumber &=& V_W + T \sum_{n} \int \frac{d^3p}{(2\pi)^3} \theta (k_\bot ^2 - \vec p^2) \ln ( k_\bot ^2 + D_0^2)\,.\end{aligned}$$ In we have used that $V_{\bot,\Lambda_{\rm UV}\to \infty}=0$ up to a constant term, and have added and subtracted the Weiss potential $V_W$ [@Weiss:1980rj], $$\label{eq:Vweiss} V_W(\varphi) = -(\tilde\varphi-\pi)^2 / (6 \beta^4) + (\tilde\varphi-\pi)^4 / (12 \pi^2 \beta^4)\,,$$ with the dimensionless $\varphi=g\beta A_0$, and $\tilde \varphi=\varphi \mod 2 \pi$. Alternatively one can simple put $\Lambda_{\rm UV}=0$, even though this seems to be counter-intuitive. We also have used that with it follows ${\mathrm{tr}}\, \Pi_\bot =2$. Performing the Matsubara sum and neglecting terms independent of the temporal gauge fields, the resulting effective potential is given by $$\begin{aligned} \label{eq:Vbot} V_{\bot,k} &=& \frac{4 T}{(2\pi)^2} \int_0^{k_\bot } dp p^2 \left\{ \mathrm{ln}\Bigl(1-2\cos(\varphi) e^{-\beta k_\bot } \right. \\ & & \hspace{-.5cm} \left. + e^{-2 \beta k_\bot }\Bigr) - \mathrm{ln}(1-2\cos(\varphi) e^{-\beta p} + e^{-2 \beta p}) \right\} + V_W \,. \nonumber \end{aligned}$$ From we deduce that the potential $ V_{k_\bot }$ approaches $V_W$ in the limit $k \to 0$ and vanishes like $e^{-\beta k_\bot } \cos(\varphi)$ for $k \to \infty$. From eq. (\[eq:FRG\_before\_intout\]) we can now extract the flow of the effective potential, by setting $V_{\mathrm{eff},k} = \Delta V_{k} + V_{\bot,k}$. Then we get $$\label{eq:flowDeltaVapp} \partial_t \Delta V_{k}=\012 \int \0{d^3 p}{(2 \pi)^3} \frac{(\eta_0(k^2-\vec p^2)+ 2k^2 )\theta (k^2 - \vec p^2) }{k^2 + g_k^2 \beta^2 (\Delta V_{k}'' + V_{\bot,k}'') }\,,$$ with the input $V_{\bot,k}$ given in and $\eta_0=\partial_t \ln Z_0$. The factor $g^2 \beta^2$ arises from the fact that we parametrise the potential in terms of $\varphi$ rather than in $A_0$, and $g_k^2=g^2/Z_0$ is nothing but the running coupling at momentum $\vec p^2\sim k_{\rm phys}^2$. Thus we estimate $g_k^2=4\pi \alpha_s(\vec p^2=k_{\rm phys}^2)$. Note that $g_k$ is an RG-invariant. The momentum integration can be performed analytically, and we are led to $$\begin{aligned} \label{eq:preflowVapp} \beta \partial_k \Delta V_k = \frac{2}{3 (2 \pi)^2} \frac{(1+\eta_0/5) k^2 }{1+\frac{ g_{k}^2 \beta^2}{ k^2 } \partial^2_{\varphi} ( V_{\bot,k} + \Delta V_k)}\,, \end{aligned}$$ where $\eta_0$ is given by $$\label{eq:eta0app} \eta_0=-\partial_t \log \alpha_s\,,$$ as the consistent choice in the given truncation. Matching scales {#app:match} =============== The flow of the temporal component of the gauge field, $A_0(\vec x)$, is computed with a three-dimensional regulator, see . In Polyakov gauge $A_0(\vec x)$ only depends on the spatial coordinates, whereas the spatial components $A_\bot(x)$ are four-dimensional fields. For cut-off scales far lower than the temperature, $k/T\ll 1$, also the spatial gauge fields are effectively three-dimensional fields as only the Matsubara zero mode propagates. Hence in this regime we can identify $k=k_\bot$. For large cut-off scales, $k/T\gg 1$, the $A_0$-flow decouples from the theory. A comparison between the two flows can only be done after the summation of the spatial flow over the Matsubara frequencies. In the asymptotic regime $k/T\gg 1$ this leads to the relation $$\label{eq:kTinf} \frac{1}{k} \simeq \sum_{n=-\infty}^{\infty} \frac{1}{\omega_n^2 + k_\bot ^2}\to \frac{1}{2 k_\bot}\,,$$ The crossover between these asymptotic regimes happens at about $k/T= 1$. This crossover is implemented with the help of an appropriately chosen interpolating function $f$, $$\begin{aligned} \label{eq:compare} \frac{T}{k^2} f(k/T) &=& T \sum_{n=-\infty}^{\infty} \frac{1}{\omega_n^2 + k_\bot ^2}\,,\end{aligned}$$ A natural choice for $f(k/T)$ is depicted in Fig. \[fig:kbotk\], and has been used in the computation. A more sophisticated adjustment of the relative scales can be performed within a comparison of the flow of momentum-dependent observables such as the wave function renormalisation $Z_0$. The peak of these flows in momentum space is directly related to the cut-off scale. Indeed, the function $f$ carries the physical information of the peak of the flow at some momentum scale. Scanning the set of $f$ gives some further access to the uncertainty in such a procedure. ![${\hat k}_\bot / \hat k$ as function of $\hat k$. []{data-label="fig:kbotk"}](kbot.eps "fig:"){width="8cm"}\ The effective cut-off scales $k_{\rm phys}(k_0)$ and $k_{\bot ,\rm phys}(k_\bot )$ in the flows of the temporal gluons and of spatial gluons respectively do not match in general. If solving the flow within a local truncation as chosen in the present work we have to identify the two effective cut-off scales, $k_{\rm phys}(k_0)=k_{\bot ,\rm phys}(k_\bot )=k_{\rm phys}$, leading to a non-trivial relation $k_0=k_0(k_\bot )$. Moreover, the effective cut-off scale has to be used in the running coupling $\alpha_s=\alpha_s(\vec p^2=k_{\rm phys}^2)$. ![$\hat k_{\rm phys}(\hat k )$ from the comparison of flows with three-dimensional regulators and four-dimensional regulators.[]{data-label="fig:kskphys"}](kphysalpha.eps "fig:"){width="8cm"}\ It is left to determine the physical cut-off scale $k_{\rm phys}$ from either the flow of the spatial gauge fields as $k_{\bot,\rm phys}(k_\bot )$ or from the temporal flow $k_{0,\rm phys}(k_0)$. We first discuss the spatial flow. For an optimised regulator depending on all momentum directions, $p^2$, we have the relation $k_{\rm phys}=k_\bot $. Hence the relation $k_{\bot,\rm phys}(k_\bot )$ can be computed if comparing the flows for a specific observable with three-dimensional regulator $R_{{\rm opt},k_\bot }(\vec p^2)$, , with flows with four-dimensional regulator $R_{{\rm opt},k_{\rm phys}}(p^2)$. Here, as a model example, we choose the effective potential of a $\phi^4$-theory. This leads to the relation $k_{\rm phys}(k_\bot )$ displayed in Fig \[fig:kskphys\]. We remark that the relation in Fig. \[fig:kskphys\] depends on the dimension $d$ of the theory, and flatten to $k_{\rm phys}(k)=k$ for $d\to\infty$. In other words, $\lim_{k\to \infty} k_{\rm phys}(k)/k$ is proportional to $d/(d-1)$. Moreover, for momentum-dependent observables the crossover rather resembles the relation $k_\bot(k_0)$ as it is more sensitive to the propagator than to the momentum integral of the propagator. Indeed, for the three-dimensional field $A_0(\vec x)$ the cut-off scale $k_0$ is another natural choice for the physical cut-off scale, $k_{0,\rm phys}(k_0)=k_0$, even though it underestimates the importance of the spatial flow for the correlations of the temporal gauge field. In summary, we take the above two extremal choices $k_{\rm phys}=k_{\bot,\rm phys}$ depicted in Fig. \[fig:kskphys\] and $k_{\rm phys}= k_{0,\rm phys}(k_0)=k_0$ as a broad estimate of the systematic error in the present computation. [99]{} B. Svetitsky, Phys. Rept.  [**132**]{} (1986) 1; J. Greensite, Prog. Part. Nucl. Phys.  [**51**]{} (2003) 1. R. Alkofer and L. von Smekal, Phys. Rept.  [**353**]{} (2001) 281; C. S. Fischer, J. Phys. G [**32**]{} (2006) R253. D. F. Litim and J. M. Pawlowski, in [*The Exact Renormalization Group*]{}, Eds. Krasnitz et al, World Sci (1999) 168. H. Gies, hep-ph/0611146. B. J. Schaefer and J. Wambach, Phys. Part. Nucl.  [**39**]{} (2008) 1025. C. S. Fischer, A. Maas and J. M. Pawlowski, arXiv:0810.1987 \[hep-ph\]; L. von Smekal, arXiv:0812.0654 \[hep-th\]. T. Schafer and E. V. Shuryak, Rev. Mod. Phys.  [**70**]{} (1998) 323; F. Lenz, Lect. Notes Phys.  [**659**]{} (2005) 7; F. Bruckmann, Eur. Phys. J. ST [**152**]{} (2007) 61. A. M. Polyakov, Phys. Lett.  B [**72**]{} (1978) 477; L. Susskind, Phys. Rev.  D [**20**]{} (1979) 2610. C. Borgs and E. Seiler, Commun. Math. Phys.  [**91**]{} (1983) 329; J. Kuti, J. Polonyi and K. Szlachanyi, Phys. Lett.  B [**98**]{} (1981) 199. H. Reinhardt, Nucl. Phys.  B [**503**]{} (1997) 505; O. Jahn and F. Lenz, Phys. Rev.  D [**58**]{} (1998) 085006. C. Ford, U. G. Mitreuter, T. Tok, A. Wipf and J. M. Pawlowski, Annals Phys.  [**269**]{} (1998) 26; C. Ford, T. Tok and A. Wipf, Phys. Lett.  B [**456**]{} (1999) 155. J. Greensite, S. Olejnik and D. Zwanziger, AIP Conf. Proc.  [**756**]{} (2005) 162. R. D. Pisarski, Phys. Rev.  D [**62**]{}, 111501 (2000); P. N. Meisinger, T. R. Miller and M. C. Ogilvie, Nucl. Phys. Proc. Suppl.  [**119**]{}, 511 (2003); K. Fukushima, Phys. Lett.  B [**591**]{}, 277 (2004); C. Ratti, M. A. Thaler and W. Weise, Phys. Rev.  D [**73**]{} (2006) 014019; E. Megias, E. Ruiz Arriola and L. L. Salcedo, Phys. Rev.  D [**74**]{} (2006) 065005; C. Sasaki, B. Friman and K. Redlich, Phys. Rev.  D [**75**]{}, 074013 (2007); B.-J. Schaefer, J. M. Pawlowski and J. Wambach, Phys. Rev.  D [**76**]{} (2007) 074023. L. McLerran and R. D. Pisarski, Nucl. Phys.  A [**796**]{} (2007) 83. J. Braun, arXiv:0810.1727 \[hep-ph\]; J. Braun, L. M. Haas, F. Marhauser, J. M. Pawlowski, work in progress. C. Wetterich, Phys. Lett. B [**301**]{}, 90 (1993). J. Berges, N. Tetradis and C. Wetterich, Phys. Rept.  [**363**]{} (2002) 223. C. Bagnuls and C. Bervillier, Phys. Rept.  [**348**]{} (2001) 91; K. Aoki, Int. J. Mod. Phys. B [**14**]{} (2000) 1249; M. Salmhofer and C. Honerkamp, Prog. Theor. Phys.  [**105**]{} (2001) 1; J. Polonyi, Central Eur. J. Phys.  [**1**]{} (2003) 1; B. Delamotte, arXiv:cond-mat/0702365. J. M. Pawlowski, Annals Phys.  [**322**]{} (2007) 2831. J. Fingberg et al., Nucl. Phys.  B [**392**]{} (1993) 493; O. Kaczmarek et al., Phys. Lett.  B [**543**]{} (2002) 41; B. Lucini et al., JHEP [**0502**]{} (2005) 033; C. Wozar, T. Kaestner, A. Wipf and T. Heinzl, Phys. Rev.  D [**76**]{} (2007) 085004 J. Braun, H. Gies and J. M. Pawlowski, arXiv:0708.2413 \[hep-th\]. D. F. Litim, Phys. Lett. B [**486**]{} (2000) 92; Phys. Rev. D [**64**]{} (2001) 105007; Int. J. Mod. Phys. A [**16**]{} (2001) 2081. D. F. Litim and J. M. Pawlowski, JHEP [**0611**]{} (2006) 026. N. Weiss, Phys. Rev. D [**24**]{}, 475 (1981); P. N. Meisinger et al., Phys. Lett.  B [**407**]{} (1997) 297; M. Engelhardt et al., Phys. Lett.  B [**430**]{} (1998) 161; H. Gies, Phys. Rev.  D [**63**]{} (2001) 025013. C. Lerche and L. von Smekal, Phys. Rev. D [**65**]{} (2002) 125006. A. Maas, J. Wambach, B. Gruter and R. Alkofer, Eur. Phys. J.  C [**37**]{} (2004) 335; J. M. Pawlowski, in preparation. L. von Smekal, R. Alkofer and A. Hauck, Phys. Rev. Lett.  [**79**]{} (1997) 3591 C. S. Fischer and R. Alkofer, Phys. Lett.  B [**536**]{} (2002) 177 J. M. Pawlowski, D. F. Litim, S. Nedelko and L. von Smekal, Phys. Rev. Lett.  [**93**]{} (2004) 152002; C. S. Fischer and H. Gies, JHEP [**0410**]{} (2004) 048. F. D. R. Bonnet, P. O. Bowman, D. B. Leinweber, A. G. Williams and J. M. Zanotti, gluon Phys. Rev.  D [**64**]{}, 034501 (2001); A. Sternbeck, E. M. Ilgenfritz, M. Mueller-Preussker and A. Schiller, Phys. Rev.  D [**72**]{}, 014507 (2005); A. Cucchieri and T. Mendes, PoS [**LAT2007**]{} (2007) 297; A. Sternbeck, L. von Smekal, D. B. Leinweber and A. G. Williams, PoS [**LAT2007**]{} (2007) 340; A. Sternbeck and L. von Smekal, arXiv:0811.4300 \[hep-lat\]. J. Braun, H. Gies and H. J. Pirner, AIP Conf. Proc.  [**775**]{} (2005) 162 \[arXiv:hep-ph/0610341\]. D. F. Litim and J. M. Pawlowski, Phys. Lett.  B [**516**]{} (2001) 197. J. M. Pawlowski, Phys. Rev.  D [**58**]{} (1998) 045011; H. Gies and C. Wetterich, Phys. Rev.  D [**69**]{} (2004) 025001; J. Braun and H. Gies, Phys. Lett.  B [**645**]{} (2007) 53; JHEP [**0606**]{} (2006) 024; J. Braun, H. Gies, F. Marhauser and J. M. Pawlowski, work in progress [^1]: A reformulation in terms of the Polyakov loop variable only along the lines outlined in [@Pawlowski:2005xe] is in progress.

--- abstract: 'Large number of weights in deep neural networks makes the models difficult to be deployed in low memory environments such as, mobile phones, IOT edge devices as well as “inferencing as a service" environments on cloud. Prior work has considered reduction in the size of the models, through compression techniques like pruning, quantization, Huffman encoding etc. However, efficient inferencing using the compressed models has received little attention, specially with the Huffman encoding in place. In this paper, we propose efficient parallel algorithms for inferencing of single image and batches, under various memory constraints. Our experimental results show that our approach of using variable batch size for inferencing achieves 15-25% performance improvement in the inference throughput for AlexNet, while maintaining memory and latency constraints.' author: - bibliography: - 'ms.bib' title: Efficient Inferencing of Compressed Deep Neural Networks --- Introduction ============ Discussion on use cases and challenges {#motivation} ====================================== Preliminaries {#sec:prelims} ============= Inferencing using Compressed Models {#sec:inference} =================================== Experimental Results with Blocking {#sec:expt1} ================================== Inferencing with Variable Batch Size {#sec:dp} ------------------------------------ Experimental Results with Batch Size {#sec:expt2} ==================================== Concluding Remarks and Future work {#sec:conc} ==================================

[Optimal Networks]{} [A.O. Ivanov and A.A. Tuzhilin]{} > [*The aim of this mini-course is to give an introduction in Optimal Networks theory. Optimal networks appear as solutions of the following natural problem: How to connect a finite set of points in a metric space in an optimal way? We cover three most natural types of optimal connection: spanning trees [(]{}connection without additional road forks[)]{}, shortest trees and locally shortest trees, and minimal fillings.* ]{} Introduction: Optimal Connection ================================ This mini-course was given in the First Yaroslavl Summer School on Discrete and Computational Geometry in August 2012, organized by International Delaunay Laboratory “Discrete and Computational Geometry” of Demidov Yaroslavl State University. We are very thankful to the organizers for a possibility to give lectures their and to publish this notes, and also for their warm hospitality during the Summer School. The real course consisted of three 1 hour lectures, but the division of these notes into sections is independent on the lectures structure. The video of the lectures can be found in the site of the Laboratory (`http://dcglab.uniyar.ac.ru`). The main reference is our books [@ITBookWP] and [@ITBookRFFI], and the paper [@ITGromov] for Section \[sec:mf\]. Our subject is optimal connection problems, a very popular and important kind of geometrical optimization problems. We all seek what is better, so optimization problems attract specialists during centuries. Geometrical optimization problems related to investigation of critical points of geometrical functionals, such as length, volume, energy, etc. The main example for us is the length functional, and the corresponding optimization problem consists in finding of length minimal connections. Connecting Two Points --------------------- If we have to points $A$ and $B$ in the Euclidean plane $\R^2$, then, as we know from the elementary school, the shortest curve joining $A$ and $B$ is unique and coincides with the straight segment $AB$, so optimal connection problem is trivial in this case. But if we change the way of distance measuring and consider, for example, so-called Manhattan plane, i.e. the plane $\R^2$ with fixed standard coordinates $(x,y)$ and the distance function $\r_1(A,B)=|a_1-b_1|+|a_2-b_2|$, where $A=(a_1,a_2)$ and $B=(b_1,b_2)$, then it is not difficult to verify that in this case there are infinitely many shortest curves connecting $A$ and $B$. Namely, if $0\le a_1<b_1$ and $0\le a_2<b_2$, then any monotonic curve $\g(t)=\bigl(x(t),y(t)\bigr)$, $t\in[0,1]$, $\g(0)=A$, $\g(1)=B$, where functions $x(t)$ and $y(t)$ are monotonic, are the shortest, see Figure \[fig:manh\], left. Another new effect that can be observed in this example is as follows. In the Euclidean plane a curve such that each its sufficiently small piece is a shortest curve joining its ends (so-called [*locally shortest curve*]{}) is a shortest curve itself. In the Manhattan plane it is not so. The length of a locally shortest curve having the form of the letter $\Pi$, see Figure \[fig:manh\], right, can be evidently decreased. Similar effects can be observed in the surface of standard sphere $S^2\subset\R^3$. Here the shortest curve joining a pair of points is the lesser arc of the great circle (the cross-section of the sphere by a plane passing through the origin). Two opposite points are connected by infinitely many shortest curves, and if points $A$ and $B$ are not opposite, then the corresponding great circle is unique and it is partitioned into two arcs, both of them are locally shortest, one is the (unique) shortest, but the other one is not. (Really speaking, the difference with the Manhattan plane consists in the fact that for the case of the sphere any directional derivative of the length of any locally shortest arc with respect to its deformation preserving the ends is equal to zero). For a pair of points on the surface of the cube describe shortest and locally shortest curves. Find out an infinite family of locally shortest curves having pairwise distinct lengths. Connecting Many Points: Possible Approaches ------------------------------------------- Let us consider general situation, when we are given with a finite set $M=\{A_1,\ldots,A_n\}$ of points in a metric space $(X,\r)$, and we want to connect them in some optimal way in the sense of the total length of the connection. We are working under assumption that we already know how to connect pairs of points in $(X,\r)$, therefore we need just to organize the set of shortest curves in appropriate way. There are several natural statements of the problem, and we list here the most popular ones. ### No Additional Forks Case: Spanning Trees We do not allow additional forks, that is, we can switch between the shortest segments at the points from $M$ only. As a result, we obtain a particular case of Graph Theory problem about minimal spanning trees in a connected weighted graph. We recall only necessary concepts of Graph Theory, the detail can be found, for example in [@Emel]. Recall that a ([*simple*]{}) [*graph*]{} can be considered as a pair $G=(V,E)$, consisting of a finite set $V=\{v_1,\ldots,v_n\}$ of [*vertices*]{} and a finite set $E=\{e_1,\ldots,e_m\}$ of [*edges*]{}, where each edge $e_i$ is a two-element subset of $V$. If $e=\{v,v'\}$, then we say that $v$ and $v'$ are [*neighboring*]{}, edge $e$ [*joins*]{} or [*connects*]{} them, the edge $e$ and each of the vertices $v$ and $v'$ are [*incident*]{}. The number of vertices neighboring to a vertex $v$ is called the [*degree of $v$*]{} and is denoted by $\deg v$. A graph $H=(V_H,E_H)$ is said to be a [*subgraph*]{} of a graph $G=(V_G,E_G)$, if $V_H\subset V_G$ and $E_H\subset E_G$. The subgraph $H$ is called [*spanning*]{}, if $V_H=V_G$. A [*path $\g$*]{} in a graph $G$ is a sequence $v_{i_1},e_{i_1},v_{i_2}\ldots,e_{i_k}v_{i_{k+1}}$ of its vertices and edges such that each edge $e_{i_s}$ connects vertices $v_{i_s}$ and $v_{i_{s+1}}$. We also say that the path $\g$ connects the vertices $v_{i_1}$ and $v_{i_{k+1}}$ which are said to be [*ending vertices*]{} of the path. A path is said to be [*cyclic*]{}, if its ending vertices coincide with each other. A cyclic path with pairwise distinct edges is referred as a [*simple cycle*]{}. A graph without simple cycles is said to be acyclic. A graph is said to be [*connected*]{}, if any its two vertices can be connected by a path. An acyclic connected graph is called a [*tree*]{}. If we are given with a function $\om\:E\to \R$ on the edge set of a graph $G$, then the pair $(G,\om)$ is referred as a [*weighted graph*]{}. For any subgraph $H=(V_H,E_H)$ of a weighted graph $\bigr(G=(V_G,E_H),\om\bigl)$ the value $\om(H)=\sum_{e\in E_H}\om(e)$ is called the [*weight of $H$*]{}. Similarly, for any path $\g=v_{i_1},e_{i_1},v_{i_2}\ldots,e_{i_k}v_{i_{k+1}}$ the value $\om(\g)=\sum_{s=1}^k\om(e_{i_s})$ is called the [*weight of $\g$*]{}. For a weighted connected graph $\bigl(G=(V_G,E_G),\om\bigr)$ with positive weight function $\om$, a spanning connected subgraph of minimal possible weight is called [*minimal spanning tree*]{}. The positivity of $\om$ implies that such subgraph is [*acyclic*]{}, i.e. it is a tree indeed. The weight of any minimal spanning tree for $(G,\om)$ is denoted by $\mst(G,\om)$. Optimal connection problem without additional forks can be considered as minimal spanning tree problem for a special graph. Let $M=\{A_1,\ldots,A_n\}$ be a finite set of points in a metric space $(X,\r)$ as above. Consider the complete graph $K(M)$ with vertex set $M$ and edge set consisting of all two-element subsets of $M$. In other words, any two vertices $A_i$ and $A_j$ are connected by an edge in $K(M)$. By $A_iA_j$ we denote the corresponding edge. The number of edges in $K(M)$ is, evidently, $n(n-1)/2$. We define the positive weight function $\om_\r(A_iA_j)=\r(A_i,A_j)$. Then any minimal spanning tree $T$ in $K(M)$ can be considered as a set of shortest curves in $(X,\r)$ joining corresponding points and forming a network in $X$ connecting $M$ without additional forks in an optimal way, i.e. with the least possible length. Such a network is called a [*minimal spanning tree for $M$ in $(X,\r)$*]{}. Its total weight $\om_\r(T)$ is called [*length*]{} and is denoted by $\mst_X(M)$. In Section \[sec:mst\] we speak about minimal spanning trees in more details. ### Shortest tree: Fermat–Steiner Problem But already P. Fermat and C.F. Gauss understood that additional forks can be profitable, i.e. can give an opportunity to decrease the length of optimal connection. For example, see Figure \[fig:tri\], if we consider the vertex set $M=\{A_1,A_2,A_3\}$ of a regular triangle with side $1$ in the Euclidean plane, then the corresponding graph $K(M)$ consists of three edges of the same weight $1$ and each minimal spanning tree consists of two edges, so $\mst_{\R^2}(M)=2$. But if we add the center $T$ of the triangle and consider the network consisting of three straight segments $A_1T$, $A_2T$, $A_3T$, then its length is equal to $3\frac{2}{3}\frac{\sqrt3}{2}=\sqrt3<2$, so it is shorter than the minimal spanning tree. This reasoning leads to the following general definition. Let $M=\{A_1,\ldots,A_n\}$ be a finite set of points in a metric space $(X,\r)$ as above. Consider a larger finite set $N$, $M\subset N\subset X$, and a minimal spanning tree for $N$ in $X$. Then this tree contains $M$ as a subset of its vertex set $N$, but also may contain some other additional vertices-forks. Such additional vertices are referred as [*Steiner points*]{}. Further, we define a value $\smt_X(M)=\inf_{N:M\subset N\subset X}\mst_X(N)$ and call it the [*length of shortest tree connecting $M$*]{} or of [*Steiner minimal tree for $M$*]{}. If this infimum attains at some set $N$, then each minimal spanning tree for this $N$ is called a [*shortest tree*]{} or a [*Steiner minimal tree*]{} connecting $M$. Famous Steiner problem is the problem of finding a shortest tree for a given finite subset of a metric space. We will speak about Steiner problem in more details in Section \[sec:smt\]. The shortest tree for the vertex set of a regular triangle in the Euclidean plane is depicted in Figure \[fig:tri\]. ### Minimizing over Different Ambient Spaces: Minimal Fillings Shortest trees give the least possible length of connecting network for a given finite set in a fixed ambient space. But sometimes it s possible to decrease the length of connection by choosing another ambient space. Let $M=\{A_1,\ldots,A_n\}$ be a finite set of points in a metric space $(X,\r)$ as above, and consider $M$ as a finite metric space with the distance function $\r_M$ obtained as the restriction of the distance function $\r$. Consider an isometric embedding $\v\:(M,\r_M)\to (Y,\r_Y)$ of this finite metric space $(M,\r_M)$ into a (compact) metric space $(Y,\r_Y)$ and consider the value $\smt_Y\bigl(\v(M)\bigr)$. It could be less than $\smt_X(M)$. For example, the vertex set of the regular triangle with side $1$ can be embedded into Manhattan plane as the set $\bigl\{(-1/2,0),(0,1/2),(1/2,0)\bigr\}$, see Figure \[fig:tri\]. Than the unique additional vertex of the shortest tree is the origin and the length of the tree is $3/2<\sqrt3$. So, for a finite metric space $\cM=(M,\r_M)$, consider the value $\mf(\cM)=\inf_\v\smt_Y\bigl(\v(M)\bigr)$ which is referred as [*weight of minimal filling*]{} of the finite metric space $\cM$. Minimal fillings can be naturally defined in terms of weighted graphs and can be considered as a generalization of Gromov’s concept of minimal fillings for Riemannian manifolds. We speak about them in more details in Section \[sec:mf\]. Minimal Spanning Trees ====================== \[sec:mst\] In this section we discuss minimal spanning trees construction in more details. As we have already mentioned above, in this case the problem can be stated in terms of Graph Theory for an arbitrary connected weighted graph. But geometrical interpretation permits to speed up the algorithms of Graph Theory. General Case: Graph Theory Approach ----------------------------------- We start with the Graph Theory problem of finding a minimal spanning tree in a connected weighted graph. It is not difficult to verify that direct enumeration of all possible spanning subtrees of a connected graph leads to an exponential algorithm. To see that, recall well-known Kirchhoff theorem counting the number of spanning subtrees. If $G=(V,E)$ is a connected graph with enumerated vertex set $V=\{v_1,\ldots,v_n\}$, then its [*Kirchhoff matrix*]{} is defined as $(n\times n)$-matrix $B_G=(b_{ij})$ with elements $$b_{ij}=\left[\begin{array}{ll} \deg v_i & \quad\text{if $i=j$,}\\ -1 &\quad \text{if $\{v_i,v_j\}\in E$,}\\ 0 & \quad \text{otherwise.} \end{array}\right.$$ Then the following result based on elementary Graph Theory and Binet–Cauchy formula for determinant calculation is valid, see proof, for example in [@Emel]. For a connected graph $G$ with $n\ge2$ vertices, the number of spanning subtrees is equal to the algebraic complement of any element of the Kirchhoff matrix $B_G$. Let $G=K_n$ be the complete graph with $n$ vertices. Than its Kirchhoff matrix has the following form: $$B_{K_n}=\left(\begin{array}{ccccc} n-1 & -1 & -1& \cdots & -1 \\ -1 & n-1& -1& \cdots & -1 \\ \vdots& \vdots&\vdots&\ddots&\vdots \\ -1&-1&-1&\cdots&n-1 \end{array}\right).$$ The algebraic complement of the element $b_{nn}$ is equal to $$\left|\begin{array}{cccc} n-1 & -1 & \cdots & -1 \\ -1 & n-1& \cdots & -1 \\ \vdots& \vdots&\ddots&\vdots \\ -1&-1&\cdots&n-1 \end{array}\right|= \left|\begin{array}{ccc} 1 & \cdots & 1 \\ -1 & \cdots & -1 \\ \vdots& \ddots&\vdots \\ -1&\cdots&n-1 \end{array}\right|= \left|\begin{array}{cccc} 1 & 1& \cdots & 1 \\ 0 & n& \cdots & 0 \\ \vdots& \vdots&\ddots&\vdots \\ 0 & 0&\cdots & n \end{array}\right|=n^{n-2},$$ where the first equality is obtained by change of the first row by the sum of all the rows, and the second equality is obtained by change of the $i$th row, $i\ge2$, by the sum of it with the first row. The complete graph with $n$ vertices contains $n^{n-2}$ spanning trees. Notice that this result is equivalent to Cayley Theorem saying that the total number of trees with $n$ enumerated vertices is equal to $n^{n-2}$. But it is a surprising fact, that there exist polynomial algorithms constructing minimal spanning trees. Several such algorithms were discovered in 1960s. We tell about Kruskal’s algorithm. Similar Prim’s algorithm can be found in [@Emel]. So, we are given with a connected weighted graph $\bigl(G=(V,E),\om\bigr)$ with positive weight function $\om$. At the initial step of Kruckal algorithm we construct the graph $T_0=(V,\emptyset)$ and put $E_0=E$. If the graph $T_{i-1}$ and the non-empty set $E_{i-1}\subset E$, $i\le n-1$, have been already constructed, then we choose in $E_{i-1}$ an edge $e_i$ of least possible weight and construct a new graph $T_i=T_{i-1}\cup e_i$ and also a new set $E_i=\{e\in E\mid \text{$e\not\in T_i$ and $T_i\cup e$ is acyclic}\}$. Algorithm stops when the graph $T_{n-1}$ is constructed[^1]. Under the above notations, the graph $T_{n-1}$ can be constructed for any connected weighted graph $\cG=(G,\om)$, and moreover, $T_{n-1}$ is a minimal spanning tree in $\cG$. The set $E_i$ is non-empty for all $0\le i\le n-2$, because the corresponding subgraphs $T_i$ are not connected (the graph $T_i$ has $n$ vertices and $i$ edges), therefore all the graphs $T_1,\ldots,T_{n-1}$ can be constructed. Further, all these graphs are acyclic due to the construction, and $T_{n-1}$ has $n$ vertices and $n-1$ edges, so it is a tree. To finish the proof it remains to show that the spanning tree $T_{n-1}\subset G$ is minimal. Since the graph $\cG$ has a finite number of spanning trees, a minimal spanning tree does exist. Let $T$ be a minimal spanning tree. We show that it can be reconstructed to the tree $T_{n-1}$ without changing the total weight, so $T_{n-1}$ is also a minimal spanning tree. To do this, recall that the edges of the tree $T_{n-1}$ are enumerated in accordance with the work of the algorithm. Denote them by $e_1,\ldots,e_{n-1}$ as above, and assume that $e_k$ is the first one that does not belong to $T$. The graph $T\cup e_k$ contains a unique cycle $c\supset e_k$. This cycle $c$ also contains an edge $e$ not belonging to $T_{n-1}$ (otherwise $c\subset T_{n-1}$, a contradiction). Consider the graph $T'=T\cup e_k \setminus e$. It is evidently a spanning tree in $G$, and therefore its weight is not less than the weight of the minimal spanning tree $T$, hence $$\om(T')=\om(T)+\om(e_k)-\om(e)\ge \om(T),$$ and thus, $\om(e_k)\ge\om(e)$. On the other hand, all the edges $e_1,\ldots, e_{k-1}$ belongs to $T$ by our assumption. Therefore, the graph $T_{k-1}\cup e$ is a subgraph of $T$ and is acyclic, in particular. Hence, $e\in E_{k-1}$ so as $e_k\in E_{k-1}$. But the algorithm has chosen $e_k$, hence $\om(e_k)\le\om(e)$. Thus, $\om(e_k)=\om(e)$, and so $\om(T')=\om(T)$, and therefore $T'$ is a minimal spanning tree in $\cG$. But now $T'$ contains the edges $e_1,\ldots,e_k$ from $T_{n-1}$. Repeating this procedure we reconstruct $T$ to $T_{n-1}$ in the class of minimal spanning trees. Theorem is proved. For a connected weighted graph with $n$ vertices and $m$ edges the complexity of the Kruskal’s algorithm can be naturally estimated as $mn\sim n^3$. The estimation can be improved to $m \log m\sim n^2\log n$. The fastest non-randomized comparison-based algorithm with known complexity belongs to Bernard Chazelle [@Chaz]. It turns out that if the weight function is geometrical, then the algorithms can be improved. Euclidean Case: Geometrical Approach ------------------------------------ Now assume that $M$ is a finite subset of the Euclidean plane $\R^2$. It turns out that a minimal spanning tree for $M$ in $\R^2$ can be constructed faster than the one for an abstract complete graph with $n=|M|$ vertices by means of some geometrical reasonings. To do that we need to construct so called Voronoi partition of the plane, corresponding to $M$, and the Delaunay graph on $M$. It turns out that any minimal spanning tree for $M$ in $\R^2$ is a subgraph of the Delaunay graph, see Figure \[fig:vordel\], and the number of edges in this graph is linear with respect to $n$, so the standard Kruskal’s algorithm applied to it gives the complexity $n\log n$ instead of $n^2 \log n$ for the complete graph with $n$ vertices. Let us pass to details. Let $M=\{A_1,\ldots, A_n\}\subset\R^2$ be a finite subset of the plain. The [*Voronoi cell*]{} of the point $A_i$ is defined as $$\Vor_M(A_i)=\left\{x\in\R^2\mid \text{$\|x-A_i\|\le\|x-A_j\|$ for all $j$}\right\}.$$ The Voronoi cell for $A_i$ is a convex polygonal domain which is equal to the intersection of the closed half-planes restricted by the perpendicular bisectors of the segments $A_iA_j$, $j\ne i$. It is easy to verify, that the intersection of any two Voronoi cells has no interior points and that $\cup_i\Vor_M(A_i)=\R^2$. This partition of the plane is referred as [*Voronoi partition*]{} or [*Voronoi diagram*]{}. Two cells $\Vor_M(A_i)$ and $\Vor_M(A_j)$ are said to be [*adjacent*]{}, if there intersection contains a straight segment. The [*Delaunay graph $D(M)$*]{} is defined as the dual planar graph to the Voronoi diagram. More precisely, the vertex set of $D(M)$ is $M$, and to vertices $A_i$ and $A_j$ are connected by an edge, if and only if their Voronoi cells $\Vor_M(A_i)$ and $\Vor_M(A_j)$ are adjacent. The edges of the Delaunay graph are the corresponding straight segments. It is easy to verify, that if the set $M$ is generic in the sense that no three points lie at a common straight line and no four points lie at a common circle, then the Delaunay graph $D(M)$ is a triangulation, i.e. its bounded faces are triangles. In general case some bounded faces could be inscribed polygons. Anyway, the number of edges of the graph $D(M)$ does not exceed $3n$. It remains to prove the following key Lemma. Any minimal spanning tree for $M\subset\R^2$ is a subgraph of the Delaunay graph $D(M)$. Let $e=A_iA_j$ be an edge of a minimal spanning tree $T$ for $M$. We have to show that the Voronoi cells $\Vor_M(A_i)$ and $\Vor_M(A_j)$ are adjacent. The graph $T\setminus e$ consists of two connected components, and this partition generates a partition of the set $M$ into two subsets, say $M_1$ and $M_2$. Assume that $A_i\in M_1$ and $A_j\in M_2$. The minimality of the spanning tree $T$ implies that $\|A_i,A_j\|$ is equal to the distance between the sets $M_1$ and $M_2$, where $\|A_i,A_j\|$ stands for the distance between $A_i$ and $A_k$. By $u$ we denote the middle point of the straight segment $A_iA_j$, and let $A_k$ be another point from $M$. Assume that $A_k\in M_2$. Due to the previous remark, $\|A_i,A_k\|\ge\|A_i,A_j\|$, therefore $$\|u,A_k\|\ge\|A_i,A_k\|-\|A_i,u\|\ge \|A_iA_j\|-\|A_i,u\|=\|A_i,A_j\|/2=\|u,A_i\|=\|u,A_j\|.$$ On the other hand, if $\|u,A_k\|=\|u,A_i\|$, then we have equalities in both above inequalities. The first one means that $u$ lies at the straight segment $A_iA_k$, hence $A_k$ lies at the ray $A_iu$. The second equality implies $\|A_iA_k\|=\|A_iA_j\|$, and so $A_k=A_j$, a contradiction. Thus, $\|u,A_k\|>\|u,A_i\|$, that is $u$ does not belong to the cell $\Vor_M(A_k)$ for $k\not\in\{i,\,j\}$. Thus, $u\in\Vor_M(A_i)\cap\Vor_M(A_j)$, Since the inequality proved is strict, the same arguments remain valid for points lying close to $u$ on the perpendicular bisector to the segment $A_iA_j$. Therefore, the intersection of the Voronoi cells $\Vor_M(A_i)$ and $\Vor_M(A_j)$ contains a straight segment, that is the cells are adjacent. Lemma is proved. The previous arguments work in any dimension. But the trouble is that starting from the dimension $3$ the number of edges in the Delaunay graph need not be linear on the number of its vertices. Verify that the same arguments can be applied to minimal spanning trees for a finite subset of $\R^n$. Give an example of a finite subset $M\subset\R^3$ such that the Delaunay graph $D(M)$ coincides with the complete graph $K(M)$. In what metric spaces similar geometrical approach also works? It definitely works for planar polygons with intrinsic metric, see [@ITInTrees]. Steiner Trees and Locally Minimal Networks {#sec:smt} ========================================== In this section we speak about shortest trees and locally shortest networks in more details. Besides necessary definitions we discuss local structure theorems, Melzak algorithm constructing locally minimal trees in the plane, global results concerning locally minimal binary trees in the plane (so called twisting number theory) and the particular case, locally minimal binary trees with convex boundaries (language of triangular tilings). The details concerning twisting number and tiling realization theory can be found in [@ITBookRFFI] or [@ITBookWP], and also in [@ITPlane]. Fermat Problem -------------- The idea that additional forks can help to decrease the length of a connecting network had been already clear to P. Fermat and his students. It seems that Fermat was the first, who stated the following optimization problem: for given three points $A_1$, $A_2$, and $A_3$ in the plane find a point $X$ minimizing the sum of distances from the points $A_i$, i.e. minimize the function $F(X)=\sum_i\|A_i,X\|$. For the case when all the angles of the triangle $A_1A_2A_3$ are less than or equal to $120^\c$ the solution was found by E. Torricelli and later by R. Simpson. The construction of Torricelli is as follows, see Figure \[fig:Torr\]. On the sides of the triangle $A_1A_2A_3$ construct equilateral triangles $A_iA_jA'_k$, $\{i,j,k\}=\{1,2,3\}$, such that they intersect the initial triangle only by the common sides. Then, as Torricelli proved, the circumscribing circles of these three triangles intersect in a point referred as [*Torricelli point $T$ of the triangle $A_1A_2A_3$*]{}. If all the angles $A_i$ are less than or equal to $120^\c$, then $T$ lies in the triangle $A_1A_2A_3$ and gives the unique solution to the Fermat problem.[^2] Later Simpson proved that the straight segments $A_iA'_i$ also pass through the Torricelli point, and the lengths of all these three segments are equal to $F(T)$. If one of the angles, say $A_3$, is more than $120^\c$, then the Torricelli point is located outside the triangle and can not be the solution to Fermat problem. In this case the solution is $X=A_3$. So we see, that shortest tree for a triangle in the plane consists of straight segments meeting at the vertices by angles more than or equal to $120^\c$. It turns out, that this [*$120^\c$-property*]{} remains valid in much more general situation. Local Structure Theorem and Locally Minimal Networks ---------------------------------------------------- Let $M=\{A_1,\ldots,A_n\}$ be a finite subset of Euclidean space $\R^N$, and $T$ is a Steiner tree connecting $M$. Recall that we defined shortest trees as abstract graphs with vertex set in the ambient metric space. In the case of $\R^N$ it is natural to model edges of such graph as straight segments joining corresponding points in the space. The configuration obtained is referred as a [*geometrical realization of the corresponding graph*]{}. Below, speaking about shortest trees in $\R^N$ we will usually mean their geometrical realizations. The local structure of a shortest tree (more exactly of a geometrical realization of the tree) can be easily described. \[th:LStr\] Let $\G$ be a shortest tree connecting a finite subset $M=\{A_1,\ldots,A_n\}$ in $\R^N$. Then 1. all edges of $\G$ are straight segments[*;*]{} 2. any vertex $v\in\G$ of degree $1$ belongs to $M$[*;*]{} 3. any two neighboring edges of $\G$ meet in common vertex by angle more than or equal to $120^\c$[*;*]{} 4. if the degree of a vertex $v$ is equal to $2$ and $v\not\in M$, then the edges meet at $v$ by $180^\c$ angle. Let $\G$ be a shortest tree connecting a finite subset $M=\{A_1,\ldots,A_n\}$ in $\R^N$. Then the degree of any its vertex is at most $3$, and if the degree of a vertex $v$ equals to $3$, then the edges meet at $v$ by angles equal to $120^\c$. Let $M$ be the vertex set of regular tetrahedron $\D$ in $\R^3$. Then the network consisting of four straight segments joining the vertices of the tetrahedra with its center $O$ is not a shortest network. Indeed, since $\deg O=4$, then the angles between the edges meeting at $O$ are less than $120^\c$. The set $M$ is connected by three different (but isometrical) shortest networks, each of which has two additional vertices of degree $3$, see Figure \[fig:tetrah\]. Theorem \[th:LStr\] can be just “word-by-word” extended to the case of Riemannian manifolds (we only need to change straight segments by geodesic segments) [@ITBookWP] and even to the case of Alexandrov spaces with bounded curvature. The case of normed spaces turned out to be more complicated (some general results can be found in [@ITBookRFFI]). A connected graph $\G$ in $\R^N$ (in a Riemannian manifold) whose vertex set contains a finite subset $M\subset\R^N$ is called a [*locally minimal network connecting $M$*]{} or [*with the boundary $\d\G=M$*]{}, if it satisfies Conditions (1)–(4) from Theorem \[th:LStr\]. In the case of complete Riemannian manifolds such graphs are minimal “in small,” i.e. the following result holds, see [@ITBookWP]. \[th:small\] Let $\G$ be a locally minimal network connecting a finite subset $M$ of a complete Riemannian manifold $W$. Then each point $P\in \G$ possesses a neighborhood $U$ in $W$, such that the network $\G\cap U$ is a shortest network with the boundary $(\d\G\cap U)\cup(\G\cap\d U)$. In the case of normed spaces Theorem \[th:small\] is not valid even for two-point sets, see example in Figure \[fig:manh\]. Melzak Algorithm and Steiner Problem Complexity ----------------------------------------------- Let us return back to the case of Euclidean plane. It turns out that in this case the Torricelli–Simpson construction can be generalized to a geometrical algorithm, that either constructs a locally minimal tree of a given structure for a given boundary set, or reports that such a tree does not exist. This algorithm was discovered by Z. Melzak [@Melz] and improved by F. Hwang [@Hw]. Assume that we are given with a tree $G$ whose vertex degrees are at most $3$, a finite subset $M$ of the plane, and a bijection $\v\:\d G\to M$, where $\d G$ is the set of all vertices from $G$ of degrees $1$ and $2$. To start with, partition the tree $G$ into the union of so-called [*non-degenerate*]{} components $G_i$ by cutting the tree at each its vertex of degree $2$, see Figure \[fig:compon\]. To construct locally minimal network $\G$ of type $G$ spanning $M$ in accordance with $\v$ it suffices to construct each its component $\G_i$ of type $G_i$ on the corresponding boundary $M_i=\v(\d G_i)$, where $\d G_i=\d G\cap G_i$, in accordance with $\v_i=\v|_{\d G_i}$ and to verify the angles between the edges of the components at the vertices of degree $2$. All these angles must be more than or equal to $120^\c$, see Figure \[fig:compon\]. Now we pass to the case of one non-degenerate component, i.e. we assume that $G$ has no vertices of degree $2$ and that $\d G$ consists of all the vertices of degree $1$. Such trees are referred as [*binary*]{}. If $|\d G|=2$, then the corresponding locally minimal tree $\G$ is a straight segment. Otherwise, it is easy to verify that each such tree $G$ contains so-called [*moustaches*]{}, i.e. a pair of vertices of degree $1$ neighboring with a common vertex of degree $3$. Fix such moustaches $\{x, x'\}\subset\d G$, by $y$ denote their common vertex of degree $3$, and make [*a forward step of Melzak algorithm*]{}, see Figure \[fig:Mel1\], that reduces the number of boundary vertices by $1$. Namely, we reconstruct the tree $G$ by deleting the vertices $x$ and $x'$ together with the edges $xy$ and $x'y$ and adding $y$ to the boundary of new binary tree; reconstruct the set $M$ by deleting the points $\v(x)$ and $\v(x')$ and adding a new point $A_{xx'}$ which is the third vertex of a regular triangle constructed on the straight segment $\v(x)\v(x')$ in the plane; and reconstruct the mapping $\v$ putting $\v(y)=A_{xx'}$. Notice that the point $A_{xx'}$ can be constructed in two ways, because there are two such regular triangles. Thus, if the number of boundary vertices in the resulting tree is more than $2$, then we can repeat the procedure described above. And if it becomes $2$, then we can construct the corresponding locally minimal tree — the straight segment. Here the forward trace of Melzak algorithm stops. Now we have to reconstruct the initial tree, if possible. Thus, we have a straight segment $I\subset\R^2$ realizing locally minimal tree with unique edge $ab$, and at least one of its ending points has the form $A_{xx'}$, where $x$ and $x'$ are the boundary vertices of the binary tree $G$ from the previous step, neighboring with their common vertex of degree $3$. Let this common vertex be $a$, that is $a$ corresponds to $A_{xx'}$. We reconstruct $G$ by adding edges $ax$ and $ax'$. Then we restore the points $\v(x)$ and $\v(x')$ in the plane together with the regular triangle $\v(x)\v(x')A_{xx'}$, circumscribe the circle $S^1$ around it and consider the intersection of $S^1$ with the segment $I$, see Figure \[fig:mel2\]. If it does not contains a point lying at the smaller arc of $S^1$ restricted by $\v(x)$ and $\v(x')$, then the tree $G$ can not be reconstructed and we have to pass to another realization of the forward trace of the algorithm. Otherwise we put $\v(a)$ be equal to this point. The straight segments $\v(x)\v(a)$ and $\v(x')\v(a)$ meet at $\v(a)$ by $120^\c$ and together with the subsegment $\v(a)\v(b)$ form a locally minimal binary tree $\G$ of type $G$ with tree boundary vertices. We repeat this procedure until we either reconstruct the tree of type $G$, or verify all possible realizations of the forward trace and conclude that the tree of type $G$ does not exists. The Melzak algorithm described above contains an exponential number of possibilities of its forward trace realization, due to two possible locations of each regular triangle constructed by the algorithm. This complexity can be reduced by means of modification suggested by F. Hwang [@Hw]. He showed that considering a bit more complicated configurations of boundary points (four points corresponding to “neighboring moustaches” or three points corresponding to moustaches and “neighboring” degree-$1$ vertex) one always can understand which regular triangle must be chosen, see details in [@Hw]. But unfortunately even a linear time realization of Melzak algorithm does not lead to a polynomial algorithm of a shortest tree finding. The reason is a huge number of possible structures of the tree $G$ with $|\d G|=n$ together with also exponential number of different mappings $\v\:\d G\to M$ for fixed $\d G$ and $M$. Even for binary trees we have $3$ possibilities for $n=4$, see Figure \[fig:manytop\], and $15$ possibilities for $n=5$ (notice that the corresponding binary trees are isomorphic as graphs). For $n=6$ we have two non-isomorphic binary trees and the number of possibilities becomes $90$. It can be shown that the total number of possibilities can be estimated by Catalan number and grough exponentially. So, to obtain an efficient algorithms, we have to find some [*a priori*]{} restrictions on possible structures of minimal networks. In the next subsection we tell about the restrictions generated by geometry of boundary sets. Boundaries Geometry and Networks Topology ----------------------------------------- Here we review our results from [@ITUMN90] and [@ITPlane]. The goal is to find some restriction on the structure of locally minimal binary trees spanning a given boundary in the plane in terms of geometry of the boundary set. To do this we need to choose or to introduce some characteristics of the network structure and of the boundary geometry. As a characteristic of the geometry of a boundary set $M$ we take the [*number of convexity levels $c(M)$*]{}. Recall the definition. Let $M$ be a finite non-empty subset of the plane. Take the convex hull $\ch M$ of $M$ and assign the points from $M$ lying at the boundary of the polygon $\ch M$ to the [*first convexity level $M^{(1)}$ of $M$*]{}. If the set $M\setminus M^{(1)}$ is not empty, then define the [*second convexity level $M^{(2)}$ of $M$*]{} to be equal to the first convexity level of $M\setminus M^{(1)}$, and so on. As a result, we obtain the partition of the set $M$ into its convexity levels, and by $c(M)$ we denote the total number of this levels, see Figure \[fig:convex\]. Now let us pass to definition of a characteristic describing the “complexity” of planar binary trees. Assume that we are given with a planar binary tree $\G$, and let the orientation of the plane be fixed. For any its two edges, say $e_1$ and $e_2$, we consider the unique path $\g$ in $\G$ starting at $e_1$ and finishing at $e_2$. All interior vertices of $\g$ are the vertices of $\G$ having degree $3$. Let us walk from $e_1$ to $e_2$ along $\g$. Then at each interior vertex of $\g$ we make either left, or right turn in $\G$. Define the value $\tw(e_1,e_2)$ to be equal to the difference between the numbers of left and right turns we have made. In other words, assign to an interior vertex of $\g$ the label $\tau=\pm1$, where $+1$ corresponds to left turns and $-1$ to right turns. Then $\tw(e_1,e_2)$ is the sum of these values, see Figure \[fig:tw\]. Notice that $\tw(e_1,e_2)=-\tw(e_2,e_1)$. At last, we put $\tw\G=\max\tw(e_i,e_j)$, where the maximum is taken over all ordered pairs of edges of $\G$. If the tree $\G$ is locally minimal, then the twisting number between any pair of its edges has a simple geometrical interpretation, see Figure \[fig:tw\]. Namely, since the angles between any neighboring edges are equal to $2\pi/3$, then $\tw(e_i,e_j)$ is equal to the total angle which the oriented edge rotates by passing from $e_i$ to $e_j$, divided by $\pi/3$. It turns out, that the twisting number of a locally minimal binary tree with a given boundary is restricted from above by a linear function on the number of convexity levels of the boundary. Namely, the following result holds. \[th:tw\_gen\] Let $\G$ be a locally minimal binary tree connecting the boundary set $M$ that coincides with the set of vertices of degree $1$ from $\G$. Then $$\tw\G\le 12\bigl(c(M)-1\bigr)+5.$$ The important particular case $c(M)=1$ corresponds to the vertex sets of convex polygons. Such boundaries are referred as [*convex*]{}. \[th:tw5\] Let $\G$ be a locally minimal binary tree with a convex boundary. Then $\tw\G\le 5$. Conversely, any planar binary tree $\G$ with $\tw\G\le5$ is planar equivalent to a locally minimal binary tree with a convex boundary. Notice that the direct statement of Theorem \[th:tw5\] is rather easy to prove (it follows from the geometrical interpretation of the twisting number, easy remark that $\tw\G$ always attains at boundary edges, and the monotony of convex polygonal lines). But the converse statement is quite non-trivial. The proof obtained in [@ITPlane] is based on the complete description of binary trees with twisting number at most five, obtained in terms of so-called triangular tilings that will be discussed in the next subsection. Estimate the number of binary trees structures with $n$ vertices of degree $1$ and twisting number at most $k$. It is more or less clear that the number is exponential on $n$ even for $k=5$, but it is interesting to obtain an exact asymptotic. Triangular Tilings and their Applications ----------------------------------------- It turns out that the description of planar binary trees with twisting number at most five can be effectively done in the language of planar triangulations of a special type which are referred as triangular tilings. The correspondence between diagonal triangulations of planar convex polygons and planar binary trees is well-known: the planar dual graph of such triangulation is a binary tree, see Figure \[fig:dualtr\], and each binary tree can be obtained in such a way. Here the vertices of the dual graph are centers of the triangles of the triangulation (medians intersection point) and middle points of the sides of the polygon; and edges are straight segments joining either the middle of a side with the center of the same triangle, or two centers of the triangles having a common side. In the context of locally minimal binary trees, the most effective way to represent the diagonal triangulations is to draw them consisting of regular triangles. Such special triangulations are referred as [*triangular tilings*]{}. The main advantage of the tilings is that the dual binary tree constructed as described above is a locally minimal binary tree with the corresponding boundary. Therefore, tilings “feel the geometry” of locally minimal binary trees and turns out to be very useful in the description of such trees with small twisting numbers. The main difficulty in constructing a triangulation consisting of regular triangles for a given binary tree is that the resulting polygon can overlap itself. An example of such overlapping can be easily constructed from a binary tree $\G$ corresponding to the diagonal triangulation of a convex $n$-gon, $n\ge6$, all whose diagonals are incident to a common vertex. But the twisting number of such $\G$ is also at least $6$. The following result is proved in [@ITPlane]. \[th:tiling\] The triangular tiling corresponding to any planar binary tree with twisting number less than or equal to five has no self-intersections. Theorem \[th:tiling\] gives an opportunity to reduce the description of the planar binary trees with twisting number at most five to the description of the corresponding triangular tilings. To describe all the triangular tilings whose dual binary trees have the twisting number at most five, we decompose each such tiling into elementary “breaks”. The triangles of the tiling are referred as [*cells*]{}. A cell of a tiling $T$ is said to be [*outer*]{}, if two its sides lie at the boundary of $T$ considered as planar polygon. Further, a cell is said to be [*inner*]{}, if no one of its sides lies at the boundary, see Figure \[fig:inout\]. An outer cell adjacent to (i.e. intersecting with by a common side) an inner cell is referred as a [*growth of $T$*]{}. A tiling can contains as un-paired growths, so as paired growths, see Figure \[fig:scelet\]. For each inner cell we delete exactly one growth adjacent to it, providing such growths exist. As a result, we obtain a decomposition of the initial tiling into its growths and its [*skeleton*]{} (a tiling without growths). Notice, that such a decomposition is not unique. It turns out that the skeletons of the tilings whose dual binary trees have twisting number at most five can be described easily. Also, the possible location of growthes in such tilings on their skeletons also can be described. The details can be found in [@ITPlane] or [@ITBookWP]. Here we only formulate the skeletons describing Theorem and include several examples of its application. Inner cells of a skeleton $S$ are organized into so-called [*branching points*]{}, see Figure \[fig:code\]. After the branching points deleting, the skeleton is partitioned into [*linear parts*]{}. Each linear part contains at most one outer cell. Construct a graph $C(S)$ referred as the [*code of the skeleton $S$*]{} as follows: the vertex set of $C(S)$ is the set of its branching points and of the outer cells of its linear parts. The edges correspond to the linear parts, see Figure \[fig:code\]. The following result is proved in [@ITPlane]. Consider all skeletons whose dual graphs twisting numbers are at most $5$ and for each of these skeletons construct its code. Then, up to planar equivalence, we obtain all planar graphs with at most $6$ vertices of degree $1$ and without vertices of degree $2$. In particular, every such skeleton contains at most $4$ branching points and at most $9$ linear parts. All possible codes of such skeletons are depicted in Figure \[fig:allcod\]. This description of skeletons and corresponding tilings obtained in [@ITPlane], was applied to the proof of inverse (non-trivial) statement of Theorem \[th:tw5\]. In some sense, the proof obtained in [@ITPlane] is constructive: for each tiling under consideration a corresponding locally minimal binary tree with a convex boundary is constructed. Another application is a description of all possible binary trees of the skeleton type that can be realized as locally minimal binary trees connecting the vertex set of a regular polygon. It turns out, see details in [@ITBookWP], that there are $2$ infinite families of such trees and $1$ finite family. The representatives of these networks together with the corresponding skeletons are shown in Figure \[fig:ngon\]. Steiner Ratio {#sec:sr} ============= As we have already discussed in the previous Section, the problem of finding a shortest tree connecting a given boundary set is exponential even in two-dimensional Euclidean plane. On the other hand, in practice it is necessary to solve transportation problems of this kind for several thousands boundary points many times a day. Therefore, in practice some heuristical algorithms are used. One of the most popular heuristics for a shortest tree is corresponding minimal spanning tree. But using such approximate solutions instead of exact one it is important to know the value of possible error appearing under the approximation. The [*Steiner ratio of a metric space*]{} is just the measure of maximal possible relative error for the approximation of a shortest tree by the corresponding minimal spanning tree. Steiner Ratio of a Metric Space ------------------------------- Let $M$ be a finite subset of a metric space $(X, \rho)$, and assume that $|M|\ge2$. We put $\sr M=\smt(M)/\mst(M)$. Evidently, $\sr M\le 1$. The next statement is also easy to prove. For any metric space $(X,\r)$ and any its finite subset $M\subset X$, $|M|\ge2$, the inequality $\sr M>1/2$ is valid. Let $G$ be a Steiner tree connecting $M$. Consider an arbitrary embedding of the graph $G$ into the plane, walk around $G$ in the plane and list consecutive paths forming this tour and joining consecutive boundary vertices from $M$. The length of each such path $\g_{PQ}$ joining boundary vertices $PQ$, i.e. the sum of the lengthes of its edges, is more than or equal to the distance $\r(P,Q)$, due to the triangle inequality. Consider the cyclic path in the complete graph with vertex set $M$ consisting of edges formed by the pairs of consecutive vertices from the tour, and let $T$ be a spanning tree on $M$ contained in this path. It is clear, that $\r(T)<\sum_{(P,Q)}\r(\g_{PQ})$, where the summation is taken over all the pairs of consecutive vertices of the tour. On the other hand, each edge of the tree $G$ belongs to exactly two such paths, hence $\sum_{(P,Q)}\r(\g_{PQ})=2\r(G)$. So, we have $\sr(M)\ge\r(G)/\r(T)>1/2$. The Assertion is proved. The value $\sr(M)$ is the relative error appearing under approximation of the length of a shortest tree for a given set $M$ by the length of a minimal spanning tree. The [*Steiner ratio of a metric space $(X,\rho)$*]{} is defined as the value $\sr(X)=\inf_{M\subset X}\sr(M)$, where the infimum is taken over all finite subsets $M$, $|M|\ge2$ of the metric space $X$. So, the Steiner ratio of $X$ is the value of the relative error in the worse possible case. For arbitrary metric space $(X,\rho)$ the inequality $1/2\le\sr(X)\le1$ is valid. Verify, that for any $r\in[1/2,1]$ there exists a metric space $(X,\r)$ with $\sr(X)=r$, see corresponding examples in [@ITBookRFFI]. Sometimes, it is convenient to consider so-called [*Steiner ratios $\sr_n(X)$ of degree $n$*]{}, where $n\ge2$ is an integer, which are defined as follows: $\sr_n(X)=\inf_{M\subset X,|M|\le n}\sr(M)$. Evidently, $\sr_2(X)=1$. It is also clear that $\sr(X)=\inf_n\sr_n(X)$. Steiner ratio was firstly defined for the Euclidean plane in [@GilPol], and during the following years the problem of Steiner ratio calculation is one of the most attractive, interesting and difficult problems in geometrical optimization. A short review can be found in [@ITBookRFFI] and in [@CiesBookSR]. One of the most famous stories here is connected with several attempts to prove so-called [*Gilbert–Pollack Conjecture*]{}, see [@GilPol], saying that $\sr(\R^2,\r_2)=\sqrt{3}/2$, where $\r_2$ stands for the Euclidean metric, and hence $\sr(\R^2,\r_2)$ is attained at the vertex set of a regular triangle, see Figure \[fig:tri\]. In 1990s D.Z. Du and F.K. Hwang announced that they proved the Steiner Ratio Gilbert–Pollak Conjecture [@DuHwang90], and their proof was published in Algorithmica [@DuHwang]. In spite of the appealing ideas of the paper, the questions concerning the proof appeared just after the publication, because the text did not appear formal. And about 2003–2005 it becomes clear that the gaps in the D.Z. Du and F.K. Hwang work are too deep and can not be repaired, see detail in [@ITAlg]. Steiner Ratio of Small Degrees for Euclidean Plane -------------------------------------------------- Gilbert and Pollack calculated $\sr_3(\R^2,\r_2)$ in their paper [@GilPol]. We include their proof here. Since the Steiner ratio of a regular triangle is equal to $\sqrt{3}/2$, then $\sr_3(\R^2,\r_2)\le\sqrt{3}/2$, so we just need to prove the opposite inequality. To do this, consider a triangle $ABC$ in the plane. If one of its angles is more than or equal to $120^\c$, then the shortest tree coincides with minimal spanning tree, so in this case $\sr(ABC)=1$. So it suffices to consider the case when all the angles of the triangle are less than $120^\c$. Let $S$ be the Torricelli point of the triangle $ABC$. Show firstly that $|AS|\le|BS|$, if and only if $|BC|\ge|AC|$, i.e. the shortest edge of the Steiner minimal tree lies opposite with the longest side of the triangle. The proof is shown in Figure \[fig:sr3\], left. Indeed, if $|BS|<|AS|$, then we take the point $B'\in[S,B]$ with $|SB'|=|SA|$, hence $|CB'|=|CA|$ due to symmetry and $|CB'|<|CB|$ because $B'\ge120^c$. Conversely, if $|BC|>|B'C|$, then there exists $B'\in[B,S]$ with $|CB'|=|CA|$, because $|BC|>|CA|>|SC|$. Then $|AS|=|SB'|<|SB|$. Thus, the two-edges tree $T=[A,B]\cup[B,C]$ is a minimal spanning tree for $ABC$, if and only if $BC$ is the longest side of $ABC$, if and only if $|AS|\le|BS|$ and $|AS|\le|CS|$. Consider the points $E\in[B,S]$ and $D\in[C,S]$, such that $|AS|=|ES|=|DS|$, and put $x=|AC|$, $y=|AB|$, $z=|DE|=|AD|=|AE|$, and $x'=|CD|$, $y'=|EB|$. Then $|SA|=|SE|=|SD|=z/\sqrt{3}$ and $$\smt(M)=3|SA|+|DC|+|EB|=\sqrt{3}z+x'+y' \qquad\text{and}\qquad \mst(M)=x+y,$$ where $M$ stands for the set $\{A,B,C\}$. But $x\le x'+z$ and $y\le y'+z$, due to the triangle inequality, and hence $$\sr(M)=\frac{\sqrt{3}z+x'+y'}{x+y}\ge\frac{\sqrt{3}z+x'+y'}{x'+z+y'+z}=\frac{\sqrt{3}z+x'+y'}{x'+y'+2z}\ge\frac{\sqrt{3}}{2}.$$ Thus, we proved the following statement. The following relation is valid: $\sr_3(\R^2,\r_2)=\sqrt{3}/2$. For small $n$ it is already proved that $\sr_n(\R^2,\r_2)=\sqrt{3}/2$ (recently O. de Wet proved it for $n\le7$, see [@deWet]). The proof of de Wet is based on the analysis of Du and Hwand method from [@DuHwang] and understanding that it works for boundary sets with $n\le 7$ points. Also in 60th several lower bounds for $\sr(\R^2,\r_2)$ were obtained, and the best of them is worse than $\sqrt{3}/2$ in the third digit only. Very attractive problem is to prove that $\sr(\R^2,\r_2)=\sqrt{3}/2$, i.e. to prove Gilbert–Pollack Conjecture. The attempts to repair the proof of Du and Hwang have remained unsuccessful, so some fresh ideas are necessary here. Steiner Ratio of Other Euclidean Spaces and Riemannian Manifolds ---------------------------------------------------------------- The following result is evident, but useful. If $Y$ is a subspace of a metric space $X$, i.e. the distance function on $Y$ is the restriction of the distance function of $X$, then $\sr(Y)\ge\sr(X)$. This implies, that $\sr(\R^n,\r_2)\le\sr(\R^2,\r_2)\le\sqrt{3}/2$. Recall that Gilbert–Pollack conjecture implies that the Steiner ratio of Euclidean plane attains at the vertex set of a regular triangle. In multidimensional case the situation is more complicated. The following result was obtained by Du and Smith [@DuSmith] If $M\subset\R^n$ is the vertex set of a regular $n$-dimensional simplex, then $\sr(M)>\sr(\R^n,\r_2)$ for $n\ge3$. Consider the boundary set $P$ in $\R^{n+1}$, consisting of the following $1+n(n+1)$ points: one point $(0,\ldots,0)$ and $n(n+1)$ points all whose coordinates except two are zero, one is equal to $1$, and the remaining one is $-1$. It is clear that $P$ is a subset of $n$-dimensional plane defined by the next linear condition: sum of all coordinates is equal to zero. Represent $P$ as the union of the subsets $P^i=\{x\in P\mid x^i=1\}\cup \{(0,\ldots,0)\}$. Notice that each set $P^i$, $i=1,\ldots,n+1$, consists of $n+1$ points and forms the vertex set of an regular $n$-dimensional simplex (to see that it suffices to verify that all the distances between the pairs of points from $P^i$ are the same and are equal to $\sqrt{2}$). The configuration of $7$ points in $\R^3$ is shown in Figure \[fig:DuSmith\] (this case is not important for us, but it is easy to draw). Now, $\mst(P)=(n+1)\mst(P^i)$, but for $n\ge 3$ we conclude that $\smt(P)<(n+1)\smt(P^i)$, because the degree of the vertex $(0,\ldots,0)$ in the corresponding network which is the union of the shortest networks for $P^i$ is equal to $n+1\ge4$ that is impossible in the shortest network due to the Local Structure Theorem \[th:LStr\]. So, $$\sr(P)=\smt(P)/\mst(P)<\frac{(n+1)\smt(P^i)}{(n+1)\mst(P^i)}=\sr(P^i).$$ Taking as a heuristic for the length of a shortest tree connecting the vertex set of regular simplex the length of the network joining the center of the simplex with all its vertices we get the following estimate. For any $n\ge 3$ the upper estimate $$\sr(\R^n,\r_2)<\sqrt{\frac{1}2+\frac{1}{2n}}$$ is valid. One of the best general low estimates is obtained by Graham and Hwang in [@GH]. For any $n\ge2$ the lower estimate $1/\sqrt3\le\sr(\R^n,\r_2)$ is valid. The best known upper estimate for $\R^3$ is obtained by Smith and Smith [@SmSm]. It is attained at an infinite boundary set which is known as “Smith sausage” and depicted in Figure \[fig:SmSm\]. The corresponding value, obtained as the limit of the ratios for finite fragments, is as follows: $$\sqrt{\frac{283}{700}-\frac{3\sqrt{21}}{700}+\frac{9\sqrt{22-2\sqrt{21}}}{140}}.$$ Notice that the idea of an infinite set is based on a deep result of Du and Smith estimating from below the number of points in a subset $M$ of $\R^n$ such that $\sr(M)=\sr(\R^n,\r_2)$ by a function $f(n)$ rapidly increasing on $n$, see details in [@DuSmith]. For example, $f(50)=53$, but $f(200)=3\,481\,911$. Therefore, it is difficult to expect to guess a finite set $M$ in $\R^n$ with $\sr(M)=\sr(\R^n,\r_2)$ for large $n$. Recently, the Steiner ratio of the Lobachevskii plane, and hence, of any Lobachevskii space has bin calculated by Innami and Kim, see [@InKim]. Steiner ratio of Lobachevskii space $L^n$ for any $n\ge2$ is equal to $1/2$. For general Riemannian manifold Ivanov, Cieslik and Tuzhilin, see [@ITC], obtained the following general result. The Steiner ratio of $n$-dimensional Riemannian manifold is less than or equal to the Steiner ratio of the Euclidean space $\R^n$. Minimal Fillings {#sec:mf} ================ This Section is devoted to minimal fillins, the third kind of optimal connections discussed in the Introduction. This problem appeared as a result of a synthesis of two classical problems: the Steiner problem on the shortest networks (it is discussed in Sections \[sec:smt\] and \[sec:sr\]), and Gromov’s problem on minimal fillings. The concept of a minimal filling appeared in papers of Gromov, see [@Gromov]. Let $M$ be a manifold endowed with a distance function $\rho$. Consider all possible films $W$ spanning $M$, i.e., compact manifolds with the boundary $M$. Consider on $W$ a distance function $d$ that does not decrease the distances between points in $M$. Such a metric space ${\mathcal W}=(W,d)$ is called a *filling* of the metric space ${\mathcal M}=(M,\rho)$, see example in Figure \[fig:MinFil\]. The Gromov Problem consists in calculating the infimum of the volumes of the fillings and describing the spaces ${\mathcal W}$ which this infimum is achieved at (such spaces are called *minimal fillings*). In the scope of Steiner problem, it is natural to consider $M$ as a finite metric space. Then the possible fillings are metric spaces having the structure of one-dimensional stratified manifolds which can be considered as graphs whose edges have nonnegative weights. This leads to the following particular case of generalized Gromov problem. Let $M$ be an arbitrary finite set, and $G=(V,E)$ be a connected graph. We say, that $G$ *connects $M$* or [*joins $M$*]{}, if $M\subset V$. Now, let ${\mathcal M}=(M,\rho)$ be a finite metric space, $G=(V,E)$ be a connected graph joining $M$, and $\omega\colon E\to{\mathbb R}_+$ is a mapping into non-negative numbers, which is usually referred as a *weight function* and which generates the *weighted graph* ${\mathcal G}=(G,\omega)$. The function $\omega$ generates on $V$ the pseudo-metric $d_\omega$ (some distances in a pseudo-metric can be equal to zero), namely, the $d_\omega$-distance between the vertices of the graph ${\mathcal G}$ is defined as the least possible weight of the paths in ${\mathcal G}$ joining these vertices. If for any two points $p$ and $q$ from $M$ the inequality $\rho(p,q)\le d_\omega(p,q)$ holds, then the weighted graph ${\mathcal G}$ is called a *filling* of the space ${\mathcal M}$, and the graph $G$ is referred as the *type* of this filing. The value $\operatorname{mf}({\mathcal M})=\inf\omega({\mathcal G})$, where the infimum is taken over all the fillings ${\mathcal G}$ of the space ${\mathcal M}$ is the *weight of minimal filling*, and each filling ${\mathcal G}$ such that $\omega({\mathcal G})=\operatorname{mf}({\mathcal M})$ is called a *minimal filling*. Parametric Networks and Optimal Connection Problems --------------------------------------------------- Here we give a common view on Steiner problem and minimal filling problem in terms of so-called parametric networks in a general metric space. Let ${\mathcal X}=(X,d)$ be a metric space and $G=(V,E)$ be an arbitrary connected graph. Any mapping $\Gamma\colon V\to X$ is called a *network in ${\mathcal X}$ parameterized by the graph $G=(V,E)$*, or a *network of the type $G$*. The *vertices* and *edges* of the network $\Gamma$ are the restrictions of the mapping $\Gamma$ onto the vertices and edges of the graph $G$, respectively. The *length of the edge* $\Gamma\colon vw\to X$ is the value $d\bigl(\Gamma(v),\Gamma(w)\bigr)$, and the *length $d(\Gamma)$ of the network $\Gamma$* is the sum of lengths of all its edges. We shall consider various boundary value problems for graphs. To do that, we fix some subsets $\partial G$ of the vertex sets $V$ of our graphs $G=(V,E)$, and we call such $\partial G$ the *boundaries*. We always suppose that in each graph under consideration a boundary, possibly, an empty one, is chosen. The *boundary $\partial\Gamma$ of a network $\Gamma$* is the restriction of $\Gamma$ onto $\partial G$. If $M\subset X$ is finite and $M\subset\Gamma(V)$, then we say that the network $\Gamma$ *joins* or *connects the set $M$*. The vertices of graphs and networks which are not boundary ones are called *interior* vertices. The value $$\operatorname{smt}(M)=\inf\bigl\{d(\Gamma)\mid\text{$\Gamma$ is a network joining $M$}\bigr\}$$ is called the *length of shortest network for $M$*. Notice that the network $\Gamma$ which joins $M$ and satisfies $d(\Gamma)=\operatorname{smt}(M)$ may not exist, see [@ITLup] and [@Borod] for nontrivial examples. If such a network exists, it is called a *shortest network connecting $M$*, or *for $M$*. One variant of the Steiner problem is to describe the shortest networks for finite subsets of metric spaces. [^3] Now let us define minimal parametric networks in a metric space ${\mathcal X}=(X,d)$. Let $G=(V,E)$ be a connected graph with some boundary $\partial G$, and let $\varphi\colon \partial G\to X$ be a mapping. By $[G,\varphi]$ we denote the set of all networks $\Gamma\colon V\to X$ of the type $G$ such that $\partial\Gamma=\varphi$. We put $$\operatorname{mpn}(G,\varphi)=\inf_{\Gamma\in[G,\varphi]}d(\Gamma)$$ and we call this value the *length of minimal parametric network*. If there exists a network $\Gamma\in[G,\varphi]$ such that $d(\Gamma)=\operatorname{mpn}(G,\varphi)$, then $\Gamma$ is called a *minimal parametric network of the type $G$ with the boundary $\varphi$*. Let ${\mathcal X}=(X,d)$ be an arbitrary metric space and $M$ be a finite subset of $X$. Then $$\operatorname{smt}(M)=\inf\bigl\{\operatorname{mpn}(G,\varphi)\mid\varphi(\partial G)=M\bigr\},$$ where the infimum is taken over all connected graphs $G$ with a boundary $\partial G$ and all mappings $\varphi\colon\partial G\to X$ with $\varphi(\partial G)=M$. Thus, as in the case of the plane, the problem of calculating the length of the shortest network is reduced to investigation of minimal parametric networks. Let ${\mathcal M}=(M,\rho)$ be a finite metric space and $G=(V,E)$ be an arbitrary connected graph connecting $M$. In this case we always assume that the boundary of such $G$ is fixed and equal to $M$. By $\Omega({\mathcal M},G)$ we denote the set of all weight functions $\omega\colon E\to{\mathbb R}$ such that $(G,\omega)$ is a filling of the space ${\mathcal M}$. We put $$\operatorname{mpf}({\mathcal M},G)=\inf_{\omega\in\Omega({\mathcal M},G)}\omega(G)$$ and we call this value the *weight of minimal parametric filling of the type $G$ for the space ${\mathcal M}$*. If there exists a weight function $\omega\in\Omega({\mathcal M},G)$ such that $\omega(G)=\operatorname{mpf}({\mathcal M},G)$, then $(G,\omega)$ is called a *minimal parametric filling of the type $G$ for the space ${\mathcal M}$*. Let ${\mathcal M}=(M,\rho)$ be a finite metric space. Then $$\operatorname{mf}({\mathcal M})=\inf\bigl\{\operatorname{mpf}({\mathcal M},G)\bigr\},$$ where the infimum is taken over all connected graphs $G$ joining $M$. It is not difficult to show that to investigate shortest networks and minimal fillings one can restrict the consideration to trees such that all their vertices of degree $1$ and $2$ belong to their boundaries. **In what follows, we always assume that this condition holds, providing the opposite is not declared**. To be more precise, we recall the following definition. We say that a tree is a *binary* one if the degrees of its vertices can be $1$ or $3$ only, and the boundary consists just of all the vertices of degree $1$. Then each finite metric space has a binary minimal filling (possibly, with some degenerate edges), and a non-degenerate minimal filling (whose type is a tree and all whose vertices of degree $1$ and $2$ belong to its boundary in accordance with the above agreement), see [@ITGromov]. Minimal Realization {#sec:realization} ------------------- It turns out that the problem on minimal filling can be reduced to Steiner problem in special metric spaces and for special boundaries. Consider a finite set $M=\{p_1,\ldots,p_n\}$, and let ${\mathcal M}=(M,\rho)$ be a metric space. We put $\rho_{ij}=\rho(p_i,p_j)$. By $\R_\infty^n$ we denote the $n$-dimensional arithmetic space with the norm $$\bigl\|(v^1,\ldots,v^n)\bigr\|_\infty=\max\bigl\{|v^1|,\dots,|v^n|\bigr\},$$ and by $\rho_\infty$ the metric on $\R_\infty^n$ generated by $\|\cdot\|_\infty$, i.e., $\rho_\infty(v,w)=\|w-v\|_\infty$. Let us define a mapping $\varphi_{\mathcal M}\colon M\to\R_\infty^n$ as follows: $$\varphi_{\mathcal M}(p_i)={\bar p}_i=(\rho_{i1},\ldots,\rho_{in}).$$ \[prop:isom\_embedding\_ellinfty\] The mapping $\varphi_{\mathcal M}$ is an isometry with its image. This easily follows from the triangle inequality. Indeed, $$\bigl\|\bar p_i-\bar p_j\bigr\|=\max_k|\rho_{ik}-\rho_{jk}|\ge\rho_{ij},$$ because the value $\rho_{ij}$ stands at the $i$th and $j$th places of the vector $\bar p_i-\bar p_j$. On the other hand, $\rho_{ij}\ge\rho_{ik}-\rho_{jk}$ for any $k$, due to the triangle inequality, hence $\bigl\|\bar p_i-\bar p_j\bigr\|\le \rho_{ij}$, and Assertion is proved. The mapping $\varphi_{\mathcal M}$ is called the *Kuratowski isometry*. Let ${\mathcal G}=(G,\omega)$ be a filling of a space ${\mathcal M}=(M,\rho)$, where $G=(V,E)$, and $d_\omega$ be the pseudo-metric on $V$ generated by the weight function $\omega$. Denote by $E_M$ the edges set of the complete graph on $M$ and put ${\bar G}=(V,{\bar E}=E\cup E_M)$. Let ${\bar\omega}$ be the weight function on ${\bar E}$ coinciding with metric $\rho$ on $E_M$ and with $\omega$ on ${\bar E}\setminus E_M$. Recall that $d_{\bar\omega}$ denotes the pseudo-metric on $V$ generated by ${\bar\omega}$. We define the network $\Gamma_{\mathcal G}\colon V\to\R_\infty^n$ of the type $G$ as follows: $$\Gamma_{\mathcal G}(v)=\bigl(d_{{\bar\omega}}(v,p_1),\ldots,d_{{\bar\omega}}(v,p_n)\bigr).$$ This network is called the *Kuratowski network for the filling ${\mathcal G}$*. \[prop:extend\] We have $\partial\Gamma_{\mathcal G}=\varphi_{\mathcal M}$. This easily follows from the filling definition. Indeed, the mapping $\partial\Gamma_{\mathcal G}$ is defined on the set $M$ only. By definition, $$\Gamma_{\mathcal G}(p_i)=\bigl(d_{{\bar\omega}}(p_i,p_1),\ldots,d_{{\bar\omega}}(p_i,p_n)\bigr),$$ hence it suffices to show that $d_{{\bar\omega}}(p_i,p_k)=\rho_{ik}$ for any $k$. The vertices $p_i$ and $p_k$ are joined by the edge $p_ip_k$ of the weight $\rho_{ik}$ in the graph $\bar G$, and the weight of any other path in $G$ connecting $p_i$ and $p_k$ is more than or equal to $\rho_{ik}$, because $G$ is a filling. Assertion is proved. For any network $\Gamma$ in a metric space $(X,d)$ by $\omega_\Gamma$ we denote the *weight function on $G$ induced by the network $\Gamma$*, i.e., $\omega_\Gamma(vw)=d\bigl(\Gamma(v),\Gamma(w)\bigr)$. \[cor:induced\_from\_ell\] Let ${\mathcal G}=(G,\omega)$ be a minimal parametric filling of a metric space $(M,\rho)$ and $\Gamma=\Gamma_{\mathcal G}$ be the corresponding Kuratowski network. Then $\omega=\omega_\Gamma$. Let $\Gamma$ be a network in a metric space ${\mathcal X}$, let $G$ be its parameterizing graph, and ${\mathcal H}=(H,\omega)$ be a weighted graph. We say that *$\Gamma$ and ${\mathcal H}$ are isometric*, if there exists an isomorphism of the weighted graphs ${\mathcal H}$ and ${\mathcal G}=(G,\omega_\Gamma)$. Corollary \[cor:induced\_from\_ell\] and the existence of minimal parametric and shortest networks in a finite-dimensional normed space [@ITBookWP] imply the following result. \[cor:Kurat\_mpf\] Let ${\mathcal M}=(M,\rho)$ be a metric space consisting of $n$ points, and $\varphi_{\mathcal M}\colon M\to\R_\infty^n$ be the Kuratowski isometry. For any graph $G$ joining $M$ there exists a minimal parametric filling of the type $G$ of the space ${\mathcal M}$. Each minimal parametric filling of the type $G$ of the space ${\mathcal M}$ is isometric to the corresponding Kuratowski network, which is, in this case, a minimal parametric network of the type $G$ with the boundary $\varphi_{\mathcal M}$. Conversely, each minimal parametric network of the type $G$ on $\varphi_{\mathcal M}(M)$ is isometric to some minimal parametric filling of the type $G$ of the space ${\mathcal M}$. \[cor:Kur\_mf\] Let ${\mathcal M}=(M,\rho)$ be a metric space consisting of $n$ points, and $\varphi_{\mathcal M}\colon M\to\R_\infty^n$ be the Kuratowski isometry. Then there exists a minimal filling ${\mathcal G}$ for ${\mathcal M}$, and the corresponding Kuratowski network $\Gamma_{\mathcal G}$ is a shortest network in the space $\R_\infty^n$ joining the set $\varphi_{\mathcal M}(M)$. Conversely, each shortest network on $\varphi_{\mathcal M}(M)$ is isometric to some minimal filling of the space ${\mathcal M}$. Minimal Parametric Fillings and Linear Programming {#sec:exist} -------------------------------------------------- Let ${\mathcal M}=(M,\rho)$ be a finite metric space connected by a (connected) graph $G=(V,E)$. As above, by $\Omega({\mathcal M},G)$ we denote the set consisting of all the weight functions $\omega\colon E\to{\mathbb R}_+$ such that ${\mathcal G}=(G,\omega)$ is a filling of ${\mathcal M}$, and by $\Omega_m({\mathcal M},G)$ we denote its subset consisting of the weight functions such that ${\mathcal G}$ is a minimal parametric filling of ${\mathcal M}$. \[prop:opt\_weight\] The set $\Omega({\mathcal M},G)$ is closed and convex in the linear space ${\mathbb R}^E$ of all the functions on $E$, and $\Omega_m({\mathcal M},G)\subset\Omega({\mathcal M},G)$ is a nonempty convex compact. It is easy to see, that the set $\Omega({\mathcal M},G)\subset{\mathbb R}^E$ is determined by the linear inequalities of two types: $\omega(e)\ge 0$, $e\in E$, and $\sum_{e\in\g_{pq}}\omega(e)\ge\rho(p,q)$, where $\g_{pq}$ stands for the unique path in the tree $G$ connecting the boundary vertices $p$ and $q$. Therefore, $\Omega({\mathcal M},G)$ is a convex closed polyhedral subset of $\R^E$ that is equal to the intersection of the corresponding closed half-spaces. The weight functions of minimal parametric fillings correspond to minima points of the linear function $\sum_{e\in E}\om(e)$ restricted to the set $\Omega({\mathcal M},G)$. Thus, the problem of minimal parametric filling finding is a linear programming problem, and the set $\Omega_m({\mathcal M},G)$ of all minima points is a nonempty convex compact polyhedron (the boundedness and, hence, compactness of this set follows from increasing of the objective function with respect to each its variable). Generalized Fillings -------------------- Investigating the fillings of metric spaces, it turns out to be convenient to expand the class of weighted trees under consideration permitting arbitrary weights of the edges (not only non-negative). The corresponding objects are called *generalized fillings*, *minimal generalized fillings* and *minimal parametric generalized fillings*. Their weights for a metric space ${\mathcal M}$ and a tree $G$ are denoted by $\operatorname{mf}_-({\mathcal M})$ and $\operatorname{mpf}_-({\mathcal M},G)$, respectively. For any finite metric space ${\mathcal M}=(M,\rho)$ and a tree $G$ connecting $M$, the next evident inequality is valid: $\operatorname{mpf}_-({\mathcal M},G)\le\operatorname{mpf}({\mathcal M},G)$. And it is not difficult to construct an example, when this inequality becomes strict, see Figure \[fig:mf-minus\]. However, for minimal generalized fillings the following result holds, see [@IOST]. \[th:IOST\] For an arbitrary finite metric space ${\mathcal M}$, the set of all its minimal generalized fillings contains its minimal filling, i.e. a generalized minimal filling with nonnegative weight function. Hence, $\operatorname{mf}_-({\mathcal M})=\operatorname{mf}({\mathcal M})$. Formula for the Weight of Minimal Filling ----------------------------------------- Let ${\mathcal M}=(M,\rho)$ be a finite metric space, and $G$ be a tree connecting $M$. Choose an arbitrary embedding $G'$ of the tree $G$ into the plane. Consider a walk around the tree $G'$. We draw the points of $M$ consecutive with respect to this walk as a consecutive points of the circle $S^1$. Notice that each vertex $p$ from $M$ appears $\deg p$ times. For each vertex $p\in M$ of degree more than $1$, we choose just one arbitrary point from the corresponding points of the circle. So, we construct an injection $\nu\colon M\to S^1$. Define a cyclic permutation $\pi$ as follows: $\pi(p)=q$, where $\nu(q)$ follows after $\nu(p)$ on the circle $S^1$. We say that $\pi$ *is generated by the embedding $G'$* (this procedure is not unique due to different possible choices of $\nu$). Each $\pi$ generated in this manner is called a *tour of $M$ with respect to $G$*. The set of all tours on $M$ with respect to $G$ is denoted by ${\mathcal O}(M,G)$. For each tour $\pi\in{\mathcal O}(M,G)$ we put $$p({\mathcal M},G,\pi)=\frac1{2}\sum_{x\in M}\rho\bigl(x,\pi(x)\bigr)$$ and we call this value by the *half-perimeter of the space ${\mathcal M}$ with respect to the tour $\pi$*. The minimal value of $p({\mathcal M},G,\pi)$ over all $\pi\in{\mathcal O}(M,G)$ for all possible $G$ (in fact, over all possible cyclic permutations $\pi$ on $M$) is called the *half-perimeter of the space ${\mathcal M}$*. A. Ivanov and A. Tuzhilin proposed the following hypothesis. \[conj:min-fill-formula\] For an arbitrary metric space ${\mathcal M}=(M,\rho)$ the following formula is valid $$\operatorname{mf}({\mathcal M})=\min_G\max_{\pi\in{\mathcal O}(M,G)}p({\mathcal M},G,\pi),$$ where minimum is taken over all binary trees $G$ connecting $M$. A.Yu. Eremin [@Eremin] constructed a counter-example to the Conjecture \[conj:min-fill-formula\] and showed that if one changes the concept of tour by the one of multitour, introduced by him, then the Conjecture \[conj:min-fill-formula\] holds. To define the multitours, let us consider the graph in which every edge of $G$ is taken with the multiplicity $2k$, $k\ge1$. The resulting graph possesses an Euler cycle consisting of *irreducible* boundary paths — the ones which do not contain properly other boundary paths. This Euler cycle generates a bijection $\pi\colon X\to X$, where $X=\sqcup_{i=1}^kM$, which is called *multitour of $M$ with respect to $G$*, see an example in Figure \[fig:multour\]. The set of all multitours on $M$ with respect to $G$ is denoted by ${\mathcal O}_\mu(M,G)$. Let ${\mathcal M}=(M,\rho)$ be a finite metric space, and $G$ be a tree connecting $M$. As in the case of tours, for each multitour $\pi\in{\mathcal O}_\mu(M,G)$ we put $$p({\mathcal M},G,\pi)=\frac1{2k}\sum_{x\in X}\rho\bigl(x,\pi(x)\bigr).$$ \[th:eremin\] For an arbitrary finite metric space ${\mathcal M}=(M,\rho)$ and an arbitrary tree $G$ joining $M$, the weight of minimal parametric generalized filling can be calculated as follows $$\operatorname{mpf}_-({\mathcal M},G)=\max\bigl\{p({\mathcal M},G,\pi)\mid \pi\in{\mathcal O}_\mu(M,G)\bigr\}.$$ The weight of minimal filling can be calculated as follows $$\operatorname{mf}({\mathcal M})=\operatorname{mf}_-({\mathcal M})=\min_G\max\bigl\{p({\mathcal M},G,\pi)\mid \pi\in{\mathcal O}_\mu(M,G)\bigr\},$$ where minimum is taken over all binary trees $G$ connecting $M$. Minimal Fillings for Generic Metric Spaces ------------------------------------------ Theorem \[th:eremin\] gives an opportunity to get several interesting corollaries. To formulate one of them, we need to define what is a “generic” metric space. Notice that the set of all metric spaces consisting of $n$ points can be naturally identified with a convex cone in ${\mathbb R}^{n(n-1)/2}$ (it suffices to enumerate the set of all two-elements subsets of these spaces and assign to each such space the vector of the distances between the pairs of points). This representation gives us an opportunity to speak about topological properties of families of metric spaces consisting of a fixed number of points. We say, that some property holds for a [*generic metric space*]{}, if for any $n$ this property is valid for an everywhere dense set of $n$-point metric spaces. The following result can be found in [@Eremin]. Each general finite metric space has a minimal filling which is a nondegenerate binary tree. Additive Spaces and Minimal Fillings {#sec:additive} ------------------------------------ The additive spaces are very popular in bioinformatics, playing an important role in evolution theory and, more general, in an hierarchy modeling. Recall that a finite metric space ${\mathcal M}=(M,\rho)$ is called *additive*, if $M$ can be joined by a weighted tree ${\mathcal G}=(G,\omega)$ such that $\rho$ coincides with the restriction of $d_\omega$ onto $M$. The tree ${\mathcal G}$ in this case is called a *generating tree* for the space ${\mathcal M}$. Not any metric space is additive. An additivity criterion can be stated in terms of so-called *$4$ points rule*: for any four points $p_i$, $p_j$, $p_k$, $p_l$, the values $\rho(p_i,p_j)+\rho(p_k,p_l)$, $\rho(p_i,p_k)+\rho(p_j,p_l)$, $\rho(p_i,p_l)+\rho(p_j,p_k)$ are the lengths of sides of an isosceles triangle whose base does not exceed its other sides. \[prop:add-unique\] A metric space is additive, if and only if it satisfies the $4$ points rule. In the class of non-degenerate weighted trees, the generating tree of an additive metric space is unique. The next criterion solves completely the minimal filling problem for additive metric spaces. \[th:additive=minimum\] Minimal fillings of an additive metric space are exactly its generating trees. The next additivity criterion is obtained by O.V. Rubleva, a student of Mechanical and Mathematical Faculty of Moscow State University, see [@Rubleva]. \[prop:Rubleva\] The weight of a minimal filling of a finite metric space is equal to the half-perimeter of this space, if and only if this space is additive. In the scope of Assertion \[prop:Rubleva\], we conjectured that if there exists a tree connecting a metric space such that all the corresponding half-perimeters are equal to each other, then the space is additive. It turns out that it is not true. Z.N. Ovsyannikov suggested to consider a wider class of spaces, so called pseudo-additive spaces, for which our conjecture becomes true, see [@Ovs]. A finite metric space ${\mathcal M}=(M,\rho)$ is said to be *pseudo-additive*, if the metric $\rho$ coincides with $d_\omega$ for a generalized weighted tree $(G,\omega)$ (which is also called *generating*), where the weight function $\omega$ can take arbitrary (not necessary nonnegative) values. Z.N. Ovsyannikov shows that these spaces can be described in terms of so-called *weak $4$-points rule*: for any four points $p_i$, $p_j$, $p_k$, $p_l$, the values $\rho(p_i,p_j)+\rho(p_k,p_l)$, $\rho(p_i,p_k)+\rho(p_j,p_l)$, $\rho(p_i,p_l)+\rho(p_j,p_k)$ are the lengths of sides of an isosceles triangle. The generating tree is also unique in the class of non-degenerate trees. Moreover, the following result is valid, see [@Ovs]. \[th:ovs\] Let ${\mathcal M}=(M,\rho)$ be a finite metric space. Then the following statements are equivalent. - There exist a tree $G$ such that $M$ coincides with the set of degree $1$ vertices of $G$ and all the half-perimeters $p(M,G,\pi)$ of $M$ corresponding to the tours around $G$ are equal to each other. - The space ${\mathcal M}$ is pseudo-additive. Moreover, the three $G$ in this case is a generating tree for the space ${\mathcal M}$. It would be interesting to see, what role these pseudo-additive spaces could play in applications. Examples of Minimal Fillings {#sec:examp} ---------------------------- Now let us give several examples of minimal filling and demonstrate how to use the technique elaborated above. ### Triangle {#subsec:triangle} Let ${\mathcal M}=(M,\rho)$ consist of three points $p_1$, $p_2$, and $p_3$. Put $\rho_{ij}=\rho(p_i,p_j)$. Consider the tree $G=(V,E)$ with $V=M\cup\{v\}$ and $E=\{vp_i\}_{i=1}^3$. Define the weight function $\omega$ on $E$ by the following formula: $$\omega(e_i)=\dfrac{\rho_{ij}+\rho_{ik}-\rho_{jk}}2,$$ where $\{i,j,k\}=\{1,2,3\}$. Notice that $d_\omega$ restricted onto $M$ coincides with $\rho$. Therefore, ${\mathcal M}$ is an additive space, ${\mathcal G}=(G,\omega)$ is a generating tree for ${\mathcal M}$, and, due to Theorem \[th:additive=minimum\], ${\mathcal G}$ is a minimal filling of ${\mathcal M}$. Recall that the value $(\rho_{ij}+\rho_{ik}-\rho_{jk})/2$ is called by the [*Gromov product*]{} $(p_j,p_k)_{p_i}$ of the points $p_j$ and $p_k$ of the space ${\mathcal M}$ with respect to the point $p_i$, see [@GromHypGr]. ### Regular Simplex {#subsec:simplex} Let all the distances in the metric space ${\mathcal M}$ are the same and are equal to $d$, i.e. ${\mathcal M}$ is a regular simplex. Then the weighted tree ${\mathcal G}=(G,\omega)$, $G=(V,E)$, with the vertex set $V=M\cup\{v\}$ and edges $vm$, $m\in M$, of the weight $d/2$ is generating for ${\mathcal M}$. Therefore, the space ${\mathcal M}$ is additive, and, due to Theorem \[th:additive=minimum\], ${\mathcal G}$ is its unique nondegenerate minimal filling. If $n$ is the number of points in $M$, then the weight of the minimal filling is equal to $dn/2$. ### Star If a minimal filling ${\mathcal G}=(G,\omega)$ of a space ${\mathcal M}=(M,\rho)$ is a star whose single interior vertex $v$ is joined with each point $p_i\in M$, $1\le i\le n$, $n\ge 3$, then the metric space ${\mathcal M}$ is additive [@ITGromov]. In this case the weights of edges can be calculated easily. Indeed, put $e_i=vp_i$. Since a subspace of an additive space is additive itself, then we can use the results for three-points additive space, see above. So, we have $\omega(e_i)=(p_j,p_k)_{p_i}$, where $p_i$, $p_j$, and $p_k$ are arbitrary distinct boundary vertices. ### Mustaches of Degree more than 2 Let $G=(V,E)$ be an arbitrary tree, and $v\in V$ be an interior vertex of degree $(k+1)\ge3$ adjacent with $k$ vertices $w_1,\ldots,w_k$ from $\partial G$. Then the set of the vertices $\{w_1,\ldots,w_k\}$, and also the set of the edges $\{vw_1,\ldots,vw_k\}$, are referred as [*mustaches*]{}. The number $k$ is called by the [*degree*]{}, and the vertex $v$ is called by the [*common vertex of the mustaches*]{}. An edge incident to $v$ and not belonging to $\{vw_1,\ldots,vw_k\}$ is called the [*root edge*]{} of the mustaches under consideration. As it is shown in [@ITGromov], any mustaches of a minimal filling of a metric space forms an additive subspace. If the degree of such mustaches is more than $2$, then we can calculate the weights of all the edges containing in the mustaches just in the same way as in the case of a star. ### Four-Points Spaces Here we give a complete description of minimal fillings for four-points spaces, see details in [@ITGromov]. Let $M=\{p_1,p_2,p_3,p_4\}$, and $\rho$ be an arbitrary metric on $M$. Put $\rho_{ij}=\rho(p_i,p_j)$. Then the weight of a minimal filling ${\mathcal G}=(G,\omega)$ of the space ${\mathcal M}=(M,\rho)$ is given by the following formula $$\frac12\bigl(\min\{\rho_{12}+\rho_{34},\,\rho_{13}+\rho_{24},\,\rho_{14}+\rho_{23}\}+ \max\{\rho_{12}+\rho_{34},\,\rho_{13}+\rho_{24},\,\rho_{14}+\rho_{23}\}\bigr).$$ If the minimum in this formula is equal to $\rho_{ij}+\rho_{rs}$, then the type of minimal filling is the binary tree with the mustaches $\{p_i,p_j\}$ and $\{p_r,p_s\}$. We apply the obtained result to the vertex set of a planar convex quadrangle. \[cor:conv4gon\] Let $M$ be the vertex set of a convex quadrangle $p_1p_2p_3p_4\subset{\mathbb R}^2$ and $\rho(p_i,p_j)=\|p_i-p_j\|$. The weight of a minimal filling of the space $(M,\rho)$ is equal to $$\frac12\min\bigl(\rho_{12}+\rho_{34},\,\rho_{14}+\rho_{23}\bigr)+ \frac{\rho_{13}+\rho_{24}}2.$$ The topology of minimal filling is a binary tree with mustaches corresponding to opposite sides of the less total length. Ratios {#sec:ratios} ------ The Steiner ratio is discussed in Section \[sec:sr\]. Here we define some other ratios based on minimal fillings, which could be more available for calculating, and which could be useful to calculate the Steiner ratio, as we hope. Steiner–Gromov Ratio -------------------- For convenience, the sets consisting of more than a single point are referred as *nontrivial*. Let ${\mathcal X}=(X,\rho)$ be an arbitrary metric space, and let $M\subset X$ be some finite subset. Recall that by $\operatorname{mst}(M,\rho)$ we denote the length of minimal spanning tree of the space $(M,\rho)$. Further, for nontrivial $M$, we define the value $$\operatorname{sgr}(M)=\operatorname{mf}(M,\rho)/\operatorname{mst}(M,\rho)$$ and call it the *Steiner–Gromov ratio* of the subset $M$. The value $\inf\operatorname{sgr}(M)$, where the infimum is taken over all nontrivial finite subsets of ${\mathcal X}$, consisting of at most $n$ vertices is denoted by $\operatorname{sgr}_n({\mathcal X})$ and is called the *degree $n$ Steiner–Gromov ratio of the space ${\mathcal X}$*. At last, the value $\inf\operatorname{sgr}_n({\mathcal X})$, where the infimum is taken over all positive integers $n>1$ is called the *Steiner–Gromov ratio of the space ${\mathcal X}$* and is denoted by $\operatorname{sgr}({\mathcal X})$, or by $\operatorname{sgr}(X)$, if it is clear what particular metric on $X$ is considered. Notice that $\operatorname{sgr}_n({\mathcal X})$ is a non-increasing function on $n$. Besides, it is easy to see that $\operatorname{sgr}_2({\mathcal X})=1$ and $\operatorname{sgr}_3({\mathcal X})=3/4$ for any nontrivial metric space ${\mathcal X}$. \[prop:steiner\_ratio\] The Steiner–Gromov ratio of an arbitrary metric space is not less than $1/2$. There exist metric spaces whose Steiner–Gromov ratio equals to $1/2$. Recently, A. Pakhomova, a student of Mechanical and Mathematical Faculty of Moscow State University, obtained an exact general estimate for the degree $n$ Steiner–Gromov ratio, see [@Pakh]. For any metric space ${\mathcal X}$ the estimate $$\operatorname{sgr}_n({\mathcal X})\ge\frac{n}{2n-2}$$ is valid. Moreover, this estimate is exact, i.e. for any $n\ge 3$ there exists a metric space ${\mathcal X}_n$ such that $\operatorname{sgr}_n({\mathcal X}_n)=n/(2n-2)$. Also recently, Z. Ovsyannikov [@Ovs-h] investigated the metric space of all compact subsets of Euclidean plane endowed with Hausdorff metric. The Steiner ratio and the Steiner–Gromov ratio of the metric space of all compact subsets of Euclidean plane endowed with Hausdorff metric are equal to $1/2$. Steiner Subratio ---------------- Let ${\mathcal X}=(X,\rho)$ be an arbitrary metric space, and let $M\subset{\mathcal X}$ be some its finite subset. Recall that by $\operatorname{smt}(M,\rho)$ we denote the length of Steiner minimal tree joining $M$. Further, for nontrivial subsets $M$, we define the value $$\operatorname{ssr}(M)=\operatorname{mf}(M,\rho)/\operatorname{smt}(M,\rho)$$ and call it by the *Steiner subratio* of the set $M$. The value $\inf\operatorname{ssr}(M)$, where infimum is taken over all nontrivial finite subsets of ${\mathcal X}$ consisting of at most $n>1$ points, is denoted by $\operatorname{ssr}_n({\mathcal X})$ and is called the *degree $n$ Steiner subratio of the space ${\mathcal X}$*. At last, the value $\inf\operatorname{ssr}_n({\mathcal X})$, where the infimum is taken over all positive integers $n>1$, is called the *Steiner subratio of the space ${\mathcal X}$* and is denoted by $\operatorname{ssr}({\mathcal X})$, or by $\operatorname{ssr}(X)$, if it is clear what particular metric on $X$ is considered. Notice that $\operatorname{ssr}_n({\mathcal X})$ is a nonincreasing function on $n$. Besides, it is easy to see that $\operatorname{ssr}_2({\mathcal X})=1$ for any nontrivial metric space ${\mathcal X}$. $\operatorname{ssr}_3({\mathbb R}^n)=\sqrt3/2$. The next result is obtained by E.I. Filonenko, a student of Mechanical and Mathematical Department of Moscow State University, see [@Filonenko]. $\operatorname{ssr}_4({\mathbb R}^2)=\sqrt3/2$. \[conj:subrat\] The Steiner subratio of the Euclidean plane is achieved at the regular triangle and, hence, is equal to $\sqrt3/2$. Recently, A. Pakhomova obtained an exact general estimate foe the degree $n$ Steiner subratio, see [@Pakh]. For any metric space $\{\mathcal X\}$ the estimate $$\operatorname{ssr}_n({\mathcal X})\ge\frac{n}{2n-2}$$ is valid. Moreover, this estimate is exact, i.e. for any $n\ge 3$ there exists a metric space ${\mathcal X}_n$ such that $\operatorname{ssr}_n({\mathcal X}_n)=n/(2n-2)$. Also recently, Z. Ovsyannikov [@Ovs-h] investigated the metric space of all compact subsets of Euclidean plane endowed with Hausdorff metric. Let ${\mathcal C}$ be the metric space of all compact subsets of Euclidean plane endowed with Hausdorff metric. Then $\operatorname{ssr}_3({\mathcal C})=3/4$ and $\operatorname{ssr}_4({\mathcal C})=2/3$. [4]{} Branching Solutions to One-Dimensional Variational Problems. Singapore, New Jersey, London, Hong Kong, 2001. 342 p. Extreme Networks Theory. Moscow, Izhevsk, 2003. 406 p. \[In Russian.\] One-dimensional Gromov minimal filling problem. // Sbornik: Mathematics. 2012. V. 203, No 5. P. 677Ц-726 \[Matem. sb. 2012. T. 203, No 5. S. 65Ц-118\]. Lections on Graph Theory. Moscow, 1990. 384 p. \[In Russian.\] The soft heap: an approximate priority queue with optimal error rate. // Journal of the Association for Computing Machinery. 2000. V. 47, No 6. P. 1012-Ц1027. The geometry of inner spanning trees for planar polygons. // Izvestiya: Mathematics. 2012. V. 76, No 2. P. 215–244 \[Izv. RAN. Ser. Matem. 2012. T. 76, No 2. S. 3Ц-36\]. The Steiner problem in the plane or in plane minimal nets.// Mathematics of the USSR-Sbornik. 1993. V. 74, No 2. P. 555–582 \[Matem. Sb. 1991. T. 182, No 12. S. 1813Ц-1844\]. A linear time algorithm for full Steiner trees. // Oper. Res. Letter. 1986. V. 5. P. 235–237. On the problem of Steiner. // Canad. Math. Bull. 1960. V. 4. P. 143–148. Solution of the Steiner problem for convex boundaries. // Russian Mathematical Surveys. 1990. V. 45, No 2. P. 214Ц-215 \[Uspekhi Matem. Nauk. 1990. T. 45, No 2(272). S. 207-Ц208\]. “Steiner minimaltrees.” // SIAM J. Appl. Math. 1968. V. 16, No 1. P. 1–29. The Steiner ratio. Springer, 2001. 256 p. . The Steiner ratio conjecture of Gilbert–Pollak is true. // Proc. Nat. Acad. Sci. 1990. V. 87. P. 9464–9466. . A proof of Gilbert–Pollak Conjecture on the Steiner ratio. // Algorithmica. 1992. V. 7. P. 121–135. The Steiner Ratio GilbertЦPollak Conjecture Is Still Open. Clarification Statement. // Algorithmica. 2012. V. 62. No 1–2. P. 630–632. Geometric Steiner Minimal Trees. PhD Thesis. UNISA, Pretoria, 2008. Disproofs of Generailzed Gilbert–Pollack conjecture on the Steiner ratio in three and more dimensions. // Combin. Theory. 1996. V. 74. Ser. A. P. 115–130. On the Steiner ratio in $3$-Space. // J. of Comb. Theory. 1995. V. 65. Ser. A. P. 301–322. A remark on Steiner minimal trees. // Bull. of the Inst. of Math. Ac. Sinica. 1976. V. 4. P. 177–182. Steiner ratio for Hyperbolic surfaces. // Proc. Japan Acad. 2006. V. 82. Ser. A. No 6. P. 77–79. Steiner Ratio for Manifolds. // Mat. Zametki. 2003. V. 74. No 3. P. 387-Ц395 \[Mathematical Notes. 2003. V. 74. No 3. P. 367-Ц374\]. Filling Riemannian manifolds. // J. Diff. Geom. 1983. V. 18. No 1. P. 1–147. Steiner Ratio. The State of the Art. // Matemat. Voprosy Kibern. 2002. V. 11. P. 27–48 \[In Russian\]. An example of nonexistence of a steiner point in a Banach space. // Matem. Zametki. 2010. V. 87. No 4. P. 514–518 \[Mathematical Notes. 2010. V. 87. No 4. P. 485Ц-488\]. One-dimensional minimal fillings with edges of negative weight. // Vestnik MGU, ser. Mat., Mekh. 2012. No 5. P. 3–8. Formula calculating the weight of minimal filling. // Matem. Sbornik. 2012. \[To appear\]. Pseudo-additive metric spaces and minimal fillings. // Vestnik MGU. 2013. \[To appear\]. Constructing a tree on the basis of a set of distances between the hanging vertices. // Uspehi Mat. Nauk. 1965. V. 20. No 6. P. 90-Ц92. \[In Russian\]. A note on the tree realizability of a distance matrix. // J. Combinatorial Th. 1969. V. 6. P. 303–310. About a Linear Denotation of Graphs. // J. vych. matem. i matem. phys. 1962. V. 2. No 2. P. 371–372. \[In Russian\]. Distane matrix of a graph and its realizability. // Quart. Appl. Math. 1975. V. 12. P. 305–317. Additivity Criterion for finite metric spaces and minimal fillings. // Vestnik MGU, ser. matem., mekh. 2012. No 2. P. 8–11. Hyperbolic groups. // in book: [*S.M. Gersten, ed.*]{} Essays in Group Theory. 1987. Springer. Degree $4$ Steiner subratio of Euclidean plane. // Vestnik MGU, ser. matem., mekh. 2013. To appear. Estimates for Steiner–Gromov ratio and Steiner subratio. // Vestnik MGU, ser. matem., mekh. 2013. To appear. Steiner ratio, Steiner–Gromov ratio and Steiner subratio for the metric space of all compact subsets of the Euclidean space with Hausdorff metric. // Vestnik MGU, ser. matem., mekh. 2013. To appear. [^1]: Here the operation of adding an edge $e$ to a graph $G=(V,E)$ can be formally defined as follows: $G\cup e=\left(V,E\cup\{e\}\right)$. Similarly, $G\setminus e=\left(V,E\setminus\{e\}\right)$. [^2]: An elementary proof can be obtained by rotation $R$ of a copy of the triangle around its vertex, say $A_1$, by $60^\c$ and considering the polygonal line $L$ joining $A_2$, $X$, image $R(X)$ of $X$ under the rotation, and $R(A_3)$. The length of $L$ is equal to $F(X)$, and minimal value of $F(X)$ corresponds to the location of the $X$ such that $L$ is a straight segment. [^3]: The denotation $\operatorname{smt}$ is an acronym for “Steiner Minimal Tree” which is a synonym for the shortest network whose edges are non-degenerate and, thus, it must be a tree.

--- abstract: 'We show that mostly right-handed Dirac sneutrinos are a viable supersymmetric light dark matter candidate. While the Dirac sneutrino scatters with nuclei dominantly through the $Z$-boson exchange and is stringently constrained by the invisible decay width of the $Z$ boson, it is possible to realize a large enough cross section with the nucleon to account for possible signals observed at direct dark matter searches, such as CDMS II(Si) or CoGeNT. Even if the XENON100 limit is taken into account, a small part of the signal region for CDMS II(Si) events remains outside the region excluded by XENON100.' author: - 'Ki-Young Choi' - Osamu Seto title: ' Light Dirac right-handed sneutrino dark matter' --- Introduction ============ Light weakly interacting massive particles (WIMPs) with masses around 10 GeV have received a lot of attention, motivated by the results of some direct dark matter (DM) detection experiments. DAMA/LIBRA has claimed detection of the annual modulation signal by WIMPs [@DAMALIBRA]. CoGeNT has found an irreducible excess [@CoGeNT] and annual modulation [@CoGeNTan]. CRESST has observed more events than expected backgrounds can account for [@CRESSTII; @Brown:2011dp]. The CDMS II Collaboration has just announced [@CDMSIISi] that their silicon detectors have detected three events and its possible signal region overlaps with the possible CoGeNT signal region analyzed by Kelso [*et al.*]{} [@Kelso:2011gd]. However, these observations are challenged by the null results obtained by other experimental collaborations, such as CDMS II [@CDMSII; @CDMSIIGe], XENON10 [@XENON10], XENON100 [@XENON100:2011; @XENON100:2012] and SIMPLE [@SIMPLE]. Recently, Frandsen [*et al.*]{} [@Frandsen:2013cna] have pointed out that the XENON10 exclusion limit in Ref. [@XENON10] might be overconstraining. It has been stressed that the signal region due to low-energy signals in CDMS II(Si) extends outside the XENON exclusion limit [@DelNobile:2013cta]. The Fermi-LAT collaboration has derived stringent constraints on the $s$-wave annihilation cross section of WIMPs by analyzing the gamma-ray flux from dwarf satellite galaxies [@dSph]. In particular, in the light-mass region below $ {\cal O} (10)$ GeV, the annihilation cross section times relative velocity $\langle\sigma v\rangle $ of $ {\cal O}(10^{-26}) {\rm cm}^3/{\rm s}$, which corresponds to the correct thermal relic abundance $\Omega h^2\simeq 0.1$, has been excluded. Light WIMPs have been investigated as a dark matter interpretation of this positive data. In fact, very light neutralinos in the minimal supersymmetric Standard Model (MSSM) [@Hooper:2002nq; @Bottino:2002ry] and the next-to-MSSM (NMSSM) [@Cerdeno:2004xw; @Gunion:2005rw] or very light right-handed (RH) sneutrinos in the NMSSM [@Cerdeno:2008ep; @Cerdeno:2011qv; @Choi:2012ba] have been regarded as such candidates. However, these candidates hardly avoid the above Fermi-LAT constraint. [^1] In this paper, we show that mostly right-handed Dirac sneutrinos are viable supersymmetric light DM candidates and have a large enough cross section with nucleons to account for possible signals observed at direct DM searches. Dirac sneutrinos scatter off nuclei dominantly via the $Z$-boson exchange process through the suppressed coupling and mostly with neutrons rather than protons. Although this $Z$-boson-mediated scattering does not relax the tension among direct DM search experiments and its availability is limited by the invisible decay width of the $Z$ boson, a part of the signal region for CDMS II(Si) events [@CDMSIISi] remains outside the excluded region by XENON100 [@XENON100:2012]. We examine the cosmic dark matter abundance as well as the constraints from indirect dark matter searches for a viable model of Dirac sneutrino dark matter. The paper is organized as follows. In Sec. \[sneutrinoDM\], we estimate the DM-nucleon scattering cross section through the $Z$-boson exchange process and show the experimental bounds and signal regions for this case. We impose the bound from the $Z$ boson invisible decay width too. In Sec. \[other\], after a brief description of the model, we examine other cosmological, astrophysical, and phenomenological constraints. We then summarize our results in Sec. \[conclusion\]. Dirac sneutrino dark matter direct detection {#sneutrinoDM} ============================================ Invisible $Z$-boson decay ------------------------- We are going to consider light Dirac sneutrino DM scattering with nuclei through the $Z$-boson exchange process in the direct detection experiments. Since the property of the $Z$ boson is well understood, the possibility of a light sneutrino has been stringently constrained from the invisible decay width of the $Z$ boson. First, we briefly summarize the bound. The $Z$-boson invisible decay is $(20.00\pm0.06) \%$ for the total decay width of the $Z$-boson decay $\Gamma_Z=2.4952\pm0.0023{\,{\rm GeV}}$ [@PDG]. This gives a constraint on the neutrino number which couples to the $Z$ boson, given by [@PDG] [$$\begin{split} N_\nu =2.984\pm0.008, \qquad(\rm PDG). \end{split}$$]{} The LEP bound on the extra invisible decay width is given as [@ALEPH:2005ab] $$\Delta \Gamma_{\rm inv}^Z< 2.0 {\,{\rm MeV}}\qquad (95\%\, \rm C.L.). \label{ZinvBound}$$ If there is a light sneutrino which couples to the $Z$ boson, the $Z$ boson can decay into light sneutrinos. The spin-averaged amplitude is [$$\begin{split} \overline{|M|^2} = \frac{|C_{\rm eff}|^2 g^2 M_Z^2}{12 \cos^2\theta_W}\left( 1-4\frac{M_{\tilde N}^2}{M_Z^2} \right) . \end{split}$$]{} Here, $C_{\rm eff}$ parametrizes the suppression in the sneutrino-sneutrino-$Z$ boson coupling as shown in Fig. \[fig:Zvertex\]. For pure left-handed sneutrinos, $C_{\rm eff}=1$. ------------------------------------------------------------------------------------------------------------------------------------ ![The effective vertex between a sneutrino and the $Z$ boson.[]{data-label="fig:Zvertex"}](Z_vertex.eps "fig:"){width="60.00000%"} ------------------------------------------------------------------------------------------------------------------------------------ The decay width of the $Z$ boson into light sneutrino DM is given by [$$\begin{split} \Gamma_{Z\rightarrow \tilde{N} \tilde{N}^*} &= \frac{|C_{\rm eff}|^2g^2 M_Z}{192 \pi \cos^2\theta_W}\left(1- 4\frac{M_{\tilde N}^2}{M_Z^2} \right)^{3/2}, \end{split}$$]{} and we impose the upper bound (\[ZinvBound\]) on this. This bound corresponds to [$$\begin{split} C_{\rm eff} \lesssim 0.15, \end{split}$$]{} for a few ${\,{\rm GeV}}$ dark matter particle. The contour plot of the invisible decay width is also shown in Fig. \[fig:ZinvAndDD\]. Direct detection ---------------- Dirac sneutrino DM can have elastic scattering with nuclei in the direct detection experiments. The most relevant process is due to the $Z$-boson exchange as in the left diagram in Fig. \[fig:DD\]. ---------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- ---------------------------------------------------------------------------------------------------------------------------------------------------------- ![The diagrams for the elastic scattering of right-handed sneutrino dark matter with quarks.[]{data-label="fig:DD"}](DD_Z.eps "fig:"){width="30.00000%"} ![The diagrams for the elastic scattering of right-handed sneutrino dark matter with quarks.[]{data-label="fig:DD"}](DD_H.eps "fig:"){width="30.00000%"} ---------------------------------------------------------------------------------------------------------------------------------------------------------- -- -- -- -- -- ---------------------------------------------------------------------------------------------------------------------------------------------------------- The $Z$-boson exchange cross section with nuclei ${}^A_ZN$ is given by $$\begin{aligned} \sigma^Z_{\chi N}&=&|C_{\rm eff}|^2 \frac{G_F^2}{2\pi} \frac{{{M_{\rm DM}}}^2 m_N^2}{({{M_{\rm DM}}}+m_N)^2} \left[ A_N +2(2\sin^2\theta_W-1)Z_N \right]^2 \label{DDZ} \\ &\simeq& (A_N-Z_N)^2{{\left(\frac{\mu_N^2}{\mu_n^2} \right) }} \sigma^Z_{\chi n}, \label{DD_Z}\end{aligned}$$ where $M_{\rm DM}$ and $m_N$ denote the dark matter mass and nucleus mass, respectively, $A_N$ and $Z_N$ are the mass number and proton number of the nucleus, and $G_F$ is the Fermi constant [@Arina:2007tm]. Here $\mu_X$ is the reduced mass defined by [$$\begin{split} \mu_X= \frac{{{M_{\rm DM}}}m_X }{({{M_{\rm DM}}}+m_X)}, \end{split}$$]{} and $m_n$ stands for the neutron mass. In the expression (\[DD\_Z\]), $\sigma^Z_{\chi n}$ denotes the DM scattering cross section with a neutron, and we have used the fact that the $Z$ boson dominantly couples with a neutron (as opposed to a proton) as $(1-4\sin^2\theta_W)\simeq 0.076$, and hence we have neglected the contribution from scattering with a proton. Usually the bound or signal of the direct detection experiments is given to the WIMP-nucleon scattering cross section, assuming the isospin-conserving case. This is true for a conventional WIMP such as a neutralino, where Higgs boson-exchange processes are dominant. For the $Z$-boson-mediated case, the DM interacts dominantly with a neutron, and thus the bound should be modified according to this. Using [Eq. (\[DD\_Z\])]{}, the corresponding WIMP-neutron cross section, $\sigma_n^{(Z)} $, for the $Z$-boson-mediated case is related to the isospin-conserving (IC) WIMP-nucleon scattering cross section, $\sigma_n^{\rm (IC)}$, by [$$\begin{split} \sigma_n^{(Z)} =\sigma_n^{\rm (IC)}{{\left(\frac{A}{A-Z} \right) }}^2. \end{split}$$]{} For Xenon $A\simeq130, Z=54$, and for Si in CDMS II $A=28, Z=14$. These factors give enhancement on the cross section by factors 4 and 3, respectively. In Fig. \[fig:ZinvAndDD\], we show the contour of the $Z$-boson extra invisible decay width and the WIMP-neutron scattering cross section in the plane of $C_{\rm eff}$ and the dark matter mass $M_{\rm DM}$. The contours of the predicted scattering cross section with a neutron (blue) are given in units of $10^{-40} {\rm cm}^2$ with those of the extra $Z$-boson invisible decay width (red). The red region is disallowed by the LEP bound on the $Z$-boson extra invisible decay given in [Eq. (\[ZinvBound\])]{}. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ![ The contours of the predicted scattering cross section with a neutron (blue) in $10^{-40} {\rm cm}^2$ and those of the extra $Z$-boson invisible decay width (red) as a function of sneutrino mass and $C_{\rm eff}$. The red region is disallowed by the LEP bound, $ \Delta \Gamma_{\rm inv}^Z< 2.0 {\,{\rm MeV}}$ [@ALEPH:2005ab].[]{data-label="fig:ZinvAndDD"}](ZinvAndDD.eps "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ In Fig. \[fig:Xsection\], we show the WIMP-neutron scattering cross section versus dark matter mass. We show the constraint from XENON100 [@XENON100:2012], and the signals measured by CDMSII-Si [@CDMSIISi] and CoGeNT [@Kelso:2011gd] with the contour of the $Z$-boson extra invisible decay width. Following Ref. [@Frandsen:2013cna], we do not include the XENON10 limit in this paper to keep our discussion conservative. We find that a still barely compatible region exists for a dark matter mass around $6{\,{\rm GeV}}$ and the WIMP-nucleon cross section $\sigma_n^{(Z)} \simeq 10^{-40 } {\rm cm}^2$. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![ The signal region and excluded region from direct dark matter searches \[XENON100 (almost vertical line with black solid color), CDMS II(Si) (big closed loop of crosses with purple color), and CoGeNT (small closed loop of crosses with turquoise color)\], and the magnitude of the corresponding Z-boson invisible decay width denoted by $\Delta \gamma = 1,2,4 $ MeV (red color).[]{data-label="fig:Xsection"}](Xsection.eps "fig:"){width="50.00000%"} ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Other constraints {#other} ================= The discussion and conclusion in the previous section are model independent and were made applicable for any scalar DM scattering with a nucleon dominantly through $Z$-boson exchange by introducing the coefficient $C_{\rm eff}$. In this section, we discuss other DM phenomenologies and experimental constraints. To do this, we need to specify the particle model for Dirac sneutrino dark matter. One model has been constructed with nonconventional supersymmetry (SUSY)-breaking mediation [@ArkaniHamed:2000bq]. Light sneutrino DM has been studied in Refs. [@Belanger:2010cd; @Dumont:2012ee] and has unfortunately turned out to be hardly compatible with LHC data, mainly due to the SM-like Higgs boson invisible decay width [@Dumont:2012ee]. There is another available model proposed by us [@Choi:2012ap] in the context of the neutrinophilic Higgs doublet model [@Ma; @Wang; @Ma:2006km; @Nandi]. Therefore in the rest of this section, as an example, we discuss other DM phenomenologies based on this model. Brief description of the model in Ref. [@Choi:2012ap] ----------------------------------------------------- The neutrinophilic Higgs model is based on the concept that the smallness of the neutrino mass might not come from a small Yukawa coupling but rather from a small vacuum expectation value (VEV) of the neutrinophilic Higgs field $H_{\nu}$. As a result, neutrino Yukawa couplings can be as large as of the order of unity for a small enough VEV of $H_{\nu}$. Other aspects-for instance, collider penomenology [@Davidson:2009ha; @Logan:2010ag; @Haba:2011nb], astrophysical and cosmological consequences [@Choi:2012ap; @HabaSeto; @Sher:2011mx; @HSY], vacuum structure [@Haba:2011fn], and variant models [@HabaHirotsu; @Haba:2012ai; @Morita:2012kh], -have also been studied. The supersymmetric neutrinophilic Higgs model has a pair of neutrinophilic Higgs doublets $H_{\nu}$ and $H_{\nu'}$ in addition to up- and down-type two-Higgs doublets $H_u$ and $H_d$ in the MSSM [@HabaSeto]. A discrete $Z_2$ parity is also introduced to discriminate $H_u (H_d)$ from $H_{\nu}(H_{\nu'})$, and the corresponding charges are assigned in Table \[Table\]. Fields $Z_{2}$ parity Lepton number ----------------------------------------- ---------------- --------------- MSSM Higgs doublets, $H_u, H_d$ $+$ 0 New Higgs doublets, $H_{\nu}, H_{\nu'}$ $-$ 0 Right-handed neutrinos, $N$ $-$ $1$ : The assignment of $Z_2$ parity and lepton number.[]{data-label="Table"} Under this discrete symmetry, the superpotential is given by $$\begin{aligned} W &=&y_{u} Q \cdot H_u U_{R} +y_d Q \cdot {H_d}D_{R}+ y_l L \cdot H_d E_{R} +y_{\nu} L \cdot H_{\nu} N \nonumber \\ && +\mu H_u \cdot H_d + \mu' H_\nu \cdot H_{\nu'} +\rho H_u \cdot H_{\nu'} + \rho' H_\nu \cdot H_d , \label{superpotential}\end{aligned}$$ where we omit generation indices and dots represent the SU(2) antisymmetric product. The $Z_2$ parity plays a crucial role in suppressing tree-level flavor-changing neutral currents and is assumed to be softly broken by tiny parameters of $\rho$ and $\rho' (\ll \mu, \mu' )$. Here, we do not introduce lepton-number-violating Majorana mass for the RH neutrino $N$ to realize a Dirac (s)neutrino. By solving the stationary conditions for the Higgs fields, one finds that tiny soft $Z_2$-breaking parameters $\rho, \rho' $ generate a large hierarchy of $v_{u,d} (\equiv \langle H_{u,d}\rangle) \gg v_{\nu, \nu'}(\equiv \langle H_{\nu, \nu'}\rangle)$ expressed as $$v_{\nu} = {\cal O}\left(\frac{\rho}{\mu'}\right) v.$$ It is easy to see that neutrino Yukawa couplings $y_{\nu}$ can be large for small $v_{\nu}$ using the relation of the Dirac neutrino mass $m_{\nu} = y_{\nu} v_{\nu}$. For $v_{\nu} \sim 0.1 $ eV, it gives $y_{\nu} \sim 1$. At the vacuum of $v_{\nu, \nu'}\ll v_{u,d}$, physical Higgs bosons originating from $H_{u, d}$ are almost decoupled from those from $H_{\nu,\nu'},$ except for a tiny mixing of the order of ${\cal O}\left(\rho/M_{\rm SUSY}, \rho'/M_{\rm SUSY} \right)$, where $M_{\rm SUSY} ( \sim 1 $ TeV) denotes the scale of soft SUSY-breaking parameters. The former $H_{u,d}$ doublets almost constitute Higgs bosons in the MSSM – two $CP$-even Higgs bosons $h$ and $H$, one $CP$-odd Higgs boson $A$, and a charged Higgs boson $H^\pm$ – while the latter, $H_{\nu, \nu'}$, constitutes two $CP$-even Higgs bosons $H_{2,3}$, two $CP$-odd bosons $A_{2,3}$, and two charged Higgs bosons $H^\pm_{2,3}$. Thus, our model does not suffer from a large invisible decay width of SM-like an Higgs boson $h$ even for a large $y_{\nu}$ and a light lightest-supersymmetric-particle (LSP) dark matter. At the vacuum, the mixing between left- and right-handed sneutrinos is estimated as $$\sin \theta_{\tilde{\nu}} = {\cal O}\left( \frac{m_\nu}{M_{\rm SUSY}} \right) . \label{LRmixing}$$ We find that the RH sneutrino $\tilde{N}$ has very suppressed interactions with the SM-like Higgs boson or $Z$ boson at tree level, since they are proportional to the mixing of left-handed and RH neutrinos’ $\sin \theta_{\tilde{\nu}} $ in [Eq. (\[LRmixing\])]{}. However, radiative corrections induce a sizable coupling between RH sneutrinos and the $Z$ boson. We have parametrized the effective interaction between the RH sneutrino DM and $Z$ boson by $C_{\rm eff}$; then, the vertex induced by the scalar ($H_{\nu}$-like Higgs boson and $\tilde{\nu}_L$) loop [^2] is given as $$\begin{aligned} {\rm Vertex} &=& \frac{g}{2\cos\theta_W}(k_1^\mu+ k_2^\mu) C_{\rm eff},\end{aligned}$$ with $$\begin{aligned} C_{\rm eff} &=& \frac{(-i)(y_\nu A_\nu)^2}{12(4\pi)^2 M^2}, \label{Ceff}\end{aligned}$$ where $k_1^\mu$ and $k_2^\mu$ are the ingoing and outgoing momenta of the RH sneutrino and for simplicity we take equal masses for particles in the loop, $M = M_{H_\nu}= M_{\tilde{\nu}_L}$. By comparing Fig. \[fig:Xsection\] and [Eq. (\[DD\_Z\])]{} with Eq. (\[Ceff\]), we find the parameter set [$$\begin{split} y_\nu \,A_\nu \simeq 14.4 \, M \qquad {\rm and} \qquad M_{\rm DM} \simeq 6 \,{\rm GeV}, \label{set} \end{split}$$]{} can explain the CDMS II Si result. Annihilation cross section -------------------------- The dominant tree-level annihilation mode of $\tilde{N}$ in the early Universe is the annihilation into a lepton pair $\tilde{N} \tilde{N}^*\rightarrow \bar{f_1} f_2 $ mediated by the heavy $H_{\nu}$-like Higgsinos as described in Fig. \[fig:Tree\]. The final states $f_1$ and $f_2$ are charged leptons for the $t$-channel $\tilde{H}_{\nu}$-like charged Higgsino ($\tilde{H}_{\nu}^\pm$) exchange, while thy are neutrinos for the $t$-channel $\tilde{H}_{\nu}$-like neutral Higgsino ($\tilde{H}_{\nu}^0$) exchange. --------------------------------------------------------------------------------------------------------------------------- ![Tree-level diagram for the annihilation of RH sneutrinos.[]{data-label="fig:Tree"}](Tree.eps "fig:"){width="50.00000%"} --------------------------------------------------------------------------------------------------------------------------- The thermal averaged annihilation cross section for this mode in the early Universe when using the partial wave expansion method is given by [@Lindner:2010rr] $$\langle \sigma v\rangle_{f\bar{f}} = \sum_f \left( \frac{y_\nu^4}{16\pi} \frac{m_f^2}{(M^2_{\tilde N} + M^2_{\tilde{H}_\nu} )^2} + \frac{ y_{\nu}^4}{8 \pi } \frac{M^2_{\tilde{N}}}{ ( M_{\tilde{N}}^2 + M_{\tilde{H_{\nu}}}^2 )^2 }\frac{T}{ M_{\tilde{N}} } + ... \right) , \label{treelevel}$$ where we used ${\langle v_{\rm rel}^2 \rangle}=6T/M_{\rm DM}$ with $v_{\rm rel}$ being the relative velocity of annihilating dark matter particles, $m_f$ is the mass of the fermion $f$, and $M_{{\tilde H}_\nu} \simeq \mu'$ denotes the mass of the $\tilde{H}_{\nu}$-like Higgsino. For simplicity we have assumed that Yukawa couplings are universal for each flavor. Since the $s$-wave contribution of the first term on the right-hand side is helicity suppressed, the $p$-wave annihilation cross section of the second term is relevant for the dark matter relic density at the freeze-out epoch. In the neutrinophilic Higgs model, the sneutrino has – in addition to the tree-level processes – a sizable annihilation cross section into two photons through a one-loop diagram, which has been pointed out in Ref. [@Choi:2012ap]. The charged components of the $H_\nu$ scalar doublet and charged scalar fermions make the triangle or box loop diagram, and the two photons can be emitted from the internal charged particles. For the mass spectrum we are interested in now, $M_{H_{\nu}}, M_{\tilde l}\gg M_{\tilde N}$, we obtain the annihilation cross section to two photons via one loop as $$\begin{aligned} {\langle \sigma v \rangle}_{2\gamma} &\simeq& \frac{ \alpha^2_{\rm em}}{8\pi^3}\frac{y_\nu^4 (A_\nu^2+\mu'^2)^2}{M_{\rm ch}^4} \frac{4}{M^2_{\tilde N}} \nonumber \\ &=& 2.8\times10^{-8} \, {\,{\rm GeV}}^{-2} {{\left(\frac{6{\,{\rm GeV}}}{M_{\tilde N}} \right) }}^2 \frac{y_\nu^4 (A_\nu^2+\mu'^2)^2}{M_{\rm ch}^4} ,\end{aligned}$$ where we have used $M_{H_\nu}=M_{H_\nu'}=M_{\tilde l}\equiv M_{\rm ch}$ for simplicity. Therefore for the total annihilation cross section of RH sneutrino DM, we obtain $${\langle \sigma v \rangle} = {\langle \sigma v \rangle}_{f\bar{f}} + {\langle \sigma v \rangle}_{2\gamma} .$$ Now if we attempt to reproduce the latest CDMS II-Si data by taking a parameter set given by Eq. (\[set\]), we find that two-photon production via one loop is dominant and thus [$$\begin{split} {\langle \sigma v \rangle} \simeq {\langle \sigma v \rangle}_{2\gamma} \simeq 10^{-3} \, {\,{\rm GeV}}^{-2}, \label{AnnihXsection} \end{split}$$]{} for the given parameters in [Eq. (\[set\])]{}. This loop-induced annihilation does not only dominate the tree-level annihilation but also exceeds the standard value ${\langle \sigma v \rangle} \simeq 10^{-9} \, {\,{\rm GeV}}^{-2}$. This DM appears to not have the correct thermal relic abundance if the relic density is determined from its thermal freeze-out. Dark matter relic abundance and indirect DM search constraints -------------------------------------------------------------- As stated above, from Eq. (\[AnnihXsection\]) we see that the standard thermal relic density of $\tilde{N}$ with zero chemical potential leads to a too small value for $\Omega h^2 \ll 0.1$. However, we know that our Universe is baryon asymmetric. Hence, we expect that lepton asymmetry is also nonvanishing. In fact, the sphaleron process, which interchanges baryons and leptons, plays an important role in many baryogenesis mechanism and leaves a similar amount of baryon asymmetry and lepton asymmetry. Because our model is supersymmetric, a promising mechanism would be Affelck-Dine (AD) baryo(lepto)genesis [@AffleckDine]. Candidates for a promissing AD field $\phi$ are, e.g., $\bar{u}\bar{d}\bar{d}$ or $LL\bar{e}$ directions with the nonrenormalizable superpotential $\Delta W = \phi^6/M^3$, where $M$ is a high cutoff scale for this operator. The generated baryon $(q=B)$ or lepton $(q=L)$ asymmetry for those directions have been studied by many authors and evaluated as [@ADpapers; @EMD; @FH; @Seto; @RS; @SY] $$\frac{n_q}{s} \simeq 10^{-10} q \sin\delta\left(\frac{A_{\phi}}{1 \rm{TeV}}\right) \left(\frac{1 \rm{TeV}}{m_{\phi}}\right)^{3/2}\left(\frac{T_R}{10 \rm{TeV}}\right) \left(\frac{M}{10^{-2} M_P}\right)^{3/2},$$ for a relatively low reheating temperature after inflation $T_R$ in gravity-mediated SUSY-breaking models, where $m_\phi$ and $A_{\phi}$ are soft SUSY-breaking mass and A term for the AD field, $\delta$ is an effective $CP$ phase, $M_P$ is the reduced Planck mass, and $M$ is taken to be around the grand unification scale. Then, the charge of $Q$-balls, even if they are formed, is small enough for a $Q$-ball to evapolate quickly [@Banerjee:2000mb] and to not affect the dark matter density. [^3] To be precise, this generated $B-L$ asymmetry is related to the baryon and lepton asymmetry generated by the sphaleron process [@Kuzmin:1985mm], $$\begin{aligned} \frac{n_B}{s} \sim \frac{n_L}{s} = {\mathcal O}(10^{-10}) .\end{aligned}$$ Since a Dirac sneutrino carries a lepton number and has a large annihilation cross section \[as in Eq. (\[AnnihXsection\])\], our sneutrino is one of the natural realizations of the so-called asymmmetric dark matter (ADM) [@Barr:1990ca; @Barr:1991qn; @Kaplan:1991ah; @Thomas:1995ze; @Hooper:2004dc; @Kitano:2004sv; @Kaplan:2009ag], and in our model only $\tilde N$ remains after annihilation with $\tilde N^*$. Thus, the relic abundance is actually determined by its asymmetry and the mass. For a novanishing sneutrino asymmetry similar to the baryon asymmetry, [$$\begin{split} Y_{\tilde N}\equiv \frac{n_{\tilde N} -n_{\tilde N^*}}{s} = {\cal O}( 10^{-10}), \end{split}$$]{} and a mass of about $5-6$ GeV, the correct relic density for dark matter $\Omega_{\tilde N}h^2\simeq 0.1$ is obtained. Finally, we note that our model is free from any indirect search for DM annihilation; in other words, DM annihilation cannot produce any signal because of the ADM property, namely, the absence of anti-DM particles in our Universe. Conclusion ========== We have shown that mostly right-handed Dirac sneutrinos are a viable supersymmetric light DM candidate and have a large enough cross section with nucleons to account for possible signals observed at direct DM searches. The $Z$-boson-mediated scattering does not relax the tension among direct DM search experiments and is constrained by the invisible decay width of the $Z$ boson. Nevetherless, we have found that a part of the signal region for CDMS II(Si) events remains outside the excluded region by XENON100. As an example of specific particle models, we have shown that a Dirac right-handed sneutrino with neutrinophilic Higgs doublet fields is a viable light dark matter candidate. Acknowledgments {#acknowledgments .unnumbered} =============== K.-Y.C. was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology Grant No. 2011-0011083. K.-Y.C. acknowledges the Max Planck Society (MPG), the Korea Ministry of Education, Science and Technology (MEST), Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at the Asia Pacific Center for Theoretical Physics (APCTP). [99]{} R. Bernabei [*et al.*]{} \[DAMA and LIBRA Collaborations\], Eur. Phys. J. C [**67**]{}, 39 (2010). C. E. Aalseth [*et al.*]{} \[CoGeNT Collaboration\], Phys. Rev. Lett.  [**106**]{}, 131301 (2011). C. E. Aalseth [*et al.*]{} \[CoGeNT Collaboration\], Phys. Rev. Lett.  [**107**]{}, 141301 (2011). G. Angloher [*et al.*]{}, Eur. Phys. J. C [**72**]{}, 1971 (2012). A. Brown, S. Henry, H. Kraus and C. McCabe, Phys. Rev. D [**85**]{}, 021301 (2012). R. Agnese [*et al.*]{} \[CDMS Collaboration\], \[arXiv:1304.4279 \[hep-ex\]\]. C. Kelso, D. Hooper and M. R. Buckley, Phys. Rev. D [**85**]{}, 043515 (2012). D. S. Akerib [*et al.*]{} \[CDMS Collaboration\], Phys. Rev. D [**82**]{}, 122004 (2010). Z. Ahmed [*et al.*]{} \[CDMS Collaboration\], Phys. Rev. Lett. [**106**]{}, 131302 (2011). J. Angle [*et al.*]{} \[XENON10 Collaboration\], Phys. Rev. Lett. [**107**]{}, 051301 (2011). E. Aprile [*et al.*]{} \[XENON100 Collaboration\], Phys. Rev. Lett. [**107**]{}, 131302 (2011). E. Aprile [*et al.*]{} \[XENON100 Collaboration\], Phys. Rev. Lett.  [**109**]{}, 181301 (2012). M. Felizardo [*et al.*]{}, Phys. Rev. Lett.  [**108**]{}, 201302 (2012). M. T. Frandsen, F. Kahlhoefer, C. McCabe, S. Sarkar and K. Schmidt-Hoberg, JCAP [**1307**]{}, 023 (2013). E. Del Nobile, G. B. Gelmini, P. Gondolo and J. -H. Huh, arXiv:1304.6183 \[hep-ph\]. M. Ackermann [*et al.*]{} \[Fermi-LAT Collaboration\], Phys. Rev. Lett. [**107**]{}, 241302 (2011). D. Hooper and T. Plehn, Phys. Lett. B [**562**]{}, 18 (2003). A. Bottino, N. Fornengo and S. Scopel, Phys. Rev. D [**67**]{}, 063519 (2003). D. G. Cerdeno, C. Hugonie, D. E. Lopez-Fogliani, C. Munoz and A. M. Teixeira, JHEP [**0412**]{}, 048 (2004). J. F. Gunion, D. Hooper and B. McElrath, Phys. Rev. D [**73**]{}, 015011 (2006). D. G. Cerdeno, C. Munoz and O. Seto, Phys. Rev. D [**79**]{}, 023510 (2009). D. G. Cerdeno, J. -H. Huh, M. Peiro and O. Seto, JCAP [**1111**]{}, 027 (2011). K. -Y. Choi, E. J. Chun and C. S. Shin, Phys. Lett. B [**723**]{}, 90 (2013). R. Allahverdi, B. Dutta, R. N. Mohapatra and K. Sinha, arXiv:1305.0287 \[hep-ph\]. Z. Kang, J. Li, T. Li, T. Liu and J. Yang, arXiv:1102.5644 \[hep-ph\]. N. Okada and O. Seto, arXiv:1304.6791 \[hep-ph\]. J. Beringer [*et al.*]{} \[Particle Data Group Collaboration\], Phys. Rev. D [**86**]{}, 010001 (2012). S. Schael [*et al.*]{} \[ALEPH and DELPHI and L3 and OPAL and SLD and LEP Electroweak Working Group and SLD Electroweak Group and SLD Heavy Flavour Group Collaborations\], Phys. Rept.  [**427**]{} 257 (2006). C. Arina and N. Fornengo, JHEP [**0711**]{} 029 (2007). N. Arkani-Hamed, L. J. Hall, H. Murayama, D. Tucker-Smith and N. Weiner, Phys. Rev. D [**64**]{}, 115011 (2001). G. Belanger, M. Kakizaki, E. K. Park, S. Kraml and A. Pukhov, JCAP [**1011**]{}, 017 (2010). B. Dumont, G. Belanger, S. Fichet, S. Kraml and T. Schwetz, JCAP [**1209**]{}, 013 (2012). K. -Y. Choi and O. Seto, Phys. Rev. D [**86**]{} 043515 (2012) \[Erratum-ibid. D [**86**]{} 089904 (2012)\]. E. Ma, Phys. Rev. Lett. [**86**]{}, 2502 (2001). F. Wang, W. Wang and J. M. Yang, Europhys. Lett. [**76**]{}, 388 (2006). E. Ma, Phys. Rev.  D [**73**]{}, 077301 (2006). S. Gabriel and S. Nandi, Phys. Lett. B [**655**]{}, 141 (2007). S. M. Davidson and H. E. Logan, Phys. Rev. D [**80**]{}, 095008 (2009); Phys. Rev. D [**82**]{}, 115031 (2010). H. E. Logan and D. MacLennan, Phys. Rev.  D [**81**]{}, 075016 (2010). N. Haba and K. Tsumura, JHEP [**1106**]{}, 068 (2011). N. Haba and O. Seto, Prog. Theor. Phys. [**125**]{}, 1155 (2011); Phys. Rev. D [**84**]{}, 103524 (2011). N. Haba, O. Seto and Y. Yamaguchi, Phys. Rev. D [**87**]{}, 123540 (2013). M. Sher and C. Triola, Phys. Rev. D [**83**]{}, 117702 (2011);\ S. Zhou, Phys. Rev. D [**84**]{}, 038701 (2011). N. Haba and T. Horita, Phys. Lett. B [**705**]{}, 98 (2011);\ T. Morozumi, H. Takata and K. Tamai, Phys. Rev. D [**85**]{}, 055002 (2012). N. Haba and M. Hirotsu, Eur. Phys. J.  C [**69**]{}, 481 (2010). N. Haba, K. Kaneta and Y. Shimizu, Phys. Rev. D [**86**]{}, 015019 (2012). Y. Morita, H. Nakano and T. Shimomura, arXiv:1212.4304 \[hep-ph\]. M. Lindner, A. Merle and V. Niro, Phys. Rev. D [**82**]{}, 123529 (2010). I. Affleck and M. Dine, Nucl. Phys. B [**249**]{}, 361 (1985). K. Enqvist and A. Mazumdar, Phys. Rept.  [**380**]{}, 99 (2003) and references therein. K. Enqvist and J. McDonald, Phys. Lett. B [**425**]{}, 309 (1998) and Nucl. Phys. B [**538**]{}, 321 (1999). M. Fujii and K. Hamaguchi, Phys. Rev. D [**66**]{}, 083501 (2002). O. Seto, Phys. Rev. D [**73**]{}, 043509 (2006). L. Roszkowski and O. Seto, Phys. Rev. Lett.  [**98**]{}, 161304 (2007). O. Seto and M. Yamaguchi, Phys. Rev. D [**75**]{}, 123506 (2007). R. Banerjee and K. Jedamzik, Phys. Lett. B [**484**]{}, 278 (2000). V. A. Kuzmin, V. A. Rubakov and M. E. Shaposhnikov, Phys. Lett. B [**155**]{}, 36 (1985). S. M. Barr, R. S. Chivukula and E. Farhi, Phys. Lett.  B [**241**]{}, 387 (1990). S. M. Barr, Phys. Rev.  D [**44**]{}, 3062 (1991). D. B. Kaplan, Phys. Rev. Lett.  [**68**]{}, 741 (1992). S. D. Thomas, Phys. Lett.  B [**356**]{}, 256 (1995). D. Hooper, J. March-Russell and S. M. West, Phys. Lett.  B [**605**]{}, 228 (2005). R. Kitano and I. Low, Phys. Rev.  D [**71**]{}, 023510 (2005). D. E. Kaplan, M. A. Luty and K. M. Zurek, Phys. Rev.  D [**79**]{}, 115016 (2009). [^1]: If we give up the standard thermal relic, we may consider a WIMP with an annihilation cross section small enough to satisfy the Fermi-LAT bound [@Allahverdi:2013tca] or the nonvanishing dark matter asymmetry [@Kang:2011wb; @Okada:2013cba]. [^2]: The fermion loop contribution is suppressed by helicity. [^3]: If unstable $Q$-balls survive the evapolation, the decay generates LSP dark matter, as discussed in Ref. [@EMD; @FH] for neutalino and Refs. [@RS; @SY] for axino.

--- abstract: 'We present an automatic moment capture system that runs in real-time on mobile cameras. The system is designed to run in the viewfinder mode and capture a burst sequence of frames before and after the shutter is pressed. For each frame, the system predicts in real-time a “goodness” score, based on which the best moment in the burst can be selected immediately after the shutter is released, without any user interference. To solve the problem, we develop a highly efficient deep neural network ranking model, which implicitly learns a “latent relative attribute" space to capture subtle visual differences within a sequence of burst images. Then the overall goodness is computed as a linear aggregation of the goodnesses of all the latent attributes. The latent relative attributes and the aggregation function can be seamlessly integrated in one fully convolutional network and trained in an end-to-end fashion. To obtain a compact model which can run on mobile devices in real-time, we have explored and evaluated a wide range of network design choices, taking into account the constraints of model size, computational cost, and accuracy. Extensive studies show that the best frame predicted by our model hit users’ top-1 (out of 11 on average) choice for $64.1\%$ cases and top-3 choices for $86.2\%$ cases. Moreover, the model(only 0.47M Bytes) can run in real time on mobile devices, e.g. only 13ms on iPhone 7 for one frame prediction.' author: - 'Baoyuan Wang, Noranart Vesdapunt, Utkarsh Sinha, Lei Zhang' bibliography: - 'egbib.bib' title: 'Real-time Burst Photo Selection Using a Light-Head Adversarial Network ' --- 18SubNumber[960]{} {width="\linewidth"} Introduction ============ This paper addresses the problem of how to take pictures of the best moment using mobile phones. With the recent advances in hardware, such as Dual-Lens camera on iPhone 7 Plus, the quality of the pictures taken on mobile phones has been dramatically improved. However, capturing a great “moment" is still quite challenging for common users, because anticipating the subject movements patiently while keeping the scene framed in viewfinder requires lots of practices and professional training. For example, taking spontaneous shots for children could be extremely hard as they may easily run out of the frame by the time you press the shutter. As a result, one may not only miss the desired moment, but also get a blurry photo due to the camera or subject motion. Taking another common example in portrait photography, keeping a perfect facial expression for long time without blinking eyes is nearly impossible. Therefore, it is likely that one has to replicate his pose and expression multiple times in order to capture a perfect shot, or one can use the burst mode to shot dozens of photos and then manually select the best one to keep and discard the rest. Although this method works for some people, it is less efficient due to the fact of wasting storage space and intensive manual selection. In this paper, we introduce a real-time system that automates the best frame (great moment) selection process “during" the capture stage without any post-capture manual operations. Specifically, we propose to buffer a few frames before and another few frames after the shutter press, we then apply an efficient photo ranking model to surface the best moment and automatically remove the rest of them to save storage space. We argue that having a real-time capture system would dramatically lower the bar of high quality moment capture for memory keeping or social sharing. To our best knowledge, there is no prior work in academia that directly targets at building automatic moment capture system during the *capture* stage, not to say on mobile phones. This is mainly due to the following challenges. First, such a system needs to run during the capture stage in the viewfinder, the ranking model has to be compact enough to be deployed on mobile phones and fast enough to run in real-time. Second, learning such an efficient and robust ranking model is challenging because the visual differences within a sequence of burst images are usually very subtle, yet the criteria for relative ranking could range from low-level image quality assessment, such as blur and exposure, to high-level image aesthetics, such as the attractiveness of facial expression or body pose, requiring a holistic way of learning all such representations in one unified model. Last but not least, due to the uniqueness of this problem, there is no available burst image sequences to serve as our training data, and it is also unclear how to collect such supervision signals in an effective way. For the same reasons, we cannot leverage related works developed for automatic photo selection from personal photo albums, because their photo selection criteria primarily focus on *absolute* attributes such as low-level image quality [@wang2004image], memorability [@IsolaParikhTorralbaOliva2011], popularity [@Khosla:2014:MIP], interestingness [@Fu_interestingnessprediction], and aesthetics [@Aesthetics2011]. In contrast, we are more interested in learning *relative* attributes that can rank a sequence of burst images with subtle differences. To address these challenges, we first created a novel burst dataset by manually capturing 15k bursts covering a rich set of common categories including selfies, portrait, landscaping, pets, action shots and so on. We sample image pairs from each burst and then conducted crowd-sourcing through Amazon Mechanical Turk (AMT) to get their overall relative goodness label (i.e., which looks better?) for each image pair. We consolidate the label information by a simple average voting. Second, considering a pair of images sampled from a burst, the visual content is largely overlapped, indicating the high-level features of a convolution network pre-trained for image classification may not be suitable for relative ranking, because classification network generally tries to achieve certain translation and rotation invariance and be robust to certain degree of image quality variations for the same object. However, those variances are the key information used for photo ranking. Therefore, in order to leverage the transfer learning from an existing classification net, one can only borrow the weights of the backbone net[^1] and must re-design a new head to tailer for our photo ranking problem. In addition to this, we observed that the relative ranking between a pair of images is determined by a few relative attributes such as sharpness, eye close or open, attractiveness of body pose or overall composition. And the overall ranker should be an aggregation of all such relative attributes. To enforce this observation, also inspired by recent advances in Generative Adversarial Networks(GANs)[@GaN0; @GAN1; @CGAN14], we introduce another generator (denoted as “G”) that can enhance the representation of the latent attributes so as to augment more training pairs in the feature space for improving the ranking model. Although we do not have the attribute level label information during the training, we expect the ranking network with a novel head can learn latent attribute values implicitly, so that it can minimize the ranking loss more easily. Motivated by the above facts and observations, we explored various choices for the backbone network and head (the final multi-layer module for ranking) design, and proposed a compact fully convolution network that can achieve good balance among model size, runtime speed, and accuracy. To sum up, we made the following contributions: - [We propose an automatic burst moment capture system running in real-time on mobile devices. The system can save significant storage space and manual operations of photo selection or editing for end users.]{} - [We explored various network backbone and head design choices, and developed a light-head network to learn the ranking function. We further applied the idea of Generative Adversarial Networks(GANs) into our framework to perform feature space augmentation, which consistently improves the performance for different configurations. ]{} - [We deployed and evaluated our system on several mobile phones. Extensive ablation studies show that our model can hit $64.1\%$ user’s top-1 accuracy(out of 11 on average). Moreover, the model(0.47M Bytes) can run in real time on mobile devices, e.g. only 13ms on iPhone 7 for one frame prediction.]{} Related Works ============= ### Automatic Photo Triage Automatic photo selection from personal photo collections has been actively studied for years [@Chu:2008:ASR; @Ceroni:2015:KKE; @Walber:2014; @Sinha:2011:SPP]. The selection criteria, however, are primarily focused on low-level image quality, representativeness, diversity, as well as coverage. Recently, there has been an increasing interest in understanding and learning various high-level image attributes, including memorability [@IsolaParikhTorralbaOliva2011; @Isola2011; @GygliICCV13; @ICCV15_Khosla; @Dubey_2015_ICCV], popularity [@Khosla:2014:MIP], interestingness [@Fu_interestingnessprediction; @GygliICCV13; @Aesthetics2011; @Dhar:2011:HLD], aesthetics [@Aesthetics2011; @Lu:2014:RRP; @Datta:2006:SAP; @Dhar:2011:HLD], importance [@Importance2012] and specificity [@Jas_2015_CVPR]. So technically, photo triage could be alternatively solved by assessing each of those image attributes. Although these prior works are relevant, our work is distinct in a number of ways: (1) we are interested in learning the ranking function that only runs “locally" within the burst rather than globally across all bursts. We do not expect the ranker to perform well between different bursts, because images coming from different bursts may not even be comparable; (2) we are interested in learning the “relative" attribute values. For example, both image A and B are blurry, but A is still relatively less blurry compared with B. However, all these prior works target for learning the attributes in an absolute manner; (3) Our ranker learns all the latent attributes (sharpness, smile, eye open/close etc) holistically in a weakly supervised manner while prior works all deal with each individual attribute with full supervision. There are a few interesting works along the line of relative attribute learning, such as [@conf/iccv/ParikhG11; @Xiao-iccv2015; @DBLP:journals/corr/SouriNM15; @Relative_Parts]. Yet they are not designed to rank photos for moment capture and require full supervision to train each individual attribute independently. Technically, the automatic photo triage system proposed by Chang etc  [@Chang:2016:ATF] might be the only work close to ours. However, it does not support burst photos for moment selection, as the differences are too subtle to treat each burst session as a “photo series” as defined in their setting. Moreover, we argue that their proposed network design is less efficient for real-time burst moment capture, as the ranker always need to feed the feature differences of an image pair to return the winner, and the winner is then recursively paired with the next frame until it loops over the whole burst. Clearly, this process can’t be easily ran in parallel. Especially, the complexity of getting the full rank is $O(n^2)$ ($n$ is the number of frames within the burst). In contrast, our ranker requires only one frame as input and directly predict its goodness score for ranking. ### Learning to Rank In the domain of image retrieval [@Image_ReRank11; @ImageRank; @ImageRetrieval07], the ranking functions generally associate with a query image and the goal is to rank the relevance of the resulting images with respect to the query. Whereas in our setting, as there is no query or reference image associated, our ranking function has to learn the degree of image goodness that can be determined by a few latent relative attributes . ### Generative Adversarial Networks(GANs) GAN [@GaN0] and its variants have become hot in research due to their ability of generating new samples from training data. However, most of the prior works[@GaN0; @GAN1; @CGAN14; @pix2pix2016] focus on realistic generating tasks, while in our work we use the concept of GAN to perform feature space augmentation to improve a ranking model. We will discard the Generator during the runtime. ### Burst Moment Capture To our best knowledge, there is no public prior work that directly targets for building an automatic moment capture system during the “capture” stage. Commercial product such as Microsoft Pix claims to have similar feature, it is however unclear how they implemented it technically. Most popular native cameras on smart phones such as iPhone, Google Pixel support burst capture by letting users keep pressing the shutter or holding it down for a while. The users then navigate to the photo album and manually compare all frames to pick the best and discard the rest ones. Although the system usually marks the best frame(s), the results are not always satisfactory. Overview ======== Problem Formulation {#Sec:formulation} ------------------- As mentioned above, we formulate the burst moment capture as a local relative ranking problem. Precisely, given an image burst with $l$ frames, denoted as $S=\{I_0,I_1,...,I_l\}$, our goal is to find a scoring function $f(x)$ that only takes a single frame as input and output its corresponding goodness score for ranking, and the best moment can be simply found by $\operatorname*{argmax}_{x \in \mathcal{S}} f(x)$. Note that, as $f(x)$ is trained to rank image pairs only within the same burst, it does not need to worry about the comparison across different burst. Therefore, $f(x)$ is forced to learn the relative attributes that can rank one image higher than the other in an image pair. Data Set -------- ### Burst Sequences As there is no public burst dataset available for our purpose, to train the ranking function $f$, we have to collect our own dataset. One potential idea is to sample continuous frames from existing videos to mimic burst sequences which seems to be fairly easy. However, we have found it nontrivial to collect a large video set with diverse categories to approximate the distribution of generic photographs. Furthermore, the defects existing in burst capture mode may not necessarily be the same as in video mode. Nevertheless, we leave this as the future work for dataset augmentation. To start with, we hired ten people to perform the data collection work. They were asked to always use the burst mode to capture each moment they want. Three mobile phones including iPhone SE, Google Pixel and Samsung S8 were used to collect bursts that range from a rich set of categories on purpose. In total, we have collected 14,769 bursts (246,715 images). We then randomly split the whole burst set into three subsets for training(8627), validation(1000) and testing(5142) respectively. On average, there are 11 images from each burst. {width="0.95\linewidth"} ### Crowd-Scouring We sampled all the image pairs from each burst and conducted an Amazon Mechanical Turk(AMT) study to get pair-wise labels. Instead of annotating the overall image quality in absolute manner, our pair-wise comparison encourages Turkers to pay attention to the subtle differences. We present one image pair side-by-side each time and ask the Turkers to select the options from (1) “The Left(Right) image is significantly better than the Right(Left) image" (2) “The Left(Right) image is marginally better than the Right(Left) image” (3) “They are equally good”. Given a sample pair $(A,B)$, we use $ A \succeq B$ to represent that $A$ is equal or better than $B$ (we can switch if B is better), then the label information $Y$ for $ A \succeq B$ can be defined as $$\label{eq1} Y(A,B) = \begin{cases} 0 &\mbox{if A, B are equal}\\ 1 &\mbox{if A is marginally better} \\ 2 &\mbox{if A is significantly better} \end{cases}$$ Each image pair was judged by 5 different Turkers, the final label was consolidated by averaging $Y$ and then rounded to 0, 1, or 2. Other more advanced consolidation method, such as the modeling probability distribution of the annotations, could be explored in the future work. In summary, our dataset consists of 838,038 pairs as well as their corresponding labels. Overall, around $35\%$ of the pairs have equal relative goodness ($\Delta(y)==0$), which is not surprising, because when there is no camera motion or subject motion, such as in still landscape photography, the frames within the same burst tend to be very close to each other. Margin Rescaling Based Ranking Model ------------------------------------ We propose to use a fully convolution network to learn both the discriminative features as well as the scoring function $f$ for ranking. As discussed in Sec. \[Sec:formulation\], we need to learn $f(x)$ that can predict the overall goodness score for an input frame $x$. Although we do not have the direct score label to train a regression function for $f(x)$, we have the pair-wise label that collected though AMT as discussed above. For a given pair $(A,B)$ from the training set, let us denote $x^i_A$ ($x^i_B$) as the feature of image A (B) in the $i$-th pair, and $\Delta(y_i)=Y(A,B)$ as its pair-wise label ground-truth. Then we need to learn the function $f$ so that for $f(x^i_A)-f(x^i_B) \geq \Delta(y_i)$ if $\Delta(y_i) \neq 0$, otherwise $|f(x^i_A)-f(x^i_B)| \leq \gamma_i$ where $\gamma_i > 0$. Then the loss for the ranking model $\mathcal{R_{\phi}}$ of each pair can be defined as $$\label{equ:loss} \begin{split} \mathcal{R_{\phi}}(A \succeq B,\Delta(y)) =\max(0, \Delta(y_i)-(f(x^i_A)-f(x^i_B))) +|f(x^j_A)-f(x^j_B)| \end{split}$$ Note that, different from the traditional rank SVM (i.e., [@conf/iccv/ParikhG11]), we use the margin rescaling technique ([@Joachims:2009:PSO:1592761.1592783; @Taskar:2003:MMN:2981345.2981349]) to enforce the ranking function to respect the degree of difference represented as $\Delta(y)$. In our experiments, we observe a slight improvement on our current test set. Attribute-Aware Head Design =========================== Intuitively, as we have discussed before, the relative goodness of a pair $(A,B)$ is attributed to a combination of a few relative attributes. For example, from the sharpness perspective, A is slightly better than B, but from the facial expression and image composition perspectives, B is more preferred. Depending on the degree of gap between A and B for each relative attribute, the final relative goodness is determined by a linear combination of multiple relative attributes. Mathematically, $f(x^i_A) - f(x^i_B) = W^T(x^i_A - x^i_B) = \sum w_k(x^{ik}_A-x^{ik}_B)$, where $x^{ik}_A$ is the $k$-th attribute value of image A in the $i$-th pair and $W$ is the combination weighting vector. Even though, in the context of a fully convolution network, the attribute feature vector (i.e., $x_A, x_B$) and its weighting vector $(i.e., W)$ are naturally integrated and can be trained end-to-end, we still expect the network design to be able to respect this simple intuition. We argue that, by adhering to the attribute-aware design guideline, it will be more feasible to learn the intrinsic relative attributes even with a network of small capacity and less computational overhead, which is crucial for real-time moment capture. One straightforward idea we could try is to customize the head of a typical classification network to output a single score. Following the design principle of several successful fully convolution networks, such as SqueezeNet [@squeezenet], Network in Network [@NIN], one can apply 1x1 point-wise convolution filter first to reduce all the feature maps into one single map with the same spatial size, and then conduct global average pooling to get the final score, as can be illustrated as “Head-A" in Figure \[fig:head\_design\]. Alternatively, we can flip the operations by first doing global average pooling and then conduct a 1x1 point-wise convolution (same as a fully connected layer) to output the final score[^2]. Such design is also popular in classification networks such as Xception [@xception] and ResNet [@resnet]. We call this design as “Head-B” as shown in the middle of Figure \[fig:head\_design\]. Intuitively speaking, the design of “Head-A" encourages to encode spatial features whereas “Head-B" encodes the channel features. However, both “Head-A" and “Head-B" reduce their spatial feature or channel feature too quickly, which potentially lead to the loss of certain informative features that represent the relative attributes used for subsequent comparison. Adversarial Ranking Loss Regularization --------------------------------------- To learn a better ranker, the network needs to output a compact set of attribute features before reducing to one final score. Inspired by the prior works on latent topic modeling[@TopicModelling] and Generative Adversarial Networks(GANs) [@GaN0; @GAN1], we propose a third head design that specially tailored for our relative ranking problem, which is called “Head-C" as illustrated in Figure \[fig:head\_design\]. Compared with “Head-A" and “Head-B", we first add an extra layer (Conv: $C'\times 1\times 1$) to project the original features into a low-dimensional subspace (denoted as $x$ in Head-C of Figure \[fig:head\_design\]) which can be regarded as a topic space or attribute space. Our hypothesis is that having an intermediate layer before outputting the final score would preserve more informative attribute features. We argue that one can safely assume the final convolution layer (equivalent to a FC layer) in Head-C serves as a linear ranker for all the attribute features from $x$. If we can learn discriminative and robust attribute features, then the ranker becomes easier to train. Another observation is, for any given image pair $(A,B)$ sampled from a burst, assume $ A \succeq B$ ($f(x_A) \geq f(x_B)$), if the learned intermediate layer in $x$ is indeed attribute aware, it is likely that by tweaking some attribute values in $x_B$ to get $x_B'$, one could flip the ranking relationship to $ B' \succeq A$ ($f(x_B') \geq f(x_A)$). An intuitive interpretation is that if the reason why $ A \succeq B$ is only caused by the degree of blurriness, then one can just reduce the value of the corresponding blurriness attribute in $x_B$ so that to flip their rank. This inspires us to synthesize more pairs in the attribute space as additional regularization to train the final ranking (scoring) network. To do this, we introduce another network which is called “G" during the training to synthesize a new attribute feature $x'$ for each $x$, as shown in Figure \[fig:head\_design\], by asking G to output a sparse residual vector $e$ so that $x'=x+e$. Like conditional-GAN [@CGAN14], G takes both $x$ and a randomly generated Gaussian noise vector as input, and then feed into a MLP subnet to output a residual vector $e$. During the training, we add a L1 norm loss constraint to encourage sparsity in $e$ to reduce the risk of over-fitting when training G. So the enhanced $x'_B$ can be regarded as the corresponding attribute feature for a new synthesized image $B'$. If G is well trained, we want to inject a new pair-wise loss for $B' \succeq B$ when training the main network in Head-C. Compared with traditional Generative Adversarial Network(GAN)[@GaN0; @GAN1; @CGAN14], we use a ranker instead of a discriminator to drive how we learn the enhanced feature, so training an accurate ranker is our final objective. The purpose of the generator is primarily feature augmentation during the training to help the ranker, whereas in traditional GAN the generator is the major learning objective. Nevertheless, as in traditional GAN, we train both our ranker and G iteratively. Specifically, for each pair $A \succeq B$, when we train G, we want to minimize the loss for enforcing both $f(x_B') \geq f(x_A)$ and the sparsity in $e$; when we train the ranker, in addition to original loss for $f(x_A) \geq f(x_B)$ , we have to add two more losses that enforce both $f(x_B') \geq f(x_B)$ and $f(x_A) \geq f(x_B')$. Mathematically, we will iteratively minimize the following two losses, namely $L(\mathcal{R_{\phi}})$ and $L(\mathcal{G_{\theta}})$ for ranker and G respectively: $$\min_{\phi} L(\mathcal{R_{\phi}}) = \mathbf{E}_{(A,B) \sim\mathcal{P}}[\mathcal{R_{\phi}}(A \succeq B,\Delta(y)) + \mathcal{R_{\phi}}(B' \succeq B,2) +\gamma\mathcal{R_{\phi}}(A \succeq B',2)]$$ $$\label{equ:g} \min_{\theta} L(\mathcal{G_{\theta}}) =\mathbf{E}_{(A,B')\sim\mathcal{G_{\theta}}}[\mathcal{R_{\phi}}(B' \succeq A, 2) + \lambda\|e\|_1]$$ where $\mathcal{P}$ represents the set of all real pairs in our dataset. $\lambda$ is empirically set to $0.1$, $\gamma$ is initialized to 1 at beginning. We will first train ranker for several epochs without adding the synthetic pairs(i.e.,($A \succeq B'$) or ($B' \succeq B$)), and train another few epochs for G while fixing the ranker by minimizing Equ. \[equ:g\]. We then start to train both ranker and G iteratively in a more frequent way, say each 25 mini-batch iterations. In our current experiment, we only take the lower-quality image, i.e, image B, in the pair and feed into G to get an enhanced $B'$. Note that, to enforce the margin, we set $\Delta(y)$ to 2 for all pairs that involved with a synthetic feature. After convergent, we discard G, and continue to fine-tune $\mathcal{R_{\phi}}$ by setting $\gamma$ to zero, assuming $B'$ can be safely ranker higher than $B$. We argue that our proposed “Head-C” with GAN loss is general enough that can sit on top of any backbone, even though we are more interested in studying its effectiveness for small backbone models considering the piratical applications. Experiments =========== Evaluation Metric ----------------- ### Pair-wise Level Since we train our system using pair-wise loss, we first measure the pair-wise level accuracy. For each pair $(x^A_i,x^B_i)$ and its ground-truth label $\Delta(y_i)$ in the test set, if $(f(x^A_i)-f(x^B_i))*\Delta(y_i) >0$ we regard the prediction as correct. Note that we have not taken the pairs with equal label into evaluation, as numerically it is infeasible to let $f(x^A_i)=f(x^B_i)$. We care more about the pairs whose $\Delta(y_i)\neq 0$. ### Burst Level The goal of our system is to predict the best frame from the burst sequence, thus the most interesting metric is to measure the ranking position of our predicted best frame in the list that is sorted based on user’s pair-wise label. This motivates us to use Top-K accuracy (with $K=1,2,3 $ in our current experiment). Intuitively, for each burst, according to the labeling, our prediction hits the Top-K accuracy if its rank is less or equal to K in the list sorted based on user label. Specifically, for a burst that contains $l$ frames, the Top-K accuracy is hit, if and only if $\sum_{A_i\neq A_{best}} \bm{1} \{Y(A_i,A_{Best}) > 0\} \leq (K-1)$, where $A_{best}$ represents the predicted best frame and $\bm{1}\{\cdot\}$ is the indicator function. We show the percentage of bursts among the whole test set that hit Top-K respectively. Implementation Detail --------------------- Our system was implemented using the Caffe framework, trained on a NVIDIA Titan X GPU. We use standard SGD with the momentum 0.9 and weight decay 0.0005. Initial learning rate is set to 0.001, dropping by a 0.1 gamma every 12000 iterations, and a total of 100000 iterations. Each mini-batch contains 35 image pairs sampled randomly from different bursts. Aside from the layers in the head, all other layers from the backbone network are fine-tuned from ImageNet pre-trained weights. Following standard technique, during the training we do random cropped sampling and augment the training set by simple mirroring. During testing, only center cropping is used. The training typically takes 6 hours to converge when either the learning rate drops below $1e^{-8}$ or the validation accuracy stay the same for a few epochs. **Backbone** **Head** **C’** **Top 1** **Top 2** **Top 3** **GFlop** -------------- ---------- -------- ------- ----------- ----------- ----------- ------ ----------- ResNet-152 A 0 222.5 65.5 79.1 85.6 76.7 11.4 GoogleNet A 0 39.3 65.0 79.5 86.1 76.0 1.6 A 0 2.8 63.9 78.6 85.4 75.6 B 0 2.8 64.1 78.2 85.2 75.2 C 5 2.8 64.2 79.1 86.0 75.6 C+G 5 2.8 65.6 80.3 86.8 77.2 C 20 2.8 64.3 79.0 85.9 75.4 C+G 20 2.8 65.9 80.3 87.0 77.2 C 50 2.9 65.3 79.6 86.5 76.2 C+G 50 2.9 65.9 80.3 86.9 77.1 C 100 3.0 64.6 78.8 85.8 76.0 C+G 100 3.0 65.7 80.0 86.9 77.4 A 0 0.46 60.9 76.1 83.72 72.2 B 0 0.46 60.8 75.8 83.1 73.6 C 5 0.47 62.3 77.3 85.1 74.5 C+G 5 0.47 64.1 78.8 86.2 75.9 C 20 0.48 62.8 77.4 84.9 74.3 C+G 20 0.48 63.9 78.7 86.0 75.8 C 50 0.51 63.4 78.4 85.7 74.8 C+G 50 0.51 64.0 78.8 86.0 76.0 C 100 0.56 62.5 77.1 84.7 74.1 C+G 100 0.56 64.1 78.9 86.2 75.7 A 0 0.10 60.3 75.2 83.2 72.1 B 0 0.10 60.3 75.6 83.4 72.3 C 5 0.10 60.3 75.3 83.1 72.1 C+G 5 0.10 61.4 76.4 83.8 73.1 C 20 0.11 60.9 75.9 83.7 72.3 C+G 20 0.11 61.5 76.5 84.0 73.1 C 50 0.12 61.0 75.9 83.6 73.2 C+G 50 0.12 61.9 76.8 84.2 73.5 C 100 0.15 61.1 75.9 83.5 72.4 C+G 100 0.15 62.1 76.9 84.4 73.5 : Results of the detailed ablation studies for the proposed head between **With** and **Without** adversarial regularization (G) when varying the number of relative attributes $C'$. All the metrics are measured as percentages ($\%$). As can be seen, adding G always improves the performance for all backbone and all $C'$, indicating the effectiveness of the adversarial regularization loss when training the ranking model[]{data-label="table:heads"} Ablation Study -------------- ### The effects of attribute-aware head design To study the efficiency and effectiveness of “Head-C", we need to choose a foundation layer where it sits on. We tried three different versions of SqueezeNet [@squeezenet] by varying the number of trimmed “Fire" layers. “SqNet-4” denotes a trimmed SqueezeNet with the top 4 fire layers removed. Like-wise, “SqNet-6” only keeps the layers from bottom to the third “Fire” layer block. We let “SqNet" denote the full SqueezeNet that keeps the layers from bottom up to the last ‘’Fire" layer block. Clearly, because of the max-pooling layer, the top layer of these three backbone networks are different in terms of the shape. For example, “SqNet-4” outputs a feature map of size $[N$x$256$x$28$x$28]$. For each of the three SqueezeNet versions, we trained different models for each different head. In “Head-C”, we also empirically set 4 different values for $C'$ and train separate models accordingly to study the effects of the attribute number in the intermediate layer. For this ablation[^3] study only, we use C+G to represent “Head-C” was trained with the adversarial regularization loss, and use C to represent its counterpart without G. As expected, for any attribute number $C'$ and whichever backbone it sits on, compared with “Head-A" and “Head-B",“Head-C" only adds negligible extra FLOPs and model size but significantly boost the performance of accuracy under all different metrics, for both with and without GAN. This can be seen in Table \[table:heads\]. For example, under “SqNet-4” and when $C'=5$, the Top-1 accuracy improves $3.2\%$ and $3.3\%$ from “Head-A" and “Head-B” to “Head-C" respectively, while only adding $2$K GFlops and $5$KB model size, which is negligible compared with the backbone. Even without the adversarial regularization loss(implemented by G), compared with Head-A and Head-B, adding the intermediate layer in Head-C seems to be always better for whichever $C'$, as shown in Table \[table:heads\]. This validated our hypothesis that quickly reducing the features to one final score could loss much useful information, indicating adding the intermediate layer to preserve the relative attributes for ranking is effective. However, a larger $C'$ is not necessarily always better. To see how important the adversarial loss for the final performance in “Head-C", we trained all the counterpart models without GAN. As can be seen in Table \[table:heads\], adding the adversarial loss during the training consistently improves the performance for all three back-bone nets and all the $C'$(100,50,20,5) we tried. For example, when $C'$ is fixed to 100, the gains that come from adding G are $1.1\%$,$1.6\%$ and $1\%$ for SqNet, SqNet-4 and SqNet-6 respectively. Interestingly, when $C'$ is reduced to 5, the gain by adding GAN is even more, the improvements are $1.4\%$, $1.8\%$ and $1.1\%$ respectively. All those studies indicate that GAN seems to be more effective for small backbone models and small number of latent attributes($C'$) in “Head-C”. Another interesting trend we can find is that the gain coming from the head optimization seems to be more economic compared with the gain coming by increasing the network capacity. For example, when $C'=50$, “SqNet-6” with “Head-C” ($61.9\%$) performs even better than “SqNet-4” with “Head-B”($60.8\%$). However, the later model is 4 times larger and 1.7 times more computationally costly in terms of FLOPS. Similarly, “SqNet-4” with “Head-C” also performs slightly better than the full“SqNet" with “Head-A” ($64.1\%$ VS $63.9\%$), but again the later model is 4 times larger and 1.7 times more computationally costly. This may indicate that attribute-aware head design indeed encodes more intrinsic relative attributes that make the ranking function easier to learn. {width="0.98\linewidth"} {width="0.98\linewidth"} ### The effects of back-bone For big networks such as GoogleNet[@GoogleNet] and ResNet[@resnet], it seems to be infeasible to make it real-time on mobile devices not to say its huge model size(i.e.,$~222$M for ResNet-152). Hence, our main focus is to study the performance of our design on small backbone network such as SqueezeNet. Nevertheless, we still train both GoogleNet and ResNet as the backbones for comparisons. As shown in Table \[table:heads\], when using the same “Head-A", GoogleNet and ResNet indeed hit higher accuracy compared with SqNet-4 and SqNet-6, the gain however is very minor given the capacity difference. However, a SqNet with “Head-C" can even achieve a higher accuracy than ResNet-152 with “Head-A", even though SqNet is only 50x smaller. We have noticed that, a deeper network does not always generate better performance of accuracy for our photo ranking problem. We argue that, unlike image classification task, relative attribute learning for ranking may not require very fine-grained features to distinguish between a cat versus a dog. So a very large capacity network, such as ResNet-152 may have high risk of over-fitting. However, when the capacity of backbone is less than a threshold, for example 3M for SqNet, the depth of backbone starts to be relatively important, for example, SqNet-6 is almost $4\%$ worse than SqNet. We argue that, 4M could be a reasonable model size for an camera application on mobile devices, therefore, it is more critical to design a novel head to improve the performance for the small backbone models, which is exactly our focus in this work. ### What features we learned? Figure \[fig:teaser\] shows one typical burst as well as the ranking result predicted by our model. Figure \[fig:gallery\] shows a gallery of testing results where our predictions all hit the Top-1 accuracy. For the sake of comparison, we only show the best and worse frame within each burst. Clearly, our model favors more opened eyes vs. closed eyes, more saliency subject (like the girl), better body pose (i.e., kid), and sharpness. We chose SqNet-4, with $C'=5$ in “Head-C” as the final model, and visualize the attributes difference gap histogram for image pairs of test set. In Figure \[fig:cluster\], we show a few such examples, from where we can find a clear clustering effect, i.e, the fifth dimension in the attribute space looks like focuses more on the attribute of eye openness, while the second and third dimension focus more on sharpness and human pose respectively. Comparison with Prior Work -------------------------- **Pairwise** **Flop** -- -------------- -------------- ---------- ------ VGG-16 514.2 73.2 15.5 VGG-16$^*$+C 47.2 73.1 15.5 SqNet-4+C 0.51 72.9 0.17 : Comparison with [@Chang:2016:ATF] on their dataset. VGG-16$^*$ represents a trimmed VGG with all fully connected layers removed. Note that, our model is about 90x smaller than the trimmed VGG net and 1000x smaller than the baseline model used in [@Chang:2016:ATF].[]{data-label="tab:triagle"} In photo triage work [@Chang:2016:ATF], they have shown that CNN-based relative learning is very efficient, and can beat all the competitors including the ones that rely on hand-crafted features. So we only chose to compare with their CNN approach. Although they target a different yet related problem and they do not support burst session data, technically, the approach can still be used for burst photo ranking problem. Their head design can be seen as the “baseline" head in Figure \[fig:head\_design\]. Although, the backbone is shared, the ranking always needs to take a feature vector pair as input to get the result, which may not be very efficient for real-time capture, as we cannot run all the frames within a burst in parallel. The time complexity of getting the full rank is $O(N^2)$. Whereas in our design, during the runtime, each image can be run independently to get the final score. Following their design principle, we conducted side-by-side comparison by training another model that sits on “SqNet-4” and using their header that shows in Figure \[fig:head\_design\]. On our dataset, we get $62.0\%$ Top-1 accuracy, which is $2.1\%$ worse compared with our design “Head-C” as shown in Table \[table:heads\]. We further trained another two models with our “Head-C" that sits on a trimmed VGG (with FC layers removed) and “SqNet-4” respectively on their Triage dataset. Compared with their VGG-16 baseline, our proposed model is up to 1000x smaller in model size and 90x faster, with relatively the same accuracy, as shown in Table \[tab:triagle\]. This again indicates that our design is both effective and efficient, and is general for photo ranking beyond burst data. Comparison with Native Cameras ------------------------------ We further conducted an informal user study though comparing our model with the built-in best of burst algorithm in Sam-sung Galaxy S8 Plus, Google Pixel and iPhone SE respectively We used the burst capture mode in the system camera app to capture around 200 bursts of ten to thirty frames. Post processing on these devices’ native camera application picks the best frame from the burst automatically. We ran the same bursts through our technique and picked the top scoring frame. To quantify the quality of the results of our technique, we conducted a five person blind A/B test to find if a user likes the system default best frame or the best frame from our technique. We averaged the responses per image-pair and rounded to the nearest option (better, equal or worse). As shown in Figure \[fig:pairwise\_label\_dist\], we see that results from our technique are clearly preferred by users when compared to the native best of burst algorithm, on both Android and iOS. We noticed that the native algorithms mostly take into account image blur without accounting for facial expressions and pose. We also deployed our system into both iPhone 7 and Google Pixel phones without aggressively engineering low level optimization. For the model “SqNet-4" and “Head-C", the runtime only takes 13ms on iPhone 7 and about 26ms on Google Pixel phone. ![Comparison with existing best of burst algorithms[]{data-label="fig:pairwise_label_dist"}](img/chart-devices_small.png){width="0.5\linewidth"} Conclusion ========== In this work, we have presented a real-time burst moment capture system based on deep learning. We formulate the problem as a relative learning problem for ranking. Currently, we consolidate the annotation of human label by simple averaging. Thus we only expect the model to learn general preferences. As a future work, one may consider to apply advanced techniques to learn personalized models for photo ranking. [^1]: Those weights in backbone will be fine-tuned when training the ranking net [^2]: In our current experiment, except the last scoring layer, there is always a “ReLu" operation followed after each convolution layer. [^3]: Unless otherwise noted, our “Head-C” is by default alway trained with “G”

--- abstract: 'With the development of high-resolution fingerprint scanners, high-resolution fingerprint-based biometric recognition has received increasing attention in recent years. This paper presents a pore feature-based approach for biometric recognition. Our approach employs a convolutional neural network (CNN) model, DeepResPore, to detect pores in the input fingerprint image. Thereafter, a CNN-based descriptor is computed for a patch around each detected pore. Specifically, we have designed a residual learning-based CNN, referred to as PoreNet that learns distinctive feature representation from pore patches. For verification, a matching score is generated by comparing the pore descriptors, obtained from a pair of fingerprint images, in a bi-directional manner using the Euclidean distance. The proposed approach for high-resolution fingerprint recognition achieves 2.91% and 0.57% equal error rates (EERs) on partial (DBI) and complete (DBII) fingerprints of the benchmark PolyU HRF dataset. Most importantly, it achieves lower FMR1000 and FMR10000 values than the current state-of-the-art approach on both the datasets.' author: - 'Vijay Anand,  and Vivek Kanhangad, [^1]' bibliography: - 'IEEEabrv.bib' - 'vijay\_pore\_match.bib' title: 'PoreNet: CNN-based Pore Descriptor for High-resolution Fingerprint Recognition' --- High-resolution fingerprints, fingerprint recognition, pore descriptor, convolutional neural network, cross-sensor fingerprints. Introduction {#intro} ============ is one of the most widely explored biometric traits, mainly due its distinctiveness and permanence [@maltoni2009handbook]. The features extracted from a fingerprint image are broadly classified into level-1, level-2 and level-3 features. Level-1 features, which include global ridge orientation, are commonly used for fingerprint classification. Level-2 fingerprint features include finer details such as ridge endings and ridge bifurcations, which are collectively called minutiae [@maltoni2009handbook]. Level-3 fingerprint features, on the other hand, include very fine details such as pores, incipient ridges, dots, and ridge contours. Level-1 and level-2 features can be observed in 500 dpi fingerprint images, while level-3 features are generally observable in fingerprint images having a resolution greater than 800 dpi [@HEF_resolution]. Commercially available automated fingerprint recognition systems (AFRS) and a majority of the methods reported in the literature employ level-1 and level-2 features. However, with the advent of high-resolution fingerprint sensors, there has been a focus shift and several methods that employ level-3 features have been developed for fingerprint recognition. In addition to enhancing the recognition performance, level-3 features provide higher level of security, as they are difficult to forge. Further, the level-3 features have also been included in the extended feature set for fingerprint recognition [@extended_feature]. Over the last few years, there has been growing interest in level-3 fingerprint features, especially the pores and several methods have been proposed for pore feature based automated fingerprint recognition [@stosz_pore; @roddy1997fingerprint; @kryszczuk2004extraction; @kryszczuk2004study; @jain2007pores; @zhao2009direct; @Zhao20102833; @ZHAO_partial; @sparse_fing; @LIU_PR; @Lemes; @segundo; @vijay_pore]. A pore-based AFRS typically consists of two major steps namely, pore detection in high-resolution fingerprint images and matching fingerprints using the detected pores. Stosz and Alyea [@stosz_pore] in their pioneering work proposed a fingerprint recognition approach that uses both pores and minutiae. Their approach involves detecting pores by tracing the ridges in skeletonized fingerprint image, followed by a multi-level matching using pores and minutiae. Roddy and Stosz [@roddy1997fingerprint] provided a detailed discussion on the statistics of the pores and examined its effectiveness in improving the performance of the existing AFRS. Krysczuk *et al.* [@kryszczuk2004extraction; @kryszczuk2004study] demonstrated the efficacy of pore features for fragmentary fingerprint recognition. In their approach, closed pores are detected by applying a set of thresholds to the binarized fingerprint image and open pores are detected by skeletonizing the valleys and finding the spurs having a sufficient number of white pixels in the neighbourhood. Their experimental results demonstrated that pore features are vital in recognizing partial fingerprint. The early studies [@stosz_pore; @roddy1997fingerprint; @kryszczuk2004extraction; @kryszczuk2004study] employed skeletonization-based approaches to detect pores. Such approaches are suitable only for very high-resolution ($\sim$ 2000 dpi) fingerprint images and their performance is likely to be adversely affected by fingerprint degradation caused by skin conditions. To circumvent these challenges, Jain *et al.* [@jain2007pores] presented a hierarchical fingerprint recognition approach that utilizes features from all the three levels. In their approach, pores are detected by applying Mexican-hat wavelet transform on the linear combination of the original and the enhanced fingerprint image. The fingerprints are first matched using minutiae and level-3 features are extracted in the neighbourhood of the matched minutiae points. The extracted level-3 features are then matched using the iterative closest point (ICP) algorithm. Later on, Zhao *et al.* [@zhao2009direct] proposed an approach, in which the pores are extracted using the adaptive pore filtering [@zhao_ICPR]. For each pore, a descriptor is formed by considering the pixel intensities in the neighbourhood. The initial correspondences are established through dot product and are refined using the random sample consensus (RANSAC) algorithm. The authors have demonstrated the usefulness of pores for biometric recognition using partial fingerprint images, which may not contain sufficient level-2 features [@Zhao20102833; @ZHAO_partial]. Liu *et al.* [@sparse_fing] proposed an improved direct pore matching approach, which employs the same pore descriptor as in [@zhao2009direct]. The coarse pore correspondences obtained through sparse representation are refined using the weighted RANSAC (WRANSAC) [@WRANSAC]. This work has been extended in [@LIU_PR], which employs the tangent distance and sparse representation to compare the pores extracted from the reference and probe fingerprint images. Recently, Lemes *et al.* [@Lemes] proposed a pore detection approach with a low computational cost. Their approach is adaptive and handles variations in the pore size. Firstly, a binary fingerprint image is obtained through global thresholding. For every white pixel, the average valley width is then estimated by computing the distance to neighbouring dark pixels in each of the four directions. The average valley width is used to define the size of a mask centered on each white pixel. Bright pixels inside the mask are then used to define a local threshold $T_{local}$ and a local radius $r_{local}$. Finally, a circle centered at each bright pixel with its local radius $r_{local}$ is used to determine whether the bright pixel is part of a pore or not. Segundo and Lemes [@segundo] improved the dynamic pore filtering approach [@Lemes] by considering the average ridge width in place of the average valley width to obtain the global and local radii, which are used in the same manner as in [@Lemes] to estimate the pore coordinates. The authors in [@segundo] performed ridge reconstruction from the detected pores by employing Kruskal’s minimum spanning tree algorithm. In the matching stage, a scale invariant feature transform (SIFT) based descriptor is obtained for each pore and the pores with bidirectional correspondences are used to compute the matching score. The ridge structure and ridge consistency of the corresponding pores are also used to generate the matching score. Most recently, Dahia and Segundo [@CNN_SIFT] presented an approach to generate pore annotation by aligning fingerprint images in the training set, followed by learning a descriptor for each of the pore patches by using an existing CNN-based patch matching model, HardNet [@HardNet2017]. The two-step method [@Pore_EA] to compare fingerprint pores aligns the fingerprint images using a data-driven descending algorithm. The alignment process utilizes the fingerprint ridges and orientation field. Once the fingerprint images are aligned, the pores present in the overlapping area between the two fingerprint images are matched using a graph comparison method. A review of the literature indicates that there is room for improvement in level-3 feature detection and the subsequent matching. The objectives of this work is to explore deep pore-descriptors and to advance the state-of-the-art in high-resolution fingerprint recognition. To this end, we have utilized CNN-based deep learning, which has proven successful for various computer vision problems [@facenet; @Deepface; @deep_conv]. The key contribution of this paper is a residual learning-based convolutional neural network, referred to as PoreNet, that learns distinctive feature representation from pore patches in high-resolution fingerprint images. In addition, we have developed an automated approach to generate labels for the pores that are common to different impressions of a finger belonging to the training set. We have also studied the effect of cross-sensor data on the proposed approach. Specifically, this is the first study that examines the performance of a learning based fingerprint recognition approach by testing the model on cross-sensor fingerprint images. The in-house high-resolution fingerprint dataset used in this study will be made publicly available to further research in this area. The rest of this paper is organized as follows: Section \[PM\] presents an introduction of the proposed methodology followed by a detailed description of the pore label generation approach and the pore descriptor learning approach. Experimental results and discussion are presented in Section \[results\]. Finally, our concluding remarks are presented in Section \[conclude\]. Proposed Method {#PM} =============== The proposed method employs a CNN model, DeepResPore [@vij_cnn], to detect pores in the input fingerprint image. It generates a pore intensity map, which is processed to obtain a binary pore map. Thereafter, another CNN model is used to compute a deep descriptor for a patch around each detected pore. This residual learning-based [@Resnet] CNN model, referred to as PoreNet, has been trained to compute distinctive feature representation from pore patches. For verification, a Euclidean distance-based matching score is generated by comparing the pore descriptor sets obtained from the probe and reference fingerprint images. The schematic diagram of the proposed method is presented in Fig. \[blockdiagram\]. \ Pore label generation {#pore_label} --------------------- Given a training dataset, we first detect pores in the fingerprint images using DeepResPore model, as shown in Fig. \[fig:pore\_det\]. For a given input fingerprint image, DeepResPore generates a pore intensity map, from which a binary pore map is obtained through local maxima filtering. To train a CNN model in a supervised manner, one requires a labeled dataset. Specifically, to train PoreNet, we require pore patches and their corresponding labels. To this end, we have first obtained labels for the pores that are common to different impressions of a finger. We have generated pore labels using a handcrafted key-point descriptor namely, DAISY [@DAISY], which has been shown to be very effective in representing pores [@Anand2019]. To obtain DAISY descriptor for a given image $I$, firstly, a set of orientation maps $O_{n}$, $1\leq n \leq N$, are computed as follows [@DAISY]: $$\label{eq1} \centering O_{n}(i,j)=max\Bigg(0, \frac{\partial I}{\partial n}\Bigg) \vspace{-0.5em}$$ where $\frac{\partial I}{\partial n}$ is the image derivative computed at $(i,j)$ along the direction $n$. The orientation maps $O_n$ are then convolved with a set of Gaussian kernels having different standard deviation ($\sigma$) values. Each pixel’s neighborhood in the convolved orientation map is then divided into overlapping circular regions on a series of concentric rings around the center pixel. Next, a normalized histogram of values from the convolved orientation maps is computed for each of the circular regions. Finally, the histograms computed from each of the circular regions are concatenated to obtain the DAISY descriptor. A detailed description of the DAISY descriptor can be found in [@DAISY]. The steps involved in our pore-label generation process are as follows: firstly, we identify a reference fingerprint image for each of the fingers present in the training dataset. The objective is to select a reference fingerprint image that has the maximum number of pores common to all other impressions of that finger. To this end, we treat the detected pore coordinates as key-points and compute DAISY-based descriptor for each of the detected pores. Since the training set contains multiple impressions of a finger, we consider one impression ($i$) at a time and perform pair-wise comparisons with rest of the impressions ($j$) using the DAISY-based pore descriptor. The match score (i.e. the number of matched descriptors) generated from each comparison is stored in $S_{ij}$. This is repeated for every impression of a finger and the index $R$ of the reference fingerprint image is determined as follows: $$\label{err1} R=\operatorname*{arg\,max}_{i \in\{1, 2, \hdots, N\}} \bigg (\sum_{j=1, j \neq i}^{N}S_{ij} \bigg )$$ where $N$ is the number of impressions of a finger. The impression, for which the sum of match scores is maximum, is considered to be the reference image. Once the reference fingerprint image is identified, the remaining impressions are aligned with the reference fingerprint to find the common pores that are present in all the impressions of that finger. The affine transformation [@Hartley:2003:MVG:861369; @TORR_MLESAC] for alignment is estimated using pore correspondences established based on the earlier comparison of DAISY-based pore descriptors. After aligning all impressions of a finger with its reference image, we identify the reference fingerprint image pore coordinates for which there are corresponding pores within the image boundaries of every transformed impression. We consider them to be common pores. At the end of this process, we have the coordinates of the common pores that are present in all impressions of each finger. Finally, a patch of $41\times 41$ pixels centered at each of the common pores is extracted and assigned a unique label. It is important to note that a finger having $P$ pores common to all its $r$ impressions will have $P\times r$ pore patches having $P$ unique labels in the training set. {width="90.00000%"} Pore descriptor learning ------------------------ [On completion of the pore-label generation process, we have a set of pore patches $X=\{x_{1},x_{2},\ldots,x_{T}\}$ and the corresponding set of labels $Y=\{y_{1},y_{2},\ldots,y_{T}\}$ for each of the fingerprint images in the training set. To learn a pore-patch descriptor, we have designed a customized residual learning-based CNN model, referred to as PoreNet.]{} [ The detailed architecture of PoreNet is presented in Table \[CNN\]. As can be seen, the proposed model has 14 learnable layers with 4 residual blocks consisting of 4 shortcut connections. The PoreNet takes a pore-patch $I_{p}$ of size $41\times41$ pixels as input and converts it into a 1681-dimensional pore descriptor. As shown in Fig. \[blockdiagram\_porenet\], the PoreNet does not perform any pooling operation in order that the size of the output feature map remains the same as that of the input. ]{} At the end of the network, a convolutional layer containing a single filter is introduced to generate the final feature map, which is flattened and normalized such that $l_2$-norm of the output embedding is equal to one [@facenet]. All convolution operations in PoreNet are performed with a stride of one. At every stage, zero padding is employed to maintain the size of the feature map. Furthermore, all convolutional layers, except the last one, perform convolution followed by batch normalization [@BN] and ReLU activation. In the training phase, the PoreNet is trained from the scratch. It is trained end-to-end in a supervised learning manner with the objective of minimizing the value of a triplet loss function. A batch of pore patches are fed to PoreNet, which generates the corresponding high-dimensional embeddings. Next, an online triplet mining scheme namely, batch-hard triplet mining [@facenet], is applied to the obtained embeddings. In this method, for each anchor sample $(a)$, we first obtain the hardest positive sample $(p)$ having the same label as that of the anchor and the hardest negative sample $(n)$ having a label different from the anchor. Finally, we form the following triplet loss [@facenet]: $$\label{loss} L_{triplet}= \max \big \{ d(a,p) - d(a,n) + margin , 0\big \}$$ where $d(a,p)$ is the distance between $a$ and $p$ and $d(a,n)$ is the distance between $a$ and $n$. The PoreNet is trained in an end-to-end manner to minimize the above triplet loss. **Layer name** **Output shape** **Kernel** ------------------ ------------------- ------------------------------------------ conv1 $41\times41, 16$ $3\times3$, $16$, stride 1, padding same conv2\_x $41\times41, 64$ $\times 2$ conv3\_x $41\times41, 128$ $\times 2$ conv4 $41\times41, 1$ $3\times3$, $1$ Flatten $1681$ $-$ $l_2$-norm $1681$ $-$ Total parameters 142,881 $-$ : Detailed architecture of PoreNet[]{data-label="CNN"} Once the PoreNet is trained, the task is to match a pair of fingerprint images using the generated pore-patch descriptors. Considering two fingerprint images $I_1$ and $I_{2}$ containing $n$ and $m$ number of pore patches, respectively, the PoreNet will generate two sets of pore descriptors $P_1 \in \mathds{R}^{n\times 1681}$ and $P_2 \in \mathds{R}^{m\times 1681}$. The pore descriptors in $P_1$ are compared with those in $P_2$ using the Euclidean distance and the pairs of descriptors matched bidirectionally are retained. Finally, the matches are refined using a distance ratio ($d_{r}$) criterion involving the distance to the second nearest neighbour [@Lowe_sift]. \[!ht\] [|&gt;M[2cm]{}|M[2cm]{}|M[2cm]{}|c|M[3cm]{}|c|]{} &Resolution (dpi)&Image size (pixels)&Fingers&Images per finger per session&Total images\ DBI:Training set&1200 &$320\times 240$&35& 3 &210\ DBI:Test set&1200 &$320\times 240$&148& 5 &1480\ DBII&1200 &$640\times 480$&148& 5 &1480\ Experimental results and discussion =================================== In this section, we first present the details of preparation of the dataset for training the PoreNet model, followed by results of our experiments. \[results\] Dataset preparation ------------------- We have performed experiments on the publicly available PolyU HRF dataset [@POLYU], which contains high-resolution fingerprint images of resolution 1200 dpi in two different sets DBI and DBII. DBI contains two subsets: DBI-train containing 210 partial fingerprint images from 35 fingers, each contributing 6 impressions and DBI-test containing 1480 partial fingerprint images. The size of these partial fingerprint images is $320\times 240$ pixels. On the other hand, DBII contains 1480 complete fingerprint images of size $640\times480$ pixels. Fingerprints images in DBI-test and DBII are collected from 148 fingers in two different sessions, with each finger contributing 5 impressions per session. The detailed description of PolyU HRF dataset is provided in Table \[DB\_table\]. While the pore patches for training PoreNet have been obtained from DBI-train, the datasets DBI-test and DBII have been used to evaluate the proposed approach using the experimental protocol adopted in the previous works [@segundo], [@CNN_SIFT], [@Pore_EA]. A training set $(80\%)$ and a validation set $(20\%)$ have been obtained by randomly partitioning DBI-train. We have obtained pore patches from every image in the training set and generated their labels using the method detailed in Section \[pore\_label\]. We have also employed data augmentation techniques to increase the amount of training data. Specifically, we have generated twenty (through ten rotations with angles randomly selected between $-20^{\circ}$ and $20^{\circ}$ and ten translations with shifts randomly selected between $-5$ and $5$ pixels) geometrically transformed images from every training sample. Further, we have generated an additional ten images by varying the contrast using the gamma transformation with gamma value selected in the range 0.45 to 0.9 in steps of 0.05. Overall, we have generated 815,982 labelled pore patches to train the PoreNet. Experimental results {#ER} -------------------- We have trained PoreNet for 100 epochs using a batch size of 256 and learning rate of 0.0001. The loss function has been optimized using the adaptive moment estimation (ADAM) [@ADAM]. The margin for the triplet loss function was set to 0.8. All our experiments have been performed on a computer with 3.60 GHz Intel core i7-6850K processor, 48 GB RAM and Nvidia GTX 1080 8 GB GPU. While the pore patches have been generated in MATLAB environment, the PoreNet has been trained and tested using TensorFlow [@Tensorflow] in Python environment. [|M[3cm]{}|M[2cm]{}|M[2cm]{}|]{} &\ &DBI& DBII\ Jain *et al.* [@jain2007pores]&$30.45\%$&$7.83\%$\ Zhao *et al.* [@zhao2009direct]&$15.42\%$&$7.05\%$\ Liu *et al.* [@sparse_fing] & $6.59\%$&$0.97\%$\ Liu *et al.* [@LIU_PR] & $3.25\%$&$0.53\%$\ Segundo and Lemes [@segundo]& $3.74\%$ & $0.76\%$\ Dahia and Segundo [@CNN_SIFT]& $4.18\%$ & [1.14%]{}\ Xu *et al.* [@Pore_EA]& **1.73%** & **0.51%**\ Proposed approach & $2.91\%$ & $0.57\%$\ \[results\_polyu\] \ To make a fair comparison with the existing approaches, we have followed the same experimental protocol as in [@segundo; @CNN_SIFT; @Pore_EA]. A set of genuine scores has been obtained by comparing every fingerprint image from the second session with all fingerprint images belonging to the same finger collected in the first session. On the other hand, impostor scores have been generated by comparing the first sample of every finger from the second session with the first sample of all other fingers from the first session. Thus, we have a total of 3700 $(148\times 25)$ genuine scores and 21,756 $(148\times 147)$ impostor scores. We report the following performance metrics: equal error rate (EER), FMR1000 and FMR10000 [@FMR_EER]. In addition, we present the detection error trade-off (DET) curve to help ascertain the performance of the proposed approach. The $d_r$ value was empirically set to 0.8 using the validation set. Table \[results\_polyu\] [^2] presents EERs of the proposed and the existing approaches on PolyU datasets. Since the recent methods show comparable performance in terms of the EER, we have further analysed their performance using FMR10000 and FMR1000. These results are presented in Table \[FMR\_values\]. As can be seen, the proposed approach achieves lower FNMRs on both the datasets. To ascertain this superior performance, we have plotted DET curves (please see Fig. \[ROC\_DBI\]). These curves clearly show that the proposed approach achieves better FNMR for low FMRs, specifically, FMR in the range $10^{-4}$ to $10^{-3}$. The performance improvement in this region translates into increased security and convenience for users. Furthermore, the proposed approach requires on average $1.41$ seconds and $2.80$ seconds to compare a pair of fingerprint images from DBI and DBII, respectively. On the other hand, the existing approach [@Pore_EA] takes $8.46$ seconds to compare a pair of fingerprint images. Overall, the experimental results presented in this section indicate that the proposed PoreNet model provides improvement in performance over the current state-of-the-art approaches. Specifically, it achieves lower FMR10000 and FMR1000 on both benchmark datasets. [|M[2.5cm]{}|M[1cm]{}|M[1cm]{}|M[1cm]{}|M[1cm]{}|]{} &&\ &DBI& DBII & DBI & DBII\ Dahia and Segundo [@CNN_SIFT]&$16.20\%$&$3.33\%$ & $10.82\%$ & $2.25\%$\ Xu *et al.* [@Pore_EA]&$6.94\%$&$2.01\%$ & $5.92\%$ & $1.31\%$\ Proposed approach &**6.47%**& **1.24%** & **5.42%** & **0.96%**\ \[FMR\_values\] Performance in a cross-sensor scenario {#DA} -------------------------------------- In this section, we present results of the experiments conducted to study the effect of cross-sensor fingerprint data on the performance of the proposed approach. As discussed previously, our model has been trained on PolyU HRF dataset [@POLYU]. The in-house IITI-HRF high-resolution fingerprint (1000 dpi resolution) dataset [@Anand2019], which has been expanded to include fingerprint images of 100 subjects, is used for testing. The expanded IITI-HRF high-resolution fingerprint dataset contains images of 8 fingers (all fingers except the little fingers on both the hands), each contributing 8 impressions. These images were acquired using the commercially available Biometrika HiScan-Pro fingerprint scanner. IITI-HRF dataset is partitioned into two subsets. The first one contains partial fingerprint images of size $320\times240$ pixels, while the second one contains full fingerprint images of size $1000\times1000$ pixels. The details of IITI-HRF dataset is presented in Table \[IITI\_table\]. \[!ht\] [|&gt;M[1.5cm]{}|M[1cm]{}|M[1.5cm]{}|c|c|]{} &Resolution (dpi)&Image size (pixels)&Fingers&Total images\ IITI-HRFP&1000 &$320\times 240$&800&6400\ IITI-HRFC&1000 &$1000\times 1000$&800&6400\ [|M[3cm]{}|M[2cm]{}|M[2cm]{}|]{} &\ &IITI-HRFP& IITI-HRFC\ Segundo and Lemes [@segundo]& $8.37\%$ & $2.90\%$\ Proposed approach & $9.58\%$ & $4.28\%$\ \[results\_IITI\] \ To make a comparison, we have also evaluated the performance of the hand-crafted feature based approach proposed by Segundo and Lemes [@segundo] on IITI-HRF dataset. For both these approaches, genuine and the impostor scores on IITI-HRF dataset have been generated using the following protocol: for each finger, 16 genuine scores have been generated by comparing the last four impressions with each of the first four impressions of the same finger. On the other hand, impostor scores have been generated by comparing the fifth impression of each finger with the first impression of all other fingers. This way, we have a total of 12,800 ($800\times16$) genuine scores and 639,200 ($800\times799$) impostor scores. The EERs and DET curves of both the approaches are presented in Table \[results\_IITI\] and Fig. \[ROC\_IITI\], respectively. As can be observed, the hand-crafted feature-based approach [@segundo] provides lower EER values as compared to the proposed learning-based approach on IITI-HRFP as well as IITI-HRFC. This may be due to the fact that our approach has been trained on PolyU HRF dataset and there exists a considerable variation in terms of resolution and the quality of pores between fingerprint images belonging to PolyU HRF and IIT-HRF. Therefore, the proposed learning based approach appears to suffer from the problem of domain adaptability. The results presented in this section underline the key challenge facing learning-based fingerprint recognition approaches, specifically, to overcome domain variability in cross-sensor scenarios. Employing domain adaptation techniques such as the one proposed in [@Deep_DA] in the training phase is likely to improve the cross-sensor performance of the learning-based approaches. Conclusion {#conclude} ========== In this paper, we have presented a deep pore-descriptor based method for high-resolution fingerprint recognition. Specifically, we have developed a residual learning-based CNN named PoreNet to learn a descriptor from pore patches in fingerprint images. The trained PoreNet generates deep embeddings from a given fingerprint image. To train PoreNet efficiently, we have also developed a method that generates pore labels by transforming the images based on the matched DAISY-based pore descriptors and finding the common pores. The results of our evaluations on the benchmark PolyU HRF datasets demonstrate the effectiveness of the proposed PoreNet in generating pore descriptors for high-resolution fingerprint recognition. Most importantly, the proposed PoreNet model achieves state-of-the-art performance in terms of FMR10000 and FMR1000. In future, we plan to work on the domain adaptability of the PoreNet model for cross-sensor fingerprint matching. Acknowledgment {#acknowledgment .unnumbered} ============== We thank Maurício P. Segundo for sharing the source codes of their approach. We also thank Yuanrong Xu for providing us with the data points of the DET curve of their approach. [^1]: V. Anand and V. Kanhangad are with the Discipline of Electrical Engineering, Indian Institute of Technology Indore, Indore 453552, India. e-mail: phd1401202011@iiti.ac.in (V. Anand), kvivek@iiti.ac.in (V. Kanhangad). [^2]: The EERs of the existing approaches [@jain2007pores], [@zhao2009direct],[@sparse_fing], [@segundo] are taken directly from the results presented in [@segundo] and those of the approaches [@LIU_PR] and [@Pore_EA] are taken from [@Pore_EA]. EERs values of [@CNN_SIFT] are obtained from the source code provided by the authors.

--- author: - | [^1]\ University of Glasgow\ E-mail: bibliography: - 'bibliography.bib' title: 'Indirect $\boldsymbol{\CP}$ Violation in $\boldsymbol{\decay{\Dz}{\hphm}}$ Decays at ' --- Introduction ============ Similarly to the and systems, the mass eigenstates of the system, , with masses $m_{1,2}$ and widths $\Gamma_{1,2}$, are superpositions of the flavour eigenstates , where $p$ and $q$ are complex and satisfy . This causes mixing between the and states, and allows for “indirect” in mixing, and in interference between mixing and decay, when decaying to a eigenstate. Indirect asymmetries in the system can be significantly enhanced beyond Standard Model (SM) predictions by new physics [@Bobrowski_indirectCPVCharm2010]. In decays of mesons to a eigenstate $f$, indirect can be probed using [@aGammaYCPTheory] $$\agamma \equiv \agammadefnot \approx \agammaexp,$$ where is the inverse of the effective lifetime of the decay, is the eigenvalue of $f$, , , , , with $\optionalBar{A}_{f}$ the decay amplitude, and . The effective lifetime is defined as the average decay time of a particle with an initial state of or , that obtained by fitting the decay-time distribution of signal with a single exponential. The detector at the , , is a forward-arm spectrometer, specifically designed for high precision measurements of decays of $b$ and ḩadrons [@JINST_LHCb]. During 2011 the experiment collected collisions at corresponding to an integrated luminosity of 1.0 . Due to the large production cross section [@lhcb_promptCharmProduction2013], the decay-time resolution of approximately 50 for decays [@lhcb_veloPerformance2014] and the excellent separation of $\pi$ and achieved by the detector [@lhcb_richPerformance2014], it is very well suited to measure with high precision. Methodology =========== ![Fits to (left) the invariant mass distribution and (right) the distribution for candidates from the data subset with magnet polarity down, recorded in the earlier of the two running periods.[]{data-label="fig:massfits"}](Massfit_KK_log.pdf "fig:"){width="35.00000%"} ![Fits to (left) the invariant mass distribution and (right) the distribution for candidates from the data subset with magnet polarity down, recorded in the earlier of the two running periods.[]{data-label="fig:massfits"}](Deltamfit_KK_log.pdf "fig:"){width="35.00000%"} The decay chain is used to determine the flavour of the candidates at production, via the charge of the meson. The -even and final states are used to calculate [@lhcb_agamma2014]. The predominant candidate selection criteria require the or tracks to have large impact parameter (IP), large transverse momentum (), invariant mass within 50 of the world average mass, and for the vector sum of their momenta to point closely back to the position of the collision. Using data corresponding to an integrated luminosity of 1.0 , 4.8M candidates and 1.5M candidates are selected. The data are divided by flavour, the polarity of the dipole magnet, and two separate running periods. Combinatorial and partially reconstructed backgrounds are discriminated using a simultaneous fit to the distributions of mass and . Examples of these fits are shown in Fig. \[fig:massfits\] for candidates, for data recorded with the magnet polarity down during the earlier of the two running periods. A fit to the decay-time distribution of the candidates is then used to determine the effective lifetimes of the and signal. Only candidates for which the is produced directly at the collision are considered as signal. The background from decays is discriminated by simultaneously fitting the distributions of the decay time and the natural logarithm of the of the hypothesis that the candidate originates directly from the collision ($\ln(\chisq_{\text{IP}})$). The selection efficiency as a function of decay time is obtained from data using per-candidate acceptance functions, as described in detail in Ref. [@lhcb_yCPAGamma]. The decay-time and $\ln(\chisq_{\text{IP}})$ distributions for combinatorial and specific backgrounds are obtained from the data using the discrimination provided by the mass and fits to employ the $_{\text{s}}$Weights technique [@sPlots] with kernel density estimation [@scott_densityEstimation]. Figure \[fig:timefits\] shows fits to the distributions of decay time and $\ln(\chisq_{\text{IP}})$ for candidates, using the same data subset as Fig. \[fig:massfits\]. Inaccuracies in the fit model are examined as a source of systematic uncertainty, as discussed in the following section. ![Fits to (left) the decay-time distribution and (right) the $\ln(\chisq_{\text{IP}})$ distribution for candidates from the data subset with magnet polarity down, recorded in the earlier of the two running periods.[]{data-label="fig:timefits"}](Timefit_KK_log.pdf "fig:"){width="35.00000%"} ![Fits to (left) the decay-time distribution and (right) the $\ln(\chisq_{\text{IP}})$ distribution for candidates from the data subset with magnet polarity down, recorded in the earlier of the two running periods.[]{data-label="fig:timefits"}](LogIPfit_D0_KK_log.pdf "fig:"){width="35.00000%"} Results and systematics ======================= The fits detailed in the previous section find $$\begin{aligned} \agamma(\pion\pion) &= \xtene{(+0.33 \pm 1.06 \pm 0.14)}{-3}, \nonumber \\ \agamma(\kk) &= \xtene{(-0.35 \pm 0.62 \pm 0.12)}{-3}, \nonumber\end{aligned}$$ where the uncertainties are statistical and systematic, respectively. These are the most precise measurements of their kind to date, and show no evidence of . The dominant systematic uncertainties arise from the modelling of the selection efficiency as a function of decay time, and the modelling of the background from decays. Figure \[fig:averages\] (left) shows the world average of , which is dominated by these measurements and is consistent with zero. Figure \[fig:averages\] (right) shows the combined fit to measurements of direct and indirect in decays, which yields a p-value for zero of 5.1 % [@HFAG2014]. ![The world averages of (left) and (right) direct vs. indirect in decays, reproduced from [@HFAG2014].[]{data-label="fig:averages"}](a_gamma_31aug13.png "fig:"){height="0.18\textheight"} ![The world averages of (left) and (right) direct vs. indirect in decays, reproduced from [@HFAG2014].[]{data-label="fig:averages"}](deltaACP_AGamma_fit_May2014.png "fig:"){height="0.18\textheight"} The precision of these measurements will be improved by the addition of 2.1 of data already collected during 2012. Together with data to be recorded in run II, and, in time, following the upgrade, measurements with precisions of approximately are possible, giving great potential for the discovery of indirect in the system. [^1]: On behalf of the collaboration.

--- abstract: 'We investigate the physics of a Tomonaga-Luttinger liquid of spin-polarized fermions superimposed on an ion chain. This compound system features (attractive) long- range interspecies interactions. By means of density matrix renormalization group techniques we compute the Tomonaga-Luttinger-liquid parameter and speed of sound as a function of the relative atom/ion density and the two quantum defect parameters, namely, the even and odd short-range phases which characterize the short-range part of the atom-ion polarization potential. The presence of ions is found to allow critical tuning of the atom-atom interaction, and the properties of the system are found to depend significantly on the short-range phases due to the atom-ion interaction. These latter dependencies can be controlled, for instance, by manipulating the ions’ internal state. This allows modification of the static properties of the quantum liquid via external driving of the ionic impurities.' author: - 'A. B. Michelsen' - 'M. Valiente' - 'N. T. Zinner' - 'A. Negretti' bibliography: - 'zotero.bib' title: ' Ion-induced interactions in a Tomonaga-Luttinger liquid ' --- Introduction ============ The quantum physics of one-dimensional (1D) interacting systems is rather peculiar as quantum fluctuations are strong and only collective excitations are possible, i.e. there are no single-particle excitations typical of Fermi liquids. Because of this, when the transverse degrees of freedom are frozen and a system acts as if one-dimensional, counterintuitive phenomena occur, such as fermionisation (bosonisation) of bosons (fermions) [@GirardeauJMP60; @GirardeauPRL06], perfect “collisional transparency" of particles [@OlshaniiPRL98] (equivalent to zero crossing of the two-body coupling constant), enhanced inter-particle interactions in a ballistic expansion [@LiebPR63], and unusual cooling mechanisms [@RauerPRL16; @SchemmerPRL18], to mention a few. While decades ago such manifestations were regarded as mere mathematical curiosities, the advent of degenerate atomic quantum gases has allowed the verification of such predictions, as the atomic confinement can be designed via optical laser fields [@GRIMM200095] or, alternatively, magnetic field landscapes can be engineered by means of tailored configurations of current-carrying wires in atom chips [@QIP:ACbook11]. The understanding of the fundamental underlying mechanisms behind such phenomenology is not only of academic interest, but also has important practical applications, as the progressive miniaturisation of electronic devices is such that, for instance, any quantitative description of transport in extremely reduced spatial dimensions and extremely low temperatures must be quantum mechanical. Very recently, experimental advances in bringing different atomic systems together to form a hybrid quantum system have opened new possibilities for quantum physics research [@tomzaCold2019a]. For instance, Rydberg [@SchmidtPRL16; @CamargoPRL18; @SchmidtPRA18] or other neutral impurities [@SpethmannPRL12; @CataniPRA12; @CetinaPRL15; @CetinaS16; @Jorgensen2016; @Hu2016; @Hohmann2015; @WideraPRL18] in quantum gases allow us to study the dressing of the atomic impurities with gas excitations and of mediated interactions [@ChenPRL18; @ChenPRA18; @KinnunenPRL18; @CamachoPRX18; @CamachoPRL18; @mistakidis2018; @DehkharghaniPRL18] as well as to utilize impurities to probe bath correlations and temperature [@SherkunovPRA09; @KollathPRA07; @RodriguezPRB18; @Mehboudi2018]. In addition, charged or dipolar impurities in degenerate atomic gases allow us to study polarons in the strong coupling regime [@CasteelsJLTP11], to quantum simulate Fröhlich-type Hamiltonians [@BissbortPRL13] as well as extended Hubbard [@Pupillo2008; @Ortner2009; @NegrettiPRB14; @Baier201] and lattice gauge theories [@DehkharghaniPRA17]. Experiments with an ion immersed in a Bose-Einstein condensate [@ZipkesNature10; @SchmidPRL10; @HartePRL12; @Kleinbach2018; @Engel2018; @MeirPRL16; @Meir2018], and in a Fermi gas [@RatschbacherPRL13; @Furst2017; @Joger2017; @ewald19] have been realised in recent years, albeit not yet in the deep quantum regime of atom-ion collisions. Specifically low dimensional quantum physics with impurities exhibits a variety of unusual quantum phenomena. A few examples of this are: Bloch oscillations experienced by a moving impurity in a strongly correlated bosonic gas without the presence of an optical lattice potential [@MeinertS17], quantum flutters [@Mathy2012] (namely injected supersonic impurities that never come to a full stop), so-called infrared-dominated dynamics [@KantianPRL14] and clustering of impurities [@DehkharghaniPRL18]. Motivated by these advances and by recent experiments that combine ytterbium ions with fermionic lithium atoms [^1] [@Furst2017; @Joger2017], we investigate the ground state properties of a spin-polarised fermionic quantum gas that is superimposed on an ion chain (see Fig. \[fig:diagram\]), where the latter is treated statically. Given the fact that the motion of the ions and their internal states can be precisely controlled in experiments, atom-ion scattering properties can thus be manipulated. This can be useful e.g. for inducing macroscopic self-trapping or tunneling dynamics in a bosonic Josephson junction [@GerritsmaPRL12; @SchurerPRA16; @ebghaCompound2019]. Here we are interested in the impact of the long-ranged atom-ion polarization potential on the 1D quantum fluid statical properties. Specifically, we employ density matrix renormalisation group techniques to extract the Tomonaga-Luttinger liquid (TLL) parameter and the speed of sound, which fully characterise the low energy physics of the atomic fluid. We find that these quantities have a significant dependence on the short-range physics of the atom-ion scattering (i.e., short-range phases), which can be controlled, for instance, by so-called confinement-induced [@IdziaszekPRA07; @MelezhikPRA16; @melezhikImpact2019] or Fano-Feshbach resonances [@IdziaszekPRA09; @tomza15]. Thus, our findings demonstrate that the quantum fluid properties not only can be tuned by manipulating the ion quantum state, but also that this dependence is strong. As has been previously discussed, TLL’s of 1D Bose-Fermi mixtures reveal a rich phase diagram [@MatheyPRL04] and our goal is to understand how long-ranged interactions can affect the picture. ![Sketch of the physical system considered in this work. A linear ion crystal, whose ions are positively charged (big blue spheres) and separated by a distance $D$, and a Tomonaga-Luttinger liquid of ultracold atoms (indicated by the red cloud with small spheres) that overlaps the crystal. []{data-label="fig:diagram"}](systemSketch){width="\columnwidth"} Theoretical framework ===================== In this section we describe the system we study, the interaction between the two atomic species and between the fermionic atoms, as well as provide the basic ingredients of TLL theory that will be used later in the paper. System Hamiltonian ------------------ We consider an ensemble of identical ultracold atoms, which are spin-polarised fermions, confined to one spatial dimension in the background of an ion chain with the ions organised as an evenly-spaced Coulomb crystal. The ions are considered static, namely their motion is neglected, e.g., because of tight confinement or heavy ions and light atoms. We use realistic atom-ion interactions via an accurate mapping of quantum defect theory (QDT) to an effective interaction potential that also includes the asymptotic power-law tail of the atom-ion forces. For the atom-atom interactions, we use instead effective field theory (EFT), which is valid at low energies and amenable to numerical treatment [@valiente15]. The Hamiltonian for $N_{\ensuremath{\text{A}}}$ atoms in the presence of an ion chain with $N_{\ensuremath{\text{I}}}$ ions takes the form $$\begin{aligned} \label{eq:Hmicro} \hat H &= \sum_{k=1}^{N_{\ensuremath{\text{A}}}} \left[ \frac{\hat p_k^2}{2 m_A} + U(x_k) \right. \nonumber\\ \phantom{=}& + \left. \sum_{j=1}^{N_{\ensuremath{\text{I}}}} V_{{\ensuremath{\text{A}}}{\ensuremath{\text{I}}}}(x_k - X_j) + \sum_{j=1}^{N_{\ensuremath{\text{A}}}} V_{{\ensuremath{\text{A}}}{\ensuremath{\text{A}}}}(x_k - x_j) \right],\end{aligned}$$ where $m_{\ensuremath{\text{A}}}$ is the atom mass, $\hat p_k$ is the atomic momentum, $U(x)$ is the external trap (specifically a box-like potential), $V_{{\ensuremath{\text{A}}}{\ensuremath{\text{I}}}}(x_k - X_j)$ is the atom-ion interaction with $x_k$ and $X_j$ denoting the $k$’th atom position and the $j$’th ion position, respectively, and $V_{{\ensuremath{\text{A}}}{\ensuremath{\text{A}}}}(x_k - x_j)$ is the atom-atom interaction. The atom-ion polarization potential is caused by the interaction between the ion electric field and the induced electric dipole of the atom. At long distances and in quasi 1D, it can be shown that the interaction takes the form [@IdziaszekPRA07] $$\begin{aligned} V_{AI}(x - X) = - \frac{\alpha e^2}{2 (x - X)^4}, \label{eq:VAI}\end{aligned}$$ where $e$ is the electron charge and $\alpha$ is the static polarisability of the atom. The potential, which is attractive and supports ion-bound atom states, is characterised by a characteristic length $R^*$ and energy $E^*$ $$\begin{aligned} R^* = \sqrt{\frac{\alpha e^2 \mu}{\hbar^2}}, \qquad E^* = \frac{\hbar^2}{2 \mu (R^*)^2}, \label{eq:units}\end{aligned}$$ where $\mu$ is the reduced atom-ion mass. Hereafter, all lengths are rescaled with respect to $R^*$. As we already pointed out, we focus on the static ion scenario and a very favourable choice for the atom-ion pair is $^6\text{Li}$ / $^{174}\text{Yb}^+$. This pair appears to be the most promising to attain the ultracold regime in radio-frequency traps [@cetina12; @Joger2014], i.e. $s$-wave collisions between atoms and ions. For this pair we have $E^*/h \simeq 178.6 \text{ kHz}$, $R^* \simeq 69.8 \text{ nm}$, and $m_{\ensuremath{\text{A}}}/m_{\ensuremath{\text{I}}}\simeq 0.035.$ Finally, the atom-atom interaction can be treated as short range two-body interaction with lattice EFT [@valiente15], where the first natural non-zero term affecting spin-polarized fermions is the lowest order odd-wave interaction [@barlette01]. The lattice, with a finite spacing, provides a regularization of the Cheon-Shigehara interaction [@cheon99; @valiente15], and its coupling constant is renormalised by fixing the atom-atom odd-wave scattering length $a_p$. Model atom-ion potential ------------------------ The previously introduced polarization potential, \[eq:VAI\] is state-independent, in that its form does not depend on the internal electronic configuration of the atom and the ion, only the polarisability. However, at short distances, below a few nanometers, the form of the interaction changes to a generally unknown form. At that spatial range the electronic configurations of the two particles enter into play and render the interaction state-dependent. Such a reliance is included theoretically by assuming that the only effect of the short-range part of the potential on the atom-ion wavefunction is to induce phase shifts. This effect is accounted for by introducing short-range phases $\phi_{e,o}$, which correspond to quantum defect parameters in the context of quantum defect theory [@IdziaszekPRA07; @idziaszek11]. Practically, this is handled by imposing appropriate boundary conditions in the limit $\vert x - X\vert \rightarrow 0$. In this limit, the polarization potential becomes extremely dominant so that all other energies can be neglected. In 1D such conditions are given by ($X=0$) [@IdziaszekPRA07] $$\begin{aligned} \psi_e(x) &= \vert x \vert \sin(1/\vert x \vert + \phi_e) ~,~ x \ll (R^*q)^{-1/2}, \label{eq:evenBound} \\ \psi_o(x) &= x \sin(1/\vert x \vert + \phi_o) ~,~ x \ll (R^*q)^{-1/2}, \label{eq:oddBound}\end{aligned}$$ with $\psi_{e,o}(x)$ being the even (e) and odd (o) solution of the scattering, respectively, and $q = \sqrt{2 \mu E/\hbar^2}$ with $E$ being the collisional energy at threshold. The short-range phases are free parameters which must be fixed to reproduce the scattering phase shifts found in experiment. Furthermore, the short-range phases fix the values of the even and odd-wave scattering lengths as $$\begin{aligned} a_{1D}^{e,o} = R^* \cot(\phi_{e,o}).\end{aligned}$$ Hence, tuning the short-range phases means to control the above scattering lengths, and therefore the effective 1D atom-ion interaction strength. The above QDT is cumbersome to implement in a many-body Hamiltonian formalism. In order to circumvent this difficulty, we use an effective interaction that faithfully reproduces the long-distance tail of the atom-ion potential, as well as the low-energy phase shifts. In particular, we use the model potential [@schurer14] $$\begin{aligned} V_{{\ensuremath{\text{A}}}{\ensuremath{\text{I}}}} (x) = v_0 e^{- \gamma x^2} - \frac{1}{x^4 + 1/\omega},\end{aligned}$$ which is characterised by three parameters: $v_0$, $\gamma$, and $\omega$. We fix $v_0$ at $3 \omega$ so that the atom wave function (almost) vanishes at $x = 0$, and $\gamma$ is chosen such that $$\begin{aligned} \gamma \geq \gamma_\text{min} = 4 \sqrt{10 \omega}\,.\end{aligned}$$ In this way, the Gaussian is kept from interfering with the long-range part. We can systematically map the free parameter $\omega$ and the semi-restricted parameter $\gamma$ to the quantum defect parameters $(\phi_e,\phi_o)$ (see Appendix A for more details). This means that we can use this potential for numerical modeling, while still considering the quantum defect parameters the tunable parameters of the system. Atom-atom interaction and discretisation ---------------------------------------- We shall solve the many-body problem by discretising it in an equally spaced grid with $N_s$ sites and spacing $d$, giving a total system length of $L = d(N_s-1)$. This will be evaluated in the continuum limit where $L$ remains finite while $d \rightarrow 0$. The discrete Hamiltonian $H_d$ is chosen so that $$\begin{aligned} \hat{H} = \lim_{d\rightarrow 0} \hat{H}_d. \label{eq:limiting}\end{aligned}$$ On the lattice (grid), the kinetic part $\hat H_{0,d}$ becomes (as for a Hubbard-like model) $$\begin{aligned} \hat H_{0,d} = -t(d) \sum_{j=1}^{L-1} \left(\hat c_{j}^\dagger \hat c_{j+1} + \hat c_{j+1}^\dagger \hat c_{j}\right),\end{aligned}$$ where we have in the continuum limit $$\begin{aligned} t(d) = \frac{\hbar^2}{2 m_{\ensuremath{\text{A}}}d^2}, \label{eq:t}\end{aligned}$$ and $\hat c_j (\hat c^\dagger_j)$ is the fermionic annihilation (creation) operator at position $x_j$, respectively. We will consider the atoms as interacting through van der Waals forces. These can be treated as short range two-body interactions with lattice EFT, where the first natural non-zero term affecting spin-polarized fermions is the lowest order odd-wave interaction  [@valiente15]. In our choice of lattice discretisation this corresponds to a nearest neighbor interaction between the atoms, $$\begin{aligned} \frac{V_{\text{AA},d}}{t(d)} = \frac{-2}{1 - d/a_p } \sum_{j = 1}^{N_s-1} \hat n_j^\text{A} \hat n_{j+1}^\text{A}, \label{eq:VAA}\end{aligned}$$ where $\hat n^{{\ensuremath{\text{A}}}/{\ensuremath{\text{I}}}}_j$ is the number operator for atoms/ions. The interaction strength is related to the tunneling rate, the lattice spacing and the p-wave (odd-wave) scattering length $a_p$ via $$\begin{aligned} V_{{\ensuremath{\text{A}}}{\ensuremath{\text{A}}}}(d) = \frac{- 2 t(d)}{1 - d/a_p}.\end{aligned}$$ In our calculations we work with $a_p = -0.1 R^*$, corresponding to an attractive interaction without bound states which has strength $V_{{\ensuremath{\text{A}}}{\ensuremath{\text{A}}}}/t \simeq -1.7$, see Appendix B for details. This particular value was chosen since it gives significant effects while keeping numerical stability. Note that odd-wave interactions may be tuned through e.g. Feshbach resonances or confinement induced resonances [@astrakharchik04; @granger04; @chin10; @zhang04; @OlshaniiPRL98; @saeidian15]. To evaluate the ground state of the discrete Hamiltonian we will employ numerical variational calculations using the density matrix renormalisation group (DMRG) [@white92; @schoellwock2005]. For such calculations it is convenient to express the Hamiltonian in the characteristic energy $t(d)$, where we combine \[eq:units,eq:t\] to find the conversion factor $$\begin{aligned} \frac{E^*}{t(d)} = \frac{(1 + m_{\ensuremath{\text{A}}}/ m_{\ensuremath{\text{I}}}) d^2}{(R^*)^2} = 1.03456 \frac{d^2}{(R^*)^2}.\end{aligned}$$ The effective atom-ion potential is discretised by introducing $x_{ij} = d|i-j|$ and thus the full discretised Hamiltonian is $$\begin{aligned} \frac{\hat{H}_d}{t(d)} &= - \sum_{j=1}^{L-1} \left(\hat{c}_{j}^\dagger \hat{c}_{j+1} + \hat{c}_{j+1}^\dagger \hat{c}_{j}\right) + \frac{-2}{1- d/a_p} \sum_{j = 1}^{N-1} \hat{n}_j^\text{A} \hat{n}_{j+1}^\text{A} \nonumber\\ \phantom{=} &+ \frac{E^*}{t(d)} \sum_{i,j} \hat{n}^\text{I}_{i} \hat{n}^\text{A}_{j} \left( v_0 e^{-\gamma x_{ij}^2} - \frac{1}{x_{ij}^4 + 1/\omega} \right), \label{eq:simHam}\end{aligned}$$ which satisfies \[eq:limiting\] up to a constant energy shift. ![A diagram of the effective potential generated by two ions. The black line shows the total potential, in this case a box with two ions as described by the model potential, whereas the blue horizontal lines indicate the energy of a specific eigenstate. A blue circle on a blue line indicates that this eigenstate has been occupied by a fermion. The number of ion-bound states per ion depends on the model parameters. The upper diagram shows the ion-bound (IB) and trap-bound (TB) filling types for the case of one ion-bound state, the lower for the case of two ion-bound states. []{data-label="fig:energylevels2"}](energyLevels){width="\columnwidth"} For the range of QDT parameters we investigate, the atom-ion interaction supports one or two two-body bound states. For ions in a finite lattice with open boundaries, this means we have two type of states (see \[fig:energylevels2\]): states deep in the effective atom-ion potential which would not exist in a flat potential, corresponding to ion-bound (IB) atoms, and a discrete set of states above the IB states similar to those found in a 1D quantum well, which we will call trap-bound (TB) since the discretisation is due to the presence of the (box-like) trap. Note that the TB states are still affected by the presence of the ions. We will consider two different $N_A/N_I$ fillings of the system. An $f_\text{IB}$ filling, where $N_{\ensuremath{\text{A}}}= N_{\ensuremath{\text{I}}}$, and each atom will occupy an IB state, and two $f_\text{TB}$ fillings, where all IB states are filled and $N_{\ensuremath{\text{I}}}$ atoms are added, which, because of quantum statistics, occupy $N_{\ensuremath{\text{I}}}$ TB states. Two such fillings must be considered to take into account the difference in the number of IB states. Tomonaga-Luttinger liquid theory {#sec:TLL} -------------------------------- A system of interacting fermions in one dimension is fully characterised at low energy by the renormalized speed of sound $u$ and the TLL parameter $K$, which is a dimensionless parameter with $K < 1$ for repulsive fermions, $K = 1$ for non-interacting fermions, and $K > 1$ for attractive fermions. The goal of our study is to investigate the impact of an ion lattice on such parameters. From the previous discussion on fillings we would expect the ions to act as attractive wells in the low filling cases, $f_\text{IB}$. As the filling rises, the attraction becomes screened by the atoms, and at high filling, $f_\text{TB}$, the atom-ion potentials effectively become soft barriers as shown pictorially in \[fig:screening\]. The expected effects are that in the $f_\text{IB}$ case, the atoms are forced closer together, effectively increasing their mutual attraction, i.e. the value of $K$ would rise. In the $f_\text{TB}$ case, the atom-ion potential is repulsive, and the expected effect is an induced repulsion between atoms, corresponding to a lowering of the value of $K$. In all cases we would expect a lowering of the speed of sound due to the introduction of barriers in the fluid, corresponding to a higher effective atomic mass. However, none of these behaviors follow trivially from the shape of the potential. Note that the degree of all these effects will depend on the nature of the atom-ion interaction as determined through the short-range phases $\phi_{e,o}$. ![Filling of the lower energy states will correspond to a screening of the attractive part of the atom-ion potential. As shown on this diagram, this can be understood microscopically as an effective cancellation of the wells on either side of the ion, ultimately only leaving a soft barrier. The expected effect in TLL terms would be a raising of the value of $K$ for low filling, corresponding to induced attraction, and a lowering of the value of $K$ for high fillings, corresponding to induced repulsion.[]{data-label="fig:screening"}](screening2){width="\columnwidth"} The ground state properties of the quantum fluid can be analyzed through the bosonized Hamiltonian $$\begin{aligned} \hat H &= \frac{1}{2 \pi} \int \left[u K (\partial_x \theta)(x)^2 + \frac{u}{K}(\partial_x \phi)(x)^2 \right] dx, \label{eq:LLHamil}\end{aligned}$$ where $\theta$ and $\phi$ are the standard bosonic fields. This effective Hamiltonian is a linearisation of \[eq:Hmicro\] around the Fermi points. We will extract $u$ and $K$ as functions of the quantum defect parameters, by treating the microscopic Hamiltonian, \[eq:Hmicro\], in a DMRG calculation and evaluating the ground state properties of systems with varying quantum defect parameters. This will allow us to extract the TLL parameters using the methods outlined below. Let us stress here that the discretisation has no physical significance, but it is done merely to allow a numerical treatment of the continuous system. Specifically, we consider that on the $N_s$ sites of our system there are $N_{\ensuremath{\text{A}}}$ atoms and $N_{\ensuremath{\text{I}}}$ ions. When $N_{\ensuremath{\text{A}}}, N_{\ensuremath{\text{I}}}\ll N_s$ and $d \ll R^*$ (i.e. low filling factor), we can use DMRG on the discretised system to approximate the $d \rightarrow 0$ continuum limit [@bellotti17; @dehkharghani17]. When we approach the thermodynamic limit numerically $$\begin{aligned} N_s \rightarrow \infty ~~,~~ d = \text{const.},~~ N_{{\ensuremath{\text{A}}}/{\ensuremath{\text{I}}}}/N_s = \text{const.},\end{aligned}$$ we can extract $K$ from the momentum space density-density correlation function for the minimum lattice momentum $k_0 = 2 \pi/N_s$ as [@ejima05] $$\begin{aligned} K = \lim_{N_s \rightarrow \infty} 2 \Big( \big\langle \hat n\left(k_0 \right) \hat n\left(-k_0\right) \big\rangle - \big\langle \hat n(k_0) \big\rangle \big\langle \hat n(k_0) \big\rangle \Big). \label{eq:K}\end{aligned}$$ Here the expectation value is with respect to the ground state $\psi_0$ of the fermionic system. We have used the Fourier transformed number operator $$\begin{aligned} \hat n(k) = \hat n^\dagger(-k) = \sum_{j=1}^{N_s} e^{-ik(j-j_c)} \hat c_j^\dagger \hat c_j,\end{aligned}$$ with $k$ being lattice momentum, $j$ being the lattice site index and $j_c$ being the central site. To reach this limit we use \[eq:K\] on a number of finite systems with increasing size and constant lattice spacing, atom density and ion density. We then extrapolate $K$ to the infinite size limit using a linear fit, see Appendix B for further details. In order to find $u$ we estimate the compressibility $\kappa$ of the system, whose inverse is related to TLL theory, \[eq:K\], as [@giamarchi03_3] $$\begin{aligned} \frac{1}{\kappa} &= \frac{u \pi}{K} = \frac{L}{2} {\frac{d^2 E}{d N_{\ensuremath{\text{A}}}^2}} \nonumber\\ \phantom{=}&\simeq \frac{L}{2} \left( \frac{E(N_{\ensuremath{\text{A}}}+2) + E(N_{\ensuremath{\text{A}}}-2) - 2E(N_{\ensuremath{\text{A}}})}{4} \right), \label{eq:kappa}\end{aligned}$$ where $E(N_A)$ is the energy of a system with $N_A$ atoms. The factor $1/2$ in the second line of \[eq:kappa\] accounts for the spin polarization. The derivative must be approximated as a finite difference, since the number of particles is discrete, and we use a difference of two atoms to avoid any effects which might arise due to the differences between having an odd and an even number of particles. By computing the ground state energy of the system for different numbers $N_A$ of fermions, we can thus calculate both TLL parameters by using \[eq:kappa,eq:K\]. Since the ions in our systems are equally spaced, effectively forming a periodic potential, the non-interacting variant can be accurately described using Bloch waves and band theory [@NegrettiPRB14] in the thermodynamic limit. Such a system contains gaps between the bands at integer fillings, i.e. $N_{\ensuremath{\text{A}}}= n N_{\ensuremath{\text{I}}}$ where $n$ is an integer. If our system is in such a gapped state it cannot be modelled using TLL theory. However, we are considering a system of interacting atoms, where the lattice model of the atom-atom interaction \[eq:VAA\] is inversely proportional to the lattice spacing. By approximating the continuum with a small lattice spacing, the interaction becomes comparably large, which can lead to a closing of said gaps, and ensure non-insulating behavior. However, an interacting system might still be a Mott insulator. To classify the behavior of the systems treated, we have calculated $\kappa$ for each system and extrapolated it to the thermodynamic limit. In this limit, $\kappa \rightarrow 0$ for any type of insulator since the energy gap causes the energy difference in \[eq:kappa\] to remain finite at infinite length. It was found that none of the systems treated exhibited such behavior, with the smallest extrapolated value being $\kappa = 0.16(3)(R^*E^*)^{-1}$. From this we conclude that all systems considered can be accurately modelled using TLL theory. In the rest of the paper we will assume an ion density of $N_{\ensuremath{\text{I}}}/L = 0.25/R^*$. This means that in the thermodynamic limit, the ions have a separation of $D = 4 R^*$, which for the atom-ion pair $^6\text{Li}$ / $^{174}\text{Yb}^+$ corresponds to 279.2 nm. In the case of $N_{\ensuremath{\text{I}}}= 7$, the ion spacing is $D=4.6 R^*$, corresponding to 321.1 nm. For the atom-ion pair $^{40}\text{K}$ / $^{174}\text{Yb}^+$ the ion spacing would correspond to 1.1 $\mu$m. For instance, for a $^{174}\text{Yb}^+$ ion chain with $N_{\ensuremath{\text{I}}}= 7$ ions and a radiofrequency of 2$\pi\times$ 2 MHz, the minimal separation is about 1.2 $\si{\micro\meter}$, whereas with a radiofrequency of 2$\pi\times$ 10 MHz it is 398.22 nm [@James1998]. Although the latter frequency is higher than typical values encountered in experiments, the quoted separations can be obtained by just generating time-dependent fields of higher frequency. Attempts at reaching ion separations that are currently attained in trapped ion experiments is beyond the capabilities of our DMRG calculations. Nonetheless, since the smaller ion separation we have considered, i.e. $D = 4 R^*$, is large enough that the atom-ion potentials have negligible overlap (see also Fig. \[fig:energylevels2\]), we do not expect any qualitative differences from increasing the separation. Results ======= The following results were obtained by using the DMRG algorithm as outlined above. Errors on $K$ are the $2 \sigma$ confidence intervals in the linear fits used for extrapolation. To ensure the correct implementation of our method we tested the calculation without ions. For $a_p=0$ we find the free fermion limit $K = 1.0000(2)$, as expected for non-interacting atoms, while the slightly attractive interaction $a_p = -0.1 R^*$ gives $K=1.0525(9)$. This is similar to the result we get by approximating the fermions as hard rods [@mazzanti08] with length $a_p$ in a system with fermionic density $\rho$, $K_\text{hs} = (1 - \rho a_p)^2 = 1.0506$. When comparing the calculated speed of sound for a free fermion gas with the Fermi velocity $v_F$ of the same system, we find $u/v_F = 1.03(5)$, where the error is due to discretisation. The parameter space which gave significant effects while being numerically feasible was found to be [@michelsen2018] $$\begin{aligned} \label{eq:range} 1 \leq \frac{\omega}{(R^*)^{-4}}, \frac{\gamma}{\gamma_\text{min}} \leq 10,\end{aligned}$$ where the combinations $$\begin{aligned} \label{eq:omega-gamma} \frac{\omega}{(R^*)^{-4}} = 2, 4, 6, 8, 10 ~~,~~ \frac{\gamma}{\gamma_\text{min}} = 1, 2, 5, 10\end{aligned}$$ give a relatively even spread of quantum defect parameters. Importantly, there is a transition in the number of bound states per ion within this parameter space, see \[tab:bound\_states\]. In the rest of the text the systems with two bound states will be said to be in the “strong" ion domain (since the potential has deeper wells and higher central Gaussian), while the systems with one bound state will be said to be in the “weak" ion domain. In the QDT parameter plots, \[fig:KmapIBa,fig:umapIBa,fig:KmapTBaWeak,fig:KmapTBaStrong\], this transition is schematically marked with a dashed line. -- --- --- --- --- ---- 1 2 5 10 4 1 1 1 1 6 1 2 2 2 8 2 2 2 2 -- --- --- --- --- ---- : Number of ion-bound atomic states per ion for those of the model parameter combinations considered in this study involved in the transition from one to two such states. This transition is marked schematically by a dashed line in \[fig:KmapIBa,fig:umapIBa,fig:KmapTBaWeak,fig:KmapTBaStrong\] \[tab:bound\_states\] This is particularly relevant for the investigation of TB states, and will be discussed further in \[sec:TB\] below. Note that ions located on an edge site will always have one bound state, which is a finite size effect. Further technical details can be found in Appendix B. Finally, for all plots in the following section, the points signify calculation results and the surface is a linear interpolation. Ion-bound atomic states ----------------------- ![Speed of sound for ion-bound states of a system of ions and interacting fermionic atoms with $N_A = N_I$. All points have errors of $\pm 0.05$. The dashed line schematically marks the transition from the weak ion domain (above the dashed line) the strong ion domain (below the dashed line), see the text for further details. The ions significantly lower the speed, corresponding to a hindering of collective excitations, especially in the strong domain (below the dashed line).[]{data-label="fig:umapIBa"}](speedOfSound){width="0.9\columnwidth"} ![TLL parameter for a range of QDT parameters in systems of ions and interacting fermionic atoms with filling $N_{\ensuremath{\text{A}}}= N_{\ensuremath{\text{I}}}$. All results have errors less than $ \pm 0.02$.[]{data-label="fig:KmapIBa"}](K-ionBound){width="0.9\columnwidth"} In \[fig:umapIBa\] we show the speed of sound $u$ of the system of interacting atoms and ions. Generally, the presence of the ions lowers this speed considerably compared to the Fermi velocity of a free fermion system, with the clearest effects in the strong domain. Since we are effectively introducing potential wells and barriers into the system, it is to be expected that collective excitations across the systems will be damped by these “obstacles", corresponding to a lower speed of sound, or equivalently a higher effective mass of the fermions. The introduction of ions into our system of interacting fermions induces a significant effective attraction between the interacting atoms, as shown in \[fig:KmapIBa\], where the $K$-parameter varies approximatively from 1.20 to 1.58, with a dip to 1 in the deep weak domain (above the dashed line). This depends mostly on $\phi_e$, and peaks for $-0.3 < \phi_e/\pi < -0.2$. For values larger than this, we see hints at a sharp dip towards the non-interacting limit. Trap-bound atomic states {#sec:TB} ------------------------ ![Luttinger liquid parameter for a range of QDT parameters in systems of ions and interacting fermionic atoms with filling $N_{\ensuremath{\text{A}}}= 2 N_{\ensuremath{\text{I}}}$. All results have errors less than $\pm 0.02$.[]{data-label="fig:KmapTBaWeak"}](K-trapBound-f1){width="0.9\columnwidth"} ![Luttinger liquid parameter for a range of QDT parameters in systems of ions and interacting fermionic atoms with filling $N_{\ensuremath{\text{A}}}= 3 N_{\ensuremath{\text{I}}}- 2$. All results have errors less than $\pm 0.05$.[]{data-label="fig:KmapTBaStrong"}](K-trapBound-f2){width="0.9\columnwidth"} Due to the previously mentioned transition in the number of IB states per ion, in order to study the behavior of a system of TB states we must consider different fillings in the different domains. In the weak domain we consider the $f_\text{TB,1}$ filling $N_{\ensuremath{\text{A}}}= 2 N_{\ensuremath{\text{I}}}$, while in the strong domain we consider the $f_\text{TB,2}$ filling $N_{\ensuremath{\text{A}}}= 3 N_{\ensuremath{\text{I}}}-2$, where two states are subtracted due to the fact that the ions at the edges can only host one odd-wave bound state. \[fig:KmapTBaWeak\] shows the $f_\text{TB,1}$ filling over both domains, and with $K$ varying between approximatively between 1.03 and 1.08, we see that the ions barely tune it away from the $1.05$ value from the system with no ions, with slightly induced attraction in the weak domain. \[fig:KmapTBaStrong\] shows the $f_\text{TB,2}$ filling over both domains, and with $K$ varying from 0.56 to 0.95 we can see a strong induced repulsion. Remarkably, there is a smooth transition between domains for both fillings, but drastically different $K$-values between the fillings, suggesting that the deciding factor in the value of $K$ is not the density of TB or IB states, but rather the total number of atoms per ion. The smooth transition between domains indicate that the difference between a low TB state and a shallow IB state has very little influence on the physics of our system. Discussion ---------- Taken together, \[fig:KmapIBa,fig:KmapTBaWeak,fig:KmapTBaStrong\] indicate that the effect of the ions on the atom-atom interaction can be separated into three different categories 1. *Stronger attraction.* This is the case when , and the atoms are bound relatively deep in the atom-ion potential. 2. *No effect or slightly stronger attraction*. This is the case when , with slightly more attraction in the weak domain. 3. *Shift to repulsive interaction.* This is the case when in the strong ion domain, with tendency towards the non-interacting limit in the weak domain. As predicted in \[sec:TLL\], we have induced attraction at low fillings (cat. 1), which transitions over a cancellation of attractive and repulsive effect (cat. 2) to induced repulsion at high fillings (cat. 3). A remarkable result is the shift of $K$ from $K > 1$ to $K < 1$, meaning that the introduction of ions causes the initially attractive atoms to have an effectively repulsive interaction. These results are in clear contrast to the behavior of a TLL in a flat potential, where changing the atomic density cannot tune the effective interaction across the free fermion limit [@giamarchi03_5]. To tune an initially attractive TLL into a repulsive TLL through a change in the atom density thus requires an inhomogeneous potential such as the one generated by an ion chain. The fact that the ions induce repulsion between the fermions indicates that the atomic gas has a tendency to form a so-called charge density wave, i.e. an ordered state. In our setting this means a density wave of fermionic atoms. A similar phenomenon has been observed for a 1D Fermi gas coupled parallel to an ion chain [@BissbortPRL13; @giamarchi03_4], where the (transverse) atom-phonon coupling induces a Peierls instability below a critical separation between the two quantum systems. Current experiments with ytterbium ions and lithium atoms [@Furst2017] show very low Langevin collisional rates, thus indicating that atoms do not occupy bound states within the ions, and so the points 2 and 3 above are the most experimentally relevant. Note that these effects are genuinely induced by the atom-ion scattering physics, that is, the occurrence of one or two bound states at threshold is a physical effect tuneable by Feshbach or confinement-induced resonances. Conclusions and outlooks ======================== We have investigated the ground state properties of a fermionic quantum fluid superimposed on a uniform ion chain. Particularly, we have assessed the Luttinger liquid parameters $K$ and $u$, which fully characterise the ground state of the spin-polarised Fermi gas and its low-energy excitations. Our goal was to analyse the reliance of the TLL parameters on the short-range phases of the atom-ion scattering. To this aim, we performed numerical density matrix renormalisation group calculations on a high-resolution discretised fermionic Hamiltonian modeling a static linear ion chain. Thus, we have been able to map the Luttinger liquid parameters to the two short-range phases characterising the atom-ion polarization potential. By changing these scattering parameters, e.g. via external driving of the ionic impurities, we have shown that the Luttinger liquid parameters can be tuned within a broad range of values. While the speed of sound is generally decreased, corresponding to a hindering of collective excitations by the ions, the interaction as measured by $K$ has a more intricate behavior. Depending on the density of the initially weakly attractive atoms, changing the ion scattering parameters can tune the interaction within a repulsive regime, an attractive regime or have completely negligible effect. The result of most immediate experimental relevance is the induced repulsion. Finally, future work could address the dimensional crossover by replacing the setup we investigated purely in 1D with an atomic waveguide, where the motional transverse degrees of freedom are taken into account, too. Recently the analytical solution of the 3D scattering problem of a trapped atom interacting with an array of contact potentials, i.e. representing the static scattering centres akin to the ions, was presented [@MartaPRA18]. Hence, one could solve the many-particle problem using this analytical solution and investigate the impact of the transverse confinement of the atoms on the TLL parameters and excitation spectrum of the liquid in order to understand the interplay between external confinement and impurity-atom scattering characteristics. An alternative approach could be by means of bosonisation techniques, where the transverse modes are coupled [@Kamar2019]. Moreover, another interesting research direction is to study the role of spatial inhomogeneities in the impurity-atom interaction strength, thus adding controlled disorder in the system. Acknowledgements ================ ABM would like to thank Rafael Barfknecht and Satoshi Ejima for fruitful discussions. This work was supported by the Aarhus University Research Foundation under a JCS Junior Fellowship and the Carlsberg Foundation through a Carlsberg Distinguished Associate Professorship grant, by the Cluster of Excellence projects “The Hamburg Centre for Ultrafast Imaging" of the Deutsche Forschungsgemeinschaft (EXC 1074, Project No. 194651731) and “CUI: Advanced Imaging of Matter" of the Deutsche Forschungsgemeinschaft (EXC 2056, Project No. 390715994), and the National Research Fund, Luxembourg, under grant ATTRACT 7556175. The numerical work presented in this paper was partially carried out using the HPC facilities of the University of Luxembourg [@VBCG_HPCS14] [– see <https://hpc.uni.lu>]{}. The model potential =================== For the sake of numerical efficiency, we have chosen the atom-ion model potential parameters within the range $$\begin{aligned} 1 \leq \frac{\omega}{(R^*)^{-4}}, \frac{\gamma}{\gamma_\text{min}} \leq 10.\end{aligned}$$ The mapping between the QDT parameters, i.e. short-range phases, and model potential parameters is performed by following this procedure: 1. We choose some values for $\omega$ and $\gamma$ within the range outlined above as well as $E = k^2$ (all parameters are in units of $E^*$ and $R^*$), the latter of which must be small (i.e. in the low-energy limit), but positive. We then use the Numerov method [@johnson77] to solve the Schrödinger equation for the two-body atom-ion problem for this potential by iterating the wavefunction from $x = 0$ to $x \gg R^*$. 2. We determine the phase shifts $\xi_{e,o}$ of the solution at large distances, i.e. far from the ion, by comparing the logarithmic derivative of the solution to a plane wave solution at $x = x_0 \gg R^*$, $$\begin{aligned} \cot(\xi_{e,o}) &= \frac{k + A_{e,o} \cot(k x_0)}{-A_{e,o} + k \cot(k x_0)}, \\ A_{e,o} &= \left.\frac{d\psi_{e,o}(x)/dx}{\psi_{e,o}(x)}\right\vert_{x=x_0}.\end{aligned}$$ 3. We test QDT solutions of different $\phi_{e,o}$ and determine the corresponding phase shifts as in the previous step 2. 4. We compare the phase shift, $\xi_{e,o} (\phi_{e,o})$, obtained via QDT for a certain pair of short-range phases $\phi_{e,o}$ with the sample of phase shifts, $\xi_{e,o} (\omega, \gamma)$, obtained with the model potential for various parameters $\omega, \gamma$. The one that is most similar to $\xi_{e,o} (\phi_{e,o})$ gives the mapping. We note that the last step of this procedure always yields a numerical error, i.e. the difference between the QDT result and the model potential will be around $10^{-12}$. We also note that for perfect precision in the mapping, the atom-ion wave-function would have to be zero at the ion position. This is only true for the model potential to a good approximation, since the model parameter $v_0$ is finite. DMRG calculations and extrapolation =================================== ![An example of values of $K(N_s)$ for trap-bound systems with constant atom and ion densities, constant lattice separation and different sizes (as measured by number of lattice points $N_s$). To calculate $K = \lim_{N_s \rightarrow \infty} K(N_s)$ we apply to finite systems and extrapolate the results to the infinite size limit $1/N_s \rightarrow 0$ using a linear fit. The value of $K$ in the limit is shown with an error which is the $2 \sigma$ confidence interval on the fit. This example has , $\gamma = \gamma_\text{min}$ and $a = 0.1 R^*$ and $N_{\ensuremath{\text{A}}}= 3 N_{\ensuremath{\text{I}}}- 2$.[]{data-label="fig:extrp"}](extrapolation){width="0.9\columnwidth"} The DMRG solutions were found by using the implementation from the <span style="font-variant:small-caps;">ITensor</span> library [@miles18]. The time taken and accuracy achieved depends on a number of supplied parameters: Sweeps : The number of sweeps to achieve convergence depends heavily on the size and complexity of the system of interest, ranging from $\sim 100$ for small atom-only systems with simple interactions, to 1000-2000 for large atom-ion systems with many atoms and all interactions turned on. Cutoff : DMRG uses a singular value decomposition (SVD) procedure, where all singular values below this cutoff value are truncated. The value was kept similar to that of Refs. [@dehkharghani17; @bellotti17], namely $\sim 10^{-13}$. Maximum bond dimension : It was found that setting this value at $1000$ gives a good convergence time. The <span style="font-variant:small-caps;">ITensor</span> implementation automatically converts common operators into matrix product operators (MPOs). This renders the implementation of the Hamiltonian as well as the extraction of the ground state energy and the density profile $$\begin{aligned} {\left< \hat{\rho}(x_j) \right>} = {\left< \psi_0 \right|} \frac{\hat{c}^\dagger_j\hat{c}_j}{d} {\left| \psi_0 \right>}.\end{aligned}$$ rather simple. A straightforward way to confirm that the algorithm has converged is to check the symmetry of the density profile. The true ground state will be completely symmetric around the center of the trap, but it was found that the DMRG algorithm would only return states with symmetric density profiles once it had completely converged. The parameter space which gave significant effects while being numerically feasible was found to be (\[eq:range\]), whereas the combinations (\[eq:omega-gamma\]) give a nice spread of quantum defect parameters. Smaller parameters would make the features of the potential too weak, while larger parameters tend to give an non-smooth potential, requiring a finer lattice to properly resolve. Within this parameter range it was found that a lattice constant of $d \sim 0.01 R^*$ with $\sim 400$ sites per ion was a minimum for reliable calculations. Extrapolation was done from the results of calculations with 5 to 12 ions, a density of $N_{\ensuremath{\text{I}}}/L = 0.25/R^*$ and a lattice separation of $d = 0.01667 R^*$, see \[fig:extrp\]. Since the data points cluster closer together towards $1/N_s \rightarrow 0$, and to have more efficient calculations, it was chosen to only extrapolate using $N_{\ensuremath{\text{I}}}= 5,6,7,9,12$, which still gives a reliable extrapolation. For $\omega/(R^*)^{-4} < 5$ the $N_I=5$ results were found to be unreliable and had to be excluded from the extrapolation. The remaining points sufficed for reliable extrapolation. The main system of interest in this paper is that with trap-bound filling and non-zero atom-atom interactions. It is however noteworthy that the extrapolation procedure failed for the ion-bound filling when atom-atom interactions were neglected (i.e. $V_{{\ensuremath{\text{A}}}{\ensuremath{\text{A}}}} = 0$). One would expect this to be a simpler system to work with, but our numerical procedure failed in this case. Extraction of $K$ using \[eq:K\] can be readily done by seeing that the sum is symmetric around $j_c$, meaning the imaginary parts of the exponential cancel, and one is left with $$\begin{aligned} \hat n(k) = \hat n(-k) = \sum_{j=1}^{N_s} \cos[ k (j-j_c) ] \hat c^\dagger_j \hat c_j,\end{aligned}$$ which is real and even, and which can be converted to an MPO and applied to the ground state before calculating the overlap in \[eq:K\]. [^1]: We note that currently the atom-ion species Li/Ca$^+$ is also under intense experimental investigations [@HazePRL18].

--- abstract: 'We show that a derivator is stable if and only if homotopy finite limits and homotopy finite colimits commute, if and only if homotopy finite limit functors have right adjoints, and if and only if homotopy finite colimit functors have left adjoints. These characterizations generalize to an abstract notion of “stability relative to a class of functors”, which includes in particular pointedness, semiadditivity, and ordinary stability. To prove them, we develop the theory of derivators enriched over monoidal left derivators and weighted homotopy limits and colimits therein.' author: - Moritz Groth and Michael Shulman bibliography: - 'stability.bib' title: Generalized stability for abstract homotopy theories --- Introduction {#sec:intro} ============ In classical algebraic topology we have the following pair of adjunctions relating topological spaces $\mathrm{Top}$ to pointed spaces $\mathrm{Top}_\ast$ and spectra $\mathrm{Sp}$: $$(\Sigma^\infty_+,\Omega^\infty_-)\colon\mathrm{Top}\rightleftarrows\mathrm{Top}_\ast\rightleftarrows\mathrm{Sp}$$ Abstractly, each of these two steps universally improves certain *exactness properties* of a homotopy theory. In the first step we pass in a universal way from a general homotopy theory to a *pointed* homotopy theory, i.e., a homotopy theory admitting a zero object. The second step realizes the universal passage from a pointed homotopy theory to a *stable* homotopy theory, i.e., to a pointed homotopy theory in which homotopy pushouts and homotopy pullbacks coincide. With this in mind, our first goal in this paper is to collect additional answers to the following question. **Question:** Which exactness properties of the homotopy theory of spectra already *characterize* the passage from (pointed) topological spaces to spectra? To put it differently, starting with the homotopy theory of (pointed) topological spaces, for which exactness properties is it true that if one imposes these properties in a universal way then the outcome is the homotopy theory of spectra? To make this question precise, we need a definition of an “abstract homotopy theory”; here we choose to work with derivators. (However, similar arguments should also apply to $\infty$-categories.) For the introduction it suffices to know that derivators provide some framework for the calculus of homotopy limits, colimits, and Kan extensions as it is available in typical situations arising in homological algebra and abstract homotopy theory (see e.g. [@groth:intro-to-der-1] for more details). A derivator is by definition *stable* if it admits a zero object (i.e. it is pointed) and if the classes of pullback squares and pushout squares coincide. Typical examples are given by derivators of unbounded chain complexes in Grothendieck abelian categories (like derivators associated to fields, rings, or schemes), and homotopy derivators of stable model categories or stable $\infty$-categories (see [@gst:basic §5] for many explicit examples). The “universal” example is the derivator of spectra, which is obtained by stabilizing the derivator of spaces [@heller:stable]. It is known that stability can be reformulated by asking that the derivator is pointed and that the suspension-loop adjunction or the cofiber-fiber adjunction is an equivalence [@gps:mayer]. Alternatively, by [@gst:basic] a pointed derivator is stable exactly when the classes of strongly cartesian $n$-cubes (in the sense of Goodwillie [@goodwillie:II]) and strongly cocartesian $n$-cubes agree for all $n\geq 2$. Our first new characterization in this paper is that stable derivators are precisely those derivators in which homotopy finite limits and homotopy finite colimits commute. (A category is “homotopy finite” if it is equivalent to a category which is finite, skeletal, and has no non-trivial endomorphisms, i.e., to a category whose nerve is a finite simplicial set.) Since Kan extensions in derivators are pointwise, these characterizations admit various improvements in terms of the commutativity of Kan extensions. This gives : The following are equivalent for a derivator . 1. The derivator is stable. 2. The derivator is pointed and the cone morphism $C\colon{\sD}^{[1]}\to{\sD}$ preserves fibers. (Here, ${\sD}^{[1]}$ denotes the derivator of morphisms in .) 3. Homotopy finite colimits and homotopy finite limits commute in . 4. Left homotopy finite left Kan extensions commute with arbitrary right Kan extensions in .\[item:il\] 5. Arbitrary left Kan extensions commute with right homotopy finite right Kan extensions in . \[item:ir\] Since the derivator of spectra is the stabilization of the derivator of spaces, these abstract characterizations of stability specialize to answers to the above question. **Answer \#1:** The homotopy theory of spectra is obtained from that of spaces if one forces homotopy finite limits and homotopy finite colimits to commute in a universal way. Characterizations \[item:il\] and \[item:ir\] in the above theorem suggest a natural generalization: if $\Phi$ is any class of functors between small categories, we define a derivator to be *left $\Phi$-stable* if left Kan extensions along functors in $\Phi$ commute with arbitrary right Kan extensions in , and dually *right $\Phi$-stable*. For instance, stable derivators are precisely the left $\mathsf{FIN}$-stable derivators and also the right $\mathsf{FIN}$-stable derivators, where $\mathsf{FIN}$ is the class of homotopy finite categories (more precisely, the class of the corresponding functors to the terminal category). But other interesting stability properties also arise in this way; for instance, pointed derivators are precisely the left or right $\{\emptyset\}$-stable ones (i.e. initial objects commute with right Kan extensions, or terminal objects commute with left Kan extensions). And semi-additive derivators are precisely the left or right $\mathsf{FINDISC}$-stable ones, where $\mathsf{FINDISC}$ is the class of finite discrete categories. In general, this notion of “relative stability” yields a Galois connection between collections of derivators and classes of functors. To understand relative stability better, we introduce *enriched* derivators and weighted colimits. These build on the theory of monoidal derivators developed in [@gps:additivity; @ps:linearity], extending the classical theory of enriched categories to the context of derivators. Just as every ordinary category is enriched over the category of sets, every derivator is enriched[^1] over the derivator of spaces; whereas pointed derivators are automatically enriched over pointed spaces, and stable ones over spectra. For any -enriched derivator we have a notion of limit or colimit weighted by “profunctors” in , which includes the ordinary homotopy Kan extensions that exist in any derivator. With the technology of enriched derivators, we can prove the following general characterization of relative stability (\[thm:stab-op\]): The following are equivalent for a derivator and a class $\Phi$ of functors. 1. is left $\Phi$-stable, i.e. left Kan extensions along functors in $\Phi$ commute with arbitrary right Kan extensions in . 2. is right $\Phi\op$-stable, i.e. right Kan extensions along functors in $\Phi\op$ commute with arbitrary left Kan extensions in . 3. Left Kan extension functors $u_! : {\sD}^A \to {\sD}^B$ for functors $u\in \Phi$ are right adjoint morphisms of derivators. 4. Right Kan extension functors $(u\op)_\ast : {\sD}^{A\op} \to {\sD}^{B\op}$ for functors $u\in \Phi$ are weighted *colimit* functors relative to some over which is enriched.\[item:ie\] This gives some additional conceptual explanations for why certain limits and colimits commute: if a colimit functor is a right adjoint, then of course it commutes with all limits; whereas if a limit functor can be identified with a (weighted) *colimit* functor, then of course it commutes with all other colimits. It also explains the left-right duality in the first theorem as due to the fact that the class $\mathsf{FIN}$ of finite categories is closed under taking opposites. Thus we can say: **Answer \#2:** The homotopy theory of spectra is obtained from that of spaces by universally forcing homotopy finite limits to be weighted *colimits*, and dually. There is one fly in the ointment: the “enrichment” in \[item:ie\] is rather weak: it has only tensors and not cotensors or hom-objects (so it is more properly called simply a “-module” rather than a “-enriched derivator”), and moreover is not itself a derivator, only a “left derivator” (having left homotopy Kan extensions but not right ones). This can be remedied by working with locally presentable $\infty$-categories rather than derivators, which we plan to do in [@gs:enriched]. However, this depends on rather more technical machinery, so it is interesting how much can be done purely in the realm of derivators. In [@gs:enriched] we will also show more, namely that given $\Phi$ there is a *universal* choice of ${\sV}$ in \[item:ie\], with pointed spaces and spectra being particular examples. The construction again depends on the good behavior of local presentability, so it seems unlikely to hold in general for derivators. However, as noted above, in particular cases such a universal derivator does exist, such as pointed spaces and spectra for the cases $\Phi=\{\emptyset\}$ and $\Phi=\mathsf{FIN}$ respectively. For $\Phi=\mathsf{FINDISC}$ we expect that the universal consists of $E_\infty$-spaces, though we have not proven this. This paper belongs to a project aiming for an abstract study of stability, and can be thought of as a sequel to [@groth:ptstab; @gps:mayer; @groth:can-can; @ps:linearity] and as a prequel to [@gs:enriched]. This abstract study of stability was developed in a different direction in the series of papers on abstract representation theory [@gst:basic; @gst:tree; @gst:Dynkin-A; @gst:acyclic] which will be continued in [@gst:acyclic-Serre]. The perspective from enriched derivator theory offers additional characterizations of stability, and these together with a more systematic study of the stabilization will appear in [@gs:enriched]. It is worth noting that in [@ps:linearity], what we here call “$\Phi$-stable monoidal derivators” are shown to admit a linearity formula for the traces and Euler characteristics of $\Phi$-colimits, so the abstract study of stability has computational as well as conceptual importance. The content of the paper is as follows. In §\[sec:char\] we characterize pointed and stable derivators by the commutativity of certain (co)limits or Kan extensions. In \[sec:galois\] we define the Galois correspondence of relative stability. In \[sec:enriched-derivators\] we define enriched derivators and weighted colimits, and in \[sec:stab-via-wcolim\] we use them to give the second class of characterizations of stability. Finally, in §\[sec:fun\] we study further the characterizations in terms of iterated adjoints to constant morphism morphisms. **Prerequisites.** We assume *basic* acquaintance with the language of derivators, which were introduced independently by Grothendieck [@grothendieck:derivators], Heller [@heller:htpythies], and Franke [@franke:adams]. Derivators were developed further by various mathematicians including Maltsiniotis [@maltsiniotis:seminar; @maltsiniotis:k-theory; @maltsiniotis:htpy-exact] and Cisinski [@cisinski:direct; @cisinski:loc-min; @cisinski:derived-kan] (see [@grothendieck:derivators] for many additional references). Here we stick to the notation and conventions from [@gps:mayer]. For a more detailed account of the basics we refer to [@groth:intro-to-der-1]. Stability and commuting (co)limits {#sec:char} ================================== In this section we obtain characterizations of pointed and stable derivators in terms of the commutativity of certain left and right Kan extensions. It turns out that a derivator is stable if and only if homotopy finite colimits and homotopy finite limits commute, and there are variants using suitable Kan extensions. We begin by collecting the following characterizations which already appeared in the literature. \[thm:stable-known\] The following are equivalent for a pointed derivator . 1. The adjunction $(\Sigma,\Omega)\colon{\sD}\rightleftarrows{\sD}$ is an equivalence. 2. The derivator is $\Sigma$-stable, i.e., a square in is a suspension square if and only if it is a loop square. 3. The adjunction $({\mathsf{cof}},{\mathsf{fib}})\colon{\sD}^{[1]}\rightleftarrows{\sD}^{[1]}$ is an equivalence. 4. The derivator is cofiber-stable, i.e., a square in is a cofiber square if and only if it is a fiber square. 5. The derivator is stable, i.e., a square in is cocartesian if and only if it is cartesian. 6. An $n$-cube in , $n\geq 2,$ is strongly cocartesian if and only if it is strongly cartesian. The equivalence of the first five statements is [@gps:mayer Thm. 7.1] and the equivalence of the remaining two is [@gst:tree Cor. 8.13]. As a preparation for a minor variant we include the following construction. In every pointed derivator there are canonical comparison maps $$\label{eq:sigma-f-c} \Sigma F\to C\colon{\sD}^{[1]}\to{\sD}\qquad\text{\and}\qquad F\to \Omega C\colon{\sD}^{[1]}\to{\sD}.$$ In fact, starting with a morphism $(f\colon x\to y)\in{\sD}^{[1]}$ we can pass to the coherent diagram encoding both the corresponding fiber and cofiber square, $$\xymatrix{ Ff\ar[r]\ar[d]\pullbackcorner&x\ar[d]^-f\ar[r]&0\ar[d]\\ 0\ar[r]&y\ar[r]&Cf.\pushoutcorner }$$ More formally, let $i\colon[1]\to\boxbar=[2]\times[1]$ classify the vertical morphism in the middle and let $$i\colon [1]\stackrel{i_1}{\to}A_1\stackrel{i_2}{\to}A_2\stackrel{i_3}{\to}A_3\stackrel{i_4}{\to}\boxbar$$ be the fully faithful inclusions which in turn add the objects $(2,0),(2,1),(0,1),$ and $(0,0)$. In every pointed derivator we can consider the corresponding Kan extension morphisms $${\sD}^{[1]}\stackrel{(i_1)_\ast}{\to}{\sD}^{A_1}\stackrel{(i_2)_!}{\to}{\sD}^{A_2}\stackrel{(i_3)_!}{\to}{\sD}^{A_3}\stackrel{(i_4)_\ast}{\to}{\sD}^\boxbar.$$ The first two functors add a cofiber square and homotopy (co)finality arguments (for example based on [@groth:ptstab Prop. 3.10]) show that the remaining two morphisms add the fiber square. Forming the composite square, we obtain a coherent square looking like $$\label{eq:F-C-square} \vcenter{ \xymatrix{ Ff\ar[r]\ar[d]&0\ar[d]\\ 0\ar[r]&Cf. } }$$ The canonical comparison maps result from considering suitable loop and suspension squares. \[prop:stable-known-mod\] The following are equivalent for a pointed derivator . 1. The derivator is stable.\[item:sk1\] 2. For every $f\in{\sD}^{[1]}$ the canonical comparison maps $\Sigma F\to C$ and $F\to \Omega C$ as in are isomorphisms.\[item:sk2\] 3. For every $f\in{\sD}^{[1]}$ the square is bicartesian.\[item:sk3\] If is a stable derivator, then the composition property of bicartesian squares [@groth:ptstab Prop. 3.13] implies that is bicartesian, and it follows from [@groth:can-can Prop. 2.16] that the canonical transformations $\Sigma F\to C$ and $F\to\Omega C$ are invertible. It remains to show that \[item:sk2\] implies \[item:sk1\], and we hence assume that $\Sigma F\toiso C$ is invertible. Associated to $x\in{\sD}$ there is by [@groth:ptstab Prop. 3.6] the morphism $1_!(x)=(0\to x)\in{\sD}^{[1]}$. The natural isomorphism $\Sigma F\toiso C$ evaluated at $1_!(x)$ yields a natural isomorphism $\Sigma\Omega x\toiso x$. Dually, we deduce $\id\toiso\Omega\Sigma$ and concludes the proof. While unrelated left Kan extensions always commute [@groth:can-can Cor. 4.3], it is, in general, not true that unrelated left and right Kan extensions commute. More specifically, given functors $u\colon A\to A'$ and $v\colon B\to B'$, recall that **left Kan extension along $u$ and right Kan extension along $v$ commute** in a derivator if the canonical mate $$\begin{aligned} (u\times\id)_!(\id\times v)_\ast&\stackrel{\eta}{\to} (u\times\id)_!(\id\times v)_\ast (u\times\id)^\ast(u\times\id)_!\\ &\toiso (u\times\id)_!(u\times\id)^\ast(\id\times v)_\ast (u\times\id)_!\\ & \stackrel{\varepsilon}{\to} (\id\times v)_\ast (u\times\id)_!\end{aligned}$$ is an isomorphism in . This is to say that the morphism $u_!\colon{\sD}^A\to{\sD}^{A'}$ preserves right Kan extensions along $v$ or that the morphism $v_\ast\colon{\sD}^B\to{\sD}^{B'}$ preserves left Kan extensions along $u$ [@groth:can-can Lem. 4.8]. For the purpose of a simpler terminology, we also say that $u_!$ and $v_\ast$ commute in . In general, these canonical mates are not invertible as is for example illustrated by the following characterization of pointed derivators. \[prop:ptd-comm\] The following are equivalent for a derivator . 1. The derivator is pointed.\[item:pc1\] 2. Empty colimits and empty limits commute in .\[item:pc2\] 3. Left Kan extensions along cosieves and right Kan extensions along sieves commute in .\[item:pc3\] Left Kan extensions along cosieves and arbitrary right Kan extensions commute in . \[item:pc4a\] Arbitrary left Kan extensions and right Kan extensions along sieves commute in .\[item:pc4b\] For the equivalence of the first two statements we consider the empty functor $\emptyset\colon\emptyset\to\bbone$. Correspondingly, for every derivator there is the canonical mate $$\xymatrix{ {\sD}^{\emptyset\times\emptyset}\ar[r]^-{(\id\times\emptyset)_\ast}\ar[d]_-{(\emptyset\times\id)_!}\drtwocell\omit{}& {\sD}^{\emptyset\times\bbone}\ar[d]^--{(\emptyset\times\id)_!}\\ {\sD}^{\bbone\times\emptyset}\ar[r]_--{(\id\times\emptyset)_\ast}&{\sD}^{\bbone\times\bbone} }$$ detecting if empty colimits and empty limits commute. By construction of initial and final objects in derivators (see [@groth:ptstab §1.1]), the source of this canonical mate is given by initial objects in while the target is given by final objects. Hence, is pointed if and only if empty colimits and empty limits commute in . Obviously, each of the statements \[item:pc4a\] or \[item:pc4b\] implies statement \[item:pc3\]. Moreover, since the empty functor is a sieve and a cosieve, statement \[item:pc3\] implies \[item:pc2\]. By duality, it remains to show that \[item:pc1\] implies \[item:pc4a\]. Given a functor $u\colon A\to B$, the morphism $u_\ast\colon{\sD}^A\to{\sD}^B$ is a right adjoint and, as a pointed morphism of pointed derivators, $u_\ast$ preserves left Kan extensions along cosieves [@groth:can-can Cor. 8.2]. We now turn to the stable context. Let us recall that a category $A\in\cCat$ is **strictly homotopy finite** if it is finite, skeletal, and it has no non-trivial endomorphisms (equivalently the nerve $NA$ is a finite simplicial set). A category is **homotopy finite** if it is equivalent to a strictly homotopy finite category. \[thm:stable-lim-I\] Homotopy finite colimits and homotopy finite limits commute in stable derivators. Let be a stable derivator and let $A\in\cCat$. Denoting by $\pi_A\colon A\to\bbone$ the unique functor, there are defining adjunctions $$(\colim_A,\pi_A^\ast)\colon{\sD}^A\rightleftarrows{\sD}\qquad\text{and}\qquad (\pi_A^\ast,\mathrm{lim}_A)\colon{\sD}\rightleftarrows{\sD}^A,$$ and these exhibit $\colim_A,\mathrm{lim}_A\colon{\sD}^A\to{\sD}$ as exact morphisms of stable derivators [@groth:can-can Cor. 9.9]. Hence, by [@ps:linearity Thm. 7.1], $\colim_A$ preserves homotopy finite limits and $\lim_A$ preserves homotopy finite colimits. For the converse to this theorem we collect the following lemma. \[lem:lim-comm\] Let be a derivator such that homotopy finite colimits and homotopy finite limits commute in . 1. The derivator is pointed. 2. The morphisms ${\mathsf{cof}}\colon{\sD}^{[1]}\to{\sD}^{[1]}$ and $C\colon{\sD}^{[1]}\to{\sD}$ preserve homotopy finite limits. 3. The morphism ${\mathsf{fib}}\colon{\sD}^{[1]}\to{\sD}^{[1]}$ and $F\colon{\sD}^{[1]}\to{\sD}$ preserve homotopy finite colimits. By assumption on , empty colimits and empty limits commute and this implies that is pointed (). Hence, by duality, it remains to take care of the second statement. Denoting by $i\colon[1]\to\ulcorner$ the sieve classifying the horizontal morphism $(0,0)\to (1,0)$ and by $k'\colon[1]\to\square$ the functor classifying the vertical morphism $(1,0)\to (1,1)$, the cofiber morphism is given by $$\label{eq:cof} {\mathsf{cof}}\colon{\sD}^{[1]}\stackrel{i_\ast}{\to}{\sD}^\ulcorner\stackrel{(i_\ulcorner)_!}{\to}{\sD}^\square\stackrel{(k')^\ast}{\to}{\sD}^{[1]}.$$ Since the morphisms $i_\ast$ and $(k')^\ast$ are right adjoints, they preserve arbitrary right Kan extensions, hence homotopy finite limits. By assumption on , [@groth:can-can Prop. 3.9], and [@groth:can-can Lem. 4.9], also the morphism $(i_\ulcorner)_!$ preserves homotopy finite limits, and hence so does ${\mathsf{cof}}$ by [@groth:can-can Prop. 5.2]. An additional composition with the continuous evaluation morphism $1^\ast\colon{\sD}^{[1]}\to{\sD}$ establishes the corresponding result for $C$. Given a pointed derivator , the derivator ${\sD}^\square={\sD}^{[1]\times[1]}$ admits cone and fiber morphisms in the first and the second coordinate, and these are respectively denoted by $$C_1, C_2\colon{\sD}^\square\to{\sD}^{[1]}\qquad\mbox{and}\qquad F_1,F_2\colon{\sD}^\square\to{\sD}^{[1]}.$$ Since these morphisms are pointed, for $X\in{\sD}^\square$ there is by [@groth:can-can Construction 9.7] a canonical comparison map $$\label{eq:cof-fib-comm} C(F_2 X)\to F(C_1X).$$ \[cor:lim-comm\] Let be a derivator in which homotopy finite colimits and homotopy finite limits commute. Then is pointed and the canonical transformations are isomorphisms for every $X\in{\sD}^\square$. This is immediate from . As we show next, this property already implies that the derivator is stable. Together with we thus obtain the following more conceptual characterization of stability. \[thm:stable-lim-II\] A derivator is stable if and only if homotopy finite colimits and homotopy finite limits commute. By it suffices to show that a derivator is stable as soon as homotopy finite colimits and homotopy finite limits commute in . Such a derivator is pointed and for every $X\in{\sD}^\square$ the canonical morphism $$\label{eq:stable-lim-II} C(F_2X)\toiso F(C_1X)$$ is an isomorphism (). For every $x\in{\sD}$ we consider the square $$X=X(x)=(i_\lrcorner)_!\pi_\lrcorner^\ast x\in{\sD}^\square.$$ The morphism $\pi_\lrcorner^\ast\colon{\sD}\to{\sD}^\lrcorner$ forms constant cospans. Since $i_\lrcorner\colon\lrcorner\to \square$ is a cosieve, $(i_\lrcorner)_!$ is left extension by zero [@groth:ptstab Prop. 3.6] and the diagram $X\in{\sD}^\square$ looks like $$\xymatrix{ 0\ar[r]\ar[d]&x\ar[d]^-\id\\ x\ar[r]_-\id&x. }$$ We calculate $CF_2(X)\cong C(\Omega x\to 0)\cong \Sigma\Omega x$ and $FC_1(X)\cong F(x\to 0)\cong x$, showing that the canonical isomorphism induces a natural isomorphism $\Sigma\Omega\toiso\id$. Using constant spans instead one also constructs a natural isomorphism $\id\toiso\Omega\Sigma$, showing that $\Sigma,\Omega\colon{\sD}\to{\sD}$ are equivalences. It follows from that is stable. It is now straightforward to obtain the following variant of this theorem. We recall from [@groth:can-can §9] that **left homotopy finite left Kan extensions** are left Kan extensions along functors $u\colon A\to B$ such that the slice categories $(u/b),$ $b\in B,$ admit a homotopy final functor $C_b\to(u/b)$ from a homotopy finite category $C_b$. The point of this notion is that right exact morphisms of derivators preserve left homotopy finite left Kan extensions [@groth:can-can Thm. 9.14]. \[thm:stable-lim-III\] The following are equivalent for a derivator . 1. The derivator is stable.\[item:sl1\] 2. Homotopy finite colimits and homotopy finite limits commute in .\[item:sl2\] Left homotopy finite left Kan extensions and arbitrary right Kan extensions commute in .\[item:sl3a\] Arbitrary left Kan extensions and right homotopy finite right Kan extensions commute in .\[item:sl3b\] Every left exact morphism ${\sD}^A\to{\sD}^B,A,B\in\cCat,$ preserves left homotopy finite left Kan extensions.\[item:sl4a\] Every right exact morphism ${\sD}^A\to{\sD}^B,A,B\in\cCat,$ preserves right homotopy finite right Kan extensions.\[item:sl4b\] 3. The derivator is pointed and $C\colon{\sD}^{[1]}\to{\sD}$ preserves right homotopy finite right Kan extensions.\[item:sl5\] 4. The derivator is pointed and $C\colon{\sD}^{[1]}\to{\sD}$ preserves homotopy finite limits.\[item:sl6\] 5. The derivator is pointed and $C\colon{\sD}^{[1]}\to{\sD}$ preserves $F$.\[item:sl7\] If is stable, then also the shifted derivators ${\sD}^A,A\in\cCat,$ are stable [@groth:ptstab Prop. 4.3]. Consequently, every left exact morphism ${\sD}^A\to{\sD}^B$ is also right exact [@groth:can-can Prop. 9.8] and it hence preserves left homotopy finite left Kan extensions [@groth:can-can Thm. 9.14]. This and a dual argument shows that statement \[item:sl1\] implies statements \[item:sl4a\] and \[item:sl4b\]. Since right Kan extension morphisms are right adjoint morphisms and hence left exact, the implications \[item:sl4a\] implies \[item:sl3a\] and \[item:sl3a\] implies \[item:sl2\] are immediate. Moreover, \[item:sl2\] implies \[item:sl1\] by , and, by duality, it remains to incorporate the three final statements. Statement \[item:sl1\] implies statement \[item:sl5\] since $C$ is left exact in this case and it hence preserves right homotopy finite right homotopy Kan extensions [@groth:can-can Thm. 9.14]. The implications \[item:sl5\] implies \[item:sl6\] and \[item:sl6\] implies \[item:sl7\] being trivial, it remains to show that \[item:sl7\] implies \[item:sl1\] which is already taken care of by the proof of . There are, of course, various additional minor variants of the characterizations in obtained, for example, by replacing $C$ by ${\mathsf{cof}}\colon{\sD}^{[1]}\to{\sD}^{[1]}$. \[rmk:interpretation\] A typical slogan is that spectra are obtained from pointed topological spaces if one forces the suspension to become an equivalence. This slogan is made precise by and the fact that the derivator of spectra is the stabilization of the derivator of pointed topological spaces [@heller:stable]. and make precise various additional slogans saying, for instance, that spectra are obtained from spaces or pointed spaces by forcing certain colimit and limit type constructions to commute. We illustrate this by two examples. 1. If one forces homotopy finite colimits and homotopy finite limits to commute in the derivator of spaces, then one obtains the derivator of spectra. 2. If one forces partial cones and partial fibers of squares to commute in the derivator of pointed spaces, then this yields the derivator of spectra. \[rmk:stable-rep-triv\] The phenomenon that certain colimits and limits commute is well-known from ordinary category theory. To mention an instance, let us recall that filtered colimits are exact in Grothendieck abelian categories, i.e., filtered colimits and finite limits commute in such categories. Additional such statements hold in locally presentable categories, Grothendieck topoi, and algebraic categories. Now, the phenomenon of stability is invisible to ordinary category theory; in fact, a represented derivator is stable if and only if the representing category is trivial (this follows from since the suspension morphism is trivial in pointed represented derivators). As a consequence the commutativity statements in have no counterparts in ordinary category theory. Stability versus absoluteness {#sec:galois} ============================= The close family resemblance between \[prop:ptd-comm,thm:stable-lim-III\] suggests the following definition. Let $\Phi$ be a class of functors between small categories. A derivator is **left $\Phi$-stable** if for every $(u\colon A\to B)\in\Phi$, left Kan extensions along $u$ in ${\sD}$ commute with arbitrary right Kan extensions. Dually, ${\sD}$ is **right $\Phi$-stable** if ${\sD}\op$ is left $\Phi$-stable, i.e. right Kan extensions along each $u\in \Phi$ in ${\sD}$ commute with arbitrary left Kan extensions. If ${\sD}$ is left (resp. right) $\Phi$-stable, we say that $\Phi$ is left (resp. right) **-absolute**. We may take $\Phi$ to be a class of categories instead of functors, in which case we identify a category $A$ with the unique functor $A\to\bbone$. In the next section we will show that left $\Phi$-stability coincides with right $\Phi\op$-stability. A derivator is pointed if and only if it is left $\emptyset$-stable, if and only if it is right $\emptyset$-stable, and if and only if it is left stable for the class of cosieves, if and only if it is right stable for the class of sieves. Similarly, is stable if and only if it is left stable for the class of homotopy finite categories, if and only if it is right stable for the same class, if and only if it is left stable for the class of left homotopy finite functors, if and only if it is right stable for the class of right homotopy finite functors. This notion of relatively stable derivators allows us to construct the following Galois correspondence. Given a class $\Phi$ of functors between small categories, we write ${\mathsf{Stab}_L}(\Phi)$ for the collection of left $\Phi$-stable derivators. Dually, given a collection $\Upsilon$ of derivators, we write ${\mathsf{Abs}_L}(\Upsilon)$ for the class of left $\Upsilon$-absolute functors (i.e. functors that are left -absolute for all ${\sD}\in\Upsilon$). Then ${\mathsf{Stab}_L}$ and ${\mathsf{Abs}_L}$ are a Galois correspondence (a contravariant adjunction of partial orders) between the classes of functors and collections of derivators. In particular, we have $$\Phi \subseteq {\mathsf{Abs}_L}(\Upsilon) \iff \Upsilon \subseteq {\mathsf{Stab}_L}(\Phi)$$ and $$\Phi \subseteq {\mathsf{Abs}_L}({\mathsf{Stab}_L}(\Phi)) \qquad \Upsilon \subseteq {\mathsf{Stab}_L}({\mathsf{Abs}_L}(\Upsilon))$$ $${\mathsf{Stab}_L}(\Phi) = {\mathsf{Stab}_L}({\mathsf{Abs}_L}({\mathsf{Stab}_L}(\Phi))) \qquad {\mathsf{Abs}_L}(\Upsilon) = {\mathsf{Abs}_L}({\mathsf{Stab}_L}({\mathsf{Abs}_L}(\Upsilon))).$$ Dually, we have ${\mathsf{Stab}_R}$ and ${\mathsf{Abs}_R}$. \[prop:ptd-comm\] can now be restated by saying that ${\mathsf{Stab}_L}(\{\emptyset\})$ and ${\mathsf{Stab}_R}(\{\emptyset\})$ are the collection $\mathsf{POINT}$ of pointed derivators, while ${\mathsf{Abs}_L}(\mathsf{POINT})$ contains all cosieves and ${\mathsf{Abs}_R}(\mathsf{POINT})$ contains all sieves. In particular, $\mathsf{POINT}$ is a fixed point of both Galois correspondences. Similarly, \[thm:stable-lim-III\] can be restated by saying that ${\mathsf{Stab}_L}(\mathsf{FIN})$ and ${\mathsf{Stab}_R}(\mathsf{FIN})$, for $\mathsf{FIN}$ the class of homotopy finite categories, are both the collection $\mathsf{STABLE}$ of stable derivators; while ${\mathsf{Abs}_L}(\mathsf{STABLE})$ contains all left homotopy finite functors and ${\mathsf{Abs}_R}(\mathsf{STABLE})$ contains all right homotopy finite functors. The cone functor $C\colon{\sD}^{[1]} \to {\sD}$ is not a colimit (though it is a weighted colimit, in the sense to be defined in \[con:wcolim\], for a suitable enrichment), so we cannot consider “${\mathsf{Stab}_L}(\{C\})$”. However, if the pushout functor ${\sD}^{\ulcorner} \to {\sD}$ is continuous, then so is $C$, since $C$ is the composite of a pushout, a right Kan extension, and an evaluation morphism. Thus, we can say that $\mathsf{STABLE} = {\mathsf{Stab}_L}(\{\emptyset,\ulcorner\})$ and similarly $\mathsf{STABLE} = {\mathsf{Stab}_R}(\{\emptyset,\lrcorner\})$. Of course, ${\mathsf{Stab}_L}(\emptyset)$ and ${\mathsf{Stab}_R}(\emptyset)$ are the collection $\mathsf{DERIV}$ of all derivators, while ${\mathsf{Abs}_L}(\emptyset)$ and ${\mathsf{Abs}_R}(\emptyset)$ are the class $\mathsf{FUNC}$ of all functors. However, ${\mathsf{Abs}_L}(\mathsf{DERIV})$ and ${\mathsf{Abs}_R}(\mathsf{DERIV})$ are nonempty; for instance, ${\mathsf{Abs}_L}(\mathsf{DERIV})$ contains all left adjoint functors, ${\mathsf{Abs}_R}(\mathsf{DERIV})$ all right adjoint functors, and they both include the splitting of idempotents. On the other hand, ${\mathsf{Stab}_L}(\mathsf{FUNC})$ and ${\mathsf{Stab}_R}(\mathsf{FUNC})$ include only the trivial derivator, by [@ps:linearity Remark 9.4]. Let $\Phi=\mathsf{FINDISC}$ be the class of finite discrete categories. Since $\emptyset\in\mathsf{FINDISC}$, any left or right $\Phi$-stable derivator is pointed. It is easy to see that ${\mathsf{Stab}_L}(\mathsf{FINDISC}) = {\mathsf{Stab}_L}(\{\emptyset,2\})$, where $2$ denotes the discrete category with two objects, and similarly for ${\mathsf{Stab}_R}$. In fact, we have ${\mathsf{Stab}_L}(\mathsf{FINDISC}) = {\mathsf{Stab}_R}(\mathsf{FINDISC}) = \mathsf{SEMIADD}$, the collection of semiadditive derivators. For since ${\sD}^2 \simeq{\sD}\times{\sD}$ by one of the derivator axioms, the left and right Kan extensions along $2\to \bbone$ are just binary coproducts and products. Then if ${\sD}$ is pointed and binary coproducts preserve all limits, then in particular they preserve binary products, which means that $$(X\times Z) + (Y\times W) \cong (X+Y)\times (Z+W)$$ canonically. Taking $Y=Z=0$, we see that $X+W \cong X\times W$ canonically, so that ${\sD}$ is semiadditive. Conversely, if ${\sD}$ is semiadditive, then the coproduct and product functors ${\sD}\times {\sD}\to{\sD}$ coincide, and in particular the coproduct is a right adjoint and so preserves all limits. Thus ${\sD}$ is left $\mathsf{FINDISC}$-stable if and only if it is semiadditive, and dually for right $\mathsf{FINDISC}$-stability. There are a number of natural questions suggested by this phrasing of the characterization theorems: 1. By definition, ${\sD}$ is left $u$-stable if and only if $u_!\colon {\sD}^A\to {\sD}^B$ is continuous. But a continuous functor is crying out to be a right adjoint, for instance if there is an adjoint functor theorem. General derivators have no adjoint functor theorem, but does $u_!$ happen to be a right adjoint anyway? 2. \[prop:ptd-comm,thm:stable-lim-III\] are self-dual, and in particular $\mathsf{POINT}$ and $\mathsf{STABLE}$ are fixed points of both Galois connections. Is there an abstract explanation for this? 3. We have seen that interesting collections of derivators like $\mathsf{POINT}$ and $\mathsf{STAB}$ can be generated as ${\mathsf{Stab}_L}(\Phi)$ for very small classes $\Phi$ of functors such as $\{\emptyset\}$ and $\{\emptyset,\ulcorner\}$. Can they also be generated as ${\mathsf{Stab}_L}({\mathsf{Abs}_L}(\Upsilon))$ for “manageable” collections $\Upsilon$ of derivators? For instance, are there “universal” pointed or stable derivators that suffice to detect whether a given functor is absolute for all pointed or stable derivators? To attack these questions, we use the technology of enriched derivators and weighted limits. We will see that it suffices to answer the first two questions positively, but it is not quite adequate for the third in general, although in particular cases the answer is yes. In [@gs:enriched] we will use a better technology to answer the third question positively in general as well. Enriched derivators {#sec:enriched-derivators} =================== We begin by defining the basic notions of enriched derivators. We freely make use of the language and techniques established in [@gps:additivity], in particular the language of *monoidal derivators* as it is developed in detail in [@gps:additivity §3]. In that paper there is also a detailed discussion of *two-variable adjunctions of derivators* [@gps:additivity §§8-9]. A monoidal derivator is a pseudo-monoid object in $\cDER$ (the 2-category of derivators and pseudonatural transformations) such that the monoidal structure $\otimes\colon{\sV}\times{\sV}\to{\sV}$ preserves colimits separately in both variables. The pseudo-monoid structure precisely amounts to a lift of ${\sV}\colon\cCat\op\to\cCAT$ against the forgetful functor from the $2$-category of monoidal categories, strong monoidal functors, and monoidal transformations. The resulting monoidal structures are denoted by $({\sV}(A),\otimes_A,\lS_A)$. We will also have occasion to consider the following weaker notions. A **left derivator** is a prederivator satisfying all the axioms of a derivator except the existence of right Kan extensions. A morphism of left derivators is again a pseudonatural transformation, giving a 2-category . We can define two-variable morphisms of left derivators, and (separate) preservation of colimits, just as for derivators. A **monoidal left derivator** is a left derivator with a pseudo-monoid structure that preserves colimits separately in both variables. If is a monoidal left derivator, a **-module** is a cocontinuous pseudo-module, i.e. a left derivator with an action ${\sV}\times{\sD}\to {\sD}$ that is coherently associative and unital and preserves colimits separately in both variables. We say that is a **-opmodule** if ${\sD}\op$ is a -module. A **closed -module**, or **-enriched derivator**, is a -module whose action is part of a two-variable adjunction (hence, in particular, it is also a -opmodule). Now recall that derivator morphisms of two variables come in three different forms; see [@gps:additivity §3 and §5]. We right away specialize to the situation of an action as above. 1. In the *internal form* $\otimes_A\colon{\sV}(A)\times{\sD}(A)\to{\sD}(A)$ which naively is given by $(W\otimes_A X)_a=W_a\otimes X_a$, where $\otimes\colon{\sV}(\bbone)\times{\sD}(\bbone)\to{\sD}(\bbone)$ denotes the underlying functor of two variables. 2. In the *external form* $\otimes\colon{\sV}(A)\times{\sD}(B)\to{\sD}(A\times B)$, which we think of as being defined by $(W\otimes X)_{a,b}=W_a\otimes X_b$. 3. Finally, in the *canceling form* $\otimes_{[A]}\colon{\sV}(A\op)\times{\sD}(A)\to{\sD}(\bbone)$ which is obtained from the external form by composing it with the coend functor $$\int^A\colon{\sD}(A\op\times A)\to{\sD}(\bbone).$$ For the notion of (co)ends in derivators we refer to [@gps:additivity §5 and Appendix A]. Note the different notation used for these three variants; the notation for internal versions was already used for the monoidal categories $({\sV}(A),\otimes_A,\lS_A)$. Every monoidal left derivator is, of course, a module over itself. If it is a closed module over itself, we call it a **closed monoidal left derivator**. More generally, if is a monoidal left derivator, then any shift ${\sV}^A$ is also a -module. We also have the following universal construction: For any left derivators ${\sD},{\sE}$, define ${\mathsf{HOM}}({\sD},{\sE})$ by $${\mathsf{HOM}}({\sD},{\sE})(A) = \cDER({\sD},{\sE}^A)$$ where a functor $u:A\to B$ induces the restriction functor $${\mathsf{HOM}}({\sD},{\sE})(B) = \cDER({\sD},{\sE}^B) \to \cDER({\sD},{\sE}^A) = {\mathsf{HOM}}({\sD},{\sE})(A)$$ by postcomposition with $u^* \colon {\sE}^B \to {\sE}^A$. This makes ${\mathsf{HOM}}({\sD},{\sE})$ into a left derivator, and indeed a derivator if is one; its Kan extension functors are also simply given by postcomposition. In this way becomes a cartesian closed 2-category in an appropriate weak sense. In particular, ${\mathsf{HOM}}({\sD},{\sD})$ is a pseudo-monoid under composition, and there is a canonical action ${\mathsf{HOM}}({\sD},{\sD}) \times {\sD}\to {\sD}$. However, this monoidal structure and action do not preserve colimits in the right variable, hence do not make into a ${\mathsf{HOM}}({\sD},{\sD})$-module. Thus, we define a new left derivator ${\mathsf{HOM}\ccsub}({\sD},{\sE})$, for which ${\mathsf{HOM}\ccsub}({\sD},{\sE})(A)$ is the category of *cocontinuous* morphisms ${\sD}\to{\sE}^A$. Since restriction and left Kan extension are cocontinuous morphisms, this is again a left derivator. The endomorphism object ${\mathsf{HOM}\ccsub}({\sD},{\sD})$, which we denote ${\mathsf{END}\ccsub}({\sD})$, *is* a monoidal left derivator under composition, and its action ${\mathsf{END}\ccsub}({\sD})\times{\sD}\to{\sD}$ does make into an ${\mathsf{END}\ccsub}({\sD})$-module. Explicitly, the external monoidal product of $F\colon{\sD}\to{\sD}^A$ and $G\colon{\sD}\to{\sD}^B$ is the morphism $GF\colon{\sD}\to{\sD}^{A\times B}$ whose component ${\sD}(C) \to {\sD}(C\times A\times B)$ is the composite ${\sD}(C) \xto{F^C} {\sD}(C\times A) \xto{G^{C\times A}} {\sD}(C\times A\times B)$. Similarly, the external action of $F\colon{\sD}\to{\sD}^A$ on $X\in {\sD}(B)$ is the image of $X$ under $F^B \colon{\sD}(B) \to {\sD}(B\times A)$. Both of these preserve colimits in both variables, on the left because colimits there are defined by postcomposition, and on the right because $F$ and $G$ preserve colimits. This construction is universal in the sense that if is a monoidal left derivator, then to make into a -module is equivalent to giving a cocontinuous monoidal morphism ${\sV}\to{\mathsf{END}\ccsub}({\sD})$. Specifically, the latter assigns to each $X\in {\sV}(A)$ a morphism ${\sD}\to {\sD}^A$, which is the external tensor product with $X$. Monoidality of the morphism ${\sV}\to{\mathsf{END}\ccsub}({\sD})$ gives the associativity and unitality of the action, while its cocontinuity gives left cocontinuity of the action; right cocontinuity of the action comes from the fact that this morphism lands in ${\mathsf{END}\ccsub}({\sD}) = {\mathsf{HOM}\ccsub}({\sD},{\sD})$ rather than ${\mathsf{HOM}}({\sD},{\sD})$. Note that unlike ${\mathsf{HOM}}({\sD},{\sE})$, the left derivator ${\mathsf{HOM}\ccsub}({\sD},{\sE})$ is not a derivator even if is: since limits and colimits do not in general commute, the limit in ${\mathsf{HOM}}({\sD},{\sE})$ of cocontinuous morphisms need no longer be cocontinuous. However, we can say; \[thm:ldh-ran\] If $u\colon A\to B$ is such that has right Kan extensions along $u$ that commute with arbitrary left Kan extensions, then so does ${\mathsf{HOM}\ccsub}({\sD},{\sE})$. Right Kan extensions in ${\mathsf{HOM}}({\sD},{\sE})$ are defined by postcomposition; if $u_*$ is cocontinuous then ${\mathsf{HOM}\ccsub}({\sD},{\sE})$ is closed under such postcomposition. Since left Kan extensions are also defined by postcomposition, commutativity follows. We now introduce the notion of weighted colimits. First note that the internal, external, and canceling versions of morphisms of two-variables can be combined. In particular, given a monoidal derivator and $A,B,C\in\cCat$, there is the **(homotopy) tensor product of functors** $$\otimes_{[B]}\colon {\sV}(A\times B\op)\times{\sV}(B\times C\op)\stackrel{\otimes}{\to}{\sV}(A\times B\op\times B\times C\op)\stackrel{\int^B}{\to}{\sV}(A\times C\op),$$ and also this operation enjoys associativity and unitality properties. \[thm:bicategory\] If is a monoidal left derivator, then there is a bicategory $\cProf({\sV})$ described as follows: - Its objects are small categories. - Its hom-category from $A$ to $B$ is ${\sV}(A\times B\op)$. - Its composition functors are the external-canceling tensor products $$\otimes_{[B]} \colon {\sV}(A\times B\op) \times {\sV}(B\times C\op) \too {\sV}(A\times C\op).$$ - The identity 1-cell of a small category $B$ is $$\lI_B\;=\;(t,s)_! \lS_{{\ensuremath{\operatorname{tw}}}(B)} \;\cong\; (t,s)_! \pi_{{\ensuremath{\operatorname{tw}}}(B)}^* \lS_{\bbone} \; \in {\sV}(B\times B\op).\label{eq:unit}$$ The notation related to the identity $1$-cells $\lI_B\in{\sV}(B\times B\op)$, also called **identity profunctors**, is as follows. ${\ensuremath{\operatorname{tw}}}(B)$ is the **twisted arrow category** of $B$, i.e., the category of elements of $\hom_B$, and the functor $(t,s)\colon{\ensuremath{\operatorname{tw}}}(B)\to B\times B\op$ sends a morphism to its target and source (see [@gps:additivity §5]). We refer to $\cProf({\sV})$ as the **bicategory of profunctors** in . \[con:wcolim\] Let be a monoidal left derivator and let be a -module with tensors $\otimes\colon{\sV}\times{\sD}\to{\sD}$. The external-canceling version of this morphism yields functors $$\otimes_{[B]}\colon{\sV}(A\times B\op)\times {\sD}(B\times C\op)\to{\sD}(A\times C\op).$$ Passing to parametrized versions of these functors, we obtain an external-canceling tensor morphism $$\otimes_{[B]}\colon{\sV}^{A\times B\op}\times{\sD}^{B\times C\op}\to{\sD}^{A\times C\op}.$$ In particular, plugging in a fixed $W\in{\sV}(A\times B\op)$ and specializing to $C=\bbone$, we obtain an induced partial morphism $$\colim^W=(W\otimes_{[B]}-)\colon{\sD}^B\to{\sD}^A,$$ the **weighted colimit morphism with weight** $W\in{\sV}(A\times B\op)$. We abuse terminology and refer to a morphism as a weighted colimit if it is naturally isomorphic to $\colim^W$ for some $W$. In a dual way, if ${\sD}$ is a -opmodule, one defines **weighted limits** $$\mathrm{lim}^W=(-\lhd_{[A]}W)\colon{\sD}^A\to{\sD}^B,$$ Moreover, if is a closed -module, then weighted colimits and weighted limits are always adjoint to each other: $$\label{eq:wcolim-adj} (\colim^W,\mathrm{lim}^W)\colon{\sD}^B\rightleftarrows{\sD}^A.$$ \[lem:wcolim\] Let be a monoidal left derivator and let be a -module. 1. The morphism $\otimes_{[B]}\colon{\sV}^{A\times B\op}\times{\sD}^{B\times C\op}\to{\sD}^{A\times C\op}$ preserves colimits in each variable separately. In particular, any weighted colimit functor is cocontinuous.\[item:wc1\] 2. If is a monoidal derivator, and is a derivator and a closed -module, then $\otimes_{[B]}$ is a left adjoint of two variables.\[item:wc2\] 3. The morphism $(\lI_B\otimes_{[B]}-)\colon{\sD}^B\to{\sD}^B$ is naturally isomorphic to the identity morphism.\[item:wc3\] Statements \[item:wc1\] and \[item:wc2\] are true for the external-canceling variant of any cocontinuous two-variable morphism, while \[item:wc3\] follows from the same argument used to prove unitality of the bicategory $\cProf({\sV})$. \[thm:wcolim\] Let be a monoidal left derivator and let be a -module. Then: 1. Restriction morphisms $u^\ast\colon{\sD}^B\to{\sD}^A$ are -weighted colimits.\[item:wcl1\] 2. Left Kan extension morphisms $u_!\colon{\sD}^A\to{\sD}^B$ are -weighted colimits.\[item:wcl2\] 3. If and are pointed derivators, then right Kan extension morphisms $u_\ast\colon{\sD}^A\to{\sD}^B$ along sieves are weighted colimits.\[item:wcl3\] 4. If and are stable derivators, then right homotopy finite right Kan extension morphisms $u_\ast\colon{\sD}^A\to{\sD}^B$ are weighted colimits.\[item:wcl4\] For every fixed $X\in{\sD}(B)$ and $u\colon A\to B$, pseudo-naturality of the partial morphism $(-\otimes_{[B]}X)\colon{\sV}^{B\op}\to {\sD}$ and yields $$u^\ast(X)\cong u^\ast(\lI_B\otimes_{[B]}X)\cong\big((u\times\id)^\ast\lI_B\big)\otimes_{[B]}X.$$ This defines a natural isomorphism $u^\ast\cong \big((u\times\id)^\ast\lI_B\big)\otimes_{[B]}-\colon{\sD}^B\to{\sD}^A$, thereby exhibiting $u^\ast$ as a weighted colimit. Similarly, if we fix $X\in{\sD}(A)$, then by the partial morphism $$\label{eq:par-mor-wcolim} (-\otimes_{[A]}X)\colon{\sV}^{A\op}\to{\sD}$$ is cocontinuous. Given a functor $u\colon A\to B$ we obtain natural isomorphisms $$u_!(X)\cong u_!(\lI_A\otimes_{[A]}X)\cong \big((u\times\id)_!\lI_A\big)\otimes_{[A]}X,$$ hence a natural isomorphism $u_!\cong\big(\big((u\times\id)_!\lI_A\big)\otimes_{[A]}-\big)\colon{\sD}^A\to{\sD}^B$, identifying $u_!$ as a weighted colimit. If and are pointed, then is a cocontinuous morphism of pointed derivators and hence automatically pointed, hence preserves right Kan extensions along sieves [@groth:can-can Cor. 8.2]. Thus, a similar calculation as above yields for every such $u\colon A\to B$ a natural isomorphism $$u_\ast\cong \big(\big((u\times\id)_\ast\lI_A\big)\otimes_{[A]}-\big) \colon{\sD}^A\to{\sD}^B,$$ exhibiting $u_\ast$ as a weighted colimit. Similarly, if and are stable derivators, we note that is an exact morphism of stable derivators (by and [@groth:can-can Cor. 9.9]) and it hence preserves right homotopy finite right Kan extensions [@groth:can-can Thm. 9.14]. Applying \[thm:wcolim\] to the -module ${\sV}^{C\op}$, we find that for any $X\in {\sV}(B\times C\op)$ and $Y\in {\sV}(A\times C\op)$ we have $$\begin{aligned} \big((u\times\id)^\ast\lI_B\big) \otimes_{[B]} X &\cong (u\times\id)^\ast X\\ \big((u\times\id)_!\lI_A\big) \otimes_{[A]} Y &\cong (u\times\id)_! Y\end{aligned}$$ Note that in this case, $\otimes_{[B]}$ and $\otimes_{[A]}$ are the composition in $\cProf({\sV})$; thus restriction and left Kan extension in can both be described using composition in $\cProf({\sV})$. The special objects $(u\times\id)^\ast\lI_B$ and $(u\times\id)_!\lI_A$ are sometimes called **base change objects**. Dually, for any $X\in {\sV}(E\times B\op)$ and $Y\in {\sV}(E\times A\op)$ we have $$\begin{aligned} X \otimes_{[B]} \big((\id\times u\op)^\ast\lI_B\big) &\cong (\id\times u\op)^\ast X\\ Y \otimes_{[A]} \big((\id\times u\op)_!\lI_A\big) &\cong (\id\times u\op)_! Y\end{aligned}$$ In fact, these dual base change objects are actually isomorphic to the first two swapped: $$\begin{aligned} (\id\times u\op)^\ast\lI_B &\cong (u\times\id)_!\lI_A\\ (\id\times u\op)_!\lI_A &\cong (u\times\id)^\ast\lI_B\end{aligned}$$ This all follows from the fact that $\cProf({\sV})$ is actually a “framed bicategory”; see [@shulman:frbi] and [@ps:linearity (15.2)]. Let be a monoidal left derivator and a -module. For $u\colon A\to B$ in $\cCat$ we obtain an isomorphism $$u_!\cong ((\id\times u\op)^\ast\lI_B)\otimes_{[A]}-\colon{\sD}^A\to{\sD}^B.$$ Specializing to $u=\pi_A\colon A\to\bbone$ we deduce that colimits are weighted colimits with constant weight $\pi_{A\op}^\ast\lS_\bbone$. More generally, the weight for $u_!$ has components $$((\id\times u\op)^\ast\lI_B)_{b,a}\cong\coprod_{\hom_B(ua,b)}\lS_\bbone,$$ and the isomorphism $u_!X\cong((\id\times u\op)^\ast\lI_B)\otimes_{[A]}X$ is hence a left derivator version of the usual coend formula for left Kan extensions in sufficiently cocomplete categories ([@maclane Thm. X.4.1]). Stability via weighted colimits {#sec:stab-via-wcolim} =============================== \[thm:wcolim\]\[item:wcl3\] and \[item:wcl4\] cry out for a generalization to $\Phi$-stability. If $\Phi$ is a class of functors $u\colon A\to B$, we define a left derivator to be **right $\Phi$-stable** if it *has* right Kan extensions along each $u\in \Phi$ which moreover commute with arbitrary left Kan extensions. By \[thm:ldh-ran\], if ${\sD}$ is right $\Phi$-stable, then so is ${\mathsf{END}\ccsub}({\sD})$. \[thm:stable-dual\] Let be a monoidal left derivator and $u\colon A\to B$ a functor. The following are equivalent: 1. ${\sV}$ is right $u\op$-stable.\[item:sd0\] 2. The base change profunctor $(u\times\id)_!\lI_A\in \cProf({\sV})(B,A)$ has a right adjoint in $\cProf({\sV})$.\[item:sd1\] 3. The base change profunctor $(\id\times u)_!\lI_{A\op}\in \cProf({\sV})(A\op,B\op)$ has a left adjoint in $\cProf({\sV})$.\[item:sd1op\] 4. The morphism $u_!\colon{\sV}^A\to{\sV}^B$ has a left adjoint that is a weighted colimit functor.\[item:sd2\] 5. The right Kan extension $(u\op)_\ast\colon {\sV}^{A\op} \to {\sV}^{B\op}$ exists and is a -weighted *colimit* functor.\[item:sd2op\] We first show that \[item:sd1\] and \[item:sd1op\] are equivalent. The right adjoint in \[item:sd1\] would be an object $Z\in {\sV}(A\times B\op)$, whereas the left adjoint in \[item:sd1op\] would be an object $Z'\in {\sV}(B\op\times (A\op)\op)$; but of course these are equivalent categories. The unit and counit in \[item:sd1\] would be morphisms $$\begin{aligned} \eta &: \lI_B \to (u\times\id)_!\lI_A \otimes_{[A]} Z \cong (u\times\id)_! Z\\ \ep &: (\id\times u\op)^\ast Z \cong Z \otimes_{[B]} (u\times\id)_!\lI_A \to \lI_A \end{aligned}$$ whereas the unit and counit in \[item:sd1op\] would be morphisms $$\begin{aligned} \eta' &: \lI_{B\op} \to Z'\otimes_{[A\op]} (\id\times u)_!\lI_{A\op} \cong (\id\times u)_! Z'\\ \ep' &: (u\op\times \id)^\ast Z' \cong (\id\times u)_!\lI_{A\op} \otimes_{[B\op]} Z' \to \lI_{A\op}. \end{aligned}$$ Thus, to give $\eta$ is the same as to give $\eta'$, since $\lI_{B\op}$ corresponds to $\lI_B$ under the equivalence ${\sV}(B\times B\op) \simeq {\sV}(B\op \times (B\op)\op)$, and so on. We leave it to the reader to check that the triangle identities likewise correspond. Now we show that \[item:sd0\] implies \[item:sd1\]. We take the right adjoint to be $(\id\times u\op)_\ast \lI_A \in \cProf({\sV})(A,B)$. Then morphisms $Y\to (\id\times u\op)_\ast \lI_A$ are equivalent to morphisms $(\id\times u\op)^\ast Y \to \lI_A$, i.e. morphisms $Y\otimes_{[B]} (u\times\id)_!\lI_A \to \lI_A$. In bicategorical language, $(\id\times u\op)_\ast \lI_A$ is a *right lifting* of $\lI_A$ along $(u\times\id)_!\lI_A$. In general, a right lifting of the identity along a 1-cell $X$ is a right adjoint as soon as it is preserved by precomposition with $X$ (see for instance [@maclane Theorem X.7.2] or [@maysig:pht 16.4.12]). In our case when $X = (u\times\id)_!\lI_A$, precomposition with $X$ is just left Kan extension along $u$, which by our assumption of $u\op$-stability preserves the right Kan extension $(\id\times u\op)_\ast$. Thus, $(u\times\id)_!\lI_A \otimes_{[A]} (\id\times u\op)_\ast \lI_A \cong (\id\times u\op)_\ast \big((u\times\id)_!\lI_A\big)$, so it has an analogous universal property, as desired. Now, if \[item:sd1\] holds, then since weighted colimits are contravariantly functorial on profunctors, the adjunction $(u\times\id)_!\lI_A \adj Z$ yields an adjunction $\colim^Z \adj \colim^{(u\times\id)_!\lI_A} = u_!$. This gives \[item:sd2\]. Conversely, if $Z\in{\sV}(A\times B\op)$ is such that $\colim^Z \adj u_! = \colim^{(u\times\id)_!\lI_A}$, then since composition in $\cProf({\sV})$ is a special case of weighted colimits, we have natural adjunctions $(Z\otimes_{[B]} -) \adj ((u\times\id)_!\lI_A \otimes_{[A]} -)$, which by the bicategorical Yoneda lemma induce an adjunction $(u\times\id)_!\lI_A\adj Z$ in $\cProf({\sV})$. Similarly, \[item:sd2op\] is equivalent to \[item:sd1op\], since $\colim^{(\id\times u)_!\lI_{A\op}} \cong (u\op)^\ast$. Finally, if \[item:sd2op\] holds then $(u\op)_\ast$, being a weighted colimit, commutes with all left Kan extensions, so that ${\sV}$ is right $u\op$-stable. Note that \[thm:stable-dual\]\[item:sd2op\] is a generalization of \[thm:wcolim\]\[item:wcl3\] and \[item:wcl4\]. This can be regarded as an explanation of “why” $\Phi$-limits in a right $\Phi$-stable derivator commute with all colimits: they are themselves weighted colimits. (If is not symmetric, then arbitrary weighted colimits need not commute with arbitrary other *weighted* colimits. However, left Kan extensions always commute with all weighted colimits, by \[lem:wcolim\]\[item:wc1\]. If we express left Kan extensions as weighted colimits themselves, then they are in the “center” of . If *is* symmetric, then the duality $A\mapsto A\op$ extends to a self-duality of the bicategory $\cProf({\sV})$, from which the equivalence of \[item:sd1\] and \[item:sd1op\] follows formally; the proof given above shows that this equivalence remains true even in the non-symmetric case, due to this “centrality”.) Now we can answer our first two questions from \[sec:galois\]. \[thm:stab-op\] For a derivator ${\sD}$ and a class of functors $\Phi$, the following are equivalent. 1. ${\sD}$ is left $\Phi$-stable, i.e. $\Phi$-colimits in commute with arbitrary limits.\[item:so1\] 2. For each $u\in\Phi$, the morphism $u_! :{\sD}^A \to{\sD}^B$ has a left adjoint.\[item:so2\] 3. ${\sD}$ is right $\Phi\op$-stable, i.e. $\Phi\op$-limits in commute with arbitrary colimits.\[item:so3\] 4. For each $u\in\Phi\op$, the morphism $(u\op)_\ast :{\sD}^{A\op} \to{\sD}^{B\op}$ has a right adjoint.\[item:so4\] We have \[item:so2\] implies \[item:so1\], since all right Kan extensions exist in a derivator (as opposed to a left derivator), and are preserved by any right adjoint morphism. Dually, \[item:so4\] implies \[item:so3\]. We will prove that \[item:so3\] implies \[item:so2\]; by duality then also \[item:so1\] implies \[item:so4\] and we are done. If ${\sD}$ is right $\Phi\op$-stable, then we remarked above that ${\mathsf{END}\ccsub}({\sD})$ is right $\Phi\op$-stable, and ${\sD}$ is a ${\mathsf{END}\ccsub}({\sD})$-module. Therefore, by \[thm:stable-dual\], $u_!$ has a left adjoint (that is even a weighted colimit functor) for each $u\in\Phi$. If $\Phi=\Phi\op$, then ${\mathsf{Stab}_L}(\Phi)={\mathsf{Stab}_R}(\Phi)$. This explains the self-dual nature of pointedness, semiadditivity, and stability as due to the fact that $\Phi=\{\emptyset\}$, $\Phi=\mathsf{FINDISC}$, and $\Phi=\mathsf{FIN}$ are self-dual. Similarly, it explains the identity ${\mathsf{Stab}_L}(\{\emptyset,\ulcorner\}) = {\mathsf{Stab}_R}(\{\emptyset,\lrcorner\}) = \mathsf{STABLE}$, since $(\ulcorner)\op = \lrcorner$. Stability via iterated adjoints {#sec:fun} =============================== In particular, \[thm:stab-op\] implies that we can characterize $\Phi$-stability in terms of iterated adjoints to constant morphism morphisms. In this section we describe what this looks like more concretely in the pointed and stable cases. \[prop:char-ptd\] The following are equivalent for a derivator . 1. The derivator is pointed.\[item:p1\] 2. The morphism $\emptyset_!\colon{\sD}^\emptyset\to{\sD}$ is a right adjoint.\[item:p2\] 3. The left Kan extension morphism $1_!\colon{\sD}\to{\sD}^{[1]}$ along the universal cosieve $1\colon\bbone\to[1]$ is a right adjoint.\[item:p3\] 4. For every cosieve $u\colon A\to B$ the left Kan extension morphism $u_!\colon{\sD}^A\to{\sD}^B$ is a right adjoint.\[item:p4\] 5. The morphism $\emptyset_\ast\colon{\sD}^\emptyset\to{\sD}$ is a left adjoint.\[item:p5\] 6. The right Kan extension morphism $0_\ast\colon{\sD}\to{\sD}^{[1]}$ along the universal sieve $0\colon\bbone\to[1]$ is a left adjoint.\[item:p6\] 7. For every sieve $u\colon A\to B$ the right Kan extension morphism $u_\ast\colon{\sD}^A\to{\sD}^B$ is a left adjoint.\[item:p7\] By duality it suffices to show the equivalence of the first four statements. The implication \[item:p1\] implies \[item:p4\] is [@groth:ptstab Cor. 3.8]. Since the empty functor $\emptyset\colon\emptyset\to\bbone$ is a cosieve it remains to show that \[item:p2\] or \[item:p3\] imply \[item:p1\]. The case of \[item:p2\] is taken care of by the proof of [@groth:ptstab Cor. 3.5]. In the remaining case, if $1_!$ is a right adjoint it preserves all limits and hence terminal objects. Since the terminal object in ${\sD}([1])$ looks like $(\ast\to\ast)$, this has by [@groth:ptstab Prop. 1.23] to be isomorphic to $1_!(\ast)\cong(\emptyset\to\ast)$. Evaluating this isomorphism at $0$ shows that is pointed. These additional adjoint functors are sometimes referred to as **(co)exceptional inverse image functors** (see [@groth:ptstab §3]). \[rmk:C-inverse\] In [@groth:ptstab] the cone $C\colon{\sD}^{[1]}\to{\sD}$ and the fiber $F\colon{\sD}^{[1]}\to{\sD}$ is defined in pointed derivators only, but the same formulas make perfectly well sense in arbitrary derivators. It turns out that a derivator is pointed if and only if $C$ is a left adjoint if and only if $F$ is a right adjoint. In that case, there are adjunctions $C\dashv 1_!$ and $0_\ast\dashv F$, exhibiting $C$ and $F$ as (co)exceptional inverse image functors; see [@groth:ptstab Prop. 3.22]. In \[thm:stable-fun\] we will characterize stable derivators with a simliar list of conditions, essentially by combining \[thm:stable-lim-III,thm:stab-op\]. We could similarly have proven \[prop:char-ptd\] by combining \[prop:ptd-comm,thm:stab-op\], but we chose instead to give a proof with a closer connection to previous literature. Let be a derivator and let $1\colon\bbone\to[1]$ classify the terminal object $1\in[1]$. In every derivator there are Kan extension adjunctions $(1_!,1^\ast)\colon{\sD}\rightleftarrows{\sD}^{[1]}$ and $(1^\ast,1_\ast)\colon{\sD}\rightleftarrows{\sD}^{[1]}$, and we hence have an adjoint triple $$1_!\dashv 1^\ast\dashv 1_\ast.$$ Similarly, associated to the functor $0\colon\bbone\to[1]$ there is the adjoint triple $$0_!\dashv 0^\ast\dashv 0_\ast.$$ \[prop:univ-sieve\] Let be a derivator and let $0,1\colon\bbone\to[1]$ classify the objects $0,1\in[1]$. 1. The morphisms $0_!,1_\ast\colon{\sD}\to{\sD}^{[1]}$ are fully faithful and induce an equivalence onto the full subderivator spanned by the isomorphisms. This equivalence is pseudo-natural with respect to arbitrary morphisms of derivators. 2. There are canonical isomorphism $0_!\cong \pi_{[1]}^\ast\cong 1_\ast\colon{\sD}\to{\sD}^{[1]}$. Both morphisms $0_!$ and $1_\ast$ are fully faithful and the essential image consists precisely of the isomorphisms by [@groth:ptstab Prop. 3.12]. Since derivators are invariant under equivalences of prederivators, the subprederivator of isomorphisms is a derivator. The equivalence is pseudo-natural with respect to arbitrary morphisms since all morphisms preserve left Kan extensions along left adjoint functors (see [@groth:can-can Prop. 5.7] and [@groth:can-can Rmk. 6.11]). As for the second statement, there is an adjoint triple $0\dashv\pi_{[1]}\dashv 1$ and hence an induced adjoint triple $1^\ast\dashv \pi_{[1]}^\ast\dashv 0^\ast$. This yields canonical isomorphisms $1_\ast\cong\pi_{[1]}^\ast$ and $0_!\cong\pi_{[1]}^\ast$. We refer to $\pi_{[1]}^\ast\colon{\sD}\to{\sD}^{[1]}$ as the **constant morphism morphism**. In every derivator there is an adjoint $5$-tuple $$\label{eq:5tuple} 1_!\dashv 1^\ast\dashv \pi_{[1]}^\ast\dashv 0^\ast\dashv 0_\ast.$$ This is immediate from . A derivator is pointed if and only if the adjoint $5$-tuple extends to an adjoint $7$-tuple, which is then given by $$\label{eq:7tuple} C\dashv 1_!\dashv 1^\ast\dashv \pi_{[1]}^\ast\dashv 0^\ast\dashv 0_\ast\dashv F.$$ This is immediate from and . While in any pointed derivator there is by [@groth:ptstab Prop. 3.20] an adjunction $$({\mathsf{cof}},{\mathsf{fib}})\colon{\sD}^{[1]}\rightleftarrows{\sD}^{[1]},$$ in pointed derivators the morphism $C$ is the sixth left adjoint of $F$. \[thm:stable-fun\] The following are equivalent for a pointed derivator ${\sD}$. 1. The derivator ${\sD}$ is stable.\[item:sf1\] 2. The cone morphism $C\colon{\sD}^{[1]}\to{\sD}$ is a right adjoint.\[item:sf2\] 3. For any homotopy finite category $A$, the colimit morphism $\colim : {\sD}^A \to {\sD}$ is a right adjoint.\[item:sf3\] 4. For any left homotopy finite functor $u:A\to B$, the left Kan extension morphism $u_!: {\sD}^A \to {\sD}^B$ is a right adjoint.\[item:sf4\] 5. The fiber morphism $F\colon{\sD}^{[1]}\to{\sD}$ is a left adjoint.\[item:sf2a\] 6. For any homotopy finite category $A$, the limit morphism $\lim : {\sD}^A \to {\sD}$ is a left adjoint.\[item:sf3a\] 7. For any right homotopy finite functor $u:A\to B$, the right Kan extension morphism $u_*: {\sD}^A \to {\sD}^B$ is a left adjoint.\[item:sf4a\] 8. The adjoint $7$-tuple extends to a doubly-infinite chain of adjoint morphisms.\[item:sf5\] Combining \[thm:stable-lim-III,thm:stab-op\], we see that \[item:sf1\] implies \[item:sf4\], which clearly implies \[item:sf3\], while \[item:sf3\] implies \[item:sf2\] since the cone is a composite of a right extension by zero with a pushout. And \[item:sf2\] implies \[item:sf1\] by \[thm:stable-lim-III\]\[item:sl6\], since right adjoints preserve all limits, so the first four statements are equivalent. The equivalence of \[item:sf1\] with \[item:sf2a\], \[item:sf3a\], and \[item:sf4a\] is dual. Evidently \[item:sf5\] implies \[item:sf2\]. And conversely, if is a stable derivator, then by there are natural isomorphisms $$\Sigma F\toiso C\qquad\text{and}\qquad F\toiso\Omega C.$$ Since $\Sigma$ and $\Omega$ are equivalences in stable derivators (), this shows that the outer morphisms in the adjoint $7$-tuple match up to an equivalence. This implies that the adjoint $7$-tuple can be extended to a doubly-infinite chain of adjoint morphisms and that this chain has period six (in the obvious sense). We conclude by offering a first interpretation and visualization of this chain of morphisms. Let be a stable derivator. Then a few additional adjoint morphisms in the doubly-infinite sequence extending are given by: $$\ldots\dashv\pi^\ast\Omega\dashv \Sigma 0^\ast\dashv 0_\ast\Omega\dashv C\dashv 1_!\dashv 1^\ast\dashv \pi^\ast\dashv 0^\ast\dashv 0_\ast\dashv F\dashv 1_!\Sigma\dashv \Omega 1^\ast\dashv \pi^\ast\Sigma\dashv\ldots$$ In fact, this is immediate from the proof of . In order to not get lost in all these morphisms, let us recall that Barratt–Puppe sequences in a stable derivator can be thought of as refinements of the more classical distinguished triangles. More precisely, associated to $(f\colon x\to y)\in{\sD}^{[1]}$ there is the Barratt–Puppe sequence $BP(f)$ generated by $f$. This is a coherent diagram as in which vanishes on the boundary stripes and which makes all squares bicartesian. $$\vcenter{ \xymatrix@-1pc{ \ar@{}[dr]|{\ddots}&\ar@{}[dr]|{\ddots}&\ar@{}[dr]|{\ddots}&&&&&&\\ \ar@{}[dr]|{\ddots}&\Omega Ff\ar[r]\ar[d]\pullbackcorner&\Omega x\ar[r]\ar[d]\pullbackcorner&0\ar[d]&&&&&\\ &0\ar[r]&\Omega y\ar[r]\ar[d]\pushoutcorner\pullbackcorner&Ff\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&&\\ &&0\ar[r]&x\ar[r]^-f\ar[d]\pushoutcorner\pullbackcorner&y\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&\\ &&&0\ar[r]&Cf\ar[r]\ar[d]\pushoutcorner\pullbackcorner&\Sigma x\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]\ar@{}[rd]|{\ddots}&&\\ &&&&0\ar[r]\ar@{}[dr]|{\ddots}&\Sigma y\ar[r]\ar@{}[rd]|{\ddots}\pushoutcorner&\Sigma Cf\ar@{}[dr]|{\ddots}\pushoutcorner&\\ &&&&&&&& } }$$ (It turns out that $BP$ defines an equivalence of derivators (see [@gst:Dynkin-A Thm. 4.5]).) Now, one half of the morphisms in the doubly-infinite chain simply amount to traveling in the Barratt–Puppe sequence in . If we imagine to sit on the morphism $f$ in $BP(f)$, then for every $n\geq 1$ an application of the $(2n\text{-}1)$-th left adjoint of $\pi^\ast$ to $f$ amounts to traveling $n$ steps in the positive direction. For low values this yields $y,Cf,\Sigma x,\Sigma y,$ and so on. There is a similar interpretation of the iterated right adjoints to $\pi^\ast$. In order to obtain a similar visualization of the remaining adjoints, let us consider the Barratt–Puppe sequence $BP(\pi_{[1]}^\ast x), x\in{\sD}$, of a constant morphism which then looks like . $$\vcenter{ \xymatrix@-1pc{ \ar@{}[dr]|{\ddots}&\ar@{}[dr]|{\ddots}&\ar@{}[dr]|{\ddots}&&&&&&\\ \ar@{}[dr]|{\ddots}&0\ar[r]\ar[d]\pullbackcorner&\Omega x\ar[r]\ar[d]\pullbackcorner&0\ar[d]&&&&&\\ &0\ar[r]&\Omega x\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&&\\ &&0\ar[r]&x\ar[r]^-\id\ar[d]\pushoutcorner\pullbackcorner&x\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]&&&\\ &&&0\ar[r]&0\ar[r]\ar[d]\pushoutcorner\pullbackcorner&\Sigma x\ar[r]\ar[d]\pushoutcorner\pullbackcorner&0\ar[d]\ar@{}[rd]|{\ddots}&&\\ &&&&0\ar[r]\ar@{}[dr]|{\ddots}&\Sigma x\ar[r]\ar@{}[rd]|{\ddots}\pushoutcorner&0\ar@{}[dr]|{\ddots}\pushoutcorner&\\ &&&&&&&& } }$$ While $\pi^\ast$ points at the constant morphism in the middle of , for every $n$ the remaining $2n$-th adjoints to $\pi^\ast$ classify suitable iterated rotations of this morphism. [^1]: In a weak sense; see below.

[**On more general forms of proportional fractional operators**]{} .20in Fahd Jarad$^{a}$, Manar A. Alqudah$^{b}$, Thabet Abdeljawad$^{c,d}$\ $^{a}$Department of Mathematics, Çankaya University, 06790 Ankara, Turkey\ email: fahd@cankaya.edu.tr\ $^{b}$ Department of Mathematical Sciences, Princess Nourah Bint Abdulrahman University\ P.O. Box 84428, Riyadh 11671, Saudi Arabia.\ email:maalqudah@pnu.edu.sa.\ $^{c}$Department of Mathematics and Physical Sciences, Prince Sultan University\ P. O. Box 66833, 11586 Riyadh, Saudi Arabia\ email:tabdeljawad@psu.edu.sa\ $^d$ Department of Medical Research, China Medical University, 40402, Taichung, Taiwan .2in Introduction ============ The fractional calculus, which is engaged in integral and differential operators of arbitrary orders, is as old as the conceptional calculus that deals with integrals and derivatives of non-negative integer orders. Since not all of the real phenomena can be modeled using the operators in the traditional calculus, researchers searched for generalizations of these operators. It turned out that the fractional operators are excellent tools to use in modeling long-memory processes and many phenomena that appear in physics, chemistry, electricity, mechanics and many other disciplines. Here, we invite the readers to read [@podlubny; @Samko; @f1; @f222; @f2; @f3] and the reference cited in these books. However, for the sake of better understanding and modeling real world problems, researchers were in need of other types of fractional operators that were confined to Riemann-Liouville fractional operators. In the literature, one can find many works that propose new fractional operators. We mention [@had; @Kat1; @Kat2; @fahd3; @fahd1; @fahd11]. Nonetheless, the fractional integrals and derivatives which were proposed in these works were just particular cases of what so called fractional integrals/derivatives of a function with respect to another function [@Samko; @f2; @fahd10]. There are other types of fractional operators which were suggested in the literature. On the other hand, due to the singularities found in the traditional fractional operators which are thought to make some difficulties in the modeling process, some researches recently proposed new types of non-singular fractional operators. Some of these operators contain exponential kernels and some of them involve the Mittag-Leffler functions. For such types of fractional operators we refer to [@FCaputo; @Losada; @TD; @ROMP; @Abdon; @TD; @JNSA]. All the fractional operators considered in the references in the first and the second paragraphs are non-local. However, there are many local operators found in the literature that allow differentiation to a non-integer order and these are called local fractional operators. In [@kh], Khalil et al. introduced the so called conformable (fractional) derivative. The author in [@T11] presented other basic concepts of conformable derivatives. We would like to mention that the fractional operators proposed in [@Kat1; @Kat2] are the non-local fractional version of the local operators suggested in [@kh]. In addition, the non-local fractional version of the ones in [@T11] can be seen in [@fahd11]. It is customary that any derivative of order 0 when performed to a function should give the function itself. This essential property is dispossessed by the conformable derivatives. Notwithstanding, in [@Anderson1; @Anderson2], the authors introduced a newly defined local derivative that tend to the original function as the order tends to zero and hence improved the conformable derivatives. In addition to this, the non-local fractional operators that emerge from iterating the above-mentioned derivative were held forth in [@fahd12]. Motivated by the above mentioned background, we extend the work done in [@fahd12] introduce a new generalized fractional calculus based on the proportional derivatives of a function with respect to another function in paralel with the definition discussed in [@Anderson1]. The kernel obtained in the fractional operators which will be proposed contains an exponential function and is function dependent. The semi–group properties will be discussed. The article is organized as follows: Section 2 presents some essential definitions for fractional derivatives and integrals. In Section 3 we present the general forms of the fractional proportional integrals and derivatives. In section 4, we present the general form of Caputo fractional proportional derivatives. In the end, we conclude our results. Preliminaries ============= In this section, we present some essential definitions of some fractional derivatives and integrals. We first present the traditional fractional operators and then the fractional proportional operators. The conventional fractional operators and their general forms ------------------------------------------------------------- For $\alpha \in \mathbb{C},~Re(\alpha)>0$, the left Riemann–Liouville fractional integral of order $\alpha $ has the f form $$\label{001} (_{a}I^\alpha f)(x)=\frac{1}{\Gamma(\alpha)}\int_a^x (x-u)^{\alpha-1}f(u)du.$$ The right Riemann–Liouville fractional integral of order $\alpha >0$ is defined by $$\label{002} (I_b^\alpha f)(x)=\frac{1}{\Gamma(\alpha)}\int_x^b (u-x)^{\alpha-1}f(u)du.$$ The left Riemann–Liouville fractional derivative of order $\alpha, Re(\alpha)\geq 0 $ is given as $$\label{003} (_{a}D^\alpha f)(x)=\Big(\frac{d}{dx}\Big)^n(_{a}I^{n-\alpha} f)(x),~~n=[\alpha]+1.$$ The right Riemann–Liouville fractional derivative of order $\alpha, Re(\alpha)\geq 0 $ reads $$\label{004} (D_b^\alpha f)(t)=\Big(-\frac{d}{dt}\Big)^n(I_b^{n-\alpha} f)(t).$$ The left Caputo fractional derivative has the following form $$\label{005} (_{a}^{C}D^\alpha f)(x)=\big(_{a}I^{n-\alpha} f^{(n)}\big)(x),~~n=[\alpha]+1.$$ The right Caputo fractional derivative becomes $$\label{006} (^CD_b^\alpha f)(x)=\big(I_b^{n-\alpha}(-1)^nf^{(n)}\big)(x).$$ The generalized left and right fractional integrals in the sense of Katugampola [@Kat1] are given respectively as $$\label{015} (_{a}\textbf{I}^{\alpha,\rho} f)(x)=\frac{1}{\Gamma(\alpha)}\int_a^x(\frac{x^\rho-u^\rho}{\rho})^{\alpha-1} f(u)\frac{du}{u^{1-\rho}}$$ and $$\label{016} (\textbf{I}_{b}^{\alpha,\rho}f)(x)=\frac{1}{\Gamma(\alpha)}\int_t^b (\frac{u^\rho- x^\rho}{\rho})^{\alpha-1} f(u)\frac{du}{u^{1-\rho}}.$$ The generalized left and right fractional derivatives in the sense of Katugampola [@Kat2] are defined respectively as $$\begin{aligned} \label{017}\nonumber (_{a}\textbf{D}^{\alpha,\rho} f)(x)&=&\gamma^n(_{a}\textbf{I}^{n-\alpha,\rho} f)(t)\\&=&\frac{\gamma^n}{\Gamma(n-\alpha)}\int_a^x(\frac{x^\rho-u^\rho}{\rho})^{n-\alpha-1} f(u)\frac{du}{u^{1-\rho}}\end{aligned}$$ and $$\begin{aligned} \label{018}\nonumber (\textbf{D}_{b}^{\alpha,\rho} f)(x)&=& (-\gamma)^n(\textbf{I}_b^{n-\alpha,\rho} f)(x)\\ &=&\frac{(-\gamma)^n}{\Gamma(n-\alpha)}\int_x^b(\frac{u^\rho-x^\rho}{\rho})^{n-\alpha-1} f(u)\frac{du}{u^{1-\rho}}, \end{aligned}$$ where $\rho>0$ and $\gamma=x^{1-\rho}\frac{d}{dx}$. The Caputo modification of the left and right generalized fractional derivatives in the sense of Jarad et al. [@fahd3] are presented respectively as $$\begin{aligned} \label{019}\nonumber (_{a}^C\textbf{D}^{\alpha,\rho} f)(x)&=&(_{a}\textbf{I}^{n-\alpha,\rho}\gamma^n f)(x)\\&=&\frac{1}{\Gamma(n-\alpha)}\int_a^x(\frac{x^\rho-u^\rho}{\rho})^{n-\alpha-1}\gamma^n f(u)\frac{du}{u^{1-\rho}},\end{aligned}$$ and $$\begin{aligned} \label{020}\nonumber (^C\textbf{D}_{b}^{\alpha,\rho} f)(x)&=& (_{a}\textbf{I}^{n-\alpha,\rho}(-\gamma)^n f)(x)\\ &=&\frac{1}{\Gamma(n-\alpha)}\int_x^b(\frac{u^\rho-x^\rho}{\rho})^{n-\alpha-1} (-\gamma)^nf(u)\frac{du}{u^{1-\rho}}.\end{aligned}$$ For $\alpha \in \mathbb{C},~Re(\alpha)>0$ the left Riemann-Liouville fractional integral of order $\alpha $ of $f$ with respect to a continuously differentiable and increasing function $g$ has the following form [@Samko; @f2] $$\label{3} ~_{a}I^{\alpha,g} f(x)=\frac{1}{\Gamma(\alpha)}\int_{a}^x \Big(g(x)-g(u)\Big)^{\alpha-1}f(u)g'(u)du.$$ For $\alpha \in \mathbb{C},~Re(\alpha)>0$ the right Riemann-Liouville fractional integral of order $\alpha $ of $f$ with respect to a continuously differentiable and increasing function $g$ has the following form [@Samko; @f2] $$\label{333} I_b^{\alpha,g} f(x)=\frac{1}{\Gamma(\alpha)}\int_{x}^b \Big(g(u)-g(x)\Big)^{\alpha-1}f(u)g'(u)du.$$ For $\alpha \in \mathbb{C},~Re(\alpha)\geq 0$, the generalized left and right Riemann-Liouville fractional derivative of order $\alpha $ of $f$ with respect to a continuously differentiable and increasing function $g$ have respectively the form [@Samko; @f3] $$\begin{aligned} \label{4}\nonumber ~_{a}D^{\alpha,g} f(x)&=&\Big(\frac{1}{g'(x)}\frac{d}{dx}\Big)^n(~_{a}I^{n-\alpha,g} f)(x)\\&=&\frac{\Big(\frac{1}{g'(x)}\frac{d}{dx}\Big)^n}{\Gamma(n-\alpha)}\int_{a}^x \Big(g(x)-g(u)\Big)^{n-\alpha-1}f(u)g'(u)du \end{aligned}$$ and $$\begin{aligned} \label{444}\nonumber D_b^{\alpha,g} f(x)&=&\Big(-\frac{1}{g'(x)}\frac{d}{dx}\Big)^n(I_b^{n-\alpha,g} f)(x)\\&=&\frac{\Big(-\frac{1}{g'(x)}\frac{d}{dx}\Big)^n}{\Gamma(n-\alpha)}\int_{a}^x \Big(g(x)-g(u)\Big)^{n-\alpha-1}f(u)g'(u)du, \end{aligned}$$ where $n=[\alpha]+1$. It is easy to observe that if we choose $g(x)=x$, the integrals in (\[3\]) and (\[333\]) becomes the left and right Riemann-Liouville fractional integrals respectively and (\[4\]) and (\[444\]) becomes the left and right Riemann-Liouville fractional derivatives. When $g(x)=\ln x$, the Hadamard fractional operators are obtained [@Samko; @f2]. While if one considers $g(x)=\frac{x^\rho}{\rho}$, the fractional operators in the settings of Katugampola [@Kat1; @Kat2] are derived. In left and right generalized Caputo derivatives of a function with respect to another function are presented respectively as [@fahd10] $$\label{555} ~_{a}^CD^{\alpha,g} f(x)=\Big(~_aI^{n-\alpha,g} f^{[n]}\Big)(x)$$ and $$\label{666} ^CD_b^{\alpha,g} f(x)=\Big(~_aI^{n-\alpha,g} (-1)^nf^{[n]}\Big)(x),$$ where $\displaystyle f^{[n]}(x)=\Big(\frac{1}{g'(x)}\frac{d}{dx}\Big)^nf(x)$. The proportional derivatives and their fractional integrals and derivatives --------------------------------------------------------------------------- The conformable derivative was first introduced by Khalil et al. in [@kh] and then explored by the current author in [@T11]. In his distinctive paper [@Anderson1], Anderson et al. modified the conformable derivative by using the proportional derivative. Indeed, he gave the following definition. \[D1\] *(Modified conformable derivatives)* For $\rho \in [0,1]$, let the functions $\kappa_0, \kappa_1:[0,1]\times \mathbb{R}\rightarrow [0,\infty)$ be continuous such that for all $t \in \mathbb{R}$ we have $$\lim_{\rho\rightarrow 0^+}\kappa_1(\rho,t)=1,~\lim_{\rho\rightarrow 0^+}\kappa_0(\rho,t)=0, \lim_{\rho\rightarrow 1^-}\kappa_1(\rho,t)=0,~\lim_{\rho\rightarrow 1^-}\kappa_0(\rho,t)=1,$$ and $\kappa_1(\rho,t)\neq 0,~~\rho \in [0,1),~~\kappa_0(\rho,t)\neq 0,~~\rho \in (0,1]$. Then, the modified conformable differential operator of order $\rho$ is defined by $$\label{anndy} D^\rho f(t)=\kappa_1(\rho,t) f(t)+\kappa_0(\rho,t) f^\prime(t).$$ The derivative given in (\[anndy\]) is called a proportional derivative. For more details about the control theory of the proportional derivatives and its component functions $\kappa_0$ and $\kappa_1$, we refer the reader to [@Anderson1; @Anderson2]. Of special interest, we shall restrict ourselves to the case when $\kappa_1(\rho,t)=1-\rho$ and $\kappa_0(\rho,t)=\rho$. Therefore, (\[anndy\]) becomes $$\label{prop derivative} D^\rho f(t)=(1-\rho) f(t)+\rho f^\prime(t).$$ Notice that $\lim_{\rho \rightarrow 0^+}D^\rho f(t)= f(t)$ and $\lim_{\rho \rightarrow 1^-}D^\rho f(t)= f^\prime(t)$. It is clear that the derivative (\[prop derivative\]) is somehow more general than the conformable derivative which does not tend to the original function as $\rho$ tends to $0$. The associated fractional proportional integrals are defined as [@fahd12] \[left and right integrals\]For $\rho>0$ and $\alpha \in \mathbb{C},~~Re(\alpha)>0$, the left fractional proportional integral of $f$ reads $$\label{LPI} (_{a}I^{\alpha,\rho} f)(x)= \frac{1}{\rho^\alpha \Gamma(\alpha)}\int_a^x e^{\frac{\rho-1}{\rho}(x-\tau)} (x-\tau)^{\alpha-1} f(\tau)d\tau$$ and the right one reads $$\label{RPI} (I_b^{\alpha,\rho} f)(x)= \frac{1}{\rho^\alpha \Gamma(\alpha)}\int_x^b e^{\frac{\rho-1}{\rho}(\tau-x)} (\tau-x)^{\alpha-1} f(\tau)d\tau.$$ \[Prop fractional derivatives\][@fahd12] For $\rho>0$ and $\alpha \in \mathbb{C},~~Re(\alpha)\geq 0$, the left fractional proportional derivative is defined as $$\begin{aligned} \label{LPD}\nonumber (_{a}D^{\alpha,\rho}f)(x)&=& D^{n,\rho} ~_{a}I^{n-\alpha,\rho} f(x)\\&=&\frac{D_x^{n,\rho}}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_a^x e^{\frac{\rho-1}{\rho}(x-\tau)}(x-\tau)^{n-\alpha-1} f(\tau)d \tau.\end{aligned}$$ The right proportional fractional derivative is defined by [@fahd12] $$\begin{aligned} \label{RPD}\nonumber (D_b^{\alpha,\rho}f)(x)&=& ~_{\ominus}D^{n,\rho} I_b^{n-\alpha,\rho} f(x)\\&=&\frac{~_{\ominus}D^{n,\rho}}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_x^b e^{\frac{\rho-1}{\rho}(\tau-x)}(\tau-x)^{n-\alpha-1} f(\tau)d \tau, \end{aligned}$$ where $n=[Re(\alpha)]+1$ and $\displaystyle (_{\ominus}D^\rho f)(t)=(1-\rho)f(t)-\rho f^\prime(t)$. Lastly, the left and right fractional proportional derivatives in the Caputo settings respectively read [@fahd12] $$\begin{aligned} \label{LPDC}\nonumber (_{a}^CD^{\alpha,\rho}f)(x)&=& \Big(~_{a}I^{n-\alpha,\rho}D^{n,\rho}f\Big)(x)\\\nonumber &=&\frac{1}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_a^x e^{\frac{\rho-1}{\rho}(x-\tau)}(x-\tau)^{n-\alpha-1} (D^{n,\rho}f)(\tau)d \tau \\\end{aligned}$$ and $$\begin{aligned} \label{RPDC}\nonumber (^CD_b^{\alpha,\rho}f)(x)&=& \Big( I_b^{n-\alpha,\rho}{~_\ominus}D^{n,\rho}f\Big)(x)\\\nonumber &=&\frac{1}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_x^b e^{\frac{\rho-1}{\rho}(\tau-x)}(\tau-x)^{n-\alpha-1} (~_{\ominus}D^{n,\rho}f)(\tau)d\tau.\\ \end{aligned}$$ The fractional proportional derivative of a function with respect to another function ===================================================================================== \[D2\] *(The proportional derivative of a function with respect to anothor function)*\ For $\rho \in [0,1]$, let the functions $\kappa_0, \kappa_1:[0,1]\times \mathbb{R}\rightarrow [0,\infty)$ be continuous such that for all $t \in \mathbb{R}$ we have $$\lim_{\rho\rightarrow 0^+}\kappa_1(\rho,t)=1,~\lim_{\rho\rightarrow 0^+}\kappa_0(\rho,t)=0, \lim_{\rho\rightarrow 1^-}\kappa_1(\rho,t)=0,~\lim_{\rho\rightarrow 1^-}\kappa_0(\rho,t)=1,$$ and $\kappa_1(\rho,t)\neq 0,~~\rho \in [0,1),~~\kappa_0(\rho,t)\neq 0,~~\rho \in (0,1]$. Let also $g(t)$ be a strictly increasing continuous function. Then, the proportional differential operator of order $\rho$ of $f$ with respect to $g$ is defined by $$\label{eq1} D^{\rho,g} f(t)=\kappa_1(\rho,t) f(t)+\kappa_0(\rho,t)\frac{f^\prime(t)}{g'(t)}.$$ we shall restrict ourselves to the case when $\kappa_1(\rho,t)=1-\rho$ and $\kappa_0(\rho,t)=\rho$. Therefore, (\[eq1\]) becomes $$\label{eq2} D^{\rho,g} f(t)=(1-\rho) f(t)+\rho \frac{f^\prime(t)}{g'(t)}.$$ The corresponding integral of $$\label{eq3} _{a}I^{1,\rho,g}f(t)=\frac{1}{\rho}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(s))}f(s)g'(s)ds,$$ where we accept that $~_{a}I^{0,\rho}f(t)=f(t)$. To produce a generalized type fractional integral depending on the proportional derivative, we proceed by induction through changing the order of integrals to show that $$\begin{aligned} \label{eq4} \nonumber (_{a}I^{n,\rho,g} f)(t) &=& \frac{1}{\rho}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau_1))} g'(\tau_1)d \tau_1 \frac{1}{\rho} \int_a^{\tau_1} e^{\frac{\rho-1}{\rho}(g(\tau_1)-g(\tau_2))}g'(\tau_2)d \tau_2\cdot\cdot\cdot\\\nonumber&\cdot\cdot\cdot&\frac{1}{\rho} \int_a^{\tau_{n-1}} e^{\frac{\rho-1}{\rho}(g(\tau_{n-1})-g(\tau_n))}f(\tau_n)g'(\tau_n) d\tau_n\\ &=& \frac{1}{\rho^n \Gamma(n)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} (g(t)-g(\tau))^{n-1} f(\tau)g'(\tau)d\tau. \end{aligned}$$ Based on (\[eq4\]), we can present the following general proportional fractional integral. \[general left and right integrals\]For $\rho \in (0,1]$, $\alpha \in \mathbb{C},~~Re(\alpha)>0$, we define the left fractional integral of $f$ with respect to $g$ by $$\label{eq5} (_{a}I^{\alpha,\rho,g} f)(t)= \frac{1}{\rho^\alpha \Gamma(\alpha)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} (g(t)-g(\tau))^{\alpha-1} f(\tau)g'(\tau)d\tau.$$ The right fractional proportional integral ending at $b$ can be defined by $$\label{eq6} (I_b^{\alpha,\rho,g} f)(t)= \frac{1}{\rho^\alpha \Gamma(\alpha)}\int_t^b e^{\frac{\rho-1}{\rho}(g(\tau)-g(t))} (g(\tau)-g(t))^{\alpha-1} f(\tau)g'(\tau)d\tau.$$ To deal with the right proportional fractional case we shall use the notation $$(_{\ominus}D^{\rho,g} f)(t):=(1-\rho)f(t)-\rho \frac{f^\prime(t)}{g'(t)}.$$ We shall also write $$(_{\ominus}D^{n,\rho,g} f)(t)= (\underbrace{_{\ominus}D^{\rho,g}~ _{\ominus}D^{\rho,g}\ldots~_{\ominus}D^{\rho,g}}_{\texttt{n times}} f)(t).$$ \[general left and right derivatives\] For $\rho>0$, $\alpha \in \mathbb{C},~~Re(\alpha)\geq 0$ and $g\in C[a,b]$, where $g'(t)>0$, we define the general left fractional derivative of $f$ with respect to $g$ as $$\begin{aligned} \label{eq7}\nonumber (_{a}D^{\alpha,\rho,g}f)(t)&=& D^{n,\rho,g} ~_{a}I^{n-\alpha,\rho,g} f(t)\\\nonumber &=&\frac{D_t^{n,\rho,g}}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))}(g(t)-g(\tau))^{n-\alpha-1} f(\tau)g'(\tau)d \tau\\ \end{aligned}$$ and the general right fractional derivative of $f$ with respect to $g$ as $$\begin{aligned} \label{eq8}\nonumber (D_b^{\alpha,\rho,g}f)(t)&=& ~_{\ominus}D^{n,\rho,g} I_b^{n-\alpha,\rho,g} f(t)\\\nonumber &=&\frac{~_{\ominus}D_t^{n,\rho,g}}{\rho^{n-\alpha}\Gamma(n-\alpha)} \int_t^b e^{\frac{\rho-1}{\rho}(g(\tau)-g(t))}(g(\tau)-g(t))^{n-\alpha-1} f(\tau)g'(\tau)d \tau,\\ \end{aligned}$$ where $n=[Re(\alpha)]+1$. \[reduction\] Clearly, if we let $\rho=1$ in Definition \[general left and right integrals\] and Definition \[general left and right derivatives\], we obtain the - the Riemann-Liouville fractional operators , , and if $g(t)=t$. - the fractional operators in the Katugampola setting, , and if $\displaystyle g(t)=\frac{t^{\mu}}{\mu}$. - The Hadamard fractional operators if $g(t)=\ln t$ [@Samko; @f2]. - The fractional operators mentioned in [@fahd11] if $\displaystyle g(t)=\frac{(t-a)^{\mu}}{\mu}$. \[2.4\] Let $\alpha, \beta \in \mathbb{C}$ be such that $Re(\alpha)\geq 0$ and $Re(\beta)>0$. Then, for any $\rho>0$ we have - \(a)  $\big(_{a}I^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\big)(t)=\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)\rho^\alpha}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\alpha+\beta-1},$\ $~~~Re(\alpha)>0.$ - (b)  $\big(I_b^{\alpha,\rho,g} e^{-\frac{\rho-1}{\rho}g(x)} (g(b)-g(x))^{\beta-1}\big)(t)=\frac{\Gamma(\beta)}{\Gamma(\beta+\alpha)\rho^\alpha}e^{-\frac{\rho-1}{\rho}g(t)}(g(b)-g(t))^{\alpha+\beta-1},$\ $~~~Re(\alpha)>0.$ - (c)  $\big(_{a}D^{\alpha,\rho} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\big)(t)=\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha},$\ $~~~Re(\alpha)\geq 0.$ - (d) $\big(D_b^{\alpha,\rho,g} e^{-\frac{\rho-1}{\rho}g(x)} (g(b)-g(x))^{\beta-1}\big)(t)=\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{-\frac{\rho-1}{\rho}g(t)}(g(b)-G g(t))^{\beta-1-\alpha},$\ $~~~Re(\alpha)\geq 0.$ The proofs of relations (a) and (b) are very easy to handle. We will prove (c) while the proof of (d) is analogous. By the definition of the left proportional fractional derivative and relation (a), we have $$\begin{aligned} &\Big(_{a}D^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\Big)(t)\\ &=D^{n,\rho,g} \Big(_{a}I^{n-\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\Big)(t)\\ &=D^{n,\rho,g}\frac{\Gamma(\beta)}{\Gamma(\beta+n-\alpha)\rho^{n-\alpha}}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{n-\alpha+\beta-1} \\ &=\frac{\rho^n\Gamma(\beta)(n-\alpha+\beta-1)(n-\alpha+\beta-1)\cdot \cdot \cdot (\beta-\alpha)}{\rho^{n-\alpha}\Gamma(n-\alpha+\beta)} \times e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha}\\ & =\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha}.\end{aligned}$$ Here, we have used the fact that $\displaystyle D^{\rho,g} \Big(h(t) e^{\frac{\rho-1}{\rho}g(t)} \Big)=\rho \frac{h'(t)}{g^\prime (t)} e^{\frac{\rho-1}{\rho}g(t)} $. Below we present the semi–group property for the general fractional proportional integrals of a function with respect to another function. \[THM1\] Let $\rho\in (0,1],~Re(\alpha)>0$ and $Re(\beta)>0$. Then, if $f$ is continuous and defined for $t \geq a$ or $t\le b$, we have $$\label{Left Semi integrals} ~_aI^{\alpha,\rho,g} (_{a}I^{\beta,\rho,g} f)(t)= ~_aI^{\beta,\rho,g} (_{a}I^{\alpha,\rho} f)(t)=(~_{a}I^{\alpha+\beta,\rho,g} f)(t)$$ and $$\label{Right Semi integrals} I_b^{\alpha,\rho,g} (I_b^{\beta,\rho,g} f)(t)= ~I_b^{\beta,\rho,g} (I_b^{\alpha,\rho} f)(t)=(I_b^{\alpha+\beta,\rho,g} f)(t).$$ We will prove . is proved similarly. Using the definition, interchanging the order and making the change of variable $z=\frac{g(u)-g(\tau)}{g(t)-g(\tau)}$, we get $$\begin{aligned} &~_aI^{\alpha,\rho,g} (_{a}I^{\beta,\rho,g} f)(t)\\ &= \frac{1}{\rho^{\alpha+\beta}\Gamma(\alpha)\Gamma(\beta)} \int_a^t \int_a^ue^{\frac{\rho-1}{\rho}(g(t)-g(u))}e^{\frac{\rho-1}{\rho}(g(u)-g(\tau))}(g(t)-g(u))^{\alpha-1}\\ & \times(g(u)-g(\tau))^{\beta-1}f(\tau)g'(\tau)d\tau g'(u)du\\ &=\frac{1}{\rho^{\alpha+\beta}\Gamma(\alpha)\Gamma(\beta)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} f(\tau) \int_\tau^t (g(t)-g(u))^{\alpha-1} (g(u)-g(\tau))^{\beta-1}\\ &\times g'(u)dug'(\tau) d\tau \\ &= \frac{1}{\rho^{\alpha+\beta}\Gamma(\alpha)\Gamma(\beta)} \int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))}(g(t)-g(\tau))^{\alpha+\beta-1} f(\tau)g'(u)d\tau \\ &\times \int_0^1 (1-z)^{\alpha-1} z^{\beta-1} dz\\ &=\frac{1}{\rho^{\alpha+\beta}\Gamma(\alpha+\beta)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))}(g(t)-g(\tau))^{\alpha+\beta-1} f(\tau)g'(\tau)d\tau\\ &=(_{a}I^{\alpha+\beta,\rho} f)(t).\end{aligned}$$ \[THM2\]Let $0\leq m< [Re(\alpha)]+1$. Then, we have $$\label{D on L} D^{m,\rho,g} (_{a}I^{\alpha,\rho,g}f)(t)=(_{a}I^{\alpha-m,\rho,g}f)(t)$$ and $$\label{LD on L} ~_{\ominus}D^{m,\rho,g} (I_b^{\alpha,\rho,g}f)(t)=(I_b^{\alpha-m,\rho,g}f)(t)$$ Here we prove , while one can prove likewise. Using the fact that $D_t^{\rho,g}e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} =0$), we have $$\begin{aligned} & D^{m,\rho,g} (_{a}I^{\alpha,\rho,g}f)(t) D^{m-1,\rho,g} (D^{\rho,g}~_{a}I^{\alpha,\rho,g}f)(t) \\ &= D^{m-1,\rho,g} \frac{1}{\rho^{\alpha-1}\Gamma(\alpha-1)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(\tau))} (g(t)-g(\tau))^{\alpha-2}f(\tau)g'(\tau)d \tau.\end{aligned}$$ Proceeding $m-$times in the same manner we obtain (\[D on L\]). \[D on I\] Let $0<Re(\beta) < Re(\alpha)$ and $m-1<Re(\beta)\leq m$. Then, we have $$\label{LD on LI} _{a}D^{\beta, \rho,g} ~_{a}I^{\alpha,\rho,g} f(t)=~_{a}I^{\alpha-\beta,\rho,g} f(t)$$ and $$\label{RD on RI} D_b^{\beta, \rho,g} I_b^{\alpha,\rho,g} f(t)=I_b^{\alpha-\beta,\rho,g} f(t).$$ By the help of Theorem \[THM1\] and Theorem \[THM2\], we have $$\begin{aligned} ~_{a}D^{\beta, \rho,g} ~_{a}I^{\alpha,\rho,g} f(t)&=& D^{m,\rho,g} _{a}I^{m-\beta,\rho,g} _{a}I^{\alpha,\rho,g}f(t)\\ &=& D^{m,\rho,g}~_{a}I^{m-\beta+\alpha,\rho,g} f(t)=~_{a}I^{\alpha-\beta,\rho,g} f(t).\end{aligned}$$ This was the proof of . One can prove in a similar way. \[THM4\] Let $f$ be integrable on $t\geq a$ or $t\le b$ and $Re[\alpha]>0, ~\rho \in (0,1],~~n=[Re(\alpha)]+1$. Then, we have $$\label{LD on LI sameorder} ~_{a}D^{\alpha, \rho,g} ~_{a}I^{\alpha, \rho,g} f(t)=f(t)$$ and $$\label{RD on RI sameorder} D_b^{\alpha, \rho,g} I_b^{\alpha, \rho,g} f(t)=f(t).$$ By the definition and Theorem \[THM1\], we have $$~_{a}D^{\alpha, \rho,g} ~_{a}I^{\alpha, \rho,g} f(t)=D^{n,\rho,g}~_{a}I^{n-\alpha, \rho,g} ~_{a}I^{\alpha, \rho,g} f(t)= D^{n,\rho,g}~_{a}I^{n, \rho,g} f(t)=f(t).$$ The Caputo fractional proportional derivative of a function with respect to another function ============================================================================================= For $\rho \in (0,1]$ and $\alpha \in \mathbb{C}$ with $Re(\alpha)\geq 0$ we define the left derivative of Caputo type as $$\begin{aligned} \label{CFP} &(^{C}_{a}D^{\alpha,\rho,g} f)(t)=_{a}I^{n-\alpha,\rho,g} (D^{n,\rho,g}f)(t)\\\nonumber &=\frac{1}{\rho^{n-\alpha}\Gamma(n-\alpha)}\int_a^t e^{\frac{\rho-1}{\rho}(g(t)-g(s))}(g(t)-g(s))^{n-\alpha-1}(D^{n,\rho,g}f)(s)g'(s)ds.\end{aligned}$$ Similarly, the right derivative of Caputo type ending is defined by $$\begin{aligned} \label{rCFP} &(^{C}D_b^{\alpha,\rho} f)(t)= I_b^{n-\alpha,\rho,g} (_{\ominus}D^{n,\rho,g}f)(t)\\\nonumber &= \frac{1}{\rho^{n-\alpha}\Gamma(n-\alpha)}\int_t^b e^{\frac{\rho-1}{\rho}(g(s)-g(t))}(g(s)-g(t))^{n-\alpha-1}(~_{\ominus}D^{n,\rho,g}f)(s)g'(s)ds,\end{aligned}$$ where $n=[Re(\alpha)]+1$. \[4.2\] Let $\alpha, \beta \in \mathbb{C}$ be such that $Re(\alpha)> 0$ and $Re(\beta)>0$. Then, for any $\rho \in (0,1]$ and $n=[Re(\alpha)]+1$ we have 1. $\big(^{C}_{a}D^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1}\big)(t)=\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha},$\ $~~~Re(\beta)> n.$ 2. $\big(^{C}D_b^{\alpha,\rho,g} e^{-\frac{\rho-1}{\rho}g(x)} (g(b)-g(x))^{\beta-1}\big)(t)=\frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{-\frac{\rho-1}{\rho}g(t)}(g(b)-g(t))^{\beta-1-\alpha},$\ $~~~Re(\beta)> n.$ For $k=0,1,\ldots,n-1$, we have $$\big(^{C}_{a}D^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a)^{k}\big)(t)=0\quad \mbox{and}\quad \big(^{C}D_b^{\alpha,\rho,g} e^{-\frac{\rho-1}{\rho}g(x)} (g(b)-g(x))^{k}\big)(t)=0.$$ In particular, $(~^{C}_{a}D^{\alpha,\rho} e^{\frac{\rho-1}{\rho}g(x})(t)=0$ and $(^{C}D_b^{\alpha,\rho} e^{-\frac{\rho-1}{\rho}g(x)})(t)=0$. We only prove the first relation. The proof of the second relation is similar. We have $$\begin{aligned} &(^{C}_{a}D^{\alpha,\rho,g} e^{\frac{\rho-1}{\rho}g(x)} (g(x)-g(a))^{\beta-1})(t)= ~_{a}I^{n-\alpha,\rho,g} D^{n,\rho,g} \left[e^{\frac{\rho-1}{\rho}g(t)} (g(t)-g(a))^{\beta-1} \right]\\ &=~_{a}I^{n-\alpha,\rho,g} \left[ \rho^n (\beta-1)(\beta-2)\ldots(\beta-1-n) (g(t)-g(a))^{\beta-n-1} e^{\frac{\rho-1}{\rho}g(t)}\right] \\ &= \frac{\rho^n (\beta-1)(\beta-2)\ldots(\beta-1-n)\Gamma(\beta-n)} {\Gamma(\beta-\alpha)\rho^{n-\alpha}} (g(t)-g(a))^{\beta-\alpha-1} e^{\frac{\rho-1}{\rho}g(t)}\\ &= \frac{\rho^\alpha\Gamma(\beta)}{\Gamma(\beta-\alpha)}e^{\frac{\rho-1}{\rho}g(t)}(g(t)-g(a))^{\beta-1-\alpha}.\end{aligned}$$ Conclusions =========== We have used the proportional derivatives of a function with respect to another to obtain left and right generalized type of fractional integrals and derivatives involving two parameters $\alpha$ and $\rho$ and depending on a kernel function . The Riemann–Liouville and Caputo fractional derivatives in classical fractional calculus can obtained as $\rho$ tends to $1$ and by choosing $g(x)=1$. The integrals have the semi–group property and together with their corresponding derivatives have exponential functions as part of their kernels. It should be noted that other properties of these new operators can be obtained by using the Laplace transform proposed in [@fahd10]. Moreover, for a specific choice of $g$, the proportional fractional operators in the settings of Hadamard and Katugamplola can be extracted. **Funding** This research was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.\ **Availability of data and materials** Not applicable.\ **Competing interests** The authors have no competing interest regarding this article.\ **Authors contributions** All authors have done equal contribution in this article. All authors read and approved the last version of the manuscript. [99]{} I. Podlubny, *Fractional Differential Equations* (Academic Press, San Diego CA, 1999). S. G. Samko, A. A. Kilbas, O. I. Marichev, *Fractional Integrals and Derivatives: Theory and Applications* (Gordon and Breach, Yverdon, 1993). R. Hilfer, *Applications of Fractional Calculus in Physics* (Word Scientific, Singapore, 2000). L. Debnath, *Recent applications of fractional calculus to science and engineering*, Int. J. Math. Math. Sci. **(2003), Issue 54**, 3413–3442. A. Kilbas, H. M. Srivastava, J. J. Trujillo, *Theory and Application of Fractional Differential Equations* (North Holland Mathematics Studies 204, 2006). R.L. Magin, *Fractional Calculus in Bioengineering* (Begell House Publishers, 2006). A. A. Kilbas, *Hadamard-type fractional calculus*, J. Korean Math. Soc. [**38**]{}(6) (2001), 1191–1204 U. N. Katugampola, *New approach to generalized fractional integral*, Appl. Math. Comput. [**218**]{} (2011), 860–865. U. N. Katugampola, *A new approach to generalized fractional derivatives*, Bul. Math. Anal. Appl. [**6** ]{} (2014), 1–15. F. Jarad, T. Abdeljawad, D. Baleanu, *On the generalized fractional derivatives and their Caputo modification*, J. Nonlinear Sci. Appl. **10 (5)** (2017), 2607-2619. F. Jarad, T. Abdeljawad,D. Baleanu, *Caputo–type modification of the Hadamard fractional derivative*, Adv. Difference Equ. [**2012**]{}, 2012:142. F. Jarad, E. Uğurlu, T. Abdeljawad,D. Baleanu, *On a new class of fractional operators*, Adv. Difference Equ. [**2018**]{}, 2018:142. F Jarad, T Abdeljawad, Generalized fractional derivatives and Laplace transform, Discrete and Continuous Dynamical Systems-S, doi:10.3934/dcdss.2020039. R. Khalil, M. Al Horani, A. Yousef, and M. Sababheh, *A new Definition Of Fractional Derivative*, J. Comput. Appl. Math. [**264**]{} (2014), 65–70. T. Abdeljawad, *On conformable fractional calculus* , J. Comput. Appl. Math. [**279**]{} (2013), 57–66. M. Caputo, M. Fabrizio, *A new definition of fractional derivative without singular kernel*, Progr. Fract. Differ. Appl. [**1**]{} (2015), 73–85. J. Losada, J. J. Nieto, *Properties of a new fractional derivative without singular kernel*, Progr. Fract. Differ. Appl. [**1**]{}(2015), 87–92. T. Abdeljawad, D. Baleanu, *On fractional derivatives with exponential kernel and their discrete versions*, Rep. Math. Phys. [**80**]{} 1 (2017), 11–27. A. Atangana, D. Baleanu, *New fractional derivative with non-local and non–singular kernel*, Thermal Sci. [**20**]{} (2016), 757–763. T. Abdeljawad, D. Baleanu, *Integration by parts and its applications of a new nonlocal fractional derivative with Mittag–Leffler nonsingular kernel*, J. Nonlinear Sci. Appl. **10 (3)** (2017), 1098–1107. D. R. Anderson, D. J. Ulness, *Newly defined conformable derivatives*, Adv. Dyn. Sys. App. **10 (2)** (2015) 109–137. D. R. Anderson, *Second–order self-adjoint differential equations using a proportional–derivative controller*, Comm. Appl. Nonlinear Anal. **24** (2017) 17–48. F. Jarad, T. Abdeljawad, J. Alzabut, *Generalized fractional derivatives generated by a class of local proportional derivatives*, Eur. Phys. J. Special Topics [**226**]{} (2017), 3457-3471.

--- abstract: | The continuous-time random walk (CTRW) model is useful for alleviating the computational burden of simulating diffusion in actual media. In principle, isotropic CTRW only requires knowledge of the step-size, $P_l$, and waiting-time, $P_t$, distributions of the random walk in the medium and it then generates presumably equivalent walks in free space, which are much faster. Here we test the usefulness of CTRW to modelling diffusion of finite-size particles in porous medium generated by loose granular packs. This is done by first simulating the diffusion process in a model porous medium of mean coordination number, which corresponds to marginal rigidity (the loosest possible structure), computing the resulting distributions $P_l$ and $P_t$ as functions of the particle size, and then using these as input for a free space CTRW. The CTRW walks are then compared to the ones simulated in the actual media. In particular, we study the normal-to-anomalous transition of the diffusion as a function of increasing particle size. We find that, given the same $P_l$ and $P_t$ for the simulation and the CTRW, the latter predicts incorrectly the size at which the transition occurs. We show that the discrepancy is related to the dependence of the effective connectivity of the porous media on the diffusing particle size, which is not captured simply by these distributions. We propose a correcting modification to the CTRW model – adding anisotropy – and show that it yields good agreement with the simulated diffusion process. We also present a method to obtain $P_l$ and $P_t$ directly from the porous sample, without having to simulate an actual diffusion process. This extends the use of CTRW, with all its advantages, to modelling diffusion processes of finite-size particles in such confined geometries. author: - Shahar Amitai - Raphael Blumenfeld date: 'Received: date / Accepted: date' title: 'Modifying continuous-time random walks to model finite-size particle diffusion in granular porous media' --- Introduction ============ Diffusion plays a key role in a wide range of natural and technological processes. A textbook modelling of such processes is the consideration of the diffusion of a single memory-free particle in a given medium. The nature of such a random walk is governed by three probability density functions (PDFs): of the step size, $P_l(l_i)$; of the step direction, $P_n(\hat{n}_i)$; and of the waiting time between steps, $P_t(t_i)$. These PDFs are, in principle, position dependent, but it is standard practice to derive (or postulate) them assuming position-independence and that $P_n(\hat{n}_i)$ is uniform. The diffusion is then modelled as a continuous-time random walk (CTRW) in free space. Specifically, the CTRW is constructed by adding vectors of uniformly random orientations, whose lengths are chosen from $P_l$, at time intervals chosen from $P_t$. Averaging over sufficiently many such independent processes, the dependence of the mean square distance (MSD) on time satisfies ${\langle \vec{x}^2 \rangle}= Dt^\alpha$. In normal diffusion $\alpha = 1$ and $D$ is the standard diffusion coefficient. But when $P_l$ and/or $P_t$ are very wide, the diffusion might become anomalous ($\alpha \ne 1$). In particular, when $P_t$ has a slowly decaying algebraic tail and $P_l$ does not, the random walk is sub-diffusive ($\alpha < 1$) [@Scher1975; @Scher1991]. Alternatively, if $P_l$ has a slowly decaying algebraic tail and $P_t$ does not, the random walk is super-diffusive ($\alpha > 1$), resembling a Lévy flight [@Mandelbrot1983]. Diffusion processes that have the same value of $\alpha$ are said to be in the same *universality class* [@Kadanoff1966]. Anomalous diffusion can arise from different sources, which can only be identified by going beyond the MSD. When single particle tracking is possible, the movement can be evaluated by the time-averaged MSD (TAMSD), $\delta^2(t, T)$ [@Metzler2014].While the MSD is the ensemble average of the squared distance, made during a time interval $t$, over different realisations, the TAMSD, $\delta^2(t, T)$, is the average of the same quantity along [*a single trajectory*]{} of length $T$. Within the model of sub-diffusive CTRW, the TAMSD satisfies $\langle \delta^2 \rangle \sim t \cdot T^{\alpha - 1}$, where the angular brackets denote a further ensemble average. In contrast, the MSD is sub-linear in $t$, which makes CTRW non-ergodic – the time-average and ensemble-average differ. In particular, the dependence of the TAMSD on $T$ points to the ageing nature of CTRW [@Metzler2014]. A key feature of sub-diffusive CTRW is the randomness of its TAMSD. Since $P_t$ is scale free, the longest waiting times each individual trajectory encounters vary significantly, as do the amplitudes of the individual TAMSDs. To quantify this, we define the amplitude scatter, $\xi = \delta^2 / \langle \delta^2 \rangle$. For ergodic processes (e.g. $\alpha = 1$) its PDF is $P(\xi) = \delta(\xi - 1)$ for sufficiently long trajectory times. But within CTRW this PDF broadens as $\alpha$ decreases. Defining the ergodicity breaking (EB) parameter, ${{\rm EB}}= \langle \xi^2 \rangle - \langle \xi \rangle^2$, it can be derived analytically for CTRW processes as a monotonically decreasing function of $\alpha$. Another cause for sub-diffusion is walking in a fractal-like environment [@Gefen1983; @Pandey1984]. Such environment is characterised by a network of narrow passages and dead ends at different length scales, which hinder the walk. Unlike CTRW, this process is stationary and therefore ergodic. The TAMSD, like the MSD, is sub-linear in $t$, independent of $T$ and its ${{\rm EB}}$ parameter vanishes. Using CTRW to model diffusion in confined geometries, such as porous media formed by either sintered or unconsolidated granular materials, is very attractive [@Berkowitz2006; @Bijeljic2006; @Wong2004; @DeAnna2013] because it alleviates the need to simulate directly the dynamics of particles within the pore space, reducing significantly the computational burden. In addition, it alleviates finite-size errors due to finite samples. This practice is based on the common assumption that $P_l$, $P_t$ and $P_n$ alone control the random walk’s universality class. The common procedure is to find first the forms of these distributions in a specific medium, using either small simulations or analytic derivation under some assumptions, and then use these to carry out a many-step CTRW in free space. It is then presumed that the CTRW yields the same universality class as the diffusion in the confined geometry. The first aim of this paper is to demonstrate that this does not apply when the size of the diffusing particle is comparable to throat sizes. We do so by analysing trajectories of individual particles diffusing in a porous sample and show statistical deviations from CTRW predictions. We also compare these simulations with an equivalent CTRW model. We show that the effective change in the medium’s connectivity with varying particle size affects directly the nature and universality class of the diffusion process. We conclude that the sub-diffusion is the result of CTRW on a percolation cluster. Indeed, a combination of underlying mechanisms, leading to sub-diffusion, has also been observed in [@Tabei2013; @Weigel2011; @Jeon2011; @Yamamoto2014]. The second aim of the paper is to propose a method to correct for the topological effect, which makes it possible to still use CTRW, with its advantages, to model diffusion of any finite size particle in confined geometries. To maximise the range of validity of our results (see discussion below), we consider very high porosity porous media. These correspond to marginally rigid assemblies of frictional particles, whose mean coordination number is four [@Blumenfeld2015]. The least confined of these are model systems whose each particle has exactly four contacts. The structure of this paper is the following. In section \[sec:sample\] we describe the simulated porous samples. In section \[sec:diffusion\_in\_sample\] we describe the diffusion process, and discuss the effects of particle size. We perform statistical analysis of the particle trajectories and show disagreements with some predictions of the CTRW model. In section \[sec:diffusion\_in\_free\_space\] we describe the equivalent CTRW simulations and show that they yield different behaviour in spite of having the same step-length and waiting-time distributions. We propose an explanation for this discrepancy. In section \[sec:memory\] we propose a modification to the conventional CTRW model to alleviate this problem, making it more suitable for modelling diffusion of finite size particles in confined geometries. We conclude in section \[sec:conclusions\] with a discussion of the results. The porous sample {#sec:sample} ================= To simulate a three-dimensional porous granular assembly of coordination number four, we first generate an open-cell structure, using Surface Evolver as follows [@Brakke1992; @Wang2006]. Initially, $N$ seed points are distributed randomly and uniformly within a cube, and the cube’s space is Voronoï-tessellated to determine the cell associated with each point. A cell around a point consists of the volume of all spatial coordinates closest to it. The resulting cellular structure is then evolved with Surface Evolver to minimise the total surface area of the cell surfaces. This procedure is used commonly to model dry foams and cellular materials whose dynamics are dominated by surface tension. The result is an equilibrated foam-like structure, comprising cells, membranes, edges and vertices. A membrane is a surface shared by two neighbouring cells, an edge is the line where three membranes coincide, and a vertex is the point where four edges coincide. ![Pseudo-grains around the foam vertices.[]{data-label="fig:pseudo_grains"}](pseudo_grain.png){width="40.00000%"} Next, we construct a tetrahedron around every vertex by connecting the mid-points of the four edges emanating from it [@Frenkel2009]. Neighbouring tetrahedra are in contact in the sense that they share the mid-point of an edge. This construction results in a pseudo-granular structure of volume fraction $\phi = 34\%$, in which every tetrahedron represents a pseudo-grain in contact with exactly four others [@Blumenfeld2006] (see fig. \[fig:pseudo\_grains\]). Since neighbouring pseudo-grains share the mid-point of the edge between them, the tetrahedra structure is topologically homeomorphic to the original structure. The void space surrounded by the pseudo-grains is still cellular, but a cell surface now consists of triangular facets – the faces of the pseudo-grains surrounding it – and throats – the skewed polygons remaining of the original cell membranes. The membranes over the throats are disregarded, resulting in an open-cell porous structure, in which the throats are the openings between neighbouring cells. The pseudo-grains volumes are smaller than those of real convex grains, which curve out into the cells of this structure. This increases the pore volume and forms a limiting case, which establishes the validity of our results for any porous medium, as will be discussed in the concluding section. Diffusion in the porous sample {#sec:diffusion_in_sample} ============================== We model the diffuser as a sphere of radius $r$, measured in units of the average effective throat radius, $r_0$. We start by considering particles that are considerably smaller than the smallest throat in the structure. The particle cannot enter the tetrahedral pseudo-grains, but only move from cell $c$ to cell $c'$ through their shared throat. The simulation progresses by moving the particle from one cell, $c$, to a neighbouring cell, $c'$. Each such an event is a step, $\vec{l}_{c,c'}$, namely a vector extending between the centres of these cells. We define a waiting time, $t_c$, which is the number of time steps spent in cell $c$ before a jump occurs. Inside a cell, the particle is assumed to undergo Brownian motion and $t_c$ is proportional to (i) the square of the effective cell radius, $R_c \equiv \left(\frac{3v_c}{4\pi}\right)^{1/3}$, where $v_c$ is the cell volume, and (ii) the inverse of the fraction of the cell’s open surface through which the particle can pass to neighbouring cells, $S_c\equiv \frac{{A^{\rm (t)}_c}}{{A^{\rm (t)}_c}+ {A^{\rm (f)}_c}}$. This is since the particle, on average, has to make $1/S_c$ journeys of length $R_c$ until it goes through a throat rather than hits a facet of a pseudo-grain. The effective area of a throat is the area through which a particle of radius $r$ can pass and ${A^{\rm (t)}_c}$ (${A^{\rm (f)}_c}$) is the total area of throats (facets) that make the surface of the cell. ${A^{\rm (t)}_c}$ is then the sum of the effective throat areas accessible for the particle to go through. Thus, the waiting time within a cell is $$\begin{aligned} \label{eq:waiting_time} t_c \equiv \frac{R_c^2}{2 d S_c} = \frac{1}{2d} \left( \frac{3v_c}{4\pi} \right)^{2/3} \left(1 + \frac{{A^{\rm (f)}_c}}{{A^{\rm (t)}_c}} \right) {{\hspace{0.25cm}},}\end{aligned}$$ where $d$ is the local diffusion coefficient, which, using the Stokes-Einstein relation, is inversely proportional to the particle radius, $d = (r_0 / r) d_0$. The probability to exit cell $c$ into $c'$, $P(c' \mid c)$, is proportional to the area of the throat between them. To reduce finite-size effects, we wrap the sample around with periodic boundary conditions and let the particle travel larger distances by re-entering the sample. This means that the same cell may occur at different locations along the random walk. To avoid distorting the statistics by using the same cell too many times, we stop the process once a cell has occurred at more than five different locations. We emphasise that we do not simulate the diffusion within the cell – each step in our simulation corresponds to a transition of the particle from one cell to another, and the waiting time associated with the step is calculated from the cell properties. This process constitutes a random walk on a graph, whose nodes are the cell centres. After waiting for a period of time $t_c$ in cell $c$, determined by eq. (\[eq:waiting\_time\]), the particle makes a step to a neighbour cell, according to $P(c' \mid c)$. Before continuing, it is instructive to put the problem in thermodynamic context. The cell can be regarded as a potential well of height $\Delta E$, and the probability to escape from it is $P(t) = P_0 e^{-t / \tau}$. We can then use Kramer’s escape rate formula, $$\begin{aligned} \label{eq:kramer} \tau = \frac{2 \pi k T}{d \sqrt{U"(a)U"(b)}} e^{\Delta E / kT} {{\hspace{0.25cm}},}\end{aligned}$$ where $k$ is the Boltzmann constant, $T$ is the temperature and $U"(a)$ and $U"(b)$ are the second derivative of the potential at the bottom and top of the well, respectively. Interpreting $t_c$ as the half-life of the particle in the cell, $t_c = \tau \ln{2}$, we can combine eq. (\[eq:waiting\_time\]) and (\[eq:kramer\]) to get: $$\begin{aligned} \frac{2 \ln(2) \pi k T}{\sqrt{U"(a)U"(b)}} e^{\Delta E / kT} &= \frac{1}{2 S_c(r)} \left( \frac{3v_c}{4\pi} \right)^{2/3} .\end{aligned}$$ Assuming that all cells have the same effective potential $U$, we get: $$\begin{aligned} \frac{\Delta E}{kT} = {\rm Const.} - \ln{T} + \ln{ \frac{v_c^{2 / 3}}{S_c(r)} } {{\hspace{0.25cm}}.}\end{aligned}$$ This equation establishes the height of the effective barrier in terms of the cell volume and the fraction of its surface through which the particle can escape. Note that the particle’s mean free path within a cell is assumed to be well smaller than the cell size, regardless of the particle size, and therefore that Knudsen diffusion [@Knudsen1909; @Clausing1930] need not be considered. However, even if this assumption is not borne out, this would only modify the coefficient $d$ in eq. (\[eq:waiting\_time\]), which is arbitrary anyway in our simulations. Also note that the above assumption, $P(t) \sim e^{-t/\tau}$, means that typical escape times do not deviate much from the mean or half-life time. This justifies our choice of taking $t_c$ as a representative. \ For each walk we calculate the particle’s position at time $t$, relative to the origin, $\vec{x}(t) = \sum_{n=1}^{N(t)} \vec{l}_n$, where $N(t)$ is the number of steps made before time $t$. We then calculate the MSD, ${\langle \vec{x}^2 \rangle}$, as a function of $t$, where the angular brackets denote average over 1000 walks. Figure \[fig:small\_particle\_in\_sample\] shows ${\langle \vec{x}^2 \rangle}$ vs. $t$ for a particle of size $r = r_0 / 100$. The linear relation indicates normal diffusion, with a diffusion coefficient of $D = (0.65 \pm 0.03)d$   ($d = 100 d_0$). The inset figure shows a narrowly bounded $P_t$. #### Large particles We next consider particles of sizes comparable to $r_0$. Such particles diffuse differently due to two effects. One is delay and trapping inside cells. The larger $r$ is the lower the probability of passing through any particular throat, since the effective area, which the particle can pass through, is smaller. This reduces the overall probability to exit a cell, increasing the waiting times spent inside cells. As a result, while the waiting times are narrowly bounded for a small particle, $P_t(t_i)$ develops a power-law tail for sufficiently large particles, $P_t(t_i > t^{(0)}) \sim t_i^{-\beta}$, with $\beta$ a function of $r$. The second effect is that the topology changes with particle size; as it increases, the probability of passing through some throats vanishes identically, changing the system’s connectivity for this particle. Fig. \[fig:large\_particles\_waiting\_time\] shows $P_t$ for a few large particles. We see that $\beta$ decreases with $r$, corresponding to longer waiting times. Fig. \[fig:large\_particles\_msd\] shows the MSD vs. time for the same particles. We see that beyond a certain particle size ($r \sim 1.2r_0$) $\alpha$ starts to decrease – the diffusion becomes anomalous. To quantify this relation, we choose a larger set of radii and plot $\alpha(r)$ vs. $\beta(r)$ (blue dots in fig. \[fig:alpha\_vs\_beta\]). We see that for smaller particles $\beta \gg 2$ and $\alpha=1$, corresponding to normal diffusion with narrow waiting-time PDFs. As $r$ increases, $\beta$ decreases and the walks eventually become sub-diffusive with $\alpha < 1$. We measure a transition at $\beta_t^{\rm (sample)} = 2.53 \pm 0.03$. A short comment on scaling windows is due – $\alpha$ is evaluated along $t \in (10^{3.5}, 10^5)$ (see fig. \[fig:large\_particles\_msd\]). At much longer times, the diffusion is normal for all particle sizes. This is merely a consequence of the finite system size – there are no cells with longer waiting times than $\sim \! 10^5$. ![Two sets of simulations – diffusion in a porous sample (blue dots) and CTRW in unconfined space (green triangles). Each simulation (dot) represents a particular particle size, and is described by the power-law of the waiting-time distribution, $\beta$, and the anomaly parameter, $\alpha$. Small (large) particles appear at the top right (bottom left) corner of the graph – they experience a narrow (wide) waiting-time distribution and undergo normal (sub-) diffusion. The blue and green lines are fits of the form $\alpha = 1 - \frac{1}{2} \exp \{ - (\beta - \beta_t - \frac{1}{2}) / \tau \}$, with ($\beta_t^{\rm (sample)} = 2.53$, $\tau=0.42$) and ($\beta_t^{\rm (CTRW)} = 2.01$, $\tau=0.40$), respectively. The theoretical CTRW prediction of a universality class transition at $\beta=2$ is denoted by the black dashed line. The red lines mark the range of $\beta$’s that corresponds to the percolation threshold of the sample. This range matches the universality-class transition for confined diffusion. Error bars denote $2 \sigma$ ranges, and were established by the variance of 20 independent measurements.[]{data-label="fig:alpha_vs_beta"}](alpha_vs_beta.pdf){width="50.00000%"} To investigate the diffusion process further, we calculate the TAMSD, $\delta^2$, and its average over 1000 walks, $\langle \delta^2 \rangle$. Fig. \[fig:tamsd\_vs\_delta\] and \[fig:tamsd\_vs\_time\] show $\langle \delta^2(t, T) \rangle$ vs. the time lag, $t$, and vs. the overall trajectory time, $T$, respectively. $\langle \delta^2 \rangle$ is sub-linear in $t$, deviating from the linear $t$-dependence in the CTRW model, and it is nearly independent of $T$. This behaviour is the same as for a random walk on a fractal. In contrast, the amplitude scatter, $\xi$, follows the CTRW prediction: Fig. \[fig:normalised\_tamsd\] shows the PDF of $\xi$ for three different particle sizes of order $r_0$. The dramatic broadening of the PDF as $r$ increases indicates that the process is non-ergodic and that there is ageing [@Metzler2014]. Fig. \[fig:ergodicity\_breaking\] shows that the corresponding EB parameters, i.e. the variance of these PDFs, follow the theoretical expectation from CTRW. Modelling the diffusion as isotropic CTRW {#sec:diffusion_in_free_space} ========================================= Next we show that an attempt to simulate this process with straightforward isotropic CTRW fails. This may not come as a surprise, as the statistical analysis showed some deviations from the traditional CTRW, but it is still constructive to describe the CTRW simulation to better understand its adjustments in section \[sec:memory\]. For such a simulation we use the $r$-dependent PDFs, $P_l$ and $P_t$, that the particle experiences while diffusing in the confined structure. Recall that these PDFs refer to the transition between cells, rather than the movement within a cell. One way to obtain these PDFs is to measure them empirically during a diffusion process. However, more efficient is to compute them directly from the structural statistics of the porous medium, as we outline next. A derivation of $P_l$ and $P_t$ from structural statistics should be made cautiously because a straight-forward histogram of the waiting times of all cells in the sample ignores the inherent correlation between the probability to visit a cell, $P(c)$, and the waiting time [@Edery2013]. Cells that are difficult to get out of (long waiting times) tend to have a lower probability of getting into and are therefore visited less frequently. In particular, some cells are completely inaccessible for particles above a certain size. To this end we use the observation that, to a very good accuracy in our diffusion process, $P(c)$ is linear in the cell’s total effective throat area, ${A^{\rm (t)}_c}$. This observation, which is independent of the cell volume, can be seen over 3.5 orders of magnitude in fig. \[fig:visits\_vs\_total\_throat\_area\]. Furthermore, this holds for all particle sizes, both well smaller and comparable to $r_0$. This allows us to estimate $P(c)$ for a particular particle size by using the effective ${A^{\rm (t)}_c}$ – a direct structural characteristic of the medium. In principle, one expects the visiting probability to be correlated with the visiting probabilities of neighbouring cells, but fig. \[fig:visits\_vs\_total\_throat\_area\] shows that this effect is negligible. ![Visiting probability, $P(c)$, vs. the cell’s total effective throat area, ${A^{\rm (t)}_c}$, in a diffusion process with $r = r_0$.[]{data-label="fig:visits_vs_total_throat_area"}](visits_vs._throat_area.pdf){width="50.00000%"} We can now estimate more accurately $P_t$, and, particularly, the power-law $\beta$, for every particle size, by manipulating the waiting-time histogram as follows. For every bin consisting of the waiting times of cells $\{c_1, ... , c_n\}$, we multiply the bin’s height by $\sum_1^n P(c_i)$ and normalise the histogram into a PDF. This suppresses long waiting times in $P_t$. Collecting the statistics of all possible steps in the sample, we get a PDF described well by the Gaussian $P_l(l_i) \sim \exp\{-(l_i - \bar{l})^2/2\sigma^2\}$, with $\sigma / \bar{l} = 0.15 \pm 0.02$ (see fig. \[fig:step\_length\_dist\]). As $P_l$ is narrow, it does not affect the universality class of the walk. $P_l$ stays narrow, and indeed nearly constant, for all particle sizes, as the cells positions are fixed. Note that it is also possible to derive $P_l$ analytically, given the cell volume distribution and nearest neighbour volume-volume correlations. This, however, is somewhat downstream from the main thrust of this paper. We are now able to obtain accurate step-length and waiting-time PDFs for every particle size, which could be used as input into unconstrained CTRW. Fig. \[fig:alpha\_vs\_beta\] shows results of CTRW simulations (green triangles) using these PDFs. The fit to this curve (solid green line) differs from the theoretical prediction [@Montroll1965], $\alpha = \beta - 1$, due to finite time and size effects. Our measured transition for CTRW is $\beta_t^{\rm (CTRW)} = 2.01 \pm 0.02$, in agreement with the theoretical prediction. A key observation is that the CTRW exhibits a normal-to-anomalous transition at a lower values of $\beta$ than in the actual sample. Since both processes have the same step-length and waiting-time distributions, but one is performed on a graph and the other in free space, the discrepancy must stem from the connectivity of the sample, which the CTRW model cannot account for. This is supported by the statistical analysis of the diffusion in the porous sample (section \[sec:diffusion\_in\_sample\]), that show behaviours typical to random walks on a fractal. As mentioned above, the size of the diffusing particle determines the effective throat sizes, and hence the connectivity of the porous sample. Moreover, above some size there is no path percolating between the sample’s boundaries. Fig. \[fig:cluster\_size\_vs\_particle\_radius\] shows the dependence of the percolating accessible volume on $r$, where a range of radii around the percolation threshold is marked by red lines. The same range is marked in fig. \[fig:alpha\_vs\_beta\]. We see that the universality class transition in the sample occurs within this range. This is more evidence that the connectivity plays an important role in determining the universality class. Specifically, around the percolation threshold the incipient cluster assumes a fractal-like structure, further inducing sub-diffusion [@Gefen1983; @Pandey1984]. We conclude that our simulations of diffusion in porous media are best described as CTRW on a percolation cluster. ![The percentage of cells belonging to the incipient-cluster vs. the particle radius (in units of $r_0$). The percolation range is marked in red. The corresponding $\beta$ range is marked in fig. \[fig:alpha\_vs\_beta\].[]{data-label="fig:cluster_size_vs_particle_radius"}](cluster_size_vs._r.pdf){width="50.00000%"} Anisotropic CTRW {#sec:memory} ================ To retain the usefulness of the CTRW model, it would be desirable to modify it to capture the effect of connectivity. We next propose such a modification, inspired by the PDF of the angle between successive steps in the porous sample (fig. \[fig:direction\_pdf\]). There is a finite probability to make a backward step, i.e. go back through the throat the particle entered a cell, $P_{\rm back} \equiv P_\theta(\theta = \pi)$. For very small particles ${P_{\rm back}}= 0.087\pm 0.001$. This is in contrast to the conventional CTRW model, where the next step direction is uniformly distributed. Moreover, we see that ${P_{\rm back}}> 1/13.7 \approx 0.073$, which is the inverse of the average number of throats per cell, and is the expected value for ${P_{\rm back}}$ when all steps are equiprobable. This is because the mean size of an entrance throat is larger than the mean size of all the throats. The enhanced backward step probability can be regarded as a correlation between successively visited throats. As $r$ increases, the total available throat area decreases and ${P_{\rm back}}$ increases, as can be seen in the right panel of fig. \[fig:direction\_pdf\]. For very large particles, ${P_{\rm back}}$ dominates the walk. This can be seen as the ‘lowest order’ effect of connectivity, which we next try to capture within CTRW. ![Left: the PDF of $\cos(\theta)$, with $\theta$ the angle between successive steps of a diffusion process in the porous sample for $r = r_0 / 100$. The singular point at $cos(\theta) = -1$ marks the finite probability of a backward step ${P_{\rm back}}= 0.087 \pm 0.001$. In addition, $P(-1 < \cos(\theta) < -0.88) = 0$. Right: the dependence of ${P_{\rm back}}$ on $r$. For large particles, backward steps dominate the walk.[]{data-label="fig:direction_pdf"}](relative_angle_and_p_back.pdf){width="50.00000%"} We examine two methods to modify the CTRW model, both introducing a backward bias. In the first, we add to $P_l$ and $P_t$ a third distribution, $P_\theta$, of step direction relative to the previous step. In this method, the particle ‘remembers’ the direction of the previous step, and a relative angle is chosen from the non-uniform distribution $P_\theta(\theta)$, e.g. the one in fig. \[fig:direction\_pdf\]. $P_\theta$ can be calculated directly from the porous structure for any particle radius $r$, similarly to $P_l$ and $P_t$ (see section \[sec:diffusion\_in\_free\_space\]). A random step is then made at the angle $\theta$ relative to the previous step direction. The waiting time and step length at each step are chosen as before. This method introduces a correlation between consecutive steps, which is present unavoidably in the diffusion of large particles in the sample. Testing this method by a set of simulations, we find that the universality class transition occurs at $\beta_t^{\rm (anisotropic)} = 2.1 \pm 0.03$ – closer to the one measured in the porous sample. The second method is more drastic: at every step of the CTRW we construct a new cell, according to the structural statistics of the porous sample. We choose the cell’s volume, the number of throats, and the throats’ areas from the corresponding distributions, derived from the sample. Using these and the particle size, we calculate the waiting time for the new cell. Then we choose randomly an accessible exit throat. The probability to exit through the throat is proportional to its effective area. The particle then makes a step in the direction of the exit throat. Another cell is then constructed around it. The step length is the sum of the radii of these two cells. This process is then iterated as many times as required. A key feature of this method is that the particle ‘remembers’ the exit direction, the last used cell and the exit throat. The latter is used as one of the throats in constructing the next cell. If this throat is chosen again then the last step is retraced. We refer to this method as DA for the two variables that the particle ‘remembers’ – step direction and throat area. Within the DA process, the particle may pass back and forth several times through a large throat before it moves on and loses memory of this throat. This process correlates not only successive steps, as the previous method does, but also successive backward steps. Using the DA model, we obtain a universality-class transition at $\beta_t^{\rm (DA)} = 2.50 \pm 0.03$, in excellent agreement with the original simulation results. Thus, this model captures much better the particle size-driven topological change. An important feature of the DA model is that ${P_{\rm back}}$ is higher than in the porous sample. To understand this, consider two cells in the porous sample, connected by a large throat, and connected to the rest of the structure by smaller throats. Once the particle enters one of these cells, it is likely to move back and forth several times before it emerges. However, the smaller the throats leading to this pair of cells, the less likely is the particle to enter in the first place. This means that the occurrence frequency of such sub-structures, which increase ${P_{\rm back}}$, is suppressed. In contrast, once a particle passes through a relatively large throat in the DA model, then, other than this throat, an entire new cell is generated for each step. This results in a higher probability that the particle oscillates across such a throat. This feature appears to compensate for other, more complex, missing topological features, making the DA a better model for the diffusion process. As a further investigation of the proposed methods, we present the step-length correlation function for the different models, all using $r = 1.2r_0$ (fig. \[fig:step\_length\_corr\]). The correlation in the sample is mainly due to the fact that each two consecutive steps enter and exit a certain cell, $c$. If $c$ is small, then the two steps will tend to be short, and vice versa, leading to positive correlation. In addition, a high ${P_{\rm back}}$ means that many consecutive steps are of exactly the same length. This further increases the correlation for larger particles. As expected, the traditional CTRW exhibits no correlations. Our first adjustment to CTRW, introducing the possibility of a backward step, adds correlation. However, since this is only a one-step correlation it decays exponentially. The increase in correlation within the DA process is because of the increased probability for a long sequence of backward steps, as discussed above. As a result, the DA correlation function does not decay exponentially, agreeing better with the diffusion simulation in the sample. The correlation of the DA at a step distance of one is higher than that in the simulated diffusion process because its ${P_{\rm back}}$ is higher, yet the DA correlation decays faster with the number of steps since it lacks the more involved topological correlations. ![Step-length correlation function for the four types of simulations discussed in the paper, all with $r = 1.2r_0$. Solid lines are decaying exponentials. The correlation function of the first adjustment to CTRW decays exponentially, and the second adjustment (DA) – more slowly.[]{data-label="fig:step_length_corr"}](step_length_corr_r=1.2.pdf){width="50.00000%"} Conclusions {#sec:conclusions} =========== To conclude, we compared between two numerical models of diffusion of a finite size particle in porous media: a direct simulation of the diffusion process in a computer-generated sample, and what is commonly believed to be an equivalent CTRW. We first presented a method to construct the representative PDFs of the step-length and waiting-time, $P_l$ and $P_t$, given the particle size and statistical information about the structure of the porous media alone. Using the same particle size dependent $P_l$ and $P_t$ in both models, we analysed the transition from regular to anomalous diffusion. We showed that the the two models give different results – while the CTRW simulations follow the theoretical prediction, up to finite-time effects, with a transition to sub-diffusion at $\beta \approx 2$, the diffusion simulations in the confined geometry exhibit a transition at $\beta \approx 2.5$. We established that the difference stems from changes to the effective connectivity available to the particle with increasing size. This particle size-driven change in connectivity is not taken into consideration in the CTRW model. We supported this conclusion by investigation of the time average MSD and by showing that the transition in universality class occurs at the same range of particle sizes that corresponds to the percolation transition. Our findings show unequivocally that the discrepancy between the two models is *not* due to different waiting-time distributions, since using identical distributions do lead to different universality classes. It is important to comment on the range of validity of our results. Increasing the particle size can be regarded, alternatively, as shrinking the porous structure, while keeping the particle size unchanged. Evidently, the smaller the pore space the more restricted the diffusing particle is and the larger the discrepancy between the simulated diffusion and the CTRW. Thus, by starting from a medium with a very large pore space, we established that our results hold true for a wide range of porous media with lower porosity. Wong et al. [@Wong2004] studied experimentally a related process of trace particles diffusing in biological networks of entangled F-actin filaments. There, the diffusion of particles, of size comparable to the typical network mesh size, is sub-linear and $P_t$ decays algebraically. The universality class they observe is a function of the particle-to-mesh size ratio, in agreement with our results. However, they do not observe size-driven topological effects and their values of $\alpha$ and $\beta$ are in good agreement with the CTRW model. This is because of the flexibility of the gel-like network, which allows trapped particles to eventually escape by deforming the filament network, a phenomenon also modelled recently in [@Godec2014]. The rigidity of the structure considered here precludes this particle escape mechanism and is the reason for this difference. A potentially related process of large particles, diffusing in rigid porous media, was studied experimentally in [@Skaug2015]. That study focused on hydrodynamic in-pore effects, which might be interesting to eventually include in our model. The main advantages of the CTRW model are that it overcomes potential finite-size problems and is less demanding computationally. However, as we have demonstrated here, this is achieved at the expense of ignoring topological information about local connectivity. To preserve the advantages of CTRW, these need to be taken into consideration. To this end, we introduced two anisotropic CTRW models. One includes memory of the last step direction and a non-uniform distribution of step direction. The other, the DA model, adds memory of the area of the last throat visited, effectively correlating successive backward steps. The DA model shows a universality-class transition at $\beta_t^{\rm (DA)} = 2.50 \pm 0.03$, in good agreement with the one measured in our simulations of the diffusion process in the confined geometry of a porous medium. We conclude that the DA model is a better alternative to the traditional CTRW for modelling diffusion of finite size particles in such media. It combines the CTRW advantages, overcoming the finiteness of the sample and convenience of application, with a better capturing of topological effects.\ [**Acknowledgement**]{}: SA is grateful for support from the Alan Howard Scholarship.\ [**Conflict of interest**]{}: The authors declare that they have no conflict of interest. [99]{} H. Scher, E.W. Montroll, Phys. Rev. [**B 12**]{}(6), 2455 (1975) H. Scher, M.F. Shlesinger, J.T. Bendler, Physics Today [**44**]{}(1), 26 (1991) B.B. Mandelbrot, [*The fractal geometry of nature*]{}, vol. 173 (Macmillan, 1983) L.P. Kadanoff, Physics [**2**]{}(6), 263 (1966) R. Metzler, J.H. Jeon, A.G. Cherstvy, E. Barkai, [*Phys. Chem. Chem. Phys.*]{}, [**16**]{}(44), 24128 (2014) Y. Gefen, A. Aharony, S. Alexander, Phys. Rev. Lett. [**50**]{}, 77 (1983) R. Pandey, D. Stauffer, A. Margolina, J. Zabolitzky, J. Stat. Phys. [**34**]{}(3-4), 427 (1984) B. Berkowitz, A. Cortis, M. Dentz, H. Scher, Reviews of Geophysics [**44**]{}(2) (2006) B. Bijeljic, M.J. Blunt, Water Resources Research [**42**]{}(1) (2006) I.Y. Wong, M.L. Gardel, D.R. Reichman, E.R. Weeks, M.T. Valentine, A.R. Bausch, D.A. Weitz, Phys. Rev. Lett. [**92**]{}(17), 178101 (2004) P. De Anna, T. Le Borgne, M. Dentz, A.M. Tartakovsky, D. Bolster, P. Davy, Phys. Rev. Lett. [**110**]{}(18), 184502 (2013) S.M.A. Tabei et al., PNAS [**110**]{} 4911 (2013) A.V. Weigel, B. Simon, M.M. Tamkun, D. Krapf, PNAS [**108**]{} 6438 (2011) J.H. Jeon et al., Phys. Rev. Lett. [**106**]{}, 048103 (2011) E. Yamamoto, T. Akimoto, M. Yasui, K. Yasuoka, Scientific Reports [**4**]{} 4720 (2014) R. Blumenfeld, S.F. Edwards, S.M. Walley, Granular systems, in [*The Oxford Handbook of Soft Condensed Matter*]{}, Eds. E.M. Terentjev and D.A. Weitz, (Oxford University Press, Oxford, UK, 2015) K.A. Brakke, Experimental Mathematics [**1**]{}(2), 141 (1992) Y. Wang, S. Neethling, Minerals Engineering [**19**]{}(10), 1069 (2006). Selected papers from Computational Modelling 05, Cape Town, South Africa G. Frenkel, R. Blumenfeld, P.R. King, M.J. Blunt, Adv. Eng. Materials [**11**]{}(3), 169 (2009) R. Blumenfeld, S.F. Edwards, The Euro. Phys. J. E: Soft Matter and Biological Physics [**19**]{}(1), 23 (2006) M. Knudsen, Ann. Phys. 4F [**28**]{}, 75 (1909) P. Clausing, Ann. Phys. 5F [**4**]{}, 567 (1930) Y. Edery, A. Guadagnini, H. Scher, B. Berkowitz, Water Resources Research [**50**]{}(2), 1490 (2014) E.W. Montroll, G.H. Weiss, J. Math. Phys. [**6**]{}(2), 167 (1965) A. Godec, M. Bauer, R. Metzler, [*New J. Phys.*]{} [**16**]{}, 092002 (2014) M.J Skaug, L. Wang, Y. Ding, D.K. Schwartz, ACS Nano [**9**]{} 2148 (2015)

--- abstract: '**Abstract:** The Fields Medal, often referred as the Nobel Prize of mathematics, is awarded to no more than four mathematician under the age of 40, every four years. In recent years, its conferral has come under scrutiny of math historians, for rewarding the existing elite rather than its original goal of elevating mathematicians from under-represented communities [@barany2018fields; @barany2015myth]. Prior studies of elitism focus on citational practices [@hirsch2005index] and sub-fields [@rossi2017genealogical; @gargiulo2016classical]; the structural forces that prevent equitable access remain unclear. Here we show the flow of elite mathematicians between countries and lingo-ethnic identity, using network analysis and natural language processing on 240,000 mathematicians and their advisor-advisee relationships. We found that the Fields Medal helped integrate Japan after WWII, through analysis of the elite circle formed around Fields Medalists. Arabic, African, and East Asian identities remain under-represented at the elite level. Through analysis of inflow and outflow, we rebuts the myth that minority communities create their own barriers to entry. Our results demonstrate concerted efforts by international academic committees, such as prize giving, are a powerful force to give equal access. We anticipate our methodology of academic genealogical analysis can serve as a useful diagnostic for equality within academic fields.' author: - 'Ho-Chun Herbert Chang' - Feng Fu bibliography: - 'references.bib' date: February 2020 title: Elitism in Mathematics and Inequality --- Although mathematics is often framed as objective and egalitarian, its access is not equally conferred. Recent attention has been given to the Fields Medal, one of the most prestigious awards in math, and its elite community. When the award was first conceived in 1930, it was in part designed to assuage international tensions [@barany2018fields]. The award was intentionally given to individuals that would otherwise not receive any recognition, rather than the best young mathematician. Using social network analysis (SNA) and neural-based natural language processing (NLP), this paper analyses the flow of elite mathematicians between nations and lingo-ethnic categories. Analysis was performed on the Mathematics Genealogy Project, one of the most complete advisor-advisee databases maintained today with more than 240,000 mathematicians. Results demonstrates the self-reinforcing behavior among the elite level in mathematics. This contrasts with prior conferral of the Fields Medal, which was a positive force in mending international relations, such as integrating Japan and Germany after World War II [@parshall2009internationalization]. We propose the Fields Medal can be used today to improve accessibility of mathematics to minority groups. The classifier for lingo-ethnic identity is textual, it would be more accurate to say we classify specific languages that overlap significantly with ethnic or cultural identity. While the use of lingo-ethnic categorization as identity is shallow, our principal aim is to show, even at the most basic definitions of ethnicity or culture through language, we find evidence of inequality. This paper also offers a methodological contribution. We show that combining network analysis, neural-based natural language processing (NLP), and well-maintained academic databases can serve as a powerful diagnosis for access and equity, and improve the practice of science. Several studies on elitism within the production of mathematical knowledge have been conducted. Methods draw predominantly from the complex network perspective [@zeng2017science], leveraging network repositories such as citation and bibliometric networks. Gargiulo et al. studied the entire, connected giant component of the mathematical genealogy project, enriching the data using data mining techniques [@gargiulo2016classical]. They work focused on integrating math history with temporal network analysis, noting the fields evolution based on country, discipline, and the structure of scientific families. Prior investigated about the relationship between scientific mentorship and winning the Fields Medal or Wolf Prize, but results were inconclusive. Rossi et al. studied the role of advisor-advisee relationships  [@rossi2017genealogical]. They propose the *genealogy index*, adapted from the *h-index* which was initially developed by Hirsh [@hirsch2005index]. Malmgren et al. studied the role of mentorship on protégé performance, focused on metrics of academic success like publication record [@malmgren2010role]. Beyond scholarship, studies have also considered hiring practices [@clauset2015systematic] and departmental prestige [@myers2011mathematical]. The lack of metadata in these genealogies has limited the scope of investigation. This paper places elite community network flow as the focal point, contrasting the historical focus on the nation-state with the modern focus of identity. Historical Networks of Elite Migration {#historical-networks-of-elite-migration .unnumbered} ====================================== We begin with a sketch of history. Figure 1a) captures the migration of elite mathematicians between five key countries. The subgroup of elites was created by aggregating the shortest paths between Fields Medalists. This ensures that the full graph is connected, and conceptually, denotes a minimal graph that connects all the medalists together. Here, migration is determined by comparing where a mathematician earned their Ph.D. and where their students earned their Ph.D. It is reasonable to assume primary advisors have moved to the same country as their advisees. Prior to WWII, Western European countries were the strong-holds of mathematical thought. Notably, France and Germany contained the highest proportion of elite mathematicians. Many Japanese mathematicians studied in Germany, before returning to Japan, as part of modernization during the Meiji restoration. Examples inclue Rikitaro Fujisawa, who studied at the Unviersity of Strasbourg with Elwin Christofeel, before returning [@chikara2013intersection]. He was instrumental to reforming mathematics education in Japan. The flow chart reveals mass flows of researchers due to historical events. By 1932, the Holocaust led to mass migration from Germany to the United States and other European countries, which accounts for the drop in green volume, including prominent scientist Albert Einstein. Similarly, we observe large amounts of outflow from Russia after the cold war, greatly diminishing the presence of Russia mathematicians after the 1990s, and the second Italian mass diaspora after WWII. Beyond forced immigration, flow analysis also reveals the movement of reintegration. Japanese mathematicians immigrated to the United States following WWII, and continued throughout the 60s to the 90s. Twenty years later, Japanese mathematicians flowed back toward Japan. France is not shown in the Sankey flow chart (1a), but is historically one of the countries that produces the most elite mathematicians. The chord graph in 1b) shows the net flow of mathematicians over all time, with the color of the chord indicating net exports. The USA-GER chord is orange, which indicates a net outflow from USA to Germany. Only France exports more American mathematicians than it imports from the USA. In all other cases, the USA exports more to other countries. Figure 1c) shows the flow dynamics on a country level. In-flow is defined as the number of incoming edges, out-flow as the number of outgoing edge, and self-flow the number of loops. These results are similar to Gargiulo et al.  [@gargiulo2016classical] with two striking differences. First, the United States is a selfish and importing country at the elite level, whereas in general it is selfish and exporting. Secondly, there are many more importing countries compared to the general case, where most countries are exporting and selfish. Notice, many of the countries that are exporting and selfish are Western or part of the Soviet Union, where there were strong programs in mathematics. Other countries appear to import more at the elite level, because their “exports” are not as competitive as mathematicians exported from other countries. These two points allow us to infer three things. First, elite mathematicians have more mobility, and in many cases can begin work in foreign countries. Second, the United States imports more compared to the general case, attracting more elite members. Third, countries considered traditional mathematics strong-holds can be observed in the lower left corner. What this analysis tells us, beyond an exposé of diasporic history, is the fields medal served as a way to mediate tensions. In a similar way that the Olympics was held in Rome, Berlin, and Tokyo, the inclusion of internationally marginalized nations. The Flow of Marginalized Identities {#the-flow-of-marginalized-identities .unnumbered} ==================================== Upon analyzing the history of elite communities in mathematics, we turn to the present. As 1a) shows, today, there is significant flow between countries. lingo-ethnic categories of identity serve as a useful construct for understanding network flow. Figure 2a) shows the representation of identities, within three subgroups: all mathematicians (blue), mathematicians within the medalist subgroup, (green) and the medalists themselves (red). Fig. \[fig:ethnicity\] compares elite representation of subgroups relative to their actual proportions. For instance, there is a higher proportion of French medalists (14%) compared to the general proportion (8%). In contrast, there is a significant number of East Asian mathematicians (14%) but very low representation in both the medalist family and medalists themselves (5% each). Upward sloping bars (left to right) mean medalists and medalist families are *over-represented*; downward sloping bars indicate *under-representation*. Over-represented groups include British, French, Japanese, East European, and Nordic names. Underrepresented groups include East Asian and Germanic. Mathematicians with Arabic names are non-existent in Medalists and underrepresented in the elite community. On the level of flow, Figure 2b) characterizes identities in terms of in-flow, out-flow, and self-flow. High in-flow means a higher likelihood of being mentored. High out-flow then corresponds to a greater likelihood to mentor others. High self-flow means higher likelihood of mentoring your own identity. The identity with the most self-flow is Japanese. However, when all mathematicians are considered, the Japanese are shown as green, that is to say opposite of selfish. This indicates reinforcing behavior only occurs at elite levels. However, once these groups are aggregated into larger groups— Greater European, Asian, African and Arabic— then differences become evident. European names has high self-reinforcing behavior, whereas Asians names and African and Arabic names are much lower in the number of self-loops. This dispels a common myth that minority groups, due to homophily, tend to group together. This myth insinuate that barriers to entry are self-inflicted. However, as we see from 2b), most minority groups are far away from the selfish pole, with a healthy balance of in-flow and out-flow. Rather, increases in the quantity of self-loops occurs in the greater European subgroup. Old Strongholds, New Possibilities {#old-strongholds-new-possibilities .unnumbered} ================================== It is understandable that, when considering all mathematicians, that there is a high levels of self-flow— studying in elite and often foreign institutions is a privilege. However, the fact that high self-flow in identity at the *elite level* suggests institutions can do more to open access, given their greater access to resources. This has been the case for Japan. Japan is unique among Asian countries and identities in that there are many Japanese Fields Medalists (3), with high representation in elite levels. Japan has been known for its rapid westernization during the Meiji restoration relative to other Asian counterparts. As early as 1872, their traditional form of math *wasan* was replaced by western science. Prussia, rather than the United Kingdom, was the primary source of westernization, and led directly to the establishment of the University of Tokyo [@parshall2009internationalization]. After WWII, mathematicians sought to re-establish international ties and formed the International Congress of Mathematicians and a new International Mathematics Union (IMU). Marshall Stone, a proponent of this movement, said it clearly: “in considering American adherence to a Union, it must be borne in mind that we want nothing to do with an arrangement which excludes Germans and Japanese as such.” Indeed, we find the ten founding members well-represented in the ternary diagrams, and not long after founding, the Soviet Union joined. Revisiting Fig. 1a), we discover the density of elite mathematicians in Japan increases after 1945. What this says, is the Fields Medal can improve the status of marginalized populations. Mathematics historian Barany captures this aspiration, believing the fields medal should help “sculpt the future, rather than reward the past [@barany2018fields].” What we observe is the opposite, where the elite perpetuate the elite. Fig. 3 demonstrates this clearly, showing French Fields Medalist Laurent Schwartz and his lineage. Within 5 generations after Schwartz, 7 Fields Medalists emerge. In particular, Schwartz-Grothendieck-Deligne form a direct chain, as do Lions-Villani-Figalli. Note, Lions’ father Jacque -Louis Lions was also a student of Schwartz. In other words, 13.3% of all Fields Medalists descended directly from Schwartz. Broadly, each of these all made contributions to some form of algebraic geometry or functional analysis. Fig. \[fig:all-trees\] further shows that all medalists can be traced to 9 connected components, with the largest one holding 44 out of 60 listed Fields Medalists. These observations are not meant to diminish the achievements of great mathematicians. They do however show the Fields Medal has deviated from its commitment to elevate under-represented mathematicians. Fig. \[fig:heat\] shows this succinctly in a tabular heatmap, which shows the power ratios. The power ratio (defined in Equation \[eq:power-ratio\]) is the conditional likelihood of being in the Fields Medalist Subgroup over the average probability of being in the group ($P = 0.00759$). $$\label{eq:power-ratio} PR = \frac{P(\textit{Fields} ~|~ \textit{Institute \& Identity} )}{P(\textit{Fields})}$$ A mathematician that is French and attends a Top 50 institution means they are 6.4 times more likely to gain membership into the elite circle. Here, the top 50 is defined as the top institutions attended by those in the elite group. Note, we defined our Fields Medalist subgroup minimally, such that any other definition of subgroup would yield a higher power ratio. On the other hand, being East Asian and attending a Top 50 institution only affords you 1.4 times the likelihood of gaining membership into this elite circle. From this diagram, we infer that institution plays a large role in elite membership. However, notice an East Asian mathematician a top 50 school is 4.5 times less likely to be included than a French mathematician attending a top 50 school. An Indian mathematician educated outside top 50 schools are 6 times less likely to be included than a French mathematician with the same education. Amongst non-elite institutions, being Japanese gives the best chance of inclusion, an after-effect of the efforts by the IMU. Conclusion {#conclusion .unnumbered} ========== In 2014, the late Iranian mathematician Maryam Mirzakhani won the Fields Medal. A talented star herself, her groundbreaking work on dynamics and geometry was encouraged by her Ph.D. advisor Curtis McMullen, also a Fields Medalist, at the elite institution Harvard University. This is by no means downplaying her achievements; rather, it serves to show the power recognition and elite communities have—all of which membership she rightly earned. Although the Fields Medal should serve to recognize under-represented researchers, the proper cultivation of talent through mentorship and institutional support should be the starting point. In our evaluation of the present, there is a large under-representation of minority groups in not just Field Medalists, but also in the elite circle for mathematics. While institutional prestige a big factor, lingo-ethnic identity is also found to be highly relevant, the widest gap being 4.5 times the power ratio even at elite institutions. Given that elite institutions have more resources, they can take a bigger role in generating higher access for marginalized groups. Flow analysis also dispels the myth that under-representation arises from homophily-driven self-selection. Although the French stronghold shows the old forces that govern mathematical knowledge remain strong, the presence of Japanese scholars also shows concerted effort can be used as an integrating force. Concerted efforts by international academic committees, such as prize giving, are a powerful force to confer equal rights for knowledge production to traditionally marginalized groups. Beyond analysis, this network analytical methodology is a call for scientific communities to use advisor-advisee databases to open knowledge production and scientific access. Methods {#methods .unnumbered} ======= Graph Construction {#graph-construction .unnumbered} ------------------ The graph was constructed using the Mathematics Genealogy database. Nodes are mathematicians, and directed edges represent advisor-advisee relationships. The data set contained information (listed in order of completeness) on the academic, advisor-advisee links, school, PhD graduation year, country, and dissertation title and topic. The ID’s of medalists were identified, then the shortest path was computed in a pairwise fashion. Analysis was conducted primarily using the Networkx package [@hagberg2008exploring]. The subgroup of elites was created by taking the union of shortest paths between Fields Medalists. Then, the full graph is connected, and denotes some form of minimal graph that connects all the medalists together. While it is possible to produce a minimal spanning tree, given the forest like structure of the genealogy, the shortest paths has more interpretive value. Identity Classifier {#identity-classifier .unnumbered} ------------------- Since lingo-ethnic identity is not included in the Mathematics Genealogy Project, a separate classifier is required. The identity categories were labeled using the *ethnicolr* package, which is a long-short term neural network (LSTM) trained on Wikipedia and the census [@sood2018predicting]. Specifically, the LSTM was based off the seminal work of Graves and Schmidhuber [@graves2005framewise]. This package has found use in evaluating under-representation in other STEM fields such as biomedicine [@marschke2018last]. It achieves between 78% to 81% accuracy. Potential shortcomings of neural methods for categorization is the accuracy levels. However, for 13 individual categories (which would result in 7.7% accuracy if truly random), 81% is quite high. Additionally, since we are interested in comparison within individual demographics, any bias would be carried forward since the group of all mathematicians supersets the medalist subgroup and medalists themselves. The goal of using this classifier is not to flatten definitions of identity, but to use the best available tools for inference, in absence of concrete data. Flow Analysis {#flow-analysis .unnumbered} ------------- Meso-graphs were constructed on attributes of each mathematician. To turn attributes into nodes, we constructed a mapping from mathematician to the meso-categories (lingo-ethnic and nationality of doctoral degree). Edges between meso-categories were simply the original directed-edges between mathematicians. Each edge is then weighed by the number of advisor-advisee relations between meso-categories. ### Constructing Ternary Diagrams {#constructing-ternary-diagrams .unnumbered} We constructed the ternary diagrams through analysis of the meso-network. Every meso-network can be represented by a its adjacency matrix, which we denote $M$. The diagonal then accounts for self-loops, the rows excluding the diagonal elements the out going edges, and the columns excluding the diagonal element the incoming edges. Explicitly, for meso-category indexed by $i$, we have the following definitions for in-flow (IF), out-flow (OF), and self-flow (SF). $$\begin{aligned} SF_i &= M_{i,i} \\ IF_i &= \sum_{j \neq i} M_{i,j} \\ OF_i &= \sum_{j \neq i} M_{j,i} \end{aligned}$$ We then normalize these values to represent each meso-category as a point in three dimensional space. $$P_i = \Big( \frac{IF_i}{K_i}, \frac{OF_i}{K_i} , \frac{SF_i}{K_i} \Big) \in [0,1]^3 \qquad \text{with } K_i = IF_i + OF_i + SF_i$$ Note, all points lie on the plane described by $x + y + z = 1$. We then transform this planar section onto the two dimensional plane using a translation and two rotations. $$\begin{aligned} P_i' &= P_i - (0,0,1) \\ P_i'' &= R_2 \circ R_1 (P_i') \end{aligned}$$ where $R_1$ rotates the plane up to the XY-plane, and $R_2$ aligns the simplex to the x-axis. Acknowledgements {#acknowledgements .unnumbered} ================ Both authors thanks the Mathematics Genealogy Project for generously providing data from its database for use in this research. H.C gratefully acknowledges the Annenberg Fellowship from the Annenberg School of Communication and Journalism, USC. F.F. gratefully acknowledges the Dartmouth Faculty Startup Fund, the Neukom CompX Faculty Grant, Walter & Constance Burke Research Initiation Award and NIH Roybal Center Pilot Grant.

--- abstract: 'Star formation in galaxies is triggered by a combination of processes, including gravitational instabilities, spiral wave shocks, stellar compression, and turbulence compression. Some of these persist in the far outer regions where the column density is far below the threshold for instabilities, making the outer disk cutoff somewhat gradual. We show that in a galaxy with a single exponential gas profile the star formation rate can have a double exponential with a shallow one in the inner part and a steep one in the outer part. Such double exponentials have been observed recently in the broad-band intensity profiles of spiral and dwarf Irregular galaxies. The break radius in our model occurs slightly outside the threshold for instabilities provided the Mach number for compressive motions remains of order unity to large radii. The ratio of the break radius to the inner exponential scale length increases for higher surface brightness disks because the unstable part extends further out. This is also in agreement with observations. Galaxies with extended outer gas disks that fall more slowly than a single exponential, such as $1/R$, can have their star formation rate scale approximately as a single exponential with radius, even out to 10 disk scale lengths. H$\alpha$ profiles should drop much faster than the star formation rate as a result of the rapidly decreasing ambient density.' author: - 'Bruce G. Elmegreen' - 'Deidre A. Hunter' title: Radial Profiles of Star Formation in the Far Outer Regions of Galaxy Disks --- Introduction ============ The outer disks of spiral galaxies have a low level of star formation (Ferguson et al. 1998; LeLièvre & Roy 2000; Cuillandre, et al. 2001; de Blok & Walter 2003; Thilker et al. 2005; Gil de Paz et a. 2005), even though the gas is gravitationally stable by the Kennicutt (1989) condition. Triggering by other mechanisms, such as turbulence compression (Mac Low & Klessen 2004), supernovae, and extragalactic cloud impacts (Tenorio-Tagle 1981), might be the reason. As a result, radial light profiles should not drop suddenly at the stability threshold, but should taper slowly as various star formation processes get more and more unlikely and the gas supply diminishes. The purpose of this paper is to investigate a simple model of star formation with generalized triggering in a smoothly varying gas disk. We seek to determine what the overall radial light profile might be. The radial light profiles of spiral and irregular galaxies are typically exponential over 3 to 5 scale lengths (van der Kruit 2001) with rare examples, particularly among low-inclination spirals, going further (Courteau 1996; Barton & Thompson 1997; Weiner et al. 2001; Erwin, Pohlen, & Beckman 2005; Bland-Hawthorn et al. 2005). Some galaxies have another, steeper exponential in the inner disk bulge region (Courteau, de Jong & Broeils 1996), which does not concern us here as it may be the result of gas inflow or bar formation (Kormendy & Kennicutt 2004). Many galaxies also have a steep exponential in the far outer disk (Näslund & Jörsäter 1997; de Grijs, Kregel, & Wesson 2001; Pohlen et al. 2002). This outer exponential is the focus of our discussion. As the outer disk is significantly below the sky brightness and generally difficult to observe, its properties are not well known; it may not even be exponential. van der Kruit (1988) suggested that disk asymmetries can make what is really a sharp outer truncation appear much smoother when the light profiles are azimuthally averaged; he noted that very deep exposures of edge-on disks tend to show sharp edges instead of smooth outer exponentials. Florido et al. (2001) showed how a sharp function could be fitted to outer disk cutoffs. The outer disk profile also depends critically on the level and uniformity of the sky brightness subtracted from the image. The transition from the main disk exponential to the outer disk profile has several observed characteristics. The outer disk scale length is about half that of the inner disk for both spiral and dwarf irregular galaxies (Hunter & Elmegreen 2006, hereafter Paper I). The ratio of the transition, or “break,” radius, $R_{br}$, to the main disk scale length, $R_D$, is 3 to 4 for spiral galaxies (van der Kruit & Searle 1981; Barteldrees & Dettmar 1994; Pohlen, Dettmar, & Lütticke 2000; Schwarzkopf & Dettmar 2000; Kregel, van der Kruit & de Grijs 2002) and $\sim2$ for dwarf and spiral Irregulars (Paper I). There is a slight increase in this ratio for decreasing $R_D$ among spirals (Pohlen, Dettmar, & Lütticke 2000; Kregel, van der Kruit & de Grijs 2002; Kregel & van der Kruit 2004), and another slight increase for increasing central surface brightness among spirals (Kregel & van der Kruit 2004). The first of these two correlations does not hold for dwarf Irregulars, which have both small disk scale lengths and small ratios $R_{br}/R_D$. The second correlation does hold for dwarf Irregulars. If there is a universal reason for outer disk transitions (as in the present model), then correlations which apply to both spirals and irregulars would seem to be most important. Thus the second correlation, in which $R_{br}/R_D$ increases with central surface brightness, should be viewed as fundamental, and the first simply a result of the second along with the independent correlation between scale length and central surface brightness found by de Jong (1996) and Beijersbergen, de Blok, & van der Hulst (1999). The apparent ratio $R_{br}/R_D$ should also depend slightly on galaxy inclination as a result of a tendency to overestimate $R_D$ for edge-on spirals where central extinction flattens the radial profile. Exponential light profiles in galaxies have been attributed to several things. Cosmological collapse during galaxy formation, starting with a nearly uniform spheroid, can produce profiles that resemble exponentials out to $\sim 2-6$ scale-lengths (Freeman 1970; Fall & Efstathiou 1980). Exponential disks also arise through radial flows in viscously evolving disks if the star formation rate is proportional to the viscosity (e.g., Lin & Pringle 1987; Yoshii & Sommer-Larsen 1989; Zhang & Wyse 2000; Ferguson & Clarke 2001). Double exponential profiles have no previous explanation (see review in Pohlen et al. 2004). van der Kruit (1987) proposed that outer disk truncations arise during galaxy formation and the break radius is determined by the maximum angular momentum of the proto-galactic cloud. Kennicutt (1989) suggested that truncation arises where the gas disk drops below the threshold for gravitational instabilities. Elmegreen & Parravano (1994) and Schaye (2004) proposed it arises when the ISM converts to a mostly warm phase, as observed in the outer regions of spirals (Dickey, Hanson & Helou 1990; Braun 1997) and dwarfs (Young & Lo 1996, 1997). Dalcanton et al. (1997), Firmani & Avila-Reese (2000), Van den Bosch (2001), Abadi, et al. (2003), Governato et al. (2004) and Robertson et al. (2004, 2005) simulated galaxy formation with threshold star formation and obtained exponential profiles with an outer disk cutoff. None of these models actually obtained double exponentials, only sharp outer disk truncations. The theory of disk truncation is highly uncertain, however. The angular momentum in the outer parts of a galaxy can change over time during interactions. The gravitational stability threshold may not be sharp if the ISM cools (Elmegreen 1991) or magnetic forces remove angular momentum (Kim, Ostriker & Stone 2002) during compression. The phase transition may not occur if the outer gas disk tapers slowly, like $1/R$ (Wolfire et al. 2003). All of these uncertainties suggest that refined models may eventually obtain more gradual outer disk truncations. The presence of double exponentials in dwarf galaxies (Paper I) places immediate constraints on the models. Most dwarfs have nearly solid body rotation curves throughout a large fraction of their optical disks. This means there is little shear, so viscous evolution should not play a significant role in structuring disk profiles. There is also no correlation in our Paper I sample between the break radius and the radius where the rotation curve changes from near solid body in the inner regions to near flat in the outer regions. Thus even the outer exponential is not likely to result from radial migration and evolution related to shear. Collapse models could in principle be arranged to give the desired radial profiles, but the collapse models in cosmological simulations so far have just given inner exponential disks with relatively sharp outer cutoffs. There have been no suggestions yet about how conditions during galaxy formation could be tuned to give outer double exponentials. One possibility is that galaxy collapse gives a single exponential disk and then subsequent accretion of gas makes the far-outer disk with a different profile (Bottema 1996). This may explain a sudden decrement in the rotation speed at the optical disk edge of NGC 4013 (Bottema 1995; see also van der Kruit 2001), but the decrement could also come from the prominent warp in that galaxy. If the outer disk is accreted, then there is no obvious reason why the ratio of outer to inner scale lengths should be about the same from galaxy to galaxy, including the dwarfs (Paper I). Here we consider a star formation model that includes turbulence and other compressions as cloud formation mechanisms, in addition to spontaneous gravitational instabilities (see also Kravtsov 2003). Observations of dwarf galaxies have shown that star formation is not simply regulated by a threshold column density (see review in Paper I). Star formation clearly occurs in clouds that stand above the threshold even if the average column density is below the threshold, and it persists far out in the outer disks of dwarfs as it does in spirals. It has also become clear that the ISM in both dwarfs and spiral galaxies is highly structured into clouds of all sizes, presumably as a result of turbulence and other processes. For the dwarf galaxies, this conclusion follows from the log-normal shape of the probability density function of H$\alpha$ emission (Hunter & Elmegreen 2004), and from the power-law power spectra of H$\alpha$ and stellar emissions (Willett, Elmegreen & Hunter 2005). The same power laws for star formation are seen in spiral galaxies (Elmegreen, Elmegreen, & Leitner 2003; Elmegreen et al. 2003). Dwarfs also show power law or fractal structure in the HI gas, as in the M81 dwarfs (Westpfahl et al. 1999) and in the Small and Large Magellanic Clouds (Stanimirovic et al. 1999; Elmegreen, Kim & Staveley-Smith 2001). The same is observed in local HI (e.g., Dickey et al. 2001). All of these distributions resemble the characteristics of compressible turbulence as illustrated in simulations (see review in Elmegreen & Scalo 2004). These considerations lead to a model for star formation in a turbulent, self-gravitating medium. This model is more general than the instability model alone as it allows for more processes, including pressurized triggering of star formation, turbulence triggering, spiral density wave triggering, and swing-amplified instabilities. It should be useful for predictions of outer disk star formation rates and for semi-analytical models of star formation in cosmological studies. Multi-component Model of Star Formation ======================================= Many of the general properties of galaxy disks and star formation can be combined into a relatively simple model that gives the star formation rate as a function of radius. These properties lead to the basic assumptions of the model, as listed here: - Galaxies form with a smoothly distributed gas disk having an outer cutoff, as usually seen in cosmology simulations. This cutoff will enter the present discussion as the outermost point of the disk, significantly beyond any break radius that may appear. We assume in some models that the smooth gas disk is a single exponential, although other forms will have the same basic properties. It will be significant that the star formation rate takes an approximately double exponential profile even in a gas disk that is a single exponential. Other models assume an exponential gas disk with an outer $1/R$ extension. In these cases, the star formation profile can be a continuous exponential or a double exponential with the outer part flatter then the inner part. In all cases, the star formation profile will drop faster than the gas profile, but it will rarely truncate suddenly. - The ISM is turbulent and partly stirred by pressures related to existing stars. This means the velocity dispersion tends to decrease slightly with radius as the stellar disk decreases exponentially. Such a velocity dispersion decrease is observed for some spiral galaxies (Boulanger & Viallefond 1992). Theoretical discussions of the radial profile of gaseous velocity dispersion are in Jog & Narayan (2005). Equating the energy densities of turbulence and stellar energy input, this gives approximately $M^2\propto e^{-R/R_D}$ for Mach number $M$, radius $R$, and stellar exponential disk scale-length $R_D$. The precise form of this relation is not important to the model; other cases considered below use a constant Mach number and get about the same result. The exponential form assumes the gas density for stirring by supernovae and other stellar pressures is about constant with radius, as appropriate for the HI medium in the main disks of galaxies. Then the Mach number alone responds to the stellar energy density. This is in rough agreement with observations showing greater HI velocity dispersions for cool HI clouds near stellar associations (Braun 2005). This equation also emphasizes that the important Mach number for our model is the one that regulates the first step of cloud formation, i.e., the conversion of ambient gas into dense cloud complexes where stars form. Such emphasis places turbulence on an equal footing with large-scale gravitational instabilities. There may be other processes governing the radial profile of the Mach number inside individual dense clouds and the Mach number for the mass-weighted ISM as a whole. - The Mach number reaches a minimum value near unity in the outer disk as a result of either a transition to a warm-dominant thermal HI phase or a sustained low level of turbulence (Sellwood & Balbus 1999). In either case, cool clouds are still possible in the compressed regions, but turbulence compression is weak. Combined with the previous point, this means $$M^2\approx1+Ae^{-R/R_d}\label{eq:mach}$$ where $A$ is the square of the effective Mach number in the inner disk. We assume in some models below that $A=100$; the results do not depend on this value as long as the main part of the inner disk is Toomre unstable. The most important point for the model is that some level of turbulence remains in the outer disk so that turbulence-induced compression makes clouds even where the average disk is Toomre-stable. Thus $A=0$ gives acceptable results too. For the models shown below, Equation \[eq:mach\] will be used with either $A=100$ or $A=0$; a more detailed treatment might have the coefficient $A$ depend on the SFR per unit gas mass, or on other processes related to interstellar turbulence. - Isothermal turbulence produces clouds with a log-normal distribution of column density, as observed in simulations by Padoan et al. (2000), Ostriker et al. (2001), and Vázquez-Semadeni & García (2001). Then the probability of a region having a local column density $\Sigma_g$ is $$P(\Sigma_g)d\ln\Sigma_g=P_0 \exp\left(-0.5\left[\ln \Sigma_g/\Sigma_p\right]^2 /\sigma^2\right)d\ln \Sigma_g.$$ The column density at the peak of this distribution, $\Sigma_p$, will be determined at each radius to give the appropriate average column density (see below). The dispersion of the log-normal may scale with the Mach number of the turbulence, $$\sigma=\left(\ln\left[1+0.5M^2\right]\right)^{1/2} \label{eq:disp}$$ as in simulations by Nordlund & Padoan (1999). The log-normal is consistent with the pixel-to-pixel distribution of H$\alpha$ intensity in Im galaxies (Hunter & Elmegreen 2004). These last two points (with $A>1$) make the ISM more clumpy in the inner regions than in the outer regions. For the unstable inner part of the disk, this clumpiness does not matter much for the star formation rate because it is relatively easy for $\Sigma_g$ to exceed $\Sigma_c$ and also because the instabilities themselves drive turbulence and cloudy structure. In the stable outer parts, however, the turbulence-formed clumps and any outward propagating spirals from the inner disk are the primary regions where $\Sigma_g>\Sigma_c$ and star formation occurs only in them. This makes star formation very patchy in outer disks, and it proceeds at a low average rate. The rate is not zero even though the average gas column density, $<\Sigma_g>$, is significantly less than $\Sigma_c$ because star formation persists in the tail of the $P(\Sigma_g)$ function. The log-normal form for $P\left(\Sigma_g\right)$ is not critical for the double exponential radial profile. It is used here primarily for convenience and because of its presumed connection with turbulence. The important point is that $P\left(\Sigma_g\right)$ has a tail at high $\Sigma_g$ that gets wider with increasing Mach number, and that some low level of turbulence compression remains in the gravitationally stable outer disk. - The critical column density for gravity to overcome Coriolis and pressure forces is the Toomre value appropriate for gas. The general concept that galaxy edges result from below-threshold $\Sigma_g/\Sigma_c$ dates back to Fall & Efstathiou (1980), Quirk (1972), Zasov & Simakov (1988) and Kennicutt (1989). We write $\Sigma_c$ here in terms of the epicyclic frequency $\kappa$ and the Mach number $M$ instead of the velocity dispersion, $$\Sigma_c=C M\kappa/\left(\pi G\right). \label{eq:thres}$$ The constant of proportionality, C (units of velocity dispersion), absorbs the fixed rms speed and effective adiabatic index that is in the usual expression because we replaced the dispersion with the radial-varying Mach number. In our model, $C$ determines where the break radius might occur in the original exponential; whether it breaks or not depends also on the run of Mach number with radius. The instability condition does not indicate only the onset of swing-amplified or shear instabilities in thin disks, or the onset of ring instabilities, as originally devised by Safronov (1960) and Toomre (1964). It is also the condition for the stability of giant expanding shells (Elmegreen, Palous, & Ehlerova 2002) and most likely relevant to the collapse of turbulence-compressed regions too (Elmegreen 2002). This is because all of these processes involve gravity, rotation, and pressure. When $\Sigma_g>\Sigma_c$, gravity overcomes the Coriolis force during the contraction of the largest cloud that is initially in pressure-gravity equilibrium. The origin of the cloud does not matter. If $\Sigma_g<\Sigma_c$, then Coriolis forces disrupt collapsing spiral arms, expanding shells, turbulence-compressed clouds, and ISM structures before much star formation begins in them. Thus, the Toomre condition should be a general condition for star formation, independent of the detailed triggering processes, which may be quite varied (Elmegreen 2002). The situation is the same if the ambient medium cools during the compression, but then $\Sigma_c$ should be set equal to $C \gamma_{eff}^{1/2} M\kappa/\left(\pi G\right)$ for effective adiabatic index $\gamma_{eff}=c^{-2}dP/d\rho$ (Elmegreen 1991), considering pressure $P$, density $\rho$, and velocity dispersion $c$. Most likely this occurs in the outer disks where compression can convert warm HI into cool diffuse clouds. We do not consider this additional factor here. - The star formation rate is proportional to some power of the [*local*]{} gas column density $\Sigma_g$ when the threshold is exceeded. A 1.4 power was observed by Kennicutt (1998) for a wide range of conditions. A lower limit to the power is $\sim1$, which also fits the data in some models (Boissier et al. 2003; Gao & Solomon 2004). Note that these power law observations differ significantly from what one would get for the Toomre instability alone, where the maximum growth rate, $\kappa\left(\Sigma_g^2/\Sigma_c^2-1\right)^{1/2},$ increases from zero rapidly as $\Sigma_g$ begins to exceed $\Sigma_c$, and then asymptotically levels off to a dependence on $\Sigma_g^1$. This makes the star formation rate per unit area, which is $\Sigma_g$ times the growth rate, proportional to $\Sigma_g$ raised to a power greater than or equal to 2. If $\Sigma_g/\Sigma_c\sim1.5,$ for example, then the star formation rate should scale with $\Sigma_g^{2.8}$. The Toomre condition alone is not appropriate for star formation because all of the other dynamical processes that are involved (such as turbulence driven by young stars and thermal cooling inside compressed regions) change both $\Sigma_g$ and $\Sigma_c$ locally. The Toomre condition assumes an isothermal uniform gas. If this isothermal assumption is relaxed, then galaxy disks can become unstable for a wider range of conditions (e.g., Elmegreen 1991). The origin of the observed power law is not fully understood, but it is probably related to star formation processes that operate at the local dynamical rate in a medium that is structured by turbulence (Elmegreen 2002). It should follow naturally from a full hydrodynamical model that includes these effects (Kravtsov 2003; Li, Mac Low & Klessen 2005). These points incorporate the main processes that are believed to be involved with galactic-scale star formation: ISM turbulence, pressurized shell formation and other pressurized triggering, thermal equilibrium, and general gravitational instabilities. Turbulence and other compressions make the disk cloudy and this cloudy structure persists in the outer disk even where $<\Sigma_g>$ is less than $\Sigma_c$, allowing star formation to continue at large radii. The decrease in the Mach number means the cloudiness decreases with radius, so the ISM becomes less turbulent and more smooth in the outer regions, as observed for HI (Braun 1997). This combination of cloud-forming turbulence with a Mach number that converges asymptotically to near-unity, along with cloud-forming instabilities that become decreasingly important with radius, produces the transition from an inner near-exponential to an outer near-exponential in our model. The outer exponential is where the disk is Toomre-stable and the Mach number is of order unity. Both of these conditions are satisfied at about the same place when a clear double exponential appears (see below). If there are spiral arms in the outer disk, even if they are generated in the inner disk and radiate dissipatively to the outer disk, then this model should not change much because these spirals provide only one more possible source of cloudy structure and triggered star formation. As long as the gas becomes gradually less compressive in the outer regions, the star formation rate tapers off smoothly until the physical edge of the disk (or ionized edge of the gas) is reached. Thus the Mach number in our model should be interpreted as the ratio of rms bulk motion to sound speed, regardless of whether the bulk motions occur in spiral shocks, turbulence, or pressurized shells. Figure \[fig-sfr\] shows radial profiles of various quantities from models based on these principles. The models calculate the results in radial steps of $dR=0.1$ (arbitrary units) for a disk exponential scale length of $R_D=2.5$ and an outer disk cutoff of 20. At each radius, $R$, the average gas column density is determined from the initial exponential, $<\Sigma_g>=\Sigma_{g0}e^{-R/R_D}$, the Mach number is determined from Equation \[eq:mach\], and the dispersion of the probability distribution function for local column density is determined from the Mach number using Equation \[eq:disp\]. Then the peak column density in this distribution, $\Sigma_p$, is determined from the integral over $\Sigma_g P\left(\Sigma_g\right)$ by setting the average column density that results from this integral equal to $<\Sigma_g>$. The threshold column density, $\Sigma_c$, is also determined at this radius, from Equation \[eq:thres\]. After this setup for the average quantities, the model makes clouds and determines star formation rates. The local column density is determined by randomly sampling from the distribution function $P\left(\Sigma_g\right)$, and then the star formation rate is set equal to this local column density raised to a power of 1 or 1.5 in the two cases shown, provided the local column density exceeds $\Sigma_c$. If the local column density is less than $\Sigma_c$, then the star formation rate at this position is set to zero. To adequately sample the random assignments of column densities, we consider a number of azimuthal points at each radius equal to $R/dR$. That is, the size of a cloud is assumed to be constant with radius. When $R/dR$ random column densities and resulting star formation rates are determined at each $R$, we average together these column densities and star formation rates to give the plotted quantities. Figure \[fig-sfr\] shows results for a rotation curve appropriate for most galaxies (the rotation curve affects only $\kappa$). This rotation curve is rising in the inner part and flat in the outer part: $V = V_0 (r/R_D)/ [ 1+ r/R_D]$. The star formation rate is shown on the left with dashed lines tracing two exponential profiles to guide the eye. Other quantities for the same models are shown on the right: critical column density, $\Sigma_c$ (magenta), Mach number (black dashed), average gas surface density, $<\Sigma_g>$ (red), and local gas column density, $\Sigma_g$ (green). Five cases are considered. In the top two and bottom two panels, the star formation rate scales with the local column density to the 1.5 power, while in the middle panel, the rate scales with $\Sigma_g$ to the first power. The difference is that when SFR$\propto\Sigma_g^{1.5}$, the inner exponential in star formation is steeper than the inner exponential in gas; otherwise the SFR and the gas have the same profiles. The break radius does not depend noticeably on whether the SFR scales with $\Sigma$ or $\Sigma^{1.5}$. The bottom two panels show the difference between models with high and low critical column densities. When $\Sigma_g/\Sigma_c$ is lower (bottom panel), less of the disk is unstable and the break radius is smaller. This is consistent with our observation that the relative break radius is smaller in dwarfs than in spirals (Paper I). It occurs because the average surface density is lower compared to the critical value in dwarfs than in spirals. The top two panels in Figure \[fig-sfr\] show cases where the Mach number has constant values with radius: 1 (top) and 10 (second from top). When the Mach number is 10 throughout, cloud formation continues at a high rate in the outer part of the disk (i.e., $\sigma$ in the dispersion of $P(\Sigma_g)$ stays large), and there is no significant drop in SFR there. Consequently, there is no clear double exponential. When the Mach number is 1 throughout, the average SFR profile is almost exactly the same as for the exponential Mach number (compare to the second panel up from the bottom which has the same parameters except for the Mach number), but the rms scatter is much lower when $M=1$ than when $M\sim10$ in the inner disk. This illustrates how the Mach number is unimportant for the average star formation rate in the unstable inner part of the disk. That region is “saturated” with star formation from spiral shocks and instabilities and unable to produce more star formation even with more compression. However the Mach number in the inner disk is important for the detailed structure of star formation, i.e., for the variability of it and for the geometry of cloud structure. These models illustrate how a combination of increasing disk stability and moderate Mach number can create an approximate double exponential in the star formation rate when the overall gas distribution is more uniform. The break radius occurs slightly beyond the point where $<\Sigma_g>\sim\Sigma_c$ if the Mach number is of order unity there. It can occur further out if the Mach number is still high. The detailed profile of the star formation rate in regions where the average column density exceeds the threshold does not depend much on the rotation curve or Mach number. This is because once the threshold is exceeded, the threshold no longer enters into the star formation rate for the simple power law model. Two other types of radial profiles are found in spiral and irregular galaxies: those which continue in an exponential fashion out to the largest measured radius and those which have a shallower exponential in the outer part (Erwin, Beckman, & Pohlen 2005; Paper I). Galaxies of the first type, with a single exponential extending out to $\sim10$ scale lengths (Weiner et al. 2001; Bland-Hawthorn, et al. 2005), are difficult to understand with single-component star formation models because the outer disk should be far below the Kennicutt (1989) threshold. Our multi-component model can reproduce the observation, but the outer gas disk has to fall more slowly than an extrapolation of the inner exponential. The bottom panels in Figure \[fig-sfr2\] show an example. The average gas disk is exponential out to $6R_D$ and then it tapers beyond that as $1/R$ out to $10R_D$. This is consistent with the shallow outer HI profile in NGC 300 (Puche et al. 1990), which has its stellar disk extend continuously to $10R_D$ (Bland-Hawthorn et al. 2005). The profiles of Mach number (using $A=100$) and rotation speed are the same as in the previous examples, and the local star formation rate is $\propto\Sigma_g^{1.5}$. In the left-hand panel, the star formation rate becomes very patchy in the outer part, but the average rate (solid blue line) follows an overall exponential profile. In the right panel, the ratio of $\Sigma_c$ to $<\Sigma_g>$, which is the Toomre stability parameter, $Q$, is $5.7$ in the outer regions, indicating great stability on average. Still, there is a lot of cloud and star formation from local compressions. The top part of Figure \[fig-sfr2\] shows the case if the $1/R$ part of the gas disk begins at a smaller radius, $5.2R_D$, with all else being the same. Then the profile in the outer disk can be shallower than in the inner disk. This is the second type of profile mentioned above. This explanation differs from that in Paper I, where we suggested that dwarf galaxies of this second type, with relatively flat outer parts, could have their steepening in the central regions because of enhanced star formation there. This was because the central regions tended to be bluer than the outer regions; most BCD galaxies were examples of this. The origin of the flat-outer exponential profiles in barred S0 galaxies (Erwin, et al. 2005) cannot be due to intense inner-disk star formation because their Hubble types are too early. In these S0 galaxies the isophotal contours tend to become more round with distance beyond the break radius. Radial profiles determined from deprojected circular averages at all radii could then introduce false inflections. Other flat-outer profiles in the Erwin et al. sample have break radii associated with outer rings or outer Lindblad resonances; these would not be connected with star formation changes either. More observations of the various types of radial profiles and their associated gas and star formation properties are necessary before the relative importance of these models can be understood. Our discussion so far has concerned the radial profile of the star formation rate but not the radial profile of the resulting stars that form over a Hubble time. Our comparisons between the predicted star formation profiles and the observed surface brightness profiles are therefore premature. The next step in this analysis should be an integration of the star formation rate over time, but this requires some knowledge of the gas accretion rate, both as a function of radius and time. Two limiting cases may be discussed at this point. At the end of the star formation process, after all of the gas has been converted into stars, the stellar mass profile should reflect the total accreted gas profile, altered, if need be, by radial gas motions, stellar migration, minor mergers and tidal stripping. There is no reason to believe that this final stellar profile will resemble the star formation profile. The second limiting case is at the beginning of the star formation process, when the outer disk is still dominated by gas. Then the stellar component is only a small perturbation to the outer disk and the star formation profile should be about the same as the accumulated stellar mass profile, altered again by any accreted stars, radial migrations, and tidal effects. Fortunately the outer parts of late-type galaxy disks, beyond $R_{br}$, are usually in this second limit, i.e., gas-dominated. For dwarf galaxies, this was shown by Hunter, Elmegreen & Baker (1998). Thus we believe the star formation profiles derived here for the outer disk are a suitable explanation for the stellar surface density profiles. To check this possibility, we ran two of the SFR models shown in Figure \[fig-sfr\] (the top panel and the second one up from the bottom) over a sequence of timesteps. At each timestep and at each radius, we deducted 1% of the instantaneous SFR from the average gas mass, and then added this mass to the integrated star mass. The initial conditions were the same as in Figure \[fig-sfr2\], with no accumulated stars at first. Figure \[fig-evol\] shows the resultant SFR, average gas surface density, and average stellar surface density at three times: the beginning of the run (blue curves), after 20 timesteps (green curves), and after 50 timesteps (red curves). We label these timesteps as 20% and 50% of the gas consumption time, respectively, where this consumption time is considered to be the inverse of the percentage (1%) deducted each step. There is no gas accretion from outside the galaxy. The panels on the left assume the same exponential Mach number profile as the second panel up from the bottom in Figure \[fig-sfr2\] (i.e., A=100 in Eq. \[eq:mach\]) and the panels on the right have the same constant Mach number profile as the top panel in Figure \[fig-sfr2\] (i.e., $A=0$ and $M=1$). Evidently, the gas gets depleted most quickly in the inner regions, as expected, and the stars build up an exponential profile over time. The double exponential is still present in the accumulated stars. The Mach number profile does not matter much, as discussed previously. At 50 timesteps, the star mass in the steep exponential part of the outer disk is still much less than the gas mass, in agreement with the observations cited in the previous paragraph. We have not commented yet on the relation between star formation rate and H$\alpha$ surface brightness. The H$\alpha$ surface brightness should become an inadequate tracer of star formation in the very low-density regions of outer galaxy disks because the emission measure drops below detectability. The emission measure through the diameter of a classical Stromgren sphere with uniform density $n$ is $\left(6S_0n^4/\left[\pi\alpha\right]\right)^{1/3}$ for Lyman continuum photon luminosity $S_0$ and recombination rate $\alpha$. The scaling with $n^{4/3}$ implies that even if the $H\alpha$ flux does not escape the galaxy, the intensity of an HII region is at least $e^{16/3}=207$ times fainter at a position four scale lengths further out compared to the inner disk. This factor assumes the density is smaller by only the factor $e^{-4}$ without a corresponding increase in scale height. An increase in scale height with decreasing disk self-gravity, as $H=c^2/\left[\pi G\Sigma_T\right]$ for velocity dispersion $c$ and total disk column density $\Sigma_T$, makes the midplane density roughly proportional to $\Sigma^2$, in which case the H$\alpha$ intensity can drop by a factor $e^{32/3}\sim4\times10^4$ in four scale lengths of an exponential disk. Thus there should be a significant drop in the radial profiles of H$\alpha$ even with outer disk star formation of the type discussed here. This type of $H\alpha$ drop is in agreement with observations (e.g., Thilker et al. 2005; Paper I). The multi-component star formation proposed here also explains some of the irregularities with the Kennicutt (1989) prediction that were found in our previous papers on dwarf Irregulars. For example, star formation in dwarfs often occurs where the average column density is less than $\Sigma_c$, in contradiction to the Kennicutt model, as long as there are cool cloudy regions that locally have $\Sigma_g>\Sigma_c$. This peculiarity was noted before in many studies of dwarf galaxies (van der Hulst et al. 1993; Taylor et al. 1994; van Zee et al. 1997; Meurer et al. 1998; Hunter, Elmegreen, & van Woerden 2001). In the current star formation model, these cloudy regions form by turbulence and other processes (pressurized shells, external cloud impacts, end-of-bar flows, gaseous spirals, etc.) even in sub-threshold regions if the Mach number for the associated flows is still relatively high. The revised model also extends the Kennicutt result by allowing for a typical decrease in velocity dispersion with radius and by associating the threshold $\Sigma_c$ with any of a variety of cloud formation processes, and not just isothermal gravitational instabilities in initially smooth disks. Conclusions {#sec-disc} =========== Many processes of star formation combine to give the radial profiles of galaxies. In the inner main-disk regions where the gas is usually gravitationally unstable in spite of the large Coriolis and pressure forces, star formation should saturate to its maximum possible rate. This is the gravitational collapse rate for the conversion of low density to high density gas, multiplied by the fraction of the high density gas that is suitable for star formation, i.e., the fraction in the form of stellar-mass globules with masses exceeding the local thermal Jeans mass (e.g., Elmegreen 2002; Kravtsov 2003). The actual dynamics involved with the first step, dense cloud formation, will be varied, involving swing-amplified spiral instabilities, spiral density wave shocks, compression or shell formation around existing star formation sites, and turbulence compression. In galaxies with strong stellar spirals, the spiral shocks may dominate dense cloud formation, making most clouds spiral-like, as in M51 (e.g., Block et al. 1997), while in galaxies without such strong spirals, another mechanism should dominate, making most clouds shell-like (as in the LMC), globular, or hierarchically fractal. In all of these cases, the same star formation rate per unit area arises, all from the saturation condition. Thus they all give the Kennicutt-Schmidt law or something like it in a regular fashion, regardless of the detailed processes involved. In the outer parts of disks, some of these processes shut down completely. There should be no strong stellar spirals beyond the outer Lindblad resonance for the main (self-amplified) modal pattern speed, and there should be few swing-amplified stellar spirals if the Toomre Q parameter is high. Cold cloud formation should also be more difficult at low ambient pressure. However, a low level of star formation may sustain itself at large radii by driving shells and turbulence and by compressing the existing clouds. Also, gaseous spiral arms can propagate there from the inner disk, as they are able to penetrate the outer Lindblad resonance unlike the stellar spirals. Gaseous arms can also form by instabilities there if there is significant cooling during compression (because that lowers $\Sigma_c$ through the $\gamma_{eff}$ parameter). These processes maintain star formation at levels much lower than the saturation rate given above, and therefore lower than the Kennicutt-Schmidt law rate, primarily because an ever-decreasing fraction of the gas can make the transition from low density to high density in the first step. In this paper, we modelled all of these processes in a general way using the few simple rules just mentioned. The transition from saturated star formation in the inner disk to unsaturated in the outer disk was followed, and radial profiles were obtained that look moderately close to real profiles. In the first case considered, the profile was exponential in the main disk and it tapered off beyond that with a form that also resembled an exponential, but steeper for several scale lengths. The ratio of the break radius to the inner scale length varied with the surface density (higher surface densities have higher ratios) because more unstable inner disks have their inner exponentials extend further out before the transition occurs. This correlation can explain the observation among both spirals and dwarfs that $R_{br}/R_D$ increases with main disk surface brightness. In two other cases with shallower outer gas profiles, the star formation profile varied between a nearly pure exponential out to $\sim10$ scale lengths and a shallow outer exponential, depending on where the transition between the inner and outer gas profiles occurred relative to the stability threshold radius. In all cases, the H$\alpha$ profile should drop much more suddenly with radius than the star formation profile as the emission measure of individual HII regions drops rapidly below the detectability limits. The main ingredients of our star formation model are: a generally smooth decline of gas column density in the disk with a cutoff in the far outer part (usually beyond the observations), a turbulent Mach number that decreases with radius and then levels off to near unity, or remains near unity throughout, a distribution function for local column density with a high column-density tail and a dispersion that increases with Mach number, a column density threshold for self-gravity to overcome Coriolis and pressure forces, and a local star formation rate that increases with the local cloud density when the threshold is exceeded. For such a model, the inner exponential occurs where the average column density exceeds the threshold, almost regardless of Mach number or Mach number gradient. The outer profile occurs where the gas column density is sub-critical and the Mach number is relatively small but non-zero, e.g., near unity. The small but non-zero Mach number gives turbulence and other dynamical processes the ability to form clouds that locally exceed the stability threshold, but these processes are not likely to do this very often. As a result, star-forming clouds become very patchy in the outer disk, making the star formation gradient significantly steeper than the gas gradient. Only the peaks of the clouds stand above the stability threshold. Gas cooling during cloud formation is also an essential ingredient. Cooling to diffuse cloud temperatures and colder is assumed to follow any significant compression, as predicted elsewhere based on studies of interstellar thermal equilibrium. We are grateful to the referee for useful comments. Funding for this work was provided by the National Science Foundation through grants AST-0204922 to DAH and AST-0205097 to BGE. Abadi, M. G., Navarro, J. F., Steinmetz, M., Eke, V.R. 2003, ApJ, 597, 21 Barteldrees, A., Dettmar, R.J. 1994, A&AS, 103, 475 Barton, I.J., & Thompson, L.A. 1997, AJ, 114, 655 Beijersbergen, M., de Blok, W. J. G., & van der Hulst, J. M. 1999, A&A, 351, 903 Bland-Hawthorn, J., Vlajić, M., Freeman, K.C., Draine, B.T. 2005, astroph/0503488 Block, D.L., Elmegreen, B.G., Stockton, A., & Sauvage, M. 1997, ApJ, 486, L95 Boissier, S., Prantzos, N., Boselli, A., & Gavazzi, G. 2003, MNRAS, 346, 1215 Bottema, R. 1995, A&A, 295, 605 Bottema, R. 1996 A&A, 306, 345 Boulanger, F., & Viallefond, F. 1992, A&A, 266, 37 Braun, R. 1997, ApJ, 484, 637 Braun, R. 2005, in The Magnetized Plasma in Galaxy Evolution, ed. K. Chy$\dot {\rm z}$y, K. Otmianowska-Mazur, M. Soida, & R.-J. Dettmar, Jagiellonian University, Kraków, p. 23 Courteau, S. 1996, ApJS, 103, 363 Courteau, S., de Jong, R.S., Broeils, A.H. 1996, ApJL, 457, 73 Cuillandre, J.-C., Lequeux, J., Allen, R.J., Mellier, Y., & Bertin, E. 2001, ApJ, 554, 190 Dalcanton, J.J., Spergel, D.N., & Summers, F.J. 1997, ApJ, 482, 659 de Blok, W.J.G., & Walter, F. 2003, MNRAS, 341, L39 de Grijs, R., Kregel, M., & Wesson, K. H. 2001, MNRAS, 324, 1074 de Jong, R. S. 1996, A&A, 313, 45 Dickey, J.M., Hanson, M.M., & Helou, G. 1990, ApJ, 352, 522 Dickey, J.M., McClure-Griffiths, N.M., Stanimirovic, S., Gaensler, B.M, & Green, A.J, 2001, ApJ, 561, 264 Elmegreen, B.G. 1991, ApJ, 378, 139 Elmegreen, B.G. 2002, ApJ, 577, 206 Elmegreen, B.G., & Parravano, A. 1994, ApJ, 435, L121 Elmegreen, B.G., Kim, S., & Staveley-Smith, L. 2001, ApJ, 548, 749 Elmegreen, B.G., Palous, J., & Ehlerova, S. 2002, MNRAS, 334, 693 Elmegreen, B.G., Elmegreen, D.M., & Leitner, S.N. 2003, ApJ, 590, 271 Elmegreen, B.G., Leitner, S.N., Elmegreen, D.M., & Cuillandre, J.-C. 2003, ApJ, 593, 333 Elmegreen, B.G., & Scalo, J. 2004, ARA&A, 42, 211 Erwin, P., Beckman, J. E., & Pohlen, M. 2005, ApJ, 626, L81 Fall, S. M., & Efstathiou, G. 1980, MNRAS, 193, 189 Ferguson, A. M. N., Wyse, R. F. G., Gallagher, J. S., & Hunter, D. A. 1998, AJ, 111, 2265 Ferguson, A.M.N., & Clarke, C.J. 2001, MNRAS, 325, 781 Firmani, C., & Avila-Reese, V. 2000, MNRAS, 315, 457 Florido, E., Battaner, E., Guijarro, A., Garzón, F., Jiménez-Vicente, J., 2001, A&A, 378, 82 Freeman, K.C. 1970, ApJ, 160, 811 Gao, Y., Solomon, P.M. 2004, ApJ, 606, 271 Gil de Paz, A., et al. 2005, ApJ, 627, L29 Governato, F., Mayer, L., Wadsley, J., Gardner, J. P., Willman, B., Hayashi, E., Quinn, T., Stadel, J., & Lake, G. 2004, ApJ, 607, 688 Hunter, D.A., Elmegreen, B.G., & Baker, A.L. 1998, ApJ, 493, 595 Hunter, D. A., & Elmegreen, B. G. 2004, AJ, 128, 2170 Hunter, D. A., Elmegreen, B. G., van Woerden, H. 2001, ApJ, 556, 773 Hunter, D.A., & Elmegreen, B.G. 2006, ApJ, submitted (Paper I) Jog, C.J., & Narayan, C.A. 2005, in preparation Kennicutt, R. C., Jr. 1989, ApJ, 344, 685 Kennicutt, R.C., Jr. 1998, ApJ, 498, 541 Kim, W.-T., Ostriker, E.C., & Stone, J.M. 2002, ApJ, 581, 1080 Kormendy, J., & Kennicutt, R. C., Jr. 2004, ARAA, 42, 603 Kravtsov, A.V. 2003, ApJ, 590, L1 Kregel, M., van der Kruit, P. C., & de Grijs, R. 2002, MNRAS, 334, 646 Kregel, M., & van der Kruit, P. C., 2004, MNRAS, 355, 143 LeLièvre, M., & Roy, J.-R. 2000, AJ, 120, 1306 Li, Y., Mac Low, M.-M., & Klessen, R. S. 2005, ApJ, 626, 823 Lin, D.N.C., & Pringle, J.E. 1987, MNRAS, 325, 781 Mac Low, M.-M., Klessen, R. S. 2004, RevModPhy, 76, 125 Meurer, G. R., Staveley-Smith, L., & Killeen, N. E. B. 1998, MNRAS, 300, 705 Näslund, M. & Jörsäter, S. 1997, A&A, 325, 915 Nordlund, A.K., & Padoan, P. 1999, in Interstellar Turbulence, Proceedings of the 2nd Guillermo Haro Conference. Eds. Jose Franco & Alberto Carraminana, Cambridge:Cambridge University Press, p.218 Ostriker, E.C., Stone, J.M., Gammie C. F. 2001, ApJ, 546, 980 Padoan, P., Juvela, M., Bally, J., & Nordlund, A. 2000, ApJ, 529, 259 Pohlen, M., Beckman, J. E., Hüttemeister, S., Knapen, J. H., Erwin, P., & Dettmar, R.-J. 2004, astroph/0405541 Pohlen, M., Dettmar, R.-J., & Lütticke, R., 2000, A&A, 357, L1 Pohlen, M., Dettmar, R.-J., Lütticke, R., & Aronica, G. 2002, A&A, 392, 807 Puche, D., Carignan, C., & Bosma, A. 1990, AJ, 100, 1468 Quirk, W.J. 1972, ApJ, 176, L69 Robertson, B., Yoshida, N., Springel, V., Hernquist, L. 2004, ApJ, 606, 32 Robertson, B., Hernquist, L., Bullock, J.S., Cox, T.J., Di Matteo, T., Springel, V., Yoshida, N. 2005, astroph/053369 Safronov, V. S. 1960, Ann.Ap., 23, 979 Schaye, J. 2004, ApJ, 609, 667 Schwarzkopf, U., & Dettmar, R.-J. 2000, A&A, 361, 451 Sellwood, J. A., & Balbus, S.A. 1999, ApJ, 511, 660 Stanimirovic, S., Staveley-Smith, L., Dickey, J.M., Sault, R.J., & Snowden, S.L. 1999, MNRAS, 302, 417 Taylor, C. L., Brinks, E., Pogge, R. W., & Skillman, E. D. 1994, AJ, 107, 971 Tenorio-Tagle, G. 1981, A&A, 94, 338 Thilker, D.A. et al. 2005, ApJ, 619, L79 Toomre, A. 1964, ApJ, 139, 1217 van den Bosch, F.C. 2001, MNRAS, 327, 1334 van der Hulst, J. M., Skillman, E. D., Smith, T. R., Bothun, G. D., McGaugh, S. S., & De Blok, W. J. G.1993, AJ, 106, 548 van der Kruit, P.C. 1987, A&A, 173, 59 van der Kruit, P.C. 1988, A&A, 192, 117 van der Kruit, P.C. 2001, in Galaxy Disks and Disk Galaxies, ed. J. G. Funes, S. J., & E. M. Corsini, PASP Conf. Series 230, p. 119 van der Kruit, P.C., & Searle, L. 1981, A&A, 110, 61 van Zee, L., Haynes, M. P., Salzer, J. J., & Broeils, A. H. 1997, AJ, 113, 1618 Vázquez-Semadeni, E., García, N. 2001, ApJ, 557, 727 Weiner, B.J., Williams, T. B., van Gorkom, J. H., & Sellwood, J. A. 2001, ApJ, 546, 916 Westpfahl, D.J., Coleman, P.H., Alexander, J., Tongue, T. 1999, AJ, 117, 868 Willett, K. W., Elmegreen, B. G., & Hunter, D. A. 2005, AJ, in press Wolfire, M. G., McKee, C. F., Hollenbach, D., & Tielens, A. G. G. M. 2003, ApJ, 587, 278 Yoshii, Y. & Sommer-Larsen, J. 1989, MNRAS, 236, 779 Young, L. M., & Lo, K. Y. 1996, ApJ, 462, 203 Young, L. M., & Lo, K. Y. 1997, ApJ, 490, 710 Zasov, A. V., & Simakov, S. G. 1989, Astrop., 29, 518 Zhang, B., Wyse, R.F.G. 2000, MNRAS, 313, 310

--- abstract: 'Let $F$ be a totally real field in which $p$ is unramified. We study the Goren-Oort stratification of the special fibers of quaternionic Shimura varieties. We show that each stratum is a $(\PP^1)^N$-bundle over other quaternionic Shimura varieties (for some appropriate $N$).' author: - Yichao Tian and Liang Xiao title: 'On Goren-Oort Stratification for quaternionic Shimura varieties' --- Introduction ============ This paper is intended as the first in a series [@tian-xiao2; @tian-xiao3], in which we study the Goren-Oort stratification for quaternionic Shimura varieties. The purpose of this paper is to give a global description of the strata, saying that they are in fact $(\PP^1)^r$-bundles over (the special fiber of) other quaternionic Shimura varieties for a certain integer $r$. We fix $p > 2$ a prime number. A baby case: modular curves {#S:modular curve} --------------------------- Let $N \geq 5$ be an integer prime to $p$. Let $\calX$ denote the modular curve with level $\Gamma_1(N)$; it admits a smooth integral model $\bfX$ over $\ZZ[1/N]$. We are interested in the special fiber $X: = \bfX \otimes_{\ZZ[1/N]} \FF_p$. The curve $X$ has a natural stratification by the supersingular locus $X^\mathrm{ss}$ and the ordinary locus $X^{\mathrm{ord}}$. In concrete terms, $X^\mathrm{ss}$ is defined as the zero locus of the Hasse-invariant $h \in H^0(X, \omega^{\otimes(p-1)})$, where $\omega^{\otimes(p-1)}$ is the sheaf for weight $p-1$ modular forms. The following deep result of Deuring and Serre (see e.g. [@serre]) gives an intrinsic description of $X^\mathrm{ss}$. \[T:Deuring-Serre\] Let $\AAA^\infty$ denote the ring of finite adèles over $\QQ$, and $\AAA^{\infty, p}$ its prime-to-$p$ part. We have a bijection of sets: $$\big\{\overline \FF_p\textrm{-points of } X^\mathrm{ss} \big\} \longleftrightarrow B^\times_{p,\infty} \backslash B_{p,\infty}^\times(\AAA^{\infty}) / K_1(N) B_{p,\infty}^\times({\ZZ_p})$$ equivariant under the prime-to-$p$ Hecke correspondences, where $B_{p, \infty}$ is the quaternion algebra over $\QQ$ which ramifies at exactly two places: $p$ and $\infty$, $B_{p, \infty}^\times({\ZZ_p})$ is the maximal open compact subgroup of $B_{p, \infty}^\times({\QQ_p})$, and $K_1(N)$ is an open compact subgroup of ${\mathrm{GL}}_2(\AAA^{\infty,p}) = B^\times_{p, \infty}(\AAA^{\infty,p})$ given by $$K_1(N) = \big\{\big(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\big) \in {\mathrm{GL}}_2(\widehat \ZZ^{(p)}) \; \big|\; c \equiv 0, d\equiv 1 \pmod N \big\}, \textrm{ where } \widehat \ZZ^{(p)} = \prod_{l \neq p} \ZZ_l.$$ The original proof of this theorem uses the fact that all supersingular elliptic curves over $\overline\FF_p$ are isogenous and the quasi-endomorphism ring is exactly $B_{p, \infty}$. We however prefer to understand the result as: certain special cycles of the special fiber of the Shimura variety for ${\mathrm{GL}}_2$ is just the special fiber of the Shimura variety for $B_{p, \infty}^\times$. The aim of this paper is to generalize this theorem to the case of quaternionic Shimura varieties. For the purpose of simple presentation, we focus on the case of Hilbert modular varieties. We will indicate how to modify the result to adapt to general cases. Goren-Oort Stratification {#S:stratification of HMV} ------------------------- Let $F$ be a totally real field, and let $\calO_F$ denote its ring of integers. We assume that $p$ is *unramified* in $F$. Goren and Oort [@goren-oort] defined a stratification of the special fiber of the Hilbert modular variety $X_{{\mathrm{GL}}_2}$. More precisely, let $\AAA_F^\infty$ denote the ring finite adèles of $F$ and $\AAA_F^{\infty, p}$ its prime-to-$p$ part. We fix an open compact subgroup $K^p \subset {\mathrm{GL}}_2(\AAA_F^{\infty, p})$. Let $\calX_{{\mathrm{GL}}_2}$ denote the [*Hilbert modular variety*]{} (over $\QQ$) with tame level $K^p$. Its complex points are given by $$\calX_{{\mathrm{GL}}_2}(\CC) = {\mathrm{GL}}_2(F)\; \backslash\; \big({\gothh^\pm}^{[F:\QQ]} \times {\mathrm{GL}}_2(\AAA_F^{\infty})\big) \;/\; \big(K^p \times {\mathrm{GL}}_2(\calO_{F, p})\big),$$ where $\gothh^\pm :=\CC \backslash \RR$ and $\calO_{F, p} := \calO_F \otimes_\ZZ {\ZZ_p}$. The Hilbert modular variety $\calX_{{\mathrm{GL}}_2}$ admits an integral model $\bfX_{{\mathrm{GL}}_2}$ over $\ZZ_{(p)}$ and let $X_{{\mathrm{GL}}_2}$ denote its special fiber over $\overline \FF_p$. Since $p$ is unramified in $F$, we may and will identify the $p$-adic embeddings of $F$ with the homomorphisms of $\calO_F$ to $\overline \FF_p$, i.e. $\operatorname{Hom}(F, \overline \QQ_p) \cong \operatorname{Hom}(\calO_F, \overline \FF_p)$. Let $\Sigma_\infty$ denote this set. (We shall later identify the $p$-adic embeddings with the real embeddings, hence the subscript $\infty$.) Under the latter description, the absolute Frobenius $\sigma$ acts on $\Sigma_\infty$ by taking an element $\tau$ to the composite $\sigma \tau:\calO_F \xrightarrow{\tau} \overline \FF_p \xrightarrow{x\mapsto x^p} \overline \FF_p$. This action decomposes $\Sigma_\infty$ into a disjoint union of cycles, parametrized by all $p$-adic places of $F$. Let $\calA$ denote the universal abelian variety over $X_{{\mathrm{GL}}_2}$. The sheaf of invariant differential 1-forms $\omega_{\calA/X_{{\mathrm{GL}}_2}}$ is then locally free of rank one as a module over $$\calO_F \otimes_\ZZ \calO_{X_{{\mathrm{GL}}_2}} \cong \bigoplus_{\tau \in \Sigma_\infty} \calO_{X_{{\mathrm{GL}}_2, \tau}},$$ where ${\mathcal{O}}_{X_{{\mathrm{GL}}_2, \tau}}$ is the direct summand on which $\calO_F$ acts through $\tau : \calO_F \to \overline \FF_p$. We then write accordingly $\omega_{\calA/X_{{\mathrm{GL}}_2}} = \bigoplus_{\tau \in \Sigma_\infty} \omega_\tau$; each $\omega_\tau$ is locally free of rank one over ${\mathcal{O}}_{X_{{\mathrm{GL}}_2}}$. The Verschiebung map induces an $\calO_F$-morphism $\omega_{A/X_{{\mathrm{GL}}_2}} \to \omega_{A^{(p)}/ X_{{\mathrm{GL}}_2}}$, which further induces a homomorphism $h_\tau: \omega_\tau \to \omega^{\otimes p}_{\sigma^{-1}\tau}$ for each $\tau \in \Sigma_\infty$. This map then defines a global section $h_\tau \in H^0(X_{{\mathrm{GL}}_2}, \omega_\tau^{\otimes -1} \otimes \omega^{\otimes p}_{\sigma^{-1}\tau})$; it is called the *partial Hasse invariant at* $\tau$. We use $X_\tau$ to denote the zero locus of $h_\tau$. For a subset $\ttT \subseteq \Sigma_\infty$, we put $X_\ttT = \bigcap_{\tau \in \ttT} X_\tau$. These $X_\ttT$’s give the *Goren-Oort stratification* of $X_{{\mathrm{GL}}_2}$. An alternative definition of $X_\ttT$ is given as follows: $z \in X_\ttT(\overline \FF_p)$ if and only if $\operatorname{Hom}(\alpha_p, A_z[p])$ under the action of $\calO_F$ has eigenvalues given by those embeddings $\tau \in \ttT$. We refer to [@goren-oort] for the proof of equivalence and a more detailed discussion. It is proved in [@goren-oort] that each $X_\tau$ is a smooth and proper divisor and they intersect transversally. Hence $X_\ttT$ is smooth of codimension $\#\ttT$ for any subset $\ttT \subseteq \Sigma_\infty$; it is proper if $\ttT \neq \emptyset$. Description of the Goren-Oort strata {#S:intro GO-strata} ------------------------------------ The goal of this paper is to give a *global* description of the Goren-Oort strata. Prior to our paper, most works focus on the $p$-divisible groups of the abelian varieties, which often provides a good access to the local structure of Goren-Oort strata, e.g. dimensions and smoothness. Unfortunately, there have been little understanding of the global geometry of $X_\ttT$, mostly in low dimension. We refer to the survey article [@andreatta-goren] for a historical account. Recently, Helm made a break-through progress in [@helm; @helm-PEL] by taking advantage of the moduli problem; he was able to describe the global geometry of certain analogous strata of the special fibers of Shimura varieties of type $U(2)$. Our proof of the main theorem of this paper is, roughly speaking, to complete Helm’s argument to cover all cases for the $U(2)$-Shimura varieties and then transfer the results from the unitary side to the quaternionic side. Rather than stating our main theorem in an abstract way, we prefer to give more examples to indicate the general pattern. When $F=\QQ$, this is discussed in Section \[S:modular curve\]. For $F \neq \QQ$, we fix an isomorphism $\CC \cong \overline \QQ_p$ and hence identify $\Sigma_\infty = \operatorname{Hom}(F, \overline \QQ_p)$ with the set of real embeddings of $F$. ### $F$ real quadratic and $p$ inert in $F$ Let $\infty_1$ and $\infty_2$ denote the two real embeddings of $F$, and $\tau_1$ and $\tau_2$ the corresponding $p$-adic embeddings (via the fixed isomorphism $ \CC \cong \overline \QQ_p$). Then our main theorem says that each $X_{\tau_i}$ is a $\PP^1$-bundle over the special fiber of discrete Shimura variety $\overline{\mathrm{Sh}}_{B_{\infty_1, \infty_2}^\times}$, where $B_{\infty_1, \infty_2}$ stands for the quaternion algebra over $F$ which ramifies at both archimedean places. The intersection $X_{\tau_1} \cap X_{\tau_2}$ is isomorphic to the special fiber of the discrete Shimura variety $\overline{\mathrm{Sh}}_{ B_{\infty_1, \infty_2}^\times}({\mathrm{Iw}}_p)$, where $({\mathrm{Iw}}_p)$ means to take Iwahori level structure at $p$ instead. The two natural embeddings of $X_{\tau_1} \cap X_{\tau_2}$ into $X_{\tau_1}$ and $X_{\tau_2}$ induces two morphisms $\overline{\mathrm{Sh}}_{ B_{\infty_1, \infty_2}^\times}({\mathrm{Iw}}_p) \to \overline{\mathrm{Sh}}_{ B_{\infty_1, \infty_2}^\times}$; this gives (certain variant of) the Hecke correspondence at $p$ (see Theorem \[T:link and Hecke operator\]). We remark here that it was first proved in [@bachmat-goren] that the one-dimensional strata are disjoint unions of $\PP^1$ and the number of such $\PP^1$’s is also computed in [@bachmat-goren]. This computation relies on the intersection theory and does not provide a natural parametrization as we gave above. Our proof will be completely different from theirs. One can easily recover their counting from our natural parametrization. ### Quaternionic Shimura varieties Before proceeding, we clarify our convention on quaternionic Shimura varieties. For $\ttS$ an even subset of archimedean and $p$-adic places of $F$, we use $B_\ttS$ to denote the quaternion algebra over $F$ which ramifies exactly at the places in $\ttS$. We fix an identification $B_\ttS^\times(\AAA^{\infty, p}) \cong {\mathrm{GL}}_2(\AAA^{\infty, p})$. We fix a maximal open compact subgroup $B_\ttS^\times(\calO_{F,p})$ of $B_\ttS^\times(F \otimes_\QQ {\QQ_p})$. We use $\ttS_\infty$ to denote the subset of archimedean places of $\ttS$. The Shimura variety $\calS h_{B_\ttS^\times}$ for the algebraic group $\operatorname{Res}_{F/\QQ} B_\ttS^\times$ has complex points $$\calS h_{B_\ttS^\times}(\CC) = B_\ttS^\times(F) \; \backslash \; \big({\gothh^\pm}^{[F:\QQ]-\#\ttS} \times B_\ttS^\times(\AAA_F^{\infty}) \big)\; / \; \big( K^p \times B_\ttS^\times(\calO_{F,p})\big).$$ Here and later, the tame level $K^p$ is uniformly matched up for all quaternionic Shimura varieties. Unfortunately, $\calS h_{B_\ttS^\times}$ itself does not possess a moduli interpretation. We follow the construction of Carayol [@carayol] to relate it with a unitary Shimura variety $Y$ and “carry over" the integral model of $Y$. The assumption $p >2$ comes from the verification of the extension property for the integral canonical model following Moonen [@moonen96 Corollary 3.8]. In any case, we use $\overline{\mathrm{Sh}}_{B_\ttS^\times}$ to denote the special fiber of the Shimura variety over $\overline \FF_p$. When we take the Iwahori level structure at $p$ instead, we write $\overline{\mathrm{Sh}}_{B_\ttS^\times}({\mathrm{Iw}}_p)$. ### $F$ real quartic and $p$ inert in $F$ Let $\infty_1, \dots, \infty_4$ denote the four real embeddings of $F$, labeled so that the corresponding $p$-adic embeddings $\tau_1, \dots, \tau_4$ satisfies $\sigma \tau_i = \tau_{i+1}$ with the convention that $\tau_i = \tau_{i\,(\mathrm{mod}\; 4)}$. We list the description of the strata as follows. Strata Description ------------------------------------------------------- ---------------------------------------------------------------------------------------------------------- $X_{\tau_i}$ for each $i$ $\PP^1$-bundle over $\overline{\mathrm{Sh}}_{B^\times_{\{\infty_{i-1}, \infty_i\}}}$ $X_{\{\tau_{i-1},\tau_i\}}$ for each $i$ $\overline{\mathrm{Sh}}_{B^\times_{\{\infty_{i-1}, \infty_i\}}}$ $X_{\{\tau_1, \tau_3\}}$ and $X_{\{\tau_2, \tau_4\}}$ $(\PP^1)^2$-bundle over $\overline{\mathrm{Sh}}_{B^\times_{\{\infty_1, \infty_2, \infty_3, \infty_4\}}}$ $X_{\ttT}$ with $\#\ttT = 3$ $\PP^1$-bundle over $\overline{\mathrm{Sh}}_{B^\times_{\{\infty_1, \infty_2, \infty_3, \infty_4\}}}$ $X_{\{\tau_1, \tau_2, \tau_3, \tau_4\}}$ $\overline{\mathrm{Sh}}_{B^\times_{\{\infty_1, \infty_2, \infty_3, \infty_4\}}}({\mathrm{Iw}}_p)$ In particular, we point out that for a codimension $2$ stratum, its shape depends on whether the two chosen $\tau_i$’s are adjacent in the cycle $\tau_1 \to \dots \to \tau_4 \to \tau_1$. ### $F$ general totally real of degree $g$ over $\QQ$ and $p$ inert in $F$ {#SS:p inert case} As before, we label the real embeddings of $F$ by $\infty_1, \dots, \infty_g$ such that the corresponding $p$-adic embeddings $\tau_1, \dots, \tau_g$ satisfy $\sigma \tau_i = \tau_{i+1}$ with the convention that $\tau_i = \tau_{i \,(\mathrm{mod}\; g)}$. The general statement for Goren-Oort strata takes the following form: for a subset $\ttT \subseteq \Sigma_\infty$, the strata $X_\ttT$ is isomorphic to a $(\PP^1)^N$-bundle over the special fiber of some quaternion Shimura variety $\overline {\mathrm{Sh}}_{B_{\ttS(\ttT)}^\times}$. We now explain, given $\ttT$, what $\ttS(\ttT)$ and $N$ are. - When $\ttT \subsetneq \Sigma_\infty$, we construct $\ttS(\ttT)$ as follows: if $\tau \notin \ttT$ and $\sigma^{-1}\tau, \dots, \sigma^{-r}\tau \in \ttT$, we put $\sigma^{-1}\tau, \dots, \sigma^{-2 \lceil r/2 \rceil}\tau$ into $\ttS(\ttT)$. In other words, we always have $\ttT \subseteq \ttS(\ttT)$, and $\ttS(\ttT)$ contains the additional element $\sigma^{-r-1}\tau$ if and only if the corresponding $r$ is odd. The number $N$ is the cardinality of $\ttS(\ttT) - \ttT$. - When $\ttT = \Sigma_\infty$, $N$ is always $0$; for $\ttS(\ttT)$, we need to distinguish the parity: - if $\#\Sigma_\infty$ is odd, we put $\ttS(\ttT) = \Sigma \cup \{p\}$; - if $\#\Sigma_\infty$ is even, we put $\ttS(\ttT) = \Sigma$ and we put an Iwahori level structure at $p$. ### $F$ general totally real and $p$ is unramified in $F$ {#S:F general p unramified} The general principle is: *different places above $p$ work “independently’ in the recipe of describing the strata (e.g. which places of the quaternion algebra are ramified); so we just take the “product" of all recipes for different $p$-adic places.* More concretely, let $p\calO_F = \gothp_1 \cdots \gothp_d$ be the prime ideal factorization. We use $\Sigma_\infty$ to denote the set of all archimedean embeddings of $F$, which is identified with the set of $p$-adic embeddings. We use $\Sigma_{\infty/\gothp_i}$ to denote those archimedean embeddings or equivalently $p$-adic embeddings that give rise to the $p$-adic place $\gothp_i$. Given any subset $\ttT \in \Sigma_\infty$, we put $\ttT_{\gothp_i} = \ttT \cap \Sigma_{\infty/\gothp_i}$. Applying the recipe in \[SS:p inert case\] to each $\ttT_{\gothp_i}$, we get a set of places $\ttS(\ttT_{\gothp_i})$ and a nonnegative number $N_{\gothp_i}$. We put $\ttS(\ttT) = \cup_{i=1}^d \ttS(\ttT_{\gothp_i})$ and $N = \sum_{i=1}^d N_{\gothp_i} = \sum_{i=1}^d \#(\ttS(\ttT_{\gothp_i}) - \ttT_{\gothp_i})$. Then $X_\ttT$ is a $(\PP^1)^N$-bundle over $\overline{\mathrm{Sh}}_{B_{\ttS(\ttT)}^\times}$ (with possibly some Iwahori level structure at appropriate places above $p$). We also prove analogous result on the global description of the Goren-Oort strata on general quaternionic Shimura varieties (Theorem \[T:main-thm\]). We refer to the content of the paper for the statement. The modification we need to do in the general quarternionic case is that one just “ignores" all ramified archimedean places and apply the above recipe formally to the set $\Sigma_\infty$ after “depriving all ramified archimedean places". Method of the proof ------------------- We briefly explain the idea behind the proof. The first step is to translate the question to an analogous question about (the special fiber of) unitary Shimura varieties. We use $X'$ to denote the special fiber of the unitary Shimura variety we start with, over which we have the universal abelian variety $A'$. Similar to the Hilbert case, we have naturally defined analogous Goren-Oort stratification given by divisors $X'_\tau$. We consider $X'_\ttT = \cap_{\tau \in \ttT}X'_\tau$. The idea is to prove the following sequence of isomorphisms $X'_\ttT \xleftarrow\cong Y'_\ttT \xrightarrow{\cong} Z'_\ttT$, where $Z'_\ttT$ is the $(\PP^1)$-power bundle over the special fiber of another unitary Shimura variety; it comes with a universal abelian variety $B'$; $Y'_\ttT$ is the moduli space classifying both $A'$ and $B'$ together with a quasi-isogeny $A' \to B'$ of certain fixed type (with very small degree); the two morphisms are just simply forgetful morphisms. We defer the characterization of the quasi-isogeny to the content of the paper. To prove the two isomorphisms above, we simply check that the natural forgetful morphisms are bijective on closed points and induce isomorphisms on the tangent spaces. We point out that we have been deliberately working with the special fiber over the algebraic closure $\overline \FF_p$. This is because the description of the stratification is *not* compatible with the action of the Frobenius. In fact, a more rigorous way to formulate theorem is to compare the special fiber of the Shimura variety associated to ${\mathrm{GL}}_2(F) \times_{F^\times} E^\times$ and that to $B_{\ttS(\ttT)}^\times \times_{F^\times} E^\times$. The homomorphism from the Deligne torus into the two $E^\times$ are in fact different, causing the incompatibility of the Frobenius action. (See Corollary \[C:main-thm-product\] for the corresponding statement.) The result about quaternionic Shimura variety is obtained by comparing geometric connected components of the corresponding Shimura varieties, in which we lose the Frobenius action. See Remark \[R:quaternionic Shimura reciprocity not compatible\] for more discussion. Ampleness of automorphic line bundle ------------------------------------ An immediate application of the study of the global geometry of the Goren-Oort stratification is to give a necessary condition (hopefully also sufficient) for an automorphic line bundle to be ample. As before, we take $F$ to be a totally real field of degree $g$ in which $p$ is inert for simplicity. Let $X^*_{{\mathrm{GL}}_2}$ denote the special fiber of the minimal compactification of the Hilbert modular variety. We label all $p$-adic embeddings as $\tau_1, \dots, \tau_g$ with subindices considered modulo $g$, such that $\sigma \tau_i = \tau_{i+1}$. We put $\omega_i = \omega_{\tau_i}$; they form a basis of the group of automorphic line bundles. The class $[\omega_i]$ in $\operatorname{Pic}(X)_\QQ : = \operatorname{Pic}(X) \otimes_\ZZ {\mathbb{Q}}$ of each $\omega_i$ extends to a class in $\operatorname{Pic}(X^*_{{\mathrm{GL}}_2})_\QQ$, still denoted by $[\omega_i]$. For a $g$-tuple $\underline k = (k_1, \dots, k_g) \in \ZZ^g$, we put $[\omega^{\underline k}] = \sum_{i=1}^g k_i[\omega_i]$. Propobably slightly contrary to the common intuition from the case of modular forms, we prove the following. \[T:introduction ample\] If the rational class of line bundle $[\omega^{\underline k}]$ is ample, then $$\label{E:ampleness condition GL2} p k_i > k_{i-1} \quad \textrm{for all }i; \quad \textrm{(and all }k_i >0).$$ Here we put the second condition in parentheses because it automatically follows from the first condition. This theorem is proved in Theorem \[T:ampleness\]. When $F$ is a real quadratic field, Theorem \[T:introduction ample\] is proved in [@andreatta-goren Theorem 8.1.1]. To see that the condition is necessary, we simply restrict to each of the GO-strata $X_{\tau_i}$, which is a $\PP^1$-bundle as we discussed before. Along each of the $\PP^1$-fiber, the line bundle $\omega^{\underline k}$ restricts to $\calO(pk_i - k_{i-1})$. The condition  is clear. We do expect the condition in Theorem \[T:introduction ample\] to be necessary; but we are not able to prove it due to a combinatorics complication. Forthcoming works in this series -------------------------------- We briefly advertise the other papers of this series to indicate the potential applications of the technical result in this paper. In the subsequent paper [@tian-xiao2], we discuss an application to the classicality of overconvergent Hilbert modular forms, following the original proof of R. Coleman. In the third paper [@tian-xiao3], we show that certain generalizations of the Goren-Oort strata realize Tate classes of the special fiber of certain Hilbert modular varieties, and hence verify the Tate Conjecture under some mild hypothesis. Structure of the paper ---------------------- In Section \[Section:Sh Var\], we review some basic facts about integral models of Shimrua varieties, which will be used to relate the quarternionic Shimura varieties with the unitary Shimura varieties. One novelty is that we include a discussion about the “canonical model" of certain discrete Shimura varieties, this can be treated uniformly together with usual Shimura varieties. In Section \[Section:Integral-model\], we construct the integral canonical model for quaternionic Shimura varieties, following the Carayol [@carayol]. However, we tailor many of the choices (e.g. the auxiliary CM field, signatures) for our later application. In Section \[Section:defn of GOstrata\], we define the Goren-Oort stratification for the unitary Shimura varieties and transfer them to the quaternionic Shimura varieties; this is a straightforward generalization of the work of Goren and Oort [@goren-oort]. In Section \[Section:GO-geometry\], we give the global description of Goren-Oort stratification. The method is very close to that used in [@helm]. In Section \[Section:GO divisors\], we give more detailed description for Goren-Oort divisors, including a necessary condition for an automorphic line bundle to be ample, and a structure theorem relating the Goren-Oort stratification along a $\PP^1$-bundle morphism provided by Theorem \[T:main-thm-unitary\]. In Section \[Section:links\], we further study some structure of the Goren-Oort stata which will play an important role in the forthcoming paper [@tian-xiao3]. Acknowledgements {#acknowledgements .unnumbered} ---------------- We thank Ahmed Abbes, Matthew Emerton, and David Helm for useful discussions. We started working on this project when we were attending a workshop held at the Institute of Advance Study at Hongkong University of Science and Technology in December 2011. The hospitality of the institution and the well-organization provided us a great environment for brainstorming ideas. We especially thank the organizers Jianshu Li and Shou-wu Zhang, as well as the staff at IAS of HKUST. We also thank Fields Institute and the Morningside Center; the authors discussed the project while both attending conferences at these two institutes. The second author is partially supported by a grant from the Simons Foundation \#278433. Notation {#S:Notation-for-the-paper} -------- ### For a scheme $X$ over a ring $R$ and a ring homomorphism $R \to R'$, we use $X_{R'}$ to denote the base change $X \times_{\operatorname{Spec}R} \operatorname{Spec}R'$. For a field $F$, we use $\operatorname{Gal}_F$ to denote its Galois group. For a number field $F$, we use $\AAA_F$ to denote its ring of adèles, and $\AAA_F^\infty$ (resp. $\AAA_F^{\infty, p}$) to denote its finite adèles (resp. prime-to-$p$ finite adèles). When $F = \QQ$, we suppress the subscript $F$ from the notation. We use superscript ${\mathrm{cl}}$ to mean closure in certain topological groups; for example, $F^{\times, {\mathrm{cl}}}$ means the closure of $F^\times$ inside $\AAA_F^{\infty, \times}$ or $ \AAA_F^{\infty, p}$ (depending the situation). We put $\widehat \ZZ = \prod_l \ZZ_l$ and $\widehat \ZZ^{(p)} = \prod_{l \neq p} \ZZ_l$. For each finite place $\gothp$ of $F$, let $F_\gothp$ denote the completion of $F$ at $\gothp$ and $\calO_\gothp$ its valuation ring, which has uniformizer $\varpi_\gothp$ and residue field $k_\gothp$. (When $F_\gothp$ is unramified over $\QQ_p$, we take $\varpi_\gothp$ to be $p$.) We normalize the Artin map $\operatorname{Art}_F: F^\times \backslash \AAA^\times_F \to \operatorname{Gal}_F^{\mathrm{ab}}$ so that for each finite prime $\gothp$, the local uniformizer at $\gothp$ is mapped to a geometric Frobenius at $\gothp$. ### For $A$ an abelian scheme over a scheme $S$, we denote by $A^{\vee}$ the dual abelian scheme, by $\operatorname{Lie}(A)$ the Lie algebra of $A$, and by $\omega_{A/S}$ the module of *invariant $1$-differential forms* of $A$ relative to $S$. We sometimes omit $S$ from the notation when the base is clear. For a finite $p$-group scheme or a $p$-divisible group $G$ over a perfect field $k$ of characteristic $p$, we use $\calD(G)$ to denote its *covariant* Dieudonné module. For an abelian variety $A$ over $k$, we write $\calD_A$ for $\calD(A[p])$ and write $\tilde \calD_A$ for $\calD(A[p^\infty])$. ### {#SS:notation-F} Throughout this paper, we fix a totally real field $F$ of degree $g>1$ over $\QQ$. Let $\Sigma$ denote the set of places of $F$, and $\Sigma_\infty$ the subset of archimedean places, or equivalently, all real embeddings of $F$. We fix a prime number $p$ which is unramified in $F$. Let $\Sigma_p$ denote the set of places of $F$ above $p$. We fix an isomorphism $\iota_p: \CC \xrightarrow{\cong} \overline \QQ_p$. For each $\gothp \in \Sigma_p$, we use $\Sigma_{\infty/\gothp}$ to denote the subset of $\tau\in \Sigma_{\infty}$ for which $\tau'$ induces the $p$-adic place $\gothp$. Since $p$ is unramified, each $\tau'$ induces an embedding $\calO_F \hookrightarrow W(\overline \FF_p)$. Post-composition with Frobenius $\sigma$ on the latter induces an action of $\sigma$ on the set of $p$-adic embeddings and hence make each $\Sigma_{\infty/\gothp}$ into one cycle; we use $\sigma \tau$ to denote this action, i.e. $\sigma \circ \tau' = (\sigma \tau)'$. ### {#SS:notation for E} We will consider a CM extension $E$ over $F$, in which all places above $p$ are unramified. Let $\Sigma_{E, \infty}$ denote the set of complex embeddings of $E$. For $\tau \in \Sigma_\infty$, we often use $\tilde \tau$ to denote a/some complex embedding of $E$ extending $\tau$; we write $\tilde \tau^c$ for its complex conjugation. Using the isomorphism $\iota$ above, we write $\tilde \tau': = \iota_p\circ \tilde \tau$ for the corresponding $p$-adic embedding of $E$; again $\tilde \tau'^c$ for the $p$-adic embedding of $E$ corresponding to $\tilde \tau ^c$. Under the natural two-to-one map $\Sigma_{E, \infty} \to \Sigma_\infty$, we use $\Sigma_{E, \infty/\gothp}$ to denote the preimage of $\Sigma_{\infty/\gothp}$. In case when $\gothp$ splits as $\gothq \gothq^c$ in $E/F$, we use $\Sigma_{E, \infty/\gothq}$ to denote the set of complex embeddings $\tilde \tau$ such that $\iota_p \circ \tilde \tau$ induces the $p$-adic place $\gothq$. ### {#SS:notation-S} For $\ttS$ an even subset of places of $F$, we denote by $B_\ttS$ the quaternion algebra over $F$ ramified at $\ttS$. Let ${\mathrm{Nm}}_{B_\ttS/F}: B_\ttS \to F$ denote the reduced norm and ${\mathrm{Tr}}_{B_\ttS/F}: B_\ttS \to F$ the reduced trace. We will use the following lists of algebraic groups. Let $G_\ttS$ denote the algebraic group $\operatorname{Res}_{F/{\mathbb{Q}}}B_\ttS^\times$. Let $E$ be the CM extension of $F$ above and we put $T_{E,\tilde \ttS} =\operatorname{Res}_{E/{\mathbb{Q}}}\GG_m$; see Subsection \[S:CM extension\] for the meaning of subscript $\tilde \ttS$. We put $\widetilde G_{\tilde \ttS} = G_\ttS \times T_{E, \tilde \ttS}$ and $G''_{\tilde \ttS} = G_\ttS \times_Z T_{E,\tilde \ttS}$, which is the quotient of $\widetilde G_{\tilde \ttS}$ by the subgroup $Z = \operatorname{Res}_{F/{\mathbb{Q}}}\GG_m$ embedded as $z \mapsto (z, z^{-1})$. let $G'_{\tilde \ttS}$ denote the subgroup of $G''_{\tilde \ttS}$ consisting of elements $(g, e)$ such that ${\mathrm{Nm}}_{B_\ttS/F}(g)\cdot {\mathrm{Nm}}_{E/F}(e) \in \GG_m$. We put $\ttS_{\infty}=\Sigma_{\infty}\cap \ttS$. For each place $\gothp\in \Sigma_p$, we set $\ttS_{\infty/\gothp}=\Sigma_{\infty/\gothp}\cap \ttS$. Basics of Shimura Varieties {#Section:Sh Var} =========================== We first collect some basic facts on integral canonical models of Shimura varieties. Our main references are [@deligne1; @deligne2; @milne-book; @kisin]. (Our convention follows [@milne-book; @kisin].) We focus on how to transfer integral canonical models of Shimura varieties from one group to another group. This is mostly well-known to the experts; we include the discussion here mostly for completeness. One novelty of this section, however, is that we give an appropriate definition of “canonical model" for certain discrete Shimura varieties, so that the construction holds uniformly for both regular Shimura varieties and these zero-dimensional ones. This will be very important for later application to transferring description of Goren-Oort strata between Shimura varieties for different groups. \[N:condition on G\] Fix a prime number $p$. Fix an isomorphism $\iota: \CC \xrightarrow{\cong }\overline \QQ_p$. We use $\overline \QQ$ to denote the algebraic closure of $\QQ$ inside $\CC$ (which is then identified with the algebraic closure of $\QQ$ inside $\overline \QQ_p$ via $\iota$). In this section, let $G$ be a connected reductive group over $\QQ$. We use $G(\RR)^+$ to denote the neutral connected component of $G(\RR)$. We put $G(\QQ)^+ = G(\RR)^+ \cap G(\QQ)$. We use $G^{\mathrm{ad}}$ to denote the adjoint group and $G^{\mathrm{der}}$ its derived subgroup. We use $G(\RR)_+$ to denote the preimage of $G^{\mathrm{ad}}(\RR)^+$ under the natural homomorphism $G(\RR) \to G^{\mathrm{ad}}(\RR)$; we put $G(\QQ)_+ = G(\RR)_+ \cap G(\QQ)$. For $S$ a torus over ${\QQ_p}$, let $S({\ZZ_p})$ denote the maximal open compact subgroup of $S({\QQ_p})$. Shimura varieties over $\CC$ {#A:Shimura varieties} ---------------------------- Put $\SSS =\operatorname{Res}_{\CC/\RR}\GG_m$. For a real vector space $V$, a *Deligne homomorphism* $h: \SSS_\RR \to {\mathrm{GL}}(V)$ induces a direct sum decomposition $V_\CC = \oplus_{a, b \in \ZZ} V^{a,b}$ such that $z \in \SSS(\RR) \cong \CC^\times$ acts on $V^{a,b}$ via the character $z^{-a} \bar z^{-b}$. Let $r$ denote the $\CC$-homomorphism $\GG_{m, \CC} \to \SSS_\CC$ such that $z^{-a} \bar z^{-b} \circ r = (x \mapsto x^{-a})$. A *Shimura datum* is a pair $(G, X)$ consisting of a connected reductive group $G$ over $\QQ$ and a $G(\RR)$-conjugacy class $X$ of homomorphisms $h: \operatorname{Res}_{\CC/\RR}\GG_m \to G_\RR$ satisfying the following conditions: - for $h \in X$, only characters $z/\bar z, 1, \bar z / z$ occur in the representation of $\SSS(\RR) \cong \CC^\times$ on $\operatorname{Lie}(G^{\mathrm{ad}})_\CC$ via $\mathrm{Ad} \circ h$; - for $h \in X$, $\mathrm{Ad}(h(i))$ is a Cartan involution on $G^{\mathrm{ad}}_\RR$; and - $G^{\mathrm{ad}}$ has no $\QQ$-factor $H$ such that $H(\RR)$ is compact. The $G(\RR)$-conjugacy class $X$ of $h$ admits the structure of a complex manifold. Let $X^+$ denote a fixed connected component of $X$. A pair $(G,X)$ satisfying only (SV1) (SV2) and the following (SV3)’ is called a *weak Shimura datum*. - $G^{\mathrm{ad}}(\RR)$ is compact (and hence connected by [@borel p.277]; this forces the image of $h$ to land in the center $Z_\RR$ of $G_\RR$). For an open compact subgroup $K \subseteq G(\AAA^\infty)$, we define the *Shimura variety* for $(G, X)$ with level $K$ to be the quasi-projective variety ${\mathrm{Sh}}_K(G,X)_\CC$, whose $\CC$-points are $${\mathrm{Sh}}_K(G, X)(\CC) := G(\QQ) \backslash X \times G(\AAA^\infty) / K \cong G(\QQ)_+ \backslash X^+ \times G(\AAA^\infty) / K.$$ When $(G,X)$ is a weak Shimura datum, ${\mathrm{Sh}}_K(G,X)_\CC$ is just a finite set of points. Reflex field ------------ Let $(G,X)$ be a (weak) Shimura data. The *reflex field*, denoted by $E= E(G, X)$, is the field of definition of the conjugacy class of the composition $h \circ r: \GG_{m, \CC} \to \SSS_\CC \to G_\CC$. It is a subfield of $\CC$, finite over $\QQ$. We refer to [@deligne1] for the definition of the canonical model ${\mathrm{Sh}}_K(G,X)$ of ${\mathrm{Sh}}_K(G,X)_\CC$ over this reflex field $E$. We assume from now on, all (weak) Shimura varieties we consider in this section admit canonical models. (In fact, [*loc. cit.*]{} excludes the case when $(G,X)$ is a weak Shimura datum; we will give the meaning of the canonical model in this case in Subsection \[S:integral model weak Shimura datum\] later.) We will always assume that $K$ is the product $K^p K_p$ of an open compact subgroup $K^p$ of $G(\AAA^{\infty, p})$ and an open compact subgroup $K_p$ of $G({\QQ_p})$. Taking the inverse limit over the open compact subgroups $K^p$, we have ${\mathrm{Sh}}_{K_p}(G, X) := \varprojlim_{K^p} {\mathrm{Sh}}_{K^pK_p}(G, X)$. This is actually a scheme locally of finite type over $E$ carrying a natural (right) action of $G(\AAA^{\infty,p})$. Extension property {#S:extension property} ------------------ Let $\calO$ be the ring of integers in a finite extension of ${\mathbb{Q}}_p$. A scheme $X$ over $\calO$ is said to have the *extension property* over $\calO$ if for any smooth $\calO$-scheme $S$, a map $S \otimes \mathrm{Frac}(\calO) \to X$ extends to $S$ (Such an extension is automatically unique if it exists by the normality of $S$.) Note that this condition is weaker than the one given in [@kisin 2.3.7] but is enough to ensure the uniqueness. The chosen isomorphism $\CC \cong \overline \QQ_p$ identifies $E$ as a subfield of $\overline \QQ_p$; let $E_\wp$ denote the $p$-adic completion of $E$, $\calO_\wp$ its valuation ring with $\FF_\wp$ as the residue field. Let $E_\wp^{\mathrm{ur}}$ be the maximal unramified extension of $E_\wp$ and $\calO_\wp^{\mathrm{ur}}$ its valuation ring. An *integral canonical model* ${\mathrm{Sh}}_{K_p}(G, X)_{\calO_\wp}$ of ${\mathrm{Sh}}_{K_p}(G,X)$ over $\calO_{\wp}$ is an ${\mathcal{O}}_{\wp}$-scheme ${\mathrm{Sh}}_{K_p}(G,X)_{\calO_\wp}$, which is an inverse limit of smooth ${\mathcal{O}}_{\wp}$-schemes ${\mathrm{Sh}}_{K_{p}K^p}(G,X)_{\calO_{\wp}}$ with finite étale transition maps as $K^p$ varies, such that - there is an isomorphism ${\mathrm{Sh}}_K(G,X)_{\calO_\wp} \otimes_{\calO_\wp} E_\wp \cong {\mathrm{Sh}}_K(G,X) \otimes_{E} E_\wp$ for each $K$, and - ${\mathrm{Sh}}_{K_p}(G,X)_{\calO_\wp}=\varprojlim_{K^p}{\mathrm{Sh}}_{K^pK_p}(G,X)_{\calO_{\wp}}$ satisfies the extension property. Existence of integral canonical model of Shimura varieties of abelian type with hyperspecial level structure was proved by Kisin [@kisin]. Unfortunately, our application requires, in some special cases, certain non-hyperspecial level structures, as well as certain ramified groups. We have to establish the integral canonical model in two steps: we first prove the existence for some group $G'$ with the same derived and adjoint groups as $G$ (as is done in Section \[Section:Integral-model\]); we then reproduce a variant of an argument of Deligne to show that the integral canonical model for the Shimura variety for $G'$ gives that of $G$. The second step is well-known at least for regular Shimura varieties when $K_p$ is hyperspecial ([@kisin]); our limited contribution here is to include some non-hyperspecial case and to cover the case of discrete Shimura varieties, in a uniform way. \[H:hypo on G\] Let $(G,X)$ be a (weak) Shimura datum. From now on, we assume that the derived subgroup $G^\mathrm{der}$ is simply-connected, which will be the case when we apply the theory later. Let $Z$ denote the center of $G$. Let $\nu: G \twoheadrightarrow T$ denote the maximal abelian quotient of $G$. We fix an open compact subgroup $K_p$ of $G({\QQ_p})$ such that $\nu(K_p) = T({\ZZ_p})$ and $K_p \cap Z({\QQ_p}) = Z({\ZZ_p})$. Geometric connected components {#A:geometric connected components} ------------------------------ We put $T(\RR)^\dagger = \mathrm{Im}(Z(\RR) \to T(\RR))$ and $T(\QQ)^\dagger = T(\RR)^\dagger \cap T(\QQ)$. Put $T(\QQ)^{?, (p)} = T(\QQ)^? \cap T({\ZZ_p})$ for $? = \emptyset$ or $\dagger$. Let $Y$ denote the finite quotient $T(\RR) / T(\RR)^\dagger$, which is isomorphic to $T(\QQ) / T(\QQ)^\dagger$ because $T(\QQ)$ is dense in $T(\RR)$. The morphism $\nu: G \to T$ then induces a natural map $$\begin{aligned} \label{E:nu map} & \xymatrix@C=15pt{\nu\colon G(\QQ)_+ \backslash X^+ \times G(\AAA^\infty) / K \ar[r]^-\nu & T(\QQ)^\dagger \backslash T(\AAA^\infty) / \nu(K) \cong T(\QQ)^{\dagger, (p)} \backslash T(\AAA^{\infty, p}) / \nu(K^p) }\\ \nonumber &\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\cong T(\QQ)^{(p)}\backslash Y \times T(\AAA^{\infty, p}) / \nu (K^p).\end{aligned}$$ If $(G, X)$ is a Shimura datum, this map induces an isomorphism [@milne-book Theorem 5.17] on the set of geometric connected components $\pi_0\big( {\mathrm{Sh}}_K(G, X)_{\overline \QQ} \big)\cong T(\QQ)^{(p)}\backslash Y \times T(\AAA^{\infty, p}) / \nu (K^p)$. Taking inverse limit gives a bijection $$\pi_0({\mathrm{Sh}}_{K_p}(G, X)_{\overline \QQ}) \cong T(\QQ)^{\dagger, (p), {\mathrm{cl}}} \backslash T(\AAA^{\infty,p}) \cong T(\QQ)^{(p), {\mathrm{cl}}} \backslash Y \times T(\AAA^{\infty,p}),$$ Here and later, the superscript ${\mathrm{cl}}$ means taking closure in $T(\AAA^{\infty,p})$ (or in appropriate topological groups). Reciprocity law {#A:reciprocity law} --------------- Let $(G,X)$ be a (weak) Shimura datum. The composite $$\xymatrix{ \nu hr: \GG_{m, \CC} \ar[r]^-{r} & \SSS_\CC \ar[r]^-{h} & G_\CC \ar[r]^-{\nu} & T_\CC }$$ does not depend on the choice of $h$ (in the conjugacy class) and is defined over the reflex field $E$. The *Shimura reciprocity map* is given by $$\operatorname{Rec}(G, X):\ \operatorname{Res}_{E/\QQ}(\GG_m) \xrightarrow{\operatorname{Res}_{E/\QQ}(\nu hr)} T_E \xrightarrow{N_{E/\QQ}} T.$$ We normalize the Artin reciprocity map $\operatorname{Art}_E: \AAA^\times_E / (E^{\times}E_\RR^{\times, +})^{\mathrm{cl}}\xrightarrow{\cong} \operatorname{Gal}_E^{\mathrm{ab}}$ so that the local parameter at a finite place $\gothl$ is mapped to a geometric Frobenius at $\gothl$, where $E_\RR^{\times, +}$ is the identity connected component of $E_\RR^\times$. We denote the unramified Artin map at $\wp$ by $\operatorname{Art}_\wp: E_\wp^\times / \calO_\wp^\times \to \operatorname{Gal}_{E_\wp}^{{\mathrm{ab}}, \mathrm{ur}}$ (again normalized so that a uniformizer is mapped to the geometric Frobenius). The morphism $\operatorname{Rec}(G, X)$ induces a natural homomorphism $$\xymatrix@C=10pt{{\gothR\mathrm{ec}}= {\gothR\mathrm{ec}}(G, X): \ \operatorname{Gal}_E^{{\mathrm{ab}}} && \ar[ll]_-{\operatorname{Art}_E}^-\cong (E^\times E_\RR^{\times, +})^{\mathrm{cl}}\backslash \AAA^\times_E\ar[rrr]^-{\operatorname{Rec}(G,X)} &&& T(\QQ)^{\mathrm{cl}}\backslash Y \times T(\AAA^{\infty}). }$$ When $(G,X)$ is a Shimura datum, the Shimura reciprocity law [@milne-book Section 13] says that the action of $\sigma \in \operatorname{Gal}_E$ on $\pi_0({\mathrm{Sh}}_{K_p}(G,X)_{\overline \QQ}) \cong T(\QQ)^{\mathrm{cl}}\backslash Y \times T(\AAA^\infty) / T({\ZZ_p})$ is given by multiplication by ${\gothR\mathrm{ec}}(G,X)(\sigma)$. As a corollary, $\pi_0({\mathrm{Sh}}_{K_p}(G,X)_{\overline \QQ}) = \pi_0({\mathrm{Sh}}_{K_p}(G,X)_{E_\wp^{\mathrm{ur}}})$, i.e. the geometric connected components are seen over an unramified extension of $E_\wp$. The action of $\operatorname{Gal}_{E_\wp}^{{\mathrm{ab}}, {\mathrm{ur}}} = \operatorname{Gal}_{\FF_\wp}$ on the geometric connected component is then given by multiplication by the image of the Galois group element under the following map: $$\label{E:reciprocity-at-p} \xymatrix@C=10pt{ {\gothR\mathrm{ec}}_\wp = {\gothR\mathrm{ec}}_\wp(G, X): \ \operatorname{Gal}_{\FF_\wp}&& \ar[ll]_-{\operatorname{Art}_\wp}^-\cong \widehat{E_\wp^\times / \calO_\wp^\times} \to (E^\times E_\RR^{\times, +})^{\mathrm{cl}}\backslash \AAA^\times_E / \calO_\wp^\times \ar[rrr]^-{\operatorname{Rec}(G,X)} &&& T(\QQ)^{(p), {\mathrm{cl}}} \backslash Y \times T(\AAA^{\infty,p}), }$$ where $\widehat{E_\wp^\times / \calO_\wp^\times}$ denotes the profinite completion of $E_\wp^\times / \calO_\wp^\times$. Integral canonical model for weak Shimura datum {#S:integral model weak Shimura datum} ----------------------------------------------- When $(G,X)$ is a weak Shimura datum, the associated Shimura variety is, geometrically, a finite set of points; we define its canonical model by specifying the action of $\operatorname{Gal}_E$. The key observation is that condition (SV3)’ ensures that the morphism $\operatorname{Rec}(G,X)$ factors as $$\xymatrix@C=50pt{ \operatorname{Res}_{E/\QQ}(\GG_m) \ar[r]^-{\operatorname{Res}_{E/\QQ}(hr)} \ar@{-->}[dr]_{\operatorname{Rec}_Z(G,X)} & Z_E \ar[r]^{\operatorname{Res}_{E/\QQ}(\nu)} \ar[d]^{N_{E/\QQ}} & T_E \ar[d]^{N_{E/\QQ}} \\ & Z \ar[r]^{\nu} & T. }$$ We consider the natural homomorphism $$\xymatrix@C=10pt{{\gothR\mathrm{ec}}_Z: \ \operatorname{Gal}_E^{{\mathrm{ab}}} && \ar[ll]_-{\operatorname{Art}_E}^-\cong (E^\times E_\RR^{\times, +})^{\mathrm{cl}}\backslash \AAA^\times_E\ar[rrr]^-{\operatorname{Rec}_Z(G,X)} &&& Z(\QQ)^{\mathrm{cl}}Z(\RR) \backslash Z(\AAA) \cong Z(\QQ)^{\mathrm{cl}}\backslash Z(\AAA^{\infty}). }$$ We define the *canonical model* ${\mathrm{Sh}}_K(G,X)$ to be the (pro-)$E$-scheme whose base change to $\CC$ is isomorphic to ${\mathrm{Sh}}_K(G,X)_\CC$, such that every $\sigma \in \operatorname{Gal}_E$ acts on its $\overline \QQ$-points by multiplication by ${\gothR\mathrm{ec}}_Z(\sigma)$. In comparison to Subsection \[A:reciprocity law\], we have $\nu(\sigma(x)) = {\gothR\mathrm{ec}}(G,X)(\sigma)\cdot \nu( x)$ for any $x \in {\mathrm{Sh}}_K(G,X)(\overline \QQ)$. Since ${\mathrm{Sh}}_K(G,X)$ is just a finite union of spectra of some finite extensions of $E$, it naturally admits an integral canonical model over $\calO_\wp$ by taking the corresponding valuation rings. With the map ${\gothR\mathrm{ec}}_\wp$ as defined in , we have $\nu(\sigma(x)) = {\gothR\mathrm{ec}}_\wp(G,X)(\sigma)\cdot \nu(x)$ for any closed point $x \in {\mathrm{Sh}}_K(G,X)_{\calO_\wp}$ and $\sigma \in \operatorname{Gal}_{\FF_\wp}$. \[N:hypo on G\] We put $K^{\mathrm{der}}_p = K_p \cap G^{\mathrm{der}}({\QQ_p})$; let $K^{\mathrm{ad}}_p$ denote the image of $K_p$ in $G^{\mathrm{ad}}({\QQ_p})$. Set $G^?(\QQ)^{(p)} = G^?(\QQ) \cap K^?_p$ for $? = \emptyset, {\mathrm{ad}}$ and ${\mathrm{der}}$; they are the subgroups of $p$-integral elements. Put $G^{\mathrm{ad}}(\QQ)^{+, (p)} = G^{\mathrm{ad}}(\RR)^+ \cap G^{\mathrm{ad}}(\QQ)^{(p)}$ and $G^?(\QQ)_+^{(p)} = G^?(\RR)_+ \cap G^?(\QQ)^{(p)}$ for $? = \emptyset$ or ${\mathrm{der}}$. A group theoretic construction ------------------------------ Before proceeding, we recall a purely group theoretic construction. See [@deligne2 § 2.0.1] for more details. Let $H$ be a group equipped with an action $r$ of a group $\Delta$, and $\Gamma \subset H$ a $\Delta$-stable subgroup. Suppose given a $\Delta$-equivariant map $\varphi: \Gamma \to \Delta$, where $\Delta$ acts on itself by inner automorphisms, and suppose that for $\gamma \in \Gamma$, $\varphi(\gamma)$ acts on $H$ as inner conjugation by $\gamma$. Given the data above, we can first define the semi-product $H \rtimes \Delta$ using the action $r$. The conditions above imply that the natural map $\gamma \mapsto (\gamma, \varphi(\gamma)^{-1})$ embeds $\Gamma$ as a normal subgroup of $H \rtimes \Delta$. We define the *star extension* $H \ast_\Gamma \Delta$ to be the quotient of $H \rtimes \Delta$ by this subgroup. Two typical examples we will encounter later are $$G^{\mathrm{der}}(\AAA^{\infty, p}) \ast _{G^{\mathrm{der}}(\QQ)^{(p)}} G(\QQ)^{(p)} \cong G^{\mathrm{der}}(\AAA^{\infty, p}) \cdot G(\QQ)^{(p)} \quad \textrm{and} \quad G^{\mathrm{der}}(\AAA^{\infty,p}) \ast _{G^{\mathrm{der}}(\QQ)^{(p)}} G^{\mathrm{ad}}(\QQ)^{(p)}.$$ The connected components of the integral model {#A:connected integral model} ---------------------------------------------- Let $(G,X)$ be a (weak) Shimura datum. Suppose that there exists an integral canonical model ${\mathrm{Sh}}_{K_p}(G,X)_{\calO_\wp}$. For $K^p$ an open compact subgroup of $G(\AAA^{\infty, p})$, let ${\mathrm{Sh}}_{K^pK_p}(G, X)^\circ_{\calO_\wp^{\mathrm{ur}}}$ denote the open and closed subscheme whose $\CC$-points consists of the preimage of $\{1\}$ under the $\nu$-map in . When $(G,X)$ is a Shimura datum, this gives a connected component of ${\mathrm{Sh}}_{K^pK_p}(G,X)_{\calO_\wp^{\mathrm{ur}}}$. We put $$\label{E:connected components of Shimura varieties} {\mathrm{Sh}}_{K_p}(G, X)^\circ_{\calO_\wp^{\mathrm{ur}}} = \varprojlim_{K^p} {\mathrm{Sh}}_{K^pK_p}(G,X)^\circ_{\calO_\wp^{\mathrm{ur}}} \quad \textrm{and} \quad {\mathrm{Sh}}_{K_p}(G, X)^\circ_{\overline \FF_\wp} = {\mathrm{Sh}}_{K_p}(G, X)^\circ_{\calO_\wp^{\mathrm{ur}}} \otimes_{\calO_\wp^{\mathrm{ur}}} \overline \FF_\wp.$$ Note that the set of $\CC$-points of ${\mathrm{Sh}}_{K_p}(G, X)^\circ_{\calO_\wp^{\mathrm{ur}}}$ is nothing but $G^{\mathrm{der}}(\QQ)^{(p), {\mathrm{cl}}}_+ \backslash (X^+ \times G^{\mathrm{der}}(\AAA^{\infty, p}))$; when $(G,X)$ is a Shimura datum, strong approximation shows that this is a projective limit of connected complex manifold. In any case, this implies that ${\mathrm{Sh}}_{K_p}(G,X)^\circ_{\calO_\wp^{\mathrm{ur}}}$ depends only on $X$, the groups $G^{\mathrm{der}}$ and $G^{\mathrm{ad}}$ (as opposed to the group $G$), and the subgroups $K_p^{\mathrm{der}}$ and $K_p^{\mathrm{ad}}$ (as opposed to $K_p$). We also point out that gives rise to a natural map $$\label{E:nu-map on Sh var} \nu\colon \pi_0({\mathrm{Sh}}_{K_p}(G, X)_{\calO_\wp^{\mathrm{ur}}}) = \pi_0({\mathrm{Sh}}_{K_p}(G, X)_{\overline \FF_\wp}) = \pi_0({\mathrm{Sh}}_{K_p}(G,X)_{\overline \QQ}) \longrightarrow T(\QQ)^{(p), {\mathrm{cl}}} \backslash Y \times T(\AAA^{\infty, p}).$$ By abuse of language, we call the (geometric) connected components of the Shimura varieties, and the target of the set of connected components (although this is not the case if $(G,X)$ is a weak Shimura datum). The Shimura varieties ${\mathrm{Sh}}_{K_p}(G,X)_{{\boldsymbol ?}}$ for ${\boldsymbol ?} = \calO_\wp^{\mathrm{ur}}$ and $\overline \FF_\wp$ admit the following actions. 1. The natural right action of $G(\AAA^{\infty, p})$ on ${\mathrm{Sh}}_{K_p}(G,X)_{\overline \QQ}$ extends to a right action on ${\mathrm{Sh}}_{K_p}(G,X)_\textbf{?}$. The subgroup $Z(\QQ)^{(p)} := Z(\QQ) \cap G({\ZZ_p})$ acts trivially. So the right multiplication action above factors through $ G(\AAA^{\infty, p})\big/ Z(\QQ)^{(p), {\mathrm{cl}}}$. The induced action on the set of connected components is given by $\nu: G(\AAA^{\infty,p})\big/ Z(\QQ)^{(p),{\mathrm{cl}}}\to T(\QQ)^{\dagger,(p),{\mathrm{cl}}} \backslash T(\AAA^{\infty,p})$. 2. There is a right action $\rho$ of $G^{\mathrm{ad}}(\QQ)^{+,(p)}$ on ${\mathrm{Sh}}_{K_p}(G,X)_{\calO_\wp^{\mathrm{ur}}}$ such that the induced map on $\CC$-points is given by, for $g \in G^{\mathrm{ad}}(\QQ)^{+,(p)}$, $$\xymatrix@C=40pt@R=0pt{ \rho(g)\colon G(\QQ)_+^{\mathrm{cl}}\backslash X^+ \times G(\AAA^\infty) / K_p \ar[r] & G(\QQ)_+^{\mathrm{cl}}\backslash X^+ \times G(\AAA^\infty) /K_p \\ [x, a] \ar@{|->}[r] & [g^{-1}x, \mathrm{int}_{g^{-1}}(a)]. }$$ Here note that $K_p$ is stable under the conjugation action of $K^{\mathrm{ad}}_p$ and hence of $G^{\mathrm{ad}}(\QQ)^{+,(p)}$. One extends the action $\rho(g)$ to the integral model and hence to the special fiber using the extension property. Moreover, this action preserves the connected component ${\mathrm{Sh}}_{K_p}(G, X)^\circ_{\boldsymbol ?}$. 3. For an element $g \in G(\QQ)_+^{(p)}$, the two actions above coincide. Putting them together, we have a right action of the group $$\label{E:calG} \calG := \big(G(\AAA^{\infty,p}) \big/ Z(\QQ)^{(p), {\mathrm{cl}}}\big) \ast_{G(\QQ)^{(p)}_+/Z(\QQ)^{(p)}} G^{\mathrm{ad}}(\QQ)^{+,(p)}$$ on ${\mathrm{Sh}}_{K_p}(G, X)^\circ_{\boldsymbol ?}$. The induced action on the set of connected components is given by $$\nu \ast \mathrm{triv}: \calG \twoheadrightarrow T(\QQ)^{\dagger,(p),{\mathrm{cl}}} \backslash T(\AAA^\infty),$$ i.e., $\nu$ on the first factor and trivial on the second factor. 4. The Galois group $\operatorname{Gal}(E_\wp^{\mathrm{ur}}/E_\wp)$ acts on ${\mathrm{Sh}}_{K_p}(G)_{\boldsymbol ?}$, according to (and Subsection \[S:integral model weak Shimura datum\]). Let $\calE_{G, \wp}$ denote the subgroup of $\calG \times \operatorname{Gal}(E_\wp^{\mathrm{ur}}/ E_\wp)$ consisting of pairs $(g, \sigma)$ such that $(\nu \ast \mathrm{triv})(g)$ is equal to ${\gothR\mathrm{ec}}_\wp(\sigma)^{-1}$ in $T(\QQ)^{\dagger,(p),{\mathrm{cl}}} \backslash T(\AAA^{\infty,p})$. Then by the discussion above, the group $\calE_{G,\wp}$ acts on the connected component ${\mathrm{Sh}}_{K_p}(G,X)^\circ_{\boldsymbol ?}$. Conversely, knowing ${\mathrm{Sh}}_{K_p}(G,X)^\circ_{\boldsymbol ?}$ together with the action of $\calE_{G,\wp}$, we can recover the integral model ${\mathrm{Sh}}_{K_p}(G,X)_{\calO_\wp}$ or its special fiber ${\mathrm{Sh}}_{K_p}(G,X)_{\FF_\wp}$ of the Shimura variety as follows. We consider the (pro-)scheme ${\mathrm{Sh}}_{K_p}(G,X)^\circ_{\calO_\wp^{\mathrm{ur}}} \times_{\calE_{G,\wp}} \big(\calG \times \operatorname{Gal}(E_\wp^{\mathrm{ur}}/ E_\wp) \big)$. Since this is a projective limit of *quasi-projective* varieties, by Galois descent, it is the base change of a projective system of varieties ${\mathrm{Sh}}_{K_p}(G,X)_{\calO_\wp}$ from $\calO_\wp$ to $\calO_\wp^{\mathrm{ur}}$. The same argument applies to the special fiber. In general, for a finite unramified extension $\tilde E_{\tilde \wp}$ of $E_\wp$, we put $\calE_{G, \tilde E_{\tilde \wp}}$ to be the subgroup of $\calE_{G, \wp}$ consisting of elements whose second coordinate lives in $\operatorname{Gal}(E_\wp^{\mathrm{ur}}/\tilde E_{\tilde \wp})$. Knowing the action of $\calE_{G, \tilde E_{\tilde \wp}}$ on ${\mathrm{Sh}}_{K_p}(G,X)^\circ_{\calO_\wp^{\mathrm{ur}}}$ or ${\mathrm{Sh}}_{K_p}(G,X)^\circ_{\overline \FF_\wp}$ allows one to descend the integral model to ${\mathrm{Sh}}_{K_p}(G,X)_{\calO_{\tilde E_{\tilde \wp}}}$. Transferring mathematical objects {#S:transfer math obj} --------------------------------- One can slightly generalize the discussion above to $\calE_{G, \tilde E_{\tilde \wp}}$-equivariant mathematical objects over the Shimura variety. More precisely, for $? = \FF_\wp, \calO_\wp$, by a *mathematical object* $\calP$ over ${\mathrm{Sh}}_{K^p}(G,X)_?$, we mean, for each sufficiently small open compact subgroup $K^p$ of $G(\AAA^{\infty,p})$, we have a (pro-)scheme or a vector bundle (with a section) $\calP_{K^p}$ over ${\mathrm{Sh}}_{K_pK^p}(G,X)_{\boldsymbol ?}$, such that, for any subgroup $K_1^p \subseteq K_2^p$, $\calP_{K_1^p}$ is the base change of $\calP_{K_2^p}$ along the natural morphism ${\mathrm{Sh}}_{K_pK_1^p}(G,X)_{\boldsymbol ?} \to {\mathrm{Sh}}_{K_pK_2^p}(G,X)_{\boldsymbol ?}$. We say $\calP$ is *$\calG \times \operatorname{Gal}(E_\wp^{\mathrm{ur}}/\tilde E_{\tilde \wp})$-equivariant* if $\calP$ carries an action of $\calG \times \operatorname{Gal}(E_\wp^{\mathrm{ur}}/\tilde E_{\tilde \wp})$ that is compatible with the actions on the Shimura variety. Similarly, a *mathematical object* $\calP^\circ$ over ${\mathrm{Sh}}_{K^p}(G, X)^\circ_{?^{\mathrm{ur}}}$ is a (pro-)scheme or a vector bundle (with a section) as above, over the connected Shimura variety ${\mathrm{Sh}}_{K^p}(G, X)^\circ_{?^{\mathrm{ur}}}$, viewed as a pro-scheme. It is called $\calE_{G, \tilde E_{\tilde \wp}}$-equivariant, if it carries an action of the group compatible with the natural group action on the base Shimura variety. Similar to the discussion above, we have the following. \[C:mathematical objects equivalence\] There is a natural equivalence of categories between $\calG \times \operatorname{Gal}(E^{\mathrm{ur}}_\wp / \tilde E_{\tilde \wp})$-equivariant mathematical objects $\calP$ over the tower of Shimura varieties ${\mathrm{Sh}}_{K_pK^p}(G,X)_?$, and the category of mathematical objects $\calP^\circ$ over ${\mathrm{Sh}}_{K^p}(G,X)^\circ_{?^{\mathrm{ur}}}$, equivariant for the action of $\calE_{G, \tilde E_{\tilde \wp}}$. As above, given $\calP$, we can recover $\calP^\circ$ by taking inverse limit with respect to the open compact subgroup $K^p$ and then restricting to the connected component ${\mathrm{Sh}}_{K_p}(G,X)^\circ_{?^{\mathrm{ur}}}$. Conversely, we can recover $\calP$ from $\calP^\circ$ through the isomorphism $ \calP_{?^{\mathrm{ur}}} \cong \calP^\circ \times_{\calE_{G, \tilde E_{\tilde \wp}}} (\calG \times \operatorname{Gal}(E_\wp^{\mathrm{ur}}/\tilde E_{\tilde \wp}))$ and then using Galois descent if needed. \[R:no Galois action\] If one does not consider the Galois action, then Theorem \[T:structure of calE\_G,p\] below implies that $${\mathrm{Sh}}_{K_p}(G,X)_{\calO_\wp^{\mathrm{ur}}} \cong {\mathrm{Sh}}_{K_p}(G,X)_{\calO_\wp^{\mathrm{ur}}}^\circ \times_{\big(G^{\mathrm{der}}(\AAA^{\infty,p}) \ast_{G^{\mathrm{der}}(\QQ)^{(p)}_+} G^{\mathrm{ad}}(\QQ)^{+,(p)} \big)}\calG,$$ and the same applies to the mathematical objects. \[L:nu(G(Q)-&gt;&gt;T(Q)\] We have $\nu(G(\QQ)_+^{(p)}) = T(\QQ)^{\dagger, (p)}$. By Subsection \[A:geometric connected components\], we have $\nu(G(\QQ)_+) = T(\QQ)^\dagger$. The lemma follows from taking the kernels of the following morphism of exact sequences $$\xymatrix{ 1 \ar[r] & G^{\mathrm{der}}(\QQ)_+ \ar[r] \ar@{->>}[d] & G(\QQ)_+ \ar[r] \ar[d] & T(\QQ)^\dagger \ar[r] \ar[d] & 1\\ 1 \ar[r] & G^{\mathrm{der}}(\QQ_p)/K^{\mathrm{der}}_p \ar[r] & G(\QQ_p)/K_p \ar[r] & T({\QQ_p}) / T({\ZZ_p}) \ar[r] & 1 }$$ Here, the left vertical arrow is surjective by the strong approximation theorem for the simply-connected group $G^{\mathrm{der}}(\QQ)$. The bottom sequence is exact because the corresponding sequences are exact both for ${\QQ_p}$ (because $H^1({\QQ_p}, G^{\mathrm{der}})=0$) and for ${\ZZ_p}$ (by Hypothesis \[H:hypo on G\]). The following structure theorem for $\calE_{G, \wp}$ is the key to transfer integral canonical models of Shimura varieties for one group to that for another group. \[T:structure of calE\_G,p\] For a finite unramified extension $\tilde E_{\tilde \wp}$ of $E_\wp$, we have a natural short exact sequence. $$\label{E:structure of E_p} 1 \longrightarrow G^{\mathrm{der}}(\AAA^{\infty,p}) \ast_{G^{\mathrm{der}}(\QQ)^{(p)}_+} G^{\mathrm{ad}}(\QQ)^{+,(p)} \longrightarrow \calE_{G, \tilde E_{\tilde \wp}} \longrightarrow \operatorname{Gal}(E_\wp^{\mathrm{ur}}/ \tilde E_{\tilde \wp}) \longrightarrow 1.$$ By the definition of $\calE_{G,\tilde E_{\tilde \wp}}$, it fits into the following short exact sequence $$1 \longrightarrow \operatorname{Ker}\Big (\widetilde G_p \to T(\QQ)^{\dagger,(p), {\mathrm{cl}}} \big \backslash T(\AAA^{\infty,p}) \Big) \longrightarrow \calE_{G, \tilde E_{\tilde \wp}} \longrightarrow \operatorname{Gal}(E_\wp^{\mathrm{ur}}/ \tilde E_{\tilde \wp}) \longrightarrow 1.$$ By Lemma \[L:nu(G(Q)-&gt;&gt;T(Q)\], the kernel above is isomorphic to $$\label{E:expression of kernel} \big(\, \big(G(\QQ)_+^{(p)} G^{\mathrm{der}}(\AAA^{\infty,p})\big)^{\mathrm{cl}}\big/ Z(\QQ)^{(p), {\mathrm{cl}}} \big) \ast_{G(\QQ)_+^{(p)} / Z(\QQ)^{(p)}} G^{\mathrm{ad}}(\QQ)^{+,(p)},$$ where both closures are taken inside $G(\AAA^{\infty, p})$. We claim that we can remove the two completions. Indeed, put $Z' = Z \cap G^{\mathrm{der}}$ and $Z'(\QQ)^{(p)} = Z'(\QQ) \cap Z(\QQ)^{(p)}$; the latter is a finite group. Consider the commutative diagram of exact sequences $$\xymatrix@R=15pt@C=10pt{ 1 \ar[r] & Z'(\QQ)^{(p)} \ar[r] \ar@{=}[d] & Z(\QQ)^{(p)} \times G^{\mathrm{der}}(\AAA^{\infty,p}) \ar[d] \ar[r] & G(\QQ)_+^{(p)} G^{\mathrm{der}}(\AAA^{\infty,p}) \ar[d] \ar[r] & T(\QQ)^{\dagger,(p)} \big/ \mathrm{Im}(Z(\QQ)^{(p)} \to T(\QQ)^{(p)}) \ar[r] \ar[d] & 1 \\ 1 \ar[r] & Z'(\QQ)^{(p), {\mathrm{cl}}} \ar[r] &{Z(\QQ)^{(p), {\mathrm{cl}}}} \times G^{\mathrm{der}}(\AAA^{\infty,p}) \ar[r] & \big(G(\QQ)^{(p)}_+ G^{\mathrm{der}}(\AAA^{\infty,p})\big)^{\mathrm{cl}}\ar[r] & T(\QQ)^{\dagger, (p), {\mathrm{cl}}} \big/\mathrm{Im}({Z(\QQ)^{(p), {\mathrm{cl}}}} \to {T(\QQ)^{(p), {\mathrm{cl}}}}) \ar[r] & 1 }$$ By diagram chasing, it suffices to prove that the right vertical arrow is an isomorphism. Since the kernel of $Z \to T$ is finite, [@deligne2 § 2.0.10] implies that $\mathrm{Im}({Z(\QQ)^{\mathrm{cl}}} \to {T(\QQ)^{\mathrm{cl}}}) \cong (\mathrm{Im}(Z(\QQ) \to T(\QQ)))^{\mathrm{cl}}$ and the right vertical arrow is an isomorphism. Now, the exact sequence follows from a series of tautological isomorphisms $$\begin{aligned} &\big( G(\QQ)_+^{(p)} \cdot G^{\mathrm{der}}(\AAA^{\infty,p}) / Z(\QQ)^{(p)} \big) \ast_{G(\QQ)^{(p)}_+ / Z(\QQ)^{(p)}} G^{\mathrm{ad}}(\QQ)^{+,(p)} \\ \cong\ & \Big[ \big(G^{\mathrm{der}}(\AAA^{\infty,p}) \ast_{G^{\mathrm{der}}(\QQ)_+^{(p)}} G(\QQ)_+^{(p)}) / Z(\QQ)^{(p)} \Big] \ast_{G(\QQ)_+^{(p)} / Z(\QQ)^{(p)}} G^{\mathrm{ad}}(\QQ)^{+,(p)} \\ \cong\ & \Big[ G^{\mathrm{der}}(\AAA^{\infty,p}) \ast_{G^{\mathrm{der}}(\QQ)_+^{(p)}} \big( G(\QQ)_+^{(p)} / Z(\QQ)^{(p)} \big) \Big] \ast_{G(\QQ)_+^{(p)} / Z(\QQ)^{(p)}} G^{\mathrm{ad}}(\QQ)^{+,(p)} \\ \cong \ & G^{\mathrm{der}}(\AAA^{\infty,p}) \ast_{G^{\mathrm{der}}(\QQ)_+^{(p)}} G^{\mathrm{ad}}(\QQ)^{+, (p)}.\end{aligned}$$ \[C:Sh(G)\^circ\_Zp independent of G\] Let $G \to G'$ be a homomorphism of two reductive groups satisfying Hypothesis \[H:hypo on G\], which induces isomorphisms between the derived and adjoint groups as well as their $p$-integral elements. A $G^{\mathrm{ad}}(\RR)^+$-conjugacy class $X^+$ of homomorphisms $h: \SSS \to G_\RR$ induces a $G'^{\mathrm{ad}}(\RR)^+$-conjugacy class $X'^+$ of homomorphisms $h': \SSS \to G'_\RR$. Put $X = G(\RR) \cdot X^+$ and $X' = G'(\RR) \cdot X'^+$. Then, for any field $\tilde E_{\tilde \wp}$ containing both $E_\wp$ and $E'_{\wp'}$ and unramified over them, there exist a natural isomorphism of groups $\calE_{G, \tilde E_{\tilde \wp}} \xrightarrow{ \cong} \calE_{G', \tilde E_{\tilde \wp}}$ and a natural isomorphism of geometric connected components of Shimura varieties ${\mathrm{Sh}}_{K_p}(G, X)_{\tilde E_{\tilde \wp}^{\mathrm{ur}}}^\circ \cong {\mathrm{Sh}}_{K'_p}(G', X')_{\tilde E_{\tilde \wp}^{\mathrm{ur}}}^\circ$ equivariant for the natural actions of the groups. As a corollary, if the Shimura variety for one of $G$ or $G'$ admits an integral canonical model and both $E_\wp$ and $E'_{\wp'}$ are unramified extensions of $\QQ_p$, then the other Shimura variety admits an integral canonical model. Moreover, when there are canonical integral models, we have an equivalence of categories between the category of $\calG \times \operatorname{Gal}(E_\wp^{\mathrm{ur}}/\tilde E_{\tilde \wp})$-equivariant mathematical objects $\calP$ over the tower of Shimura varieties ${\mathrm{Sh}}_{K_pK^p}(G,X)_?$ (for $? = \calO_{\tilde \wp}$ or $\FF_{\tilde \wp}$) and the categories of $\calG' \times \operatorname{Gal}(E_\wp^{\mathrm{ur}}/\tilde E_{\tilde \wp})$-equivariant mathematical objects $\calP'$ over the tower of Shimura varieties ${\mathrm{Sh}}_{K'_pK'^p}(G',X')_{?'}$ (for $?' = \calO_{\tilde \wp'}$ or $\FF_{\tilde \wp'}$). The first part follows from Theorem \[T:structure of calE\_G,p\] and the discussion in Subsection \[A:connected integral model\]. For the second part, the existence of integral canonical model over $\tilde E_{\tilde \wp}$ follows from the first part and the discussion at the end of Subsection \[A:connected integral model\]. The extension property allows one to further descend the integral canonical model to $\calO_\wp$ (or $\calO_{\wp'}$). The last part follows from Corollary \[C:mathematical objects equivalence\]. Integral canonical model of quaternionic Shimura varieties {#Section:Integral-model} ========================================================== Classically, the integral model for a quaternionic Shimura variety is defined by passing to a unitary Shimura variety, as is done in the curve case by Carayol [@carayol]. As pointed out earlier that we will encounter some groups which are not quasi-split at $p$, Kisin’s general work [@kisin] unfortunately does not apply. We have to work out the general Carayol’s construction for completeness; this will also be useful later when discussing the construction of the Goren-Oort stratification. We tailor the choice of the unitary group to our application of Helm’s isogeny trick later; in particular, we will assume that certain places above $p$ to be inert in the CM extension. Quaternionic Shimura varieties {#S:quaternionic-shimura-varieties} ------------------------------ Recall the notation from \[SS:notation-F\]. Let $\ttS$ be an even subset of places of $F$. Put $\ttS_\infty = \ttS \cap \Sigma_\infty$ and $\ttS_p = \ttS\cap \Sigma_p$. Let $B_\ttS$ be the quaternion algebra over $F$ ramified presicsely at $\ttS$. Let $G_{\ttS}$ denote the reductive group $\operatorname{Res}_{F/\QQ}(B^\times_\ttS)$. Then $G_{\ttS,\RR}$ is isomorphic to $$\prod_{\tau \in \ttS_\infty} \HH^\times \times \prod_{\tau \in\Sigma_\infty -\ttS_\infty} {\mathrm{GL}}_{2, \RR}.$$ We define the [Deligne homomorphism]{} to be $h_\ttS: \SSS \to G_{\ttS,\RR}$, sending $z = x +\ii y$ to $(z_{G_\ttS}^\tau)_{\tau \in \Sigma_\infty}$, where $z_{G_\ttS}^\tau = 1$ if $\tau \in \ttS_\infty$ and $z_{G_\ttS}^\tau = \big(\begin{smallmatrix} x&y\\ -y&x \end{smallmatrix}\big)$ if $\tau \in \Sigma_\infty -\ttS_\infty$. Let $\gothH_\ttS$ denote the $G_\ttS(\RR)$-conjugacy class of the homomorphism $h_\ttS$; it is isomorphic to the product of $\#(\Sigma_\infty - \ttS_\infty)$ copies of $\gothh^\pm = \PP^1(\CC) - \PP^1(\RR)$. We put $\gothH_\ttS^+ = (\gothh^+)^{\Sigma_\infty - \ttS_\infty}$, where $\gothh^+$ denotes the upper half plane. We will consider the following type of open compact subgroups of $G_\ttS(\AAA^\infty)$: $K = K^pK_p$, where $K^p$ is an open compact subgroup of $B^\times_\ttS(\AAA_F^{\infty, p})$ and $K_p = \prod_{\gothp \in \Sigma_p} K_\gothp$ with $K_\gothp$ an open compact subgroup of $B_\ttS^\times(F_\gothp)$. From this point onward, we write ${\mathrm{Sh}}_K(G)$ instead of ${\mathrm{Sh}}_K(G,X)$ for Shimura varieties when the choice of $X$ is clear. Associated to the data above, there is a Shimura variety ${\mathrm{Sh}}_K(G_\ttS)$ whose $\CC$-points are $${\mathrm{Sh}}_K(G_\ttS)(\CC) = G_\ttS(\QQ) \backslash ( \gothH_\ttS \times G_\ttS(\AAA^\infty))/ K.$$ The *reflex field* $F_\ttS$ is a subfield of $\CC$ characterized as follows: an element $\sigma \in \operatorname{Aut}(\CC/\QQ)$ fixes $F_\ttS$ if and only if the subset $\ttS_\infty$ of $\Sigma_\infty $ is preserved under the action of $\sigma$ by post-composition. Following Subsection \[A:Shimura varieties\], we put ${\mathrm{Sh}}_{K_p}(G_\ttS) = \varprojlim_{K^p} {\mathrm{Sh}}_{K^pK_p}(G_\ttS)$. (Note that the level structure at $p$ is fixed in the inverse limit.) Put $T_F = \operatorname{Res}_{F/\QQ}\GG_m$. The reduced norm on $B_\ttS$ induces a homomorphism ${\mathrm{Nm}}= {\mathrm{Nm}}_{B_\ttS / F}: G_\ttS \to T_F$. This homomorphism induces a map $$\pi_0^{\mathrm{geom}}({\mathrm{Sh}}_K(G_\ttS)) \longrightarrow T_F(\QQ) \backslash ( T_F(\AAA^\infty) \times \{\pm 1\}^g) / {\mathrm{Nm}}(K),$$ which is an isomorphism if $\ttS_\infty \subsetneq \Sigma_\infty$. We will make the Shimura reciprocity law (Subsection \[A:reciprocity law\]) explicit for ${\mathrm{Sh}}_K(G_\ttS)$ later when it is in use. Level structure at $p$ {#S:level-structure-at-p} ---------------------- We fix an isomorphism $\iota_p: \CC \simeq \overline \QQ_p$. For each $\gothp \in \Sigma_p$, let $\Sigma_{\infty/\gothp}$ denote the subset of $\Sigma_\infty$ consisting of real embeddings which, when composed with $\iota_p$, induce the $p$-adic place $\gothp$. We put $\ttS_{\infty/\gothp} = \ttS \cap \Sigma_{\infty/\gothp}$. Similarly, we can view the reflexive field $F_\ttS$ above as a subfield of $\overline \QQ_p$ via $\iota_p$, which induces a $p$-adic place $\wp$ of $F_\ttS$. We use $\calO_\wp$ to denote the valuation ring and $k_\wp$ the residue field. In this paper, we always make the following assumption on $\ttS$: \[H:B\_S-splits-at-p\] If $B_\ttS$ does not split at a $p$-adic place $\gothp$ of $F$, then $\ttS_{\infty/\gothp}= \Sigma_{\infty/\gothp}$. For each $\gothp\in \Sigma_{p}$, we now specify the level structure $K_{\gothp}\subset B_{\ttS}^\times(F_\gothp)$ of ${\mathrm{Sh}}_{K}(G_{\ttS})$ to be considered in this paper. We distinguish four types of the prime $\gothp \in \Sigma_p$: - [**Types $\alpha$ and $\alpha^\sharp$:**]{} $B_{\ttS}$ splits at $\gothp$ and the cardinality $\# (\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp})$ is even. We fix an identification $B^{\times}_{\ttS}(F_{\gothp})\simeq {\mathrm{GL}}_2(F_{\gothp})$. We take $K_{\gothp}$ to be - either ${\mathrm{GL}}_2({\mathcal{O}}_{\gothp})$, or - ${\mathrm{Iw}}_\gothp = \big( \begin{smallmatrix} \calO_\gothp ^\times & \calO_\gothp\\ \gothp\calO_\gothp & \calO_\gothp^\times \end{smallmatrix} \big)$ which we allow only when $\Sigma_{\infty/\gothp} = \ttS_{\infty/\gothp}$. We name the former case as type $\alpha$ and the latter as type $\alpha^\sharp$. (Under our definition, when $\Sigma_{\infty/\gothp} = \ttS_{\infty/\gothp}$, the type of $\gothp$ depends on the choice of the level structure.) - [**Type $\beta$:**]{} $B_{\ttS}$ splits at $\gothp$ and the cardinality $\#(\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp})$ is odd. We fix an identification $B_{\ttS}^{\times}(F_{\gothp})\simeq {\mathrm{GL}}_2(F_{\gothp})$. We take $K_{\gothp}$ to be ${\mathrm{GL}}_2({\mathcal{O}}_{\gothp})$. - [**Type $\beta^\sharp$:**]{} $B_\ttS$ ramifies at $\gothp$ and $\ttS_{\infty/\gothp} = \Sigma_{\infty/\gothp}$. In this case, $B_{\ttS}\otimes_{F}F_{\gothp}$ is the division quaternion algebra $B_{F_{\gothp}}$ over $F_{\gothp}$. Let ${\mathcal{O}}_{B_{F_{\gothp}}}$ be the maximal order of $B_{F_{\gothp}}$. We take $K_{\gothp}$ to be ${\mathcal{O}}_{B_{F_{\gothp}}}^{\times}$. The aim of this section is to construct an integral canonical model of ${\mathrm{Sh}}_K(G_{\ttS})$ over $\calO_\wp$ with $K_{p}=\prod_{\gothp|p}K_{\gothp}$ specified above. For this, we need to introduce an auxiliary CM extension and a unitary group. Auxiliary CM extension {#S:CM extension} ---------------------- We choose a CM extension $E$ over $F$ such that - every place in $\ttS$ is inert in $E/F$; and - a place $\gothp \in \Sigma_p$ is split in $E/F$ if it is of type $\alpha$ or $\alpha^\sharp$, and is inert in $E/F$ if it is of type $\beta$ or $\beta^\sharp$. We remark that our construction slightly differs from [@carayol] in that Carayol requires all places above $p$ to split in $E/F$. For later convenience, we fix some totally negative element $\gothd \in \calO_F$ coprime to $p$ so that $E = F(\sqrt{\gothd})$. (The construction will be independent of such choice.) Let $\Sigma_{E, \infty}$ denote the set of complex embeddings of $E$. We have a natural two-to-one map $\Sigma_{E, \infty} \to \Sigma_\infty$. For each $\tau \in \Sigma_\infty$, we often use $\tilde \tau $ to denote a complex embedding of $E$ extending $\tau$; its complex conjugate is denoted by $\tilde \tau^c$. We fix a choice of a subset $\tilde \ttS_\infty \subseteq \Sigma_{E,\infty}$ which consists of, for each $\tau \in \ttS_\infty$, a choice exactly one lift $\tilde \tau\in \Sigma_{E,\infty}$. This choice is equivalent to a collection of the numbers $s_{\tilde \tau} \in \{0,1,2\}$ for all $\tilde \tau \in \Sigma_{E, \infty}$ such that - if $\tau \in \Sigma_\infty - \ttS_\infty$, we have $s_{\tilde \tau} = 1$ for all lifts $\tilde \tau $ of $\tau$; - if $\tau \in \ttS_\infty$ and $\tilde \tau$ is the lift in $\tilde \ttS_\infty$, we have $s_{\tilde \tau} = 0$ and $s_{\tilde \tau^c} = 2$. We put $\tilde\ttS=(\ttS,\tilde\ttS_{\infty})$. Consider the torus $T_{E, \tilde \ttS} = \operatorname{Res}_{E/{\mathbb{Q}}}\GG_m$ together with the following choice of the Deligne homomorphism: $$\xymatrix@R=0pt@C=50pt{ h_{E, \tilde \ttS}\colon \SSS(\RR) = \CC^\times \ar[r] & T_{E, \tilde \ttS}(\RR) = \bigoplus_{\tau \in \Sigma_\infty} (E \otimes_{F, \tau}\RR)^\times \simeq \bigoplus_{\tau\in\Sigma_\infty} \CC^\times\\ z\ar@{|->}[r] & (z_{E, \tau})_\tau. }$$ Here $z_{E, \tau} = 1$ if $\tau \in \Sigma_\infty -\ttS_\infty$ and $z_{E, \tau} = z$ otherwise, where, in the latter case, the isomorphism $(E \otimes_{F, \tau}\RR)^\times \simeq \CC^\times$ is given by the lift $\tilde \tau \in \tilde \ttS^c_\infty$. The reflex field $E_{\tilde \ttS}$ is the subfield of $\CC$ corresponding to the subgroup of $\operatorname{Aut}({\mathbb{C}}/{\mathbb{Q}})$ which stabilizes the set $\tilde \ttS_\infty \subset \Sigma_{E, \infty}$; it contains $F_\ttS$ as a subfield. The isomorphism $\iota_p: \CC \simeq \overline \QQ_p$ determines a $p$-adic place $\tilde \wp$ of $E_{\tilde \ttS}$; we use $\calO_{\tilde \wp}$ to denote the valuation ring and $k_{\tilde \wp}$ the residue field. Note that $[k_{\tilde \wp}: \FF_p]$ is always even whenever there is a place $\gothp\in \Sigma_p$ of type $\beta$. We take the level structure $K_E$ to be $K_E^pK_{E,p}$, where $K_{E, p} = (\calO_E \otimes_\ZZ \ZZ_p)^\times$, and $K_E^p$ is an open compact subgroup of $\AAA_E^{\infty,p,\times}$. This then gives rise to a Shimura variety ${\mathrm{Sh}}_{K_E}(T_{E, \tilde \ttS})$ and its limit ${\mathrm{Sh}}_{K_{E,p}}(T_{E, \tilde \ttS}) = \varprojlim_{K_E^p}{\mathrm{Sh}}_{K_{E,p}K_E^p}(T_{E, \tilde \ttS})$; they have integral canonical models ${\mathbf{Sh}}_{K_E}(T_{E, \tilde \ttS})$ and ${\mathbf{Sh}}_{K_{E,p}}(T_{E, \tilde \ttS})$ over $\calO_{\tilde \wp}$ as specified in \[S:integral model weak Shimura datum\]. We also consider the product group $ G_\ttS \times T_{E, \tilde \ttS}$ with the product Deligne homomorphism $$\tilde h_{\tilde \ttS} = h_\ttS \times h_{E, \tilde \ttS} \colon \SSS({\mathbb{R}}) = \CC^\times \longrightarrow (G_\ttS\times T_{E, \tilde \ttS})(\RR).$$ This gives rise to the product Shimura varieties: $$\begin{aligned} {\mathrm{Sh}}_{K \times K_E}(G_\ttS \times T_{E,\tilde \ttS}) &= {\mathrm{Sh}}_K(G_\ttS) \times_{F_{\ttS,\wp}} {\mathrm{Sh}}_{K_E}(T_{E, \tilde \ttS});\\ {\mathrm{Sh}}_{K_p \times K_{E,p}}(G_\ttS \times T_{E,\tilde \ttS}) &= {\mathrm{Sh}}_{K_p}(G_\ttS) \times_{F_{\ttS,\wp}} {\mathrm{Sh}}_{K_{E,p}}(T_{E, \tilde \ttS}) .\end{aligned}$$ Let $Z = \operatorname{Res}_{F/\QQ}\GG_m$ denote the center of $G_\ttS$. Put $G''_{\tilde \ttS} = G_\ttS \times_Z T_{E, \tilde \ttS}$ which is the quotient of $G_\ttS \times T_{E, \tilde \ttS}$ by $Z$ embedded anti-diagonally as $z \mapsto (z, z^{-1})$. The corresponding Deligne homomorphism $h''_{\tilde \ttS}: \SSS(\RR) \to G''_{\tilde \ttS}({\mathbb{R}})$ is the one induced by $\tilde h_{\tilde \ttS}$. We will consider open compact subgroups $K'' \subseteq G''_{\tilde \ttS}(\AAA^\infty)$ of the form $K''^pK''_p$, where $K''^p$ is an open compact subgroup of $G''_{\tilde \ttS}(\AAA^{\infty,p})$ and $K''_p $ is an open compact subgroup of $G''_{\tilde \ttS}({\QQ_p})$. Finally, the $G''_{\tilde \ttS}(\RR)$-conjugacy class of $h''_{\tilde \ttS}$ can be canonically identified with $\gothH_\ttS$. We then get the Shimura variety ${\mathrm{Sh}}_{K''}(G''_{\tilde \ttS})$ and its limit ${\mathrm{Sh}}_{K''_p}(G''_{\tilde \ttS})$ over the reflex field $E_{\tilde \ttS}$. The set of $\CC$-points of ${\mathrm{Sh}}_{K''}(G''_{\tilde \ttS})$ is $${\mathrm{Sh}}_{K''}(G''_{\tilde \ttS})(\CC) = G''_{\tilde \ttS}(\QQ)\backslash (\gothH_\ttS \times G''_{\tilde \ttS}(\AAA^\infty) ) / K''.$$ Unitary Shimura varieties {#S:unitary-shimura} ------------------------- We now introduce the unitary group. Consider the morphism $$\xymatrix@R=0pt@C=10pt{ \nu = {\mathrm{Nm}}_{B/F} \times {\mathrm{Nm}}_{E/F}: & G''_{\tilde \ttS}=G_\ttS \times_Z T_E \ar[rr]&& T\\ & (g, z) \ar@{|->}[rr]&& {\mathrm{Nm}}(g) z\bar z. }$$ Viewing $\GG_m$ naturally as a subgroup of $T = \operatorname{Res}_{F/\QQ}\GG_m$, we define $G'_{\tilde \ttS}$ to be the reductive group $\nu^{-1}(\GG_m)$; this will be our auxiliary unitary group, whose associated Shimura variety will provide ${\mathrm{Sh}}_K(G_{ \ttS})$ an integral canonical model. We will occasionally use the algebraic group $G'_{{\tilde \ttS}, 1} = \operatorname{Ker}\nu$; but we view it as a reductive group over $F$. Note that the Deligne homomorphism $h''_{\tilde \ttS} : \SSS(\RR) \to G''_{\tilde \ttS}({\mathbb{R}})$ factors through a homomorphism $h'_{\tilde \ttS}: \SSS({\mathbb{R}}) \to G'_{\tilde \ttS}({\mathbb{R}})$. The $G'_{\tilde \ttS}(\RR)$-conjugacy classes of $h'_{\tilde \ttS}$ is canonically isomorphic to $\gothH_\ttS$. We will consider open compact subgroups of $G'_{\tilde \ttS}(\AAA^\infty)$ of the form $K' = K'_pK'^p$, where $K'_p$ is an open compact subgroup of $G'_{\tilde \ttS}(\QQ_p)$ (to be specified later in Subsection \[S:level-structure\]) and $K'^p$ is an open compact subgroup of $G'_{\tilde \ttS}(\AAA^{\infty, p})$. We will always take $K'^p$ to be sufficiently small so that $K'$ is *neat* and hence the moduli problem we encounter later would be representable by a fine moduli space. Given the data above, we have a Shimura variety ${\mathrm{Sh}}_{K'}(G'_{\tilde \ttS})$ whose $\CC$-points are given by $${\mathrm{Sh}}_{K'}(G'_{\tilde \ttS})(\CC) = G'_{\tilde \ttS}(\QQ) \backslash (\gothH_\ttS \times G'_{\tilde \ttS}(\AAA^\infty) )/ K'.$$ The Shimura variety ${\mathrm{Sh}}_{K'}(G'_{\tilde \ttS})$ is defined over the reflex field $E_{\tilde \ttS}$. We put ${\mathrm{Sh}}_{K'_p}(G'_{\tilde \ttS}) = \varprojlim_{K'^p} {\mathrm{Sh}}_{K'_pK'^p}(G'_{\tilde \ttS})$. The upshot is the following lemma, which verifies the conditions listed in Corollary \[C:Sh(G)\^circ\_Zp independent of G\]. This allows us to bring the integral canonical models of the unitary Shimura varieties to that of the quaternionic Shimura varieties. \[L:compatibility of derived group and adjoint group\] The natural diagram of morphisms of groups $$\label{E:morphism-of-groups} G_{\ttS}\leftarrow G_{\ttS}\times T_{E, \tilde \ttS}{\rightarrow}G''_{\tilde \ttS} = G_{\ttS}\times_{Z}T_{E, \tilde \ttS}\leftarrow G'_{\tilde \ttS}$$ - are compatible with the Deligne homorphisms; and - induce isomorphisms on their associated derived and adjoint groups. Straightforward. PEL Shimura data {#S:PEL-Shimura-data} ---------------- We put $D_\ttS = B_\ttS \otimes_F E$; it is isomorphic to $\rmM_2(E)$ under Hypothesis \[H:B\_S-splits-at-p\]. This is a quaternion algebra over $E$ equipped with an involution $l \to \bar l$ given by the tensor product of the natural involution on $B_\ttS$ and the complex conjugation on $E$. Let $D_\ttS^{\mathrm{sym}}$ denote the subsets of *symmetric* elements, i.e. those elements $\delta \in D_\ttS$ such that $\delta = \bar \delta$. For any element $\delta\in (D_{\ttS}^{\mathrm{sym}})^{\times}$, we can define a new involution on $D_\ttS$ given by $l \mapsto l^* = \delta^{-1}\bar l \delta$. In the following Lemma \[L:property-PEL-data\], we will specify a convenient choice of such $\delta$. Let $V$ be the underlying $\QQ$-vector space of $D_\ttS$, with the natural left $D_\ttS$-module structure. Define a pairing $\psi_{E}: V\times V{\rightarrow}E$ on $V$ by $$\label{Equ:pairing-E} \psi_E(v, w) = {\mathrm{Tr}}_{D_\ttS/E}(\sqrt{\gothd} \cdot v \delta w^*), \quad \quad v, w \in V.$$ It is easy to check that $\psi_E$ is skew-hermitian over $E$ for $*$, i.e. $\overline{\psi_{E}(v,w)}=-\psi_E(w,v)$ and $\psi_E(lv, w) = \psi_E(v, l^*w)$ for $l \in D_\ttS$ and $v, w \in V$. We define the bilinear form $$\psi={\mathrm{Tr}}_{E/{\mathbb{Q}}}\circ \psi_{E}\colon V\times V{\mathrm{\longrightarrow}}{\mathbb{Q}}.$$ which is skew-symmetric and hermitian for $*$. One checks easily that the subgroup consisting of elements $l \in D^\times_\ttS$ satisfying $\psi(vl, wl) = c(l)\psi(v, w)$ for some $c(l) \in \QQ^\times$ is exactly the subgroup $G'_{\tilde \ttS} \subset D^\times_\ttS$. *We make the above right action of $G'_\ttS$ on $V$ into a left action by taking the inverse action.* Then the group $G'_{\tilde \ttS}$ is identified with the $D_\ttS$-linear unitary of group of $V$ with similitudes in $\QQ^\times$, i.e. for each ${\mathbb{Q}}$-algebra $R$, we have $$\label{Equ:description-G'} G'_{\tilde \ttS}(R)=\{g\in \operatorname{End}_{D_{\ttS}\otimes_{{\mathbb{Q}}}R}(V\otimes_{{\mathbb{Q}}}R)\;|\; \psi(gv, gw)=c(g)\psi(v,w)\; \text{with }c(g)\in R^{\times}\}.$$ We describe $D_{\ttS,\gothp} = D_\ttS\otimes_{F}F_{\gothp}$ by distinguishing three cases according to the types of $\gothp\in \Sigma_{p}$ in Subsection \[S:level-structure-at-p\]: - [**Types $\alpha$ or $\alpha^\sharp$:**]{} In this case, the place $\gothp$ splits into two primes $\gothq$ and $\bar{\gothq}$ in $E$. We have natural isomorphisms $F_\gothp \cong E_{\gothq} \cong E_{\bar \gothq}$. We fix an isomorphism $B_{\ttS}\otimes_{F}F_{\gothp}\simeq \rmM_2(F_{\gothp})$ as above, then $D_{\ttS,\gothp} \simeq \rmM_2(E_{\gothq})\oplus \rmM_2(E_{\bar{\gothq}})$. Under these identification, we put ${\mathcal{O}}_{B_{\ttS}, \gothp}=\rmM_2({\mathcal{O}}_{\gothp})$ and ${\mathcal{O}}_{D_{\ttS, \gothp}}=\rmM_2({\mathcal{O}}_{\gothq})\oplus \rmM_2({\mathcal{O}}_{\bar\gothq})$. - [**Type $\beta$:**]{} In this case, the place $\gothp$ is inert in $E/F$ and let $\gothq$ denote the unique place in $E$ above $\gothp$. Using the fixed isomorphism $B_{\ttS}\otimes_{F}F_{\gothp}\simeq \rmM_2(F_{\gothp})$, we have $D_{\ttS,\gothp}\simeq \rmM_2(E_{\gothq})$. We put ${\mathcal{O}}_{B_{\ttS}, \gothp}=\rmM_2({\mathcal{O}}_{\gothp})$ and ${\mathcal{O}}_{D_{\ttS}, \gothp}=\rmM_2({\mathcal{O}}_{\gothq})$. - [**Type $\beta^\sharp$:**]{} Let $\gothq$ be the unique place in $E$ above $\gothp$. The division quaternion algebra $B_{F_{\gothp}}=B_{\ttS}\otimes_{F}F_{\gothp}$ over $F_{\gothp}$ is generated by an element $\varpi_{B_{F_\gothp}}$ over $E_{\gothq}$, with the relations $\varpi_{B_{F_\gothp}}^2 = p$ and $\varpi_{B_{F_\gothp}} a = \bar a \varpi_{B_{F_\gothp}}$ for $a \in E_\gothq$. We identify $B_{F_\gothp} \otimes_{F_\gothp} E_\gothq$ with $ \rmM_2(E_\gothq)$ via the map $$\label{E:involution-on-quaternion-embedding} (a+b\varpi_{B_{F_\gothp}}) \otimes c \longmapsto \big( \begin{smallmatrix} ac & bc\\ p\bar b c & \bar a c \end{smallmatrix} \big).$$ This also identifies $D_{\ttS ,\gothp}$ with $\rmM_2(E_\gothq)$. We put ${\mathcal{O}}_{B_{\ttS}, \gothp}={\mathcal{O}}_{B_{F_\gothp}}$, and take ${\mathcal{O}}_{D_{\ttS}, \gothp}$ to be the preimage of $\rmM_2({\mathcal{O}}_\gothq)$ in $D_{\ttS}\otimes_F F_{\gothp}$. We put $\calO_{D_\ttS, p} = \prod_{\gothp \in \Sigma_p} \calO_{D_\ttS, \gothp}$. \[L:property-PEL-data\] - We can choose the symmetric element $\delta \in (D_\ttS^{\mathrm{sym}})^\times$ above such that - $\delta \in \calO_{D_\ttS, \gothp}^\times$ for each $\gothp \in \Sigma_p$ not of type $\beta^\sharp$, and $\delta \in \big( \begin{smallmatrix} p^{-1} &0\\0&1 \end{smallmatrix} \big) \calO_{D_\ttS, \gothp}^\times$ for each $\gothp \in \Sigma_p$ of type $\beta^\sharp$, and - the following (symmetric) bilinear form on $V_\RR$ is positive definite. $$(v, w) \mapsto \psi\big(v, w\cdot h'_{\tilde \ttS}(\ii)^{-1} \big).$$ - Through $h'_{\tilde \ttS}: \SSS({\mathbb{R}}) \to G'_{\tilde \ttS}({\mathbb{R}})$, $h'_{\tilde \ttS}(\ii)$ acts on the vector space $V_\RR$ and gives it a Hodge structure of type $\{(-1, 0), (0, -1)\}$. For $l \in D_\ttS$, we have $${\mathrm{tr}}(l; V_\CC / F^0V_\CC) = \big( \sum_{\tilde \tau \in \Sigma_{E,\infty}}s_{\tilde \tau} \tilde \tau \big) ({\mathrm{Tr}}_{D_\ttS/E}(l)).$$ The reflex field $E_{\tilde \ttS}$ is the subfield of $\CC$ generated by these traces for all $l \in D_\ttS$. - With the choice of $\delta$ in (1), the group $G'_{\tilde \ttS,1}$ is unramified at $\gothp \in \Sigma_p$ not of type $\beta^\sharp$ and is ramified at $\gothp \in \Sigma_p$ of type $\beta^\sharp$. Moreover, $\calO_{D_\ttS, p}$ is a maximal $*$-invariant lattice of $D_\ttS(\QQ_p)$. \(1) Since $F$ is dense in $F \otimes_\QQ \QQ_p \oplus F \otimes_\QQ \RR$, the symmetric elements in $V$ is dense in the symmetric elements in $V \otimes_\QQ {\QQ_p}\oplus V \otimes_\QQ \RR$. The conditions at places above $p$ is clearly open and non-empty; so are the conditions at archimedean places, which follows from the same arguments in [@carayol 2.2.4]. \(2) This follows from the same calculation as in [@carayol 2.3.2]. \(3) We first remark that $G'_{\tilde\ttS, 1, F_\gothp}$ does not depend on the choice of $\delta$, and hence we may take a convenient $\delta$ to ease the computation. We discuss each of the types separately. If $\gothp$ is of type $\alpha$ or $\alpha^\sharp$, $G'_{\tilde\ttS,1}(F_\gothp)$ is isomorphic to the kernel of ${\mathrm{GL}}_2(F_\gothp) \times_{F_\gothp^\times}(E_{\gothq}^\times \times E_{\bar \gothq}^\times) \to F_\gothp^\times$ given by $(l, x, y) \mapsto {\mathrm{Nm}}(l)xy$. Hence $l \mapsto (l, {\mathrm{Nm}}(l)^{-1}, 1)$ induces an isomorphism ${\mathrm{GL}}_2(F_\gothp) \to G'_{\ttS,1}(F_\gothp)$. They are of course unramified. If $\gothp$ is of type $\beta$, when we identify $D_{\ttS, \gothp}$ with $\rmM_2(E_\gothq)$, the convolution $l \mapsto \bar l$ is given by $\big(\begin{smallmatrix} a& b\\ c& d \end{smallmatrix} \big) \mapsto \big(\begin{smallmatrix} \bar d& -\bar b\\ -\bar c& \bar a \end{smallmatrix} \big)$ for $a, b, c, d \in E_\gothq$. We take the element $\delta$ to be $\big( \begin{smallmatrix} 0&1\\ 1&0 \end{smallmatrix} \big)$. The Hermitian form on $\rmM_2(E_\gothq)$ is then given by $$\langle v,w \rangle = {\mathrm{tr}}_{\rmM_2(E_\gothq) / E_\gothq}(v\bar w\delta) = -a\bar b' + b \bar a' +c \bar d' - d \bar c', \quad v= \big(\begin{smallmatrix} a& b\\ c& d \end{smallmatrix} \big)\textrm{ and } w= \big(\begin{smallmatrix} a'& b'\\ c'& d' \end{smallmatrix} \big).$$ One checks easily that $\gothe = \big(\begin{smallmatrix} 1 &0 \\0&0 \end{smallmatrix} \big)$ is invariant under the $*$-involution. So $D_{\ttS, \gothp}$ is isomorphic to $(\gothe D_{\ttS,\gothp})^{\oplus 2}$ as a $*$-Hermitian space and $G'_{\tilde \ttS,1,F_\gothp}$ is the unitary group for $\gothe D_{\ttS, \gothp}$. It is clear from the expression above that $\gothe D_{\ttS, \gothp}$ is a hyperbolic plane ([@minguez Example 3.2]). Hence $G'_{\tilde \ttS,1, F_\gothp}$ being the unitary group of such Hermitian space is unramified. If $\gothp$ is of type $\beta^\sharp$, the identification of $D_{\ttS, \gothp}$ with $\rmM_2(E_\gothq)$ using implies that the convolution $l \mapsto \bar l$ is given by $$\big(\begin{smallmatrix} a& b\\ c& d \end{smallmatrix} \big) \mapsto \big(\begin{smallmatrix} \bar a& -\bar c/p\\ -p\bar b& \bar d \end{smallmatrix} \big) \quad \textrm{ for }a, b, c, d \in E_\gothq.$$ We take the element $\delta$ to be $\big( \begin{smallmatrix} p^{-1} &0\\0&1 \end{smallmatrix} \big)$. The Hermitian form on $\rmM_2(E_\gothq)$ is then given by $$\label{E:Hermitian-type-gamma} \langle v,w\rangle = {\mathrm{Tr}}_{\rmM_2(E_\gothq) / E_\gothq}( v\bar w\delta) = a\bar a'/p -b \bar b' - c\bar c'/p + d \bar d', \quad v= \big(\begin{smallmatrix} a& b\\ c& d \end{smallmatrix} \big)\textrm{ and } w= \big(\begin{smallmatrix} a'& b'\\ c'& d' \end{smallmatrix} \big).$$ Similar to above, $\gothe = \big( \begin{smallmatrix} 1&0\\0&0 \end{smallmatrix} \big)$ is invariant under $*$-involution; and $D_{\ttS, \gothp}$ is isomorphic to $(\gothe D_{\ttS, \gothp})^{\oplus 2}$ as $*$-Hermitian spaces. The unitary group $G'_{\tilde \ttS,1,F_\gothp}$ is just the usual unitary group of $\gothe D_{\ttS, \gothp}$. But the Hermitian form there takes the form of $a\bar a'/p - b\bar b'$, which is a typical example of anisotropic plane ([@minguez Example 3.2]). So $G'_{\tilde \ttS,1,F_\gothp}$ is a ramified unitary group. To see that $\calO_{D_\ttS,p}$ is a maximal $*$-stable lattice, it suffices to prove it for $\calO_{D_\ttS, \gothp}$ for each $\gothp \in \Sigma_p$. When $\gothp$ is of type $\alpha, \alpha^\sharp$, or $\beta$, this is immediate. When $\gothp$ is of type $\gamma$, we write $\delta$ as $\big( \begin{smallmatrix} p^{-1} &0\\0&1 \end{smallmatrix} \big) u$ for $u \in \calO_{D_\ttS, \gothp}^\times$. The involution $*$ is given by $$\big(\begin{smallmatrix} a& b\\ c& d \end{smallmatrix} \big) \mapsto u^{-1}\big( \begin{smallmatrix} p &0\\0&1 \end{smallmatrix} \big) \big(\begin{smallmatrix} \bar a& -\bar c/p\\ -p\bar b& \bar d \end{smallmatrix} \big) \big( \begin{smallmatrix} p^{-1} &0\\0&1 \end{smallmatrix} \big) u = u^{-1} \big(\begin{smallmatrix} \bar a& -\bar c\\ -\bar b& \bar d \end{smallmatrix} \big) u \quad \textrm{ for }a, b, c, d \in E_\gothq.$$ It is then clear that $\calO_{D_\ttS,\gothp}$ is a maximal $*$-stable lattice. Level structures at $p$ in the unitary case {#S:level-structure} ------------------------------------------- We specify our choice for $K'_p$ corresponding to the level structure $K_p=\prod_{\gothp|p}K_{\gothp}\subset \prod_{\gothp|p} (B_{\ttS}\otimes_{F}F_{\gothp})^{\times}$ considered in Subsection \[S:level-structure-at-p\]. By , giving an element $g_p\in G_{\tilde \ttS}'({\mathbb{Q}}_p)$ is equivalent to giving tuples $(g_{\gothp})_{\gothp\in \Sigma_{p}}$ with $g_{\gothp}\in \operatorname{End}_{D_{\ttS}\otimes_{F}F_{\gothp}}(V\otimes_{F}F_{\gothp})$ such that there exists $\nu(g_p)\in {\mathbb{Q}}_{p}^{\times}$ independent of $\gothp$ satisfying $$\psi_{E, \gothp}(g_{\gothp}v, g_{\gothp}w)=\nu(g_p)\psi_{E, \gothp}(v,w), \quad \forall v,w \in V\otimes_F F_{\gothp},$$ where $\psi_{E, \gothp}$ is the base change of $\psi_E$ to $V\otimes_F F_{\gothp}= D_{\ttS,\gothp}$. In the following, we will give a chain of lattices $\Lambda_{\gothp}^{(1)}\subseteq \Lambda_{\gothp}^{(2)}$ in $D_{\ttS,\gothp}$ for each $\gothp$, and define $K_p'\subseteq G_{\tilde \ttS}'({\mathbb{Q}}_p)$ to be the subgroup consisting of the elements $(g_{\gothp})_{\gothp\in \Sigma_{p}}$ with $g_{\gothp}$ belonging to the stablizer of $\Lambda^{(1)}_{\gothp}\subseteq \Lambda_{\gothp}^{(2)}$ and with $\nu(g_p)\in \ZZ_{p}^{\times}$ independent of $\gothp$. - When $\gothp$ is of type $\alpha$, we take $\Lambda_\gothp^{(1)}=\Lambda_{\gothp}^{(2)}$ to be $\calO_{D_\ttS, \gothp}$. - When $\gothp$ is of type $\alpha^\sharp$, we take $$\Lambda_\gothp^{(1)} = \left( \begin{smallmatrix} \gothq & \calO_\gothq \\ \gothq & \calO_\gothq \end{smallmatrix} \right)\oplus\left( \begin{smallmatrix} \calO_{\bar \gothq} & \calO_{\bar \gothq} \\ \calO_{\bar \gothq} & \calO_{\bar \gothq} \end{smallmatrix}\right) \quad \textrm{ and } \quad \Lambda_\gothp^{(2)} =\left( \begin{smallmatrix} \calO_\gothq & \calO_\gothq \\ \calO_\gothq & \calO_\gothq \end{smallmatrix}\right) \oplus\left( \begin{smallmatrix} \calO_{\bar \gothq} & \bar \gothq^{-1} \\ \calO_{\bar \gothq} &\bar \gothq^{-1} \end{smallmatrix}\right).$$ - When $\gothp$ is of type $\beta$, we take $\Lambda_{\gothp}^{(1)} =\Lambda_{\gothp}^{(2)} =\calO_{D_{\ttS},\gothp}$. - When $\gothp$ is of type $\beta^\sharp$, we take $$\Lambda_\gothp^{(1)} = \left( \begin{smallmatrix} \gothq & \calO_\gothq\\ \gothq & \calO_\gothq \end{smallmatrix} \right) \subseteq \Lambda_\gothp^{(2)} = \left( \begin{smallmatrix} \calO_\gothq & \calO_\gothq\\ \calO_\gothq & \calO_\gothq \end{smallmatrix} \right).$$ Note that, these two lattices are dual of each other under the Hermitian form . Similarly, we give the level structure at $p$ for the Shimura variety associated to the group $G''_{\tilde \ttS}$: take $K''_{ p}$ to be the image of $K_p \times K_{E,p} $ under the natural map $(G_\ttS \times T_{E,\tilde \ttS})({\QQ_p}) \to G''_{\tilde \ttS}({\QQ_p})$. \[L:compatibility of derived group and adjoint group2\] The Shimura data for $G_\ttS, G_\ttS \times T_{E,\tilde \ttS}, G''_{\tilde \ttS},$ and $G'_{\tilde \ttS}$ satisfy Hypothesis \[H:hypo on G\]. Moreover, The natural diagram of morphisms of groups $$\label{E:morphism-of-groups2} G_{\ttS}\leftarrow G_{\ttS}\times T_{E, \tilde \ttS}{\rightarrow}G''_{\tilde \ttS} = G_{\ttS}\times_{Z}T_{E,\tilde \ttS}\leftarrow G'_{\tilde \ttS}$$ induce isomorphisms on the $p$-integral points of the derived and adjoint groups. This is straightforward from definition. In fact, both $K_p^{\mathrm{ad}}= \prod_{\gothp \in \Sigma_p}K_\gothp^{\mathrm{ad}}$ and $K_p^{\mathrm{der}}= \prod_{\gothp \in \Sigma_p} K_\gothp^{\mathrm{der}}$ are products and we give the description case by case: - if $\gothp$ is of type $\alpha$ or $\beta$, then $K^{\mathrm{der}}_\gothp = {\mathrm{SL}}_{2, \calO_\gothp}$ and $K^{\mathrm{ad}}_\gothp = {\mathrm{PGL}}_{2, \calO_\gothp}$; - if $\gothp$ is of type $\alpha^\sharp$, then $K^{\mathrm{der}}_\gothp = {\mathrm{SL}}_{2, \calO_\gothp} \cap \big( \begin{smallmatrix} \calO_\gothp^\times & \calO_\gothp\\ \gothp & \calO_\gothp^\times \end{smallmatrix} \big)$ and $K^{\mathrm{ad}}_\gothp = \big( \begin{smallmatrix} \calO_\gothp^\times & \calO_\gothp\\ \gothp & \calO_\gothp^\times \end{smallmatrix} \big) / \calO_\gothp^\times$; and - if $\gothp$ is of type $\beta^\sharp$, $K^{\mathrm{der}}_\gothp$ and $K^{\mathrm{ad}}_\gothp$ are the maximal compact open subgroups of $(B_\ttS^\times)^{\mathrm{der}}(F_\gothp)$ and $(B_\ttS^\times)^{\mathrm{ad}}(F_\gothp)$, respectively. \[C:comparison of shimura varieties\] The natural morphisms between Shimura varieties $$\label{E:morphisms of Shimura varieties} {\mathrm{Sh}}_{K_p}(G_\ttS)\longleftarrow{\mathrm{Sh}}_{K_p \times K_{E, p}}(G_\ttS \times T_{E,\tilde \ttS}) \longrightarrow {\mathrm{Sh}}_{K''_p}(G''_{\tilde \ttS}) \longleftarrow{\mathrm{Sh}}_{K'_p}(G'_{\tilde \ttS})$$ induce isomorphisms on the geometric connected components. Moreover, the groups $\calE_{G, \tilde \wp}$ defined in \[A:connected integral model\] (and made explicit below) are isomorphic for each of the groups; and is equivariant for the actions of $\calE_{G, \tilde \wp}$’s on the geometric connected components. Moreover, if one of the Shimura varieties admits an integral canonical model; so do the others. This follows from Corollary \[C:Sh(G)\^circ\_Zp independent of G\] for which the conditions are verified in Lemmas \[L:compatibility of derived group and adjoint group\] and \[L:compatibility of derived group and adjoint group2\]. Structure groups for connected Shimura varieties {#S:structure group} ------------------------------------------------ In order to apply the machinery developed in Section \[Section:Sh Var\], we now make explicit the structure groups $\calG$ in and $\calE_{G, \tilde \wp}$ in Subsection \[A:connected integral model\] in the case of our interest. We use $\calG_\ttS$ (resp. $\calG'_{\tilde \ttS}$, $\calG''_{\tilde \ttS}$) to denote the group defined in for $G = G_\ttS$ (resp. $ G'_{\tilde \ttS}$, $G''_{\tilde \ttS}$). Explicitly, since the center of $G_\ttS$ is $\operatorname{Res}_{F/\QQ}\GG_m$, we have $G^{\mathrm{ad}}_\ttS(\QQ) = B_\ttS^\times / F^\times$. Taking the positive and $p$-integral part as in Lemma \[L:nu(G(Q)-&gt;&gt;T(Q)\], we have $G_\ttS^{\mathrm{ad}}(\QQ)^{+, (p)} = B_\ttS^{\times, >0, (p)} / \calO_{F, (p)}^\times$ where the superscript $>0$ means to take the elements whose determinant is positive for all real embeddings. It follows that $\calG_\ttS = G_\ttS(\AAA^{\infty, p}) / \calO_{F, (p)}^{\times, {\mathrm{cl}}}$. The same argument applies to $G''_{\tilde \ttS}$ whose center is $\operatorname{Res}_{E/\QQ}\GG_m$, and shows that $\calG''_{\tilde \ttS} = G''_{\tilde \ttS}(\AAA^{\infty, p}) \big/ \calO_{E,(p)}^{\times, {\mathrm{cl}}}$. Determination of $\calG'_{\tilde \ttS}$ is more subtle. By Lemmas \[L:compatibility of derived group and adjoint group\] and \[L:compatibility of derived group and adjoint group2\], we have $(G'_{\tilde \ttS})^{\mathrm{ad}}(\QQ)^{+,(p)} = (G''_{\tilde \ttS})^{\mathrm{ad}}(\QQ)^{+,(p)}$. So if we use $Z'_{\tilde \ttS}$ to denote the center of $G'_{\tilde \ttS}$, then we have $$\begin{aligned} \label{E:structure group description} \calG'_{\tilde \ttS} &= \big(G'_{\tilde \ttS}(\AAA^{\infty,p}) \big/ Z'_{\tilde \ttS}(\QQ)^{(p), {\mathrm{cl}}}\big) \ast_{G'_{\tilde \ttS}(\QQ)^{(p)}_+/Z'_{\tilde \ttS}(\QQ)^{(p)}} (G'_{\tilde \ttS})^{\mathrm{ad}}(\QQ)^{+,(p)} \\ \nonumber &= \big(G'_{\tilde \ttS}(\AAA^{\infty,p}) \big/ Z'_{\tilde \ttS}(\QQ)^{(p), {\mathrm{cl}}}\big) \ast_{G'_{\tilde \ttS}(\QQ)^{(p)}_+/Z'_{\tilde \ttS}(\QQ)^{(p)}} \big( G''_{\tilde \ttS}(\QQ)_+^{(p)} / \calO_{E, (p)}^\times \big) \\ \nonumber &= G'_{\tilde \ttS}(\AAA^{\infty, p}) G''_{\tilde \ttS}(\QQ)_+^{(p)} \big/ \calO_{E, (p)}^{\times, {\mathrm{cl}}}.\end{aligned}$$ The subgroup $G'_{\tilde \ttS}(\AAA^{\infty, p}) G''_{\tilde \ttS}(\QQ)_+^{(p)}$ can be characterized by the following commutative diagram of exact sequence as the pull back of the right square. $$\label{E:description of G'' G'} \xymatrix{ 1 \ar[r] & \ar@{=}[d] G'_{\tilde \ttS,1}(\AAA^{\infty,p}) \ar[r] &G''_{\tilde \ttS}(\QQ)_+^{(p)} G'_{\tilde \ttS}(\AAA^{\infty,p}) \ar[r] \ar@{^{(}->}[d] & \calO_{F, (p)}^\times (\AAA^{\infty,p})^\times\ar[r] \ar@{^{(}->}[d] & 1 \\1 \ar[r]& G'_{{\tilde \ttS},1}(\AAA^{\infty,p}) \ar[r] & G''_{\tilde \ttS}(\AAA^{\infty,p}) \ar[r] & (\AAA_F^{\infty,p})^\times \ar[r] &1. }$$ We use $\calE_{G, \ttS, \wp}$ to denote the group $\calE_{G, \tilde \wp}$ defined in Subsection \[A:connected integral model\]. As an abstract group, it is isomorphic for all groups $G_\ttS$, $G'_{\tilde \ttS}$, and $G''_{\tilde \ttS}$. But we point out that it is important (see Remark \[R:quaternionic Shimura reciprocity not compatible\]) to know how they sit as subgroups of $\calG_\ttS \times \operatorname{Gal}_{k_\wp}$, $\calG'_{\tilde \ttS} \times \operatorname{Gal}_{k_{\tilde \wp}}$ and $\calG''_{\tilde \ttS} \times \operatorname{Gal}_{k_{\tilde \wp}}$, respectively, according to the Shimura reciprocity map. Integral models of unitary Shimura varieties {#S:integral-unitary} -------------------------------------------- We choose a finite extension $k_0$ of $k_{\tilde \wp}$ that contains all residual fields $k_\gothq$ for any $p$-adic place $\gothq$ of $E$. Then the ring of Witt vectors $W(k_0)$ may be viewed as a subring of $\overline \QQ_p$, containing $\calO_{\tilde \wp}$ as a subring. We fix an order $\calO_{D_\ttS}$ of $D_\ttS$ stable under the involution $l \mapsto l^*$ such that $\calO_{D_\ttS} \otimes_{\calO_F} \calO_{F,p} \simeq \calO_{D_\ttS, p}$. Recall that $V$ is the abstract $\QQ$-vector space $D_\ttS$; we choose and fix an $\calO_{D_\ttS}$-lattice $\Lambda$ of $V$ such that, - for each $\gothp \in \Sigma_p$, we have $\Lambda \otimes_{\calO_F} \calO_\gothp \cong \Lambda_\gothp^{(1)}$, and - if we put $\widehat{\Lambda}^{(p)} : = \Lambda \otimes_\ZZ \widehat \ZZ^{(p)}$ as a lattice of $V \otimes_\QQ \AAA^{\infty, p}$, we have $$\label{E:Lambda-dual} \widehat \Lambda^{(p)} \subseteq \widehat \Lambda^{(p),\vee}\textrm{ under the bilinear form } \psi, \textrm{ or equivalently, } \psi(\widehat \Lambda^{(p)}, \widehat \Lambda^{(p)}) \subseteq \widehat{\ZZ}^{(p)}.$$ We call such $\Lambda$ *admissible*. \[T:unitary-shimura-variety-representability\] Let $K'_p$ be the open compact subgroup of $G_{\tilde \ttS}'({\mathbb{Q}}_p)$ considered in Subsection \[S:level-structure\], and $K'^p\subset G_{\tilde \ttS}'(\AAA^{\infty, p})$ sufficiently small so that $K'=K'^pK'_p$ is neat. Then there exists a unique *smooth* quasi-projective scheme ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ over $W(k_0)$ representing the functor that sends a locally noetherian $W(k_0)$-scheme $S$ to the set of isomorphism classes of tuples $(A, \iota, \lambda, \alpha_{K'})$, as described as follows. - $A$ is an abelian scheme over $S$ of dimension $4g$ equipped with an embedding $\iota: \calO_{D_\ttS} \to \operatorname{End}_S(A)$ such that the characteristic polynomial of the endomorphism $\iota(b)$ on $\operatorname{Lie}(A/S)$ for $b \in \calO_E$ is given by $$\prod_{\tilde \tau \in \Sigma_{E,\infty}} \big(x - \tilde \tau(b)\big) ^{2s_{\tilde \tau}}.$$ - $\lambda:A \to A^\vee$ is a polarization of $A$, such that - the Rosati involution associated to $\lambda$ induces the involution $l \mapsto l^*$ on $\calO_{D_\ttS}$, - $(\operatorname{Ker}\lambda)[p^\infty] $ is a finite flat closed subgroup scheme contained in $\prod_{\gothp \textrm{ of type } \beta^\sharp}A[\gothp]$ of rank $\prod_{\gothp \textrm{ of type } \beta^\sharp} (\#k_{\gothp})^{4}$ and - the cokernel of $\lambda_*: H_1^{\mathrm{dR}}(A / S) \to H_1^{\mathrm{dR}}(A^\vee/S)$ is a locally free module of rank two over $$\bigoplus_{\gothp \textrm{ of type }\beta^\sharp} \calO_S \otimes_{\ZZ_p} (\calO_E \otimes_{\calO_F} k_\gothp).$$ - $ \alpha_{K'}$ is a pair $( \alpha^p_{K'^p}, \alpha_p)$ defined as follows: - For each connected component $S_i$ of $S$, we choose a geometric point $\bar{s}_i$, and let $T^{(p)}(A_{\bar{s}_i})$ be the product of $l$-adic Tate modules of $A$ at $\bar{s}_i$ for all $l\neq p$. Then $\alpha^p_{K'^p}$ is a collection of $\pi_1(S_i, \bar{s}_i)$-invariant $K'^p$-orbit of pairs $(\alpha^p_i, \nu(\alpha^p_i))$, where $\alpha^p_i$ is an $\calO_{D_\ttS}\otimes_\ZZ \widehat{\ZZ}^{(p)}$-linear isomorphism $\widehat \Lambda^{(p)} \xrightarrow{\sim} T^{(p)}( A_{\bar s_i})$ and $\nu(\alpha^p_i)$ is an isomorphism $\widehat{{\mathbb{Z}}}^{(p)}{\xrightarrow}{\sim} \widehat {\mathbb{Z}}^{(p)}(1)$ such that the following diagram commute: $$\xymatrix{ \widehat \Lambda^{(p)}\times \widehat \Lambda^{(p)}\ar[rr]^-{\psi}\ar[d]_{\alpha^p_i\times \alpha_i^p} &&\widehat{{\mathbb{Z}}}^{(p)}\ar[d]^{\nu(\alpha_i^p)}\\ T^{(p)}(A_{\bar s_i})\times T^{(p)}(A_{\bar s_i}) \ar[rr]^-{\lambda-\mathrm{Weil}} && \widehat{{\mathbb{Z}}}^{(p)}(1). }$$ - For each prime $\gothp\in \Sigma_p$ of type $\alpha^\sharp$, let $\gothq$ and $\bar\gothq$ be the two primes of $E$ above $\gothp$. Then $\alpha_p$ is a collection of $\calO_{D_\ttS}$-stable closed finite flat subgroups $\alpha_{\gothp}=H_{\gothq}\oplus H_{\bar{\gothq}} \subset A[\gothq]\oplus A[\bar\gothq]$ of order $(\#k_\gothp)^4$ such $H_\gothq$ and $H_{\bar\gothq}$ are dual to each other under the perfect pairing $$A[\gothq]\times A[\bar\gothq]{\rightarrow}\mu_p$$induced by the polarization $\lambda$. By Galois descent, the moduli space ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ can be defined over $\calO_{ \tilde \wp}$. Moreover, if the ramification set $\ttS_{\infty}$ is non-empty, ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ is projective. We will postpone the proof of this theorem after Notation \[N:notation-reduced\]. Deformation theory {#S:deformation} ------------------ We recall briefly the crystalline deformation theory of abelian varieties due to Serre-Tate and Grothendieck-Messing. This will be used in the proof of Theorem \[T:unitary-shimura-variety-representability\]. We start with a general situation. Let $S$ be a ${\mathbb{Z}}_p$-scheme over which $p$ is locally nilpotent, and $S_0{\hookrightarrow}S$ be a closed immersion whose ideal sheaf $\calI$ is equipped with a divided power structure compatible with that on $p{\mathbb{Z}}_p$, e.g. $S_0= \operatorname{Spec}k{\hookrightarrow}S= \operatorname{Spec}k[\epsilon]/ (\epsilon^2)$ with $k$ a perfect field of characteristic $p$. Let $(S_0/{\mathbb{Z}}_p)_{{\mathrm{cris}}}$ be the crystalline site of $S_0$ over $\operatorname{Spec}{\mathbb{Z}}_p$, and ${\mathcal{O}}^{\mathrm{cris}}_{S_0/{\mathbb{Z}}_p}$ be the structure sheaf. Let $A_0$ be an abelian scheme over $S_0$, and $H^{{\mathrm{cris}}}_1(A_0/S_0)$ be the *dual* of the relative crystalline cohomology $H^1_{{\mathrm{cris}}}(A_0/S_0)$ (or isomorphically $H^{{\mathrm{cris}}}_1(A_0/S_0)=H^1_{{\mathrm{cris}}}(A_0^\vee/S_0)$). Then $H^{{\mathrm{cris}}}_1(A_0/S_0)$ is a crystal of locally free ${\mathcal{O}}^{\mathrm{cris}}_{S_0/{\mathbb{Z}}_p}$-modules whose evaluation $H^{{\mathrm{cris}}}_1(A_0/S_0)_{S}$ at the pd-embedding $S_0{\hookrightarrow}S$ is a locally free ${\mathcal{O}}_S$-module. We have a canonical isomorphism $H^{{\mathrm{cris}}}_1(A_0/S_0)_{S}\otimes_{{\mathcal{O}}_S}{\mathcal{O}}_{S_0}\simeq H^{{\mathrm{dR}}}_1(A_0/S_0)$, which is the *dual* of the relative de Rham cohomology of $A_0/S_0$. For each abelian scheme $A$ over $S$ with $A\times_S S_0\simeq A_0$, we have a canonical Hodge filtatrion $$0{\rightarrow}\omega_{A^\vee/S}{\rightarrow}H^{{\mathrm{cris}}}_1(A_0/S_0)_{S}{\rightarrow}\operatorname{Lie}(A/S){\rightarrow}0.$$ Hence, $\omega_{A^\vee/S}$ gives rise to a local direct factor of $H^{{\mathrm{cris}}}_1(A_0/S_0)_{S}$ that lifts the subbundle $\omega_{A_0^\vee/S_0}\subseteq H_1^{{\mathrm{dR}}}(A_0/S_0)$. Conversely, the theory of deformations of abelian schemes says that knowing this lift of subbundle is also enough to recover $A$ from $A_0$. More precisely, let ${\mathtt{AV}}_{S}$ be the category of abelian scheme over $S$, ${\mathtt{AV}}^+_{S_0}$ denote the category of pairs $(A_0, \omega)$, where $A_0$ is an abelian scheme over $S_0$ and $\omega$ is a subbundle of $H_1^{\mathrm{cris}}( A_0/S_0)_{S}$ that lifts $\omega_{A_0^\vee/S_0} \subseteq H_1^{\mathrm{dR}}(A_0/S_0)$. The main theorem of the crystalline deformation theory (cf. [@grothendieck pp. 116–118], [@mazur-messing Chap. II §1]) says that *the natural functor ${\mathtt{AV}}_{S} \to {\mathtt{AV}}_{S_0}^+$ given by $A\mapsto (A\times_{S}S_0, \omega_{A^\vee/S})$ is an equivalence of categories.* Let $A$ be a deformation of $A_0$ corresponding to a direct factor $\omega\subseteq H_1^{{\mathrm{cris}}}(A_0/S_0)_S$ that lifts $\omega_{A_0^\vee/S_0}$. If $A_0$ is equipped with an action $\iota_0$ by a certain algebra $R$, then $\iota_0$ deforms to an action $\iota$ of $R$ on $A$ if and only if $\omega_{S}\subseteq H_1^{{\mathrm{cris}}}(A_0/S_0)_S$ is $R$-stable. Let $\lambda_0:A_0{\rightarrow}A_0^\vee$ be a polarization. Then $\lambda_0$ induces a natural alternating pairing [@bbm 5.1] $$\langle\ ,\ \rangle_{\lambda_0}\colon H^{{\mathrm{cris}}}_1(A_0/S_0)_S\times H^{{\mathrm{cris}}}_1(A_0/S_0)_S{\rightarrow}{\mathcal{O}}_S,$$ which is perfect if $\lambda_0$ is prime-to-$p$. Then there exists a (necessarily unique) polarization $\lambda: A{\rightarrow}A^\vee$ that lifts $\lambda_0$ if and only if $\omega_S$ is isotropic for $\langle\ ,\ \rangle_{\lambda_0}$ by [@lan 2.1.6.9, 2.2.2.2, 2.2.2.6]. \[N:notation-reduced\] Before going to the proof of Theorem \[T:unitary-shimura-variety-representability\], we introduce some notation. Recall that we have an isomorphism ${\mathcal{O}}_{D_{\ttS},p}\simeq \rmM_2({\mathcal{O}}_{E}\otimes {\mathbb{Z}}_p)$. We denote by $\gothe\in {\mathcal{O}}_{D_{\ttS},p}$ the element corresponding to $\bigl( \begin{smallmatrix}1 &0\\0&0\end{smallmatrix} \bigr)$ in $\rmM_2({\mathcal{O}}_{E}\otimes {\mathbb{Z}}_p)$. For $S$ a $W(k_0)$-scheme and $M$ an ${\mathcal{O}}_S$-module locally free of finite rank equipped with an action of ${\mathcal{O}}_{D_{\ttS},p}$, we call $M^{\circ}:=\gothe M$ *the reduced part* of $M$; we have $M=(M^{\circ})^{\oplus 2}$ by Morita equivalence. Moreover, the $\calO_E$-action induces a canonical decomposition $$M^{\circ}=\bigoplus_{\tilde \tau\in \Sigma_{E, \infty}} M^{\circ}_{\tilde\tau},$$ where $\calO_E$ acts on each factor $M^\circ_{\tilde \tau}$ by $\tilde \tau': \calO_E \to W(k_0)$. Let $A$ be an abelian scheme over $S$ carrying an action of ${\mathcal{O}}_{D_{\ttS}}$. The construction above gives rise to locally free $\calO_S$-modules $\omega^\circ_{A/S}$, $\operatorname{Lie}(A/S)^\circ$ and $ H_1^{\mathrm{dR}}(A/S)^\circ $, which are of rank $\frac12\dim A$, $\frac12\dim A$ and $\dim A$, respectively. We call them the *reduced invariant differential $1$-forms*, *reduced Lie algebra* and the *reduced de Rham homology* of $A$ respectively. For each $\tilde \tau \in \Sigma_{E, \infty}$, we have a *reduced Hodge filtration* in $\tilde \tau$-component $$\label{Equ:reduced-Hodge} 0{\rightarrow}\omega_{A^\vee/S, \tilde \tau}^{\circ}{\rightarrow}H_1^{\mathrm{dR}}(A/S)^\circ_{\tilde \tau}{\rightarrow}\operatorname{Lie}(A/S)^{\circ}_{\tilde \tau}{\rightarrow}0.$$ The representability of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ by a quasi-projective scheme over $W(k_0)$ is well-known (cf. for instance [@lan 1.4.13, 2.3.3, 7.2.3.10]). To show the smoothness of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$, it suffices to prove that it is formally smooth over $W(k_0)$. Let $R$ be a $W(k_0)$-algebra, $I\subset R$ be an ideal with $I^2=0$, and $R_0=R/I$. We need to show that, every point $x_0=(A_0, \iota_0, \lambda_0, \alpha_{K',0})$ of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ with values in $R_0$ lifts to an $R$-valued point $x$ of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$. We apply the deformation theory recalled in \[S:deformation\]. The relative crystalline homology $H_1^{{\mathrm{cris}}}(A_0/R_0)$ is naturally equipped with an action of ${\mathcal{O}}_{D_{\ttS}}\otimes {\mathbb{Z}}_p$. Let $H_1^{{\mathrm{cris}}}(A_0/R_0)^{\circ}:=\gothe H_1^{{\mathrm{cris}}}(A_0/R_0)$ be its reduced part, and $H_1^{{\mathrm{cris}}}(A_0/R_0)^{\circ}_{R}$ be its evaluation on $R$. This is a free $R\otimes{\mathcal{O}}_E$-module of rank $4[F:{\mathbb{Q}}]$, and we have a canonical decomoposition $$H_1^{{\mathrm{cris}}}(A_0/R_0)^{\circ}_{R}= \bigoplus_{\tilde \tau\in \Sigma_{E,\infty}} H_1^{{\mathrm{cris}}}(A_0/R_0)^{\circ}_{R,\tilde \tau}.$$ The polarization $\lambda_0$ on $A_0$ induces a pairing $$\label{Equ:pairing-R} H_1^{{\mathrm{cris}}}(A_0/R_0)_{R, \tilde \tau}^{\circ}\times H_1^{{\mathrm{cris}}}(A_0/R_0)_{R,\tilde \tau^c}^{\circ}{\mathrm{\longrightarrow}}R,$$ which is perfect for $\tilde \tau\in \Sigma_{E, \infty/\gothp}$ with $\gothp$ not of type $\beta^\sharp$. By the deformation theory \[S:deformation\], giving a deformation of $(A_0, \iota_0)$ to $R$ is equivalent to giving, for each $\tilde \tau \in \Sigma_{E, \infty}$, a direct summand $\omega_{R, \tilde \tau}^{\circ}\subseteq H_1^{{\mathrm{cris}}}(A_0/R_0)^{\circ}_{R, \tilde \tau}$ which lifts $\omega_{A_0^\vee/R_0, \tilde \tau}^{\circ}$. Let $\gothp\in \Sigma_{p}$ with $\tilde \tau\in \Sigma_{E, \infty/\gothp}$. We distinghish several cases: - If $\tilde \tau$ restricts to $\tau \in \ttS_\infty$, $\operatorname{Lie}(A_0/R_0)^{\circ}_{\tilde \tau}$ has rank $s_{\tilde \tau} \in \{0,2\}$ by the determinant condition (a). By duality or the Hodge filtration , $\omega_{A_0^\vee/R_0, \tilde \tau}^\circ$ has rank $2-s_{\tilde{\tau}}$, i.e. $\omega_{A_0^\vee/R_0, \tilde{\tau}^\circ}=0$ when $s_{\tilde \tau} =2$ and $\omega_{A_0^\vee/R_0, \tilde \tau}^{\circ}\cong H_1^{{\mathrm{dR}}}(A_0/R_0)^{\circ}_{\tilde \tau}$ when $s_{\tilde \tau} = 0$. Therefore, $\omega_{R, \tilde \tau}^{\circ}=0$ or $\omega_{R, \tilde \tau}^{\circ}=H_1^{{\mathrm{cris}}}(A_0/R_0)_{R, \tilde \tau}^{\circ}$ is the unique lift in these cases respectively. - If $\tilde \tau$ restricts to $\tau\in \Sigma_{\infty}-\ttS_{\infty}$, then $\omega_{A_0^\vee/R_0,\tilde \tau}^{\circ}$ and $\omega_{A_0^\vee/R_0, \tilde \tau^c}^{\circ}$ are both of rank 1 over $R_0$, and we have $\omega_{A_0^\vee/R_0, \tilde \tau}^\circ=(\omega_{A_0^\vee/R_0, \tilde \tau^c}^\circ)^{\perp}$ under the perfect pairing between $H_1^{{\mathrm{dR}}}(A_0/R_0)^{\circ}_{\tilde \tau}$ and $H_1^{{\mathrm{dR}}}(A_0/R_0)_{\tilde \tau^c}$ induced by $\lambda_0$. (Note that $\tau \in \Sigma_{\infty/\gothp}- \ttS_\infty$ means that $\gothp$ is not of type $\beta^\sharp$ and hence the Weil pairing is perfect.) Within each pair $\{\tilde \tau, \tilde \tau^c\}$, we can take an arbitrary direct summand $\omega_{R,\tilde \tau }^{\circ} \subseteq H_1^{{\mathrm{cris}}}(A_0/R_0)_{R, \tilde \tau}^\circ$ which lifts $\omega_{A_0^\vee/R_0, \tilde \tau}^{\circ}$, and let $\omega_{R, \tilde \tau^c}^\circ$ be the orthogonal complement of $\omega_{R, \tilde \tau^c}^\circ$ under the perfect pairing . By the Hodge filtration , such choices of $(\omega_{R, \tilde \tau}^\circ, \omega_{R, \tilde \tau^c}^{\circ})$ form a torsor under the group $$\operatorname{Hom}_{R_0}(\omega_{A_0^\vee/R_0, \tilde \tau}^{\circ}, \operatorname{Lie}(A_0)_{\tilde \tau}^\circ)\otimes I\cong \operatorname{Lie}(A_0)_{\tilde \tau}^{\circ}\otimes_{R_0} \operatorname{Lie}(A_0)_{\tilde \tau^c}^\circ \otimes I,$$ where in the second isomorphism, we have used the fact that $\operatorname{Lie}(A_0^\vee)^\circ_{\tilde \tau}\simeq \operatorname{Lie}(A_0)^\circ_{\tilde \tau^c}$. We take liftings $\omega_{R, \tilde \tau}^\circ$ for each $\tilde \tau \in \Sigma_{E, \infty}$ as above, and let $(A, \iota)$ be the corresponding deformation to $R$ of $(A_0, \iota_0)$. It is clear that $\bigoplus_{\tau\in \Sigma_{\infty}}(\omega_{R,\tilde{\tau}}^{\circ}\oplus \omega_{R, \tilde{\tau}^c}^{\circ})$ is isotropic for the pairing on $H^{{\mathrm{cris}}}_1(A_0/R_0)^{\circ}_R$ induced by $\lambda_0$. Hence, the polarization $\lambda_0$ lifts uniquely to a polarization $\lambda: A{\rightarrow}A^\vee$ satisfying condition (b1) in the statement of the Theorem. By the criterion of flatness by fibres [@ega 11.3.10], $\operatorname{Ker}(\lambda)$ is a finite flat group scheme over $R$, and the condition (b2) is thus satisfied. Condition (b3) follows from the fact that the morphism $\lambda_*:H_1^{{\mathrm{dR}}}(A/R){\rightarrow}H_1^{{\mathrm{dR}}}(A/R)$ is the same as $\lambda_{0,*}: H_1^{{\mathrm{cris}}}(A_0/R_0)_R{\rightarrow}H_1^{{\mathrm{cris}}}(A_0^\vee/R_0)_R$ under the canonical isomorphism $H_1^{{\mathrm{dR}}}(B/R)\simeq H_1^{{\mathrm{cris}}}(B_0/R_0)_R$ for $B=A_0, A_0^\vee$. We have to show moreover that the level structure $\alpha_{K',0}=(\alpha^p_0, \alpha_{p,0})$ extends unique to $A$. It is clear for $\alpha^{p}_0$. For $\alpha_{p,0}$, let $H_0=\prod_{\gothp\text{ of type } \alpha^\sharp} \alpha_{\gothp}$ be the product of the closed subgroup in the data of $\alpha_{p,0}$. Let $f_0:A_0{\rightarrow}B_0=A_0/H_0$ be the canonical isogeny. It suffices to show that $B_0$ and $f_0$ deform to $R$. The abelian variety $B_0$ is equipped with an induced action of ${\mathcal{O}}_{D_{\ttS}}$, a polarization $\lambda_{B_0}$ satisfying the same conditions (a) and (b). The isogeny $f_0$ induces canonical isomorphisms $H^{{\mathrm{cris}}}_{1}(A_0/R_0)_{R, \tilde \tau}\cong H_1^{{\mathrm{cris}}}(B_0/R_0)_{R,\tilde \tau}$ for $\tilde \tau \in \Sigma_{E,\infty/\gothp}$ with $\gothp$ not of type $\alpha^\sharp$. So for such primes $\gothp$ and $\tilde \tau \in \Sigma_{\infty/\gothp}$, the liftings $\omega^{\circ}_{R,\tilde \tau}$ chosen above give the liftings of $\omega_{B_0^\vee/R_0, \tilde \tau}^\circ\subset H_1^{\mathrm{dR}}(B_0/R_0)_{\tilde \tau}$. For $\tilde \tau\in \Sigma_{\infty/\gothp}$ with $\gothp$ of type $\alpha^\sharp$, we note that at each closed point $x$ of $R_0$, $\omega_{B_0^\vee/k_x, \tilde \tau}^\circ$ is either trivial or isomorphic to the whole $H_1^{{\mathrm{dR}}}(B_0/k_x)_{\tilde \tau}^{\circ}$ as in the case for $A_0$; hence the same holds for $R_0$ in place of $k_x$. Therefore, $\omega_{B_0^\vee/R_0, \tilde \tau}^\circ$ admits a unique lift to a direct summand of $H^{{\mathrm{cris}}}_1(B_0/R_0)_{R, \tilde \tau}^{\circ}$. Such choices of liftings of $\omega_{B_0^\vee/R_0,\tilde \tau}^{\circ}$ give rise to a deformation $B/R$ of $B_0/R_0$. It is clear that $f_0 : A_0 \to B_0$ also lifts to an isogeny $f: A \to B$. Then the kernel of $f$ gives the required lift of $H_0$. This concludes the proof of the smoothness of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$. The dimension of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ follows from the calculation of the tangent bundle of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ as the following corollary shows. When the ramification set $\ttS_{\infty}$ is non-empty, it is a standard argument to use valuative criterion to check that ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ is proper. We will postpone the proof to Proposition \[Prop:smoothness\], where a more general statement is proved. \[C:deformation\] Let $S_0{\hookrightarrow}S$ be a closed immersion of ${\overline{\FF}_p}$-schemes with ideal sheaf $\calI$ such that $\calI^2=0$. Let $x_0=(A_0, \iota_0, \lambda_0, \bar{\alpha}_{K',0} )$ be an $S_0$-valued point of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$. Then the set-valued sheaf of local deformations of $x_0$ to $S$ form a torsor under the group $$\bigoplus_{\tau\in \Sigma_{\infty}-\ttS_{\infty}} \bigl(\operatorname{Lie}(A_0)_{\tilde \tau}^{\circ}\otimes \operatorname{Lie}(A_0)_{\tilde \tau ^c}^{\circ}\bigr)\otimes \calI $$ In particular, the tangent bundle $\calT_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})}$ of ${\mathbf{Sh}}_{K'}(G_{\tilde \ttS}')$ is canonically isomorphic to $$\bigoplus_{\tau\in \Sigma_{\infty}-\ttS_{\infty}} \operatorname{Lie}(\bfA')^\circ_{\tilde \tau}\otimes \operatorname{Lie}(\bfA')^\circ_{\tilde \tau^c}$$ where $\bfA' = \bfA'_{{\tilde \ttS}, K'}$ denotes the universal abelian scheme over ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$. A deformation of $x_0$ is determined by the liftings $\omega^\circ_{S, \tilde \tau}\subseteq H^{{\mathrm{cris}}}_1(A_0/S_0)_{S, \tilde \tau}^\circ$ of $\omega_{A_0^\vee/S_0, \tilde \tau}^\circ$ for $\tilde \tau\in \Sigma_{E,\infty}$. From the proof of Theorem \[T:unitary-shimura-variety-representability\], we see that the choices for $\omega_{S, \tilde \tau}^\circ$ are unique if $\tilde \tau$ restricts to $\tau\in \ttS_{\infty}$. For $\tau\in \Sigma_{\infty}-\ttS_{\infty}$, the possible liftings $\omega_{S, \tilde \tau}$ and $\omega_{S, \tilde \tau^c}$ determines each other, and form a torsor under the group $$\operatorname{Hom}_{{\mathcal{O}}_{S_0}}(\omega_{A_0^\vee/S_0, \tilde \tau}^\circ, \operatorname{Lie}(A_0)_{\tilde \tau}^\circ) \otimes_{{\mathcal{O}}_{S_0}}\calI\simeq \operatorname{Lie}(A_0)_{\tilde \tau}^\circ \otimes \operatorname{Lie}(A_0)^\circ_{\tilde \tau^c}\otimes_{{\mathcal{O}}_{S_0}} \calI.$$ The statement for the local lifts of $x_0$ to $S$ follows immediately. Applying this to the universal case, we obtain the second part of the Corollary. We remark that the moduli space ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ does not depend on the choice of the admissible lattice $\Lambda$ in Subsection \[S:integral-unitary\]; but the universal abelian scheme $\bfA'$ does in the following way. If $\Lambda_1$ and $\Lambda_2$ are two admissible lattices, we put $\widehat{\Lambda}_i^{(p)}: = \Lambda_i \otimes_\ZZ \widehat{\ZZ}^{(p)}$, and we use $\bfA'_i$ to denote the corresponding universal abelian variety over ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ and $\bar {\boldsymbol{\alpha}}^p_{K'^p, i}$ to denote the universal level structure (away from $p$), for $i = 1,2$. Then there is a natural prime-to-$p$ quasi-isogeny $\eta: \bfA'_1 \dashrightarrow \bfA'_2$ such that $$\xymatrix@C=50pt{ \widehat \Lambda^{(p)}_1 \ar@{-->}[d] \ar[r]^-{\bar {\boldsymbol{\alpha}}^p_{K'^p,1}}_-\cong & T^{(p)}( \bfA'_1) \ar@{-->}[d]^{T^{(p)}(\eta)}\\ \widehat \Lambda^{(p)}_2 \ar[r]^-{\bar {\boldsymbol{\alpha}}^p_{K'^p, 2}}_-\cong & T^{(p)}( \bfA'_2) }$$ is a commutative diagram up the action of $K'^p$, where the left vertical arrow is the isogeny of lattices inside $V \otimes_\QQ \AAA^{\infty,p}$. (For more detailed discussion, see [@lan 1.4.3].) \[C:integral-model-quaternion\] The integral model ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ defined in Theorem \[T:unitary-shimura-variety-representability\] gives an integral canonical model ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$ of ${\mathrm{Sh}}_{K'_p}(G'_\ttS)$. Consequently, the quaternionic Shimura variety ${\mathrm{Sh}}_{K_p}(G_\ttS)$ admits an integral canonical model over $\calO_{\wp}$. Similarly, the Shimura varieties ${\mathrm{Sh}}_{K_p \times K_{E,p}}(G_\ttS \times T_{E, \tilde \ttS})$ and ${\mathrm{Sh}}_{K''_p}(G''_{\tilde \ttS})$ both admit integral canonical models over $\calO_{\tilde \wp}$. The geometric connected component of these integral canonical models are canonically isomorphic. We first assume that $\ttS_\infty \neq \Sigma_\infty$. We need to verify that for any smooth ${\mathcal{O}}_{\tilde \wp}$-scheme $S$, any morphism $s_0:S\otimes_{{\mathcal{O}}_{\tilde \wp}} E_{\tilde \ttS, \tilde \wp}{\rightarrow}{\mathrm{Sh}}_{K'_p}(G'_{\tilde \ttS})$ extends to a morphism $s: S{\rightarrow}{\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$. Explicitly, we have to show that a tuple $(A, \iota,\lambda_, \alpha^p\alpha_p)$ over $S\otimes_{{\mathcal{O}}_{\tilde \wp}}E_{{\tilde \ttS},\tilde \wp}$ extends to a similar tuple over $S$. Here, $\alpha^p\alpha_p$ is the projective limit in $K'^{p}$ of level structures $\alpha^p_{K'^p} \alpha_p$ as in Theorem \[T:unitary-shimura-variety-representability\](c). The same arguments of [@moonen96 Corollary 3.8] apply to proving the existence of extension of $A$, $\iota$, $\lambda$, and the prime-to-$p$ level structure $\alpha^p$. It remains to extend the level structure $\alpha_p$. Let ${\mathbf{Sh}}_{\widetilde {K}'_{p}}(G'_{\tilde \ttS})$ denote the similar moduli space as ${\mathbf{Sh}}_{K'_{p}}(G_{\tilde \ttS}')$ by forgetting the $p$-level structure $\alpha_p$. The discussion above shows that ${\mathbf{Sh}}_{\widetilde {K}'_{p}}(G'_{\tilde \ttS})$ satisfies the extension property. We have seen in the proof of Theorem \[T:unitary-shimura-variety-representability\] that there is no local deformation of $\alpha_p$, which means the forgetful map ${\mathbf{Sh}}_{K'_{p}}(G'_{\tilde \ttS}){\rightarrow}{\mathbf{Sh}}_{\widetilde {K}'_{p}}(G'_{\tilde \ttS})$ is finite and étale. By the discussion above, there exists a morphism $\tilde s: S{\rightarrow}{\mathbf{Sh}}_{\widetilde{K}'_p}(G'_{\tilde \ttS})$ such that the square of the following diagram $$\xymatrix{ S\otimes_{{\mathcal{O}}_{\tilde \wp}}E_{\tilde \ttS,\tilde \wp}\ar[r]^{s_0}\ar[d]& {\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})\ar[d]\\ S\ar@{-->}[ur]^{s} \ar[r]^{\tilde s} &{\mathbf{Sh}}_{\widetilde{K}'_p}(G'_{\tilde \ttS}) }$$ is commutative. We have to show that there exists a map $s$ as the dotted arrow that makes whole diagram commutative. Giving such a map $s$ is equivalent to giving a section of the finite étale cover $S\times_{{\mathbf{Sh}}_{ \widetilde{K}'_{\ttS}}(G'_{\tilde \ttS})}{\mathbf{Sh}}_{{K}'_{\ttS}}(G'_{\tilde \ttS}){\rightarrow}S$ extending the section corresponding to $s_0$. Since a section of a finite étale cover of separated schemes is an open and closed immersion, the existence of $s$ follows immediately. The existence of integral canonical models for ${\mathrm{Sh}}_{K_p}(G_{\ttS})$, ${\mathrm{Sh}}_{K''_p}(G''_{\tilde \ttS})$ and ${\mathrm{Sh}}_{K_p\times K_{E,p}}(G_{\ttS}\times T_{E, \tilde \ttS})$ follows from Corollary \[C:Sh(G)\^circ\_Zp independent of G\]. When $\ttS_\infty = \Sigma_\infty$, we need to show that the geometric Frobenius ${\mathrm{Frob}}_{\tilde \wp}$ acts on the moduli space ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ by appropriate central Hecke action specified by reciprocity map. Put $n_{\tilde \wp} = [k_{\tilde \wp}: \FF_p]$. Let ${\gothR\mathrm{ec}}_{Z'}: \operatorname{Gal}_{E_{\tilde \ttS}} \to Z'(\QQ)^{\mathrm{cl}}\backslash Z'(\AAA^\infty)/Z'({\ZZ_p})$ denote the reciprocity map defined in Subsection \[S:integral model weak Shimura datum\], where $Z'$ is the center of $G'_{\tilde \ttS}$ which is the algebraic group associated to the subgroup of $E^\times$ consisting of elements with norm to $F^{\times}$ lying in ${\mathbb{Q}}^{\times}$. By definition, ${\gothR\mathrm{ec}}_{Z'}({\mathrm{Frob}}_{\tilde \wp})$ is the image of $\varpi_{\tilde \wp}$ under the composite of $${\gothR\mathrm{ec}}_{Z', \tilde \wp}\colon E_{\tilde\ttS,\tilde \wp}^\times / \calO_{\tilde \wp}^\times \xrightarrow{\operatorname{Rec}_{Z'}(G'_{\tilde \ttS}, \gothH_\ttS)} Z'({\QQ_p}) /Z'({\ZZ_p})$$ and the natural map $Z'({\QQ_p})/Z'({\ZZ_p}) {\rightarrow}Z'(\QQ)^{\mathrm{cl}}\backslash Z'(\AAA^\infty)/Z'({\ZZ_p})$. Explicitly, $$Z'(\QQ_p) = \big\{ \big( (x_\gothp)_{\gothp \in \Sigma_p}, y\big)\,\big|\, y \in \QQ_p^\times,\ x_\gothp \in E_{\gothp}^\times, \textrm{ and }{\mathrm{Nm}}_{E_\gothp/F_\gothp}(x_\gothp) = y\big\}.$$ We note that there is no $p$-adic primes of $F$ of type $\beta^{\sharp}$, and the valuation of $y$ determines the valuation of $x_\gothp$ for $\gothp$ of type $\beta^{\sharp}$. For each prime $\gothp\in \Sigma_p$ of type $\alpha$ or $\alpha^{\sharp}$, choose a place $\gothq$ of $E$ above $\gothp$, then the map $((x_{\gothp})_{\gothp}, y)\mapsto (\mathrm{val}_p(y), (\mathrm{val}_{p}(x_{\gothq}))_{\gothp})$ defines an isomorphism $$\xi:Z'({\QQ_p})/Z'({\ZZ_p})\cong \ZZ \times \prod_{\gothp \textrm{ of type $\alpha$ or $\alpha^{\sharp}$}} \ZZ,$$ where we have written $x_{\gothp}=(x_{\gothq},x_{\bar\gothq})$ for each prime $\gothp\in \Sigma_p$ of type $\alpha$ or $\alpha^{\sharp}$. By definition of ${\gothR\mathrm{ec}}_{Z', \tilde \wp}$ in Subsection \[S:integral model weak Shimura datum\] using $h'_{\tilde \ttS}$, we see that $\xi\circ{\gothR\mathrm{ec}}_{Z', \tilde \wp}(\varpi_{\tilde \wp})$ is equal to $$\label{E:image of rec} \big(n_{\tilde \wp}, (\#\tilde \ttS_{\infty / \gothq} \cdot n_{\tilde \wp} / f_\gothp)_{\gothp \in \Sigma_p}\big),$$ where $n_{\tilde \wp} = [\FF_{\tilde \wp}: \FF_p]$ and $f_\gothp$ is the inertia degree of $\gothp $ in $F/\QQ$. On the other hand, ${\mathrm{Frob}}_{\tilde \wp}$ takes a closed point $x = (A, \iota, \lambda, \alpha_{K'})$ of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{\overline \FF_p}$ to ${\mathrm{Frob}}_{\tilde \wp}(x) = ({\mathrm{Frob}}_{\tilde \wp}^*(A), \iota^{{\mathrm{Frob}}_{\tilde \wp}}, \lambda^{{\mathrm{Frob}}_{\tilde \wp}}, \alpha_{K'} \circ {\mathrm{Frob}}_{\tilde \wp})$. For a $p$-adic prime $\gothp$ of $F$ (or of $E$), denote by ${\tilde{{\mathcal{D}}}}(A)_{\gothp}$ the covariant Dieudonné module of $A[\gothp^{\infty}]$. We observe that, if $\gothp$ is a prime of $F$ of type $\beta^{\sharp}$, then $$\tilde \calD({\mathrm{Frob}}^*_{\tilde \wp}(A))_\gothp = p^{n_{\tilde \wp}/2} \tilde \calD(A)_\gothp;$$ if $\gothp$ is of type $\alpha$ or $\alpha^{\sharp}$ with $\gothq$ a place of $E$ above $\gothp$, then $$\tilde \calD({\mathrm{Frob}}_{\tilde \wp}(A))_\gothq = p^{\#\tilde \ttS_{\infty / \gothq} \cdot n_{\tilde \wp} / f_\gothp} \tilde \calD(A)_\gothq$$ This agrees with the computation of ${\gothR\mathrm{ec}}_{Z', \tilde \wp}(\varpi_{\tilde \wp})$ above. The rest of this section is devoted to understanding how to pass the universal abelian varieties on ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$ to other Shimura varieties, as well as natural partial Frobenius morphisms among these varieties and their compatibility with the abelian varieties. Actions on universal abelian varieties in the unitary case {#S:abel var in unitary case} ---------------------------------------------------------- We need to extend the usual tame Hecke algebra action on the universal abelian variety $\bfA'_{K'_p}$ over ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$ to the action of a slightly bigger group $\widetilde G_{\tilde \ttS}: = G'_{\tilde \ttS}(\AAA^{\infty, p}) G''_{\tilde \ttS}(\QQ)_+^{(p)}$. Take an element $\tilde g \in \widetilde G_{\tilde \ttS}$; let $K'^p_1$ and $K'^p_2$ be two open compact subgroups of $G'_{\tilde \ttS}(\AAA^{\infty, p})$ such that $\tilde g^{-1} K'^p_1 \tilde g \subseteq K'^p_2$ (note that $G''_{\tilde \ttS}$ normalizes $G'_{\tilde \ttS}$). We put $K'_i = K'^p_iK'_p$ for $i=1,2$. Then starting from the universal abelian variety $\bfA'_{K'_1}$ together with the tame level structure $\bar {\boldsymbol{\alpha}}_{K'^p_1}^p: \widehat \Lambda^{(p)} \xrightarrow{\cong} T^{(p)} \bfA'_{K'_1}$, we may obtain an abelian variety $\bfB'$ over ${\mathbf{Sh}}_{K'_1}(G'_{\tilde \ttS})$, together with a prime-to-$p$ quasi-isogeny $\eta: \bfA'_{K'_1} \to \bfB' $ and a tame level structure such that the following diagram commutes $$\xymatrix@C=40pt{ & \widehat \Lambda^{(p)} \ar[r]^-{\bar {\boldsymbol{\alpha}}^p_{K'^p}}_-\cong \ar@{-->}[d] & T^{(p)}( \bfA'_{K'_1}) \ar@{-->}[d]^{T^{(p)}(\eta)} \\ \widehat \Lambda^{(p)} \ar[r]^-{\cdot \tilde g^{-1}}_-\cong & \tilde g\widehat \Lambda^{(p)} \ar[r]^-\cong & T^{(p)}(\bfB'), }$$ where the left vertical arrow is the natural quasi-isogeny as lattices inside $V \otimes_\QQ \AAA^{\infty,p}$. Since $\tilde g^{-1}K'^p_1 \tilde g \subseteq K'^p_2$, we may take the $K'^p_2$-orbit of the composite of the bottom homomorphism as the tame level structure. One can easily transfer other data in the moduli problem of Theorem \[T:unitary-shimura-variety-representability\] to $\bfB'$, *except for the polarization* which we make the modification as follows: since $\tilde g \in G'_{\tilde \ttS}(\AAA^{\infty, p}) G''_{\tilde \ttS}(\QQ)_+^{(p)}$, we have $\nu(\tilde g) \in \calO_{F, (p)}^{\times,>0} \cdot \AAA_\QQ^{\infty, p,\times} = \calO_{F, (p)}^{\times,>0} \cdot \widehat \ZZ^{(p),\times}$. We can then write $\nu(\tilde g)$ as the product $\nu^+_{\tilde g} \cdot u$ for $\nu^+_{\tilde g} \in \calO_{F, (p)}^{\times,>0}$ and $u \in \widehat \ZZ^{(p),\times}$. In fact, $\nu^+_{\tilde g}$ is uniquely determined by this restriction. We take the polarization on $\bfB'$ to be the composite of a sequence of quasi-isogenies: $$\lambda_{\bfB'}: \bfB' \xleftarrow{\quad} \bfA'_{K'_1} \xrightarrow{\nu^+_{\tilde g} \lambda_{\bfA'}} \big(\bfA'_{K'_1}\big)^\vee \xrightarrow{\quad} \bfB'^\vee.$$ Such modification ensures that $\bfB'$ satisfies condition (c1) of Theorem \[T:unitary-shimura-variety-representability\]. The moduli problem then implies that $\bfB' \cong (H_{ \tilde g})^*(\bfA'_{K'_2})$ for a uniquely determined morphism $H_{\tilde g}:{\mathbf{Sh}}_{K'_1}(G'_{\tilde \ttS})\to {\mathbf{Sh}}_{K'_2}(G'_{\tilde \ttS})$; this gives the action of $\widetilde G$. Moreover, we have a quasi-isogeny $$H_{\tilde g}^\#: \bfA'_{K'_1} \xrightarrow{\eta} \bfB' \cong (H_{\tilde g})^*(\bfA'_{K'_2})$$ giving rise to an equivariant action of $\widetilde G$ on the universal abelian varieties $\bfA'_{K'_p}$ over ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$. One easily checks that the action of diagonal $ \calO_{E,(p)}^\times$ on the Shimura variety ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$ is trivial, and hence we have an action of $\calG'_{\tilde \ttS}= \widetilde G_{\tilde \ttS} / \calO_{E,(p)}^{\times, {\mathrm{cl}}}$ on ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$. However, the action of $\calO_{E,(p)}^\times$ on the universal abelian variety $\bfA'_{K'_p}$ is *not* trivial. So the latter does not carry a natural action of $\calG'_{\tilde \ttS}$. So our earlier framework for Shimura varieties does not apply to this case directly. However, we observe that, by the construction at the end of Subsection \[A:connected integral model\], $${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS}) = {\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS}) \times_{\calG'_{\tilde \ttS}} \calG''_{\tilde \ttS} = {\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS}) \times_{\widetilde G_{\tilde \ttS}} G''_{\tilde \ttS}(\AAA^{\infty,p}).$$ So $$\bfA''_{K''_p}: = \bfA'_{K'_p}\times_{\calG'_{\tilde \ttS}} \calG''_{\tilde \ttS} = \bfA'_{K'_p} \times_{\widetilde G_{\tilde \ttS}} G''_{\tilde \ttS}(\AAA^{\infty,p})$$ gives a natural family of abelian variety over ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})$. We will not discuss family of abelian varieties over the quaternionic Shimura variety ${\mathbf{Sh}}_{K_p}(G_\ttS)$ (except when $\ttS =\emptyset$). Automorhpic $l$-adic systems on ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})$ and its geometric interpretation ----------------------------------------------------------------------------------------------------------- By a *multiweight*, we mean a tuple $(\underline{k}, w) =((k_\tau)_{\tau\in\Sigma_\infty}, w) \in \NN^{[F:{\mathbb{Q}}]} \times \NN$ such that $ k_\tau \geq 2$ and $w \equiv k_\tau \pmod 2$ for each $\tau$. We also fix a section of the natural map $\Sigma_{E, \infty} \to \Sigma_\infty$, that is to fix a extension $\tilde \tau$ to $E$ of each real embedding $\tau \in \Sigma_\infty$ of $F$; use $\tilde \Sigma$ to denote the image of this section. In this subsection, we use $\tilde \tau$ to denote this chosen lift of $\tau$. We fix a subfield $L$ of $\overline \QQ \subset \CC$ containing all embeddings of $E$, as the coefficient field. Let $\gothl$ be a finite place of $L$ over a prime $l$ with $l \neq p$. Fix an isomorphism $\iota_\gothl:\CC \simeq \overline L_\gothl $. Consider the injection $$G''_{\tilde \ttS} \times_\QQ L = \big( \operatorname{Res}_{F/\QQ}(B_\ttS^\times) \times_{\operatorname{Res}_{F/\QQ}\GG_m} \operatorname{Res}_{E/\QQ}\GG_m \big) \times_\QQ L \hookrightarrow \operatorname{Res}_{E/\QQ} D^\times_\ttS \times_\QQ L \cong \prod_{\tau \in \Sigma_\infty} {\mathrm{GL}}_{2, L, \tilde \tau} \times {\mathrm{GL}}_{2, L, \tilde \tau^c},$$ where $E^\times$ acts on ${\mathrm{GL}}_{2, L, \tilde \tau}$ (resp. ${\mathrm{GL}}_{2, L, \tilde \tau^c}$) through $\tilde \tau$ (resp. $\tilde \tau^c$). For a multiweight $(\underline k, w)$, we consider the following representation of $G''_{\tilde \ttS} \times_\QQ L$: $$\rho''^{(\underline{k}, w)}_{\tilde \ttS, \widetilde \Sigma} = \bigotimes_{\tau \in \widetilde \Sigma} \rho_\tau^{(k_\tau, w)} \circ \check{\mathrm{pr}}_{\tilde \tau} \quad \textrm{for }i = 1, 2 \textrm{ with} \quad \rho_\tau^{(k_\tau, w)} = {\mathrm{Sym}}^{k_\tau-2} \otimes \det{}^{\frac{w-k_\tau}{2}} ,$$ where $\tau$ is the restriction of $\tilde \tau$ to $F$, and $\check{\mathrm{pr}}_{\tilde \tau}$ is the *contragradient* of the natural projection to the $\tilde \tau$-component of $G''_{\tilde \ttS} \times_\QQ L \hookrightarrow \operatorname{Res}_{E/\QQ} D_\ttS^\times \times_\QQ L$. Note that $\rho''^{(\underline k, w)}_{\tilde \ttS, \widetilde \Sigma}$ is trivial on the maximal anisotropic ${\mathbb{R}}$-split subtorus of the center of $G''_{\tilde \ttS}$, i.e. $\operatorname{Ker}(\operatorname{Res}_{F/{\mathbb{Q}}}\GG_m \to \GG_m)$. By [@milne Ch. III, §7], $\rho''^{(\underline k, w)}_{\tilde \ttS, \widetilde \Sigma}$ corresponds to a lisse $L_\gothl$-sheaf $\scrL''^{(\underline{k}, w)}_{\tilde \ttS, \widetilde \Sigma}$ over the Shimura variety ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS})$ compatible as the level structure changes. We now give a geometric interpretation of this automorphic $l$-adic sheaf on ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS})$. For this, we fix an isomorphism $D_\ttS\simeq \rmM_2(E)$ and let $\gothe = \big( \begin{smallmatrix} 1&0\\0&0 \end{smallmatrix}\big)\in \rmM_2({\mathcal{O}}_E)$ denote the idempotent element. Let $\bfA'' = \bfA''_{\tilde \ttS, K''}$ denote the natural family of abelian varieties constructed in Subsection \[S:abel var in unitary case\]. Let $V(\bfA'')$ denote the $l$-adic Tate module of $\bfA''$. We then have a decomposition $$V(\bfA'') \otimes_{\QQ_l} L_\gothl \cong \bigoplus_{\tau \in\Sigma_\infty} \big(V(\bfA'')_{\tilde \tau} \oplus V(\bfA'')_{\tilde \tau^c} \big) = \bigoplus_{\tau \in\Sigma_\infty} \big(V(\bfA'')^{\circ, \oplus 2}_{\tilde \tau} \oplus V(\bfA'')^{\circ, \oplus 2}_{\tilde \tau^c} \big),$$ where $V(\bfA'')_{\tilde \tau}$ (resp. $V(\bfA'')_{\tilde \tau^c}$) is the component where $\calO_E$ acts through $\iota_\gothl \circ\tilde \tau$ (resp. $\iota_\gothl \circ \tilde \tau^c$), and $V(\bfA'')_{\tilde \tau}^\circ = \gothe V(\bfA'')_{\tilde \tau}$ (resp. $V(\bfA'')_{\tilde \tau^c}^\circ = \gothe V(\bfA'')_{\tilde \tau^c}$) is a lisse $L_\gothl$-sheaf of rank $2$. For a multiweight $(\underline k, w)$, we put $$\calL^{({\underline{k}},w)}_{\widetilde\Sigma}(\bfA'')=\bigotimes_{\tau \in \widetilde \Sigma} \bigg( {\mathrm{Sym}}^{k_\tau -2} V(\bfA'')_{\tilde \tau}^{\circ, \vee} \otimes (\wedge^2 V(\bfA'')_{\tilde \tau}^{\circ, \vee})^{\frac{w-k_\tau}2} \bigg).$$ Note the duals on the Tate modules mean that we are essentially taking the relative first étale *cohomology*. The moduli interpretation implies that we have a canonical isomorphism $$\scrL''^{(\underline k, w)}_{\tilde \ttS, \widetilde \Sigma} \cong \calL_{\widetilde \Sigma}^{(\underline k, w)}(\bfA''_{\tilde \ttS}).$$ Twisted Partial Frobenius {#S:partial Frobenius} ------------------------- The action of the twisted partial Frobenius and its compatibility with the GO-strata description will be the key to later applications in [@tian-xiao2]. We start with the action of the twisted partial Frobenius on the universal abelian scheme $\bfA'_{\tilde \ttS} = \bfA'_{\tilde \ttS, K'}$ over the unitary Shimura variety ${\mathbf{Sh}}_{K'}(G_{\tilde \ttS}')$. Fix $\gothp \in \Sigma_p$. We define an action of $\sigma_\gothp $ on $\Sigma_{E,\infty}$ as follows: for $\tilde\tau\in \Sigma_{E,\infty}$, we put $$\label{E:defn-sigma-gothp} \sigma_{\gothp}\tilde\tau=\begin{cases}\sigma \circ\tilde\tau & \text{if }\tilde \tau\in \Sigma_{E,\infty/\gothp},\\ \tilde\tau &\text{if }\tilde\tau\notin \Sigma_{E,\infty/\gothp}, \end{cases}$$ where $\Sigma_{E,\infty/\gothp}$ denotes the lifts of places in $\Sigma_{\infty/\gothp}$. Note that $\sigma_{\gothp}$ induces a natural action on $\Sigma_{\infty}$, and $\prod_{\gothp\in \Sigma_p}\sigma_{\gothp}=\sigma$ is the Frobenius action. Let $\sigma_{\gothp}\tilde \ttS$ denote the image of $\tilde \ttS$ under $\sigma_{\gothp}$. We fix an isomorphism $B_{\sigma_{\gothp}\ttS}\otimes\AAA^{\infty}\simeq B_{\ttS}\otimes\AAA^{\infty}$, which induces in turns an isomorphism $G'_{\sigma_\gothp\tilde \ttS}(\AAA^{\infty})\simeq G'_{\tilde \ttS}(\AAA^{\infty})$. We may thus regard $K'$ as an open subgroup of $G'_{\sigma_{\gothp}\tilde \ttS}(\AAA^{\infty})$. Note that a prime $\gothp'\in \Sigma_p$ has the same type with respect to $\tilde \ttS$ or $\sigma_{\gothp}\tilde \ttS$. We get therefore a unitary Shimura variety ${\mathbf{Sh}}_{K'}(G'_{\sigma_{\gothp}\tilde \ttS})$. We also point out that the $p$-adic completion of the reflex field at $\tilde \wp$ for $G'_{\tilde \ttS}$ and for $G'_{\sigma_\gothp \tilde \ttS}$ are the same. Let $S$ be a locally noetherian $k_{\tilde \wp}$-scheme and let $(A, \iota, \lambda, \bar \alpha_{K'})$ be an $S$-point on ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_{\tilde \wp}}$. We will define a new $S$-point $(A',\iota', \lambda', \bar{\alpha}_{K'})$ on ${\mathbf{Sh}}_{K'}(G'_{\sigma^2_\gothp \tilde \ttS})_{k_{\tilde \wp}}$ as follows. The kernel of the relative $p^2$-Frobenius ${\mathrm{Fr}}_{A}\colon A \to A^{(p^2/S)}$ carries an action of $\calO_F$, and we denote by $\operatorname{Ker}_{\gothp^2}$ its $\gothp$-component. We put $A'=(A/\operatorname{Ker}_{\gothp^2})\otimes_{{\mathcal{O}}_F} \gothp$ with its induced action by ${\mathcal{O}}_{D_{\ttS}}$. It also comes equipped with a quasi-isogeny $\eta$ given by the composite $$\eta: A \longrightarrow A/\operatorname{Ker}_{\gothp^2} \longleftarrow (A/\operatorname{Ker}_{\gothp^2})\otimes_{{\mathcal{O}}_F} \gothp =A'.$$ It induces canonical isomorphisms of $p$-divisible groups $A'[\gothq^{\infty}]\simeq A[\gothq^{\infty}]$ for $\gothq\in \Sigma_{p}$ with $\gothq\neq \gothp$, and $A'[\gothp^{\infty}]\simeq A[\gothp^{\infty}]^{(p^2)}$. From this, one can easily check the signature condition for $A'$. We define the polarization $\lambda'$ to be the quasi-isogeny defined by the composite $$\label{E:polarization for partial frob} A' \xleftarrow{\ \eta\ } A \xrightarrow{\ \lambda\ } A^\vee \xleftarrow{\ \eta^\vee\ } A'^\vee.$$ We have to check that $\lambda'$ is a genuine isogeny, and it verifies condition Theorem \[T:unitary-shimura-variety-representability\](b) on $\lambda'$ at prime $\gothp$. By flatness criterion by fibers, it suffices to do this after base change to every geometric point of $S$. We may thus suppose that $S=\operatorname{Spec}(k)$ for an algebraically closed field $k$ of characteristic $p$. Let ${\tilde{{\mathcal{D}}}}(A)_{\gothp}$ be the covariant Dieudonné module of the $p$-divisible group $A[\gothp^{\infty}]$, and similarly for ${\tilde{{\mathcal{D}}}}(A')_{\gothp}$. By definition, we have $${\tilde{{\mathcal{D}}}}(A')_{\gothp}=p{\tilde{{\mathcal{D}}}}(A/\operatorname{Ker}_{\gothp^2})_{\gothp}=pV^{-2}{\tilde{{\mathcal{D}}}}(A)_{\gothp}=p^{-1}F^2{\tilde{{\mathcal{D}}}}(A)_{\gothp}$$ where $pV^{-2}{\tilde{{\mathcal{D}}}}(A)_{\gothp}$ means the inverse image of ${\tilde{{\mathcal{D}}}}(A)_{\gothp}$ under the bijective endomorphism $V^{2}$ on ${\tilde{{\mathcal{D}}}}(A)_{\gothp}[1/p]$. Applying the Dieudonné functor to , we get $$\lambda'_*: {\tilde{{\mathcal{D}}}}(A')_{\gothp}=pV^{-2}{\tilde{{\mathcal{D}}}}(A)_{\gothp} \xleftarrow{\ \eta_*\ } {\tilde{{\mathcal{D}}}}(A)_{\gothp} \xrightarrow{\ \lambda_*\ } {\tilde{{\mathcal{D}}}}(A^\vee)_{\gothp} \xleftarrow{\ \eta^\vee_*\ }\tilde \calD(A'^\vee) = p^{-1}F^2{\tilde{{\mathcal{D}}}}(A^\vee)_{\gothp}.$$ Now it is easy to see that $\lambda'$ is an isogeny, and the condition on $\lambda'$ follows from that for $\lambda$. The tame level structure $\bar\alpha'_{K'}$ is given by the composition $$\widehat\Lambda^{(p)} \xrightarrow{\alpha_{K'}} T^{(p)}A \xrightarrow \cong T^{(p)}(A/\operatorname{Ker}_{\gothp^2}) \xleftarrow \cong T^{(p)}((A / \operatorname{Ker}_{\gothp^2}) \otimes_{\calO_F} \gothp) = T^{(p)}(A').$$ We are left to define the subgroups $\alpha'_{\gothp'}$ for all $\gothp' \in \Sigma_p$ of type $\alpha^\sharp$. The definition is clear for $\gothp'\neq \gothp$, since $A'[\gothp'^{\infty}]$ is canonically identified with $A[\gothp'^{\infty}]$. Assume thus $\gothp' = \gothp$ is of type $\alpha^\#$. In the data of $\alpha'_\gothp = H'_\gothq \oplus H'_{\bar \gothq}\subseteq A'[\gothp]$, the subgroup $H'_{\bar\gothq}$ is determined as the orthogonal complement of $H'_{\gothq}$ under the Weil-pairing on $A[\gothp]$. Therefore, it suffices to construct $H'_{\gothq}$, or equivalently an ${\mathcal{O}}_{D_{\ttS}}$-isogeny $f':A'{\rightarrow}B'=A'/H'_{\gothq}$ with kernel in $A'[\gothq]$ of degree $\#k_{\gothp}^2$. Let $f:A{\rightarrow}B=A/H_{\gothq}$ be the isogeny given by $\alpha_{\gothp}$. We write $\operatorname{Ker}_{\gothp^2, B}$ for the $\gothp$-component of the kernel of the relative $p^2$-Frobenius $B \to B^{(p^2)}$. It is easy to see that we have a natural isogeny $f_{\gothp^2}: A / \operatorname{Ker}_{\gothp^2} \to B / \operatorname{Ker}_{\gothp^2, B}$. Then $H'_\gothq$ is defined to be the kernel of $$f_{\gothp^2} \otimes 1:\ A' = (A / \operatorname{Ker}_{\gothp^2}) \otimes \gothp \longrightarrow (B / \operatorname{Ker}_{\gothp^2, B}) \otimes \gothp =: B',$$ and $\alpha'_\gothp$ is the direct sum of $H'_\gothq$ and its orthogonal dual $H'_{\bar \gothq}$. To sum up, we obtain a morphism $$\label{E:twist-partial-Frob} \gothF'_{\gothp^2}: {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_{\tilde \wp}} \to {\mathbf{Sh}}_{K'}(G'_{\sigma^2_\gothp \tilde \ttS})_{k_{\tilde \wp}}.$$ In all cases, we call the morphism $\gothF'_{\gothp^2}$ the *twisted partial Frobenius map* on the unitary Shimura varieties. Moreover, if $\bfA'_{\tilde \ttS}$ and $\bfA'_{\sigma^2_{\gothp}\tilde \ttS}$ are respectively the universal abelian schemes over ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})$ and ${\mathbf{Sh}}_{K'}(G'_{\sigma^2_{\gothp}\tilde \ttS})$, we have the following universal quasi-isogeny: $$\label{E:universal-isog-Frob} \eta'_{\gothp^2}\colon \bfA'_{\tilde \ttS, k_{\tilde \wp}} \longrightarrow \gothF'^{*}_{\gothp^2}(\bfA'_{\sigma^2_{\gothp}\tilde \ttS, k_{\tilde \wp}}).$$ It is clear from the definition that $(\gothF'_{\gothp^2}, \eta'_\gothp)$’s for different $\gothp \in \Sigma_p$ commute with each other. Let $S_p: {\mathbf{Sh}}_{K'}(G_{\tilde\ttS}'){\rightarrow}{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})$ be the automorphism defined by $(A,\iota,\lambda,\bar\alpha_{K'})\mapsto (A,\iota,\lambda, p\bar\alpha_{K'})$. It is clear that $S_p^*\bfA'_{\tilde\ttS}\cong \bfA'_{\tilde\ttS}$. Hence, $S_p$ induces an automorphism of the cohomology groups $H^{\star}_{{\mathrm{rig}}}({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS}), \scrD^{({\underline{k}},w)}_{{\widetilde{\Sigma}}}(\bfA'_{\tilde\ttS,k_0}))$, still denoted by $S_p$. If $$F^2_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_{\tilde \wp}}/k_{\tilde \wp}}\colon {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_{\tilde \wp}} \longrightarrow {\mathbf{Sh}}_{K'}(G'_{\sigma^2\tilde \ttS})_{k_{\tilde \wp}}\simeq {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_{\tilde\wp}}^{(p^2)}$$ denotes the relative $p^2$-Frobenius, then we have $ S_p^{-1}\circ F^2_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_{\tilde \wp}}/k_{\tilde \wp}} = \prod_{\gothp\in\Sigma_p} \gothF_{\gothp^2}. $ Similarly, if $[p]: \bfA'^{(p^2)}_{\tilde \ttS} {\rightarrow}\bfA'^{(p^2)}_{\tilde \ttS}$ denotes the multiplication by $p$ and $$F^2_{A} \colon \bfA'_{\tilde \ttS} \to (F^2_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_{\tilde \wp}}/k_{\tilde \wp}})^* (\bfA'_{\sigma^2\tilde \ttS}) \cong \bfA'^{(p^2)}_{\tilde \ttS}$$ the $p^2$-Frobenius homomorphism, we have $ [p]^{-1}\circ F_{A}^2 = \prod_{\gothp \in \Sigma_p} \eta_\gothp. $ Finally, we note that all the discussions above are equivariant with respect to the action of the Galois group and the action of $\widetilde G_{\tilde \ttS} = G''_{\tilde\ttS}(\QQ)^{+, (p)} G'_{\tilde\ttS}(\AAA^{\infty, p}) \simeq G''_{\sigma_\gothp^2\tilde\ttS}(\QQ)^{+, (p)} G'_{\sigma_\gothp^2\tilde\ttS}(\AAA^{\infty, p})$ when passing to the limit. (The isomorphism follows from the description of the group $\widetilde G_{\tilde \ttS}$ in .) So applying $-\times_{\widetilde G_{\tilde \ttS}} G''_{\tilde \ttS}(\AAA^{\infty, p})$ to the construction gives the following. \[P:product of partial Frobenius\] Let $\bfA''_{\tilde \ttS}$ denote the natural family of abelian varieties over ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})$. We identify the level structure for $G''_{\tilde \ttS}$ with that of $G''_{\sigma_\gothp^2\tilde \ttS}$ similarly. Then for each $\gothp \in \Sigma_p$, we have a $G''_{\tilde \ttS}(\AAA^{\infty, p})$-equivariant natural *twisted partial Frobenius morphism* and an quasi-isogeny of family of abelian varieties. $$\gothF''_{\gothp^2}\colon {\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_{\tilde \wp}} \longrightarrow {\mathbf{Sh}}_{K''_p}(G''_{\sigma_\gothp^2\tilde \ttS})_{k_{\tilde \wp}} \quad \textrm{and} \quad \eta''_{\gothp^2} \colon \bfA''_{\tilde \ttS, k_{\tilde \wp}} \longrightarrow \gothF''^{*}_{\gothp^2}(\bfA''_{\sigma^2_{\gothp}\tilde \ttS, k_{\tilde \wp}}).$$ This induces a natural $G''_{\tilde \ttS}(\AAA^{\infty, p})$-equivariant homomorphism of étale cohomology groups: $$\xymatrix{ H^*_{\mathrm{et}}\big( {\mathbf{Sh}}_{K''_p}(G''_{\sigma_\gothp^2\tilde \ttS})_{\overline \FF_p}, \calL_{\tilde \Sigma}^{(\underline k, w)}(\bfA''_{\sigma_\gothp^2 \tilde \ttS})\big) \ar[rr]^-{\gothF''^* _{\gothp^2}} \ar[drr]_{\Phi_{\gothp^2}} && H^*_{\mathrm{et}}\big( {\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{\overline \FF_p}, \calL_{\tilde \Sigma}^{(\underline k, w)}(\gothF''^* _{\gothp^2}\bfA''_{\sigma_\gothp^2 \tilde \ttS})\big) \ar[d]^{\eta''^*_{\gothp^2}} \\ && H^*_{\mathrm{et}}\big( {\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{\overline \FF_p}, \calL_{\tilde \Sigma}^{(\underline k, w)}(\bfA''_{\tilde \ttS})\big) }$$ Moreover, we have an equality of morphisms $$\prod_{\gothp \in \Sigma_p} \Phi_{\gothp^2} = S_p^{-1} \circ F^2 \colon H^*_{\mathrm{et}}\big( {\mathbf{Sh}}_{K''_p}(G''_{\sigma^2\tilde \ttS})_{\overline \FF_p}, \calL_{\tilde \Sigma}^{(\underline k, w)}(\bfA''_{\sigma^2 \tilde \ttS})\big) \longrightarrow H^*_{\mathrm{et}}\big( {\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{\overline \FF_p}, \calL_{\tilde \Sigma}^{(\underline k, w)}(\bfA''_{ \tilde \ttS})\big),$$ where $F^2$ is the relative $p^2$-Frobenius and $S_p$ is the Hecke action given by multiplication by $\underline p^{-1}$. Here $\underline p$ is the idele element which is $p$ at all places above $p$ and $1$ elsewhere. This is clear from the construction. Comparison with the Hilbert modular varieties {#S:comparison-Hilbert} --------------------------------------------- When $\ttS=\emptyset$, we have $G_{\emptyset}=\operatorname{Res}_{F/{\mathbb{Q}}}({\mathrm{GL}}_{2,F})$. Let $K_p={\mathrm{GL}}_2({\mathcal{O}}_{F}\otimes_{{\mathbb{Z}}} {\mathbb{Z}}_p)$. It is well known that ${\mathrm{Sh}}_{K_p}(G_{\emptyset})=\varprojlim_{K^p}{\mathrm{Sh}}_{K^pK_p}(G_{\emptyset})$ is a projective system of Shimura varieties defined over ${\mathbb{Q}}$, and it parametrizes polarized Hilbert-Blumenthal abelian varieties (HBAV for short) with prime-to-$p$ level structure. Using this moduli interpretation, one can construct an integral canonical model over ${\mathbb{Z}}_p$ of ${\mathrm{Sh}}_{K_p}(G_{\emptyset})$ as in [@rap; @lan]. By the uniqueness of the integral canonical model, we know that this classical integral model is isomorphic to ${\mathbf{Sh}}_{K_p}(G_{\emptyset})$ constructed in Corollary \[C:integral-model-quaternion\]. For this “abstract” isomorphism to be useful in applications, we need to relate the universal HBAV $\calA$ on ${\mathbf{Sh}}_{K_p}(G_{\emptyset})$ and the abelian scheme $\bfA''_\emptyset$ on ${\mathbf{Sh}}_{K''_p}(G''_\emptyset)$ constructed at the end of Subsection \[S:abel var in unitary case\]. Let $G_{\emptyset}^\star \subseteq G_{\emptyset}$ be the inverse image of ${\mathbb{G}}_{m,{\mathbb{Q}}}\subseteq T_{F}=\operatorname{Res}_{F/{\mathbb{Q}}}({\mathbb{G}}_{m,{\mathbb{Q}}})$ via the determinant map $\nu: G_{\emptyset}{\rightarrow}T_{F}$. The homomorphism $h_{\emptyset}: {\mathbb{C}}^{\times}{\rightarrow}G_{\emptyset}({\mathbb{R}})$ factors through $G^\star_\emptyset({\mathbb{R}})$. We can talk about the Shimura variety associated to $(G^\star_{\emptyset}, h_{\emptyset})$. We put $K^\star _p=K_p\cap G^\star_{\emptyset}({\mathbb{Q}}_p)$. Then by Corollary \[C:Sh(G)\^circ\_Zp independent of G\], ${\mathrm{Sh}}_{K_p}(G_{\emptyset})$ and ${\mathrm{Sh}}_{K_{p}^\star}(G^\star_{\emptyset})$ has isomorphic neutral connected components ${\mathrm{Sh}}_{K_p}(G_{\emptyset})^\circ_{{\mathbb{Q}}_{p}^{{\mathrm{ur}}}}\simeq {\mathrm{Sh}}_{K^*_p}(G^\star_\emptyset)^{\circ}_{{\mathbb{Q}}_p^{{\mathrm{ur}}}}$. The Shimura variety ${\mathrm{Sh}}_{K^\star_p}(G^\star_\emptyset)$ is of PEL-type, and the universal abelian scheme on ${\mathrm{Sh}}_{K^\star_p}(G^\star_\emptyset)^\circ_{{\mathbb{Q}}_p^{{\mathrm{ur}}}}$ is identified with that on ${\mathrm{Sh}}_{K_p}(G_{\emptyset})^\circ_{{\mathbb{Q}}_p^{{\mathrm{ur}}}}$ via the isomorphism above. Actually, if $K^p\subseteq {\mathrm{GL}}_2(\AAA_F^{\infty,p})$ is an open compact subgroup such that $\det(K^p\cap {\mathcal{O}}_{F}^{\times})=\det(K^p)\cap {\mathcal{O}}_{F}^{\times, +}$, where ${\mathcal{O}}^{\times, +}_{F}$ denotes the set of totally positive units of $F$, then ${\mathrm{Sh}}_{K^pK_p}(G_{\emptyset})$ is isomorphic to a finite union of ${\mathrm{Sh}}_{K^{\star p}K^\star_{p}}(G^\star_{\emptyset})$ [@hida 4.2.1]. We now describe the Hilbert moduli problem that defines an integral canonical model of ${\mathrm{Sh}}_{K^\star_p}(G_{\emptyset}^\star)$. Let $\widehat{{\mathcal{O}}}_F^{(p)}=\prod_{v\nmid p\infty}{\mathcal{O}}_{F_v}$, and put $\widehat\Lambda_{F}^{(p)}=\widehat{{\mathcal{O}}}_F^{(p)}e_1\oplus \widehat{{\mathcal{O}}}_F^{(p)}\gothd_F^{-1} e_{2}$. We endow $\widehat \Lambda_{F}^{(p)}$ with the symplectic form $$\psi_F( a_1 e_1+a_2e_2, b_1e_1+b_2e_2)={\mathrm{Tr}}_{F/{\mathbb{Q}}}(a_2b_1-a_1b_2)\in \widehat{{\mathbb{Z}}}^{(p)}\quad$$ [for ]{}$a_1, b_1\in \widehat{{\mathcal{O}}}_F^{(p)}$, and $ a_2,b_2\in \gothd_F^{-1}\widehat{{\mathcal{O}}}^{(p)}_F$. It is an elementary fact that every rank two free $\widehat{{\mathcal{O}}}_F^{(p)}$-module together with a $\widehat{{\mathbb{Z}}}^{(p)}$-linear ${\mathcal{O}}_F$-hermitian symplectic form is isomorphic to $(\widehat\Lambda^{(p)}_F, \psi_F)$. Let $K^\star_p={\mathrm{GL}}_2({\mathcal{O}}_{F}\otimes {\mathbb{Z}}_p)\cap G^\star_{\emptyset}({\mathbb{Q}}_p)$ be as above. For an open compact subgroup $K^{\star p}$, we put $K^\star=K^{\star p}K_p^\star$. Assume that $K^{\star p}$ stabilizes the lattice $\widehat \Lambda_F^{(p)}$. We consider the functor that associates to each connected locally noetherian ${\mathbb{Z}}_p$-scheme $S$ the set of isomorphism classes of quadruples $(A,\iota,\lambda, \alpha_{K^{\star p}})$, where 1. $(A,\iota)$ is a HBAV, i.e. an abelian scheme $A/S$ of dimension $g$ equipped with a homomorphism $\iota: {\mathcal{O}}_F{\hookrightarrow}\operatorname{End}_S(A)$, 2. $\lambda : A{\rightarrow}A^\vee$ is an ${\mathcal{O}}_F$-linear ${\mathbb{Z}}_{(p)}^{\times}$-polarization in the sense of [@lan 1.3.2.19], 3. $\alpha_{K^{\star p}}$ is a $\pi_1(S,\bar{s})$-invariant $K^{\star p}$-orbit of $\widehat{{\mathcal{O}}}_F^{(p)}$-linear isomorphisms $\widehat\Lambda^{(p)}_F{\xrightarrow}{\sim} T^{(p)}(A_{\bar{s}})$, sending the symplectic pairing $\psi_F$ on the former to the $\lambda$-Weil pairing on the latter. This functor is representable by a quasi-projective and smooth scheme ${\mathbf{Sh}}_{K^\star}(G^\star_{\emptyset})$ over ${\mathbb{Z}}_p$ such that ${\mathbf{Sh}}_{K^\star}(G_{\emptyset}^\star)_{{\mathbb{Q}}_p}\simeq {\mathrm{Sh}}_{K^\star}(G_{\emptyset}^\star)$ [@rap] and [@lan 1.4.1.11]. By the same arguments of [@moonen96 Corollary 3.8], it is easy to see that ${\mathbf{Sh}}_{K^\star_p}(G_{\emptyset}^\star)$ satisfies the extension property . This then gives rise to an integral canonical model ${\mathbf{Sh}}_{K_p}(G_\emptyset)$ of ${\mathrm{Sh}}_{K_p}(G_\emptyset)$. We could pull back the universal abelian variety $\calA^\star$ over ${\mathbf{Sh}}_{K^\star_p}(G^\star_\emptyset)$ to a family of abelian variety over ${\mathbf{Sh}}_{K_p}(G_\emptyset)$ using [@hida 4.2.1] cited above. But we prefer to do it more canonically following the same argument as in Subsection \[S:abel var in unitary case\]. More precisely, there is a natural equivariant action of $\widetilde G^\star := G_\emptyset({\mathbb{Q}})^{(p)}_+ \cdot G^\star(\AAA^{\infty, p})$ on the universal abelian variety $\calA^\star$ over ${\mathbf{Sh}}_{K^\star_p}(G^\star _\emptyset)$. Then $$\label{E:A from A*} \calA : = \calA^\star \times_{\widetilde G^\star} {\mathrm{GL}}_2(\AAA^{\infty,p})$$ gives a natural family of abelian variety over $ {\mathbf{Sh}}_{K_p}(G_\emptyset)$. The natural homomorphism ${\mathrm{GL}}_{2,F}{\rightarrow}{\mathrm{GL}}_{2,F}\times_{F^{\times}}E^{\times}$ induces a closed immersion of algebraic groups $G^\star_{\emptyset}{\rightarrow}G'_{\emptyset}$ compatible with Deligne’s homomorphisms $h_{\emptyset}$ and $h_{\emptyset}'$. (This does not hold in general if $\ttS_\infty \neq \emptyset$.) Therefore, one obtains a map of (projective systems of) Shimura varieties $f: {\mathrm{Sh}}_{K^\star_p}(G^\star_{\emptyset}){\rightarrow}{\mathrm{Sh}}_{K_p'}(G'_{\emptyset})$ which induces an isomorphism of neutral connected component ${\mathrm{Sh}}_{K^\star_p}(G^\star_{\emptyset})^{\circ}_{{\mathbb{Q}}_p^{{\mathrm{ur}}}}\simeq {\mathrm{Sh}}_{K'_p}(G'_{\emptyset})^{\circ}_{{\mathbb{Q}}_p^{{\mathrm{ur}}}}$. We will extend $f$ to a map of integral models ${\mathbf{Sh}}_{K^\star_p}(G^\star_{\emptyset}){\rightarrow}{\mathbf{Sh}}_{K_p'}(G'_{\emptyset})$. In the process of constructing the pairing $\psi$ on $D_\emptyset$, we may take $\delta_\emptyset$ to be $\big(\begin{smallmatrix} 0 & -1/\sqrt{\gothd} \\ 1/\sqrt{\gothd} &0 \end{smallmatrix}\big)$ which is coprime to $p$, where $\gothd$ is the totally negative element chosen in \[S:PEL-Shimura-data\]. It is easy to check that it satisfies the conditions in Lemma \[L:property-PEL-data\](1), and the $*$-involution given by $\delta_{\emptyset}$ on $D_\emptyset=\rmM_2(E)$ is given by $\big(\begin{smallmatrix} a&b\\c&d \end{smallmatrix}\big) \mapsto \big(\begin{smallmatrix} \bar a&\bar c\\\bar b&\bar d \end{smallmatrix}\big)$ for $a,b, c, d \in E$. The $\ast$-hermitian pairing on $\rmM_2(E)$ is given by $$\begin{aligned} \psi(v, w) &= {\mathrm{Tr}}_{\rmM_2(E) / \QQ}\Big( v \bar w \big(\begin{smallmatrix} 0 & -1 \\ 1 &0 \end{smallmatrix}\big) \Big), \textrm{ for } v = \big(\begin{smallmatrix} a_v & b_v \\ c_v & d_v \end{smallmatrix}\big) \textrm{ and } w = \big(\begin{smallmatrix} a_w & b_w \\ c_w & d_w \end{smallmatrix}\big) \in \rmM_2(E)\\ &={\mathrm{Tr}}_{E/\QQ} \big( b_v \bar a_w -a_v \bar b_w + d_v \bar c_w - c_v \bar d_w \big).\end{aligned}$$ In defining the PEL data for $G'_\emptyset$, we take the $\calO_{D_\emptyset} $-lattice $\Lambda$ to be $ \begin{pmatrix} \calO_E & \gothd_F^{-1}\calO_E \\ \calO_E & \gothd_F^{-1}\calO_E \end{pmatrix} $; clearly $\widehat\Lambda^{(p)}=\Lambda\otimes_{{\mathbb{Z}}}\widehat{{\mathbb{Z}}}^{(p)}$ satisfies $\widehat\Lambda^{(p)} \subseteq \widehat\Lambda^{(p),\vee}$ for the bilinear form $\psi$ above. Moreover, if we equip $\Lambda_{F}^{(p)} \otimes_{\calO_F} \calO_E$ with the symplectic form $\psi_{E}=\psi_F({\mathrm{Tr}}_{E/F}(\bullet), {\mathrm{Tr}}_{E/F}(\bullet))$, then $(\widehat\Lambda^{(p)},\psi) $ is isomorphic to $ ((\widehat{\Lambda}_{F}^{(p)} \otimes_{\calO_F} \calO_E)^{\oplus 2}, \psi_E^{\oplus 2})$ as a $\ast$-hermitian symplectic $\rmM_2(\calO_E)$-module. \[P:integral-HMV-unitary\] For any open compact subgroup $K'^p$ of $G'_\emptyset(\AAA^{\infty, p})$, we put $K^{\star p} = K'^p \cap G^*_\emptyset(\AAA^{\infty, p})$. Then we have a canonical morphism $$\bff: {\mathbf{Sh}}_{K^{\star p}K^\star_p}(G^\star _\emptyset) {\rightarrow}{\mathbf{Sh}}_{K'^pK'_p}(G'_\emptyset)$$ such that, if $\calA$ and $\bfA'$ denote respectively the universal abelian scheme on ${\mathbf{Sh}}_{K^{\star p}K^\star _p}(G^\star _{\emptyset})$ and that on ${\mathbf{Sh}}_{K'^pK'_p}(G'_{\emptyset})$, then we have an isomorphism of abelian schemes $\bff^*\bfA'\simeq (\calA\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_E)^{\oplus 2}$ compatible with the natural action of $\rmM_2({\mathcal{O}}_E)$ and polarizations on both sides. By passing to the limit, the morphism $\bff$ induces an isomorphism between the integral models of connected Shimura varieties ${\mathbf{Sh}}_{K_p^\star }(G^\star _\emptyset)_{{\ZZ_p}^{\mathrm{ur}}}^\circ \simeq {\mathbf{Sh}}_{K_p'}(G'_\emptyset )_{{\ZZ_p}^{\mathrm{ur}}}^\circ$. By Galois descent, it is enough to work over $W(k_0)$ for $k_0$ in Theorem \[T:unitary-shimura-variety-representability\]. Let $S$ be a connected locally noetherian $W(k_0)$-scheme, and $x=(A,\iota, \lambda, \alpha_{K^{\star p}})$ be an $S$-valued point of ${\mathbf{Sh}}_{K^{\star p}K^\star _p}(G_{\emptyset}^\star )$. We define its image $f(x)=(A', \iota', \lambda', \alpha_{K'^p})$ as follows. We take $A'=(A\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_E)^{\oplus 2}$ equipped with the naturally induced action $\iota'$ of $\rmM_2({\mathcal{O}}_E)$. It is clear that $\operatorname{Lie}(A')_{\tilde\tau}$ is an ${\mathcal{O}}_{S}$-module locally free of rank $1$ for all $\tilde\tau\in \Sigma_{E,\infty}$. The prime-to-$p$ polarization $\lambda'$ on $A'$ is defined to be $$\lambda': A'{\xrightarrow}{\sim} (A\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_E)^{\oplus 2}{\xrightarrow}{(\lambda\otimes 1)^{\oplus 2}} (A^\vee\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_E)^{\oplus 2}\simeq A'^\vee.$$ We define the $K'^p$-level structure to be the $K'^p$-orbit of the isomorphism $$\alpha_{K'^p}\colon \widehat{\Lambda}^{(p)}{\xrightarrow}{\cong} (\widehat{\Lambda}^{(p)}_F\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_E)^{\oplus 2}{\xrightarrow}{\alpha_{K^{\star p}}^{\oplus 2}} \big (T^{(p)}(A_{\bar s})\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_E\big)^{\oplus 2}\simeq T^{(p)}(A'_{\bar s}).$$ By the discussion before the Proposition, it is clear that $\alpha_{K'^p}$ sends the symplectic form $\psi$ on the left hand side to the $\lambda'$-Weil pairing on the right. This defines the morphism $\bff$ from ${\mathbf{Sh}}_{K^{\star p}K^\star _p}(G^\star _{\emptyset})$ to ${\mathbf{Sh}}_{K'^pK'_p}(G'_{\emptyset})$. By looking at the complex uniformization, we note that $\bff$ extends the morphism $f: {\mathrm{Sh}}_{K^{\star p}K^\star _p}(G^\star _{\emptyset})_{{\mathbb{Q}}_p}{\rightarrow}{\mathrm{Sh}}_{K'^pK'_p}(G'_{\emptyset})_{{\mathbb{Q}}_p}$ defined previously by group theory. Since both ${\mathbf{Sh}}_{K^\star _p}(G^\star _{\emptyset})$ and ${\mathbf{Sh}}_{K'_{p}}(G'_{\emptyset})$ satisfy the extension property \[S:extension property\], it follows that $\bff$ induces an isomorphism ${\mathbf{Sh}}_{K_p^\star }(G^\star _\emptyset)_{{\ZZ_p}^{\mathrm{ur}}}^\circ \simeq {\mathbf{Sh}}_{K_p'}(G'_\emptyset )_{{\ZZ_p}^{\mathrm{ur}}}^\circ$. \[C:integral-HMV-unitary\] Let $\calA$ denote the universal HBAV over ${\mathbf{Sh}}_{K_p}(G_{\emptyset})$, and $\bfA''_\emptyset$ be the family of abelian varieties over ${\mathbf{Sh}}_{K''_p}(G''_{\emptyset})$ defined in Subsection \[S:abel var in unitary case\]. Then under the natural morphisms of Shimura varieties $$\label{E:morphisms of Shimura varieties HMV} {\mathbf{Sh}}_{K_p}(G_\emptyset) \xleftarrow{\ \mathbf{pr}_1\ } {\mathbf{Sh}}_{K_p \times K_{E,p}} (G_\emptyset \times T_{E, \emptyset}) \xrightarrow{\ {\boldsymbol{\alpha}}\ } {\mathbf{Sh}}_{K''_p}(G''_\emptyset),$$ one has an isomorphism of abelian schemes over ${\mathbf{Sh}}_{K_p \times K_{E,p}} (G_\emptyset \times T_{E, \emptyset})$ $$\label{E:comparison abelian varieties over HMV and unitary} {\boldsymbol{\alpha}}^*\bfA''_\emptyset \cong (\mathbf{pr}_1^*\calA\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_E)^{\oplus 2}$$ compatible with the action of $\rmM_2({\mathcal{O}}_E)$ and prime-to-$p$ polarizations. This follows immediately from the constructions of $\calA$ and $\bfA''_\emptyset$ and the proposition above. Comparison of twisted partial Frobenius --------------------------------------- Keep the notation as in Subsection \[S:comparison-Hilbert\]. The Shimura variety ${\mathbf{Sh}}_{K^\star }(G_\emptyset^\star )_{{\FF_p}}$ also admits a twisted partial Frobenius $\Phi_{\gothp^2}$ for each $\gothp \in \Sigma_p$ which we define as follows. Let $S$ be a locally noetherian $ \FF_p$-scheme. Given an $S$-point $(A, \iota, \lambda, \alpha_{K^{\star p}})$ of ${\mathbf{Sh}}_{K^\star }(G_\emptyset^*)_{{\FF_p}}$, we associate a new point $(A', \iota', \lambda', \alpha'_{K^{\star p}})$: - $A' = A / \operatorname{Ker}_{\gothp^2} \otimes_{\calO_F} \gothp$, where $\operatorname{Ker}_{\gothp^2}$ is the $\gothp$-component of the kernel of relative Frobenius homomorphism ${\mathrm{Fr}}^2_A: A \to A^{(p^2)}$; it is equipped with the induced $\calO_F$-action $\iota'$; - using the natural quasi-isogeny $\eta: A \to A'$, $\lambda'$ is given by the composite of quasi-isogenies $A' \xleftarrow{\eta} A \xrightarrow{\lambda} A^\vee \xleftarrow{\eta^\vee} A'^\vee$ (which is a $\ZZ_{(p)}^\times$-isogeny by the same argument as in Subsection \[S:partial Frobenius\]); - $\bar \alpha'_{K^{\star p}}$ is the composite $\widehat\Lambda_F^{(p)} \xrightarrow{\bar \alpha_{K^{\star p}}} T^{(p)}(A) \xleftarrow{\eta} T^{(p)}(A')$. The construction above gives rise to a *twisted partial Frobenius morphism* $$\gothF^\star_{\gothp^2}: {\mathbf{Sh}}_{K^\star }(G^\star _\emptyset)_{\FF_p} \longrightarrow {\mathbf{Sh}}_{K^\star }(G^\star _\emptyset)_{\FF_p} \quad \textrm{and} \quad \eta_{\gothp^2}: \calA_{\FF_p} \to (\gothF^\star_{\gothp^2})^*\calA_{\FF_p}.$$ Using the formalism of Shimura varieties (Corollary \[C:Sh(G)\^circ\_Zp independent of G\] and more specifically ), it gives rise to a *twisted partial Frobenius morphism* $$\gothF^\emptyset_{\gothp^2}: {\mathbf{Sh}}_{K}(G_\emptyset)_{\FF_p} \longrightarrow {\mathbf{Sh}}_{K}(G_\emptyset)_{\FF_p} \quad \textrm{and} \quad \eta_{\gothp^2}^\emptyset: \calA_{\FF_p} \to (\gothF^\emptyset_{\gothp^2})^*\calA_{\FF_p}.$$ The twisted partial Frobenius morphism $\gothF''_{\gothp^2}$ on ${\mathbf{Sh}}_{K''_p}(G''_\emptyset)_{\FF_p}$ and the twisted partial Frobenius $\gothF^\emptyset_{\gothp^2}$ on ${\mathbf{Sh}}_{K_p}(G_\emptyset)_{{\FF_p}}$ are compatible, in the sense that there exists a morphism $\tilde \gothF_{\gothp^2}$ so that both squares in the commutative diagram are Cartesian. $$\xymatrix{ {\mathbf{Sh}}_{K_p}(G_\emptyset)_{{\FF_p}} \ar[d]^{\gothF^\emptyset_{\gothp^2}} & \ar[l]_-{\mathbf{pr}_1} {\mathbf{Sh}}_{K_p \times K_{E,p}} (G_\emptyset \times T_{E, \emptyset})_{{\FF_p}} \ar[r]^-{ {\boldsymbol{\alpha}}} \ar[d]^{\tilde \gothF_{\gothp^2}} & {\mathbf{Sh}}_{K''_p}(G''_\emptyset)_{{\FF_p}} \ar[d]^{\gothF''_{\gothp^2}} \\ {\mathbf{Sh}}_{K_p}(G_\emptyset)_{{\FF_p}} & \ar[l]_-{\mathbf{pr}_1} {\mathbf{Sh}}_{K_p \times K_{E,p}} (G_\emptyset \times T_{E, \emptyset})_{{\FF_p}} \ar[r]^-{ {\boldsymbol{\alpha}}} & {\mathbf{Sh}}_{K''_p}(G''_\emptyset)_{{\FF_p}} }$$ Moreover, $\eta^\emptyset_{\gothp^2}$ is compatible with $\eta''_{\gothp^2}$ in the sense that the following diagram commutes. $$\xymatrix{ {\boldsymbol{\alpha}}^*\bfA''_{\emptyset,\FF_p} \ar[d]^{{\boldsymbol{\alpha}}^*(\eta'' _{{\gothp^2}})} \ar[r]^-{\eqref{E:comparison abelian varieties over HMV and unitary}} & (\mathbf{pr}_1^* \calA_{\FF_p} \otimes_{\calO_F} \calO_E)^{\oplus 2} \ar[d]_{\eta^\emptyset_{\gothp^2} \otimes 1}\\ {\boldsymbol{\alpha}}^*\gothF''^*_{\gothp^2} \bfA_{\emptyset, \FF_p} \ar[r]^-{\eqref{E:comparison abelian varieties over HMV and unitary}} & \big(\mathbf{pr}_1^*\gothF_{\gothp^2}^{\emptyset,*}( \calA_{\FF_p}) \otimes_{\calO_F} \calO_E \big)^{\oplus 2} }$$ This follows from the definition of the partial Frobenii in various situations and the comparison Proposition \[P:integral-HMV-unitary\] above. Goren-Oort Stratification {#Section:defn of GOstrata} ========================= We define an analog of the Goren-Oort stratification on the special fibers of quaternionic Shimura varieties. This is first done for unitary Shimura varieties and then pulled back to the quaternionic ones. Unfortunately, the definition apriori depends on the auxiliary choice of CM field (as well as the signatures $s_{\tilde \tau}$). In the case of Hilbert modular variety, we show that our definition of the GO-strata agrees with Goren-Oort’s original definition in [@goren-oort] (and hence does not depend on the auxiliary choice of data). Notation {#S:GO-notation} -------- Keep the notation as in previous sections. Let $k_0$ be a finite extension of ${\FF_p}$ containing all residue fields of $\calO_E$ of characteristic $p$. Let $X':={\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ denote the base change to $k_0$ of the Shimura variety ${\mathbf{Sh}}_{K}(G'_{\tilde \ttS})$ considered in Theorem \[T:unitary-shimura-variety-representability\]. Recall that $\gothe \in \calO_{D_\ttS,p}$ corresponds to $\big( \begin{smallmatrix} 1&0\\0&0 \end{smallmatrix} \big)$ when identifying $\calO_{D_\ttS,p}$ with $\rmM_2(\calO_{E,p})$. For an abelian scheme $A$ over a $k_0$-scheme $S$ carrying an action of ${\mathcal{O}}_{D_{\ttS}}$, we have the reduced module of invariant differential 1-forms $\omega_{A/S}^{\circ}$, the reduced Lie algebra $\operatorname{Lie}(A/S)^\circ$, and the reduced de Rham homology $H^{{\mathrm{dR}}}_1(A/S)^{\circ}$ defined in Subsection \[N:notation-reduced\]. Their $\tilde{\tau}$-components $\omega_{A/S, \tilde{\tau}}^{\circ}$, $\operatorname{Lie}(A/S)_{\tilde{\tau}}^{\circ}$ and $H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde{\tau}}$ for $\tilde{\tau}\in \Sigma_{E,\infty}$, fit in an exact sequence, called the *reduced Hodge filtration*, $$0{\rightarrow}\omega_{A^\vee/S,\tilde{\tau}}^{\circ}{\rightarrow}H^{{\mathrm{dR}}}_1(A/S)^{\circ}_{\tilde{\tau}}{\rightarrow}\operatorname{Lie}(A/S)^\circ_{\tilde{\tau}}{\rightarrow}0.$$ Let $A^{(p)}$ denote the base change of $A$ via the absolute Frobenius on $S$. The Verschiebung ${\mathrm{Ver}}:A^{(p)}{\rightarrow}A$ and the Frobenius morphism ${\mathrm{Fr}}: A{\rightarrow}A^{(p)}$ induce respectively maps of coherent sheaves on $S$: $$F_A: H_1^{{\mathrm{dR}}}(A/S)^{\circ, (p)}{\rightarrow}H_1^{{\mathrm{dR}}}(A/S)^{\circ}\quad \text{and } V_A: H_1^{{\mathrm{dR}}}(A/S)^{\circ}{\rightarrow}H_1^{{\mathrm{dR}}}(A/S)^{\circ, (p)},$$ which are compatible with the action of ${\mathcal{O}}_E$. Here, for a coherent ${\mathcal{O}}_S$-module, $M^{(p)}$ denotes the base change $M\otimes_{{\mathcal{O}}_S, F_{\mathrm{abs}}} {\mathcal{O}}_S$. If there is no confusion, we drop the subscript $A$ from the notation and simply write $F$ and $V$ for the two maps. Moreover, we have $$\operatorname{Ker}(F)={\mathrm{Im}}(V)=(\omega_{A^\vee/S}^{\circ})^{(p)}\quad \textrm{ and } \quad {\mathrm{Im}}(F)=\operatorname{Ker}(V)\simeq \operatorname{Lie}(A^{(p)}/S)^{\circ}.$$ Let $(A, \iota, \lambda, \alpha_{K'})$ be an $S$-valued point of $X'={\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$. By Kottwitz’ determinant condition \[T:unitary-shimura-variety-representability\](a), for each $\tilde{\tau}\in \Sigma_{E,\infty}$, $\operatorname{Lie}(A/S)^{\circ}_{\tilde{\tau}}$ is a locally free ${\mathcal{O}}_{S}$-module of rank $s_{\tilde{\tau}}$. (The numbers $s_{\tilde \tau}$ are defined as in Subsection \[S:CM extension\].) By duality, this implies that $\omega_{A^\vee/S,\tilde{\tau}}^{\circ}$ is locally free of rank $s_{\tilde{\tau}^c}=2-s_{\tilde{\tau}}$. Moreover, when $\tau \in \Sigma_{\infty/\gothp}$ with $\gothp$ not of type $\beta^\sharp$, the universal polarization $\lambda$ induces an isomorphism of locally free ${\mathcal{O}}_{S}$-modules $$\label{Equ:duality-omega} \omega_{A^\vee/S,\tilde{\tau}}^\circ\simeq \omega_{A/S,\tilde{\tau}^c}^\circ.$$ Essential Frobenius and essential Verschiebung {#N:essential frobenius and verschiebung} ---------------------------------------------- We now define two very important morphisms: essential Frobenius and essential Verschiebung; we will often encounter later their variants for crystalline homology and Dieudonné modules, for which we shall simply refer the similar construction given here. Let $(A,\iota, \lambda, \alpha_{K'})$ be as above. For $\tilde \tau \in \Sigma_{E, \infty}$ lifting a place $\tau \in \ttS_\infty$, we define the *essential Frobenius* to be $$\begin{aligned} \label{E:defintion of Fes} F_{\mathrm{es}}=F_{A,{\mathrm{es}}, \tilde \tau}: (H^{{\mathrm{dR}}}_1(A/S)_{\sigma^{-1}\tilde{\tau}}^{\circ})^{ (p)}&=H_1^{{\mathrm{dR}}}(A^{(p)}/S)^{\circ}_{\tilde{\tau}}\longrightarrow H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde{\tau}} \\ \nonumber x&\longmapsto{ \left\{ \begin{array}{ll} F(x) & \textrm{when }s_{\sigma^{-1}\circ \tilde \tau} = 1\textrm{ or }2;\\ V^{-1}(x) & \textrm{when }s_{\sigma^{-1}\circ \tilde \tau} = 0.\\ \end{array} \right.}\end{aligned}$$ Note that in the latter case, the morphism $V: H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde{\tau}} {\xrightarrow}{\sim} H_1^{{\mathrm{dR}}}(A^{(p)}/S)^{\circ}_{\tilde{\tau}}$ is an isomorphism by Kottwitz’ determinant condition. Similarly, we define the *essential Verschiebung* to be $$\begin{aligned} \label{E:definition of Ves} V_{\mathrm{es}}=V_{A,{\mathrm{es}}, \tilde \tau}: H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde{\tau}}&\longrightarrow H_1^{{\mathrm{dR}}}(A^{(p)}/S)^{\circ}_{\tilde{\tau}} =(H_1^{{\mathrm{dR}}}(A/S)_{\sigma^{-1}\tilde{\tau}}^{\circ})^{(p)} \\ \nonumber x&\longmapsto{ \left\{ \begin{array}{ll} V(x) & \textrm{when }s_{\sigma^{-1}\circ \tilde \tau} = 0\textrm{ or }1;\\ F^{-1}(x) & \textrm{when }s_{\sigma^{-1}\circ \tilde \tau} = 2.\\ \end{array} \right.}\end{aligned}$$ Here, in the latter case, the morphism $F: H_1^{{\mathrm{dR}}}(A^{(p)}/S)^{\circ}_{\tilde{\tau}} \to H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde{\tau}}$ is an isomorphism. When no confusion arises, we may suppress the subscript $A$ and/or $\tilde \tau$ from $F_{A, {\mathrm{es}}, \tilde\tau}$ and $V_{A,{\mathrm{es}},\tilde\tau}$. Thus, if $s_{\sigma^{-1}\tilde \tau} =0$ or $2$, both $F_{{\mathrm{es}}, \tilde \tau}:H_1^{{\mathrm{dR}}}(A^{(p)}/S)^{\circ}_{\tilde{\tau}}{\rightarrow}H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde{\tau}}$ and $V_{{\mathrm{es}}, \tilde \tau}: H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde{\tau}}{\rightarrow}H_1^{{\mathrm{dR}}}(A^{(p)}/S)^{\circ}_{\tilde{\tau}}$ are isomorphisms, and both $F_{{\mathrm{es}}, \tilde \tau}V_{{\mathrm{es}}, \tilde \tau}$ and $V_{{\mathrm{es}}, \tilde \tau}F_{{\mathrm{es}}, \tilde \tau}$ are isomorphisms. When $s_{\sigma^{-1}\tilde \tau}=1$, we usually prefer to write the usual Frobenius and Verscheibung. We will also use composites of Frobenii and Verschiebungs: $$\begin{aligned} \label{E:Ves n} V_{{\mathrm{es}}, \tilde \tau}^n: &H_1^\mathrm{dR}(A/S)^\circ_{\tilde \tau}{\xrightarrow}{V_{{\mathrm{es}}, \tilde \tau}} H^{{\mathrm{dR}}}_1(A^{(p)}/S)_{\tilde{\tau}}^{\circ}{\xrightarrow}{V_{{\mathrm{es}}, \sigma^{-1}\tilde \tau}^{(p)}} \cdots {\xrightarrow}{V_{{\mathrm{es}}, \sigma^{1-n}\tilde \tau}^{(p^{n-1})}} H^{{\mathrm{dR}}}_1(A^{(p^{n})}/S)_{\tilde{\tau}}^{\circ}, \\ \label{E:Fes n} F_{{\mathrm{es}}, \tilde \tau}^n:& H^{{\mathrm{dR}}}_1(A^{(p^{n})}/S)_{\tilde{\tau}}^{\circ} {\xrightarrow}{F_{{\mathrm{es}}, \sigma^{1-n} \tilde \tau}^{(p^{n-1})}} H^{{\mathrm{dR}}}_1(A^{(p^{n-1})}/S)_{\tilde{\tau}}^{\circ} {\xrightarrow}{F_{{\mathrm{es}}, \sigma^{2-n}\tilde \tau}^{(p^{n-2})}} \cdots {\xrightarrow}{F_{{\mathrm{es}},\tilde\tau}} H_1^\mathrm{dR}(A/S)^\circ_{\tilde \tau}.\end{aligned}$$ Suppose now $S=\operatorname{Spec}(k)$ is the spectrum of a perfect field of characteristic $p>0$. Let ${\tilde{{\mathcal{D}}}}_{A}$ denote the *covariant* Dieudonné module of $A$. We have a canonical decomposition ${\tilde{{\mathcal{D}}}}_A=\bigoplus_{\tilde\tau\in \Sigma_{E,\infty}}{\tilde{{\mathcal{D}}}}_{A,\tilde\tau}$. We put ${\tilde{{\mathcal{D}}}}_{A,\tilde\tau}^{\circ}=\gothe\cdot {\tilde{{\mathcal{D}}}}_{A,\tilde\tau}$. Then we define the essential Frobenius and essential Verschiebung $$F_{\mathrm{es}}= F_{A,{\mathrm{es}}, \tilde \tau}: {\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-1}\tilde\tau}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau}\quad \text{and}\quad V_{{\mathrm{es}}} = V_{A, {\mathrm{es}}, \tilde \tau}:{\tilde{{\mathcal{D}}}}_{A,\tilde\tau}^{\circ}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-1}\tilde\tau}$$ in the same way as those on $H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde\tau}$, as done in and . The morphisms $F_{A,{\mathrm{es}}, \tilde \tau}$ and $V_{A,{\mathrm{es}}, \tilde \tau}$ on $H_1^{{\mathrm{dR}}}(A/S)^{\circ}_{\tilde\tau}$ can be recovered from those on ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau}$ by reduction modulo $p$. \[N:n tau\] For $\tau \in \Sigma_\infty-\ttS_\infty$, we define $n_{\tau} = n_{\tau, \ttS}\geq 1$ to be the integer such that $\sigma^{-1}\tau, \dots, \sigma^{-n_\tau+1}\tau \in \ttS_\infty$ and $\sigma^{-n_{\tau}}\tau\notin \ttS_{\infty}$. Partial Hasse invariants {#S:partial-Hasse} ------------------------ For each $\tilde \tau$ lifting a place $\tau \in \Sigma_\infty-\ttS_\infty$, we must have $s_{\tilde \tau} = 1$; so in definition of $V_{{\mathrm{es}}, \tilde \tau}^{n_\tau}$ in , all morphisms are isomorphisms except the last one; similarly, in the definition of $F_{{\mathrm{es}}, \tilde \tau}^{n_\tau}$ in , all morphisms are isomorphisms except the first one. It is clear that $V_{{\mathrm{es}}, \tilde \tau}^{n_\tau} F_{{\mathrm{es}}, \tilde \tau}^{n_\tau} =F_{{\mathrm{es}}, \tilde \tau}^{n_\tau} V_{{\mathrm{es}}, \tilde \tau}^{n_\tau}=0$, coming from the composition of $V^{(p^{n_\tau-1})}_{\sigma^{n_\tau-1}\tilde \tau}$ and $F^{(p^{n_\tau-1})}_{\sigma^{n_\tau-1}\tilde \tau}$ in both ways. Note also that the cokernels of $V_{{\mathrm{es}}, \tilde \tau}^{n_{\tau}}$ and $F_{{\mathrm{es}}, \tilde \tau}^{n_{\tau}}$ are both locally free ${\mathcal{O}}_{X'}$-modules of rank $1$. The restriction of $V_{{\mathrm{es}}, \tilde \tau}^{n_\tau}$ to the line bundle $\omega^{\circ}_{A^\vee/S,\tilde{\tau}}$ induces a homomorphism $$h_{\tilde{\tau}}(A): \omega_{A^\vee/S,\tilde{\tau}}^{\circ}{\mathrm{\longrightarrow}}\omega^\circ_{A^{\vee, (p^{n_{\tau}})}/S,\tilde{\tau}}=(\omega^{\circ}_{A^\vee/S,\sigma^{-n_{\tau}}\tilde{\tau}})^{\otimes p^{n_{\tau}}}.$$ Applied to the universal case, this gives rise to a global section $$\label{Equ:partial-hasse} h_{\tilde{\tau}}\in \Gamma(X', (\omega^{\circ}_{\bfA'^\vee/X', \sigma^{-n_{\tau}}\tilde{\tau}})^{\otimes p^{n_{\tau}}}\otimes( \omega_{\bfA'^\vee/X', \tilde{\tau}}^{\circ})^{\otimes (-1)}),$$ where $\bfA'$ is the universal abelian scheme over $X'={\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$. We call $h_{\tilde \tau}$ *the $\tilde{\tau}$-partial Hasse invariant*. With $\tilde{\tau}$ replaced by $\tilde{\tau}^c$ everywhere, we can define similarly a partial Hasse invariant $h_{\tilde{\tau}^c}$. They are analogs of the partial Hasse invariants in the unitary case. \[Lemma:partial-Hasse\] Let $(A,\iota, \lambda, \bar{\alpha}_{K'})$ be an $S$-valued point of $X'$ as above. Then the following statements are equivalent. 1. We have $h_{\tilde{\tau}}(A)=0$. 2. The image of $F_{{\mathrm{es}}, \tilde \tau}^{n_\tau}: H_{1}^{{\mathrm{dR}}}(A^{(p^{n_{\tau}})}/S)^{\circ}_{\tilde{\tau}}{\rightarrow}H_1^{{\mathrm{dR}}}(A/S)_{\tilde{\tau}}^{\circ}$ is $\omega_{A^\vee/S,\tilde{\tau}}^{\circ}$. 3. We have $h_{\tilde{\tau}^c}(A)=0$. 4. The image of $F_{{\mathrm{es}}, \tilde \tau^c}^{n_\tau}:H_{1}^{{\mathrm{dR}}}(A^{(p^{n_{\tau}})}/S)^{\circ}_{\tilde{\tau}^c}{\rightarrow}H_1^{{\mathrm{dR}}}(A/S)_{\tilde{\tau}^c}^{\circ}$ is $\omega_{A^\vee/S, \tilde{\tau}^c}^{\circ}$. The equivalences $(1)\Leftrightarrow(2)$ and $(3)\Leftrightarrow (4)$ follow from the fact that the image of $F$ coincides with the kernel of $V$. We prove now $(2)\Leftrightarrow (4)$. Let $\gothp\in \Sigma_{p}$ be the prime above $p$ so that $\tau\in \Sigma_{\infty/\gothp}$. Since $\Sigma_{\infty/\gothp}\neq \ttS_{\infty/\gothp}$, $\gothp$ can not be of type $\beta^\sharp$ by Hypothesis \[H:B\_S-splits-at-p\]. We consider the following diagram: $$\xymatrix@C=5pt{ H_1^{\mathrm{dR}}(A^{(p^{n_{\tau}})}/S)^\circ_{\tilde{\tau}} \ar@/^10pt/[d]^{F_{{\mathrm{es}}, \tilde \tau}^{n_\tau}} & \times & H_1^{\mathrm{dR}}(A^{(p^{n_{\tau}})}/S)^{\circ}_{\tilde{\tau}^c} \ar@/_10pt/[d]_{F_{{\mathrm{es}}, \tilde \tau^c}^{n_\tau}} \ar[rrr]^-{\langle\ , \ \rangle} &&& \calO_{S} \\ H_1^{\mathrm{dR}}(A/S)^{\circ}_{\tilde{\tau}} \ar@/^10pt/[u]^{V_{{\mathrm{es}}, \tilde \tau}^{n_\tau}} &\times& H_1^{\mathrm{dR}}(A/S)^{\circ}_{\tilde{\tau}^c} \ar@/_10pt/[u]_{V_{{\mathrm{es}}, \tilde \tau^c}^{n_\tau}} \ar[rrr]^-{\langle\ , \ \rangle} &&& \calO_{S}, \ar@{=}[u] }$$ where the pairings $\langle\ , \ \rangle$ are induced by the polarization $\lambda$, and they are perfect because $\gothp$ is not of type $\beta^\sharp$. We have $\langle F_{{\mathrm{es}}, \tilde \tau}^{n_\tau} x, y\rangle=\langle x,V_{{\mathrm{es}}, \tilde \tau^c}^{n_\tau} y \rangle$. It follows that $$(\omega_{A^\vee/S,\tilde{\tau}}^{\circ})^\perp=\omega_{A^\vee/S,\tilde{\tau}^c}^{\circ},\quad \text{and}\quad {\mathrm{Im}}(F_{{\mathrm{es}}, \tilde \tau}^{n_\tau})^\perp={\mathrm{Im}}(F_{{\mathrm{es}}, \tilde \tau^c}^{n_\tau}),$$ where $\perp$ means the orthogonal complement under $\langle\ ,\ \rangle$. Therefore, we have $$\begin{aligned} & \text{(2) } \omega_{A^\vee/S,\tilde{\tau}}^{\circ}={\mathrm{Im}}(F_{{\mathrm{es}}, \tilde \tau}^{n_\tau}) \Leftrightarrow (\omega_{A^\vee/S,\tilde{\tau}}^{\circ})^\perp={\mathrm{Im}}(F_{{\mathrm{es}}, \tilde \tau}^{n_\tau} )^\perp \Leftrightarrow \text{(4) } \omega_{A^\vee/S, \tilde{\tau}^c }^{\circ}= {\mathrm{Im}}(F_{{\mathrm{es}}, \tilde \tau^c}^{n_\tau}).\end{aligned}$$ \[Defn:GO-strata\] We fix a section $ \tau \mapsto \tilde{\tau}$ of the natural restriction map $\Sigma_{E,\infty}{\rightarrow}\Sigma_{\infty}$. Let $\ttT\subset \Sigma_{\infty}-\ttS_{\infty}$ be a subset. We put ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\emptyset}=X'$, and $X'_\ttT : ={\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT}$ to be the closed subscheme of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ defined by the vanishing locus of $\{h_{\tilde{\tau}}: \tau\in \ttT\}$. Passing to the limit, we put $${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})_{k_0,\ttT}:=\varprojlim_{K'^p}{\mathbf{Sh}}_{K'^pK'_p}(G'_{\tilde \ttS})_{k_0,\ttT}$$ We call $\{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\ttT}: \ttT\subset \Sigma_{\infty}-\ttS_{\infty}\}$ (resp. $\{{\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})_{k_0,\ttT}: \ttT\subset \Sigma_{\infty}-\ttS_{\infty}\}$) the *Goren-Oort stratification* (or GO-stratification for short) of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ (resp. ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})_{k_0}$). By Lemma \[Lemma:partial-Hasse\], the GO-stratum $X'_\ttT$ does not depend on the choice of the section $\tau\mapsto \tilde{\tau}$. \[Prop:smoothness\] For any subset $\ttT\subseteq \Sigma_\infty - \ttS_\infty$. The closed GO-stratum $X'_\ttT\subseteq X'$ is smooth of codimension $\#\ttT$, and the tangent bundle $\calT_{X'_\ttT}$ is the subbundle $$\bigoplus_{\tau\in \Sigma_\infty - (\ttS_\infty \cup \ttT)} \bigl( \operatorname{Lie}(\bfA')^{\circ}_{\tilde{\tau}}\otimes \operatorname{Lie}(\bfA')_{\tilde{\tau}^c}^{\circ}\bigr)|_{X'_\ttT}\subseteq \bigoplus_{\tau\in \Sigma_\infty - \ttS_\infty}\bigl(\operatorname{Lie}(\bfA')^\circ_{\tilde{\tau}}\otimes \operatorname{Lie}(\bfA')^\circ_{\tilde{\tau}^c}\bigr)|_{X'_\ttT},$$ where the latter is identified with the restriction to $X'_\ttT$ of the tangent bundle of $X'$ computed in Corollary \[C:deformation\]. Moreover, $X'_\ttT$ is proper if $\ttS_{\infty}\cup \ttT$ is non-empty. We follow the same strategy as in [@helm Proposition 3.4]. First, the same argument as [@helm Lemma 3.7] proves the non-emptyness of $X'_{\ttT}$. We now proceed as in the proof of Corollary \[C:deformation\]. Let $S_0{\hookrightarrow}S$ be a closed immersion of $k_0$-schemes whose ideal of definition $\calI$ satisfies $\calI^2=0$. Consider an $S_0$-valued point $x_0=(A_0, \iota_0, \lambda_0, \bar{\alpha}_{K'})$ of $X'_{\ttT}$. To prove the smoothness of $X'_{\ttT}$, it suffices to show that, locally for the Zariski topology on $S_0$, there exists $x\in X'_{\ttT}(S)$ lifting $x_0$. By Lemma \[Lemma:partial-Hasse\], we have, for every $\tau \in \ttT$, $$\omega_{A_0^\vee/S_0,\tilde{\tau}}^{\circ}=F_{{\mathrm{es}}, \tilde \tau}^{n_\tau} (H_1^{{\mathrm{dR}}}(A_0^{(p^{n_{\tau}})}/S_0)^{\circ}_{\tilde{\tau}}).$$ The reduced “crystalline homology” $H_1^{{\mathrm{cris}}}(A_0/S_0)_{S}^{\circ}$ is equipped with natural operators $F$ and $V$, lifting the corresponding operators on $H_1^{{\mathrm{dR}}}(A_0/S_0)^{\circ}$. We define the composite of essential Frobenius $$\tilde{F}_{{\mathrm{es}}, \tilde \tau}^{n_\tau}: H_1^{{\mathrm{cris}}}(A_0^{(p^{n_{\tau}})}/S_0)^{\circ}_{S, \tilde{\tau}}{\rightarrow}H_1^{{\mathrm{cris}}}(A_0/S_0)^{\circ}_{S, \tilde{\tau}}$$ in the same manner as $F_{{\mathrm{es}}, \tilde \tau}^{n_\tau}$ on $H^{{\mathrm{dR}}}_1(A_0^{(p^{n_{\tau}})}/S_0)^{\circ}_{\tilde{\tau}}$ in Notation \[N:essential frobenius and verschiebung\]. Let $\tilde{\omega}_{A_0^\vee/S_0, \tilde{\tau}}^{\circ}$ denote the image of $\tilde{F}_{{\mathrm{es}}, \tilde \tau}^{n_\tau}$ for $\tau\in \ttT$. This is a local direct factor of $H_1^{{\mathrm{cris}}}(A_0/S_0)_{S, \tilde{\tau}}^{\circ}$ that lifts $\omega_{A_0^\vee/S_0, \tilde{\tau}}^{\circ}$. As in the proof of Theorem \[T:unitary-shimura-variety-representability\], specifying a deformtion $x\in X'(S)$ of $x_0$ to $S$ is equivalent to giving a local direct summand $\omega_{S,\tilde{\tau}}^{\circ}\subseteq H_1^{{\mathrm{cris}}}(A_0/S_0)_{S, \tilde{\tau}}^{\circ}$ that lifts $\omega_{A_0^\vee/S_0, \tilde{\tau}}^{\circ}$ for each $\tau \in \Sigma_{\infty}-\ttS_{\infty}$. By Lemma \[Lemma:partial-Hasse\], such a deformatoin $x$ lies in $X'_{\ttT}$ if and only if $\omega_{S,\tilde{\tau}}^{\circ}=\tilde{\omega}_{A_0^\vee/S_0, \tilde{\tau}}^{\circ}$ for all $\tau \in \ttT$. Therefore, to give a deformation of $x_0$ to $S$ in $X'_{\ttT}$, we just need specify the liftings $\omega_{S,\tilde{\tau}}^{\circ}$ of $\omega_{A_0^\vee/S_0, \tilde{\tau}}^{\circ}$ for $\tau\in \Sigma_{\infty}-(\ttS_{\infty}\cup \ttT)$. Since the set-valued sheaf of liftings $\omega_{S, \tilde{\tau}}^{\circ}$ for $\tau\in \Sigma_{\infty}-(\ttS_{\infty}\cup \ttT)$ form a torsor under the group $${\mathcal{H}om}_{{\mathcal{O}}_{S_0}}(\omega_{A_0^\vee/S_0, \tilde{\tau}}^\circ, \operatorname{Lie}(A_0)_{\tilde{\tau}}^\circ)\otimes_{{\mathcal{O}}_{S_0}}\calI\simeq \operatorname{Lie}(A_0)_{\tilde{\tau}}\otimes_{{\mathcal{O}}_{S_0}} \operatorname{Lie}(A_0)_{\tilde{\tau}^c}\otimes_{{\mathcal{O}}_{S_0}}\calI.$$ Here, in the last isomorphism, we have used . The statement for the tangent bundle of $X'_{\ttT}$ now follows immediately. It remains to prove the properness of $X'_{\ttT}$ when $\ttS_{\infty}\cup \ttT$ is non-empty. The arguments are similar to those in [@helm Proposition 3.4]. We use the valuative criterion of properness. Let $R$ be a discrete valuation ring containing ${\overline{\FF}_p}$ and $L$ be its fraction field. Let $x_L=(A_L, \iota, \lambda, \bar{\alpha}_{K'})$ be an $L$-valued point of $X'_{\ttT}$. We have to show that $x_L$ extends to an $R$-valued point $x_R\in X'_{\ttT}$ up to a finite extension of $L$. By Grothendieck’s semi-stable reduction theorem, we may assume that, up to a finite extension of $L$, the Néron model $A_R$ of $A_L$ over $R$ has a semi-stable reduction. Let $\overline{A}$ be the special fiber of $A_R$, and $\TT\subset \overline{A}$ be its torus part. Since the Néron model is canonical, the action of ${\mathcal{O}}_{D_{\ttS}}$ extends uniquely to $A_R$, and hence to $\TT$. The rational cocharacter group $X_*(\TT)_{{\mathbb{Q}}}:=\operatorname{Hom}(\GG_m,\TT)\otimes_{{\mathbb{Z}}}{\mathbb{Q}}$ is a ${\mathbb{Q}}$-vector space of dimension at most $\dim(\overline A)= 4 g=\frac{1}{2}\dim_{{\mathbb{Q}}}(D_{\ttS})$, and equipped with an induced action of $D_{\ttS}\cong \rmM_{2}(E)$. By the classification of $\rmM_{2}(E)$-modules, $X_*(\TT)_{{\mathbb{Q}}}$ is either $0$ or isomorphic to $E^{\oplus 2}$. In the latter case, we have $X_*(\TT)_{{\mathbb{Q}}}\otimes L\cong \operatorname{Lie}(A_L)$, and the trace of the action of $b\in E$ on $X_*(\TT)_{{\mathbb{Q}}}$ is $2\sum_{\tilde\tau\in \Sigma_{E}}\tilde\tau(b)$, which implies that $\ttS_{\infty}=\emptyset$. Therefore, if $\ttS_{\infty}\neq \emptyset$, $\TT$ has to be trivial and $A_R$ is an abelian scheme over $R$ with generic fiber $A_L$. The polarization $\lambda$ and level structure $\bar{\alpha}_{K'}$ extends uniquely to $A_R$ by the canonicalness of Néron model. We obtain thus a point $x_R\in X'(R)$ extending $x_L$. Since $X'_{\ttT}\subseteq X'$ is a closed subscheme, we see easily that $x_R\in X'_{\ttT}$. Now consider the case $\ttS_{\infty}=\emptyset$ but $\ttT$ is non-empty. If $X_*(\TT)_{{\mathbb{Q}}}\cong E^{\oplus 2}$, then the abelian part of $\overline{A}$ is trivial. Since the action of Verschiebung on $\omega_{\TT}$ is an isomorphism, the point $x_L$ cannot lie in any $X'_{\ttT}$ with $\ttT$ non-empty. Therefore, if $\ttT\neq \emptyset$, $\TT$ must be trivial, and we conclude as in the case $\ttS_{\infty}\neq \emptyset$. It seems that $X'_{\ttT}$ is still proper if $\ttS$ is non-empty. But we can not find a convincing algebraic argument. GO-stratification of connected Shimura varieties {#S:GO-stratum connected Shimura variety} ------------------------------------------------ From the definition, it is clear that the GO-stratification on ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})_{k_0}$ is compatible with the action (as described in Subsection \[S:abel var in unitary case\]) of the group $\calG'_{\tilde \ttS}$ (introduced in Subsection \[S:structure group\]). By Corollary \[C:mathematical objects equivalence\], for each $\ttT \subseteq \Sigma_\infty -\ttS_\infty$, there is a natural scheme $${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})^\circ_{\overline \FF_p, \ttT} \subseteq {\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})^\circ_{\overline \FF_p}$$ equivariant for the action of $\calE_{G, k_0}$. We call them the *Goren-Oort stratification* for the connected Shimura variety. Using the identification of connected Shimura variety in Corollary \[C:comparison of shimura varieties\] together with Corollary \[C:mathematical objects equivalence\], we obtain *Goren-Oort stratum* ${\mathbf{Sh}}_{K_p}(G_\ttS)_{k_0, \ttT} \subseteq {\mathbf{Sh}}_{K_p}(G_\ttS)_{k_0}$ and ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0, \ttT} \subseteq {\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0}$, for each subset $\ttT \subseteq \Sigma_\infty- \ttS_\infty$. Explicitly, for the latter case, we have $${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0, \ttT}:= {\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})_{k_0, \ttT} \times_{\widetilde G_{\tilde \ttS}} G''_{\tilde \ttS}(\AAA^{\infty,p}).$$ Alternatively, in terms of the natural family of abelian varieties $\bfA''_{\tilde \ttS}$, the stratum ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0, \ttT}$ is the common zero locus of partial Hasse-invariants $$h_{\tilde \tau}: \omega^\circ_{\bfA''_{\tilde \ttS,k_0}/{\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0}, \tilde \tau} \longrightarrow \big( \omega^\circ_{\bfA''_{\tilde \ttS,k_0}/{\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0}, \sigma^{-n_\tau} \tilde\tau}\big)^{\otimes p^{n_\tau}}$$ for all $\tilde \tau$ lifting $\tau \in \ttT$. When $\ttS=\emptyset$, the GO-stratification on ${\mathbf{Sh}}_{K_p}(G_{\emptyset})_{k_0}$ defined above agrees with the original definition given in [@goren-oort]. Moreover, the for each subset $\ttT \subseteq \Sigma_\infty$, under the morphisms , we have $$\mathbf{pr}_1^*({\mathbf{Sh}}_{K_p}(G_{\emptyset})_{k_0, \ttT}) = {\boldsymbol{\alpha}}^*({\mathbf{Sh}}_{K''_p}(G''_{\emptyset})_{k_0,\ttT}),$$ where ${\mathbf{Sh}}_{K_p}(G_{\emptyset})_{k_0, \ttT}$ denote the GO-stratum for $\ttT$ defined in loc. cit. Put $X={\mathbf{Sh}}_{K_p}(G_{\emptyset})_{k_0}$ for simplicity. By Proposition \[P:integral-HMV-unitary\], we have an isomorphism of abelian varieties ${\boldsymbol{\alpha}}^*\bfA''_{\emptyset}=(\mathbf{pr}_1^*(\calA)\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_E)^{\oplus 2}$ on ${\mathbf{Sh}}_{K_p\times K_{E,p}}(G_\emptyset \times T_{E, \emptyset})$. Let $\omega_{\calA_{k_0}^\vee/X}=\bigoplus_{\tau\in \Sigma_{\infty}}\omega_{\calA_{k_0}^\vee/X, \tau}$ be the canonical decomposition, where $\omega_{\calA_{k_0}^\vee/X,\tau}$ is the local direct factor on which ${\mathcal{O}}_F$ acts via $\iota_p\circ\tau: {\mathcal{O}}_{F}{\rightarrow}\ZZ_p^{\mathrm{ur}}\twoheadrightarrow {\overline{\FF}_p}$. Then we have a canonical isomorphism of line bundles over ${\mathbf{Sh}}_{K_p\times K_{E,p}}(G_\emptyset \times T_{E, \emptyset})_{k_0}$ $${\boldsymbol{\alpha}}^*\omega_{\bfA''^\vee_{k_0}/X, \tilde{\tau}}^{\circ}\simeq \mathbf{pr}_1^*\omega_{\calA_{k_0}^\vee/X,\tau},$$ for either lift $\tilde{\tau}\in \Sigma_{E,\infty}$ of $\tau$. Via these identifications, the (pullback of) partial Hasse invariant ${\boldsymbol{\alpha}}^*(h_{\tilde{\tau}})$ defined in coincides with the pullback via $\mathbf{pr}_1$ of the partial Hasse invariant $h_{\tau}\in \Gamma(X, \omega_{\calA/X, \sigma^{-1}\tau}^{\otimes p}\otimes \omega_{\calA/X, \tau}^{\otimes -1})$ defined in [@goren-oort]. Therefore, for any $\ttT\subset \Sigma_{\infty}$, the pull back along $\mathbf{pr}_1$ of the GO-strata $X_{\ttT}\subseteq X$ defined by the vanishing of $\{h_{\tau}:\tau\in \ttT\}$ is the same as the pullback along ${\boldsymbol{\alpha}}$ of the GO-stratum defined by $\{h_{\tilde{\tau}}:\tau\in \ttT\}$. It would be interesting to know, in general, whether the GO-strata on quaternionic Shimura varieties depends on the auxiliary choice of CM field $E$. To understand the “action" of the twisted partial Frobenius on GO-strata, we need the following. \[L:partial Frobenius vs partial Hasse inv\] Let $x = (A, \iota, \lambda, \bar \alpha_{K'}) $ be a point of $X'$ with values in a noetherian $k_0$-scheme $S$, and $\gothF'_{\gothp^2}(x) = (A', \iota', \lambda', \bar \alpha'_{K'})$ be the image of $x$ under the twisted partial Frobenius at $\gothp$ (which lies on another Shimura variety). Then $h_{\tilde \tau} (x) = 0$ if and only if $h_{\sigma_{\gothp}^2\tilde \tau}(\gothF'_{\gothp^2}(x))=0$. The statement is clear if $\tilde\tau\notin \Sigma_{E,\infty/\gothp}$, since $\gothF'_{\gothp^2}$ induces a canonical isomorphism of $p$-divisible groups $A[\gothq^{\infty}]\simeq A'[\gothq^{\infty}]$ for $\gothq\in \Sigma_p$ with $\gothq\neq \gothp$. Consider the case $\tilde\tau \in \Sigma_{E,\infty/\gothp}$. We claim that there exists an isomorphism $$H^{{\mathrm{dR}}}_1(A'/S)^{\circ}_{\tilde\tau}\cong (H^{{\mathrm{dR}}}_1(A/S)^{\circ}_{\sigma^{-2}\tilde\tau})^{(p^2)}$$ compatible with the action of $F$ and $V$ on both sides with $\tilde\tau\in \Sigma_{E,\infty/\gothp}$ varying. By Lemma \[Lemma:partial-Hasse\], the Lemma follows from the claim immediately. It remains thus to prove the claim. Actually, the $\gothp$-component of de Rham homology $$H^{{\mathrm{dR}}}_1(A'/S)_{\gothp}:=\bigoplus_{\tilde\tau\in \Sigma_{E,\infty/\gothp}}H^{{\mathrm{dR}}}_1(A'/S)_{\tilde\tau}$$ is canonically isomorphic to the evaluation at the trivial pd-thickening $S{\hookrightarrow}S$, denoted by ${\mathcal{D}}(A'[\gothp^{\infty}])_S$, of the reduced covariant Dieudonné crystal of $A'[\gothp^{\infty}]$. By definition of $\gothF'_{\gothp^2}$, the $p$-divisible group $A'[\gothp^{\infty}]\cong (A/\operatorname{Ker}_{\gothp^2})[\gothp^{\infty}]$ is isomorphic to the quotient of $A[\gothp^{\infty}]$ by its kernel of $p^2$-Frobenius $A[\gothp^{\infty}]{\rightarrow}(A[\gothp^{\infty}])^{(p^2)}$. Therefore, by functoriality of Dieudonné crystals, one has ${\mathcal{D}}(A'[\gothp^{\infty}])_S={\mathcal{D}}(A[\gothp^\infty])_S^{(p^2)}$, whence the claim. One deduces immediately For ${\mathrm{Sh}}_{\tilde \ttS} = {\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})_{k_0}$ and ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0}$, the twisted partial Frobenius map $\gothF_{\gothp^2}: {\mathrm{Sh}}_{\tilde \ttS} \to {\mathrm{Sh}}_{\sigma_\gothp^2 \tilde \ttS}$ takes the subvariety ${\mathrm{Sh}}_{\tilde \ttS, \ttT}$ to ${\mathrm{Sh}}_{\sigma_\gothp^2\tilde \ttS, \sigma_\gothp^2 \ttT}$ for each $\ttT\subseteq \Sigma_{\infty}-\ttS_{\infty}$. The global geometry of the GO-strata: Helm’s isogeny trick {#Section:GO-geometry} ========================================================== In this section, we will prove that each closed GO-stratum of the special fiber of the unitary Shimura variety defined in Definition \[Defn:GO-strata\] is a $(\PP^1)^N$-bundle over the special fiber of another unitary Shimura variety for some appropriate integer $N$. This then allows us to deduce the similar result for the case of quaternionic Shimura varieties. This section is largely inspired by Helm’s pioneer work [@helm], where he considered the case when $p$ splits in $E_0/\QQ$ and $\ttS$ is “sparse” (we refer to *loc. cit.* for the definition of sparse subset; essentially, this means that, for any $\tau \in \Sigma_\infty$, $\tau$ and $\sigma\tau$ cannot belong to $\ttS$ simultaneously.) The associated quaternionic Shimura data for a GO-stratum {#S:quaternion-data-T} --------------------------------------------------------- We first introduce the recipe for describing general GO-strata. We recommend first reading the light version of the same recipe in the special case of Hilbert modular varieties, as explained in the introduction \[S:intro GO-strata\], before diving into the general but more complicated definition below. Keep the notation as in the previous sections. Let $\ttT$ be a subset of $\Sigma_{\infty}-\ttS_{\infty}$. Our main theorem will say that the Goren-Oort stratum ${\mathbf{Sh}}_K(G_\ttS)_{\overline \FF_p, \ttT}$ is a $(\PP^1)^N$-bundle over ${\mathbf{Sh}}_{K_\ttT}(G_{\ttS(\ttT)})_{\overline{\FF}_p}$ for some $N \in \ZZ_{\geq0}$ and some even subset $\ttS(\ttT)$ of places of $F$ and open compact subgroup $K_{\ttT}\subseteq G_{\ttS(\ttT)}(\AAA^{\infty})$. We describe the set $\ttS(\ttT)$ now. For each prime $\gothp\in \Sigma_{p}$, we put $\ttT_{/\gothp}=\ttT\cap \Sigma_{\infty/\gothp}$. We define first a subset $\ttT'_{/\gothp}\subseteq \Sigma_{\infty/\gothp}\cup \{\gothp\} $ containing $\ttT_{/\gothp}$ which depends on the types of $\gothp$ as in Subsection \[S:level-structure-at-p\] and we then put $$\label{Equ:defn-S-T} \ttT'=\coprod_{\gothp\in \Sigma_p}\ttT'_{/\gothp},\quad\text{and }\quad\ttS(\ttT)=\ttS \sqcup \ttT'.$$ We separate the discussion into several cases: - If $\gothp$ is of type $\alpha^\sharp$ or type $\beta^\sharp$ for ${\mathbf{Sh}}_{K}(G_{\ttS})$, we put $\ttT'_{/\gothp} = \emptyset$. - If $\gothp$ is of type $\alpha$ for $\ttS$, i.e. $(\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp})$ has even cardinality. We distinguish two cases: - (Case $\alpha 1$) $\ttT_{/\gothp}\subsetneq \Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp}$. We write $\ttS_{\infty/\gothp}\cup \ttT_{/\gothp}=\coprod C_i$ as a disjoint union of chains. Here, by a chain, we mean that there exists $\tau_i\in \ttS_{\infty/\gothp}\cup \ttT_{/\gothp}$ and an integer $m_i\geq 0$ such that $C_i=\{\sigma^{-a}\tau_i: 0\leq a\leq m_i\}$ belong to $\ttS_{\infty/\gothp} \cup \ttT_{/\gothp}$ and $\sigma\tau_i, \sigma^{-m_i-1}\tau_i\notin (\ttS_{\infty/\gothp}\cup \ttT_{/\gothp})$. We put $\ttT'_{/\gothp} = \coprod_i C'_i$, where $$C'_i: = \left\{ \begin{array}{ll} C_i \cap \ttT_{/\gothp} & \textrm{if } \#(C_i \cap \ttT_{/\gothp}) \textrm{ is even}; \\ (C_i\cap \ttT_{/\gothp})\cup \{\sigma^{-m_i-1}\tau_i\}& \textrm{if } \#(C_i \cap \ttT_{/\gothp}) \textrm{ is odd}. \end{array} \right.$$ For example, if $\Sigma_{\infty/\gothp}=\{\tau_0, \sigma^{-1}\tau_0,\dots, \sigma^{-9}\tau_0\}$, $\ttS_{\infty/\gothp}=\{\sigma^{-2}\tau_0, \sigma^{-6}\tau_0\}$, and $\ttT_{/\gothp}=\{\sigma^{-3}\tau_0, \sigma^{-5}\tau_0, \sigma^{-7}\tau_0\}$, then $\ttS_{\infty/\gothp} \cup \ttT_{/\gothp}$ is separated into two chains $C_1 = \{\sigma^{-2}\tau_0, \sigma^{-3} \tau_0\}$ and $C_2 = \{\sigma^{-5}\tau_0, \sigma^{-6}\tau_0, \sigma^{-7}\tau_0\}$; we have $ \ttT'_{/\gothp}=\{\sigma^{-3}\tau_0,\sigma^{-4}\tau_0, \sigma^{-5}\tau_0, \sigma^{-7}\tau_0\}. $ An alternative way to understand the partition of $\ttT_{/\gothp}$ is to view it as a subset of $\Sigma_{\infty/\gothp} - \ttS_{\infty/\gothp}$ with the cycle structure inherited from $\Sigma_{\infty/\gothp}$; then $C_i \cap \ttT_{/\gothp}$ is just to group elements of $\ttT_{/\gothp}$ into connected subchains. - (Case $\alpha 2$) $\ttT_{/\gothp}= \Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp}$. We put $\ttT'_{/\gothp}=\ttT_{/\gothp}$. - If $\gothp$ is of type $\beta$ for $\ttS$, i.e. $(\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp})$ has odd cardinality and $B_{\ttS}$ splits at $\gothp$. We distinguish two cases: - (Case $\beta 1$) $\ttT_{/\gothp}\subsetneq \Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp}$. In this case, we define $\ttT'_{/\gothp}$ using the same rule as in Case $\alpha 1$. - (Case $\beta 2$) $\ttT_{/\gothp}=\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp}$. We put $\ttT'_{\gothp}=\ttT_{/\gothp}\cup \{\gothp\}$. In either case, we put $\ttT'_{\infty/\gothp} = \ttT'_{/\gothp} \cap \Sigma_\infty$; it is equal to $\ttT'_{/\gothp}$ unless in case $\beta2$. It is easy to see that each $\ttT'_{/\gothp}$ has even cardinality. Therefore, $\ttS(\ttT)$ is also even, and it defines a quaternion algebra $B_{\ttS(\ttT)}$ over $F$. Note that $\ttS(\ttT)$ still satisfies Hypothesis \[H:B\_S-splits-at-p\]. Let $G_{\ttS(\ttT)}=\operatorname{Res}_{F/{\mathbb{Q}}}(B_{\ttS(\ttT)}^{\times})$ be the algebraic group over ${\mathbb{Q}}$ associated to $B_{\ttS(\ttT)}^{\times}$. We fix an isomorphism $B_{\ttS}\otimes_{F}F_{\gothl}\simeq B_{\ttS(\ttT)}\otimes_{F}F_{\gothl}$ whenever $\{\gothl\}\cap \ttS=\{\gothl\}\cap\ttS(\ttT)$. We define an open compact subgroup $K_{\ttT}=K_{\ttT}^pK_{\ttT, p}\subseteq G_{\ttS(\ttT)}(\AAA^{\infty})$ determined by $K$ as follows. - We put $K_{\ttT}^p=K^p$. This makes sense, because $B_{\ttS}\otimes_{F}F_{\gothl}\simeq B_{\ttS(\ttT)}\otimes_{F}F_{\gothl}$ for any finite place $\gothl$ prime to $p$. - For $K_{\ttT, p}=\prod_{\gothp\in \Sigma_p}K_{\ttT, \gothp}$, we take $K_{\ttT,\gothp}=K_{\gothp}$, unless we are in case $\alpha 2$ or $\beta2 $. - If $\gothp$ is of type $\alpha 2$ for ${\mathrm{Sh}}_{K}(G_{\ttS})$, we have $B_{\ttS(\ttT)}\otimes_{F}F_{\gothp}\simeq B_{\ttS}\otimes_{F}F_{\gothp}\simeq \rmM_2({\mathcal{O}}_{F_{\gothp}})$. We take $K_{\ttT, \gothp}=K_{\gothp}$ if $\ttT_{/\gothp}=(\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp})=\emptyset$, and $K_{\ttT, \gothp}={\mathrm{Iw}}_{\gothp}$ if $\ttT_{/\gothp}\neq \emptyset$. - If we are in case $\beta 2$ (and $\beta^\sharp$), $B_{\ttS(\ttT)}$ is ramified at $\gothp$. We take $K_{\ttT, \gothp}={\mathcal{O}}_{B_{F_{\gothp}}}^{\times}$, where ${\mathcal{O}}_{B_{F_{\gothp}}}$ is the unique maximal order of the division algebra over $F_{\gothp}$ with invariant $1/2$. The level $K_{\ttT}$ fits into the framework considered in Subsection \[S:level-structure-at-p\]. We obtain thus a quaternionic Shimura variety ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$, and its integral model ${\mathbf{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$ is given by Corollary \[C:integral-model-quaternion\]. Note that - if we are in case $\alpha 1$ above, then $\gothp$ is of type $\alpha$ for the Shimura variety ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$; - if we are in case $\alpha 2$ above, then $\gothp$ is of type $\alpha^\sharp$ for ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$ unless $\gothp$ is of type $\alpha$ for ${\mathrm{Sh}}_{K}(G_{\ttS})$ and $\ttT_{/\gothp}=\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp}=\emptyset$, in which case $\gothp$ remains of type $\alpha$ for ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$; - if we are in case $\beta 1$, then $\gothp$ is of type $\beta$ for ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$; - if we are in case $\beta2$ or $\beta^\sharp$ above, then $\gothp$ is of type $\beta^\sharp$ for ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$. \[T:main-thm\] For a subset $\ttT\subseteq \Sigma_{\infty}-\ttS_{\infty}$, the GO-stratum ${\mathbf{Sh}}_{K}(G_\ttS)_{\overline \FF_p, \ttT}$ is isomorphic to a $(\PP^1)^{I_{\ttT}}$-bundle over ${\mathbf{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})_{\overline \FF_p}$, where $\ttS(\ttT)$ is as described above, and the index set is given by $$I_{\ttT}=\ttS(\ttT)_{\infty}-(\ttS_{\infty}\cup \ttT)=\bigcup_{\gothp\in \Sigma_p}(\ttT'_{\infty/\gothp}-\ttT_{/\gothp}).$$ Moreover, this isomorphism is compatible with the action of $G_{\ttS}(\AAA^{\infty,p})$, if we let $K^p\subseteq G_{\ttS}(\AAA^{\infty,p})$ vary. Theorem \[T:main-thm\] will follow from the analogous statement (Theorem \[T:main-thm-unitary\] and Corollary \[C:main-thm-product\]) in the unitary case. But note Remark \[R:quaternionic Shimura reciprocity not compatible\]. The signature at infinity for the unitary Shimura varieties {#S:tilde S(T)} ----------------------------------------------------------- In order to describe the unitary Shimura data associated to ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$ as in Subsections \[S:CM extension\] and \[S:unitary-shimura\], we need to pick a lift $\tilde \ttS(\ttT)$ of the set $\ttS(\ttT)$ to embeddings of $E$. More precisely, we will define a subset $\tilde \ttS(\ttT)_\infty = \coprod_{\gothp \in \Sigma_p} \tilde \ttS(\ttT)_{\infty/\gothp}$, where $\tilde \ttS(\ttT)_{\infty/\gothp}$ consists of exactly one lift $\tilde \tau \in \Sigma_{E, \infty}$ for each $\tau \in \ttS(\ttT)_{\infty/\gothp}$. Then we put $\tilde \ttS(\ttT) = (\ttS(\ttT), \tilde \ttS(\ttT)_\infty)$. So we just need to assign such choices of lifts. - When $\tau \in \ttS(\ttT)_{\infty/\gothp}$ belongs to $\ttS_{\infty/\gothp}$, we choose its lift $\tilde \tau \in \Sigma_{E, \infty}$ be the one that belongs to $\tilde \ttS$. We now specify our choices of the lifts in $\tilde\ttS(\ttT)_{\infty/\gothp}$ for the elements of $ \ttT'_{\infty/\gothp}$, which are collectively denoted by $\tilde \ttT'_{/\gothp}$. We separate into cases and use freely the notation from Subsection \[S:quaternion-data-T\]. There is nothing to do if $\gothp$ is of type $\alpha^\sharp$ or type $\beta^\sharp$ (for $\ttS$). - $\gothp$ is of type $\alpha$ (for $\ttS$); in this case, $\gothp$ splits into two primes $\gothq$ and $\gothq^c$ in $E$. For a place $\tau \in \Sigma_{\infty/\gothp}$, we use $\tilde \tau$ to denote its lift to $\Sigma_{E, \infty}$ which corresponds to the $p$-adic place $\gothq$. - (Case $\alpha1$) For a chain $C_i=\{\sigma^{-a}\tau_i, 0\leq a\leq m_{i}\}\subseteq \ttS_{\infty/\gothp}\cup\ttT_{/\gothp}$ and the subset $C'_i =\{ \sigma^{-a_1}\tau_i, \dots, \sigma^{-a_{r_i}}\tau_i\}\subseteq C_i$ as defined in \[S:quaternion-data-T\] for some $0\leq a_{1}< \dots<a_{r_i}\leq m_i+1$ (note that $r_i$ is always even by construction), we put $$\tilde C'_i = \{ \sigma^{-a_1}\tilde \tau_i, \sigma^{-a_2}\tilde \tau^c_i, \sigma^{-a_3}\tilde \tau_i, \dots, \sigma^{-a_{r_i}}\tilde \tau^c_i\};$$ put $\tilde \ttT'_{/\gothp} = \coprod_i \tilde C'_i$. - (Case $\alpha2$) We need to fix $\tau_0 \in \ttT_{/\gothp} = \Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp}$ and write $\ttT_{/\gothp}$ as $\{\sigma^{-a_1}\tau_0, \dots, \sigma^{-a_{2r}}\tau_0\}$ for integers $0 = a_1 < \cdots < a_{2r} \leq f_\gothp-1$. We put $$\tilde \ttT'_{/\gothp} = \{\sigma^{-a_1}\tilde \tau_0, \sigma^{-a_2}\tilde \tau^c_0, \sigma^{-a_3}\tilde \tau_0, \dots, \sigma^{-a_{2r}}\tilde \tau^c_0\}.$$ - $\gothp$ is of type $\beta$ (for $\ttS$). In this case, $\gothp$ is inert in $E/F$, and we do not have a canonical choice for the lift $\tilde \tau$ of a $\tau$. - (Case $\beta1$) In this case, we fix a partition of the preimage of $C'_i$ under the map $\Sigma_{E, \infty/\gothp} \to \Sigma_{\infty/\gothp}$ into two chains $\tilde C''_i \coprod \tilde C''^c_i$, where $$\tilde C''_i = \{\sigma^{-a_1} \tilde \tau_i, \dots, \sigma^{-a_{r_i}} \tilde \tau_i\}, \quad \textrm{and} \quad \tilde C''^c_i = \{\sigma^{-a_1} \tilde \tau^c_i, \dots, \sigma^{-a_{r_i}} \tilde \tau^c_i\}.$$ Here, the choice of $\tilde\tau_i$ is arbitrary, and $r_i$ is always even by construction. We put $$\tilde C'_i :=\{ \sigma^{-a_1}\tilde \tau_i, \sigma^{-a_2}\tilde \tau^c_i, \sigma^{-a_3}\tilde \tau_i, \dots, \sigma^{-a_{r_i}}\tilde \tau^c_i\}.$$ Finally, we set $\tilde \ttT'_{/\gothp} = \coprod_i \tilde C'_i$. - (Case $\beta2$) We fix an element $\tilde \tau_0 \in \Sigma_{E, \infty/\gothp}$ lifting some element from $\Sigma_{\infty/\gothp}$. Then the preimage of $\ttT'_{/\gothp}$ under the natural map $\Sigma_{E, \infty/\gothp} \to \Sigma_{\infty/\gothp}$ can be written as $\{ \sigma^{-a_1}\tilde \tau_0, \dots, \sigma^{-a_{2r}}\tilde \tau_0\}$ (where $r = \# ( \Sigma_\infty - \ttS_\infty)$ is odd), with $0=a_1< \cdots < a_{2r}\leq 2f_\gothp-1$ and $a_{r+i} = a_i + f_\gothp$ for all $i$. We put $$\tilde \ttT'_{/\gothp} = \{ \sigma^{-a_1} \tilde \tau_0, \sigma^{-a_3} \tilde \tau_0, \dots, \sigma^{-a_{2r-1}} \tilde \tau_0\}.$$ Since $r$ is odd, this consists exactly one lift of each element of $\ttT'_{\infty/\gothp}$. Now, we can assign integers $s_{\ttT, \tilde \tau}$ according to $\tilde \ttS(\ttT)$: - if $\tau \in \Sigma_\infty - \ttS(\ttT)_\infty$, we have $s_{\tilde \tau} = 1$ for all lifts $\tilde \tau $ of $\tau$; - if $\tau \in \ttS(\ttT)_\infty$ and $\tilde \tau$ is the lift in $\tilde \ttS(\ttT)_\infty$, we have $s_{\ttT,\tilde \tau} = 0$ and $s_{\ttT,\tilde \tau^c} = 2$. We put $\tilde \ttT' = \cup_{\gothp \in \Sigma_p} \tilde \ttT'_{/\gothp}$ and $\tilde \ttT'^c$ the complex conjugations of the elements in $\tilde \ttT'$.\ Now we compare the PEL data for the Shimura varieties for $G'_{\tilde \ttS}$ and $G'_{\tilde \ttS(\ttT)}$. We fix an isomorphism $\theta_{\ttT}:D_{\ttS}{\rightarrow}D_{\ttS(\ttT)}$ that sends ${\mathcal{O}}_{D_{\ttS},\gothp}$ to ${\mathcal{O}}_{D_{\ttS(\ttT)},\gothp}$ for each $\gothp\in \Sigma_p$, where ${\mathcal{O}}_{D_{\ttS},\gothp}$ and ${\mathcal{O}}_{D_{\ttS(\ttT)},\gothp}$ are respectively fixed maximal orders of $D_{\ttS}\otimes_F F_{\gothp}$ and $D_{\ttS(\ttT)}\otimes_{F}F_{\gothp}$ as in \[S:PEL-Shimura-data\]. \[L:compare D\_S with D\_S(T)\] Let $\delta_{\ttS}\in (D_{\ttS}^{\mathrm{sym}})^{\times}$ be an element satisfying Lemma \[L:property-PEL-data\](1). Then there exists an element $\delta_{\ttS(\ttT)}\in (D_{\ttS(\ttT)}^{{\mathrm{sym}}})^{\times}$ satisfying the same condition with $\ttS$ replaced by $\ttS(\ttT)$ such that, if $*_{\ttS}: l\mapsto \delta_{\ttS}^{-1}\bar{l}\delta_{\ttS}$ and $*_{\ttS(\ttT)}: l\mapsto \delta_{\ttS(\ttT)}^{-1}\bar{l}\delta_{\ttS(\ttT)}$ denote the involutions on $D_{\ttS}$ and $D_{\ttS(\ttT)}$ induced by $\delta_{\ttS}$ and $\delta_{\ttS(\ttT)}$ respectively, then $\theta_{\ttT}$ induces an isomorphism of algebras with positive involutions $(D_{\ttS},*_{\ttS}){\xrightarrow}{\sim} (D_{\ttS(\ttT)}, *_{\ttS(\ttT)})$. We choose first an arbitrary ${\delta}'_{\ttS(\ttT)}\in (D_{\ttS(\ttT)}^{{\mathrm{sym}}})^{\times}$ satisfying Lemma \[L:property-PEL-data\](1). Let $*'_{\ttS(\ttT)}$ denote the involution $l\mapsto (\delta'_{\ttS(\ttT)})^{-1}\bar{l}\delta'_{\ttS(\ttT)}$ on $D_{\ttS(\ttT)}$. By Skolem-Noether theorem, there exists $g\in D^{\times}_{\ttS(\ttT)}$ such that $\theta_{\ttT}(x)^{*'_{\ttS(\ttT)}}=g\theta_{\ttT}(x^{*_\ttS})g^{-1}$ for all $x\in D_{\ttS}$. Since both $*_{\ttS}^2$ and $*'^2_{\ttS(\ttT)}$ are identity, we get $g^{*'_{\ttS(\ttT)}}=g\mu$ for some $\mu\in E^{\times}$ with $\bar\mu\mu=1$. By Hilbert 90, we can write $\mu={\lambda}/{\bar\lambda}$ for some $\lambda\in E^{\times}$. Up to replacing $g$ by $g\lambda$, we may assume that $g^{*'_{\ttS(\ttT)}}=g$, or equivalently, $\overline{\delta'_{\ttS(\ttT)}g}=\delta'_{\ttS(\ttT)}g$ and hence $\delta'_{\ttS(\ttT)}g \in (D_{\ttS(\ttT)}^{\mathrm{sym}})^\times$. Note that we still have the freedom to modify $g$ by an element of $F^{\times}$ without changing $*_{\ttS(\ttT)}$. We claim that, up to such a modification on $g$, $\delta_{\ttS(\ttT)}=\delta'_{\ttS(\ttT)}g$ will answer the question. Indeed, by construction, $\theta_{\ttT}$ is an $*$-isomorphism, i.e. $\theta_{\ttT}(x)^{*_{\ttS(\ttT)}}=\theta_{\ttT}(x^{*_\ttS})$. Note that $\theta_{\ttT}$ sends ${\mathcal{O}}_{D_{\ttS},\gothp}$ to ${\mathcal{O}}_{D_{\ttS(\ttT)},\gothp}$ for every $\gothp\in \Sigma_p$. Up to modifying $g$ by an element of $F^{\times}$, we may assume that $g\in {\mathcal{O}}_{D_{\ttS(\ttT)},\gothp}^{\times}$ for all $\gothp\in \Sigma_p$. Then it is clear that $\delta_{\ttS(\ttT)}$ satisfies the first part of Lemma \[L:property-PEL-data\](1), since so does $\delta'_{\ttS(\ttT)}$ by assumption. It remains to prove that, up to multiplying $g$ by an element of $\calO_{F,(p)}^{\times}$, $$(v,w)\mapsto \psi_{\delta_{\ttS(\ttT)}}(v,w h'_{\tilde\ttS(\ttT)}(\bfi)^{-1})={\mathrm{Tr}}_{D_{\ttS(\ttT),{\mathbb{R}}}/{\mathbb{R}}}(\sqrt{\gothd}vh'_{\tilde\ttS(\ttT)}(\bfi)\bar{w}\delta_{\ttS(\ttT)})$$ on $D_{\ttS(\ttT),{\mathbb{R}}}:=D_{\ttS(\ttT)}\otimes_{{\mathbb{Q}}}{\mathbb{R}}$ is positive definite, where $\psi_{\delta_{\ttS(\ttT)}}$ is the $*_{\ttS(\ttT)}$-hermitian alternating form on $D_{\ttS(\ttT)}$ defined as in Subsection \[S:PEL-Shimura-data\]. Since the elements $\delta_{\ttS}$ and $\delta'_{\ttS(\ttT)}$ satisfy similar positivity conditions by assumption, we get two semi-simple ${\mathbb{R}}$-algebras with positive involution $(D_{\ttS,{\mathbb{R}}}, *_{\ttS})$ and $(D_{\ttS(\ttT),{\mathbb{R}}}, *_{\ttS(\ttT)})$. By [@kottwitz Lemma 2.11], there exists an element $b\in D^{\times}_{\ttS(\ttT), {\mathbb{R}}}$ such that $b\theta_{\ttT}(x^{*_{\ttS}})b^{-1}=(b(\theta_{\ttT}(x)b^{-1})^{*'_{\ttS(\ttT)}}$. It follows that $g=b^{*'_{\ttS(\ttT)}}b\lambda$ with $\lambda\in (F\otimes_{{\mathbb{Q}}} {\mathbb{R}})^{\times}$. Up to multiplying $g$ by an element of $\calO_{F,(p)}^{\times}$, we may assume that $\lambda$ is totally positive so that $\lambda=\xi^2$ with $\xi\in (F\otimes_{{\mathbb{Q}}} {\mathbb{R}})^{\times}$. Then, up to replacing $b$ by $b\xi$, we have $g=b^{*'_{\ttS(\ttT)}}b$. Then the positivity of the form $\psi_{\delta_{\ttS(\ttT)}}$ follows immediately from the positivity of $\psi_{\delta'_{\ttS(\ttT)}}$, and the fact that $\psi_{\delta_{\ttS(\ttT)}}(v,wh'_{\tilde\ttS(\ttT)}(\bfi)^{-1})=\psi_{\delta'_{\ttS(\ttT)}}(bv, bwh'_{\tilde\ttS(\ttT)}(\bfi)^{-1})$. \[L:comparison of Hermitian space\] We keep the choice of $\delta_{\ttS(\ttT)}$ as in Lemma \[L:compare D\_S with D\_S(T)\]. Then there exists an isomorphism $\Theta_\ttT: D_\ttS(\AAA^{\infty,p}) \to D_{\ttS(\ttT)}(\AAA^{\infty, p})$ of skew $*$-Hermitian spaces compatible with the actions of $D_\ttS$ and $D_{\ttS(\ttT)}$, respectively. This could be done explicitly. We however prefer a sneaky quick proof. Under Morita equivalence, we are essentially working with two-dimensional Hermitian spaces and the associated unitary groups. It is well-known that, over a nonarchimedean local field, there are exactly two Hermitian spaces and the associated unitary groups are not isomorphic (see e.g. [@minguez 3.2.1]). In our situation, we know that $G'_{\tilde \ttS,1, v} \cong G'_{\tilde \ttS(\ttT),1,v}$ for any place $v \nmid p\infty$. It follows that the associated Hermitian spaces at $v$ are isomorphic. The lemma follows. \[C:identify structure groups\] The isomorphisms $\theta_\ttT$ and $\Theta_\ttT$ induce an isomorphism $\theta'_\ttT:\calG'_{\tilde \ttS} \xrightarrow{\cong}\calG'_{\tilde \ttS(\ttT)}$; moreover $\theta'_\ttT \times \mathrm{id}$ takes the subgroup $\calE_{G, \ttS, k_0} \subset\calG'_{\tilde \ttS} \times \operatorname{Gal}_{k_0}$ to the subgroup $\calE_{G, \ttS(\ttT), k_0}\subset\calG'_{\tilde \ttS(\ttT)} \times \operatorname{Gal}_{k_0}$. The first statement follows from the description of the two groups in and and the interpretation of these groups as certain automorphic groups of the skew $*$-Hermitian spaces. The second statement follows from the description of both subgroups in Subsection \[S:structure group\] and the observation that the choice of signatures in Subsection \[S:tilde S(T)\] ensures that the reciprocity map ${\gothR\mathrm{ec}}_{k_0}$ for both Shimura data are the *same* at $p$. Level structure of ${\mathrm{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})$ {#S:level structure tilde ttS(ttT)} ---------------------------------------------------------------------- We now specify the level structure $K_{\ttT}'\subseteq G'_{\tilde \ttS(\ttT)}(\AAA^{\infty})$. - For the prime-to-$p$ level, since $\Theta_\ttT$ induces an isomorphism $G_{\tilde \ttS(\ttT)}'(\AAA^{\infty,p}) \simeq G_{\tilde \ttS}'(\AAA^{\infty, p})$, the subgroup $K'^{p}\subseteq G_{\tilde \ttS}'(\AAA^{\infty,p})$ gives rise to a subgroup $K_{\ttT}'^{p}\subseteq G_{\tilde \ttS(\ttT)}'(\AAA^{\infty,p})$. - For $K'_{\ttT, p}$, we take it as the open compact subgroup of $G_{\tilde \ttS(\ttT)}'({\mathbb{Q}}_p)$ corresponding to $K_{\ttT,p}\subseteq G_{\tilde \ttS(\ttT)}({\mathbb{Q}}_p)$ by the rule in Subsection \[S:level-structure\]. According to the discussion there, it suffices to choose a chain of lattices $\Lambda^{(1)}_{\ttT,\gothp}\subset \Lambda^{(2)}_{\ttT,\gothp}$ in $D_{\ttS(\ttT)}\otimes_{F}F_{\gothp}$ for each $\gothp\in \Sigma_{p}$. Using the isomorphism $\theta_{\ttT}$, we can identify $D_{\ttS(\ttT)}\otimes_F F_{\gothp}$ with $D_{\ttS}\otimes_F F_{\gothp}$, and hence with $\rmM_2(E \otimes_F F_\gothp)$. - For $\gothp\in \Sigma_{p}$ with $K_{\ttT,\gothp}=K_{\gothp}$, we take $\Lambda_{\ttT, \gothp}^{(1)}\subseteq \Lambda_{\ttT,\gothp}^{(2)}$ to be the same as the chain $\Lambda_{\gothp}^{(1)}\subseteq \Lambda_{\gothp}^{(2)}$ for defining $K_{p}'\subset G'_{\ttS}({\mathbb{Q}}_p)$. - For $\gothp\in \Sigma_{p}$ with $K_{\ttT, \gothp}\neq K_{\gothp}$, then $K_{\ttT, \gothp}$ is either Iwahori subgroup of ${\mathrm{GL}}_2({\mathcal{O}}_{F_{\gothp}})$ or ${\mathcal{O}}_{B_{F_{\gothp}}}^{\times}$. We take then $\Lambda_{\ttT,\gothp}^{(1)}\subsetneq \Lambda_{\ttT,\gothp}^{(2)}$ to be the corresponding lattices as in Subsection \[S:level-structure\] that defines the Iwahori level at $\gothp$. Note that we have always $K_{p}'\subseteq K'_{\ttT,p}$ under the isomorphism $G_{\tilde \ttS}'({\mathbb{Q}}_p)\simeq G'_{\tilde \ttS(\ttT)}({\mathbb{Q}}_p)$ induced by $\theta_{\ttT}$. We also specify the lattices we use for both Shimura varieties: if $\Lambda_\ttS$ denotes the chosen lattice of $D_\ttS$, we choose the lattice of $D_{\ttS(\ttT)}$ to be $\Lambda_{\ttS(\ttT)} =\theta_\ttT(\Lambda_\ttS)$. With these data, we have a unitary Shimura variety ${\mathrm{Sh}}_{K_{\ttT}'}(G_{\tilde \ttS(\ttT)}')$ over the reflex field $E_{\tilde \ttS(\ttT)}$, which is the field corresponding to the Galois group fixing the subset $\tilde \ttS(\ttT) \subseteq \Sigma_{E, \infty}$. To construct an integral model of ${\mathrm{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})$, we need to choose an order ${\mathcal{O}}_{D_{\ttS(\ttT)}}$. Let ${\mathcal{O}}_{D_{\ttS}}$ be the order stable under $*$ and maximal at $p$ used to define the integral model ${\mathbf{Sh}}_{K'}(G_{\tilde \ttS}')$. We put ${\mathcal{O}}_{D_{\ttS(\ttT)}}=\theta_{\ttT}({\mathcal{O}}_{D_{\ttS}})$. For any $\gothp\in \Sigma_{p}$, both ${\mathcal{O}}_{D_{\ttS}, \gothp}$ and ${\mathcal{O}}_{D_{\ttS(\ttT)}, \gothp}$ can be identified with $\rmM_2({\mathcal{O}}_{E}\otimes_{{\mathcal{O}}_F}{\mathcal{O}}_{F_{\gothp}})$. We have now all the PEL-data needed for Theorem \[T:unitary-shimura-variety-representability\], which assures that ${\mathrm{Sh}}_{K_{\ttT}'}(G_{\tilde \ttS(\ttT)}')$ admits an integral model ${\mathbf{Sh}}_{K_{\ttT}'}(G_{\tilde \ttS(\ttT)}')$ over $W(k_0)$. Using ${\mathbf{Sh}}_{K_{\ttT}'}(G_{\tilde \ttS(\ttT)}')$, we can construct an integral model ${\mathbf{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$ of the quaternionic Shimura variety ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$. \[T:main-thm-unitary\] For a subset $\ttT\subseteq \Sigma_{\infty}-\ttS_{\infty}$, let ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\ttT}\subseteq {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ denote the GO-stratum defined in Definition \[Defn:GO-strata\]. Let $I_{\ttT}$ be as in Theorem \[T:main-thm\]. Then we have the following: - (Description) ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\ttT}$ is isomorphic to a $(\PP^1)^{I_{\ttT}}$-bundle over ${\mathbf{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})_{k_0}$. - (Compatibility of abelian varieties) Let $\pi_{\ttT}: {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\ttT}{\rightarrow}{\mathbf{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})_{k_0}$ denote the projection of the $(\PP^1)^{I_{\ttT}}$-bundle in *(1)*. The abelian schemes $\bfA'_{\tilde \ttS,k_0}$ and $\pi_{\ttT}^*\bfA'_{\tilde \ttS(\ttT),k_0}$ over ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ are isogenous, where $\bfA'_{\tilde \ttS,k_0}$ and $\bfA'_{\tilde \ttS(\ttT),k_0}$ denote respectively the universal abelian varieties over ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\ttT}$ and over ${\mathbf{Sh}}_{K'_{ \ttT}}(G'_{\tilde \ttS(\ttT)})_{k_0}$. - (Compatibility with Hecke action) When the open compact subgroup $K'^p\subseteq G'_{\tilde \ttS}(\AAA^{\infty,p})$ varies, the isomorphism as well as the isogeny of abelian varieties are compatible with the action of the Hecke correspondence given by $\widetilde G_{\tilde \ttS} = G''_{\tilde \ttS}(\QQ)^{+,(p)} G'_{\tilde \ttS}(\AAA^{\infty, p}) \cong \widetilde G_{\tilde \ttS(\ttT)}$. - (Compatibility with partial Frobenius) The description in (1) is compatible with the action of the twisted partial Frobenius (Subsection \[S:partial Frobenius\]) in the sense that we have a commutative diagram $$\xymatrix{ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT} \ar[r]_-{\xi^\mathrm{rel}} \ar[rd]_{\pi_{\ttT}} \ar@/^15pt/[rr]^-{\gothF'_{\gothp^2, \tilde \ttS}} & \gothF'^*_{\gothp^2}( {\mathbf{Sh}}_{K'_\ttT}(G'_{\sigma_\gothp^2\tilde \ttS})_{k_0, \sigma_\gothp^2\ttT}) ) \ar[d] \ar[r]_-{\gothF'^*_{\gothp^2, \tilde \ttS(\ttT)}} & {\mathbf{Sh}}_{K'_\ttT}(G'_{\sigma_\gothp^2\tilde \ttS})_{k_0, \sigma_\gothp^2\ttT} \ar[d]^{\pi_{\sigma_{\gothp}^2\ttT}} \\ & {\mathbf{Sh}}_{K'_\ttT}(G'_{\tilde \ttS(\ttT)})_{k_0} \ar[r]^-{\gothF'_{\gothp^2, \tilde \ttS(\ttT)}} & {\mathbf{Sh}}_{K'_{\sigma_\gothp^2\ttT}}(G'_{\sigma_\gothp^2(\tilde \ttS(\ttT))})_{k_0} }$$ where the square is cartesian, we added subscripts to the partial Frobenius to indicate the corresponding base scheme, and the morphism $\xi^\mathrm{rel}$ is a morphism whose restriction to each fiber $\pi_{\ttT}^{-1}(x)=(\PP^1_x)^{I_{\ttT}}$ is the product of the relative $p^2$-Frobenius of the $\PP^1$’s indexed by $I_{\ttT}\cap \Sigma_{\infty/\gothp}=\ttT'_{\infty/\gothp}-\ttT_{/\gothp}$, and the identify of the other $\PP^1_x$’s. The proof of this theorem will occupy the rest of this section and concludes in Subsection \[S:End-of-proof\]. We first state a corollary. \[C:main-thm-product\] 1. The Goren-Oort stratum ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})^\circ_{\overline \FF_p, \ttT}$ is isomorphic to a $(\PP^1)^{I_\ttT}$-bundle over ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS(\ttT)})^\circ_{\overline \FF_p}$, equivariant for the action of $\calE_{G, \ttS, \tilde \wp} \cong \calE_{G, \ttS(\ttT), \tilde \wp}$ (which are identified as in Corollary \[C:identify structure groups\]) with trivial action on the fibers. 2. The GO-stratum ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0,\ttT}$ is isomorphic to a $(\PP^1)^{I_\ttT}$-bundle over ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS(\ttT)})_{k_0}$, such that the natural projection $\pi_{\ttT}: {\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0,\ttT} \to{\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS(\ttT)})_{k_0}$ is equivariant for the tame Hecke action. 3. The abelian schemes $\bfA''_{\tilde \ttS,k_0}$ and $\pi_{\ttT}^*(\bfA''_{\tilde \ttS(\ttT),k_0})$ over ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0, \ttT}$ are isogenous. 4. The following diagram is commutative $$\xymatrix@C=50pt{ {\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0, \ttT} \ar[r]_-{\xi^\mathrm{rel}} \ar@/^15pt/[rr]^-{\gothF''_{\gothp^2,\tilde \ttS}} \ar[dr]_{\pi_{\ttT}} & \gothF_{\gothp^2, \tilde \ttS(\ttT)}^*({\mathbf{Sh}}_{K''_p}(G''_{\sigma_\gothp^2\tilde \ttS})_{k_0, \sigma_{\gothp}^2 \ttT} \ar[r]_-{\gothF''^*_{\gothp^2, \tilde \ttS(\ttT)}} \ar[d] & {\mathbf{Sh}}_{K''_p}(G''_{\sigma_\gothp^2\tilde \ttS})_{k_0, \sigma_{\gothp}^2 \ttT} \ar[d]^{\pi_{\tilde \ttS(\ttT)}} \\ & {\mathbf{Sh}}_{K''_{\ttT,p}}(G''_{\tilde \ttS(\ttT)})_{k_0} \ar[r]^-{\gothF''_{\gothp^2, \tilde \ttS(\ttT)}} & {\mathbf{Sh}}_{K''_{\sigma_\gothp^2\ttT,p}}(G''_{\sigma_\gothp^2(\tilde \ttS(\ttT))})_{k_0} }$$ where the square is cartesion, $\gothF''_{\gothp^2, \tilde \ttS}$ and $\gothF''_{\gothp^2, \tilde \ttS(\ttT)}$ denote the twisted partial Frobenii (Subsection \[S:partial Frobenius\]) on ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})_{k_0}$ and ${\mathbf{Sh}}_{K''_{\ttT,p}}(G''_{\tilde \ttS(\ttT)})_{k_0}$ respectively, and $\xi^\mathrm{rel}$ is a morphism whose restriction to each fiber $\pi_{\ttT}^{-1}(x)=(\PP^1_x)^{I_{\ttT}}$ is the product of the relative $p^2$-Frobenius of the $\PP^1_x$’s indexed by $I_{\ttT}\cap \Sigma_{\infty/\gothp}=\ttT'_{\infty/\gothp}-\ttT_{/\gothp}$, and the identify of the other $\PP^1_x$’s. This is an immediate consequence of Corollary \[C:main-thm-product\] above. The claims regarding the universal abelian varieties follows from the explicit construction of $\bfA''_{\tilde \ttS}$ and $\bfA''_{\tilde \ttS(\ttT)}$ in Subsection \[S:abel var in unitary case\]. \[R:quaternionic Shimura reciprocity not compatible\] We emphasize that the analogous of Corollary \[C:main-thm-product\](2) for quaternionic Shimura varieties only holds over $\overline \FF_p$. This is because the subgroups $\calE_{G, \ttS, \wp}$ and $\calE_{G, \ttS(\ttT), \wp}$, although abstractly isomorphic, sit in $\calG_\ttS \times \operatorname{Gal}_{k_0} \cong \calG_{\ttS(\ttT)} \times \operatorname{Gal}_{k_0}$ as *different* subgroups. The two Deligne homomorphisms are different. The rest of this section is devoted to the proof of Theorem \[T:main-thm-unitary\], which concludes in Subsection \[S:End-of-proof\]. Signature changes {#S:Delta-pm} ----------------- The basic idea of proving Theorem \[T:main-thm-unitary\] is to find a quasi-isogeny between the two universal abelian varieties $\bfA_{\tilde \ttS}$ and $\bfB: =\bfA_{\tilde \ttS(\ttT)}$ (over an appropriate base). We view this quasi-isogeny as two genuine isogenies $ \bfA_{\tilde \ttS} \xrightarrow{\phi} \bfC \xleftarrow{\phi'} \bfB $ for some abelian variety $\bfC$; each isogeny is characterized by the set of places $\tilde\tau\in \Sigma_{E,\infty}$ where the isogeny does not induce an isomorphism of the $\tilde\tau$-components of the de Rham cohomology of the abelian varieties. We define these two subsets $\tilde \Delta(\ttT)^+$ and $\tilde \Delta(\ttT)^-$ of $\Sigma_{E, \infty}$ now, as follows. As usual, $ \tilde \Delta(\ttT)^\pm = \coprod_{\gothp \in \Sigma_p} \tilde \Delta(\ttT)_{/ \gothp}^\pm$ for subsets $\tilde \Delta(\ttT)_{/\gothp}^\pm \subseteq \Sigma_{E, \infty/\gothp}$. When $\gothp$ is of type $\alpha^\sharp$ or $\beta^\sharp$ for $\ttS$, we set $\tilde \Delta(\ttT)_{/\gothp}^\pm =\emptyset$. For the other two types, we use the notation in Subsection \[S:tilde S(T)\] in the corresponding cases (in particular our convention on $\tilde \tau$ and $a_j$’s): - (Case $\alpha1$) Put $$\tilde C^-_i: = \bigcup_{\substack{j \text{ odd}\\ 1\leq j\leq r_i}} \{\sigma^{-\ell}\tilde \tau_i: a_{j}\leq \ell\leq a_{j+1}-1\}.$$ We set $\tilde \Delta(\ttT)_{/\gothp}^- = \coprod_i \tilde C^-_i$ and $\tilde \Delta(\ttT)_{/\gothp}^+ =(\tilde \Delta(\ttT)_{/\gothp}^-)^c$. - (Case $\alpha2$) Put $$\tilde \Delta(\ttT)_{/\gothp}^- : = \bigcup_{1\leq i\leq r} \big\{ \sigma^{-l}\tilde\tau_0: a_{2i-1} \leq l< a_{2i} \big\}; \quad \textrm{and}\quad \tilde \Delta(\ttT)_{/\gothp}^+: = (\tilde \Delta(\ttT)_{/\gothp}^-)^c.$$ - (Case $\beta1$) Put $$\tilde C_i^-: = \bigcup_{\substack{j \text{ odd}\\ 1\leq j\leq r_i}} \{\sigma^{-\ell}\tilde \tau_i: a_{j}\leq \ell\leq a_{j+1}-1\}.$$ We set $\tilde\Delta(\ttT)_{/\gothp}^- = \coprod_i \tilde C^-_i$ and $\tilde \Delta(\ttT)_{/\gothp}^+=(\tilde \Delta(\ttT)_{/\gothp}^-)^c$. (Formally, this is the same recipe as in case $\alpha1$, but the choice of $\tilde \tau_i$ is less determined; see Subsection \[S:tilde S(T)\].) - (Case $\beta2$) Put $$\tilde \Delta(\ttT)_{/\gothp}^- : = \bigcup_{1\leq i\leq r} \big\{ \sigma^{-l}\tilde\tau_0: a_{2i-1} \leq l< a_{2i} \big\}.$$ *Unlike in all other cases, we put $\tilde \Delta(\ttT)_{/\gothp}^+=\emptyset$.* We use $\ttT_E$ (resp. $\ttT'_E$) to denote the preimage of $\ttT$ (resp. $\ttT'$) under the map $\Sigma_{E, \infty} \to \Sigma_\infty$. The following two lemmas follow from the definition by a case-by-case check. \[L:distance to T’\] For each $\tilde \tau \in \tilde \Delta(\ttT)^+$ (resp. $\tilde \Delta(\ttT)^-$), let $n$ be the unique positive integer such that $\tilde \tau, \sigma^{-1}\tilde \tau, \dots, \sigma^{1-n} \tilde \tau$ all belong to $\tilde \Delta(\ttT)^+$ (resp. $\tilde \Delta(\ttT)^-$) but $\sigma^{-n}\tilde \tau$ does not. Then, for this $n$, $\sigma^{-n}\tilde \tau \in \ttT'_E$. Moreover, if $\tilde \tau$ also belongs to $\ttT'_E$, then $n$ equals to the number $n_\tau$ introduced in Subsection \[S:partial-Hasse\]. \[L:property of Delta\] *(1)* If both $\tilde \tau$ and $\sigma \tilde \tau$ belong to $\tilde \Delta(\ttT)^+$ (resp. $\tilde \Delta(\ttT)^-$), then $\tilde \tau|_F$ belongs to $\ttS_\infty$. *(2)* If $\tilde\tau \in \tilde \Delta(\ttT)^-$ but $\sigma \tilde \tau \notin \tilde \Delta(\ttT)^-$, then $\tilde \tau \in \tilde \ttT'$. *(3)* If $\tilde\tau \notin \tilde \Delta(\ttT)^-$ but $\sigma \tilde \tau \in \tilde \Delta(\ttT)^-$, then $\tilde \tau \in \tilde \ttT'^c$. Description of the strata ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT}$ via isogenies {#S:moduli-Y_S} ------------------------------------------------------------------------------------------ To simplify the notation, we put $X'={\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ and $X'_{\ttT}={\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\ttT}$ for a subset $\ttT\subseteq \Sigma_{\infty}-\ttS_{\infty}$. We will first prove statement (1) of Theorem \[T:main-thm-unitary\]. Following the idea of Helm [@helm], we introduce auxiliary moduli spaces $Y'_{\ttT}$ and $Z'_{\ttT}$ and establishing isomorphisms $$\label{E:sequence-moduli} \xymatrix{ X'_\ttT &Y'_\ttT\ar[l]^-{\cong}_-{\eta_1} \ar[r]^-{\eta_2}_-{\cong} &Z'_\ttT},$$ where $Z'_\ttT$ is a $(\PP^1)^{I_{\ttT}}$-bundle over the special fiber of ${\mathbf{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})_{k_0}$. Recall that we have fixed an isomorphism $\theta_{\ttT}: (D_{\ttS},*_{\ttS}){\xrightarrow}{\sim}(D_{\ttS(\ttT)}, *_{\ttS(\ttT)})$ of simple algebras over $E$ with positive involution, and put ${\mathcal{O}}_{D_{\ttS(\ttT)}}=\theta_{\ttT}({\mathcal{O}}_{D_{\ttS}})$. To ease the notation, we identify ${\mathcal{O}}_{D_{\ttS(\ttT)}}$ with ${\mathcal{O}}_{D_{\ttS}}$ via $\theta_{\ttT}$, and denote them by ${\mathcal{O}}_D$ when there is no confusions. We start now to describe $Y'_\ttT$: It is the moduli space over $k_0$ which attaches to a locally noetherian $k_0$-scheme $S$ the set of isomorphism classes of $(A, \iota_A,\lambda_A, \alpha_{K'}, B, \iota_B, \lambda_B, \beta_{K'_\ttT}, C, \iota_C; \phi_A, \phi_B)$, where - $(A, \iota_A, \lambda_A, \alpha_{K'})$ is an element in $X'_\ttT(S)$. - $(B, \iota_B, \lambda, \beta_{K'_\ttT})$ is an element in ${\mathbf{Sh}}_{K'_\ttT}(G'_{\tilde \ttS(\ttT)})(S)$. - $C$ is an abelian scheme over $S$ of dimension $4g$, equipped with an embedding $\iota_C: \calO_D \to \operatorname{End}_S(C)$, - $\phi_A: A \to C$ is an $\calO_D$-isogeny whose kernel is killed by $p$, such that the induced map $$\phi_{A,*,\tilde{\tau}}: H_1^{{\mathrm{dR}}}(A/S)^\circ_{\tilde{\tau}} \to H_1^{\mathrm{dR}}(C/S)^\circ_{\tilde{\tau}}$$ is an isomorphism for $\tilde{\tau}\in \Sigma_{E,\infty}$ *unless* $\tilde{\tau}\in \tilde\Delta(\ttT)^+$, in which case, we require that $$\label{E:condition-phi-A} \operatorname{Ker}(\phi_{A,*,\tilde{\tau}})={\mathrm{Im}}(F_{A,{\mathrm{es}}, \tilde \tau}^{n}),$$ where $n$ is as determined in Lemma \[L:distance to T’\], and $F_{A,{\mathrm{es}}, \tilde \tau}^n$ is defined in . (When $\tau$ itself belongs to $\ttT'$, the number $n$ equals to $n_\tau$ introduced in Subsection \[S:partial-Hasse\]; in this case, condition is equivalent to saying that $\operatorname{Ker}(\phi_{A,*,\tilde\tau})=\omega^\circ_{A^\vee/S,\tilde\tau}$.) - $\phi_B: B\to C$ is an $\calO_D$-isogeny whose kernel is killed by $p$ such that $\phi_{B,*,\tilde{\tau}}: H_1^{\mathrm{dR}}(B/S)_{\tilde{\tau}}^\circ \to H_1^{\mathrm{dR}}(C/S)_{\tilde{\tau}}^\circ$ is an isomorphism for $\tilde{\tau}\in \Sigma_{E,\infty}$ *unless* $\tilde{\tau}\in \tilde{\Delta}(\ttT)^-$, in which case, we require that ${\mathrm{Im}}(\phi_{B,*,\tilde{\tau}})$ is equal to $\phi_{A,*,\tilde{\tau}}({\mathrm{Im}}(F_{A,{\mathrm{es}}, \tilde \tau}^n))$, where $n$ is as determined in Lemma \[L:distance to T’\]. - The tame level structures are compatible, i.e. $T^{(p)}(\phi_A)\circ \alpha^p_{K'^p} = T^{(p)}(\phi_B) \circ \alpha^p_{K'^p_\ttT}$ as maps from $\widehat \Lambda^{(p)}_\ttS \cong \widehat \Lambda^{(p)}_{\ttS(\ttT)}$ to $T^{(p)}(C)$, modulo $K'^p$, if we identify the two lattices naturally as in Subsection \[S:level structure tilde ttS(ttT)\]. - If $\gothp$ is a prime of type $\alpha^\sharp$ for the original quaternionic Shimura variety ${\mathrm{Sh}}_{K}(G_{\ttS})$, then $\alpha_{\gothp}$ and $\beta_{\gothp}$ are compatible, i.e. $\phi_{A}(\alpha_{\gothp})=\phi_{B}(\beta_{\gothp})$, where $\alpha_{\gothp}\subseteq A[\gothp]$ denotes the closed finite flat group scheme given by Theorem \[T:unitary-shimura-variety-representability\](c2). - Let $\gothp$ be a prime in Case $\alpha2$, splitting into $\gothq \bar \gothq$ in $E$. Then $\beta_\gothp = H_\gothq \oplus H_{\bar \gothq}$. If $\phi_{\gothq}: B{\rightarrow}B'_{\gothq}=B/H_{\gothq}$ the canonical isogeny. Then the kernel of the induced map $\phi_{\gothq,*}:H^{{\mathrm{dR}}}_1(B/S)^{\circ}_{\tilde{\tau}}{\rightarrow}H^{{\mathrm{dR}}}_1(B'_{\gothq}/S)^\circ_{\tilde\tau}$ coincides with that of $\phi_{B,*}: H^{{\mathrm{dR}}}_1(B/S)^\circ_{\tilde\tau}{\rightarrow}H^{{\mathrm{dR}}}_1(C/S)^\circ_{\tilde\tau}$ for all $\tilde\tau\in \tilde \Delta(\ttT)_{/\gothp}^-$. - We have the following commutative diagram $$\xymatrix{ A \ar[r]^-{\phi_{A}}\ar[d]_{\lambda_A} & C & B \ar[l]_-{\phi_B}\ar[d]^{\lambda_B}\\ A^\vee & C^\vee\ar[l]_-{\phi_A^\vee} \ar[r]^-{\phi_B^\vee} & B^\vee. }$$ Compared to [@helm], our moduli problem appears to be more complicated. This is because we allow places above $p$ of $F$ to be inert in the CM extension $E$. It is clear that $B$ is quasi-isogenous to $A$. So when $S$ is the spectrum of a perfect field $k$, the *covariant* Dieudonné module $\tilde{\calD}_{B}$ is a $W(k)$-lattice in $\tilde{\calD}_A[1/p]$. The complicated conditions (v) and (vi) can be better understood by looking at $\tilde{\calD}_{B}$ (See the proof of Proposition \[P:Y\_S=X\_S\] below). \[P:Y\_S=X\_S\] The natural forgetful functor $$\eta_1\colon (A, \iota_A,\lambda_A, \alpha_{K'}, B, \iota_B, \lambda_B, \beta_{K'_\ttT}, C, \iota_C; \phi_A, \phi_B) \mapsto (A, \iota_A,\lambda_A, \alpha_{K'})$$ induces an isomorphism $\eta_1\colon Y'_\ttT \to X'_\ttT$. By the general theory of moduli spaces of abelian schemes due to Mumford, $Y'_{\ttT}$ is representable by an $k_0$-scheme of finite type. Hence, to prove the proposition, it suffices to show that the natural map $Y'_{\ttT}{\rightarrow}X'_{\ttT}$ induces a bijection on closed points and the tangent spaces at each closed point. The proposition will follow thus from Lemmas \[L:Y\_T=X\_T-1\] and \[L:Y\_T-X\_T-tangent\] below. This is a long and tedious book-keeping check, essentially following the ideas of [@helm Proposition 4.4]. \[L:Y\_T=X\_T-1\] Let $x=(A, \iota_{A}, \lambda_{A}, \alpha_{K'})$ be a closed point of $X'_\ttT$, with values in a perfect field $k$. Then there exist *unique* $(B,\lambda_{B}, \iota_B, \beta_{K'_\ttT}, C, \iota_C; \phi_A, \phi_B)$ such that $(A, \iota_{A}, \lambda_{A}, \alpha_{K'}, B,\lambda_{B}, \iota_B, \beta_{K'_\ttT}, C, \iota_C; \phi_A, \phi_B)\in Y'_{\ttT}(k)$. We first recall some notation regarding Dieudonné modules. Let $\tilde \calD_A$ denote the *covariant* Dieudonné module of $A[p^\infty]$. Then $\calD_A := \tilde \calD_A/p$ is the covariant Dieudonné module of $A[p]$. Given the action of $\calO_D\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_p\simeq \rmM_2({\mathcal{O}}_{E}\otimes_{{\mathbb{Z}}}{\mathbb{Z}}_p)$ on $A$, we have direct sum decompositions $${\tilde{{\mathcal{D}}}}_A^\circ := \gothe {\tilde{{\mathcal{D}}}}_A = \bigoplus_{\tilde{\tau}\in \Sigma_{E,\infty}} {\tilde{{\mathcal{D}}}}_{A, \tilde{\tau}}^\circ, \quad {\mathcal{D}}_A^\circ := \gothe {\mathcal{D}}_A = \bigoplus_{\tilde{\tau}\in \Sigma_{E,\infty}} {\mathcal{D}}_{A, \tilde{\tau}}^{\circ},$$ where $\gothe$ denotes the idempotent $\bigl(\begin{smallmatrix} 1&0\\0&0\end{smallmatrix}\bigr)$. By the theory of Dieudonné modules, we have canonical isomorphisms $$H^{{\mathrm{cris}}}_1(A/k)_{W(k)}\cong {\tilde{{\mathcal{D}}}}_{A},\quad H^{{\mathrm{dR}}}_1(A/k)\cong {\mathcal{D}}_{A},$$ compatible with all the structures. For $\tilde{\tau}\in \Sigma_{E,\infty}$, we have the Hodge filtration $ 0{\rightarrow}\omega^{\circ}_{A^\vee,\tilde{\tau}}{\rightarrow}{\mathcal{D}}_{A,\tilde{\tau}}^{\circ}{\rightarrow}\operatorname{Lie}(A)^{\circ}_{\tilde{\tau}}{\rightarrow}0. $ We use $\tilde \omega^\circ_{A^\vee, \tilde \tau}$ to denote the preimage of $\omega^\circ_{A, \tilde\tau} \subseteq {\mathcal{D}}^\circ_{A, \tilde\tau}$ under the reduction map ${\tilde{{\mathcal{D}}}}^\circ_{A, \tilde\tau} \twoheadrightarrow {\mathcal{D}}^\circ_{A, \tilde\tau}$. We first construct $C$ from $A$. For each $\tilde{\tau}\in \Sigma_{E,\infty}$, we define a $W(k)$-module $M^{\circ}_{\tilde{\tau}}$ with ${\tilde{{\mathcal{D}}}}_{A,\tilde{\tau}}\subseteq M^{\circ}_{\tilde{\tau}}\subseteq p^{-1}{\tilde{{\mathcal{D}}}}_{A,\tilde{\tau}}^\circ$ as follows. We put $M^{\circ}_{\tilde{\tau}}={\tilde{{\mathcal{D}}}}^{\circ}_{\tilde{\tau}}$ unless $\tilde{\tau}\in \tilde{\Delta}(\ttT)^+$. In the exceptional case, let $n$ be the integer as determined in Lemma \[L:distance to T’\] (or equivalently as in property (v) of $Y'_{\ttT}$ above), and put $$M^{\circ}_{\tilde{\tau}}:=p^{-1}F_{A,{\mathrm{es}}}^n({\tilde{{\mathcal{D}}}}^\circ_{A,\sigma^{-n}\tilde \tau}),$$ where $ F_{A,{\mathrm{es}}}^n$ is the $n$-iteration of the essential Frobenius on ${\tilde{{\mathcal{D}}}}^{\circ}_{A}$ defined in Notation \[N:essential frobenius and verschiebung\]. If $\tilde{\tau}\in \tilde \Delta(\ttT)^+ \cap \ttT_E$, then the number $n$ for $\tilde{\tau}$ in Lemma \[L:distance to T’\] coincides with $n_{\tau}$ introduced in Subsection \[S:partial-Hasse\]. Since the partial Hasse invariant $h_{\tilde{\tau}}(A)$ vanishes for any $\tilde{\tau}\in \ttT_E$ by the definition of $X'_{\ttT}$, we see that $M^{\circ}_{\tilde\tau}=p^{-1}\tilde\omega^\circ_{A,\tilde{\tau}}$ for $\tilde \tau \in \tilde \Delta(\ttT)^+ \cap \ttT_E$. We now check that, for any $\tilde{\tau}\in \Sigma_{E,\infty}$, $$\label{E:stability-FV} F_A(M^\circ_{\sigma^{-1}\tilde{\tau}})\subseteq M^{\circ}_{\tilde{\tau}},\quad \textrm{and} \quad V_A(M^{\circ}_{\tilde{\tau}})\subseteq M^{\circ}_{\sigma^{-1}\tilde{\tau}}.$$ Note that we are using the genuine but not essential Frobenius and Verschiebung here. We distinguish several cases: - $\tilde\tau,\sigma^{-1}\tilde\tau\notin \tilde\Delta(\ttT)^+$. Then $M^\circ_{\tilde\tau}={\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau}$ and $M^\circ_{\sigma^{-1}\tilde\tau}={\tilde{{\mathcal{D}}}}^\circ_{A,\sigma^{-1}\tilde\tau}$; hence is clear. - $\tilde{\tau}\in \tilde\Delta(\ttT)^+$ and $\sigma^{-1}\tilde{\tau}\notin \tilde\Delta(\ttT)^+$. Then we have $M^{\circ}_{\sigma^{-1}\tilde{\tau}}={\tilde{{\mathcal{D}}}}_{A,\tilde\tau}$ and $M^\circ_{\tilde{\tau}}=p^{-1}F_A({\tilde{{\mathcal{D}}}}_{A,\sigma^{-1}\tilde\tau})$. Hence $F_A(M^{\circ}_{\sigma^{-1}\tilde{\tau}})\subseteq M^\circ_{\tilde{\tau}}$ is trivial, and $V_A( M^\circ_{\tilde{\tau}})=M^{\circ}_{\sigma^{-1}\tilde{\tau}}$. - $\tilde{\tau},\sigma^{-1}\tilde{\tau}\in \tilde\Delta(\ttT)^+$. Let $n$ be positive integer for $\tilde{\tau}$ as in Lemma \[L:distance to T’\]. Then we have $$M^{\circ}_{\tilde{\tau}}=p^{-1} F_{A,{\mathrm{es}}}^n({\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-n}\tilde{\tau}})= F_{A,{\mathrm{es}}}\big( p^{-1} F_{A,{\mathrm{es}}}^{n-1}({\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-n}\tilde{\tau}})\big) = F_{A,{\mathrm{es}}}\big(M^{\circ}_{\sigma^{-1}\tilde\tau} \big).$$ The inclusions are clear from this. - $\tilde{\tau}\notin \tilde\Delta(\ttT)^+$ and $\sigma^{-1}\tilde{\tau}\in \tilde\Delta(\ttT)^+$. In this case, $\sigma^{-1}\tilde\tau$ must be in $\ttT_E$ by Lemma \[L:distance to T’\]. Hence, we have $M^{\circ}_{\tilde\tau}={\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau}$ and $M^\circ_{\sigma^{-1}\tilde{\tau}}=p^{-1}\tilde\omega^\circ_{A,\sigma^{-1}\tilde{\tau}}$ as remarked above. We see thus $F_A(M^\circ_{\sigma^{-1}\tilde{\tau}})=M^\circ_{\tilde{\tau}}$ and $V_A(M^\circ_{\tilde{\tau}})=p M^\circ_{\sigma^{-1}\tilde{\tau}}$. Consequently, if we put $M^{\circ}=\bigoplus_{\tilde\tau\in \Sigma_{E,\infty}}M^\circ_{\tilde\tau}$ and $M=(M^{\circ})^{\oplus 2}$, then $M$ is a Dieudonné module, and ${\tilde{{\mathcal{D}}}}_{A}\subseteq M\subseteq p^{-1}{\tilde{{\mathcal{D}}}}_A$ with induced $F$ and $V$ on $M$. Consider the quotient $M/{\tilde{{\mathcal{D}}}}_A$. It corresponds to a finite subgroup scheme $K$ of $A[p]$ stable under the action of ${\mathcal{O}}_D$ by the covariant Dieudonné theory. We put $C=A/K$ and let $\phi_A:A{\rightarrow}C$ denote the natural quotient. Then the induced map $\phi_{A,*}: {\tilde{{\mathcal{D}}}}_{A}{\rightarrow}{\tilde{{\mathcal{D}}}}_{C}$ is identified with the natural inclusion ${\tilde{{\mathcal{D}}}}_{A}{\hookrightarrow}M$. The morphisms $F_C$ and $V_C$ on ${\tilde{{\mathcal{D}}}}_C$ are induced from those on ${\tilde{{\mathcal{D}}}}_{A}[1/p]$. It is clear that $C$ is equipped with a natural action $\iota_C$ by ${\mathcal{O}}_D$, and $\phi_{A}$ satisfies conditions (iii) and (iv) for the moduli space $Y'_{\ttT}$. Conversely, if $C$ exists, the conditions (iii) and (iv) imply that ${\tilde{{\mathcal{D}}}}_{C,\tilde\tau}^{\circ}$ has to coincide with $M^\circ_{\tilde\tau}$. Therefore, $C$ is uniquely determined by $A$. We finally remark that, by construction, $\tilde \calD_{C, \tilde \tau}^\circ / \tilde \calD_{A, \tilde \tau}^\circ$ is isomorphic to $k$ if $\tilde \tau \in \tilde \Delta(\ttT)^+$ and trivial otherwise. We now construct the abelian variety $B$ and the isogeny $\phi_B:B{\rightarrow}C$. Similarly to the construction for $C$, we will first define a $W(k)$-lattice $N^\circ=\bigoplus_{\tilde{\tau}\in \Sigma_{E,\infty}}N^\circ_{\tilde{\tau}}\subseteq {\tilde{{\mathcal{D}}}}_{C}^{\circ}$, with $N^\circ_{\tilde\tau}={\tilde{{\mathcal{D}}}}^{\circ}_{C,\tilde{\tau}}$ unless $\tilde\tau\in \tilde\Delta(\ttT)^-$. In the exceptional case, we put $N^\circ_{\tilde{\tau}}={F}_{A,{\mathrm{es}}}^n({\tilde{{\mathcal{D}}}}^\circ_{C,\sigma^{-n}\tilde{\tau}})$, where $n$ is the positive integer given by Lemma \[L:distance to T’\]. Here, we view ${\tilde{{\mathcal{D}}}}^{\circ}_{C,\sigma^{-n}\tilde{\tau}}$ as a lattice of ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-n}\tilde\tau}[1/p]$ so that ${F}_{A,{\mathrm{es}}}^n({\tilde{{\mathcal{D}}}}^\circ_{C,\sigma^{-n}\tilde{\tau}})$ makes sense. Note once again that, if $\tilde{\tau}\in \tilde \Delta(\ttT)^- \cap \ttT_E$, then $n$ equals to $n_{\tau}$ defined in \[S:partial-Hasse\], and we have $ N^\circ_{\tilde{\tau}}=\tilde{\omega}^\circ_{C, \tilde{\tau}}\simeq \tilde{\omega}^\circ_{A, \tilde{\tau}}, $ since $h_{\tilde{\tau}}(A)$ vanishes. We now check that $N^\circ$ is stable under $F_C$ and $V_C$, i.e. $F_{C}(N^\circ_{\sigma^{-1}\tilde\tau})\subseteq N^\circ_{\tilde{\tau}}$ and $V_C(N^\circ_{\tilde\tau})\subseteq N^\circ_{\sigma^{-1}\tilde\tau}$ for all $\tilde\tau\in \Sigma_{E,\infty}$. The same arguments for $M$ above work verbatim in this case (with $\tilde \Delta(\ttT)^-$ in places of $\tilde \Delta(\ttT)^+$). Again, we point out that, by construction, $\tilde \calD_{C, \tilde \tau}^\circ / \tilde \calD_{B, \tilde \tau}^\circ$ is isomorphic to $k$ if $\tilde \tau \in \tilde \Delta(\ttT)^-$ and trivial otherwise. Therefore, $N=(N^\circ)^{\oplus 2}$ is a Dieudonné module such that the inclusions $p{\tilde{{\mathcal{D}}}}_{C}\subseteq N\subseteq {\tilde{{\mathcal{D}}}}_C$ respect the Frobenius and Verschiebung actions. In particular, the Dieudonné submodule $N/p{\tilde{{\mathcal{D}}}}_{C}\subset {\tilde{{\mathcal{D}}}}_{C}/p{\tilde{{\mathcal{D}}}}_{C}$ is the covariant Dieudonné module of a closed finite subgroup scheme $H\subset C[p]$ stable under the action of ${\mathcal{O}}_D$. We put $B=C/H$, and define $\phi_{B}:B{\rightarrow}C$ to be the isogeny such that the composite $C{\rightarrow}B=C/H{\xrightarrow}{\phi_B} C$ is the multiplication by $p$. Then the induced morphism $\phi_{B,*}:{\tilde{{\mathcal{D}}}}_{B}{\rightarrow}{\tilde{{\mathcal{D}}}}_{C}$ is identified with the natural inclusion $N\subseteq {\tilde{{\mathcal{D}}}}_{C}$. It is clear that $B$ is equipped with a natural action by ${\mathcal{O}}_D$, and the condition (v) for the moduli space $Y'_{\ttT}$ is satisfied. Conversely, if the abelian variety $B$ exists, then condition (v) implies that ${\tilde{{\mathcal{D}}}}_{B}^\circ$ has to be $N^\circ$ defined above. This means that $B$ is uniquely determined by $C$, thus by $A$. To see condition (ix) of the moduli space $Y'_{\ttT}$, we consider the quasi-isogeny: $$\lambda_B: B{\xrightarrow}{\phi_B}C\xleftarrow{\phi_A}A{\xrightarrow}{\lambda_A}A^\vee \xleftarrow{\phi_{A}^\vee}C^\vee{\xrightarrow}{\phi_B^\vee}B^\vee.$$ We have to show that $\lambda_B$ is a genuine isogeny, and verify that it satisfies conditions (b2) and (b3) in Theorem \[T:unitary-shimura-variety-representability\] for the Shimura variety ${\mathbf{Sh}}_{K'_\ttT}(G'_{\tilde \ttS(\ttT)})$. It is equivalent to proving that, when viewing ${\tilde{{\mathcal{D}}}}_{B, \tilde \tau}^\circ$ as a $W(k)$-lattice of ${\tilde{{\mathcal{D}}}}_{A, \tilde \tau}^\circ[1/p]$ via the quasi-isogeny $B{\xrightarrow}{\phi_B}C\xleftarrow{\phi_A}A$, the perfect alternating pairing $$\langle\ ,\ \rangle_{\lambda_A, \tilde \tau}: {\tilde{{\mathcal{D}}}}^\circ_{A, \tilde \tau}[1/p]\times {\tilde{{\mathcal{D}}}}^\circ_{A, \tilde \tau^c}[1/p]{\rightarrow}W(k)[1/p]$$ for $\tilde \tau \in \Sigma_{E, \infty/\gothp}$ induces a perfect pairing of ${\tilde{{\mathcal{D}}}}_{B,\tilde{\tau}}^\circ\times {\tilde{{\mathcal{D}}}}_{B,\tilde\tau^c}^\circ{\rightarrow}W(k)$ if $\gothp$ is not of type $\beta^\sharp$ for $\ttS(\ttT)$, and induces an inclusion $\tilde \calD^\circ_{B, \tilde \tau^c} \subset \tilde \calD^{\circ, \vee}_{B, \tilde \tau}$ with quotient equal to $k$ if $\gothp$ is of type $\beta^\sharp$ for $\ttS(\ttT)$. We discuss case by case. - if $\gothp$ is of type $\beta^\sharp$ for $\ttS$, then both $\phi_A$ and $\phi_B$ induce isomorphisms on the $\gothp$-divisible groups and the statement is clear in this case. - If $\gothp$ is of Case $\beta2$, $\tilde\Delta(\ttT)^+_{/\gothp}=\emptyset$. By the construction of $B$, we have ${\tilde{{\mathcal{D}}}}_{B,\tilde\tau}^{\circ}={\tilde{{\mathcal{D}}}}_{A,\tilde\tau}^{\circ}$ unless $\tilde\tau\in \tilde\Delta(\ttT)^-_{/\gothp}$; in the latter case, ${\tilde{{\mathcal{D}}}}_{B,\tilde\tau}^{\circ}=\tilde{F}^n_{{\mathrm{es}},\tilde\tau}({\tilde{{\mathcal{D}}}}_{A,\sigma^{-n}\tilde\tau}^\circ)$ is a submodule of ${\tilde{{\mathcal{D}}}}_{A,\tilde\tau}^\circ$ with quotient isomorphic to $k$. Note that $\Sigma_{E, \infty/\gothp} = \tilde \Delta(\ttT)^-_{/\gothp} \coprod (\tilde \Delta(\ttT)^{-}_{/\gothp})^c$; this then implies that the pairing $\langle \ ,\ \rangle_{\lambda_A, \tilde \tau}$ induces an inclusion $\tilde \calD^\circ_{B, \tilde \tau^c} \subset \tilde \calD^{\circ, \vee}_{B, \tilde \tau}$ with quotient equal to $k$. - In all other cases, we have $\tilde \Delta(\ttT)^+_{/\gothp} = (\tilde \Delta(\ttT)^-_{/\gothp})^c$. So $$\label{E:Dieud-B-2} {\tilde{{\mathcal{D}}}}_{B,\tilde\tau}^\circ= \begin{cases} {\tilde{{\mathcal{D}}}}_{A,\tilde{\tau}}^\circ &\text{if }\tilde{\tau}\notin (\tilde\Delta(\ttT)^+_{/\gothp}\cup \tilde\Delta(\ttT)^-_{/\gothp});\\ p^{-1}{F}_{A,{\mathrm{es}}}^n({\tilde{{\mathcal{D}}}}_{A,\sigma^{-n}\tilde\tau}^\circ) &\text{if }\tilde\tau\in \tilde\Delta(\ttT)^+_{/\gothp};\\ {F}_{A,{\mathrm{es}}}^n({\tilde{{\mathcal{D}}}}_{A,\sigma^{-n}\tilde\tau}^\circ) &\text{if }\tilde{\tau}\in \tilde\Delta(\ttT)^-_{/\gothp}. \end{cases}$$ It is clear that $\langle\ ,\ \rangle_{\lambda_A}$ induces a perfect pairing on ${\tilde{{\mathcal{D}}}}_{B,\tilde{\tau}}^\circ\times{\tilde{{\mathcal{D}}}}_{B,\tilde\tau^c}^\circ$ if $\tilde\tau\notin (\tilde\Delta(\ttT)^+_{/\gothp}\cup \tilde\Delta(\ttT)^-_{/\gothp})$. If $\tilde\tau\in (\tilde\Delta(\ttT)^+_{/\gothp}\cup \tilde\Delta(\ttT)^-_{/\gothp})$, the perfect duality between $\tilde \calD_{B, \tilde \tau}^\circ$ and $\tilde \calD_{B, \tilde \tau^c}^\circ$ follows from the equality $$\langle p^{-1} F_{A,{\mathrm{es}}}^n u, F_{A,{\mathrm{es}}}^n v\rangle_{\lambda_A, \tilde \tau} = \langle u, v \rangle_{\lambda_A, \sigma^{-n} \tilde \tau}^{\sigma^n},$$ for all $u\in {\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-n}\tilde\tau}$ and $v\in {\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-n}\tilde\tau^c}$. This completes the verification of condition (viii) of the moduli space $Y'_\ttT$ and conditions (b2) and (b3) in Theorem \[T:unitary-shimura-variety-representability\] for ${\mathbf{Sh}}_{K'_\ttT}(G'_{\tilde \ttS(\ttT)})$. It is also clear that $\lambda_B$ induces the involution $*_{\ttS(\ttT)}$ on ${\mathcal{O}}_D={\mathcal{O}}_{D_{\ttS(\ttT)}}$. We now check that the abelian variety $B$ has the correct signature required by the moduli space ${\mathbf{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})$. For convenience of future reference, we put this into the following separated lemma. \[L:dimension count\] In the setup above, that is, knowing $$\dim \operatorname{Coker}(\phi_{A, *, \tilde \tau}) = \delta_{\tilde \Delta(\ttT)^+}(\tilde \tau)\quad \dim \operatorname{Coker}(\phi_{B, *, \tilde \tau}) = \delta_{\tilde \Delta(\ttT)^-}(\tilde \tau),$$ where $\delta_\bullet(?)$ is $1$ if $? \in \bullet$ and is $0$ if $? \notin \bullet$, we have $\dim \omega_{B^\vee/k, \tilde \tau}^\circ = 2- s_{\ttT, \tilde \tau}$ for all $\tilde \tau \in \Sigma_{E, \infty}$ if and only if $\dim \omega_{A^\vee/k, \tilde \tau}^\circ = 2- s_{ \tilde \tau}$ for all $\tilde \tau \in \Sigma_{E, \infty}$, with the numbers $s_{\ttT, \tilde \tau}$ defined as in Subsection \[S:tilde S(T)\]. This is a simple dimension count. We prove the sufficiency; the necessity follows by reversing the argument. Using the signature condition for the Shimura variety $X'_\ttT$, we have $$s_{\tilde \tau} = \dim_k \big( \tilde \calD^\circ_{A, \tilde \tau} \big/ V(\tilde\calD^\circ_{A, \sigma\tilde \tau} ) \big).$$ Comparing this with the abelian variety $B$, we have $$\dim_k\frac{\tilde \calD^\circ_{B, \tilde \tau} }{ V(\tilde\calD^\circ_{B, \sigma\tilde \tau} )} = \dim_k \frac{\tilde \calD^\circ_{A, \tilde \tau}}{ V(\tilde\calD^\circ_{A, \sigma\tilde \tau} )} - \big( \dim_k \frac{ \tilde \calD^\circ_{C, \tilde \tau} }{ \tilde\calD^\circ_{B,\tilde \tau}} -\dim_k \frac{ \tilde \calD^\circ_{C, \tilde \tau} }{ \tilde\calD^\circ_{A,\tilde \tau}} \big) + \big( \dim_k \frac{ \tilde \calD^\circ_{C, \sigma\tilde \tau} }{ \tilde\calD^\circ_{B,\sigma \tilde\tau}} -\dim_k \frac{ \tilde \calD^\circ_{C, \sigma\tilde \tau} }{ \tilde\calD^\circ_{A,\sigma\tilde \tau}} \big);$$ here we used the fact that the quotient $\tilde \calD^\circ_{C, \sigma\tilde \tau} / \tilde \calD_{B, \sigma \tilde \tau}^\circ$ has the same dimension as $V(\tilde \calD^\circ_{C, \sigma\tilde \tau}) / V(\tilde \calD_{B, \sigma \tilde \tau}^\circ)$ and the same for $A$ in place of $B$ because $V$ is equivariant. Using our construction of the abelian varieties $B$ and $C$, we deduce $$\label{E:dimension count} \dim_k \big( \tilde \calD^\circ_{B, \tilde \tau} \big/ V(\tilde\calD^\circ_{B, \sigma\tilde \tau} ) \big)= s_{\tilde \tau} - \big( \delta_{\tilde \Delta(\ttT)^-}(\tilde \tau) - \delta_{\tilde \Delta(\ttT)^+}(\tilde \tau) \big) + \big(\delta_{\tilde \Delta(\ttT)^-}(\sigma\tilde \tau) - \delta_{\tilde \Delta(\ttT)^+}(\sigma\tilde \tau)\big).$$ Using the definition of $\tilde \Delta(\ttT)^\pm$, one checks case-by-case that the expression is equal to $s_{ \ttT, \tilde \tau}$. We will only indicate the proof when $\tilde \tau \in \Sigma_{E, \infty/\gothp}$ for $\gothp$ in Case $\alpha1$, and leave the other cases as an exercise for interested readers. Indeed, under the notation from Subsections  \[S:Delta-pm\], when $\gothp\in \Sigma_p$ is of type $\alpha1$, $\tilde \Delta(\ttT)_{/\gothp}^\pm = \coprod_i \tilde C_i^\pm$. Then $$\delta_{\tilde \Delta(\ttT)^+}(\tilde \tau)-\delta_{\tilde \Delta(\ttT)^+}(\sigma\tilde \tau) =\left\{ \begin{array}{ll} 1&\textrm{if } \tilde \tau \textrm{ is one of }\sigma^{-a_1}\tilde \tau_i^c, \sigma^{-a_3}\tilde \tau_i^c, \dots; \\ -1&\textrm{if } \tilde \tau \textrm{ is one of }\sigma^{-a_2}\tilde \tau_i^c, \sigma^{-a_4}\tilde \tau_i^c, \dots; \\ 0&\textrm{otherwise}; \end{array} \right.$$ $$\textrm{and}\ \delta_{\tilde \Delta(\ttT)^-}(\tilde \tau)-\delta_{\tilde \Delta(\ttT)^-}(\sigma\tilde \tau) =\left\{ \begin{array}{ll} 1&\textrm{if } \tilde \tau \textrm{ is one of }\sigma^{-a_1}\tilde \tau_i, \sigma^{-a_3}\tilde \tau_i, \dots; \\ -1&\textrm{if } \tilde \tau \textrm{ is one of }\sigma^{-a_2}\tilde \tau_i, \sigma^{-a_4}\tilde \tau_i, \dots; \\ 0&\textrm{otherwise}. \end{array} \right.$$ Putting these two formulas together and using the notation from Subsection \[S:tilde S(T)\], we have $$\big(\delta_{\tilde \Delta(\ttT)^+}(\tilde \tau) -\delta_{\tilde \Delta(\ttT)^+}(\sigma\tilde \tau) \big) -\big( \delta_{\tilde \Delta(\ttT)^-}(\tilde \tau) -\delta_{\tilde \Delta(\ttT)^-}(\sigma \tilde \tau)\big) = \delta_{\tilde \ttT'}(\tilde \tau^c)-\delta_{\tilde \ttT'}(\tilde \tau).$$ This implies that is equal to $s_{ \ttT, \tilde \tau}$, and concludes the proof of Lemma \[L:dimension count\]. We now continue our proof of Proposition \[P:Y\_S=X\_S\]. To fulfill condition (vi) of the moduli space $Y'_\ttT$, the tame level structure on $B$ is chosen and determined as the composite $$\beta^p_{K'^p_\ttT}: \widehat \Lambda_{\ttS(\ttT)}^{(p)} \xrightarrow{\theta_\ttT^{-1}} \widehat \Lambda_{\ttS}^{(p)} \xrightarrow{\alpha} T^{(p)}(A) \xrightarrow{T^{(p)}(\phi_A)} T^{(p)}(C) \xrightarrow{T^{(p)}(\phi_B)^{-1}} T^{(p)}(B),$$ where both $T ^{(p)}(\phi_A)$ and $T^{(p)}(\phi_B)$ are isomorphisms because $\phi_A$ and $\phi_B$ are $p$-isogenies. It remains to show that there exists a unique collection of subgroups $\beta_{p}$ satisfying \[T:unitary-shimura-variety-representability\](c2) for ${\mathbf{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})$ and properties (vii) and (viii) of $Y'_{\ttT}$. So the corresponding prime $\gothp \in \Sigma_p$ is either of type $\alpha^\sharp$ for $\ttS$ or in the Case $\alpha2$ of Subsection \[S:quaternion-data-T\]. In the former case, we have $\ttT_{/\gothp} = \emptyset$, which forces $\Delta(\ttT)^\pm_{/\gothp} = \emptyset$ by definition. So the induced morphisms $\phi_{A,\gothp}: A[\gothp^\infty] \to C[\gothp^\infty]$ and $\phi_{B,\gothp}: B[\gothp^\infty] \to C[\gothp^\infty]$ are isomorphisms. Now, condition (vii) of the moduli space $Y'_\ttT$ determines that the level structure $\beta_\gothp$ is taken to be $\phi_{B,\gothp}^{-1}\big( \phi_{A, \gothp}(\alpha_\gothp)\big)$. If $\gothp$ is in Case $\alpha 2$ of Subsection \[S:quaternion-data-T\], the prime $\gothp$ splits into two primes $\gothq$ and $\bar\gothq$ in $E$. Using the polarization $\lambda_B$, we just need to show that there exists a unique subgroup $H_{\gothq}\subseteq B[\gothq]$ satisfying condition (vii). Since $s_{\ttT, \tilde \tau} = 0$ or $2$ for $\tilde \tau \in \Sigma_{E,\infty/\gothp}$, both $ F_{B,{\mathrm{es}},\tilde \tau}$ and $ V_{B,{\mathrm{es}}, \tilde \tau}$ are isomorphisms. We define a one-dimensional $k$-vector subspace ${\mathcal{D}}_{H_{\gothq}}^\circ\subseteq {\tilde{{\mathcal{D}}}}_{B,\tilde\tau}^\circ/p{\tilde{{\mathcal{D}}}}_{B,\tilde\tau}^\circ$ for each $\tilde\tau\in \Sigma_{E,\infty/\gothq}$ as follows: - If $\tilde\tau\in \tilde\Delta(\ttT)^-_{/\gothp}$, then $\tilde \calD^\circ_{B, \tilde \tau}$ is contained in $ \tilde \calD^\circ_{C, \tilde \tau}\cong\tilde \calD^\circ_{A, \tilde \tau}$ with quotient isomorphic to $k$; put ${\mathcal{D}}^{\circ}_{H_{\gothq}}=p{\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau}/p{\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}$. - If $\tilde\tau\notin \tilde\Delta(\ttT)^{-}_{/\gothp}$, let $n\in \NN$ be the least positive integer such that $\sigma^{-n}\tilde\tau\in \tilde\Delta(\ttT)^-_{/\gothp}$ (such $n$ exists because $\tilde \tau_0$ in Subsection \[S:tilde S(T)\] belongs to $\tilde \Delta(\ttT)^-_{/\gothp}$); put ${\mathcal{D}}^{\circ}_{H_{\gothq},\tilde\tau}=F^{n}_{B,{\mathrm{es}}}({\mathcal{D}}^{\circ}_{H_{\gothq},\sigma^{-n}\tilde\tau})$. Put ${\mathcal{D}}_{H_{\gothq}}=\bigoplus_{\tilde\tau\in \Sigma_{E,\infty/\gothq}}{\mathcal{D}}^{\circ,\oplus 2}_{H_{\gothq}, \tilde\tau}$. Using the vanishing of the partial Hasse invariants $\{h_{\tilde\tau}(A): \tilde\tau\in {\ttT}_{E,\infty/\gothp} \}$, one checks easily that ${\mathcal{D}}_{H_{\gothq}}\subseteq {\mathcal{D}}_{B[\gothq]}$ is a Dieudonné submodule. We define $H_{\gothq}\subseteq B[\gothq]$ as the finite subgroup scheme corresponding to ${\mathcal{D}}_{H_{\gothq}}$ by covariant Dieudonné theory. Then ${\mathcal{D}}_{H_{\gothq}}$ is canonically identified with the kernel of the induced map $$\phi_{\gothq,*} \colon {\mathcal{D}}_{B}=H^{{\mathrm{dR}}}_1(B/k){\rightarrow}{\mathcal{D}}_{B/H_{\gothq}}=H^{{\mathrm{dR}}}_1((B/H_{\gothq})/k).$$ Therefore, $H_{\gothq}$ satisfies condition (viii) of the moduli space $Y'_\ttT$. This shows the existence of $H_{\gothq}$. For the uniqueness, the condition (viii) forces the choice of ${\mathcal{D}}_{H_{\gothq}, \tilde{\tau}}^{\circ}$ for $\tilde\tau\in \tilde{\Delta}(\ttT)^-_{/\gothp}$ and the stability under $F_B$ and $V_B$ forces the choice at other $\tilde \tau$’s. This concludes the proof that $Y'_{\ttT}{\rightarrow}X'_{\ttT}$ induces a bijection on closed points. \[L:Y\_T-X\_T-tangent\] The map $\eta_1: Y'_{\ttT}{\rightarrow}X'_{\ttT}$ induces an isomorphism of tangent spaces at every closed point. Let $y=(A, \iota_A,\lambda_A, \alpha_{K'}, B, \lambda_B, \iota_B, \beta_{K'_\ttT}, C, \iota_C; \phi_A, \phi_B)$ be a closed point of $Y'_{\ttT}$ with values in a perfect field $k$, and $x=(A, \iota_A,\lambda_A, \alpha_{K'})$ be its image in $X'_{\ttT}$. We have to show that $Y'_{\ttT}{\rightarrow}X'_{\ttT}$ induces an isomorphism of $k$-vector spaces between tangent spaces: $T_{Y'_{\ttT}, y}{\xrightarrow}{\cong}T_{X'_{\ttT}, x}$. Set $\II=\operatorname{Spec}(k[\epsilon]/\epsilon^2)$. By deformation theory, $T_{X'_{\ttT},x}$ is identified with the $\II$-valued points $x_{\II}=(A_{\II}, \iota_{A,\II}, \lambda_{A,\II}, \alpha_{K',\II})$ of $X'_{\ttT}$ with reduction $x\in X'_{\ttT}(k)$ modulo $\epsilon$. In the proof of Proposition \[Prop:smoothness\], we have seen that giving an $x_{\II}$ is equivalent to giving, for each $\tilde\tau\in \Sigma_{E,\infty}$, a direct factor $\omega^{\circ}_{A^\vee,\II,\tilde\tau}\subseteq H^{{\mathrm{cris}}}_1(A/k)_{\II,\tilde\tau}^\circ$ that lifts $\omega_{A^\vee,\tilde\tau}\subseteq H^{{\mathrm{dR}}}_1(A/k)^{\circ}$ and satisfies the following properties: - If $\tilde\tau\in \Sigma_{E,\infty/\gothp}$ with $\gothp$ not of type $\beta^\sharp$ for $\ttS$, then $\omega_{A^\vee,\II,\tilde\tau}^{\circ}$ and $\omega_{A^\vee,\II,\tilde\tau^c}^\circ$ are orthognal complements of each other under the perfect pairing $$H^{{\mathrm{cris}}}_1(A/k)_{\II,\tilde\tau}^\circ\times H^{{\mathrm{cris}}}_1(A/k)_{\II,\tilde\tau^c}^\circ {\rightarrow}k[\epsilon]/\epsilon^2$$ induced by the polarization $\lambda_A$. - If $\tilde\tau \in \tilde \ttS_\infty$, then $\omega_{A^\vee,\II,\tilde\tau}^\circ=0$ and $\omega_{A^\vee,\II,\tilde\tau^c}^\circ = H^{{\mathrm{cris}}}_1(A/k)^{\circ}_{\II,\tilde\tau^c}$. - If $\tilde\tau$ restricts to $\tau\in \ttT$, then $\omega_{A^\vee,\II,\tilde\tau}^{\circ}$ has to be $F_{A,{\mathrm{es}}}^{n_{\tau}}(H^{\mathrm{cris}}_1(A^{(p^{n_{\tau}})}/k)_{\II,\tilde\tau}^\circ)$, where $n_\tau$ is as introduced in Subsection \[S:partial-Hasse\] and $F_{A,{\mathrm{es}}}^{n_\tau}$ on the crystalline homology are defined in the same way as $F_{A, {\mathrm{es}}}^{n_\tau}$ on the de Rham homology as in Notation \[N:essential frobenius and verschiebung\]. Since we are in characteristic $p$, we have $F_{A,{\mathrm{es}}}^{n_{\tau}}(H^{\mathrm{cris}}_1(A^{(p^{n_{\tau}})}/k)_{\II,\tilde\tau}^\circ)=\omega_{A^\vee,\tilde\tau}\otimes k[\epsilon]/\epsilon^2$. Note also that, the crystal nature of $H^{{\mathrm{cris}}}_1(A/k)$ implies that there is a canonical isomorphism $$H^{{\mathrm{cris}}}_1(A/k)_{\II}\simeq H^{{\mathrm{dR}}}_1(A/k)\otimes_k k[\epsilon]/\epsilon^2.$$ We have to show that, given such an $x_{\II}$, or equivalently given the liftings $\omega_{A^\vee, \II,\tilde\tau}^\circ$ as above, there exists a unique tuple $(B_{\II}, \lambda_{B,\II}, \iota_{B,\II},\beta_{K'_\ttT, \II}, C_{\II}, \iota_{C, \II}; \phi_{A,\II},\phi_{B,\II}) $ over $\II$ deforming $(B,\lambda_B, \iota_B, \beta_{K'_\ttT}, C, \iota_C; \phi_A, \phi_B)$ such that $(A_{\II}, \iota_{A,\II}, \lambda_{A,\II}, \alpha_{K',\II}, B_{\II}, \lambda_{B,\II}, \iota_{B,\II},\beta_{K'_\ttT, \II}, C_{\II}, \iota_{C, \II}; \phi_{A,\II},\phi_{B,\II})$ is an $\II$-valued point of $Y'_{\ttT}$. We start with $C_{\II}$. To show its existence, it suffices to construct, for each $\tilde{\tau}\in \Sigma_{E,\infty}$, a direct factor $\omega_{C^\vee,\II,\tilde\tau}^\circ\subseteq H^{{\mathrm{cris}}}_1(C/k)_{\II,\tilde\tau}^\circ$ that lifts $\omega^\circ_{C^\vee, \tilde\tau}\subseteq{\mathcal{D}}_{C,\tilde\tau}^\circ\cong H^{{\mathrm{dR}}}_1(C/k)_{\tilde\tau}^\circ$. - When neither $\tilde \tau$ nor $\sigma \tilde \tau $ belongs to $ \tilde \Delta(\ttT)^+$, $\phi_{A, *, ?}: H^{{\mathrm{dR}}}_{1}(A/k)^\circ_{?}{\xrightarrow}{\sim}H^{{\mathrm{dR}}}_1(C/k)^\circ_{?}$ is an isomorphism for $?=\tilde\tau, \sigma\tilde\tau$. We take $\omega_{C^\vee,\II,\tilde\tau}^\circ\subseteq H^{{\mathrm{cris}}}_1(C/k)_{\II,\tilde\tau}^\circ$ to be the image of $\omega_{A^\vee, \II,\tilde\tau}\subseteq H^{{\mathrm{cris}}}_1(A/k)_{\II,\tilde\tau}^\circ$ under the induced morphism $\phi^{\mathrm{cris}}_{A, *, \tilde \tau}$ on the crystalline homology. - When either one of $\tilde\tau$ and $\sigma \tilde \tau$ belongs to $\tilde \Delta(\ttT)^+$, an easy dimension count argument similar to Lemma \[L:dimension count\] (using Lemma \[L:property of Delta\]) shows that $\omega_{C^\vee, \tilde \tau}^\circ$ is either $0$ or of rank $2$; there is a unique obvious such lift $\omega_{C^\vee,\II,\tilde\tau}^\circ$. This finishes the construction of $\omega^\circ_{C^\vee,\II,\tilde\tau}$ for all $\tilde\tau$, hence one gets a deformation $C_{\II}$ of $C$. It is clear that the map $\phi^{\mathrm{cris}}_{A,*}: H^{{\mathrm{cris}}}_1(A/k)^\circ_{\II,\tilde\tau}{\rightarrow}H^{{\mathrm{cris}}}_1(C/k)^\circ_{\II,\tilde\tau}$ sending $\omega^\circ_{A^\vee,\II,\tilde\tau}$ to $\omega^\circ_{C^\vee,\II,\tilde\tau}$. Hence, $\phi_{A}$ deforms to an isogeny of abelian schemes $\phi_{A,\II}:A_{\II}{\rightarrow}C_{\II}$ by [@lan 2.1.6.9]. We check now that $\phi_{A_\II}$ satisfies condition (iv) of the moduli space $Y'_{\ttT}$. We note that the map $\phi_{A_{\II},*}: H^{{\mathrm{dR}}}_1(A_{\II}/\II){\rightarrow}H^{{\mathrm{dR}}}_1(C_{\II}/\II)$ is canonically identified with $\phi_{A,*}^{{\mathrm{cris}}}: H^{\mathrm{cris}}_1(A/k)_{\II}\simeq H^{{\mathrm{cris}}}_1(C /k)_\II$ by crystalline theory, which is in turn isomorphic to the base change of $\phi_{A,*}:H^{\mathrm{dR}}_1(A/k){\rightarrow}H^{{\mathrm{dR}}}_1(C/k)$ via $k{\hookrightarrow}k[\epsilon]/\epsilon^2$. Let $\tilde\tau\in \tilde \Delta(\ttT)^+$. Since the Frobenius on $k[\epsilon]/\epsilon^2$ factors as $$k[\epsilon]/\epsilon^2\twoheadrightarrow k{\xrightarrow}{x\mapsto x^p} k{\hookrightarrow}k[\epsilon]/\epsilon^2,$$ we see that $$F_{A,{\mathrm{es}}}^n(H^{{\mathrm{dR}}}_1(A_{\II}^{(p^n)}/\II)^\circ_{\tilde\tau}) =F_{A,{\mathrm{es}}}^n(H^{{\mathrm{dR}}}_1(A^{(p^n)}/k)^\circ_{\tilde\tau})\otimes_k k[\epsilon]/\epsilon^2.$$ Hence, the kernel of $\phi_{A_{\II},*,\tilde\tau}: H^{{\mathrm{dR}}}_1(A_{\II}/\II)^\circ_{\tilde\tau}{\rightarrow}H^{\mathrm{dR}}_1(C_{\II}/\II)^\circ_{\tilde\tau}$ coincides with $F_{A,{\mathrm{es}}}^n(H^{{\mathrm{dR}}}_1(A_{\II}^{(p^n)}/\II)^\circ_{\tilde\tau})$, since it is the case after reduction modulo $\epsilon$. This shows that $\phi_{A_\II}$ satisfies the condition (iv). Conversely, it is clear that, if $C_{\II}$ and $\phi_{A_\II}$ satisfy condition (iv), then they have to be of the form as above. We show now that there exists a unique deformation $(B_{\II}, \phi_{B_\II})$ over $\II$ of $(B,\phi_B)$ satisfying condition (vi) of the moduli space $Y'_\ttT$. To construct $B_{\II}$, one has to specify, for each $\tilde\tau\in \Sigma_{E,\infty}$ a subbundle $\omega_{B^\vee, \II, \tilde\tau}^\circ\subseteq H^{{\mathrm{cris}}}_1(B/k)_{\II,\tilde\tau}^\circ$ lifting $\omega^\circ_{B^\vee,\tilde\tau}\subseteq H^{{\mathrm{dR}}}_1(B/k)^\circ_{\tilde\tau}$. Similar to the discussion above, - If neither $\tilde \tau$ nor $\sigma \tilde \tau$ belong to $\tilde \Delta(\ttT)^-$, then $\phi_{B, *, ?}: H^{{\mathrm{dR}}}_{1}(B/k)^\circ_{?}{\xrightarrow}{\sim}H^{{\mathrm{dR}}}_1(C/k)^\circ_{?}$ is an isomorphism for $?=\tilde\tau, \sigma\tilde\tau$. We take $\omega_{B^\vee,\II,\tilde\tau}^\circ\subseteq H^{{\mathrm{cris}}}_1(B/k)_{\II,\tilde\tau}^\circ$ to be the image of $\omega_{C^\vee, \II,\tilde\tau}\subseteq H^{{\mathrm{cris}}}_1(C/k)_{\II,\tilde\tau}^\circ$ under the induced morphism $\phi_{B, *, \tilde \tau}^{-1}$ on the crystalline homology. - If at least one of $\tilde \tau$ and $\sigma \tilde \tau$ belongs to $\tilde \Delta(\ttT)^-$, then an easy dimension count argument similar to Lemma \[L:dimension count\] (using Lemma \[L:property of Delta\]) shows that $\omega^\circ_{B^\vee, \tilde \tau}$ is either $0$ or of rank $2$; there is a unique obvious such lift $\omega^\circ_{B^\vee, \II, \tilde \tau}$. This defines $\omega_{B^\vee,\II, \tilde\tau}^\circ$ for all $\tilde\tau\in \Sigma_{E,\infty}$. Hence, one gets a deformation $B_{\II}$ of $B$ over $k[\epsilon]/\epsilon^2$. It is immediate from the construction that the action of ${\mathcal{O}}_D$ lifts to $B_{\II}$, and $\phi^{\mathrm{cris}}_{B,*,\tilde\tau}: H^{{\mathrm{cris}}}_1(B/k)^{\circ}_{\II, \tilde\tau}{\rightarrow}H^{{\mathrm{cris}}}_1(C/k)^\circ_{\II,\tilde\tau}$ sends $\omega_{B^\vee,\II, \tilde\tau}^\circ$ to $ \omega_{C^\vee, \II,\tilde\tau}^\circ$ for all $\tilde\tau\in \Sigma_{E,\infty}$. Hence, $\phi_{B}:B{\rightarrow}C$ deforms to an isogeny $\phi_{B_\II}: B_{\II}{\rightarrow}C_{\II}$. In the same way as for $\phi_{A_{\II}}$, we prove that $\phi_{B_\II}$ satisfies condition (v) of the moduli space $Y'_{\ttT}$, and conversely the condition (v) determines $B_{\II}$ uniquely. Let $\langle\ ,\ \rangle_{\lambda_{B}}:H^{{\mathrm{cris}}}_1(B/k)_{\II}^\circ\times H^{\mathrm{cris}}_1(B/k)^\circ_{\II}{\rightarrow}k[\epsilon]/\epsilon^2$ be the pairing induced by the polarization $\lambda_B$. To prove that $\lambda_{B}$ deforms (necessarily uniquely) to a polarization $\lambda_{B_\II}$ on $B_{\II}$, it suffices to check that $\langle\ ,\ \rangle_{\lambda_B}^{\mathrm{cris}}$ vanishes on $\omega_{B, \II, \tilde\tau}^\circ\times \omega_{B,\II, \tilde\tau^c}^\circ$ for all $\tilde\tau\in \Sigma_{E,\infty}$ (cf. [@lan 2.1.6.9, 2.2.2.2, 2.2.2.6]): - If $\tau=\tilde\tau|_F$ lies in $\ttS(\ttT)_\infty$, this is trivial, because one of $\omega^{\circ}_{B^\vee, \II, \tilde\tau}$ and $\omega_{B^\vee, \II, \tilde\tau^c}^\circ$ equals $0$ and the other one equals to $H^{{\mathrm{cris}}}_1(B/k)_{\II,\tilde\tau}^\circ$ by construction. - If $\tau=\tilde\tau|_F$ is not in $\ttS(\ttT)_\infty$, then the natural isomorphism $H^{\circ}_1(B/k)^{\mathrm{cris}}_{\II,\star}\cong H^{\mathrm{cris}}_1(A/k)^\circ_{\II, \star}$ sends $\omega^\circ_{B^\vee, \II, \star}$ to $\omega^\circ_{A^\vee,\II, \star}$ for $\star=\tilde\tau, \tilde\tau^c$. The vanishing of $\langle \ ,\ \rangle_{\lambda_B}$ on $\omega^\circ_{B^\vee,\II,\tilde\tau}\times \omega_{B^\vee, \II, \tilde\tau^c}^\circ$ follows from the similar statement with $B$ replaced by $A$. Therefore, we see that $\lambda_B$ deforms to a polarization $\lambda_{B_{\II}}$ on $B_\II$. Since $\lambda_{B_\II,*}^{\mathrm{dR}}:H^{{\mathrm{dR}}}_1(B/\II){\rightarrow}H^{{\mathrm{dR}}}_1(B^\vee/\II)$ is canonically identified with $\lambda_{B,*}^{\mathrm{cris}}: H^{{\mathrm{cris}}}_1(B/k)_{\II}{\rightarrow}H^{{\mathrm{cris}}}_1(B/k)_{\II}$, which is in turn identified with the base change of $\lambda^{\mathrm{dR}}_{B,*}$ via $k{\hookrightarrow}k[\epsilon ]/\epsilon^2$, it is clear that condition (ii) regarding the polarization is preserved by the deformation $\lambda_{B_{\II}}$. It remains to prove that $\beta_{K'_\ttT}$ deforms to $\beta_{K'_\ttT, \II}$. The deformation of the tame level structure is automatic; the deformation of the subgroup at $p$-adic places of type $\alpha^\sharp$ and $\alpha2$ is also unique, by the same argument as in Theorem \[T:unitary-shimura-variety-representability\]. A lift of $I_\ttT$ {#S:tilde IT} ------------------ Recall that $I_\ttT$ is the subset $\ttS(\ttT)_\infty - (\ttS_\infty \cup \ttT)$ defined in Theorem \[T:main-thm\]. We use $\tilde I_{\ttT}$ to denote the subset of complex embeddings of $E$ consisting of the unique lift $\tilde \tau$ of every element $\tau \in I_\ttT$, for which $\tilde \tau^c \in \tilde \ttS(\ttT)_\infty$. We describe this set explicitly as follows. We write $I_{\ttT/\gothp} = I_\ttT \cap \Sigma_{\infty/\gothp}$ and $\tilde I_{\ttT/\gothp} = \tilde I_\ttT \cap \Sigma_{E, \infty/\gothp}$ for $\gothp \in \Sigma_p$; then they are empty set unless $\gothp$ is of type $\alpha 1$ or $\beta 1$. When $\gothp$ is of type $\alpha1$ or $\beta1$, using the notation of Subsection \[S:quaternion-data-T\], $I_{\ttT/\gothp}$ consists of $\sigma^{-m_i-1}\tau_i$ for all $i$ such that $\#(C_i \cap \ttT_{/\gothp})$ is odd. In the notation of Subsection \[S:tilde S(T)\], the set $\tilde I_{\ttT/\gothp}$ consists of $\sigma^{-a_{r_i}}\tilde \tau_i$ for all such $i$ as above. We remark that, in either case, for any $\tilde \tau$ lifting a place $\tau \in I_{\ttT}$, $\tilde \tau \notin \tilde \Delta(\ttT)^+ \cup \tilde \Delta(\ttT)^-$. Isomorphism of $Y'_\ttT$ with $Z'_{\ttT}$ {#S:Y_T=Z_T} ----------------------------------------- Let $Z'_\ttT$ be the moduli space over $k_0$ representing the functor that takes a locally noetherian $k_0$-scheme $S$ to the set of isomorphism classes of tuples $(B, \iota_B, \lambda_B, \beta_{K'_{\ttT}}, J^\circ)$, where - $(B, \iota_B, \lambda_B, \beta_{K'_{\ttT}})$ is an $S$-valued point of ${\mathbf{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})$. - $J^\circ$ is the collection of sub-bundles $J^\circ_{\tilde\tau}\subseteq H^{{\mathrm{dR}}}_1(B/S)_{\tilde\tau}^\circ$ locally free of rank $1$ for each $\tilde\tau\in \tilde I_\ttT$. It is clear that $Z'_{\ttT}$ is a $(\PP^1)^{I_{\ttT}}$-bundle over ${\mathbf{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})$. We define a morphism $\eta: Y'_{\ttT}{\rightarrow}Z'_{\ttT}$ as follows: Let $S$ be a locally noetherian $k_0$-scheme, and $x=(A, \iota_{A},\lambda_A, \alpha_{K'}, B, \iota_B, \lambda_B, \beta_{K'_\ttT}, C, \iota_C;\phi_A, \phi_B)$ be an $S$-valued point of $Y'_{\ttT}$. We define $\eta(x)\in Z'_{\ttT}(S)$ to be the isomorphism class of the tuple $(B, \iota_B, \lambda_B, \beta_{K'_{\ttT}}, J^\circ)$, where $J^\circ_{\tilde \tau}$ is given by $\phi_{B,*,\tilde\tau}^{-1}\circ \phi_{A,*,\tilde\tau}(\omega_{A^\vee,\tilde\tau}^\circ)$ for the isomorphisms $$\xymatrix{ H^{{\mathrm{dR}}}_1(A/S)^\circ_{\tilde\tau}\ar[r]^{\phi_{A,*,\tilde\tau}}_{\cong} &H^{{\mathrm{dR}}}_1(C/S)^\circ_{\tilde\tau} & H^{{\mathrm{dR}}}_1(B/S)^\circ_{\tilde\tau}\ar[l]^{\cong}_{\phi_{B,*, \tilde\tau}}. }$$ Note that $\tilde \tau \notin \tilde \Delta(\ttT)^\pm$ implies that both $\phi_{A, *, \tilde \tau}$ and $\phi_{B, *, \tilde \tau}$ are isomorphisms. \[P:Y\_T=Z\_T\] The morphism $\eta_2: Y'_{\ttT}{\rightarrow}Z'_{\ttT}$ is an isomorphism. We note that Theorem \[T:main-thm-unitary\] follows immediately from this Proposition and Proposition \[P:Y\_S=X\_S\]. As in the proof of Proposition \[P:Y\_S=X\_S\], it suffices to prove that $\eta$ induces a bijection on the closed points and on tangent spaces. **Step I.** We show first that $\eta$ induces a bijection on closed points. Let $z=(B,\iota_B, \lambda_B, \beta_{K'_\ttT}, J^\circ)$ be a closed point of $Z'_{\ttT}$ with values in $k={\overline{\FF}_p}$. We have to show that there exists a *unique* point $y=(A, \iota_{A}, \lambda_A, \alpha_{K'}, B, \iota_B, \lambda_B, \beta_{K'_\ttT}, C, \iota_C;\phi_A, \phi_B)\in Y'_{\ttT}(k)$ with $\eta(y)=z$. To prove this, we basically reverse the construction in the proof of Lemma \[L:Y\_T=X\_T-1\]. We start by reconstructing $C$ from $B$ and $J^\circ$. We denote by ${\tilde{{\mathcal{D}}}}_B=({\tilde{{\mathcal{D}}}}^{\circ}_B)^{\oplus 2}$ the covariant Dieudonné module of $B$, and by ${\tilde{{\mathcal{D}}}}^\circ_B=\bigoplus_{\tilde\tau\in \Sigma_{E,\infty}}{\tilde{{\mathcal{D}}}}_{B,\tilde\tau}^\circ$ the canonical decomposition according to the $\calO_E$-action. We construct a Dieudonné submodule $M^\circ=\bigoplus_{\tau\in \Sigma_{E,\infty}} M^\circ_{\tilde\tau}\subseteq {\tilde{{\mathcal{D}}}}_{B}^\circ [1/p]$ with ${\tilde{{\mathcal{D}}}}^\circ_{B}\subseteq M^\circ\subseteq p^{-1}{\tilde{{\mathcal{D}}}}^\circ_{B}$ as follows. Let $\tilde\tau\in \Sigma_{E,\infty/\gothp}$ with $\gothp\in \Sigma_p$. If $\tilde\tau\notin \tilde \Delta(\ttT)^-$, we put $M^\circ_{\tilde\tau}={\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}$. To define $M^\circ_{\tilde\tau}$ in the other case, we separate the discussion according to the type of $\gothp$. - (Case $\alpha 1$ and $\beta 1$) Recall our notation from Subsections \[S:quaternion-data-T\], \[S:tilde S(T)\], \[S:Delta-pm\], and \[S:tilde IT\]. There are two subcases according to the parity of $\#(C_i \cap \ttT_{/\gothp})$, where $C_i$ is a chain of $\ttS_{\infty/\gothp}\cup \ttT_{/\gothp}$ as in Subsection \[S:quaternion-data-T\]. (It should not be confused with the abelian variety $C$.) - When $r_i=\#(C_i \cap \ttT_{/\gothp})$ is odd, $\sigma^{-m_i-1}\tilde\tau_i\in \tilde I_{\ttT/\gothp}$ so that $J^\circ_{\sigma^{-m_i-1}\tilde\tau_i}$ is defined. In this case, all $\tau = \sigma^{-\ell}\tau_i$ belong to $ \ttS(\ttT)_{\infty/\gothp}$ for $0 \leq \ell \leq m_i+1$; so $s_{\ttT, \sigma^{-\ell} \tilde \tau_i} \in \{0,2\}$ and the essential Frobenii $$\xymatrix{ F_{B,{\mathrm{es}}}^{m_i+1-\ell}:\ {\tilde{{\mathcal{D}}}}^\circ_{B,\sigma^{-m_i-1}\tilde\tau_i}\ar[r]_-{ F_{B,{\mathrm{es}}}}^-{\cong} &{\tilde{{\mathcal{D}}}}^\circ_{B,\sigma^{-m_i}\tilde\tau_i}\ar[r]^-{\cong}_-{ F_{B,{\mathrm{es}}}} &\cdots \ar[r]^-{\cong}_-{ F_{B, {\mathrm{es}}}}&{\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-\ell}\tilde\tau_i} }$$ are isomorphisms for such an $\ell$. If $a_j \leq \ell < a_{j+1}$ for some odd number $j$, we put $$M^\circ_{\sigma^{-\ell}\tilde\tau_i}=p^{-1} F_{B,{\mathrm{es}}}^{m_i+1-\ell}(\tilde J^\circ_{\sigma^{-m_i-1}\tilde\tau_i}),$$ where $\tilde J^\circ_{\sigma^{-m_i-1}\tilde\tau_i}$ denotes the inverse image in ${\tilde{{\mathcal{D}}}}_{B,\sigma^{-m_i-1}\tilde\tau_i}^\circ$ of $J^\circ_{\sigma^{-m_i-1}\tilde\tau_i}\subseteq {\mathcal{D}}^\circ_{B,\sigma^{-m_i-1}\tilde\tau_i}$ under the natural reduction map modulo $p$; otherwise, we have already defined $M^\circ_{\sigma^{-\ell}\tilde\tau_i}$ to be $\tilde \calD^\circ_{B, \sigma^{-\ell}\tilde\tau_i}$. - When $r_i=\#(C_i\cap \ttT_{/\gothp})$ is even, there is no $J^\circ$ involved in this construction. Note that all $\tau = \sigma^{-\ell}\tau_i$ belong to $\ttS(\ttT)_{\infty/\gothp}$ for $0 \leq \ell \leq m_i$; so $s_{\ttT, \sigma^{-\ell} \tilde \tau_i} \in \{0,2\}$ and in the sequence of essential Frobenii $$\xymatrix{\quad\quad F_{B,{\mathrm{es}}}^{m_i-\ell+1}:\ {\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-m_i-1}\tilde\tau_i}\ar[r]_-{F_{B}}&{\tilde{{\mathcal{D}}}}^\circ_{B,\sigma^{-m_i}\tilde\tau_i}\ar[r]_-{ F_{B,{\mathrm{es}}}}^-{\cong} &{\tilde{{\mathcal{D}}}}^\circ_{B,\sigma^{-m_i+1}\tilde\tau_i}\ar[r]^-{\cong}_-{ F_{B,{\mathrm{es}}}} &\cdots \ar[r]^-{\cong}_-{ F_{B, {\mathrm{es}}}}&{\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-\ell}\tilde\tau_i}, }$$ all the maps except the first one are isomorphisms. If $a_j \leq \ell < a_{j+1}$ for some odd number $j$, we put $$M^\circ_{\sigma^{-\ell}\tilde\tau_i}=p^{-1} F_{B,{\mathrm{es}}}^{m_i-\ell+1} ({\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-m_i-1}\tilde\tau_i});$$ then we have $\dim_k(M^\circ_{\sigma^{-\ell}\tilde\tau_i}/{\tilde{{\mathcal{D}}}}^\circ_{B,\sigma^{-\ell}\tilde\tau_i})=1$, since the cokernel of $F_B: {\tilde{{\mathcal{D}}}}^\circ_{B,\sigma^{-m_i-1}\tilde\tau_i}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-m_i}\tilde\tau_i}$ has dimension $1$, as $s_{\ttT,\sigma^{-m_i-1}\tilde\tau_i}=1$. (For other $\ell$, we have already defined $M^\circ_{\sigma^{-\ell}\tilde\tau_i}$ to be $\tilde \calD^\circ_{B, \sigma^{-\ell}\tilde\tau_i}$.) - (Case $\alpha 2$) In this case, $\gothp$ is a prime of type $\alpha^\sharp$ for ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$, and it splits into two primes $\gothq$ and $\bar\gothq$ in $E$. Let $H_{\gothq}\subseteq B[\gothq]$ be the closed subgroup scheme given in the data $\beta_{K'_\ttT}$. Let $H_{\bar \gothq}$ be its annihilator under the Weil pairing on $B[\gothq]$ induced by $\lambda_B$. (We collectively write $H_\gothp$ for $H_\gothq \times H_{\bar \gothq}$.) Let ${\mathcal{D}}^\circ_{H_\gothp}=\bigoplus_{\tilde\tau\in \Sigma_{E,\infty/\gothp}}{\mathcal{D}}^\circ_{H_\gothp,\tilde\tau} \subseteq {\mathcal{D}}_{B}^\circ$ be the reduced covariant Dieudonné module of $H_\gothp = H_{\gothq} \times H_{\bar \gothq}$. Then each ${\mathcal{D}}^\circ_{H_{\gothp},\tilde\tau}$ is necessarily one-dimensional over $k$ for all $\tilde\tau\in \Sigma_{E,\infty/\gothp}$. For $\tilde\tau\in \tilde\Delta(\ttT)^-_{/\gothp}$, we define $$M^\circ_{\tilde\tau} =p^{-1}\tilde{\mathcal{D}}^\circ_{H_\gothp,\tilde\tau},$$ where ${\tilde{{\mathcal{D}}}}^\circ_{H_\gothp,\tilde\tau}$ denotes the inverse image in ${\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}$ of the subspace ${\mathcal{D}}_{H_{\gothp},\tilde\tau}\subseteq {\mathcal{D}}_{B,\tilde\tau}^\circ$. (We have defined $M^\circ_{\tilde \tau} = \tilde D^\circ_{B, \tilde \tau}$ for $\tilde \tau \notin \tilde \Delta(\ttT)^-_{/\gothp}$ before.) - (Case $\beta 2$) In this case, $\gothp$ is a prime of type $\beta^\sharp$ for ${\mathrm{Sh}}_{K_{\ttT}}(G_{\ttS(\ttT)})$. For $\tilde\tau\in \tilde \Delta(\ttT)^-_{/\gothp}$, let $\lambda_{B,*,\tilde\tau}: {\mathcal{D}}^{\circ}_{B,\tilde\tau}{\rightarrow}{\mathcal{D}}^\circ_{B^\vee,\tilde\tau}$ be the morphism induced by the polarization $\lambda_B$. By Theorem \[T:unitary-shimura-variety-representability\](b3), $J^\circ_{\tilde\tau}:=\operatorname{Ker}(\lambda_{B,*,\tilde\tau})$ is a $k$-vector space of dimension $1$. We set $M^\circ_{\tilde\tau}=p^{-1}\tilde J^\circ_{\tilde\tau}$ for such $\tilde \tau$, where $\tilde J^\circ_{\tilde \tau}$ is the preimage of $J^\circ_{\tilde \tau}$ under the reduction map $\tilde \calD^\circ_{B, \tilde \tau} \to \calD^\circ_{B, \tilde \tau}$. Note that when viewing ${\tilde{{\mathcal{D}}}}^\circ_{B^\vee,\tilde\tau}$ as a lattice of ${\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}[1/p]$ using the polarization, we have $M^\circ_{\tilde\tau} ={\tilde{{\mathcal{D}}}}^\circ_{B^\vee,\tilde\tau}$. (We have defined $M^\circ_{\tilde \tau} = \tilde D^\circ_{B, \tilde \tau}$ for $\tilde \tau \notin \tilde \Delta(\ttT)^-_{/\gothp}$ before.) This concludes the definition of $M^\circ\subseteq p^{-1}{\tilde{{\mathcal{D}}}}^\circ_{B}$. One checks easily that $M^\circ$ is stable under $F_B$ and $V_B$. Consider the quotient Dieudonné modules $$M/{\tilde{{\mathcal{D}}}}_{B}=(M^\circ/{\tilde{{\mathcal{D}}}}^\circ_{B})^{\oplus 2}\subseteq p^{-1}{\tilde{{\mathcal{D}}}}_{B}/{\tilde{{\mathcal{D}}}}_B\cong {\mathcal{D}}_B.$$ Then $M/{\tilde{{\mathcal{D}}}}^\circ_B$ corresponds to a closed finite group scheme $G\subseteq B[p]$ stable under the action of ${\mathcal{O}}_D$. We put $C=B/G$ with the induced $\calO_D$-action, and define $\phi_B:B{\rightarrow}C$ as the canonical $\calO_D$-equivariant isogeny. Then the natural induced map $\phi_{B,*}:{\tilde{{\mathcal{D}}}}^\circ_{B}{\rightarrow}{\tilde{{\mathcal{D}}}}^\circ_{C}$ is identified with the inclusion ${\tilde{{\mathcal{D}}}}^\circ_B{\hookrightarrow}M^\circ$. We now construct $A$ from $C$. Similar to above, we first define a $W(k)$-lattice $L^\circ = \bigoplus_{\tilde \tau\in \Sigma_{E, \infty}} L^\circ_{\tilde \tau} \subseteq \tilde \calD_C^\circ$, with $L^\circ_{\tilde \tau} = \tilde \calD_{C, \tilde \tau}^\circ$ unless $\tilde \tau \in \tilde \Delta(\ttT)^+$. If $\tilde \tau \in \tilde \Delta(\ttT)^+$, then the corresponding $p$-adic place $\gothp \in \Sigma_p$ cannot be of type $\beta2$ or $\beta^\sharp$. In this case, we identify ${\tilde{{\mathcal{D}}}}_{B}^\circ[1/p]$ with ${\tilde{{\mathcal{D}}}}_C^\circ[1/p]$ so that ${\tilde{{\mathcal{D}}}}_B^\circ$ and ${\tilde{{\mathcal{D}}}}^\circ_C$ are both viewed as $W(k)$-lattices in ${\tilde{{\mathcal{D}}}}_B^\circ[1/p]$. The polarization $\lambda_B$ induces a perfect pairing $$\langle\ ,\ \rangle_{\lambda_B}\colon {\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}[1/p]\times {\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau^c}[1/p]{\rightarrow}W(k)[1/p],$$ which induces a perfect pairing between ${\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}$ and ${\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau^c}$. We put $$L^\circ_{\tilde\tau}=\tilde \calD^{\circ,\perp}_{C,\tilde\tau^c}\colon =\{v\in {\tilde{{\mathcal{D}}}}^\circ_{C,\tilde\tau}[1/p]: \langle v, w\rangle_{\lambda_B}\in W(k),\text{ for all } w\in \tilde \calD^\circ_{C,\tilde\tau^c} \}.$$ Note that $\tilde \tau \in \tilde \Delta(\ttT)^+$ always implies that $\tilde \tau^c \in \tilde \Delta(\ttT)^-- \tilde \Delta(\ttT)^+$. So $\tilde \calD^\circ_{C, \tilde \tau} = \tilde \calD^\circ_{B, \tilde \tau}$ and $\tilde \calD^\circ_{C, \tilde \tau^c} \supset \tilde \calD^\circ_{B, \tilde \tau^c}$ with quotient isomorphic to $k$. This implies that $L^\circ_{\tilde \tau} \subseteq \tilde \calD_{C, \tilde \tau}^\circ$ with quotient isomorphic to $k$. As usual, one verifies that $L^\circ$ is stable under $F_B$ and $V_B$ (because it either equals to $\tilde \calD_{C, \tilde \tau}^\circ$ or $\tilde \calD_{C, \tilde \tau^c}^{\circ, \perp}$ in various cases), and we put $L=(L^\circ)^{\oplus 2}$. The quotient Dieudonné module $L/p{\tilde{{\mathcal{D}}}}_{C}$ corresponds to a closed subgroup scheme $K\subseteq C[p]$ stable under the action of ${\mathcal{O}}_D$. We put $A=C/K$ equipped with the induced $\calO_D$-action, and define $\phi_A\colon A{\rightarrow}C$ as the canonical $\calO_D$-equivariant isogeny with kernel $C[p]/K$. Then $\phi_{A,*}:{\tilde{{\mathcal{D}}}}_{A}{\rightarrow}{\tilde{{\mathcal{D}}}}_C$ is identified with the natural inclusion $L{\hookrightarrow}{\tilde{{\mathcal{D}}}}_C$. We define $\lambda_A:A{\rightarrow}A^\vee$ to be the quasi-isogeny: $$\lambda_A: A{\xrightarrow}{\phi_A} C \xleftarrow{\phi_B} B{\xrightarrow}{\lambda_B}B^\vee\xleftarrow{\phi_{B}^\vee} C^\vee {\xrightarrow}{\phi_A^\vee}A^\vee,$$ and we will verify that $\lambda_A$ is a genuine isogeny (hence a polarization since $\lambda_B$ is) satisfying condition (b) of the moduli space ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$ as in Theorem \[T:unitary-shimura-variety-representability\]. We may identify ${\tilde{{\mathcal{D}}}}_{A}^\circ[1/p]$ and ${\tilde{{\mathcal{D}}}}^\circ_{A^\vee}[1/p]$ with ${\tilde{{\mathcal{D}}}}_{B}^\circ[1/p]$, and view both ${\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau}$ and ${\tilde{{\mathcal{D}}}}^\circ_{A^\vee,\tilde\tau}$ as lattices of ${\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}[1/p]$. It suffices to show that we have a natural inclusion $${\tilde{{\mathcal{D}}}}^\circ_{A^\vee, \tilde\tau} \subseteq ({\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau^c})^\perp= \big\{v\in {\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}[1/p]: \langle v, w\rangle_{\lambda_B}\in W(k)\text{ for all }w\in {\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau^c}\big\},$$ which is an isomorphism unless $\tilde \tau$ induces a $p$-adic place of type $\beta^\sharp$ for ${\mathrm{Sh}}_{K}(G_{\ttS})$ in which case it is an inclusion with quotient $k$. - By the construction of $A$, this is clear for $\tilde \tau \in \tilde \Delta(\ttT)^+$ and hence for all their complex conjugates (as the duality is reciprocal). - For all places $\tilde \tau \in \Sigma_{E,\infty/ \gothp}$ such that $\gothp$ is not of type $\beta2$ and $\tilde \tau \notin \tilde \Delta(\ttT)^\pm$, we know that $\tilde \tau^c \notin \tilde \Delta(\ttT)^\pm$. So $\tilde \calD^\circ_{A, ? } = \tilde \calD^\circ_{B,?}$ for $? = \tilde \tau, \tilde \tau^c$ under the identification. The statement is clear. Note that this includes the case that $\gothp$ is a prime of type $\beta^\sharp$ for ${\mathrm{Sh}}_{K}(G_{\ttS})$. - The only case left is when $\tilde \tau \in \Sigma_{E, \infty/\gothp}$ for $\gothp$ of type $\beta2$. In this case, ${\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau}={\tilde{{\mathcal{D}}}}^\circ_{C,\tilde\tau}$ which is the dual of ${\tilde{{\mathcal{D}}}}^\circ_{C,\tilde\tau^c}$ for all $\tilde\tau\in \Sigma_{E,\infty/\gothp}$ by construction. This concludes the verification of that $\lambda_A$ is isogeny satisfying condition (b) of Theorem \[T:unitary-shimura-variety-representability\] for the moduli space ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})$. We now define the level structure $\alpha_{K'}=\alpha^p\alpha_p$ on $A$. For the prime-to-$p$ level structure $\alpha^p$, we define it to be the $K_{\ttT}'$-orbit of the isomorphism class: $$\xymatrix{ \alpha^p:&\Lambda^{(p)}\ar[r]^-{\sim}_-{\beta^p}&T^{(p)}(B)\ar[r]^-{\sim}_-{\phi_{B,*}}& T^{(p)}(C) &T^{(p)}(A)\ar[l]_-{\sim}^-{\phi_{A,*}}. }$$ We take the closed subgroup scheme $\alpha_\gothp\subseteq A[\gothp]$ for each $\gothp\in \Sigma_p$ of type $\alpha^\sharp$ for $\ttS$ (and hence for $\ttS(\ttT)$) to be the subgroup scheme corresponding to $\beta_\gothp$ under the sequence of isomorphisms (note that $\tilde \Delta(\ttT)^\pm_{/\gothp} = \emptyset$ for $\gothp$ of type $\alpha^\sharp$) of $p$-divisible groups $$\xymatrix{ A[\gothp^\infty]\ar[r]^{\phi_{A}}_{\cong} &C[\gothp^\infty] &B[\gothp^\infty]\ar[l]_{\phi_B}^{\cong} }.$$ It is clear that $\alpha_{K'}$ verifies condition (c) in Theorem \[T:unitary-shimura-variety-representability\]. This finishes the construction of all the data $y=(A,\iota_A,\lambda_A,\alpha_{K'}, B, \iota_B,\lambda_B, \beta_{K'_\ttT}, C, \iota_C;\phi_A,\phi_B)$. To see that $y$ is indeed a $k$-point of $Y'_{\ttT}$, we have to check that $y$ satisfies the conditions (i)-(ix) for $Y'_{\ttT}$ in Subsection \[S:moduli-Y\_S\]. Conditions (ii), (iii), and (vi)-(ix) being clear from our construction, it remains to check (i), (iv) and (v). Moreover, the Kottwitz signature condition Theorem \[T:unitary-shimura-variety-representability\](a) follows from Lemma \[L:dimension count\] immediately. So for property (i), it remains to show that the partial Hasse invariant $h_{\tilde\tau}(A)$ vanishes if $\tau = \tilde\tau|_F\in \ttT$. We now check these properties (i), (iv), and (v) in various cases. For this, we identify $\tilde \calD_A^\circ[\frac1p]$, $\tilde \calD_B^\circ[\frac 1p]$, and $\tilde \calD_C^\circ[\frac 1p]$ via $\phi_{A, *}$ and $\phi_{B, *}$. \(1) Assume $\gothp$ is a prime of case $\alpha1$ or $\beta1$. We keep the notation as before. If $\tilde \tau$ does not lift a place belonging to some chain $C_i$ inside $\ttS_{\infty/\gothp}\cup \ttT_{/\gothp}$, the conditions (i), (iv), and (v) trivially hold. So we assume $\tilde \tau|_F \in C_i$ for some $C_i$. If $r_i=\#(\ttT_{/\gothp}\cap C_i)$ is odd, unwinding our earlier construction gives, for $0 \leq \ell \leq m_i+1$, $$\begin{aligned} &\tilde \calD_{A, \sigma^{-\ell}\tilde \tau_i}^\circ = \tilde \calD_{C, \sigma^{-\ell}\tilde \tau_i}^\circ = \left\{ \begin{array}{ll} p^{-1} F_{B,{\mathrm{es}}}^{m_i+1-\ell}(\tilde J^\circ_{\sigma^{-m_i-1}\tilde \tau_i}) & \textrm{ if }a_j \leq \ell< a_{j+1} \textrm{ for some odd }j, \\ \tilde \calD_{B, \sigma^{-\ell}\tilde \tau_i}^\circ & \textrm{ otherwise;} \end{array} \right. \\ &\tilde \calD_{A, \sigma^{-\ell}\tilde \tau_i^c}^\circ = \left\{ \begin{array}{ll} p F_{B,{\mathrm{es}}}^{m_i+1-\ell}(\tilde J^{\circ,\perp}_{\sigma^{-m_i-1}\tilde \tau_i}) & \textrm{ if }a_j \leq \ell< a_{j+1} \textrm{ for some odd }j, \\ \tilde \calD_{B, \sigma^{-\ell}\tilde \tau_i^c}^\circ & \textrm{ otherwise; and} \end{array} \right.\\ & \tilde \calD_{C, \sigma^{-\ell}\tilde \tau_i^c}^\circ = \tilde \calD_{B, \sigma^{-\ell}\tilde \tau_i^c}^\circ \textrm{ for all } \ell.\end{aligned}$$ For condition (v) in Subsection \[S:moduli-Y\_S\], it is trivial unless $\tilde \tau = \sigma^{-\ell} \tilde \tau_i$ for some $\ell \in [a_j, a_{j+1})$ with $j$ odd; and in the exceptional case, it is equivalent to proving that (for the $n$ as in condition (v)) $$\tilde \calD^\circ_{B, \sigma^{-\ell} \tilde \tau_i} = F_{A, {\mathrm{es}}}^n (\tilde \calD_{A, \sigma^{-\ell - n} \tilde \tau_i}^\circ).$$ Note that $n = a_{j+1}-\ell$, it follows that ${\tilde{{\mathcal{D}}}}_{A,\sigma^{-\ell-n}\tilde\tau_i}^{\circ}={\tilde{{\mathcal{D}}}}_{B,\sigma^{-a_{j+1}}\tilde\tau_i}^{\circ}$ by definition. As $s_{\ttT, \sigma^{-a_{j+1}}\tilde\tau_i}=2$ and $s_{\sigma^{-a_{j+1}}\tilde\tau_i}=1$, $F_{A,{\mathrm{es}}}^n(\tilde \calD_{A, \sigma^{-\ell - n} \tilde \tau_i}^\circ)$ coincides with $F_{B,{\mathrm{es}}}^n(\tilde \calD_{B, \sigma^{-a_{j+1}}\tilde\tau_i}^\circ )$ by the definition of essential Frobenius. The desired equality follows from the fact that $F_{B,{\mathrm{es}}}^n(\tilde \calD_{B, \sigma^{-a_{j+1}}\tilde\tau_i}^\circ )={\tilde{{\mathcal{D}}}}_{B,\sigma^{-\ell}\tilde\tau_i}$. Similarly, condition (iv) is trivial unless $\tilde \tau = \sigma^{-\ell} \tilde \tau_i^c$ for some $\ell \in [a_j, a_{j+1})$ with $j$ odd, in which case, it is equivalent to the following equality (for the $n$ as in condition (iv)) $$p\tilde \calD^\circ_{B, \sigma^{-\ell} \tilde \tau_i^c} = \tilde F_{A, {\mathrm{es}}}^n (\tilde \calD_{A, \sigma^{-\ell - n} \tilde \tau_i^c}^\circ).$$ But $n = a_{j+1}-\ell$ by definition; so $\tilde \calD_{A, \sigma^{-\ell - n} \tilde \tau_i^c}^\circ = \tilde \calD^\circ_{B, \sigma^{-a_{j+1}}\tilde \tau_i^c}$. Since $s_{\ttT, \sigma^{-a_{j+1}}\tilde \tau^c_i} = 0$ and $s_{\sigma^{-a_{j+1}}\tilde \tau^c_i} = 1$, the essential Frobenius of $A$ at $\sigma^{-a_{j+1}}\tilde \tau^c_i$ is defined to be $F_A$ while that of $B$ at $\sigma^{-a_{j+1}}\tilde \tau^c_i$ is defined to be $V_{B}^{-1}$. Therefore, $F_{A, {\mathrm{es}}, \sigma^{-\ell}\tilde \tau_i^c}^n$ is the same as $pF_{B, {\mathrm{es}}, \sigma^{-\ell}\tilde \tau_i^c}^n$. The equality above is now clear. We now check the vanishing of partial Hasse invariants $h_{\sigma^{a_j}\tilde\tau_i}(A)$ with $1\leq j\leq r_i-1$. By Lemma \[Lemma:partial-Hasse\], it suffices to show that, for any $j = 1, \dots, r_i-1$ and setting $a_0 = -1$, the image of $$F_{A, {\mathrm{es}}}^{a_{j+1}- a_{j-1}}: {\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-a_{j+1}} \tilde \tau_i} {\rightarrow}{\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-a_{j-1}}\tilde \tau_i}$$ is contained in $p{\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-a_{j-1}}\tilde \tau_i}$. First, regardless of the parity of $j$, we find easily that $ F_{A, {\mathrm{es}}}^{a_{j+1}- a_{j-1}}=pF_{B, {\mathrm{es}}}^{a_{j+1}- a_{j-1}}$ as maps from ${\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-a_{j+1}} \tilde \tau_i}$ to ${\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-a_{j-1}} \tilde \tau_i}$ by checking carefully the dependence of the essential Frobenii on the signatures. Now if $j$ is odd, then $${\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-\ell}\tilde\tau_i}={\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-\ell}\tilde\tau_i}\quad \text{for }\ell=a_{j+1} \textrm{ and }a_{j-1};$$ hence one gets $F_{A, {\mathrm{es}}}^{a_{j+1}- a_{j-1}}({\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-a_{j+1}}\tilde\tau_i})=p{\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-a_{j-1}} \tilde \tau_i}$, since $F_{B, {\mathrm{es}}}^{a_{j+1}- a_{j-1}}({\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-a_{j+1}}\tilde\tau_i})={\tilde{{\mathcal{D}}}}^\circ_{B, \sigma^{-a_{j-1}} \tilde \tau_i}$. If $j$ is even, then $${\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-\ell}\tilde\tau_i}=p^{-1} F_{B,{\mathrm{es}}}^{m_i+1-\ell}(\tilde J^\circ_{\sigma^{-m_i-1}\tilde \tau_i}),\quad \text{for } \ell=a_{j+1}\textrm{ and } a_{j-1};$$ now it is also obvious that $F_{A, {\mathrm{es}}}^{a_{j+1}- a_{j-1}}({\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-a_{j+1}}\tilde\tau_i})=p{\tilde{{\mathcal{D}}}}^\circ_{A, \sigma^{-a_{j-1}} \tilde \tau_i}$. If $r_i=\#(\ttT_{/\gothp}\cap C_i)$ is even, all conditions can be proved in exactly the same way, except replacing $ F_{B, {\mathrm{es}}}^{m_i+1-\ell}(\tilde J^\circ_{\sigma^{-m_i-1}\tilde \tau_i})$ by $ F_{B,{\mathrm{es}}}^{m_i-\ell} (F_B({\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-m_i-1}\tilde\tau_i}))$ and the proof of the vanishing of Hasse invariant $h_{\sigma^{-a_{r_i}}\tilde \tau}(A)$ needs a small modification. In fact, we have $${\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-\ell}\tilde\tau_i}=\begin{cases} {\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-\ell}\tilde\tau_i} & \text{for } a_{r_i}\leq \ell\leq m_i+1,\\ p^{-1} F_{B,{\mathrm{es}}}^{m_i-\ell} (F_B({\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^{-m_i-1}\tilde\tau_i})) &\text{for } a_{r_{i}-1}\leq \ell < a_{r_i}. \end{cases}$$ Note that the number $n_{\sigma^{-a_{r_{i}}}\tau}$ defined in \[S:partial-Hasse\] is equal to $m_i+1-a_{r_i}$, and the essential Frobenius $F_{A,{\mathrm{es}}}: {\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-a_{r_i}}\tilde \tau_i}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^{-a_{r_i}+1}\tilde \tau_i}$ is simply $F_A$. We have $$F_{A} F_{A, {\mathrm{es}}}^{m_i+1-a_{r_i}}(\tilde \calD^\circ_{A, \sigma^{-m_i-1}\tilde \tau_i} ) = F_{A, {\mathrm{es}}}^{m_i+2-a_{r_i}} (\tilde \calD^\circ_{A, \sigma^{-m_i-1}\tilde \tau_i} ) = p\tilde \calD^\circ_{A, \sigma^{-a_{r_i}+1}\tilde\tau_i}.$$ This verifies the vanishing of $h_{\sigma^{-a_{r_i}}\tilde \tau}(A)$. \(2) Assume that $\gothp$ is a prime of Case $\alpha2$. We write $H_\gothp = H_{\gothq}\oplus H_{\bar\gothq}$ and ${\tilde{{\mathcal{D}}}}^\circ_{H_{\gothq},\tilde\tau}\subseteq {\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}$ as before. We have $${\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau}= \begin{cases}p^{-1}{\tilde{{\mathcal{D}}}}^\circ_{H_{\gothp},\tilde\tau} &\textrm{if }\tilde\tau \in \tilde \Delta(\ttT)_{/\gothp}^-, \\ p({\tilde{{\mathcal{D}}}}^\circ_{H_{\gothp},\tilde\tau^c})^\perp &\textrm{if }\tilde\tau \in \tilde \Delta(\ttT)_{/\gothp}^+, \\ {\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau} &\text{otherwise}, \end{cases} \quad \textrm{and} \quad {\tilde{{\mathcal{D}}}}^\circ_{C,\tilde\tau}= \begin{cases}p^{-1}{\tilde{{\mathcal{D}}}}^\circ_{H_{\gothp},\tilde\tau} &\textrm{if }\tilde\tau \in \tilde \Delta(\ttT)_{/\gothp}^-, \\ {\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau} &\text{otherwise}. \end{cases}$$ The same arguments as in (1) allows us to check conditions (i), (iv) and (v). \(3) Assume now that $\gothp$ is prime of Case $\beta2$ in Subsection \[S:quaternion-data-T\]. For each $\tilde\tau\in \Sigma_{E,\infty/\gothp}$, let $\lambda_{B,*,\tilde\tau}: {\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}{\rightarrow}{\tilde{{\mathcal{D}}}}^\circ_{B^\vee,\tilde\tau}$ be the map induced by the polarization $\lambda_B$. By condition (b3) of Theorem \[T:unitary-shimura-variety-representability\], its cokernel has dimension $1$ over $k$. When viewing ${\tilde{{\mathcal{D}}}}^\circ_{B^\vee,\tilde\tau}$ as a lattice of ${\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau}[1/p]$ via $\lambda^{-1}_{B,*,\tilde\tau}$, we have $${\tilde{{\mathcal{D}}}}^\circ_{A,\tilde\tau} = {\tilde{{\mathcal{D}}}}^\circ_{C,\tilde\tau}= \begin{cases} {\tilde{{\mathcal{D}}}}^\circ_{B^\vee,\tilde\tau} &\text{if }\tilde\tau\in \tilde \Delta(\ttT)_{/\gothp}^-,\\ {\tilde{{\mathcal{D}}}}^\circ_{B,\tilde\tau} &\text{otherwise}. \end{cases}$$ The same argument as in (1) allows us to check the conditions (i), (iv) and (v). This then concludes the proof of Step I. **Step II:** Let $y=(A, \iota_A,\lambda_A, \alpha_{K'}, B, \lambda_B, \iota_B, \beta_{K'_\ttT}, C, \iota_C; \phi_A, \phi_B)\in Y'_{\ttT}$ be a closed point with values in $k={\overline{\FF}_p}$, and $z=\eta(y)=(B, \iota_B, \lambda_B, \beta_{K'_\ttT}, J^\circ)\in Z'_{\ttT}$. We prove that $\eta: Y_{T}'{\rightarrow}Z'_{\ttT} $ induces a bijection of tangent spaces $\eta_{y}: T_{Y'_{\ttT}, y}{\xrightarrow}{\cong} T_{Z'_{\ttT},z}$. We follow the same strategy as Lemma \[L:Y\_T-X\_T-tangent\]. Set $\II=\operatorname{Spec}(k[\epsilon]/\epsilon^2)$. The tangent space $T_{Z'_{\ttT},z}$ is identified with the set of deformations $z_{\II}=(B_{\II}, \iota_{B,\II}, \lambda_{B,\II}, \beta_{K'_\ttT,\II}, J^\circ_{\II})\in Z'_{\ttT}(\II)$ of $z$, where $J^\circ_{\II}$ is the collection of sub-bundles $J^\circ_{\II,\tilde\tau}\subseteq H^{\mathrm{dR}}_{1}(B_{\II}/\II)^\circ_{\tilde\tau}=H^{{\mathrm{cris}}}_{1}(B/k)_{\II,\tilde\tau}^{\circ}$ for each $\tilde\tau\in \tilde I_\ttT$. We have to show that every point $z_\II$ lifts uniquely to a deformation $y_{\II}=(A_{\II}, \iota_{A_\II}, \lambda_{A_{\II}}, \alpha_{K',\II}, B_{\II}, \lambda_{B_{\II}}, \iota_{B_{\II}}, \beta_{K'_\ttT,\II}, C_{\II}, \iota_{C_{\II}}; \phi_{A_{\II}}, \phi_{B_{\II}})\in Y'_{\ttT}(\II)$ with $\eta(y_{\II})=z_{\II}$. We start with $C_{\II}$ and $\phi_{B_{\II}}$. For $\tilde\tau\in \Sigma_{E,\infty}$, denote by $$\phi_{B,*,\tilde\tau}^{{\mathrm{cris}}}: H^{\mathrm{cris}}_1(B/k)^{\circ}_{\II,\tilde\tau}{\rightarrow}H^{\mathrm{cris}}_1(C/k)^\circ_{\II,\tilde\tau}$$ the natural morphism induced by $\phi_{B}$, and by $\phi^{{\mathrm{dR}}}_{B,*,\tilde\tau}$ the analogous map between the de Rham homology $H^{{\mathrm{dR}}}_1$. The crystalline nature of $H^{{\mathrm{cris}}}_1$ implies that $\phi^{{\mathrm{cris}}}_{B,*,\tilde\tau}=\phi^{{\mathrm{dR}}}_{B,*,\tilde\tau}\otimes_k k[\epsilon]/\epsilon^2$. To construct $C_{\II}$ and $\phi_{B,\II}$ it suffices to specify, for each $\tilde\tau\in \Sigma_{E,\infty}$, a sub-bundle $\omega^\circ_{C^\vee,\II,\tilde\tau}\subseteq H^{{\mathrm{cris}}}_1(C/k)^\circ_{\II,\tilde\tau}$ which lifts $\omega^\circ_{C^\vee,\tilde\tau}$ and satisfies $$\label{E:omega-inclusion-C} \phi^{{\mathrm{cris}}}_{B,*,\tilde\tau}(\omega^{\circ}_{B^\vee_{\II},\tilde\tau})\subseteq \omega^\circ_{C^\vee,\II,\tilde\tau}.$$ We distinguish a few cases: 1. If neither $\tilde\tau$ nor $\sigma\tilde\tau$ belong to $\tilde\Delta(\ttT)^-$, both $\phi^{{\mathrm{cris}}}_{B,*,\tilde\tau}$ and $\phi^{{\mathrm{cris}}}_{B,*,\sigma\tilde\tau}$ are isomorphisms. It follows that $\phi_{B,*,\tilde\tau}^{{\mathrm{dR}}}(\omega^\circ_{B^\vee,\tilde\tau})=\omega^\circ_{C^\vee, \tilde\tau}$; hence we have to take $\omega^{\circ}_{C^\vee,\II, \tilde\tau}=\phi^{{\mathrm{cris}}}_{B,*, \tilde\tau}(\omega^\circ_{B^\vee_{\II},\tilde\tau})$. 2. If both $\tilde \tau, \sigma \tilde \tau \in \tilde \Delta(\ttT)^-$, then $\tau=\tilde\tau|_{F}\in \ttS_{\infty/\gothp}$ by Lemma \[L:property of Delta\]. A simple dimension count similar to Lemma \[L:dimension count\] implies that $ \dim_{k}(\omega^\circ_{C^\vee, \tilde\tau})=\dim_{k}(\omega^\circ_{B^\vee,\tilde\tau})\in \{0,2\}. $ We take $\omega^{\circ}_{C^\vee,\II, \tilde\tau}$ to be $0$ or $H^{{\mathrm{cris}}}_1(C/k)^\circ_{\II, \tilde\tau}$ correspondingly, and is trivial. 3. If $\tilde \tau \in \tilde I_\ttT$, property of the morphism $\eta$ forces $\omega^{\circ}_{C^\vee, \II,\tilde\tau}=\phi^{{\mathrm{cris}}}_{B,*,\tilde\tau}(J^\circ_{B_{\II},\tilde\tau})$. 4. For all other $\tilde \tau$, $\tilde \tau|_F$ must belong to $\ttT$. Let $n$ be the number associated to $\tilde\tau $ as in Lemma \[L:distance to T’\]. By the vanishing of the partial Hasse invariant on $A$ at $\tilde \tau$, we have $\omega^\circ_{C^\vee, \tilde \tau} = F_{C,{\mathrm{es}}}^n(H^{{\mathrm{dR}}}_1(C/k)^\circ_{\sigma^{-n}\tilde \tau}).$ We take $$\omega^\circ_{C^\vee, \II, \tilde \tau} = F_{C,{\mathrm{es}}}^n(H_1^{\mathrm{cris}}(C^{(p^n)}/k)^\circ_{\II, \tilde \tau}).$$ This is not a forced choice now; but it will become one when we have constructed the lift $A_\II$ and require $A_\II$ to have vanishing partial Hasse invariant. Since $F_{B,{\mathrm{es}}}^n: H_1^{\mathrm{cris}}(B^{(p^n)}/k)^\circ_{\II, \tilde \tau} \to H_1^{\mathrm{cris}}(B/k)^\circ_{\II, \tilde \tau}$ is an isomorphism, we conclude that holds for $\tilde \tau$. We now construct $A_\II$ and the isogeny $\phi_{A_{\II}}:A_{\II}{\rightarrow}C_{\II}$. As usual, we have to specify, for each $\tilde\tau\in \Sigma_{E,\infty}$, a sub-bundle $\omega^\circ_{A^\vee,\II,\tilde\tau}\subseteq H^{{\mathrm{cris}}}_1(A/k)^\circ_{\II,\tilde\tau}$ that lifts $\omega^{\circ}_{A^\vee,\tilde\tau}$ and satisfies $\phi_{A,*,\tilde\tau}^{\mathrm{cris}}(\omega^\circ_{A^\vee,\II,\tilde\tau})\subseteq \omega^\circ_{C^\vee,\II,\tilde\tau}$. Let $\gothp\in \Sigma_p$ be the prime such that $\tilde\tau\in \Sigma_{E,\infty/\gothp}$. - If neither $\tilde \tau$ nor $\sigma \tilde \tau$ belong to $\tilde \Delta(\ttT)^+$, then $\phi_{A, *, \tilde \tau}^{\mathrm{dR}}$ and hence $\phi^{\mathrm{cris}}_{A, *, \tilde \tau}$ is an isomorphism. We are forced to take $\omega^{\circ}_{A^\vee,\II, \tilde\tau}=(\phi^{{\mathrm{cris}}}_{A,*,\tilde\tau})^{-1}(\omega^\circ_{C^\vee, \II, \tilde\tau})$. In particular, if $\tilde\tau\in \tilde I_{\ttT}$, we have $\omega_{A^\vee,\II, \tilde\tau}=(\phi^{{\mathrm{cris}}}_{A,*,\tilde\tau})^{-1}\phi^{{\mathrm{cris}}}_{B,*,\tilde\tau}(J^{\circ}_{B_{\II},\tilde\tau})$. - In all other cases, we must have $\tilde \tau \in \Sigma_{E, \infty/\gothp}$ for $\gothp$ not of type $\beta2$ or $\beta^\sharp$. Then we have to take $\omega^\circ_{A^\vee,\II,\tilde\tau}$ to be the orthogonal complement of $\omega^\circ_{A^\vee,\II, \tilde\tau^c}$ (which is already defined in the previous case) under the perfect pairing $$\langle\ ,\ \rangle_{\lambda_A}\colon H^{{\mathrm{cris}}}_1(A/k)^\circ_{\II, \tilde\tau}\times H^{{\mathrm{cris}}}_{1}(A/k)^\circ_{\II,\tilde\tau^c}{\rightarrow}k[\epsilon]/\epsilon^2$$ induced by the polarization $\lambda_A$. It is clear that $\omega^\circ_{A^\vee, \II, \tilde\tau}$ is a lift of $\omega^\circ_{A^\vee, \tilde\tau}$. It remains to show that $\phi^{\mathrm{cris}}_{A, *, \tilde \tau}(\omega^\circ_{A^\vee, \II, \tilde \tau}) \subseteq \omega^\circ_{C^\vee, \II, \tilde \tau}$. We consider the following commutative diagram $$\label{E:dualization ACB} \xymatrix@C=5pt{ H_1^{\mathrm{cris}}(A/k)^\circ_{\II,\tilde{\tau}} \ar[d]^{\phi^{{\mathrm{cris}}}_{A,*,\tilde\tau}} & \times & H_1^{\mathrm{cris}}(A/k)^\circ_{\II,\tilde{\tau}^c}\ar[d]^{\phi^{{\mathrm{cris}}}_{A,*,\tilde\tau^c}}_\cong \ar[rrrr]^-{\langle\ , \ \rangle_{\lambda_A}} &&&& \II \\ H_1^{\mathrm{cris}}(C/k)^\circ_{\II,\tilde{\tau}} & \times & H_1^{\mathrm{cris}}(C/k)^\circ_{\II,\tilde{\tau}^c} \\ H_1^{\mathrm{cris}}(B/k)^\circ_{\II,\tilde{\tau}} \ar[u]_{\phi^{{\mathrm{cris}}}_{B,*,\tilde\tau}}^\cong & \times & \ar[u]_{\phi^{{\mathrm{cris}}}_{B,*,\tilde\tau^c}} H_1^{\mathrm{cris}}(B/k)^\circ_{\II,\tilde{\tau}^c}\ar[rrrr]^-{\langle\ , \ \rangle_{\lambda_B}} &&&& \II,\ar@{=}[uu] }$$ where both duality pairings are perfect. By our choice of $\tilde \tau$, we have $\tilde \tau, \sigma \tilde \tau \notin \tilde \Delta(\ttT)^-$ and $\tilde \tau^c, \sigma \tilde \tau^c \notin \tilde \Delta(\ttT)^+$; so both $\phi^{\mathrm{cris}}_{A, *, \tilde \tau^c}$ and $\phi^{\mathrm{cris}}_{B, *, \tilde \tau}$ in are isomorphisms and they induce isomorphisms on the reduced differentials. Using the diagram of perfect duality, it suffices to prove that $\phi^{\mathrm{cris}}_{A, *, \tilde \tau}(\omega^\circ_{A^\vee, \II, \tilde \tau}) \subseteq \omega^\circ_{C^\vee, \II, \tilde \tau}$ is equivalent to $\phi^{\mathrm{cris}}_{B, *, \tilde \tau^c}(\omega^\circ_{B,\II, \tilde \tau^c}) \subseteq \omega^\circ_{C,\II, \tilde \tau^c}$, which was already checked. By construction, the $\tilde\tau$-partial Hasse invariant of $A_\II$ vanishes if $\tilde\tau\in \tilde\Delta(\ttT)^-$ and $\tilde\tau|_F\in \ttT$; the duality guarantees the vanishing of Hasse invariants at their conjugate places. This condition conversely forces the uniqueness of our choice of $C_\II$ and $A_\II$. From the construction, $\omega^\circ_{A^\vee,\II}=\bigoplus_{\tilde\tau\in \Sigma_{E,\infty}}\omega^\circ_{A^\vee,\II,\tilde\tau}$ is isotropic under the paring on $H^{\mathrm{cris}}_{1}(A/k)^\circ_{\II}$ induced by $\lambda_A$. This concludes checking condition (1) of Subsection \[S:moduli-Y\_S\]. The lift of the level structure $\alpha_{K', \II}$ is automatic for the tame part, and can be done in a unique way as in the proof of Theorem \[T:unitary-shimura-variety-representability\]. It then remains to check that $\phi_{A_\II}$ and $\phi_{B_\II}$ satisfy conditions (iv) and (v) of Subsection \[S:moduli-Y\_S\]. For condition (v), it is obvious except when $\tilde \tau \in \tilde \Delta(\ttT)^-$, in which case, $$\mathrm{Im}(\phi^{\mathrm{cris}}_{B, *, \tilde \tau}) = \mathrm{Im}(\phi^{\mathrm{dR}}_{B, *, \tilde \tau}) \otimes_k \II, \quad \textrm{and} \quad \phi_{A, *, \tilde \tau}^{\mathrm{cris}}(\mathrm{Im}(F_{{\mathrm{es}},A_\II, \tilde \tau}^n)) = \phi_{A, *, \tilde \tau}^{\mathrm{dR}}(\mathrm{Im}(F_{{\mathrm{es}},A, \tilde \tau}^n)) \otimes_k \II,$$ where $n>1$ is the number determined in Lemma \[L:distance to T’\]. So condition (v) for the lift follows from that for $\phi_A: A \to C$. (Note that $n\geq 1$ implies that the image of the essential image is determined by the reduction.) Exactly the same argument proves condition (iv). This concludes Step II of the proof of Proposition \[P:Y\_T=Z\_T\]. End of proof of Theorem \[T:main-thm-unitary\] {#S:End-of-proof} ---------------------------------------------- Statement (1) of \[T:main-thm-unitary\] follows from Proposition \[P:Y\_S=X\_S\] and \[P:Y\_T=Z\_T\]. Statements (2) and (3) are clear from the proof of (1). It remains to prove statement (4), namely the compatibility of partial Frobenius. We use $X'_{\tilde \ttS,\ttT}$, $Y_{\tilde \ttS,\ttT}'$ and $Z'_{\tilde \ttS,\ttT}$ to denote the original $X'_{\ttT}$, $Y'_{\ttT}$ and $Z'_{\ttT}$ in to indicate their dependence on $\tilde \ttS$. We will define a twisted partial Frobenius $$\gothF'_{\gothp^2,\ttS}:Y'_{\tilde \ttS,\ttT}{\rightarrow}Y'_{\sigma^2_{\gothp}\tilde \ttS,\sigma^2_{\gothp}\ttT}$$ compatible via $\eta_1$ with the $\gothF'_{\gothp^2,\tilde \ttS}$ on ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ defined in Subsection \[S:partial Frobenius\]. For an $S$-valued point $x=(A, \iota_A,\lambda_A, \alpha_{K'}, B, \iota_B, \lambda_B, \beta_p, C, \iota_C; \phi_A, \phi_B)$ of $Y'_{\tilde \ttS,\ttT}$, its image $$\gothF'_{\gothp^2,\tilde \ttS}(x)=(A', \iota_{A'},\lambda_{A'}, \alpha'_{K'}, B', \iota_{B'}, \lambda_{B'}, \beta'_p, C', \iota_{C'}; \phi_{A'}, \phi_{B'}).$$ is given as follows. Here, for $G=A,B,C$, we put $G'=(G/\operatorname{Ker}_{G,\gothp^2})\otimes_{{\mathcal{O}}_F}\gothp$ where $\operatorname{Ker}_{G',\gothp^2}$ is the $\gothp$-component of the $p^2$-Frobenius of $G$. The induced structures $(\iota_{A'},\lambda_{A'}, \alpha'_{K'},\iota_{B'},\lambda_{B'},\beta'_{p})$ are defined in the same way as in \[S:partial Frobenius\]. The isogenies $\phi_{A'}: A'{\rightarrow}C'$ and $\phi_{B'}:B'{\rightarrow}C'$ are constructed from $\phi_{A}$ and $\phi_B$ by the functoriality of $p^2$-Frobenius. We have to prove that the induced map on de Rham homologies $\phi_{A',*,\tilde\tau}$ and $\phi_{B',*,\tilde\tau}$ satisfy the required conditions in (v) and (vi) of \[S:moduli-Y\_S\]. If $\tilde\tau\in \Sigma_{E,\infty/\gothp'}$ with $\gothp'\neq \gothp$, this is clear, because the $p$-divisible groups $G'[\gothp'^{\infty}]$ are canonically identified with $G[\gothp'^{\infty}]$ for $G=A,B,C$. Now consider the case $\gothp'=\gothp$. As in the proof of Lemma \[L:partial Frobenius vs partial Hasse inv\], for $G=A,B,C$, the $p$-divisible group $G'[\gothp^{\infty}]$ is isomorphic to the base change of $G[\gothp^{\infty}]$ via $p^2$-Frobenius on $S$. One deduces thus isomorphisms of de Rham homologies $$\label{E:isom-dR-ABC} H^{{\mathrm{dR}}}_1(G'/S)^{\circ}_{\tilde\tau}=(H^{{\mathrm{dR}}}_1(G/S)^{\circ}_{\sigma^{-2}\tilde\tau})^{(p^2)},$$ which is compatible with $F$ and $V$ as $\tilde\tau\in \Sigma_{E,\infty/\gothp}$ varies, and compatible with $\phi_{A',*,\tilde\tau}$ and $\phi_{B',*,\tilde\tau}$ by functoriality. Hence, the required properties on $\phi_{A',*,\tilde\tau}$ and $\phi_{B',*,\tilde\tau}$ follow from those on $\phi_{A,*,\sigma^{-2}\tilde\tau}$ and $ \phi_{B,*,\sigma^{-2}\tilde\tau}$. This finishes the construction of $\gothF'_{\gothp^2}$ on $Y'_{\tilde \ttS,\ttT}$. Via the isomorphism $\eta_2:Y'_{\tilde \ttS,\ttT}{\xrightarrow}{\sim}Z'_{\tilde \ttS,\ttT}$ proved in \[P:Y\_T=Z\_T\], $\gothF'_{\gothp^2,\ttS}$ induces a map $\gothF'_{\gothp^2,\ttS}:Z'_{\tilde \ttS,\ttT}{\rightarrow}Z'_{\sigma^2_{\gothp}\tilde \ttS,\sigma^2_{\gothp}\ttT}$. Let $z=(B,\iota_{B},\lambda_{B},\beta_{K'_{\ttT}}, J^{\circ})$ be an $S$-valued point of $Z'_{\tilde \ttS,\ttT}$ as described in \[S:Y\_T=Z\_T\]. Then its image $\gothF'_{\gothp^2,\tilde \ttS}(z)$ is given by $(B',\iota_{B'},\lambda_{B'},\beta'_{K'_{\ttT}}, J^{\circ,\prime})\in Z'_{\sigma^2_{\gothp}\ttS,\sigma^{2}\ttT}$, where $(B',\iota_{B'},\lambda_{B'}, \beta'_{K'_{\ttT}})$ are defined as in \[S:partial Frobenius\], and $J^{\circ,\prime}$ is the collection of line bundles $J^{\circ,\prime}_{\tilde\tau}\subseteq H^{{\mathrm{dR}}}_1(B'/S)^{\circ}_{\tilde\tau}$ for each $\tilde\tau\in \bigcup_{\gothp'\in\Sigma_{p}}\sigma^2_{\gothp}(\tilde\ttT_{\infty/\gothp'}-\tilde\ttT_{/\gothp'})$ given as follows. For $\tilde\tau\in \tilde\ttT_{\infty/\gothp'}-\tilde\ttT_{/\gothp'}$ with $\gothp'\neq \gothp$, we have $J^{\circ,\prime}_{\tilde\tau}=J^{\circ}_{\tilde\tau}$ since $H^{{\mathrm{dR}}}_1(B'/S)^{\circ}_{\tilde\tau}$ is canonically identified with $H^{{\mathrm{dR}}}_1(B/S)^{\circ}_{\tilde\tau}$; for $\tilde\tau\in \sigma^2(\tilde\ttT_{\infty/\gothp}-\tilde\ttT_{/\gothp})$, we have $J^{\circ,\prime}_{\tilde\tau}=(J^{\circ}_{\sigma^{-2}\tilde\tau})^{(p^2)}$, which makes sense thanks to the isomorphism for $G=B'$. After identifying ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\ttT}=X'_{\tilde \ttS,\ttT}$ with $Z'_{\tilde \ttS,\ttT}$, the projection $\pi_{\ttT}:Z'_{\tilde \ttS,\ttT}\to {\mathbf{Sh}}_{K'_{\ttT}}(G'_{\tilde \ttS(\ttT)})_{k_0}$ is given by $(B,\iota_{B},\lambda_{B},\beta_{K'_{\ttT}}, J^{\circ})\mapsto (B,\iota_{B},\lambda_{B},\beta_{K'_{\ttT}})$. It is clear that we have a commutative diagram: $$\xymatrix{ Z'_{\tilde \ttS,\ttT}\ar[r]_-{\xi^\mathrm{rel}} \ar[rd]_{\pi_{\ttT}} \ar@/^15pt/[rr]^-{\gothF'_{\gothp^2, \tilde \ttS}} & \gothF'^*_{\gothp^2}(Z'_{\sigma^2_{\gothp}\tilde \ttS,\sigma^2_{\gothp}\ttT}) \ar[d] \ar[r]_-{\gothF'^*_{\gothp^2, \tilde \ttS(\ttT)}} & Z'_{\sigma^{2}_{\gothp}\tilde \ttS,\sigma^2_{\gothp}\ttT} \ar[d]^{\pi_{\sigma_{\gothp}^2\ttT}} \\ & {\mathbf{Sh}}_{K'_\ttT}(G'_{\tilde \ttS(\ttT)})_{k_0} \ar[r]^-{\gothF'_{\gothp^2, \tilde \ttS(\ttT)}} & {\mathbf{Sh}}_{K'_\ttT}(G'_{\sigma_\gothp^2(\tilde \ttS(\ttT))})_{k_0}, }$$ where $\xi^{\mathrm{rel}}$ is given by $(B,\iota_{B},\lambda_{B},\beta_{K'_{\ttT}}, J^{\circ})\mapsto (B,\iota_{B},\lambda_{B},\beta_{K'_{\ttT}}, J^{\circ,\prime})$ with $J^{\circ,\prime}$ defined above. This proves statement (4) immediately. Ampleness of modular line bundles {#Section:GO divisors} ================================= In this section, we suppose that $F\neq {\mathbb{Q}}$. We will apply Theorem \[T:main-thm-unitary\] to prove some necessary conditions for the ampleness of certain modular line bundles on quaternionic/unitary Shimura varieties. In this section, let $X' = {\mathbf{Sh}}_{K'}(G_{\tilde \ttS}')_{k_0}$ be a unitary Shimura variety over $k_0$ considered in Subsection \[S:GO-notation\]. This is a smooth and quasi-projective variety over $k_0$, and projective if $\ttS_{\infty}\neq\emptyset$. Let $(\bfA',\iota,\lambda, \alpha_{K'})$ be the universal abelian scheme over $X'$. For each $\tilde\tau\in \Sigma_{E,\infty}$, the ${\mathcal{O}}_{X'}$-module $\omega^{\circ}_{\bfA'^\vee/X', \tilde \tau}$ is locally free of rank $2-s_{\tilde\tau}$; it is a line bundle if $\tilde \tau|_F$ belongs to $ \Sigma_{\infty}-\ttS_\infty$. Rational Picard group --------------------- For a variety $Y$ over $k_0$, we write $\operatorname{Pic}(Y)_{\mathbb{Q}}$ for $\operatorname{Pic}(Y)\otimes_\ZZ \QQ$. For a line bundle $\calL$ on $Y$, we denote by $[\calL]$ its class in $\operatorname{Pic}(Y)_{{\mathbb{Q}}}$. \[L:omega-Pic\] *(1)* For any $\tilde\tau\in\Sigma_{E,\infty}$ lifting a place $\tau \in \Sigma_\infty-\ttS_\infty$, we have equalities $$[\omega^\circ_{\bfA'^\vee/X',\tilde\tau}]=[\omega^\circ_{\bfA'^\vee/X',\tilde\tau^c}]=[\omega^\circ_{\bfA'/X',\tilde\tau}]=[\omega^\circ_{\bfA'/X',\tilde\tau^c}].$$ *(2)* For any $\tilde\tau\in \Sigma_{E,\infty}$, we have $[\wedge^2_{{\mathcal{O}}_{X'}} H_{1}^{\mathrm{dR}}(\bfA'/X')_{\tilde\tau}]=0.$ *(3)* Let $X'^*$ denote the minimal compactification of $X'$ (which is just $X'$ if $\ttS_{\infty} \neq \emptyset$). Then the natural morphism $j: \operatorname{Pic}(X'^*) \to \operatorname{Pic}(X')$ is injective. Moreover, for each $\tilde \tau \in \Sigma_{E, \infty}$ lifting a place $\tau \in \Sigma_\infty - \ttS_\infty$, $[\omega^\circ_{\bfA'^\vee/X', \tilde \tau}]$ belongs to the image of $j_\QQ: \operatorname{Pic}(X'^*)_\QQ \to \operatorname{Pic}(X')_{\mathbb{Q}}$. \(1) Suppose that $\tau \in \Sigma_{\infty/\gothp} - \ttS_{\infty/\gothp}$ for $\gothp \in \Sigma_\gothp$. Clearly, $\gothp$ is not of type $\alpha^\sharp$ or $\beta^\sharp$. The equality $[\omega^\circ_{\bfA'^\vee/X',\tilde\tau}]=[\omega^\circ_{\bfA'/X',\tilde\tau^c}]$ follows from the isomorphism $\omega^\circ_{\bfA'^\vee/X',\tilde\tau}\cong \omega^\circ_{\bfA'/X',\tilde\tau^c} $ thanks to the polarization $\lambda$ on $\bfA'$. To prove the equality $[\omega^\circ_{\bfA'^\vee/X',\tilde\tau}]=[\omega^\circ_{\bfA'^\vee/X',\tilde\tau^c}]$, we consider the partial Hasse invariants $h_{\tilde\tau}\in \Gamma(X', (\omega^\circ_{\bfA'^\vee/X',\sigma^{-n_\tau}\tilde\tau})^{\otimes p^{n_{\tau}}}\otimes \omega^{\circ,\otimes (-1)}_{\bfA'^\vee/X',\tilde\tau})$, and $h_{\tilde\tau^c}$ defined similarly with $\tilde\tau$ replaced by $\tilde\tau^c$. By Lemma \[Lemma:partial-Hasse\] and Proposition \[Prop:smoothness\], the vanishing of $h_{\tilde\tau}$ and $h_{\tilde\tau^c}$ define the same divisor: $X'_{\tau}\subseteq X'$. Hence, for each $\tilde\tau\in \Sigma_{E,\infty}$ lifting some $ \tau \in \Sigma_{\infty/\gothp} -\ttS_{\infty/\gothp}$, we have an equality $$\label{E:equality-tau-tau-c} p^{n_{\tau}} [\omega^\circ_{\bfA'^\vee/X',\sigma^{-n_\tau}\tilde\tau}]- [\omega^{\circ}_{\bfA'^\vee/X',\tilde\tau}] =p^{n_{\tau}} [\omega^{\circ}_{\bfA'^\vee/X',\sigma^{-n_\tau}\tilde\tau^{c}}]-[\omega^{\circ}_{\bfA'^\vee/X',\tilde\tau^c}].$$ Let $C$ be the square matrix with coefficients in ${\mathbb{Q}}$, whose rows and columns are labeled by those places $\tilde \tau \in \Sigma_{E, \infty}$ lifting a place $\tau \in \Sigma_{\infty/\gothp} - \ttS_{\infty/\gothp}$, and whose $(\tilde \tau_1, \tilde \tau_2)$-entry is $$c_{\tilde \tau_1, \tilde \tau_2}=\begin{cases} -1&\text{if } \tilde \tau_1=\tilde \tau_2,\\ p^{n_{\tau_2}}&\text{if }\tilde \tau_1=\sigma^{-n_{\tau_2}}\tilde \tau_2,\\ 0&\text{otherwise}. \end{cases}$$ One checks easily that $C$ is invertible, hence it follows from that $[\omega^\circ_{\bfA'^\vee/X',\tilde\tau}]=[\omega^\circ_{\bfA'^\vee/X',\tilde\tau^c}]$. \(2) Assume first that $\tilde\tau\in \Sigma_{E,\infty}$ lifts some $\tau \in \Sigma_{\infty/\gothp}-\ttS_{\infty}$. From the Hodge filtration $0{\rightarrow}\omega^\circ_{\bfA'^\vee,\tilde\tau}{\rightarrow}H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\tilde\tau}{\rightarrow}\operatorname{Lie}(\bfA'/X')^\circ_{\tilde\tau}{\rightarrow}0$, one deduces that $$[\wedge^2_{{\mathcal{O}}_{X'}}H^{{\mathrm{dR}}}_1(\bfA'/X')_{\tilde\tau}^\circ]=[\omega^\circ_{\bfA'^\vee/X',\tilde\tau}]+[\operatorname{Lie}(\bfA'/X')_{\tilde\tau}^\circ].$$ Then statement (2) follows from (1) and that $[\operatorname{Lie}(\bfA'/X')^\circ_{\tilde\tau}]=-[\omega^\circ_{\bfA'/X',\tilde\tau}]=-[\omega^\circ_{\bfA'^\vee/X',\tilde\tau^c}]$. Consider now the case when $\tilde\tau\in \Sigma_{E,\infty}$ lifts some $\tau \in \ttS_{\infty/\gothp}$ for a place $\gothp$ of type $\alpha$ or $\beta$. Then there is an integer $m\geq 1$ such that $\sigma^m\tau\in \Sigma_{\infty}-\ttS_{\infty}$ and $\sigma^{i}\tau\in \ttS_{\infty}$ for all $0\leq i\leq m-1$, and we have a sequence of isomorphisms $$\xymatrix{ H^{\mathrm{dR}}_1(\bfA'/X')^{\circ, (p^m)}_{\tilde\tau}\ar[r]^-{F_{\bfA',{\mathrm{es}}}}_-{\cong} &H^{\mathrm{dR}}_1(\bfA'/X')^{\circ, (p^{m-1})}_{\sigma\tilde\tau}\ar[r]^-{F_{\bfA',{\mathrm{es}}}}_-{\cong} &\cdots\ar[r]^-{F_{\bfA',{\mathrm{es}}}}_-{\cong} &H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\sigma^m\tilde\tau}. }$$ From this, one gets $$p^m[\wedge^2_{{\mathcal{O}}_{X'}}H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\tilde\tau}]=[\wedge^2_{{\mathcal{O}}_{X'}}H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\sigma^m\tilde\tau}]=0.$$ Finally, if $\tilde\tau\in \Sigma_{E,\infty/\gothp}$ for a place $\gothp$ of type $\alpha^\sharp$ or $\beta^\sharp$ and if $m$ is the inertia degree of $\gothp$ over $p$, then the sequence of isomorphism $$\xymatrix{ H^{\mathrm{dR}}_1(\bfA'/X')^{\circ, (p^{2m})}_{\tilde\tau}\ar[r]^-{F_{\bfA',{\mathrm{es}}}}_-{\cong} &H^{\mathrm{dR}}_1(\bfA'/X')^{\circ, (p^{2m-1})}_{\sigma\tilde\tau}\ar[r]^-{F_{\bfA',{\mathrm{es}}}}_-{\cong} &\cdots\ar[r]^-{F_{\bfA',{\mathrm{es}}}}_-{\cong} &H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\sigma^{2m}\tilde\tau} }$$ gives rise to an equality $$p^{2m}[\wedge^2_{{\mathcal{O}}_{X'}}H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\tilde\tau}]=[\wedge^2_{{\mathcal{O}}_{X'}}H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\sigma^{2m}\tilde\tau}]=[\wedge^2_{{\mathcal{O}}_{X'}}H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\tilde\tau}],$$ as $\sigma^{2m}\tilde \tau = \tilde \tau$. This forces $[\wedge^2_{{\mathcal{O}}_{X'}}H^{\mathrm{dR}}_1(\bfA'/X')^\circ_{\tilde\tau}] = 0$. \(3) If $X'$ is a Shimura curve, then $X'^*=X'$ as $F\neq {\mathbb{Q}}$. Assume now that $X'$ has dimension at least $2$. The injectivity of $j: \operatorname{Pic}(X'^*) \to \operatorname{Pic}(X')$ follows from the fact that $X'^*$ is normal [@lan Proposition 7.2.4.3], and that the complement $X'^*-X'$ has codimension $\geq 2$. Recall from (1) that the partial Hasse invariant defines the class as described in . Note that the inverse matrix of $C$ has all entries positive. It follows that each $[\omega^\circ_{\bfA'^\vee/X', \tilde\tau_0}]$ for $\tilde \tau_0$ lifting $\tau_0 \in \Sigma_\infty- \ttS_\infty$ is a positive linear combination of $\calO_{X'}(X'_\tau)$’s. Let $X'^{\mathrm{n-ord}}=\bigcup_{\tau\in \Sigma_{\infty}-\ttS_{\infty}}X'_{\tau}\subseteq X'$ be the union of the Goren-Oort strata of codimension $1$. Since $X'^{\mathrm{n-ord}}$ is closed in $X'^*$ and is disjoint from the cusps, each line bundle $\calO_{X'}(X'_\tau)$ extends to a line bundle $\calO_{X'^*}(X'_\tau)$. By linear combination, each $[\omega^\circ_{\bfA'^\vee/X', \tilde\tau_0}]$ extends to a class in $\operatorname{Pic}(X'^*)_\QQ$. For any $\tau \in \Sigma_{\infty}-\ttS_{\infty}$, we put $[\omega_{\tau}]=[\omega^\circ_{\bfA'/X',\tilde\tau}]$ for simplicity, where $\tilde \tau$ is a lift of $\tau$. This is a well defined element in $\operatorname{Pic}(X'^*)_{\mathbb{Q}}$ by Lemma \[L:omega-Pic\]. \[P:normal bundle\] Let $\gothp$ be a $p$-adic place such that $\#(\Sigma_{\infty/\gothp} - \ttS_{\infty/\gothp}) >1$. When $\ttT$ consists of a single element $\tau \in \Sigma_{\infty/\gothp}$ with $\gothp$ not of type $\beta2$, let $n_\tau = n_{\tau,\ttS}$ be as in Subsection \[S:partial-Hasse\]. Let $N_{X'_{\ttT}}(X')$ denote the normal bundle of the embedding $X'_{\ttT} \hookrightarrow X'$. Then the equality $[N_{X'_{\ttT}}(X')]=[\calO(-2p^{n_\tau})]$ holds in $\operatorname{Pic}(X'_{\ttT})_{{\mathbb{Q}}}$, where $\calO(1)$ is the canonical quotient bundle of the $\PP^1$-bundle $\pi_\tau: X'_{\ttT}={\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT} \to {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\ttT)})_{k_0} $, and $\calO(-2p^{n_\tau})$ is the dual of $\calO(1)^{\otimes2 p^{n_\tau}}$. By the construction in Subsection \[S:Y\_T=Z\_T\], the set $\tilde I_\ttT = \{\sigma^{-n_\tau}\tilde \tau\}$ for a specific lift $\tilde \tau$ of $\tau$. We have $$J^\circ_{\sigma^{-n_\tau}\tilde \tau} = \phi_{B, *, \sigma^{-n_\tau}\tilde \tau}^{-1} \circ \phi_{A, *, \sigma^{-n_\tau}\tilde \tau}(\omega^\circ_{\bfA^\vee, \sigma^{-n_\tau}\tilde \tau}),$$ in terms of the moduli description of $Y'_\ttT\cong X'_\ttT$. So the restriction of $[\omega_{ \sigma^{-n_\tau} \tau}]$ to $X'_{\ttT}$ is $[\calO(-1)]$. The Goren-Oort stratum $X'_{\ttT}$ is defined as the zero locus of $$h_{\tilde \tau}: \omega^\circ_{\bfA'^\vee/X', \tilde \tau} \to (\omega^\circ_{\bfA'^\vee / X', \sigma^{-n_\tau} \tilde \tau})^{\otimes p^{n_\tau}}.$$ So firstly, the class of $N_{X'_{\ttT}}(X')$ in $\operatorname{Pic}(X'_{\ttT})_{{\mathbb{Q}}}$ is given by the restriction of $p^{n_\tau}[\omega_{\sigma^{-n_\tau}\tau}] - [\omega_{\tau}]$ to $X'_{\ttT}$; and secondly, on $X'_{\ttT}$ we have an isomorphism $$\omega^\circ_{\bfA'^\vee/X', \tilde \tau} \xrightarrow{\cong} H_1^{\mathrm{dR}}(\bfA'/X')^{\circ, (p^{n_\tau})}_{\tilde \tau} / (\omega^\circ_{\bfA'^\vee / X', \sigma^{-n_\tau} \tilde \tau})^{\otimes p^{n_\tau}}.$$ This implies that $[\omega_\tau]$ equals to $-p^{n_\tau}[\omega_{\sigma^{-n_\tau}\tau}]$ in $\operatorname{Pic}(X'_{\ttT})_{{\mathbb{Q}}}$. To sum up, we have equalities in $\operatorname{Pic}(X'_{\ttT})_{{\mathbb{Q}}}$: $$\begin{aligned} [N_{X'_{\ttT}}(X')]&=p^{n_\tau}[\omega_{\sigma^{-n_\tau}\tau}] - [\omega_{\tau}]\\ &=2p^{n_\tau}[\omega_{\sigma^{-n_\tau}\tau}]=[\calO(-2p^{n_\tau})].\end{aligned}$$ \[T:ampleness\] Let $\underline t=(t_{\tau})\in {\mathbb{Q}}^{\Sigma_{\infty}-\ttS_{\infty}}$. If the element $[\omega^{\underline t}]=\sum_{\tau\in \Sigma_{\infty}-\ttS_{\infty}}t_{\tau}[\omega_{\tau}]$ of $\operatorname{Pic}(X')_{{\mathbb{Q}}}$ is ample, then $$\label{E:condition-ample} p^{n_{\tau}}t_{\tau}> t_{\sigma^{-n_\tau}\tau}\quad \text{(and $t_{\tau}>0$) for all }\tau.$$ Here, we put the second condition in parentheses, because it follows from the first one. Assume that $[\omega^{\underline t}]$ is ample. Let $\tau\in \Sigma_{\infty}-\ttS_{\infty}$, and $\gothp\in \Sigma_p$ be the prime of $F$ such that $\tau\in \Sigma_{\infty/\gothp}$. We distinguish two cases: - $\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp}=\{\tau\}$. Condition for $\tau$ is simply $t_{\tau}>0$. We consider the GO-stratum $X'_{\ttT_{\tau}}$ with $\ttT_{\tau}=\Sigma_{\infty}-(\ttS_{\infty}\cup \{\tau\})$. Then $X'_{\ttT_{\tau}}$ is isomorphic to a Shimura curve by Theorem \[T:main-thm-unitary\], and let $i_{\tau}: X'_{\ttT_{\tau}}{\rightarrow}X'$ denote the canonical embedding. For any $\tilde\tau'\in \Sigma_{E,\infty}-\ttS_{E,\infty}$ with restriction $\tau'=\tilde\tau'|_{F}\neq \tau$. Let $$\label{E:definition-F-tau} F_{\bfA',{\mathrm{es}},\tilde\tau'}^{n_{\tau'}}: H^{{\mathrm{dR}}}_1(\bfA'^{(p^{n_{\tau}})}/X')^\circ_{\tilde\tau'}{\rightarrow}H^{{\mathrm{dR}}}_1(\bfA'/X')^\circ_{\tilde\tau'}.$$ be the $n_{\tau'}$-th iteration of the essential Frobenius in Subsection \[S:partial-Hasse\]. We always have $\operatorname{Ker}(F_{\bfA',{\mathrm{es}},\tilde\tau'}^{n_{\tau'}})=(\omega^\circ_{\bfA'^\vee/X',\sigma^{-n_{\tau'}}\tilde\tau'})^{(p^{n_{\tau'}})}$. The vanishing of $h_{\tilde\tau'}$ on $X'_{\ttT_{\tau}}$ is equivalent to $${\mathrm{Im}}(F_{\bfA',{\mathrm{es}},\tilde\tau'}^{n_{\tau'},\tilde\tau'})|_{X'_{\ttT_{\tau}}}=(\omega^\circ_{\bfA'^\vee/X',\tilde\tau'})|_{X'_{\ttT_{\tau}}}.$$ Therefore, one deduces an equality in $\operatorname{Pic}(X'_{\ttT_{\tau}})_{{\mathbb{Q}}}$: $$p^{n_{\tau}}i_{\tau}^*[\omega_{\sigma^{-n_{\tau'}}\tau'}]+i^*_{\tau}[\omega_{\tau'}]=0.$$ Letting $\tau'$ run through the set $\Sigma_{\infty/\gothq}-\ttS_{\infty/\gothq}$ with $\gothq\neq \gothp$, one obtains that $i_{\tau}^*[\omega_{\tau'}]=0$. Therefore, we have $i_{\tau}^*[\omega^{\underline t}]=t_{\tau} i_{\tau}^*[\omega_{\tau}]$, which is ample on $X'_{\ttT_\tau}$ since so is $[\omega^{\underline t}]$ on $X'_{\ttT_\tau}$ by assumption. By the ampleness of $\det(\omega)=\bigotimes_{\tilde\tau'\in \Sigma_{E,\infty}}\omega_{\bfA',\tilde\tau'}$ on $X'$ and hence on $X'_{\ttT_{\tau}}$, we see that $i^*_{\tau}[\omega_{\tau}]$ is ample on $X'_{\ttT_{\tau}}$. It follows that $t_{\tau}>0$. - $\Sigma_{\infty/\gothp}-\ttS_{\infty/\gothp}\neq\{\tau\}$. Consider the GO-strata $X'_{\tau}$ given by $\ttT=\{\tau\}$, then $\ttS(\ttT)=\ttS\cup \{\tau,\sigma^{-n_{\tau}}\tau\}$ in the notation of Subsection \[S:quaternion-data-T\]. By Propositions \[P:Y\_S=X\_S\] and \[P:Y\_T=Z\_T\], $X'_\tau$ is isomorphic to a $\PP^1$-bundle over ${\mathbf{Sh}}_{K'_{\tau}}(G'_{\tilde \ttS(\ttT)})_{k_0}$. Let $\pi: X'_{\tau}{\rightarrow}{\mathbf{Sh}}_{K'_{\tau}}(G'_{\tilde \ttS(\ttT)})_{k_0}$ denote the natural projection. The ampleness of $[\omega^{\underline t}]$ on $X'$ implies the ampleness of its restriction to each closed fibre $\PP^1_s$ of $\pi$. By the proof of Proposition \[P:normal bundle\], we have $$\omega^\circ_{\bfA'^\vee/X',\tilde\tau'}|_{\PP^1_{s}}\simeq \begin{cases} {\mathcal{O}}_{\PP^1_{s}}(-1) &\text{if $\tau'=\sigma^{-n_{\tau}}\tau$};\\ {\mathcal{O}}_{\PP^1_{s}}(p^{n_{\tau}}) &\text{if } \tau'=\tau;\\ {\mathcal{O}}_{\PP^1_{s}} &\text{otherwise}. \end{cases}$$ The relation  follows immediately. Since the Hilbert modular varieties and the unitary Shimura varieties have the same neutral geometric connected components, the following is an immediate corollary of Theorem \[T:ampleness\]. \[C:ampleness Hilbert\] Let $X$ denote the special fiber of the Hilbert modular variety, and $X^*$ its minimal compactification. Then for each $\tau \in \Sigma_\infty$, the class $[\omega_\tau] \in \operatorname{Pic}(X)_{\mathbb{Q}}$ uniquely extends to a class $[\omega_\tau] \in \operatorname{Pic}(X^*)_{\mathbb{Q}}$. Moreover, $[\omega^{\underline t}] = \sum_{\tau \in \Sigma_\infty}t_\tau [\omega_\tau]$ is ample only when $t_{\tau}>0$ and $p t_\tau > t_{\sigma^{-1}\tau}$ for all $\tau \in \Sigma_\infty$. For the converse to Theorem \[T:ampleness\], we have the following \[C:ampleness conjecture\] The conditions in Theorem \[T:ampleness\] and Corollary \[C:ampleness Hilbert\] are also sufficient for $[\omega^{\underline t}]$ to be ample. In the case of Hilbert modular surface, Corollary \[C:ampleness Hilbert\] and the sufficiency of the condition were proved by Andreatta and Goren [@andreatta-goren Theorem 8.1.1], which relies heavily on some intersection theory on surfaces. It seems difficult to generalize their method. Using our global geometric description, it seems possible to prove, for small inertia degrees (at least when all inertia degrees are $\leq 5$), Conjecture \[C:ampleness conjecture\] using variants of Nakai-Moishezon criterion. The combinatorics becomes complicated when the inertia degree is large. Link morphisms {#Section:links} ============== We will introduce certain generalizations of partial Frobenius morphisms, called *link morphisms*, on unitary Shimura varieties associated to quaternionic ones. These morphisms appear naturally when considering the restriction of the projection maps $\pi_{\ttT}$ in Theorem \[T:main-thm-unitary\] to other Goren-Oort strata. The explicit descriptions of these morphisms are essential for the application considered in the forthcoming paper [@tian-xiao3]. For simplicity, *we will assume that $p$ is inert of degree $g$ in the totally field $F$.* Denote by $\gothp$ the unique prime of $F$ above $p$. Let $E$ be a CM extension of $F$. If $\gothp$ splits in $E$, fix a prime $\gothq$ of $E$ above $\gothp$, and denote the other prime by $\bar \gothq$; if $\gothp$ is inert in $E$, we denote by $\gothq$ the unique prime of $E$ above $\gothp$. Links {#S:links} ----- We introduce some combinatorial objects. Let $n\geq 1$ be an integer. Put $n$ points aligned equi-distantly on a horizontal section of a vertical cylinder. We label the $n$ points by the elements of ${\mathbb{Z}}/n{\mathbb{Z}}$ so that the $(i+1)$-st point is next to the $i$th point on the right. Let $S$ be a subset of the $n$ points above. To such an $S$, we associate a graph as follows: We start from left to right with the plot labeled $0\in {\mathbb{Z}}/n{\mathbb{Z}}$, and draw a *plus sign* if the element is in $S$, and a *node* if it is in ${\mathbb{Z}}/n{\mathbb{Z}}-S$. We call such a picture a *band of length $n$* associated to $S$. For instance, if $n=5$ and $S=\{1,3\}$, then the band is $ \psset{unit=0.3} \begin{pspicture}(-.5,-0.3)(4.5,0.3) \psset{linecolor=black} \psdots(0,0)(2,0)(4,0) \psdots[dotstyle=+](1,0)(3,0) \end{pspicture}$. Let $S'$ be another subset of ${\mathbb{Z}}/n{\mathbb{Z}}$ of the same cardinality as $S$. Then a link *link* $\eta: S{\rightarrow}S'$ is a graph of the following kind: Put the band attached to $S$ on the top of the band for $S'$ in the same cylinder; draw non-intersecting curves from each of the nodes from the top band to a node on the bottom band. We say a curving is turning to the *left* (resp. to the *right*) if it is so as moving from the top band to the bottom band. If a curve travels $m$-numbers of points (of both plus signs and nodes) to the *right* (resp. *left*), we say the displace of this curve is $m$ (resp. $-m$). When both $S$ and $S'$ are equal to ${\mathbb{Z}}/n{\mathbb{Z}}$ (so that there are no nodes at all), then we say that $\eta:S{\rightarrow}S'$ is the trivial link. We define the *total displacement* of a link $\eta$ as the sum of the displacements of all curves in $\eta$. For example, if $n=5$, $S=\{1, 3\}$ and $S'=\{1, 4\}$, then $$\label{E:left turn link} \psset{unit=0.3} \eta = \begin{pspicture}(-.5,-0.3)(5,2.3) \psset{linecolor=red} \psset{linewidth=1pt} \psbezier(0,2)(1,1)(3,1)(3,0) \psbezier(2,2)(2,1)(3.5,1.5)(4.5,0.5) \psbezier(-0.5,1.3)(0.5,.3)(2,1)(2,0) \psarc{-}(-0.5,0){0.5}{0}{90} \psarc{-}(4.5,2){0.5}{180}{270} \psset{linecolor=black} \psdots(0,2)(2,2)(4,2) \psdots(0,0)(2,0)(3,0) \psdots[dotstyle=+](1,2)(3,2) \psdots[dotstyle=+](1,0)(4,0) \end{pspicture}.$$ is a link from $S$ to $S'$, and its total displacement is $v(\eta)=3+3+3=9$. For a link $\eta: S{\rightarrow}S'$, we denote by $\eta^{-1}: S'{\rightarrow}S$ the link obtained by flipping the picture about the equator of the cylinder. For two links $\eta: S{\rightarrow}S'$ and $\eta':S'{\rightarrow}S''$, we define the composition of $\eta'\circ\eta: S{\rightarrow}S''$ by putting the picture of $\eta$ on the top of the picture of $\eta'$ and joint the nodes corresponding to $\eta'$. It is obvious that $v(\eta^{-1})=-v(\eta)$ and $v(\eta'\circ\eta)=v(\eta')+v(\eta)$. Links for a subset of places of $F$ or $E$ ------------------------------------------ We return to the setup of Notation \[S:Notation-for-the-paper\], and recall that $p$ is inert in $F$. We fix an isomorphism $\Sigma_{\infty}\cong {\mathbb{Z}}/g{\mathbb{Z}}$ so that $i\mapsto i+1$ corresponds to the action of Frobenius on $\Sigma_{\infty}$. For an even subset $\ttS$ of places in $F$, we have the *band* for $\ttS$ when applying Subsection \[S:links\] to the subset $\ttS_{\infty}$ of $\Sigma_{\infty}$. Let $\ttS'$ be another even subset of places of $F$ such that $\#\ttS_{\infty}=\#\ttS'_{\infty}$ and $\ttS'$ contains the same finite places of $F$ as $\ttS$ does. A link $\eta$ from the band for $\ttS$ to that for $\ttS'$ is denoted by $\eta: \ttS{\rightarrow}\ttS'$. When $\ttS'=\ttS$ and $\ttS_{\infty}=\Sigma_{\infty}$, $\eta:\ttS{\rightarrow}\ttS'$ is necessarily the trivial link (so that there is no curves at all). The Frobenius action on $\Sigma_{\infty}$ defines a link $\sigma: \ttS{\rightarrow}\sigma(\ttS)$, in which all curves turn to the right with displacement 1; the total displacement of this link $\sigma$ is $v(\sigma)=g-\#\ttS_{\infty}$. Here, $\sigma(\ttS)$ denotes the subset of places of $F$ whose finite part is the same as $\ttS$, and whose infinite part is the image of Frobenius on $\ttS_{\infty}$. \[S:notation-n-tau\] Recall the definition of $n_\tau$ for $\tau \in \Sigma_\infty - \ttS_\infty$ from Notation \[N:n tau\]. For simplicity, we write $\tau^-$ for $\sigma^{-n_\tau} \tau$; and we use $\tau^+$ to denote the unique place in $\Sigma_\infty-\ttS_\infty$ such that $\tau = (\tau^+)^- = \sigma^{-n_{\tau^+}}\tau^+$. When there are several $\ttS$ involved, we will write $n_{\tau}(\ttS)$ for $n_{\tau}$ to emphasize its dependence on $\ttS$. Link morphisms {#S:link-morphisms} -------------- Let $\eta: \ttS{\rightarrow}\ttS'$ be a link of two even subsets of places of $F$. If $\ttS_{\infty}\neq \Sigma_{\infty}$, we denote by $m(\tau)$ the displacement of the curve at $\tau$ in the link $\eta$ for each $\tau\in \Sigma_{\infty}-\ttS_{\infty}$; and put $m(\tau) =0$ for $\tau \in \ttS_\infty$. Let ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ and ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'})_{k_0}$ denote the special fibers of some unitary Shimura varieties of type considered in Subsection \[S:PEL-Shimura-data\]. (There is no restriction on the signatures, i.e. the sets $\tilde \ttS_\infty$ and $\tilde \ttS'_\infty$ that lift $\ttS_\infty$ and $\ttS'_\infty$; but we fix them.) Here, we have fixed (compatible) isomorphisms ${\mathcal{O}}_D: = {\mathcal{O}}_{D_{\ttS}}\cong {\mathcal{O}}_{D_{\ttS'}}\cong \rmM_{2\times 2}({\mathcal{O}}_E)$ and $G'_{\tilde\ttS}(\AAA^{\infty})\cong G'_{\tilde\ttS'}(\AAA^{\infty})$, and regard $K'$ as an open compact subgroups of both of the groups; this is possible because $\ttS$ and $\ttS'$ have the same finite part, and the argument in Lemma \[L:compare D\_S with D\_S(T)\] applies verbatim in this situation. Note that $K'_p$ is assumed to be hyperspecial as in Subsection \[S:PEL-Shimura-data\]. Let $\bfA'_{\tilde\ttS, k_0}$ be the universal abelian scheme over ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$. For a point $x$ of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$ with values in a perfect field $k(x)$, we denote by $\bfA'_{\tilde\ttS,x}$ the base change of $\bfA'_{\tilde\ttS}$ to $x$, and ${\tilde{{\mathcal{D}}}}(\bfA'_{\tilde\ttS, x})^{\circ}$ the reduced part of the *covariant* Dieudonné module of $\bfA'_{\tilde\ttS,x}$ (cf. Subsection \[S:GO-notation\] and proof of Lemma \[L:Y\_T=X\_T-1\]). For each $\tilde\tau\in \Sigma_{E, \infty} - \ttS_{E,\infty}$ we have the essential Frobenius map defined in \[N:essential frobenius and verschiebung\]: $$F_{\bfA',{\mathrm{es}}}: {\tilde{{\mathcal{D}}}}(\bfA'_{\tilde \ttS,x})^{\circ}_{\sigma^{-1}\tilde\tau}{\rightarrow}{\tilde{{\mathcal{D}}}}(\bfA'_{\tilde \ttS,x})_{\tilde\tau}^{\circ}.$$ Finally, recall that a *$p$-quasi-isogeny* of abelian varieties means a quasi-isogeny of the form $f_1\circ f_2^{-1}$, where $f_1$ and $f_2$ are isogenies of $p$-power order. \[D:link-morphism\] Assume that $m(\tau)\geq 0$ for each $\tau\in \Sigma_\infty$, i.e. all curves (if any) in $\eta$ are either straight lines or all turning to the right. Let $n$ be an integer. If $\gothp$ is inert in $E$, we assume that $n=0$. A *link morphism of indentation degree $n$* associated to $\eta$ on ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$ (if exists) is a morphism of varieties $$\eta'_{(n),\sharp}: {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}{\rightarrow}{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'})_{k_0}$$ together with a $p$-quasi-isogeny of abelian varieties $$\eta'^{\sharp}_{(n)}: \bfA'_{\tilde\ttS, k_0}{\rightarrow}\eta'^{*}_{(n),\sharp}(\bfA'_{\tilde\ttS',k_0}),$$ such that the following conditions are satisfied: - $\eta'_{(n),\sharp}$ induces a bijection on geometric points. - The quasi-isogeny $\eta'^{\sharp}_{(n)}$ is compatible with the actions of ${\mathcal{O}}_D$, level structures, and the poarlizations on both abelian varieties. - There exists, for each $\tilde\tau\in \Sigma_{E, \infty} - \ttS_{E,\infty}$, some $t_{\tilde\tau}\in {\mathbb{Z}}$, such that, for every $\overline \FF_p$-point $x$ of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ with image $x'=\eta'_{(n),\sharp}(x)$, $$\eta'^{\sharp}_{(n), *}\big(F_{{\mathrm{es}},\bfA'_{\tilde\ttS,x}}^{m(\tau)}({\tilde{{\mathcal{D}}}}(\bfA'_{\tilde \ttS,x})^\circ_{\tilde\tau})\big)=p^{t_{\tilde\tau}}{\tilde{{\mathcal{D}}}}(\bfA'_{\tilde\ttS',x'})^\circ_{\sigma^{m(\tau)}\tilde\tau},$$ where $\tau\in \Sigma_{\infty}$ is the image of $\tilde\tau$. - The quasi-isogeny of $\gothq$-divisible group $$\eta'^{\sharp}_{(n),\gothq}:\bfA'_{\tilde\ttS}[\gothq^{\infty}]{\rightarrow}\eta'^{*}_{(n),\sharp}\bfA'_{\tilde\ttS'}[\gothq^{\infty}]$$ has degree $p^{2n}$. Here, our convention for $\gothq$ is as at the beginning of this section; in particular, if $\gothp$ splits in $E$, then the quasi-isogeny on the $\bar\gothq^{\infty}$-divisible groups $$\eta'^{\sharp}_{(n),\bar\gothq}:\bfA'_{\tilde\ttS}[\bar\gothq^{\infty}]{\rightarrow}\eta'^*_{(n),\sharp}\bfA'_{\tilde\ttS'}[\bar\gothq^{\infty}]$$ has necessarily degree $p^{-2n}$. Here, the exponent $2n$ is due to the fact that $\bfA'[\gothq^{\infty}]$ is two copies of its reduced part $\bfA'[\gothq^{\infty}]^{\circ}$. Let $\eta_i:\ttS_i{\rightarrow}\ttS_{i+1}$ for $i=1,2$ be two links with all curves turning to the right, and let $(\eta'_{i,\sharp},\eta'^{\sharp}_i)$ be the link morphism of indentation degree $n_i$ on ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS_i})_{k_0}$ attached to $\eta_{i}$. The composition of $(\eta'_{2,\sharp},\eta'^{\sharp}_{2})$ with $(\eta'_{1,\sharp},\eta'^{\sharp}_{1})$ defined by $$\eta'_{12,\sharp}: {\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS_1})_{k_0}{\xrightarrow}{\eta'_{1,\sharp}}{\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS_2})_{k_0}{\xrightarrow}{\eta'_{2,\sharp}}{\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS_3})_{k_0}$$ and $$\eta'^{\sharp}_{12}: \bfA'_{\tilde\ttS_1,k_0}{\xrightarrow}{\eta'^{\sharp}_{1}}\eta'^{*}_{1,\sharp}(\bfA'_{\tilde\ttS_2,k_0}){\xrightarrow}{\eta'^{*}_{1,\sharp}(\eta'^{\sharp}_{2})}\eta'^{*}_{12,\sharp}(\bfA'_{\tilde\ttS_3,k_0}),$$ is a link morphism attached to the composed link $\eta_{12} :=\eta_2\circ\eta_1$ with indentation degree $n_1+n_2$. Variants -------- The formulation of link morphisms on ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ is compatible with changing the tame level $K'^p$. By taking the inverse limit of $K'^p$, one can define a link morphism on ${\mathbf{Sh}}_{K'_p}(G'_{\tilde \ttS})_{k_0}$ associated to $\eta$ in the obvious way. One can define similarly a link morphism of indentation degree $n$ on ${\mathbf{Sh}}_{K''_p}(G''_{\tilde \ttS})$ as a pair $(\eta''_{(n), \sharp}, \eta''_{(n),\sharp})$, where $$\eta''_{(n),\sharp}: {\mathbf{Sh}}_{K''_p}(G''_{\tilde\ttS})_{k_0}{\rightarrow}{\mathbf{Sh}}_{K''_p}(G''_{\tilde\ttS'})_{k_0}$$ is a morphism of varieties and $$\eta''^{\sharp}_{(n)}: \bfA''_{\tilde\ttS, k_0}{\rightarrow}\eta''^{*}_{(n),\sharp}(\bfA''_{\tilde\ttS',k_0})$$ is a $p$-quasi-isogeny of abelian schemes such that same conditions (1)-(4) in \[D:link-morphism\] are satisfied (except the primes is replaced by double primes). Here, $\bfA''_{\tilde\ttS, k_0}$ is the family of abelian varieties constructed in Subsection \[S:abel var in unitary case\]. \(1) Consider the second iteration of the Frobenius link $\sigma^2=\sigma_{\gothp}^2: \ttS{\rightarrow}\sigma^{2}(\ttS)$. The twisted (partial) Frobenius map $$\gothF'_{\gothp^2}: {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_{0}} \to {\mathbf{Sh}}_{K'}(G'_{\sigma^2 \tilde \ttS})_{k_{0}}$$ together with the isogeny $\eta'_{\gothp^2}$ defined in is a link morphism associated to $\sigma^2$; the indentation degree is $0$ if $\gothp$ is inert in $E/F$, and is $2\#\tilde \ttS_{\infty/\bar \gothq} - 2\#\tilde \ttS_{\infty/ \gothq}$ if $\gothp$ splits in $E/F$. \(2) Assume that $\ttS_\infty = \Sigma_\infty$ and $\gothp\notin\ttS$ (so that $\gothp$ splits in $E$ by our choice of $E$). The Shimura variety ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ is just a finite union of closed points. Let $\tau_0\in \Sigma_{\infty}$, and $\tilde \tau_0\in \tilde \ttS_\infty$ be the lift of $\tau_0$ with signature $s_{\tilde\tau_0}=0$. We assume that $\sigma^{-1}\tilde \tau_0 \notin \tilde \ttS_\infty$ (so that $\sigma^{-1}\tilde \tau_0^c \in \tilde \ttS_\infty$). Let $\tilde \ttS'$ denote the subset of places of $F$ containing the same finite places as $ \tilde \ttS$ and such that $\tilde \ttS'_\infty = \tilde \ttS_\infty \cup \{\tilde \tau_0^c, \sigma^{-1}\tilde\tau_0\} \backslash \{ \tilde \tau_0, \sigma^{-1}\tilde \tau_0^c\}$. Let $\ttS'$ be the subset of places of $F$ defined by the restriction of $\tilde\ttS'_{\infty}$. Then there exists a link morphism $(\delta'_{\tau_0,\sharp}, \delta'^{\sharp}_{\tau_0})$ from ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ to ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0}$ associated to the trivial link $\ttS{\rightarrow}\ttS'$ defined as follows; its indentation is $0$ if $\gothp$ is inert in $E/F$, is $2$ if $\gothp$ splits in $E/F$ and $\tilde \tau$ induces the $p$-adic place $\gothq$, and is $-2$ if $\gothp$ splits in $E/F$ and $\tilde \tau$ induces the $p$-adic place $\bar \gothq$. It suffices to define $\delta'_{\tau_0,\sharp}$ on the geometric closed points, as both Shimura varieities are zero-dimensional. For each $\overline \FF_p$-point $x = (A, \iota ,\lambda_{A}, \alpha_{K'}) \in {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})(\overline \FF_p)$, let $\tilde \calD^\circ_{A,\tilde \tau}$ denote the $\tilde\tau$-component of the reduced covariant Dieudonné module of $A$ for each $\tilde \tau \in \Sigma_{E, \infty}$. We put $M_{\tilde \tau} = \tilde \calD^\circ_{A,\tilde \tau}$ for $\tilde \tau \neq \tilde \tau_0, \tilde \tau_0^c$ and $$M_{\tilde \tau_0^c} = p\tilde \calD^\circ_{A,\tilde \tau_0^c},\quad M_{\tilde \tau_0} = \frac 1p\tilde \calD^\circ_{A,\tilde \tau_0}\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau_0}[\frac{1}{p}].$$ We check that the signature condition implies that $M_{\tilde\tau}$’s are stable under the actions of Frobenius and Verschiebung of ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau_0}[1/p]$. As in the proof of Proposition \[P:Y\_S=X\_S\], this gives rise to an abelian variety $B$ of dimension $4g$ with an action of $\calO_D$ and an $\calO_D$-quasi-isogeny $\phi:B \to A$ such that the induced morphism on Dieudonné modules $\phi_*: {\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau}[1/p]$ is identified with the natural inclusion $M_{\tilde\tau}{\hookrightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau}[1/p]$ for all $\tilde\tau\in \Sigma_{E,\infty}$. The polarization $\lambda_{A}$ induces naturally a polarization $\lambda_{B}$ on $B$ such that $\lambda_B=\phi^\vee\circ \lambda_A\circ\phi$, since $M_{\tilde\tau}$ is the dual lattice of $M_{\tilde\tau^c}$ for every $\tilde\tau\in \Sigma_{E,\infty}$. When $\gothp$ is of type $\alpha_2$, then $K'_p$ is the Iwahoric subgroup and the level structure at $\gothp$ is equivalent to the data of a collection of submodules $L_{\tilde \tau} \subset \tilde \calD^\circ_{A,\tilde \tau}$ for $\tilde \tau \in \Sigma_{E, \infty/\gothq}$ which are stable under the action of Frobenius and Verschiebung morphisms and such that $\tilde \calD^\circ_{A,\tilde \tau} / L_{\tilde \tau}$ is a one-dimensional vector space over $\overline \FF_p$ for each $\tilde \tau$. This then gives rise to a level structure at $\gothp$ for $B_x$ by taking $L'_{\tilde \tau} = L_{\tilde \tau}$ if $\tilde \tau \neq \tilde \tau_0$, and $L'_{\tilde \tau_0} = p^{-1}L_{\tilde \tau_0}$. It is clear that other level structures of $A$ transfer to that of $B$ automatically. This then defines a morphism $$\delta'_{\tau_0,\sharp}: {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0} \to {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0}.$$ One checks easily that one can reverse the construction to recover $A$ from $B$. So $\delta'_{\tau_0}$ is an isomorphism, and there exists a $p$-quasi-isogeny $$\delta'^{\sharp}_{\tau_0}: \bfA'_{\tilde \ttS, k_0} \to (\delta'_{\tau_0})^*\bfA'_{\tilde \ttS', k_0},$$ whose base change to $x$ is $\phi^{-1}: A{\rightarrow}B$ constructed above. It is evident by construction that $(\delta'_{\tau_0,\sharp},\delta'^{\sharp}_{\tau_0})$ is a link morphism of the prescribed indentation associated to the trivial link $\ttS{\rightarrow}\ttS'$. The following proposition will play a crucial role in our application in [@tian-xiao3]. \[P:uniqueness-link-morphism\] For a given link $\eta: \ttS{\rightarrow}\ttS'$ with all curves (if any) turning to the right and an integer $n\in {\mathbb{Z}}$ (with $n=0$ if $\gothp$ is inert in $E$), there exists at most one link morphism of indentation degree $n$ from ${\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS})_{k_0}$ to ${\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS'})_{k_0}$ (or from ${\mathbf{Sh}}_{K''_p}(G''_{\tilde\ttS})_{k_0}$ to ${\mathbf{Sh}}_{K''_p}(G''_{\tilde\ttS'})_{k_0}$ ) associated to $\eta$. Since ${\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS})_{k_0}$ and ${\mathbf{Sh}}_{K''_p}(G''_{\tilde\ttS})_{k_0}$ have canonically isomorphic neutral connected component (and the restrictions of $\bfA'_{\tilde\ttS,k_0}$ and $\bfA''_{\tilde\ttS,k_0}$ to this neutral connected component are also canonically isomorphic), it suffices to treat the case of ${\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS})_{k_0}$. Let $(\eta'_{i,\sharp}, \eta'^{ \sharp}_{i})$ for $i=1,2$ be two link morphisms of indentation degree $n$ associated to $\eta$. By the moduli property of ${\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS'})_{k_0}$, it suffices to show that the $p$-quasi-isogeny of abelian varieties $$\phi: \eta'^{ *}_{1,\sharp}(\bfA'_{\tilde\ttS',k_0}) \xleftarrow{\eta'^{\sharp}_{1}}\bfA'_{\tilde\ttS, k_0}{\xrightarrow}{\eta'^{\sharp}_{2}} \eta'^{ *}_{2,\sharp}(\bfA'_{\tilde\ttS',k_0})$$ is an isomorphism. By [@rapoport-zink Proposition 2.9], the locus where $\phi$ is an isomorphism is a closed subscheme of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$. As ${\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS})_{k_0}$ is a reduced variety, $\phi$ is an isomorphism if and only if it is so after base changing to every $\overline \FF_p$-point of ${\mathbf{Sh}}_{K'_p}(G'_{\tilde\ttS})_{k_0}$. Let $x$ be an $\overline \FF_p$-point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$, and put $x_i=\eta'_{i,\sharp}(x)$ for $i=1,2$. Consider first the case $\ttS_{\infty}\neq \Sigma_{\infty}$. By condition \[D:link-morphism\](3), there exists an integer $u_{\tilde\tau}$ for each $\tilde\tau\in \Sigma_{E,\infty}-\ttS'_{E,\infty}$ such that $$\phi_{x,*}\big({\tilde{{\mathcal{D}}}}(\bfA_{\tilde\ttS',x_1})^{\circ}_{\tilde\tau}\big)=p^{u_{\tilde\tau}}{\tilde{{\mathcal{D}}}}(\bfA_{\tilde\ttS',x_2})^{\circ}_{\tilde\tau}.$$ We claim that $u_{\tilde\tau}$ must be $0$ for all $\tilde\tau$. Note that the cokernel of $$F^{n_{\tau}}_{{\mathrm{es}},\bfA_{\tilde\ttS',x_i}}: {\tilde{{\mathcal{D}}}}(\bfA_{\tilde\ttS',x_i})^{\circ}_{\sigma^{-n_{\tau}}\tilde\tau}{\rightarrow}{\tilde{{\mathcal{D}}}}(\bfA_{\tilde\ttS',x_i})^{\circ}_{\tilde\tau}$$ has dimension $1$ over $k(x_i)$ for $i=1,2$. Since $\phi_{x,*}$ commutes with $F^{n_{\tau}}_{{\mathrm{es}}}$, we see that $u_{\tilde\tau}=u_{\sigma^{-n_{\tau}}\tilde\tau}$. Consequently, for all $\tilde\tau\in \Sigma_{E,\infty/\gothq}$, $u_{\tilde\tau}$ takes the same value, which we denote by $u$. However, both $\eta'^{\sharp}_{1, \gothq}$ and $\eta'^{\sharp}_{2, \gothq}$ have degree $p^{2n}$ by condition \[D:link-morphism\](4). It follows that $$\phi_{x,\gothq}:\eta'^{ *}_{1,\sharp}(\bfA'_{\tilde\ttS',x_1})[\gothq^{\infty}]{\rightarrow}\eta'^{ *}_{2,\sharp}(\bfA'_{\tilde\ttS',x_2})[\gothq^{\infty}]$$ is a quasi-isogeny of degree $0$, which forces $u$ to be $0$. Hence $\phi_{*}$ is an isomorphism. When $\ttS_{\infty}=\Sigma_{\infty}$, we have similarly an integer $u_{\tilde\tau}$ for all $\tilde\tau\in \Sigma_{E,\infty}$ such that $\phi_{*}\big({\tilde{{\mathcal{D}}}}(\bfA_{\tilde\ttS',x_1})^{\circ}_{\tilde\tau}\big)=p^{u_{\tilde\tau}}{\tilde{{\mathcal{D}}}}(\bfA_{\tilde\ttS',x_2})^{\circ}_{\tilde\tau}$. Since ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$ and ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'})_{k_0}$ are both zero-dimensional and $$F_{{\mathrm{es}},\bfA'_{\tilde\ttS,x_i}}: {\tilde{{\mathcal{D}}}}(\bfA_{\tilde\ttS',x_i})^{\circ}_{\sigma^{-1}\tilde\tau}{\rightarrow}{\tilde{{\mathcal{D}}}}(\bfA_{\tilde\ttS',x_i})_{\tilde\tau}^{\circ}$$ is an isomorphism for all $\tilde\tau\in \Sigma_{E,\infty}$ and $i=1,2$, the commutativity of $\phi_*$ with essential Frobenii show that $u_{\tilde\tau}=u_{\sigma^{-1}\tilde\tau}$. The same arguments as above show that $\phi_*$ is an isomorphism. This Proposition does not guarantee the existence of the link morphism associated to a given link. Link morphisms and Hecke operators ---------------------------------- Assume $\ttS_\infty = \Sigma_\infty$ so that $\gothp$ is of type $\alpha$ or $\alpha^{\sharp}$ and the band associated to $\ttS$ consists of only plus signs. Let $\gothq$ and $\bar\gothq$ be the two primes of $E$ above $\gothp$. We will focus on the compatibility of link morphisms with the Hecke operators at $\gothq$, whose definition we recall now. We have the following description $$G''_{\tilde \ttS}({\mathbb{Q}}_p) \cong {\mathrm{GL}}_2(F_{\gothp})\times_{F_{\gothp}^{\times}}(E_{\gothq}^{\times}\times E_{\bar\gothq}^{\times}){\xrightarrow}{\sim} {\mathrm{GL}}_2(E_{\gothq})\times F_{\gothp}^{\times},$$ where the last isomorphism is given by $(g, (\lambda_1,\lambda_2))\mapsto(g\lambda_1, \det(g)\lambda_{1}\lambda_2)$ for $g\in {\mathrm{GL}}_2(F_{\gothp})$ and $\lambda_1\in E_{\gothq}^{\times}$ and $\lambda_2\in E_{\bar\gothq}^{\times}$. Then $G'_{\tilde \ttS}({\mathbb{Q}}_p)$ is the subgroup ${\mathrm{GL}}_2(E_{\gothq})\times {\mathbb{Q}}_p^{\times}$ of $G''_{\tilde \ttS}({\mathbb{Q}}_p)$. Let $\gamma_{\gothq}$ (resp. $\xi_\gothq$) be the element of $G'_{\tilde \ttS}(\AAA^{\infty})$ which is equal to $$(\begin{pmatrix}p^{-1}&0\\0&p^{-1}\end{pmatrix},1)\in {\mathrm{GL}}_2(E_\gothq)\times{\mathbb{Q}}_p^{\times} \quad \textrm{(resp. }(\begin{pmatrix}p^{-1} &0\\ 0&1\end{pmatrix}, 1)\in {\mathrm{GL}}_2(E_\gothq)\times{\mathbb{Q}}_p^{\times} \ )$$ at $p$ and is equal to $1$ at other places. Assume that $K'\subseteq G'(\AAA^{\infty})$ is hyperspecial at $p$, i.e. $K'_p={\mathrm{GL}}_{2}({\mathcal{O}}_{E_{\gothq}})\times \ZZ_p^{\times}$. We use $S_{\gothq}$ and $T_{\gothq}$ to denote the Hecke correspondences on ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})$ defined by $K'\gamma_\gothq K'$ and $K'\xi_\gothq K'$, respectively. Explicitly, if ${\mathrm{Iw}}'_p= {\mathrm{Iw}}_{\gothq}\times \ZZ_p^{\times}\subseteq G'_{\tilde\ttS}({\mathbb{Q}}_p)$ with ${\mathrm{Iw}}_{\gothq}\subseteq {\mathrm{GL}}_2({\mathcal{O}}_{E_{\gothq}})$ the standard Iwahoric subgroup reducing to upper triangular matrices when modulo $p$, then the Hecke correspondence $T_{\gothq}$ is given by the following diagram: $$\label{E:Hecke-T_q} \xymatrix{ & {\mathbf{Sh}}_{K'^p {\mathrm{Iw}}'_p}(G'_{\tilde\ttS}) \ar[rd]^{{\mathrm{pr}}_2}\ar[ld]_{{\mathrm{pr}}_1}\\ {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})&& {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS}), }$$ where ${\mathrm{pr}}_1$ is the natural projection, and ${\mathrm{pr}}_2$ is induced by the right multiplication by $\xi_{\gothq}$. Note that $S_{\gothq}$ is an automorphism of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})$, and there is a natural $p$-quasi-isogeny of universal abelian schemes $$\Phi_{S_{\gothq}}: \bfA'_{\tilde \ttS}{\rightarrow}S_{\gothq}^*\bfA'_{\tilde\ttS}$$ compatible with all structures such that the induced quasi-isogeny of $p$-divisible groups $\Phi_{S_{\gothq}}[\gothq^{\infty}]: \bfA'_{\tilde\ttS}[\gothq^{\infty}]{\rightarrow}(S_{\gothq}^*\bfA'_{\tilde\ttS})[\gothq^{\infty}]$ is the canonical isogeny with kernel $\bfA'_{\tilde\ttS}[\gothq]$. Similarly, the elements $\gamma_{\gothq}$ and $\xi_{\gothq}$ induce Hecke correspondences on ${\mathbf{Sh}}_{K''}(G''_{\tilde\ttS})$, which we denote still by $S_\gothq$ and $T_\gothq$ respectively. Assume that $\ttS_\infty = \Sigma_\infty$. Let $\tilde \ttS_\infty$ and $\tilde \ttS'_\infty$ be two different choices of signatures in Subsection \[S:CM extension\]. Suppose that there exists a link morphism $(\eta'_\sharp,\eta'^{\sharp})$ from $ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}$ to $ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0}$ (of some indentation) associated to the trivial link $\ttS \to \ttS'$, where $K'_p= {\mathrm{GL}}_2({\mathcal{O}}_{E_{\gothq}})\times {\mathbb{Z}}_p^{\times}\subseteq G'({\mathbb{Q}}_p)$ is hyperspecial. Then $(\eta'_{\sharp}, \eta'^{\sharp})$ lifts uniquely to a link morphism $(\eta'_{\sharp, {\mathrm{Iw}}}, \eta'^{\sharp}_{{\mathrm{Iw}}})$ on ${\mathbf{Sh}}_{K'^p{\mathrm{Iw}}'_p}(G'_{\tilde\ttS})_{k_0}$ such that the following commutative diagrams are Cartesian: $$\xymatrix{ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0} \ar[d]^{\eta'_\sharp} & {\mathbf{Sh}}_{K'^p{\mathrm{Iw}}'_p}(G'_{\tilde \ttS})_{k_0} \ar[l]_{{\mathrm{pr}}_1}\ar[d]^{\eta'_{\sharp,{\mathrm{Iw}}}} \ar[r]^{{\mathrm{pr}}_2} & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}\ar[d]^{\eta'_{\sharp}} & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}\ar[d]^{\eta'_\sharp} \ar[r]^{S_\gothq} & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0}\ar[d]^{\eta'_\sharp} \\ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0} & {\mathbf{Sh}}_{K'^p{\mathrm{Iw}}'_{p}}(G'_{\tilde \ttS'})_{k_0}\ar[l]_{{\mathrm{pr}}_1} \ar[r]^{{\mathrm{pr}}_2} & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0} & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0} \ar[r]^{S_\gothq} & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0} }$$ where the top and the bottom lines of the left diagram are the Hecke correspondences $T_{\gothq}$ defined above. The same holds for the link morphism $(\eta''_\sharp,\eta''^{\sharp})$: ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS})_{k_0}\to {\mathbf{Sh}}_{K''}(G''_{\tilde \ttS'})_{k_0}$. Note that $S_\gothq$ is in fact an isomorphism of Shimura varieties; so the compatibility with $S_\gothq$-action follows from the uniqueness of link morphism by Proposition \[P:uniqueness-link-morphism\]. We prove now the existence of the lift $(\eta'_{\sharp,{\mathrm{Iw}}},\eta'^{\sharp}_{{\mathrm{Iw}}})$, whose uniqueness is proved in \[P:uniqueness-link-morphism\]. Let $x=(A,\iota,\lambda, \alpha_{K'^p})$ be a point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}(\overline \FF_p)$. Put $x'=\eta'_{\sharp}(x)=(A',\iota',\lambda',\alpha'_{K'^p})$. By Definition \[D:link-morphism\](3), for any $\tilde\tau\in \Sigma_{E,\infty}$, there exists $t_{\tilde\tau}\in \ZZ$ independent of $x$ such that $\eta'^{\sharp}_{*}({\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau})=p^{t_{\tilde\tau}}{\tilde{{\mathcal{D}}}}^{\circ}_{A',\tilde\tau}$. Fix a $\tilde\tau_0\in \Sigma_{E,\infty/\gothq}$. Giving a point $y$ of ${\mathbf{Sh}}_{K'^p{\mathrm{Iw}}'_{p}}(G'_{\tilde\ttS})_{k_0}$ with ${\mathrm{pr}}_1(y)=x$ is equivalent to giving a $W(\overline \FF_p)$-submodule $\tilde H_{\tilde\tau_0}\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau_0}$ such that $F_{{\mathrm{es}}, A}^{g}(\tilde H_{\tilde\tau_0})=\tilde H_{\tilde\tau_0}$ and ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau_0}/\tilde H_{\tilde\tau_0}$ is one-dimensional over $\overline \FF_p$. We put $\tilde H_{\tilde\tau_0}'=p^{-t_{\tilde\tau_0}}\eta'^{\sharp}_*(\tilde H_{\tilde\tau_0})\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{A',\tilde\tau_0}$. Then one sees easily that the quotient ${\tilde{{\mathcal{D}}}}^{\circ}_{A',\tilde\tau_0}/\tilde H'_{\tilde\tau_0}$ is one-dimensional over $\overline \FF_p$ and $\tilde H'_{\tilde\tau_0}$ is fixed by $F_{{\mathrm{es}}, A'}^g$. This gives rise to a point $y'$ of ${\mathbf{Sh}}_{K'^p{\mathrm{Iw}}'_p}(G'_{\tilde\ttS'})_{k_0}$ with ${\mathrm{pr}}_1(y')=x'$. One defines thus $\eta_{\sharp,{\mathrm{Iw}}}'(y)=y'$, and the quasi-isogeny $\eta'^{\sharp}_{{\mathrm{Iw}}}$ as the pull-back of $\eta'^{\sharp}$ via ${\mathrm{pr}}_1$. It is clear by construction that $\eta'_{\sharp}\circ{\mathrm{pr}}_{1}={\mathrm{pr}}_1\circ \eta'_{\sharp,{\mathrm{Iw}}}$. It remains to prove that $\eta'_{\sharp}\circ {\mathrm{pr}}_2={\mathrm{pr}}_2\circ\eta'_{\sharp,{\mathrm{Iw}}}$. Let $y=(A,\iota, \lambda, \alpha_{K'^p}, \tilde H_{\tilde\tau_0})\in {\mathbf{Sh}}_{K'^p{\mathrm{Iw}}_p}(G'_{\tilde\ttS})_{k_0}$ be a point above $x$ as above. We put ${\tilde{{\mathcal{D}}}}_{A,\gothq}^{\circ}\colon =\bigoplus_{\tilde\tau\in \Sigma_{\infty/\gothq}} {\tilde{{\mathcal{D}}}}_{A,\tilde\tau}^{\circ}$, and we define ${\tilde{{\mathcal{D}}}}_{A,\bar\gothq}^{\circ}$ similarly with $\gothq$ replaced by $\bar\gothq$. Then $\tilde H_{\gothq}\colon =\frac{1}{p}\bigoplus_{i=0}^{g-1} F_{{\mathrm{es}}, A}^{i}(\tilde H_{\tilde\tau_0})$ is a $W(\overline \FF_p)$-lattice of ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\gothq}[1/p]$ stable under the action of $F$ and $V$. Let $\tilde H_{\bar\gothq}=\tilde H^{\vee}_{\gothq}\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{A,\bar\gothq}[1/p]$ denote the dual lattice of $\tilde H_{\gothq}$ under the perfect pairing between ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\gothq}[1/p]$ and ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\bar\gothq}[1/p]$ induced by $\lambda$. By the theory of Dieudonné modules, there exists a unique abelian variety $B$ equipped with ${\mathcal{O}}_D$-action $\iota_B$ together with an ${\mathcal{O}}_D$-linear $p$-quasi-isogeny $\phi\colon B{\rightarrow}A$ such that $\phi_*({\tilde{{\mathcal{D}}}}_{B}^{\circ})$ is identified with the lattice $\tilde H_{\gothq}\oplus \tilde H_{\bar\gothq}$ of ${\tilde{{\mathcal{D}}}}_{A}^{\circ}[1/p]$. Note that $B$ satisfies the signature condition of ${\mathbf{Sh}}_{K'}(G_{\tilde\ttS})_{k_0}$. Since $\tilde H_{\gothq}$ and $\tilde H_{\bar\gothq}$ are dual to each other, the quasi-isogeny $\lambda_B = \phi^{\vee}\circ\lambda\circ\phi\colon B{\rightarrow}B^{\vee}$ is a prime-to-$p$ polarization $\lambda_B$ on $B$. We equip moreover $B$ with the $K'^p$-level structure $\beta_{K'^p}$ such that $\alpha_{K'^p}=\phi\circ \beta_{K'^p}$. Thus $z\colon =(B,\iota_B,\lambda_B, \beta_{K'^p})$ gives rise to an $\overline \FF_p$-point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$, and we have $z={\mathrm{pr}}_2(y)$ by the moduli interpretation of ${\mathrm{pr}}_2$. Let $(B',\iota_{B'},\lambda_{B'},\beta'_{K'^p})$ denote the image $\eta'_{\sharp}(z)$. Then ${\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau_0}$ is identified via $(\eta'^{\sharp}_{*})^{-1}$ with the lattice $p^{-t_{\tilde\tau_0}} {\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau_0}$ of ${\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau_0}[1/p]$, hence with the lattice $p^{-t_{\tilde\tau_0}-1}\tilde H_{\tilde\tau_0}$ of ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau_0}[1/p]$. By our construction of $y'=\eta'_{\sharp, {\mathrm{Iw}}}(y)$, it is easy to see that, if ${\mathrm{pr}}_2(y')=(B'',\iota_{B''},\lambda_{B''},\beta''_{K'^p})$, then ${\tilde{{\mathcal{D}}}}^{\circ}_{B'',\tilde\tau_0}$ can be canonical identified with ${\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau_0}$ as lattices of ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau_0}[1/p]$. Since other components ${\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau}$ or ${\tilde{{\mathcal{D}}}}^{\circ}_{B'',\tilde\tau}$ for $\tilde\tau\in \Sigma_{E,\infty}$ are determined from ${\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau_0}$ by the same rules (i.e. stability under the essential Frobenius and the duality), we see that $B'$ is canonically isomorphic to $B''$, compatible with all structures. This concludes the proof of ${\mathrm{pr}}_2\circ\eta'_{\sharp,{\mathrm{Iw}}}=\eta'_{\sharp}\circ{\mathrm{pr}}_2$. For the rest of this paper, we discuss two topics; their proofs are nested together. One topic is to understand the behavior of the description of the Goren-Oort strata under the link morphisms; the other is to understand the restriction of the $\PP^1$-bundle description of the Goren-Oort strata to other Goren-Oort strata. \[P:restriction of GO strata\] Let $\tau \in \Sigma_{\infty} - \ttS_{\infty}$ be a place such that $\tau^-\neq \tau$, and let $\pi_\tau: {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\tau} \to {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0}$ be the $\PP^1$-bundle fibration given by Theorem \[T:main-thm-unitary\] for the Goren-Oort stratum defined by the vanishing of the partial Hasse invariant at $\tau$. Let $\ttT$ be a subset of $\Sigma_\infty - \ttS_\infty$ containing $\tau$. - If $\tau^+ \notin \ttT$, then we put $\ttT_{\tau} = \ttT\backslash \{\tau, \tau^-\}$ and we have a commutative diagram $$\label{E:good commutative diagram} \xymatrix{ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT} \ar@{^{(}->}[r] \ar[d]& {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \tau} \ar[d]^{\pi_\tau} \\ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0, \ttT_{\tau}} \ar@{^{(}->}[r]& {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0}. }$$ If $\tau^- \in \ttT$ the left vertical arrow is an isomorphism. If $\tau^- \notin \ttT$, this diagram is Cartesian. - If $\tau, \tau^- \in \ttT$ and $\tau^+\neq \tau^-$, then we put $\ttT_{\tau} = \ttT\backslash \{\tau, \tau^-\}$ and $\pi_\tau$ induces a natural isomorphism $$\label{E:easy projection} \pi_\tau\colon {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT} \to {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0, \ttT_{\tau}}$$ Moreover, all descriptions above are compatible with the natural quasi-isogenies on universal abelian varieties, and analogous results hold for ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS})_{k_0}$. The statements for ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS})_{k_0}$ follow from those analogs for ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$ by \[S:transfer math obj\] (or in this case more explicitly by \[S:abel var in unitary case\]). Thus, we will just prove the proposition for ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$. \(1) If $\tau^+ \notin \ttT$, the prime $\gothp$ must be of type $\alpha1$ or $\beta1$. By the proof of Proposition \[P:Y\_S=X\_S\], the natural quasi-isogeny $\phi: \pi_\tau^*(\bfA'_{\tilde \ttS(\tau), k_0}) \to \bfA'_{\tilde \ttS,k_0}|_{ {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}}$ induces an isomorphism on the (reduced) differential forms at $\tilde \tau'$ for all $\tilde \tau'\in \Sigma_{E,\infty}$ *not* lifting $\tau, \tau^-$. So $\pi_\tau$ induces a Cartesian square $$\xymatrix{ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT_{\tau}\cup\{\tau\}} \ar@{^{(}->}[r] \ar[d]^{\pi_{\tau}}& {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \tau} \ar[d]^{\pi_\tau} \\ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0, \ttT_{\tau}} \ar@{^{(}->}[r]& {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0} }$$ This already proves (1) in case $\tau^- \notin \ttT$. Suppose now $\tau^-\in \ttT$. By Proposition \[P:Y\_T=Z\_T\], ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT_{\tau}\cup\{\tau\}}$ is the moduli space of tuples $(B,\iota_B, \lambda_B, \beta_{K'_{\ttT_{\tau}}}; J^{\circ}_{\tilde\tau^-})$, where - $(B,\iota_B, \lambda_B, \beta_{K'_{\ttT_{\tau}}})$ is a point of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0,\ttT_{\tau}}$ with values in a scheme $S$ over $k_0$; - $J^{\circ}_{\tilde\tau^-}$ is a sub-bundle of $H^{{\mathrm{dR}}}_1(B/S)^{\circ}_{\tilde\tau^-}$ of rank 1 (here, $\tilde\tau^-\in \Sigma_{E,\infty}$ is the specific lift of $\tau^-$ defined in Subsection \[S:tilde IT\]). Then the closed subscheme ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT}$ of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT_{\tau}\cup\{\tau\}}$ is defined by the condition: $$J^\circ_{\tilde\tau^-} =F_{B,{\mathrm{es}}}^{n_{\tau^-}}\big(H_1^{\mathrm{dR}}(B^{(p^{n_{\tau^-}})} / S)^{\circ}_{\sigma^{-n_{\tau^-}}\tilde\tau^{-}}\big).$$ This shows that the restriction of $\pi_{\tau}:{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT_{\tau}\cup\{\tau\}}{\rightarrow}{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT_{\tau}}$ to ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT}$ is an isomorphism. \(2) When $\tau^+\notin \ttT$, this was proved in (1). Assume now $\tau^+ \in \ttT$. To complete the proof, it suffices to prove that, for a $k_0$-scheme $S$, the $\tau^+$-th partial Hasse invariant vanishes at an $S$-point $x=(A, \iota_A, \lambda_A, \alpha_{K'})\in {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\{\tau\}}$ if and only if it vanishes at $\pi_{\tau}(x)=(B, \iota_B, \lambda_B, \beta_{K'})\in {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$. Let $\tilde\tau$ be the lift of $\tau$ contained in $\tilde\Delta(\tau)^+$ (See Subsection \[S:Delta-pm\] for the notation $\tilde\Delta(\tau)^+$). Put $\tilde\tau^+=\sigma^{n_{\tau^+}}\tilde\tau$ and $\tilde\tau^-=\sigma^{-n_{\tau}}\tilde\tau$. By Lemma \[Lemma:partial-Hasse\], it suffices to show that $$\begin{aligned} \label{E:equivalent condition for triple Hasse invariant} & F_{A,{\mathrm{es}}}^{n_{\tau^+}} \big( H_1^{\mathrm{dR}}(A/S)^{\circ, (p^{n_{\tau^+}})}_{\tilde \tau} \big) = \omega^\circ_{A^\vee/S, \tilde \tau^+},\\ \nonumber \Leftrightarrow \ & F_{B,{\mathrm{es}}}^{n_{\tau^+}+ n_\tau + n_{\tau^-}}\big( H_1^{\mathrm{dR}}(B/S)^{\circ, (p^{n_{\tau^+}+ n_\tau + n_{\tau^-}})}_{\sigma^{-n_{\tau^-}}\tilde \tau^-} \big) = \omega^\circ_{B^\vee/S, \tilde \tau^+}.\end{aligned}$$ But this follows from the following three facts. - By the definition of essential Frobenius \[N:essential frobenius and verschiebung\], one deduces a commutative diagram $$\xymatrix{H_1^{\mathrm{dR}}(A/S)_{\tilde \tau}^{\circ,(p^{n_{\tau^+}})}\ar[rr]^{F_{A,{\mathrm{es}}}^{n_{\tau^+}}}\ar[d]_{\phi_{*,\tilde\tau}} && H^{{\mathrm{dR}}}_1(A/S)_{\tilde\tau^+}\ar[d]^{\phi_{*,\tilde\tau^+}}_{\cong}\\ H_1^{\mathrm{dR}}(B/S)_{\tilde \tau}^{\circ,(p^{n_{\tau^+}})}\ar[rr]^{F_{B,{\mathrm{es}}}^{n_{\tau^+}}} && H^{{\mathrm{dR}}}_{1}(B/S)^{\circ}_{\tilde\tau^+}, }$$ - It follows from condition (v) of the moduli description in Subsection \[S:moduli-Y\_S\] that $$\phi_{*, \tilde\tau}(H_1^{\mathrm{dR}}(A/S)_{\tilde \tau}^\circ) = F_{ B,{\mathrm{es}}}^{n_{\tau^-}+ n_\tau} \big(H_1^{\mathrm{dR}}(B/S)^{\circ, (p^{ n_\tau + n_{\tau^-}})}_{\sigma^{-n_{\tau^-}}\tilde \tau^-} \big).$$ - The condition $\tau^-\neq \tau^+$ implies that the quasi-isogeny $\phi:A{\rightarrow}B$ induces an isomorphism $\phi_{*, \tilde\tau^+}:H^{{\mathrm{dR}}}_{1}(A/S)^{\circ}_{\tilde \tau^+}\cong H^{{\mathrm{dR}}}_1(B/S)^{\circ}_{\tilde \tau^+}$ preserving the Hodge filtrations, in particular identifying the submodules $\phi_{*, \tilde \tau}(\omega^\circ_{A^\vee/S, \tilde \tau^+}) = \omega^\circ_{B^\vee/S, \tilde \tau^+}$. Compatibility of link morphisms and the description of Goren-Oort stata ----------------------------------------------------------------------- We first recall that, although the subset $\ttS(\tau)$ is completely determined by $\ttS$ and $\tau$ as in Subsection \[S:quaternion-data-T\], the lift $\tilde \ttS(\tau)_{\infty}$, which consists of all $\tilde\tau'\in \Sigma_{E,\infty}$ with signature $s_{\tilde\tau'}=0$ (see Subsection \[S:CM extension\]), depends on an auxiliary choice in Subsection \[S:tilde S(T)\]: a lift $\tilde\tau$ of $\tau$ to be contained in $\tilde\ttS(\tau)_{\infty}$. We assume that $\#(\Sigma_\infty-\ttS_\infty) \geq 2$. If $\gothp$ splits as $\gothq\bar\gothq$ in $E$ for a fixed place $\gothq$, then the $\tilde\tau$ contained in $\tilde\ttS(\tau)_{\infty}$ is always chosen to be the one in $\Sigma_{E,\infty/\gothq}$. If $\gothp$ is inert in $E$, then there are two possible choices: $\tilde\tau$ and its conjugate $\tilde\tau^c$ for a fixed lift $\tilde\tau$ of $\tau$. In the latter case, we denote by $\tilde\tau$ the lift of $\tau$ contained in $\tilde\ttS(\tau)_{\infty}$, by $\tilde\ttS(\tau)'=(\ttS(\tau),\ttS(\tau)'_{\infty})$ the lift of $\ttS(\tau)$ such that $\tilde\tau^c\in \tilde\ttS(\tau)'_{\infty}$, and let $\pi_{\tau}'\colon {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\tau} \to {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)'})_{k_0}$ be the corresponding $\PP^1$-bundle. The following proposition says that $\pi_{\tau}$ and $\pi_{\tau}'$ differ from each other by a link isomorphism. \[P:ambiguity-signature\] Assume that $\gothp$ is inert in $E$. Then there exists a link isomorphism $$(\eta'_{ \tilde\ttS(\tau), \tilde \ttS(\tau)', \sharp},\eta'^{\sharp}_{\tilde\ttS(\tau),\tilde\ttS(\tau)'})\colon {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0}{\xrightarrow}{\sim} {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)'})_{k_0}$$ (of indentation degree $0$) associated to the identity link $\eta: \ttS(\tau){\rightarrow}\ttS(\tau)$ such that the diagram $$\xymatrix{ & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \tau} \ar[dl]_{\pi_\tau} \ar[dr]^{\pi'_\tau} \\ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0} \ar[rr]_{\eta_{\tilde \ttS(\tau), \tilde \ttS(\tau)', \sharp}}^{\cong} && {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)'})_{k_0} }$$ commutes. Moreover, the link morphism $\eta_{\tilde \ttS(\tau), \tilde \ttS(\tau)', \sharp}$ satisfies the natural cocycle condition under composition. Similar statements hold for ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS(\tau)})_{k_0}$ and ${\mathbf{Sh}}_{K''}(G''_{\tilde\ttS(\tau)'})_{k_0}$. Consider the closed subvariety ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau, \tau^-\}}$ of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}$. Then by Proposition \[P:restriction of GO strata\] (2) (note that $\gothp$ being inert in $E$ implies that $\tau^+\neq \tau^-$), $\pi|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau^-, \tau\}}}$ is an isomorphism. We put $$\eta'_{\tilde \ttS(\tau),\tilde \ttS(\tau)',\sharp}\colon =\pi'_{\tau}\circ(\pi|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau^-, \tau\}}})^{-1}$$ and define $\eta'^{\sharp}_{\tilde \ttS(\tau),\tilde \ttS(\tau)'}$ as the pull-back via $(\pi_{\tau}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau^-, \tau\}}})^{-1}$ of the quasi-isonegy $$\pi_{\tau}^*\bfA'_{\tilde \ttS(\tau),k_0}{\rightarrow}\bfA'_{\tilde\ttS,k_0}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0,\{\tau,\tau^-\}}}{\rightarrow}\pi'^*_{\tau}\bfA_{\tilde\ttS(\tau)',k_0},$$ where the two quasi-isogenies given by Theorem \[T:main-thm-unitary\](2). By Proposition \[P:restriction of GO strata\](2) again, $\pi'_{\tau}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau,\tau^-\}}}$ is an isomorphism, hence so is $\eta'_{\tilde \ttS(\tau),\tilde \ttS(\tau)',\sharp}$. It remains to show that $(\eta'_{\tilde \ttS(\tau),\tilde \ttS(\tau)',\sharp}, \eta'^{\sharp}_{\tilde \ttS(\tau),\tilde \ttS(\tau)'})$ is a link morphism associated to the identity link on $\ttS$. Let $x$ be a geometric point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$, and $x'=\eta'_{\tilde \ttS(\tau),\tilde \ttS(\tau)',\sharp}(x)$. By construction, it is easy to see that the quasi-isogeny $\eta'^{\sharp}_{\tilde \ttS(\tau),\tilde \ttS(\tau)'}$ induces an isomorphism ${\tilde{{\mathcal{D}}}}(\bfA'_{\tilde\ttS(\tau), k_0,x})^{\circ}_{\tilde\tau'}{\xrightarrow}{\sim} {\tilde{{\mathcal{D}}}}(\bfA'_{\tilde\ttS(\tau)', k_0,x'})^{\circ}_{\tilde\tau'}$ for $\tilde\tau'\neq \sigma^a\tilde\tau, \sigma^a\tilde\tau^c$ with $a = 0, \dots, n_\tau-1$; and in the exceptional cases, we have $$\eta'^{\sharp}_{\tilde \ttS(\tau),\tilde \ttS(\tau)'}({\tilde{{\mathcal{D}}}}(\bfA'_{\tilde\ttS(\tau), k_0,x})^{\circ}_{\tilde\tau'})=\begin{cases} p{\tilde{{\mathcal{D}}}}(\bfA'_{\tilde\ttS(\tau)', k_0,x'})^{\circ}_{\tilde\tau'} &\text{for }\tilde\tau'=\sigma^a\tilde\tau \textrm{ for }a = 0, \dots, n_\tau-1;\\ \frac{1}{p} {\tilde{{\mathcal{D}}}}(\bfA'_{\tilde\ttS(\tau)', k_0,x'})^{\circ}_{\tilde\tau'} &\text{for }\tilde\tau'=\sigma^a\tilde\tau^c\textrm{ for }a = 0, \dots, n_\tau-1. \end{cases}$$ Hence, $(\eta'_{\tilde \ttS(\tau),\tilde \ttS(\tau)',\sharp}, \eta'^{\sharp}_{\tilde \ttS(\tau),\tilde \ttS(\tau)'})$ verifies Definition \[D:link-morphism\]. The other statements of this proposition are evident by the uniqueness of link morphisms (Proposition \[P:uniqueness-link-morphism\]). The following Lemma will be needed in the proof of the main result of this section. \[L:non-vanishing-chi\] Assume that $\ttS_\infty \neq \emptyset$. Let $\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\overline \FF_p}): =\sum_{i=0}^{+\infty}(-1)^i\dim H^{i}_{\mathrm{et}}({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\overline \FF_p}, {\mathbb{Q}}_{\ell})$ denote the Euler-Poincaré characteristic of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\overline \FF_p}$ for some fixed prime $\ell\neq p$. Then we have $\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\overline \FF_p})\neq 0$. The assumption $\ttS_\infty \neq \emptyset$ implies that all the Shimura varieties we talk about are proper. Consider the integral model ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})$ over ${\mathcal{O}}_{\tilde\wp}$, and choose an embedding ${\mathcal{O}}_{\tilde\wp}{\hookrightarrow}\CC$. By proper base change theorem and standard comparison theorems, we have $$\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\overline \FF_p})=\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC}): =\sum_{i=0}^{\infty} (-1)^{i}\dim H^{i}({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC}, \CC),$$ where $H^{i}({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC}, \CC)$ denotes the singular cohomology of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC}$ for the usual complex topology. For each $\tilde\tau$ lifting an element of $\Sigma_{\infty}-\ttS_{\infty}$, put $\omega_{\tilde\tau}^{\circ}=\omega_{\bfA'_{\tilde\ttS,\CC},\tilde\tau}^{\circ}$ and $\gotht_{\tilde\tau}^{\circ}=\operatorname{Lie}(\bfA'_{\tilde\ttS,\CC})_{\tilde\tau}=\omega^{\circ,\vee}_{\tilde\tau}$ to simplify the notation. They are line bundles over ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC}$. We have a Hodge filtration $$0{\rightarrow}\omega^{\circ}_{\tilde\tau^c}{\rightarrow}H^{{\mathrm{dR}}}_1(\bfA'_{\tilde\ttS,\CC}/{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC})^{\circ}_{\tilde\tau}{\rightarrow}\gotht_{\tilde\tau}^{\circ}{\rightarrow}0.$$ Note that $H^{{\mathrm{dR}}}_1(\bfA'_{\tilde\ttS,\CC}/{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC})_{\tilde\tau}^{\circ}$ is equipped with the integrable Gauss-Manin connection so that its Chern classes are trivial by classical Chern-Weil theory. One obtains thus $$\begin{cases} c_1(\omega_{\tilde\tau^c}^{\circ})c_1(\gotht_{\tilde\tau}^{\circ})=c_2(H^{{\mathrm{dR}}}_1(\bfA'_{\tilde\ttS,\CC}/{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC})^{\circ}_{\tilde\tau})=0,\\ c_1(\omega_{\tilde\tau^c}^{\circ})+c_1(\gotht^{\circ}_{\tilde\tau})=c_1(H^{{\mathrm{dR}}}_1(\bfA'_{\tilde\ttS,\CC}/{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC})^{\circ}_{\tilde\tau})=0, \end{cases} \Longrightarrow \begin{cases} c_1(\omega^{\circ}_{\tilde\tau})^2=0,\\ c_1(\omega^{\circ}_{\tilde\tau})=c_1(\omega_{\tilde\tau^c}^{\circ}), \end{cases}$$ where $c_i(\calE)\in H^{2i}({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC},\CC)$ denotes the $i$-th Chern class of a vector bundle $\calE$. Let $\calT$ denote the tangent bundle of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})(\CC)$, and put $\det(\omega): =\bigoplus_{\tilde\tau\in \Sigma_{E,\infty}}\omega^{\circ}_{\tilde\tau}$. By Proposition \[Prop:smoothness\],[^1], we get $$\begin{cases}c_1(\calT)=-2\sum_{\tau\in \Sigma_{\infty}-\ttS_{\infty}} c_1\big(\omega_{\tilde\tau}^{\circ}\big)=-c_1(\det(\omega)),\\ c_d(\calT)=\prod_{\tau\in \Sigma_{\infty}-\ttS_{\infty}}(-2c_1(\omega_{\tilde\tau}^{\circ})), \end{cases}$$ where $\tilde\tau\in \Sigma_{E,\infty}$ is an arbitrary lift of $\tau$, and $d=\#\Sigma_{\infty}-\#\ttS_{\infty}$ is the dimension of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC}$. Note that $c_1(\omega^{\circ}_{\tilde\tau})^2=0$ implies that $c_d(\calT)=\frac{(-1)^{d}}{d!}c_1(\det(\omega))^d$. It is well known that $\det(\omega)$ is ample (see [@lan], for instance), hence it follows that $c_d(\calT)\neq 0$. On the other hand, there exists a canonical isomorphism $${\mathrm{Tr}}\colon H^{2d}({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC},\CC){\xrightarrow}{\sim} \CC,$$ which sends the cycle class of a point to $1$. The Lemma follows immediately from the non-vanishing of $c_d(\calT)$ and the well-known fact that ${\mathrm{Tr}}(c_d(\calT))=\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{\CC})$. We state now the main result of this section, which will play a crucial role in our application to Tate cycles in [@tian-xiao3]. \[T:link and Hecke operator\] Keep the same notation as in Proposition \[P:restriction of GO strata\], that is, let $\tau \in \Sigma_{\infty} - \ttS_{\infty}$ be a place such that $\tau^-$ is different from $\tau$ and let $\ttT$ be a subset of $\Sigma_\infty - \ttS_\infty$ containing $\tau$. - If $\tau^- \notin \ttT$ and $\tau,\tau^+ \in \ttT$, we put $\ttT_{\tau^+} = \ttT \backslash \{\tau, \tau^+\}$. Let $\eta = \eta_{\tau^-{\rightarrow}\tau^+}: \ttS(\tau^+){\rightarrow}\ttS(\tau)$ be the link given by straight lines except sending $\tau^-$ to $\tau^+$ (to the right) with displacement $v(\eta)=n_{\tau}+n_{\tau^+}$: $$\psset{unit=0.3} \begin{pspicture}(-1.2,-0.4)(16,2.4) \psset{linecolor=red} \psset{linewidth=1pt} \psline{-}(0,0)(0,2) \psline{-}(14.4,0)(14.4,2) \psbezier(4.8,2)(4.8,0)(9.6,2)(9.6,0) \psset{linecolor=black} \psdots(0,0)(0,2)(4.8,2)(9.6,0)(14.4,0)(14.4,2) \psdots[dotstyle=+](1,0)(-1,0)(-1,2)(1,2)(3.8,0) (3.8,2)(4.8,0)(5.8,0)(5.8,2)(8.6,0)(8.6,2)(10.6,0)(9.6,2)(10.6,2)(13.4,0)(13.4,2)(15.4,0)(15.4,2) \psset{linewidth=.1pt} \psdots(1.7,0)(1.7,2)(2.4,0)(2.4,2)(3.1,0)(3.1,2)(6.5,0)(7.2,0)(7.9,0)(6.5,2)(7.2,2)(7.9,2)(11.3,0)(12,0)(12.7,0)(11.3,2)(12,2)(12.7,2). \end{pspicture}$$ 1. We put $\eta'_{ \sharp}: \pi_{\tau}\circ i_{\tau^+}\circ (\pi_{\tau^+|_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^+\}}}})^{-1}$, and let $\eta'^{\sharp}$ denote the natural quasi-isogeny of abelian varieties on $ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^+)})_{k_0}$ given by (the pull-back via $(\pi_{\tau^+|_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^+\}}}})^{-1,*}$ of) $$(\pi_{\tau^+}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^+\}}})^*\bfA'_{\tilde\ttS(\tau^+),k_0} \leftarrow i_{\tau^+}^*(\bfA'_{\tilde\ttS,k_0}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}}){\rightarrow}i_{\tau^+}^*\pi_{\tau}^* (\bfA'_{\tilde\ttS(\tau),k_0}).$$ Then $(\eta'_{ \sharp}, \eta'^{\sharp})$ is the link morphism associated to the link $\eta$ of indentation degree $n=n_{\tau^+}-n_{\tau}$ if $\gothp$ splits in $E/F$ and $n=0$ if $\gothp$ is inert in $E/F$. Moreover, the following diagram $$\label{E:diagram involving link} \xymatrix@C=55pt{ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \ttT} \ar@{^{(}->}[r] \ar[d]_\cong & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^+\}} \ar@{^{(}->}[r]^-{i_{\tau^+}} \ar[d]^{\pi_{\tau^+}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^+\}}}}_\cong & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \tau} \ar[d]^{\pi_\tau} \\ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^+)})_{k_0, \ttT_{\tau^+}} \ar@{^{(}->}[r]& {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^+)})_{k_0} \ar[r]^{\eta'_{ \sharp}}& {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0}, }$$ is commutative, where the two vertical isomorphisms are given by Proposition \[P:restriction of GO strata\](2). 2. For $\tilde\tau'\in \Sigma_{E,\infty}$, the quasi-isogeny $\eta'^{\sharp}:\bfA'_{\tilde\ttS(\tau^+), k_0}{\rightarrow}\eta'^{*}_{\sharp} \bfA'_{\tilde\ttS(\tau), k_0}$ induces a canonical isomorphism $$\eta'^*_{\sharp}(\operatorname{Lie}(\bfA'_{\tilde\ttS(\tau), k_0})_{\tilde\tau'}^{\circ}\cong \begin{cases} \operatorname{Lie}(\bfA'_{\tilde\ttS(\tau^+), k_0})^{\circ, (p^{v(\eta)})}_{\sigma^{-v(\eta)}\tilde\tau'} &\text{if $\tilde\tau'$ is a lifting of $\tau^+$,}\\ \operatorname{Lie}(\bfA'_{\tilde\ttS(\tau^+), k_0})^{\circ}_{\tau'} &\text{otherwise.} \end{cases}$$ 3. The morphism $\eta'_{\sharp}$ is finite flat of degree $p^{v(\eta)}$. - Assume $\Sigma_{\infty} - \ttS_{\infty} = \{\tau, \tau^-\}= \ttT$ (so that $\tau^+ = \tau^-$ and $\gothp$ is of type $\alpha2$ for $\ttT$). Then there exists a link morphism $(\eta_{\sharp},\eta^{\sharp}): {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^-)})_{k_0}{\rightarrow}{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0}$ of indentation degree $2(g-n_{\tau})=2n_{\tau^-}$ associated with the trivial link $\eta:\ttS(\tau^-){\rightarrow}\ttS(\tau)$ such that the diagram $$\xymatrix{ & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^-\}} \ar[dl]_{\pi_\tau} \ar[dr]^{\pi_{\tau^-}} \\ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0} && {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^-)})_{k_0}\ar[rr]^{\eta_\sharp}&& {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0}. }$$ coincides with the Hecke correspondence $T_{\gothq}$ if we identify ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^-\}}$ with ${\mathbf{Sh}}_{K'^p{\mathrm{Iw}}'_p}(G'_{\tilde\ttS(\tau)})_{k_0}$ via the isomorphism given by Theorem \[T:main-thm-unitary\]. All descriptions above are compatible with the natural quasi-isogenies on universal abelian varieties, and similar results apply to ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS})_{k_0}$. The statements for ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS})_{k_0}$ follow from those analogs for ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$ by \[S:transfer math obj\] (or in this case more explicitly by \[S:abel var in unitary case\]). Thus, we will just prove the theorem for ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$. (1)(a) The commutativity of is tautological. It remains to show that $\pi_\tau \circ (\pi_{\tau^+}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^+\}}})^{-1}$ is the link morphism $\eta'_{\sharp}$ on ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau^+)})_{k_0}$ associated to the link $\eta=\eta_{\tau^-{\rightarrow}\tau^+}$. Let $\tilde\tau\in \Sigma_{E,\infty}$ (resp. $\tilde\tau^+$) denote the lift of $\tau$ (resp. $\tau^+$) contained in $\tilde\ttS(\tau)_{\infty}$ (resp. $\tilde\ttS(\tau^+)_{\infty}$). By Subsection \[S:tilde S(T)\], we have $\tilde\tau=\sigma^{-n_{\tau^+}}\tilde\tau^+$ if $\gothp$ is splits in $E$. If $\gothp$ is inert in $E$, it is also harmless to assume $\tilde\tau=\sigma^{-n_{\tau^+}}\tilde\tau^+$ in view of Propositions \[P:ambiguity-signature\] and \[P:compatibility of link and GO\]. Put $\tilde \tau^- =\sigma^{-n_\tau} \tilde\tau$. Let $y=(B, \iota_{B}, \lambda_{B}, \beta'_{K'})$ be an $S$-point of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^+)})_{k_0}$ for a test $k_0$-scheme $S$. Then the pre-image of $y$ under $ \pi_{\tau^+}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \{\tau, \tau^+\}}}$ is given by $x=(A, \iota_A, \lambda_A, \alpha_{K'})\in {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau,\tau^+\}}$, for which there exists a quasi-isogeny $\phi: B \to A$ such that $\phi_{*, \tilde \tau'}: H^{{\mathrm{dR}}}_1(B/S)^{\circ}_{\tilde\tau'}{\rightarrow}H^{{\mathrm{dR}}}_1(A/S)^{\circ}_{\tilde\tau'}$ is a well-defined isomorphism for all $\tilde \tau' \in \Sigma_{E, \infty}$ except for $\tilde \tau'=\sigma^a\tilde\tau$ or $ \sigma^{a}\tilde\tau^c$ with $a = 1, \dots, n_{\tau^+}$. In the exceptional cases, $\phi_{*,\sigma^a\tilde\tau}$ and $(\phi^{-1})_{*, \sigma^a\tilde \tau^c}$ are well defined, where $\phi^{-1}\colon A{\rightarrow}B$ denotes the quasi-isogeny inverse to $\phi$, and we have (by the proof of Proposition \[P:Y\_T=Z\_T\]) $$\begin{aligned} &\quad \operatorname{Ker}(\phi_{*, \sigma^a\tilde \tau}) = F_{B,{\mathrm{es}}}^{n_\tau + a} \big( H^{\mathrm{dR}}_1(B/S)^{\circ, (p^{n_\tau+a})}_{\tilde \tau^-} \big)\cong \operatorname{Lie}(B/S)^{\circ, (p^{n_{\tau}+a})}_{\tilde\tau^-},\text{ and }\\ & {\mathrm{Im}}((\phi^{-1})_{*, \sigma^a\tilde \tau^c}) = F_{B,{\mathrm{es}}}^{n_\tau + a} \big( H^{\mathrm{dR}}_1(B/S)^{\circ, (p^{n_\tau+a})}_{\tilde \tau^{-,c}} \big)\cong \operatorname{Lie}(B/S)^{\circ,(p^{n_{\tau}+a})}_{\tilde\tau^{-,c}}.\end{aligned}$$ If $\pi_\tau$ sends $x=(A, \iota_A, \lambda_A, \alpha_{K'})$ to $z=(B', \iota_{B'}, \lambda_{B'}, \beta_{K'})$, there is a quasi-isogeny $\psi: A \to B'$ such that $\psi_{*, \tilde \tau'}$ is an isomorphism for all $\tilde \tau' \in\Sigma_{E, \infty}$ except for $\tilde \tau'=\sigma^b\tilde\tau^-$ or $\sigma^b\tilde\tau^{-,c}$ with $b =1, \dots, n_\tau$. In the exceptional cases, $\psi_{*, \sigma^b\tilde \tau^{-,c}}$ and $(\psi^{-1})_{*, \sigma^b\tilde \tau^{-}}$ are well defined, and we have (by the proof of Proposition \[P:Y\_S=X\_S\] or rather the moduli description in Subsection \[S:moduli-Y\_S\]) $$\begin{aligned} &\quad \operatorname{Ker}(\psi_{*, \sigma^b\tilde \tau^{-,c}}) = F_{A,{\mathrm{es}}}^{ b} \big( H^{\mathrm{dR}}_1(A/S)^{\circ, (p^{b})}_{\tilde \tau^{-,c}} \big),\text{ and }\\ & {\mathrm{Im}}((\psi^{-1})_{*, \sigma^b\tilde \tau^{-}}) = F_{A,{\mathrm{es}}}^{b} \big( H^{\mathrm{dR}}_1(A/S)^{\circ, (p^{b})}_{\tilde \tau^{-}} \big).\end{aligned}$$ By definition, we have $\eta'_{\sharp}(y)=z$, and the composed quasi-isogeny $\psi\circ\phi: B{\rightarrow}B'$ is nothing but the base change of $\eta'^{\sharp}$ to $S$. For later reference, we remark that $\psi$ and $\phi$ induces isomorphisms $$\label{E:Lie-algebra-equality} \operatorname{Lie}(B/S)^{\circ}_{\tilde\tau'}\cong \operatorname{Lie}(A/S)^{\circ}_{\tilde\tau'}\cong \operatorname{Lie}(B'/S)^{\circ}_{\tilde\tau'}$$ for all $\tilde\tau'$ with restriction $\tau'\in \Sigma_{\infty}-\ttS_{\infty}$ different from $\tau^-,\tau, \tau^+$, and $$\begin{aligned} \label{E:isom-Lie-algebras} \operatorname{Lie}(B/S)^{\circ,(p^{n_{\tau}+n_{\tau^+}})}_{\tilde\tau^-}&{\xrightarrow}{\sim} \operatorname{Coker}(\phi_{*,\tilde\tau^+})\cong \operatorname{Lie}(A/S)^{\circ}_{\tilde\tau^+}{\xrightarrow}{\sim} \operatorname{Lie}(B'/S)^{\circ}_{\tilde\tau^+}\\ \operatorname{Lie}(B/S)^{\circ,(p^{n_{\tau}+n_{\tau^{+}}})}_{\tilde\tau^{-,c}}&{\xrightarrow}{\sim}\operatorname{Coker}(\phi_{*,\tilde\tau^{+,c}})\cong \operatorname{Lie}(A/S)^{\circ}_{\tilde\tau^{+,c}}{\xrightarrow}{\sim} \operatorname{Lie}(B'/S)^{\circ}_{\tilde\tau^{+,c}}\nonumber\end{aligned}$$ Consider the case when $S=\operatorname{Spec}(k)$ with $k$ a perfect field containing $k_0$. Denote by ${\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau'}$ the $\tilde\tau'$-component of the reduced covariant Dieudonné module of $B$. From the discussion above, one sees easily that ${\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau}={\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau'}$ for all $\tilde\tau'\in \Sigma_{E,\infty}$ expect for $\tilde\tau'\in\{\sigma^a\tilde\tau, \sigma^{a}\tilde\tau^c\;|\; 1\leq a\leq n_{\tau^+}\}\cup \{\sigma^b\tilde\tau^-, \sigma^b\tilde\tau^{-,c}\;|\;1\leq b\leq n_{\tau}\}$. In the exceptional cases, we have $$(\psi\circ\phi)^{-1}_*{\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau'}=\begin{cases} p^{-1}F_{B,{\mathrm{es}}}^{n_{\tau}+a}({\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau^-}) &\text{if }\tilde\tau'=\sigma^a\tilde\tau;\\ F_{B,{\mathrm{es}}}^{n_{\tau}+a}({\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau^{-,c}}) &\text{if }\tilde\tau'=\sigma^a\tilde\tau^c;\\ p^{-1}F_{B,{\mathrm{es}}}^{b}({\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau^{-,c}}) &\text{if }\tilde\tau'=\sigma^b\tilde\tau^{-,c};\\ F_{B,{\mathrm{es}}}^b({\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau^{-}}) &\text{if }\tilde\tau'=\sigma^b\tilde\tau^-. \end{cases}$$ Since the essential Frobenius $F_{B,{\mathrm{es}}}$ is bijective after inverting $p$, one sees easily that ${\tilde{{\mathcal{D}}}}^{\circ}_{B}$ can be recovered from ${\tilde{{\mathcal{D}}}}^{\circ}_{B'}$. This implies immediately that $\eta'_{\sharp}: {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^+)})_{k_0} {\rightarrow}{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0} $ induces a bijection on $k$-valued points, i.e. $\eta'_{\sharp}$ verifies condition (1) in Definition \[D:link-morphism\]. By the discussion above, it is also obvious that conditions (2) and (3) of Definition \[D:link-morphism\] are also verified for $(\eta'_{\sharp},\eta'^{\sharp})$. Finally, from the formulas for ${\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau}$, one sees easily that the degree of the quasi-isogeny $$(\phi\circ\psi)_{\gothq}: B[\gothq^{\infty}]{\rightarrow}B'[\gothq^{\infty}]$$ is $2(n_{\tau^+}-n_{\tau})$ if $\gothp$ splits in $E$, and is $0$ if $\gothp$ is inert in $E$. This shows that $(\eta'_{\sharp},\eta'^{\sharp})$ is the link morphism associated to $\eta$ with the said indentation degree. Statement (1)(b) follows from the isomorphisms and applied to the case when $B$ is the universal abelian scheme $\bfA'_{\tilde\ttS(\tau^+),k_0}$. It remains to prove (1)(c). The morphism $\eta'_{\sharp}$ is clearly quasi-finite, and hence finite because ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau^+)})_{k_0}$ is proper by Proposition \[Prop:smoothness\]. Since both ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$ and ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau^+)})_{k_0}$ are regular, we conclude by [@matsumura Theorem 23.1] that $\eta'_{\sharp}$ is flat at every point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$. It remains to see that the degree of $\eta'_{\sharp}$ is $p^{v(\eta)}$. Let $\calT_{\tau}$ and $\calT_{\tau^+}$ denote respectively the tangent bundle of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$ and ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau^+)})_{k_0}$, and let $d=\#\Sigma_{\infty}-\#\ttS(\tau)_{\infty}$ be the common dimension of these Shimura varieties. Fix a prime $\ell\neq p$. For a vector bundle $\calE$ over a proper and smooth $k_0$-variety $X$, we denote by $c_i(\calE)\in H^{2i}_{{\mathrm{et}}}(X_{\overline \FF_p},{\mathbb{Q}}_{\ell})(i)$ the $i$-th Chern class of $\calE$. By Proposition \[Prop:smoothness\], we have $$c_{d}(\calT_{\tau})=\prod_{\tau'\in \Sigma_{\infty}-\ttS(\tau)_{\infty}}c_1 \bigg(\operatorname{Lie}(\bfA'_{\tilde\ttS(\tau),k_0})^{\circ}_{\tilde\tau'}\otimes\operatorname{Lie}(\bfA'_{\tilde\ttS(\tau),k_0})^{\circ}_{\tilde\tau'^c}\bigg),$$ where $\tilde\tau',\tilde\tau'^c\in \Sigma_{E,\infty}$ denote the two liftings of $\tau'$. A similar formula for $c_d(\calT_{\tau^+})$ holds with $\tau$ replaced by $\tau^+$. By (1)(b), we have $$\eta'^{*}_{\sharp}c_{d}(\calT_{\tau})=c_d(\eta'^{*}_{\sharp}\calT_{\tau})=p^{v(\eta)}c_d(\calT_{\tau^+}).$$ Let $${\mathrm{Tr}}_{?}\colon H^{2d}_{{\mathrm{et}}}({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(?)})_{\overline\FF_p},{\mathbb{Q}}_{\ell})(d){\xrightarrow}{\sim} {\mathbb{Q}}_{\ell} \quad \text{for }?=\tau,\tau^+$$ be the $\ell$-adic trace map. Then we have $$\deg(\eta'_{\sharp}) {\mathrm{Tr}}_{\tau}(c_d(\calT_{\tau}))={\mathrm{Tr}}_{\tau}(\eta'^{*}_{\sharp}c_d(\calT_{\tau}))=p^{v(\eta)} {\mathrm{Tr}}_{\tau^+}(c_d(\calT_{\tau^+})).$$ It is well known that ${\mathrm{Tr}}_{\tau}(c_d(\calT_{\tau}))=\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{\overline \FF_p})$ (see [@SGA5 Exposé VII, Corollaire 4.9]), where $$\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{\overline \FF_p}): =\sum_{i=0}^{2d}(-1)^i\dim H^{i}_{{\mathrm{et}}}({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{\overline \FF_p},{\mathbb{Q}}_{\ell})$$ is the ($\ell$-adic) Euler-Poincaré characteristic. Hence, one obtains $$\deg(\eta'_{\sharp}) \cdot \chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{\overline \FF_p})=p^{v(\eta)} \cdot \chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau^+)})_{\overline \FF_p})$$ Since $\eta'_{\sharp}$ is purely inseparable, we have $\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{\overline \FF_p})=\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau')})_{\overline \FF_p})$. By Lemma \[L:non-vanishing-chi\] proved below, we have $\chi({\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)}))\neq 0$, and hence $\deg(\eta'_{\sharp})=p^{v(\eta)}$. \(2) Note that $\gothp$ splits in $E$, and fix a prime $\gothq$ of $E$ dividing $\gothp$. We denote by $\tilde\tau$ and $\tilde\tau^-$ the liftings of $\tau$ and $\tau^-$ in $\Sigma_{E,\infty/\gothq}$ respectively. We define first a link morphism $(\eta_{\sharp},\eta^{\sharp}): {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau^-)})_{k_0}{\rightarrow}{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$ of indentation degree $p^{2(g-n_{\tau})}$ as follows. Let $y=(B',\iota_{B'},\lambda_{B'},\beta_{K'})$ be an $\overline \FF_p$-point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau^-)})_{k_0}$. We define first a lattice $M^{\circ}_{\tilde\tau'}$ of ${\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau'}[1/p]$ for each $\tilde\tau'\in \Sigma_{E,\infty}$ as follows: We put $$M^{\circ}_{\sigma^{i}\tilde\tau}= \begin{cases}\frac{1}{p}{\tilde{{\mathcal{D}}}}^{\circ}_{B',\sigma^{i}\tilde\tau} &\text{for } i=1,\cdots, n_{\tau^-}=g-n_{\tau},\\ {\tilde{{\mathcal{D}}}}^{\circ}_{B',\sigma^i\tilde\tau} &\text{for } i=n_{\tau^-}+1,\cdots,g, \end{cases}$$ and $M^{\circ}_{\sigma^i\tilde\tau^{c}}=M^{\circ,\vee}_{\sigma^i\tilde\tau}$. One checks easily that $M=\bigoplus_{\tilde\tau'\in \Sigma_{E,\infty}} M^{\circ,\oplus 2}_{\tilde\tau'}$ is stable under the action of Frobenius and Verschiebung homomorphisms, hence a Dieudonné submodule of ${\tilde{{\mathcal{D}}}}_{B'}[1/p]$. As in the proof of Proposition \[P:Y\_S=X\_S\], this gives rise to an abelian variety $B''$ equipped with an action by ${\mathcal{O}}_{D}$ with Dieudonné module ${\tilde{{\mathcal{D}}}}_{B''}\cong M$. The natural inclusion $M{\hookrightarrow}{\tilde{{\mathcal{D}}}}_{B'}[1/p]$ induces an ${\mathcal{O}}_D$-equivariant $p$-quasi-isogeny $\phi: B'{\rightarrow}B''$. Since the lattice $M\subseteq {\tilde{{\mathcal{D}}}}_{B'}[1/p]$ is self-dual by construction, the polarization $\lambda_{B'}$ induces a prime-to-$p$ polarization $\lambda_{B''}$ on $B''$ such that $\lambda_{B'}=\phi^\vee\circ \lambda_{B''}\circ \phi$. Finally, the $K'^p$-level structure $\beta_{K'}$ on $B'$ induces naturally a $K'^p$-level structure $\beta''_{K'}$ on $B''$. Moreover, an easy computation shows that the signatures of $B''$ at $\tilde\tau$ and $\tilde\tau^-$ are respectively $0$ and $2$, and those at other $\tilde\tau'$ not lifting $\tau, \tau^-$ remain the same as those of $B'$. Thus, $(B'',\iota_{B''},\lambda_{B''},\beta''_{K'})$ is a point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$. Let $$\eta_{\sharp}\colon {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau^-)})_{k_0} \to {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$$ be the map sending $y=(B',\iota_{B'},\lambda_{B'},\beta_{K'})\mapsto (B'',\iota_{B''},\lambda_{B''},\beta''_{K'})$, and let $$\eta^{\sharp}\colon \bfA'_{\tilde\ttS(\tau^-), k_0}{\rightarrow}\eta_{\sharp}^*\bfA'_{\tilde\ttS(\tau), k_0}$$ be the $p$-quasi-isogeny whose base change to each $y$ is $\phi: B'{\rightarrow}B''$ constructed above. Then it is clear by construction that $(\eta_{\sharp}, \eta^{\sharp})$ is the link morphism of indentation degree ${2(g-n_{\tau})}$ associated to the trivial link on from $\ttS(\tau)$ to $\ttS(\tau^-)$. Denote by $\pi_{\{\tau,\tau^-\}}: {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0,\{\tau,\tau^-\}}{\xrightarrow}{\sim} {\mathbf{Sh}}_{K'^p{\mathrm{Iw}}'_{p}}(G'_{\tilde\ttS(\tau)})_{k_0}$ the isomorphism given by Theorem \[T:main-thm-unitary\]. Let $x=(A, \iota_A, \lambda_A, \alpha_{K'})$ be an $\overline \FF_p$-point of ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{\{\tau, \tau^-\}}$. Then its image $(B,\iota_{B},\lambda_{B},\beta_{K'^\gothp},\beta_{\gothp})$ under $\pi_{\{\tau,\tau^-\}}$ is characterized as follows: - There exists an ${\mathcal{O}}_D$-equivariant $p$-quasi-isogeny $\phi: B{\rightarrow}A$ such that $\phi$ induces an isomorphism $\phi_{*}\colon {\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau'}{\xrightarrow}{\sim} {\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau'}$ for $\tilde\tau'$ different from $ \sigma^{i}\tilde\tau^-$ with $i=1,\dots, n_{\tau}$ and and their conjugates. In the exceptional cases, we have $$\phi_{*}({\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^i\tilde\tau^-})={\mathrm{Im}}(F_{A,{\mathrm{es}}}^{i}\colon {\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau^-}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^i\tilde\tau^-}), \quad \text{and} \quad \phi_{*}({\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^i\tilde\tau^{-,c}})=\frac{1}{p}{\mathrm{Im}}(F_{A,{\mathrm{es}}}^i\colon{\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau^{-,c}}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^i\tilde\tau^{-,c}}).$$ - We have $\lambda_{B}=\phi^\vee\circ\lambda_{A}\circ\phi$, and $\beta_{K'^p}=\alpha_{K'^p}\circ\phi$. - Let $H_{\tilde\tau}\subseteq {\tilde{{\mathcal{D}}}}_{B,\tilde\tau}^{\circ}/p{\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau}$ be the one-dimensional subspace given by the image of $p{\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau}$ via $\phi_{*}^{-1}$. Then $H_{\tilde\tau}$ is stable under $F_{B,{\mathrm{es}}}^g$, and $M\colon=\bigoplus_{i=0}^{g-1}F_{B,{\mathrm{es}}}^i(H_{\tilde\tau})^{\oplus 2}$ is a Dieudonné submodule of $\calD_{B[\gothq]}=\bigoplus_{i=0}^{g-1}{\tilde{{\mathcal{D}}}}_{B,\sigma^{i}\tilde\tau}/p{\tilde{{\mathcal{D}}}}_{B,\sigma^i\tilde\tau}$. Let $H_{\gothq}$ be the subgroup scheme of $B[\gothq]$. Then the Iwahoric level structure of $B$ at $p$ is given by $\beta_{\gothp}=H_{\gothq}\oplus H_{\bar\gothq}$, where $H_{\bar\gothq}\subseteq B[\bar\gothq]$ is the orthogonal complement of $H_{\gothq}$ under the natural duality between $B[\gothq]$ and $B[\bar\gothq]$. It is clear that the image of $x$ under $\pi_{\tau}$ is $(B,\iota_{B},\lambda_{B},\beta_{K'^p})$ by forgetting the Iwahoric level structure at $p$ of $\pi_{\{\tau,\tau^-\}}(x)$. This shows that, via the isomorphism $\pi_{\{\tau,\tau^-\}}$, the map $\pi_{\tau}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0,\{\tau,\tau^-\}}}$ coincides with the projection ${\mathrm{pr}}_1$ in . To finish the proof of (2), it remains to show that $$\eta_{\sharp}\circ\pi_{\tau^-}\circ\pi_{\{\tau,\tau^-\}}^{-1}: {\mathbf{Sh}}_{K'^p{\mathrm{Iw}}'_p}(G'_{\tilde \ttS(\tau)})_{k_0}{\rightarrow}{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$$ is the second projection ${\mathrm{pr}}_2$ in . Let $x=(A,\iota_{A},\lambda_{A},\alpha_{K'})$ be a point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0,\{\tau,\tau^-\}}$ with image $\pi_{\{\tau,\tau^-\}}(x)=(B,\iota_B,\lambda_B,\beta_{K'^\gothp},\beta_{\gothp})$, together with the $p$-quasi-isogeny $\phi\colon B{\rightarrow}A$ as described above. The image of $(A,\iota_{A},\lambda_{A},\alpha_{K'})$ under $\pi_{\tau^-}$ is given by $(B', \iota_{B'}, \lambda_{B'}, \beta_{K'}) \in {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^-)})_{k_0}$, which admits a quasi-isogeny $\psi: B'{\rightarrow}A$ compatible with all structures such that $\psi_{*}: {\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau'}\cong {\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau'}$ for all $\tilde\tau'$ except for those $\tilde\tau'$’s lifting $\sigma^i\tau$ for any $i = 1, \dots, g-n_{\tau}=n_{\tau^-}$. In the exceptional cases, we have $$\psi_{*}( {\tilde{{\mathcal{D}}}}^{\circ}_{B',\sigma^i\tilde\tau})={\mathrm{Im}}(F_{A,{\mathrm{es}}}^i\colon {\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{\sigma^i\tilde\tau})\quad\text{and}\quad \psi_*({\tilde{{\mathcal{D}}}}^{\circ}_{B',\sigma^i\tilde\tau^c})=\frac1 p{\mathrm{Im}}(F_{A,{\mathrm{es}}}^i\colon {\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau^c}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A,\sigma^i\tilde\tau^c}).$$ Consider the composed isogeny $\phi^{-1}\circ\psi\colon B'{\rightarrow}B$. Let $\tilde H_{\tilde\tau}\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau}$ be the inverse image of the one-dimensional subspace $H_{\tilde\tau}\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau}/p{\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau}$ given by the image of $p{\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau}$ as in (c) above. Then, we have $$(\phi^{-1}\circ\psi)_*({\tilde{{\mathcal{D}}}}^{\circ}_{B',\sigma^{i}\tilde\tau})= \begin{cases} F_{B,{\mathrm{es}}}^i(\tilde H_{\tilde\tau})&\text{for }i=1,\cdots, n_{\tau^-}=g-n_{\tau};\\ \frac1pF_{B,{\mathrm{es}}}^i(\tilde H_{\tilde\tau}) &\text{for }i=n_{\tau^-}+1,\cdots, g; \end{cases}$$ and $(\phi^{-1}\circ\psi)_*({\tilde{{\mathcal{D}}}}^{\circ}_{B',\sigma^{i}\tilde\tau^c})$ is the orthogonal complement of $(\phi^{-1}\circ\psi)_*({\tilde{{\mathcal{D}}}}^{\circ}_{B',\sigma^{i}\tilde\tau})$. Let $(B'',\iota_{B''},\lambda_{B''},\beta''_{K'})$ be the image of $(B', \iota_{B'}, \lambda_{B'}, \beta_{K'}) \in {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau^-)})_{k_0}$ under $\eta_{\sharp}$. Then, the composed $p$-quasi-isogeny $B{\xrightarrow}{\psi^{-1}\circ\phi} B'{\xrightarrow}{\eta^{\sharp}}B''$, identifies ${\tilde{{\mathcal{D}}}}^{\circ}_{B'',\sigma^i\tilde\tau}$ with the lattice $$\frac{1}{p}F_{B,{\mathrm{es}}}^{i}(H_{\tilde\tau})\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{B,\sigma^i\tilde\tau}[1/p],\quad \text{for all }i=1,\cdots, g.$$ Thus, one sees immediately that the map $(B,\iota_B, \lambda_B, \beta_{K'^\gothp},\beta_{\gothp})\mapsto (B'',\iota_{B''},\lambda_{B''},\beta''_{K'})$ is nothing but the second projection in . This finishes the proof of (2). Our last proposition explains the compatibility of the description of GO-divisors as in Theorem \[T:main-thm-unitary\] with respect to the link morphism, especially to the link morphism appearing in Theorem \[T:link and Hecke operator\](1). \[P:compatibility of link and GO\] Assume that $\#\Sigma_\infty - \#\ttS_\infty \geq 2$. Let $\tau_0\in \Sigma_{\infty}-\ttS_{\infty}$, and $\eta: \ttS \to \ttS'$ be a link such that all curves are straight lines except for (possibly) one curve turning to the right, linking $\tau_0\in \Sigma_{\infty}-\ttS_{\infty}$ with $\tau'_0 = \eta(\tau_0) = \sigma^{m(\tau_0)}\tau_0$ for some integer $m(\tau_0)\geq 0$. Assume that the link morphism $(\eta'_\sharp, \eta'^\sharp): {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0} \to {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0}$ of (some) indentation degree $n \in \ZZ$ associated to $\eta$ exists. The setup automatically implies that $\tau^+_0 = \tau'^+_0$. Let $\tau\in \Sigma_\infty - \ttS_\infty$. Then the following statements hold: 1. The link morphism $\eta'_\sharp$ sends the GO-divisor ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \tau}$ into the GO-divisor ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0, \eta(\tau)}$. 2. Let $\eta_{\tau}: \ttS(\tau) \to \ttS'(\eta(\tau))$ denote the link given by removing from $\eta$ the two curves attached to $\tau$ and $\tau^-$. Then there exists a link morphism $(\eta'_{\tau,\sharp},\eta'^{\sharp}_{\tau})$ (of some indentation degree $m$) from ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0}$ to ${\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'(\eta(\tau))})_{k_0}$, associated to the link $\eta_\tau$ such that we have the following commutative diagram of Shimura varieties $$\label{E:induced-link-morphism} \xymatrix@C=60pt{ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS})_{k_0, \tau} \ar[r]^-{\pi_{\tau}} \ar[d]^{\eta'_{\sharp}} & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS(\tau)})_{k_0} \ar[d]^{\eta'_{\tau,\sharp}} \\ {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'})_{k_0,\eta(\tau)} \ar[r]^-{\pi_{\eta(\tau)}} & {\mathbf{Sh}}_{K'}(G'_{\tilde \ttS'(\eta(\tau))})_{k_0}, }$$ and a similar commutative diagram of quasi-isogenies of universal abelian varieties. Moreover, the indentation degree of the link morphism $\eta'_{\tau, \sharp}$ is given by $$m=n+n_{\tau}-n_{\eta(\tau)}(\ttS')=\begin{cases} 0 &\text{if $\gothp$ is inert in $E/F$;}\\ n &\text{if $\gothp$ splits in $E/F$ and $\tau\neq \tau_0, \tau_0^+$;}\\ n-m(\tau_0) &\text{if $\gothp$ splits in $E/F$ and $\tau=\tau_0$;}\\ n+m(\tau_0) &\text{if $\gothp$ splits in $E/F$ and $\tau=\tau_0^+$.} \end{cases}$$ 3. Suppose moreover that the link $\eta$ and the link morphism $(\eta'_{\sharp}, \eta'^{\sharp})$ are those appearing in Theorem \[T:link and Hecke operator\](1) (so our $\tilde\ttS$ being $\tilde\ttS(\tau^+)$, $\ttS'$ being $\tilde\ttS(\tau)$, and $\tau_0$ being $\tau^-$ therin, respectively). If ${\mathcal{O}}_{\tau}(1)$ (resp. ${\mathcal{O}}_{\eta(\tau)}(1)$) denotes the tautological quotient line bundle on ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}$ (resp. on ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'})_{k_0, \eta(\tau)}$) for the $\PP^1$-fibration $\pi_{\tau}$ (resp. $\pi_{\eta(\tau)}$), then we have a canonical isomorphism $$\label{E:link-line-bundles} \eta'^{*}_{\sharp}({\mathcal{O}}_{\eta(\tau)}(1))\cong \begin{cases} {\mathcal{O}}_{\tau}(1) &\text{if } \tau\neq \tau_0^+;\\ {\mathcal{O}}_{\tau}(p^{m(\tau_0)}) &\text{if } \tau=\tau_0^+. \end{cases}$$ Moreover, the induced link morphism $\eta'_{\tau, \sharp}$ finite flat of degree $p^{v(\eta_{\tau})}$, i.e it is an isomorphism if $\tau\in \{\tau^{+}_0, \tau_0\}$, and it is finite flat of degree $p^{m(\tau_0)}$ if $\tau\notin \{ \tau_0, \tau_0^{+}\}$. The analogous results hold for link morphisms for ${\mathbf{Sh}}_{K''}(G''_{\tilde \ttS})_{k_0}$’s. \(1) Since ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}$ is reduced, it suffices to prove that $\eta'_{\sharp}$ sends every $\overline \FF_p$-point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}$ to ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'})_{k_0,\eta(\tau)}$. Let $x=(A,\iota, \lambda, \alpha_{K'^p})$ be an $\overline \FF_p$-point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}$, and $\eta'_{\sharp}(x)=(A',\iota',\lambda',\alpha'_{K'^p})$ be its image. Let $\tilde\tau\in \Sigma_{E,\infty}$ be a place above $\tau$, and put $\tilde\tau^-=\sigma^{-n_{\tau}}\tilde\tau$ and $\tilde\tau^+ =\sigma^{n_{\tau^+}}\tilde\tau$. By Lemma \[Lemma:partial-Hasse\], the condition $h_{\tau}(A)=0$ is equivalent to $F_{{\mathrm{es}},A}^{n_{\tau}}({\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau^-})=\tilde\omega_{A,\tilde\tau}^{\circ}$, where $\tilde\omega^{\circ}_{A^\vee,\tilde\tau}\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau}$ denotes the inverse image of $\omega^{\circ}_{A^\vee,\tilde\tau}\subseteq \calD^{\circ}_{A,\tilde\tau}$. The latter condition is in turn equivalent to that $F_{{\mathrm{es}}, A}^{n_{\tau^+}}\circ F_{{\mathrm{es}},A}^{n_{\tau}}({\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau^-})=p{\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau^+}$. We set $\eta(\tilde\tau^-)=\sigma^{m(\tau^-)}\tilde\tau^-$, where $m(\tau^-)$ the displacement of the curve in $\eta$ connecting $\tau^-$ and $\eta(\tau^-)$ (which equals to $0$ except when $\tau^-=\tau_0$); similarly, we put $\eta(\tilde\tau^+) = \sigma^{m(\tau^+)}\tilde \tau^+$. Since $\eta'^{\sharp}_{*}: {\tilde{{\mathcal{D}}}}^{\circ}_{A}[1/p]{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{A'}[1/p]$ commutes with Frobenius and Verschiebung homomorphisms, one sees easily from condition (3) in Definition \[D:link-morphism\] that $F_{{\mathrm{es}},A'}^{n_{\eta(\tau^+)}(\ttS')} \circ F_{{\mathrm{es}},A'}^{n_{\eta(\tau)}(\ttS')}({\tilde{{\mathcal{D}}}}^{\circ}_{A',\eta(\tilde\tau^-)})=p^u{\tilde{{\mathcal{D}}}}^{\circ}_{A',\eta(\tilde\tau^+)}$ for some integer $u\in {\mathbb{Z}}$. Here, $n_{\eta(\tau)}(\ttS')$ is the integer defined in Notation \[N:n tau\] associated to $\tau$ for the set $\ttS'$. But $F_{{\mathrm{es}},A'}^{n_{\eta(\tau^+)}(\ttS')} \circ F_{{\mathrm{es}},A'}^{n_{\eta(\tau)}(\ttS')}({\tilde{{\mathcal{D}}}}^{\circ}_{A',\eta(\tilde\tau^-)})$ is a $W(\overline \FF_p)$-sublattice of ${\tilde{{\mathcal{D}}}}^{\circ}_{A,\eta(\tilde\tau^+)}$ with quotient of length $2$ over $W(\overline \FF_p)$. Hence, the integer $u$ has to be $1$. By the same reasoning using Lemma \[Lemma:partial-Hasse\], this is equivalent to saying that $h_{\eta(\tau)}(A')=0$, or equivalently $\eta'_{\sharp}(x)\in {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'})_{k_0,\eta(\tau)}$. \(2) Assume first that $\#\Sigma_{\infty}-\#\ttS_{\infty}>2$. By Proposition \[P:restriction of GO strata\](2), $\pi_{\tau}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau^-, \tau\}}}$ is an isomorphism. We define $$\eta'_{\tau,\sharp}\colon= \pi_{\eta(\tau)}\circ \eta'_{\sharp}\circ (\pi_{\tau}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau^-, \tau\}}})^{-1}$$ and $\eta'^{\sharp}_{\tau}$ as the pull-back via $(\pi_{\tau}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \{\tau^-, \tau\}}})^{-1}$ of the quasi-isogeny $$\pi_{\tau}^{*}\bfA'_{\tilde\ttS(\tau), k_0}{\xrightarrow}{\phi_{\tau}^{-1}} (\bfA'_{\tilde\ttS, k_0}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}}){\xrightarrow}{\eta'^{\sharp}} \eta'^{*}_{\sharp} (\bfA'_{\tilde\ttS',k_0}|_{{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'})_{k_0,\eta(\tau)}}){\xrightarrow}{\phi_{\eta(\tau)}} \eta'^*_{\sharp}\pi_{\eta(\tau)}^*(\bfA'_{\tilde\ttS(\eta(\tau)), k_0})$$ of abelian schemes on ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0, \tau}$. Here, the first and the third quasi-isogenies are given by Theorem \[T:main-thm-unitary\](3). It is clear that the diagram is commutative. It remains to show that $(\eta'_{\tau,\sharp},\eta'^{\sharp}_{\tau})$ defines a link morphism. Conditions (1) and (2) of Definition \[D:link-morphism\] being clear, condition (3) can be verified by a tedious but straightforward computation. To see condition (4) on the indentation degree, we need only to discuss the case when $\gothp$ splits in $E/F$. In this case, $\phi^{-1}_\tau[\gothq^\infty]$ has degree $p^{2n_{\tau}}$, $\eta'^\sharp[\gothq^\infty]$ has degree $p^{2n}$, and $\phi_{\eta(\tau)}[\gothq^\infty]$ has degree $p^{-2n_{\eta(\tau)}(\ttS')}$. So the total degree of quasi-isogeny of $\eta'^{\sharp}_{\tau}$ is $m=n+n_{\tau}-n_{\eta(\tau)}(\ttS')$ A case-by-case discussion proves the condition (4) of Definition \[D:link-morphism\] on indentation degrees. Assume now $\#\Sigma_{\infty}-\#\ttS_{\infty}=2$ so that $\Sigma_{\infty}-\ttS_{\infty}=\{\tau_0, \tau_0^+=\tau_0^{-}\}$. This implies that $\gothp$ splits as $\gothq\bar\gothq$ in $E$. Since the Shimura variety ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$ is zero-dimensional, we just need to define the desired link morphism $(\eta'_{\tau,\sharp}, \eta'^{\sharp}_{\tau})$ on $\overline \FF_p$-points. For each $\tilde\tau'\in \Sigma_{E,\infty}$ with restriction $\tau'\in \{\tau_0,\tau_0^-\}$, let $t_{\tilde\tau'}\in {\mathbb{Z}}$ denote the integer as in Definition \[D:link-morphism\](3) attached to $\tilde\tau'$ for the link morphism $(\eta'_{\sharp},\eta'^{\sharp})$. Let $y=(B,\iota_{B},\lambda_B, \beta_{K'^p})$ be an $\overline \FF_p$-point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$. We now distinguish two cases: - Consider first the case $\tau=\tau_0$. Let $\tilde\tau^-_0$ be the lift of $\tau_0$ in $ \Sigma_{E,\infty/\gothq}$. We define $M_{\tilde\tau_0^-}=p^{-t_{\tilde\tau_0}} {\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau_0}$, and $M_{\sigma^i\tilde\tau_0^-}=p^{-\delta_i}F^{i}(M_{\tilde\tau_0^-})$ for each integer $i$ with $1\leq i\leq g-1$, where $\delta_i$ denotes the number of integers $j$ with $1\leq j\leq i$ such that $\sigma^{j}\tilde\tau_0^-\in \tilde\ttS'(\eta(\tau))_{\infty}$. Put $M_{\gothq}=\bigoplus_{0\leq i\leq g-1}M_{\sigma^i\tilde\tau_0^-}$, and let $M_{\bar\gothq}\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{B,\bar\gothq}[1/p]$ be the dual lattice of $M_{\gothq}$ with respect to the pairing induced by $\lambda_B$. Then $M:= M_{\gothq}\oplus M_{\bar\gothq}$ is a Dieudonné submodule of ${\tilde{{\mathcal{D}}}}^{\circ}_{B}[1/p]$. By the same argument as in the proof of Proposition \[P:Y\_S=X\_S\], there exists a unique abelian variety $B'$ equipped with an ${\mathcal{O}}_D$-action $\iota_{B'}$ together with a $p$-quasi-isogeny $\phi: B{\rightarrow}B'$ such that the induced map $\phi^{-1}_{*}: {\tilde{{\mathcal{D}}}}^{\circ}_{B'}{\rightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{B}[1/p]$ is identified with the natural inclusion $M{\hookrightarrow}{\tilde{{\mathcal{D}}}}^{\circ}_{B}[1/p]$. As usual, since $M$ is a self-dual lattice, $\lambda_{B}$ induces a prime-to-$p$ polarization $\lambda_{B'}$ such that $\phi^{\vee}\circ\lambda_{B'}\circ\phi=\lambda_{B}$. We equip $B'$ with the $K'^p$-level structure $\beta'_{K'^p}=\phi\circ\beta_{K'^p}$. By the construction, one sees also easily that $B'$ satisfies the necessary signature condition so that $y'=(B',\iota_{B'},\lambda_{B'},\beta'_{K'^p})$ is a point of ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'(\eta(\tau))})_{k_0}$. We define $$\eta'_{\tau, \sharp}\colon {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}{\rightarrow}{\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'(\eta(\tau))})_{k_0},\quad \text{and}\quad \eta'^{\sharp}_{\tau}\colon \bfA'_{\tilde\ttS(\tau),k_0}{\rightarrow}\bfA'_{\tilde\ttS'(\eta(\tau)), k_0}$$ by $\eta'_{\tau, \sharp}(y)= y'$ and $\eta'^{\sharp}_{\tau, y}=\phi$. It is evident that $(\eta'_{\tau,\sharp},\eta'^{\sharp}_{\tau})$ is a link morphism. It remains to check that the diagram is commutative. Let $x=(A,\iota_{A},\lambda_A,\alpha_{K'^p})\in {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0,\tau}$ be a point above $y$, $x'=(A',\iota_{A'},\lambda_{A'},\alpha'_{K'^p})\in {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'})_{k_0,\eta(\tau)}$ be the image of $x$ under $\eta'_{\sharp}$. We need to prove that $\pi_{\eta(\tau)}(x')=y'$. Let $y''=(B'',\iota_{B''},\lambda_{B''},\beta''_{K'^p})\in {\mathbf{Sh}}_{K'}(G'_{\tilde\ttS'(\eta(\tau))})_{k_0,\eta(\tau)}$ denote temporarily the point $\pi_{\tau}(x')$. Denote by $\psi\colon A{\rightarrow}B$ and $\psi'\colon A'{\rightarrow}B''$ be the quasi-isogenies given by Theorem \[T:main-thm-unitary\]. Then we have $\psi_* ({\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau_0^-})= {\tilde{{\mathcal{D}}}}^{\circ}_{B,\tilde\tau_0^-}$ and $\psi'_{*}({\tilde{{\mathcal{D}}}}^{\circ}_{A',\tilde\tau_0^-})={\tilde{{\mathcal{D}}}}^{\circ}_{B'',\tilde\tau_0^-}$ by Subsection \[S:moduli-Y\_S\], and $\eta'^{\sharp}_{*}({\tilde{{\mathcal{D}}}}^{\circ}_{A,\tilde\tau_0^-})=p^{t_{\tilde\tau_0^-}}{\tilde{{\mathcal{D}}}}^{\circ}_{A',\tilde\tau_0^-}$ by Definition \[D:link-morphism\](3) as $\eta$ is a straight line at $\tau_0^-$. Consider the quasi-isogeny $\psi'\circ\eta'^{\sharp}\circ\psi^{-1}\circ\phi^{-1}: B'{\rightarrow}B''$. Iit induces an isomorphism ${\tilde{{\mathcal{D}}}}^{\circ}_{B',\tilde\tau_0^-}{\xrightarrow}{\sim} {\tilde{{\mathcal{D}}}}^{\circ}_{B'',\tilde\tau_0^-}$. But the other components of the Dieudonné modules are determined by that at $\tilde\tau_0^-$. It follows that $\psi'\circ\eta'^{\sharp}\circ\psi^{-1}\circ\phi^{-1}: B'{\rightarrow}B''$ is an isomorphism compatible with all structures, i.e. $y''=y'$. The computation of the indentation degree is the same as that in the case when $\#\Sigma_\infty - \#\ttS_\infty >2$. - In the case $\tau=\tau_0^+ = \tau_0^-$, the construction is similar. Let $\tilde\tau_0$ be the lift of $\tau_0$ in $\Sigma_{E,\infty/\gothq}$, and $\eta(\tilde\tau_0)=\sigma^{m(\tau_0)}\tilde\tau_0$. Put $M_{\eta(\tilde\tau_0)}=p^{-t_{\tilde\tau_0}}F^{m(\tau_0)}({\tilde{{\mathcal{D}}}}^{\circ}_{B,\eta(\tilde\tau_0)})$, and $M_{\sigma^{i}\eta(\tilde\tau_0)}=p^{-r_i}F^{i}(M_{\eta(\tilde\tau_0)})$ for each integer $i$ with $1\leq i\leq g-1$, where $r_i$ is the number of integers $j$ with $1\leq j\leq i$ such that $\sigma^j\eta(\tilde\tau_0)\in \tilde\ttS'(\eta(\tau))_{\infty}$. Let $M_{\gothq}=\bigoplus_{0\leq i\leq g-1}M_{\sigma^{i}\eta(\tilde\tau_0)}$, and $M_{\bar\gothq}\subseteq {\tilde{{\mathcal{D}}}}^{\circ}_{B,\bar\gothq}[1/p]$ be the dual lattice. As in case (a) above, such a lattice $M:=M_\gothq \oplus M_{\bar \gothq}$ gives rise to an $\overline \FF_p$-point $y'=(B',\iota_{B'},\lambda_{B'},\beta'_{K'^p})$ together with a $p$-isogeny $\phi: B{\rightarrow}B'$ compatible with all structures. We define the desired link morphism $(\eta'_{\tau,\sharp}, \eta'^{\sharp}_{\tau})$ such that $\eta'_{\tau,\sharp}(y)=y'$ and $\eta'_{\tau,\sharp,y}=\phi$. The commutativity of is proved by the same arguments as in (a). We leave the details to the reader. \(3) We note that, if $\tilde\tau^-$ is the lifting of $\tau^-$ not contained in $\tilde\ttS(\tau)_{\infty}$, then we have a canonical isomorphism ${\mathcal{O}}_{\tau}(1)\cong \operatorname{Lie}(\bfA'_{\tilde\ttS, k_0})^{\circ}_{\tilde\tau^-}$ by the construction of $\pi_{\tau}$; similarly, one has ${\mathcal{O}}_{\eta(\tau)}(1)\cong \operatorname{Lie}(\bfA'_{\tilde\ttS',k_0})^{\circ}_{\eta(\tilde\tau^-)}$. Now the isomorphism follows immediately from Theorem \[T:link and Hecke operator\](1)(b). We prove now the second part of (3). By the construction of $\eta'_{\tau}$, it follows from Theorem \[T:link and Hecke operator\](1)(b) that $\eta'^{\sharp}_{\tau}\colon \bfA'_{\tilde\ttS(\tau),k_0}{\rightarrow}\eta'^*_{\tau,\sharp}(\bfA'_{\tilde\ttS(\eta(\tau)),k_0})$ induces, for any $\tilde\tau'\in \Sigma_{E,\infty}$ lifting an element $\Sigma_{\infty}-\ttS(\eta(\tau))$, an isomorphism $$\eta'^{*}_{\tau, \sharp}(\operatorname{Lie}(\bfA'_{\tilde\ttS(\eta(\tau)),k_0})^{\circ}_{\tilde\tau'})\cong \begin{cases} \operatorname{Lie}(\bfA'_{\tilde\ttS(\tau), k_0})^{\circ, (p^{m(\tau_0)})}_{\sigma^{-m(\tau_0)}\tilde\tau'} &\text{if $\tilde\tau'$ lifts $\eta(\tau_0)$,}\\ \operatorname{Lie}(\bfA'_{\tilde\ttS(\tau), k_0})^{\circ}_{\tilde\tau'} &\text{otherwise}, \end{cases}$$ If $\tau\in \{\tau_0, \tau_0^{+}\}$ or equivalently $\tau_0\in \ttS(\tau)$, then the first case above never happens. Therefore, by Proposition \[Prop:smoothness\], we see that $\eta'_{\tau, \sharp}$ induces an isomorphism of tangent spaces between ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\tau)})_{k_0}$ and ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS(\eta(\tau))})_{k_0}$. Since $\eta'_{\tau,\sharp}$ is bijective on closed points by definition of link morphism, $\eta'_{\tau, \sharp}$ is actually an isomorphism. If $\tau\notin \{\tau_0, \tau_0^+\}$ or equivalently $\tau_0\notin \ttS(\tau)$, we conclude by the same arguments as Theorem \[T:link and Hecke operator\](1)(d). [999999]{} F. Andreatta and E. Goren, Hilbert modular varieties of low dimension, [*Geometric Aspects of Dwork Theory*]{}, [**I**]{}:113–175. E. Bachmat and E. Z. Goren, On the non-ordinary locus in Hilbert-Blumenthal surfaces, [*Math. Annalen*]{}, [**313**]{} (1999) 3, 475–506. P. Berthelot, L. Breen, and W. Messing, *Théorie de Dieudonné cristalline II*, Lecture notes in Mathematics, **930**, Springer-Verlag, 1982. A. Borel, [*Linear algebraic groups*]{}, Vol. [**126**]{} of Graduate Texts in Mathematics, Springer-Verlag, New York. H. Carayol, Sur la mauvaise réduction des courbes de Shimura, [*Compositio Math.*]{} [**59**]{} (1986), no. 2, 151–230. P. Deligne, Travaux de Shimura. [*Séminaire Bourbaki, 23ème année*]{} (1970/71), Exp. No. [**389**]{}, pp. 123–165. [*Lecture Notes in Math.*]{}, Vol. [**244**]{}, Springer, Berlin, 1971. P. Deligne, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques, in [*Automorphic forms, representations and $L$-functions*]{}, [*Proc. Sympos. Pure Math.*]{}, XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 247–289. E. Goren and F. Oort, Stratifications of Hilbert modular varieties. [*Journal of Algebraic Geometry*]{}, v. [**9**]{}, 2000, 111–154. A. Grothendieck, [Éléments de géométrie algébrique (rédigés avec la collaboration de Jean Dieudonné):IV. Étude locale des schémas et des morphismes de schémas, Troisième partie]{}, *Publ. Math. de l’I.H.É.S.*, tome **28**(1966), p.5–255. A. Grothendieck, [*Groupes de Barsotti-Tate et cristaux de Dieudonné*]{}, [*Séminaire de Mathématiques Supérior*]{} 45, Presses de l’Université de Montréal (1974). A. Grothendieck, *Séminaire de Géométrie algébrique du Bois Marie 1965-1966, SGA 5*, avec la collaboration de I. Bucur, C. Houzel, L. Illusie, J.P. Jouanolou et J.P. Serre, *Cohomologie $\ell$-adic et Fonctions $L$*, Lect. Notes in Math. **589**, Springer-Verlag, 1977. D. Helm, Towards a geometric Jacquet-Langlands correspondence for unitary Shimura varieties, [*Duke Math. J.*]{} [**155**]{} (2010), no. 3, 483–518. D. Helm, A geometric Jacquet-Langlands correspondence for $U(2)$ Shimura varieties, [*Isr. J. Math.*]{} [**187**]{} (2012), 37–80. D. Helm, Y. Tian, L. Xiao, On Tate conjecture for the special fiber of some unitary Shimura varieties, preprint. H. Hida, *$p$-adic Automorphic Forms on Shimura Varieties*, Springer-Verlag, 2004. M. Kisin, Integral models for Shimura varieties of abelian type, [*J. Amer. Math. Soc.*]{} [**23**]{} (2010), no. 4, 967–1012. R. Kottwitz, Points on some Shimura varieties over finite fields. [*J. Amer. Math. Soc.*]{} [**5**]{} (1992), no. 2, 373–444. K.-W. Lan, *Arithmetic compactifications of PEL-type Shimura varieties*, London Mathematical Society Monographs, vol. [**36**]{}, Princeton University Press, Princeton, 2013. R. Lazarsfeld, [*Positivity in algebraic geometry. I. Classical setting: line bundles and linear series.*]{} [*Ergebnisse der Mathematik und ihrer Grenzgebiete*]{}, [**48**]{}, Springer-Verlag, Berlin, 2004. xviii+387 pp. H. Matsumura, *Commutative ring theory*, translated by M. Reid, Cambridge University Press, 1986. B. Mazur and W. Messing, [*Universal extensions and one-dimensional crystalline cohomology*]{}, [*Lecture Notes in Math.*]{} Vol. [**370**]{}, Springer-Verlag (1974). J. Milne, Canonical models of (mixed) Shimura varieties and automorphic vector bundles, in [*Shimura varieties and L-functions, I*]{} (Academic Press, New York, 1990), 284–414. J. Milne, Introduction to Shimura varieties, in [*Harmonic analysis, the trace formula, and Shimura varieties*]{}, 265–378, [*Clay Math. Proc.*]{}, [**4**]{}, Amer. Math. Soc., Providence, RI, 2005. A. Minguez, Unramified representations of unitary groups, in [*On the Stabilization of the Trace Formula*]{}, International Press. B. Moonen, Models of Shimura varieties in mixed characteristics, in [*Galois representations in arithmetic algebraic geometry*]{} (Durham, 1996), LMS Lecture Note Ser. **254**, Cambridge Univ. Press, (1998), 267–350. M. Rapoport, Compactification de l’espace de modules de Hilbert-Blumenthal, [*Compo. Math.*]{}, [**36**]{}(3), 255–335. M. Rapoport and T. Zink, *Period spaces for $p$-divisible groups*, Annals of Mathematics Studies, Vol. **141**, Princeton University Press, 1996. T. Saito, Hilbert modular forms and $p$-adic Hodge theory, *Compositio Mathematica*, [**145**]{}, (2009) 1081–1113. J. P. Serre, Two letters on quaternions and modular forms (mod $p$), *Israel J. Math.* **95** (1996), 281–299. Y. Tian and L. Xiao, $p$-adic cohomology and classicality of overconvergent Hilbert modular forms, [arXiv:1308.0779]{}. Y. Tian and L. Xiao, Tate cycles on quaternionic Shimura varieties over finite fields, in preparation. [^1]: Even though Proposition \[Prop:smoothness\] was stated for ${\mathbf{Sh}}_{K'}(G'_{\tilde\ttS})_{k_0}$, its proof also works over $\CC$.

--- author: - | Hongmin Zhu\ University of Science and Technology of China\ `zhuhm@mail.ustc.edu.cn`\ Fuli Feng\ National University of Singapore\ `fulifeng93@gmail.com`\ Xiangnan He\ University of Science and Technology of China\ `xiangnanhe@gmail.com`\ Xiang Wang\ University of Science and Technology of China\ `xiangwang@u.nus.edu`\ Yan Li\ Kuaishou Technology\ `liyan@kuaishou.com`\ Kai Zheng\ University of Electronic Science and Technology of China\ `zhengkai@uestc.edu.cn`\ Yongdong Zhang\ University of Science and Technology of China\ `zhyd73@ustc.edu.cn`\ - | **Hongmin Zhu$^1$** **Fuli Feng**$^{2}$ **Xiangnan He**$^1$**Xiang Wang**$^2$\ **Yan Li$^{3}$Kai Zheng$^4$Yongdong Zhang$^1$**\ \ $^1$University of Science and Technology of China $^2$ National University of Singapore\ $^3$ Kuaishou Technology$^4$ University of Electronic Science and Technology of China\ [zhuhm@mail.ustc.edu.cn]{}\ [liyan@kuaishou.com]{} bibliography: - 'refers.bib' title: Bilinear Graph Neural Network with Node Interactions ---

--- abstract: 'We demonstrate on the example of the dc+ac driven overdamped Frenkel-Kontorova model that an easily calculable measure of complexity can be used for the examination of Shapiro steps in presence of thermal noise. In real systems, thermal noise causes melting or even disappearance of Shapiro steps, which makes their analysis in the standard way from the response function difficult. Unlike in the conventional approach, here, by calculating the Kolmogorov complexity of certain areas in the response function we were able to detect Shapiro steps, measure their size with desired precision and examine their temperature dependence. The aim of this work is to provide scientists, particularly experimentalists, an unconventional but a practical and easy tool for examination of Shapiro steps in real systems.' author: - 'Petar Mali$^{1}$, Anela Šakota$^{1}$, Jasmina Teki'' c$^2$, Slobodan Radoševi'' c$^{1}$, Milan Panti'' c$^{1}$, Milica Pavkov - Hrvojevi'' c$^{1}$' bibliography: - 'Paper\_Mali.bib' title: Complexity of Shapiro steps --- Introduction {#intro} ============ It is well known that dynamical systems with competing frequencies can exhibit some form of frequency locking and Shapiro steps. Since their discovery in Josephson junctions, Shapiro steps have been widely studied phenomenon in all kinds of nonlinear systems from charge density waves [@Grun; @GrunPR; @Thorne; @Hund; @Sinch], and Josephson junctions [@Dub; @Sel; @Lim; @Shukrinov; @Shukrinov2] to colloidal systems [@Nature; @NJP] and superconducting nanowires [@Dins; @Bae; @Rid]. However, in experiments these systems are exposed to various environmental effects among which certainly the most significant is the thermal noise. Thermal noise greatly affects mode locking by melting and changing properties of Shapiro steps such as their amplitude [@ACT; @ACTr] and frequency dependence [@ACTr; @ACTn]. It is, therefore, very difficult or even impossible to detect, measure or control them, and often, different methods have to be developed in order to address this problem. In charge density wave systems and Josephson junctions, for example, instead of current-voltage characteristics, where steps are hardly visible due to noise, differential resistance is used for their analysis [@Hund; @Sinch; @Dub]. On the other hand, Shapiro steps are ideally suited for realizing a voltage standard in different devices that are used in various technological applications [@Buckel]. Therefore, an understanding of Shapiro steps as a phenomenon and control of their behavior in real systems is of great importance. In the meanwhile, in the theory of nonlinear dynamical systems an interesting and easily applicable tool known as complexity measure or Kolmogorov complexity (KC) [@Kolmogorov; @Ziv] has been proposed. Today it is closely related to the information theory [@Kolmogorov; @Ziv; @Kaspar; @Kov], and applied in many other fields such as hydrology [@Mih], climatology [@Mih2; @Gordan], etc. The complexity measure represents a useful quantity, which characterizes spatiotemporal patterns and provides fine measure of order, i.e., periodic motion and chaotic behavior. Nonlinear systems, which exhibit Shapiro steps, besides the quasiperiodic and periodic motion can also exhibit a transition to chaos. It is, therefore, naturally to ask whether a complexity measure could be a convenient tool for the studies of Shapiro steps. In this paper we will apply on the studies of Shapiro steps a method very different from the conventional ones, the Kolmogorov complexity. In particular, we will consider the dc+ac driven overdamped Frenkel-Kontorova (FK) model with deformable substrate potential in the presence of noise. The standard FK model represents a chain of harmonically interacting identical particles subjected to sinusoidal substrate potential [@OBBook; @ACFK]. It can describe various commensurate and incommensurate structures [@Obri1; @Obri2], which exhibit very rich dynamics under external forces. When external dc and ac forces are applied, the mode locking appears due to resonance between the frequency of particle motion over the substrate potential and the frequency of the external ac force [@ACFK; @Flor; @Falo; @FlorAP]. This effect is characterized as a staircase of resonant steps, i.e., Shapiro steps in the plot of average velocity as a function of average driving force $\bar{v}(\bar{F})$. The steps are called harmonic if the locking appears at integer values of ac frequencies or subharmonic if it appears at noninteger rational ones. Although it was successful in describing harmonic steps, the standard ac+dc driven overdamped FK model can not be used for modeling phenomena related to subharmonic steps [@ACFK; @ACDS]. Namely, subharmonic steps do not exist in commensurate structures with integer values of winding numbers while for noninteger values their size is too small, which makes their analysis very difficult [@Falo; @Rene; @Wu]. To overcome this, some generalizations of standard FK model are necessary. For instance, large subharmonic steps in staircase like response appear in FK model with asymmetric deformable potential [@Bambi; @ACr]. The fact that subharmonic steps are present even in the case of integer value of winding number $\omega = 1$ [@Sat] indicates that, by choosing this type of potential, additional effective degrees of freedom are induced in the system. Therefore, generalization of the FK model by using some form of deformable substrate potential provides a good framework for studies of subharmonic mode locking [@ACFK; @ACr; @ACP]. If noise is present in the system, at certain temperature, the response function $\bar{v}(\bar{F})$ will be substantially affected. All steps will start melting, and while some will be more robust, the others might disappear completely [@ACFK; @ACT; @ACTr]. Therefore, it is often hard to get any information about resonances just from the observation of response function $\bar{v}(\bar{F})$. We will further introduce a method, which can overcome that difficulty, and by using KC, examine in detail Shapiro steps in the presence of noise. Model and method {#model} ================ We consider the dynamics of coupled harmonic oscillators $u_l$, subjected to asymmetric deformable substrate potential [@Peyrard]: $$\label{V} V(u)=\frac{K}{4 \pi^2}\frac{(1-r^2)^2 \big[1-\cos(2\pi u) \big]}{\big[1+r^2+2r\cos(\pi u) \big]^2},$$ where $K$ is the pinning strength and $r$ is deformation parameter ($-1< r <1)$. By changing $r$, the potential can be tuned in a very fine way, from the simple sinusoidal one for $r=0$ and to a deformable one for $0<|r|<1$. The total potential energy of such system is $$H = \sum_{l} \left( V(u_l) + W (u_{l+1}-u_l) \right),$$ where $$W (u_{l+1}-u_l) = \frac 12 \left( u_{l+1}-u_l \right)^2$$ represents harmonic coupling between neighboring particles [@Obri1; @Obri2]. The system is driven by dc and ac forces, $F(t)=F_{\mathsf{dc}} + F_{\mathsf{ac}}\cos(2\pi\nu_0t)$, where $F_{\mathsf{ac}}$ and $\nu_0$ are amplitude and frequency of ac force which, in the overdamped limit, leads to the system of equations of motion: $$\label{u} \dot{u}_l=u_{l+1}+u_{l-1}-2u_l-\frac{\partial V}{\partial u_l}+F_{\mathsf{dc}}+F_{\mathsf{ac}}\cos (2\pi \nu_0 t)+L_l(t),$$ where $l=1,...,N$, and $N$ is the number of particles, and $u_{N+1}=u_1$. The noise term is chosen as Gaussian white noise which satisfies $\langle L_l(t)L_{l'}(t') \rangle=2T\delta_{l,l'}\delta(t-t')$. When the system is driven by a periodic force, the competition between the frequency $\nu _0$ of the external periodic (ac) force and the characteristic frequency of the particle motion over the periodic substrate potential driven by the average force $\bar F=F_{\mathsf{dc}}$ results in the appearance of dynamical mode locking. The solution of the system (\[u\]) is called resonant if average velocity $\bar{v}$ satisfies the relation [@ACDS]: $$\label{v} \bar{v}=\left(i \pm \frac{1}{m\pm \frac{1}{n\pm \frac {1}{p\pm ...}}} \right) \omega\nu_0,$$ where $i, m, n, p$ are integers. The first level terms, which involve only $i$, represent harmonic steps, whereas the other terms describe subharmonic steps. The system of equations (\[u\]) has been numerically integrated using periodic boundary conditions for the commensurate structure $\omega =\frac 12 $ with two particles per potential well. The time step used in the simulations was $0.001 \tau$, and a time interval of $100 \tau$ was used as a relaxation time to allow the system to reach the steady state [@ACT; @ACTr]. The dc force was varied with steps $10^{-5}$ and $10^{-6}$. Before we focus on the Shapiro steps, let us shortly introduce the Kolmogorov complexity. In general, the KC of a finite object is defined as the length of the shortest effective binary description of that object. In other words, if we have some binary sequence of the length $n$, then the measure of complexity, $c(n)$, of that sequence is the number of patterns or parts required to reproduce that sequence. Calculation of $c(n)$ is performed by Lempel–Ziv algorithm (detailed explanation can be found in [@Ziv; @Kaspar; @Mih]). We shall describe the algorithm first and then illustrate it with a specific example. If we have some sequence $(x_1,..., x_n)$ of zeros and ones, the algorithm which determines the patterns, which form this sequence, consists of the following steps: (1) for a given sequence of zeros and ones, the first digit is always the first pattern; (2) define the sequence $S=x_1,..., x_k\cdot $, which contains the first pattern and grows until the whole sequence is analyzed; (3) in order to check whether the rest of sequence $(x_{k+1},..., x_n)$ can be reconstructed by simple copying (or whether one has to insert new digits), define sequence $Q\equiv x_{k+1}$ by adding a new digit; (4) define sequence $SQ$ by adding $S$ and $Q$; (5) form the sequence $SQ\pi $ by removing the last digit of sequence $SQ$ and examine whether $Q$ is part of the vocabulary $V(SQ\pi )$, if it is not, then sequence $Q$ is the new pattern, and this new pattern is added to the list of known patterns called vocabulary $R$. Sequence $SQ$ becomes new sequence $S$, while $Q$ is emptied and ready for further testing. If on the other hand $Q$ is part of $SQ\pi $, repeat the process until the mentioned condition is satisfied. For an illustration let us consider the binary sequence $11010001$. The first digit is the first pattern $R=1\cdot $. Since the first two digit are both $1$, we have that $S=1$, $Q=1$, which leads to $SQ=11$ and $SQ\pi =1$. In the following step we ask whether $Q$ is contained in the vocabulary $V(SQ\pi)$. Since $Q=1$ is the part of $SQ\pi =1$, we add the third digit to $Q$ so that $Q=10$, while $S=1$ and repeat the procedure. The algorithm can be written as: - $S=1, Q=1, SQ=11, SQ\pi=1, \\ Q \in V(SQ\pi) \rightarrow R=1 \cdot 1$ - $S=1, Q=10, SQ=110, SQ\pi=11, \\ Q \notin V(SQ\pi) \rightarrow R=1 \cdot 10 \ \cdot $ - $S=110, Q=1, SQ=1101, SQ\pi=110, \\ Q\in V(SQ\pi) \rightarrow R=1 \cdot 10 \cdot 1$ - $S=110, Q=10, SQ=11010, SQ\pi=1101, \\ Q\in V(SQ\pi) \rightarrow R=1 \cdot 10 \cdot 10$ - $S=110, Q=100, SQ=110100, SQ\pi=11010, \\ Q\notin V(SQ\pi) \rightarrow R=1 \cdot 10 \cdot 100 \ \cdot$ - $S=110100, Q=0, SQ=1101000, SQ\pi=110100, Q \in V(SQ\pi) \rightarrow R=1 \cdot 10 \cdot 100 \cdot 0$ - $S=110100, Q=01, SQ=11010001, SQ\pi=1101000, Q \in V(SQ\pi) \rightarrow R=1 \cdot 10 \cdot 100 \cdot 01$ According to Lempel–Ziv algorithm the above sequence can be written as $1 \cdot 10 \cdot 100 \cdot 01$. The measure of complexity $c(n)$ is simply the number of parts separated by dots, i.e., the pattern counter, which in this case is $c=4$. The application of KC on the analysis of Shapiro steps in the presence of thermal noise, in principle, requires comparison of sequences which differ in length. This is due to the fact that Shapiro steps do not have equal widths. Previous remark holds for Shapiro steps observed in experiments or in numerical simulations. In our analysis of Shapiro steps we will further use the normalized measure of complexity given as [@Ziv; @Kaspar]: $$\label{CK} C_{\mathsf{K}}=\frac{c(n)}{b(n)},$$ where $b(n) = \frac{n}{\mbox{log}_2n}$ represents the asymptotic value of $c(n)$ [@Kaspar]. This means that for sequence of infinite length ($n \to \infty$), Kolomogorov complexity (\[CK\]) takes values $C_{\mathsf{K}} \in [0,1]$, i.e. $0 \leq \lim_{n \to \infty} [c(n)/b(n)] \leq 1$. However, for sequences of finite length, the values of KC can exceed 1[^1]. Since it was shown in [@Kaspar] that the limit $\lim_{n \to \infty} [c(n)/b(n)] $ is reached within $5\%$ for $n > 10^3$, all calculations of the Shapiro steps complexity in next section were performed on sequences of size $n>1000$. In numerical simulations this length can be easily reached with suitable choice of force step $\delta F_{\mathsf{dc}}$. Results {#Res} ======= Influence of temperature on the response function $\bar{v}(F_{\mathsf{dc}})$ is presented in Fig. \[Fig1\]. ![\[Fig1\] (Color online) (a) Average velocity $\bar{v}$ as a function of the average driving force $F_{\mathsf{dc}}$ for $K=4$, $\omega=1/2$, $r=0.2$, $\nu_0=0.5$, and tree different values of temperature $T=0$, $T=10^{-4}$, and $T=10^{-5}$. (b) High resolution plot of the first harmonic step in (a). $F_{\mathsf{dc}}$ was varied with step $\delta F_{\mathsf{dc}} = 10^{-5}$ and $n_1$, $n_2$,... $n_i$ denote lengths of sequences for which $C_{\mathsf{K}}$ was calculated. Dashed black line at $\bar{v}=0.25$ represents the first harmonic step at $T=0$ and the blue line represents the first harmonic step at $T=10^{-5}$. ](Fig1.eps){width="8"} The presence of thermal noise causes melting of Shapiro steps, which become more and more rounded as the temperature increases as can be seen in Fig. \[Fig1\](a); meanwhile the critical depinning force decreases [@ACFK; @ACT; @ACTr]. At very high temperature the pinning potential has no influence and the system becomes the system of free particles [@ACFK; @ACT]. At $T=0$, the step can be analyzed easily, however, if $T\neq 0$, just a simple task such as measuring the step size can not be done accurately enough since it is impossible to determine exactly at which value of $F_{\mathsf{dc}}$ the step begins or ends. In conventional approach, some criterion needs to be always introduced in order to determine the beginning and the end of each step. In our previous works we considered the system to be on the step if the changes of $\bar v(\bar F)$ are less than $0.1\% $ [@ACT; @ACTr; @ACTn]. When steps are large and the temperature is low this does not represent a difficulty. However, for smaller subharmonic steps or at the higher temperature the error could be the order of step or the order of the fluctuations of resonant value $\bar v (T\neq 0)$ around $\bar v (T=0)$, which makes analysis of subharmonic steps challenging. From Fig. \[Fig1\](b) at $T\neq 0$ it is evident that the function $\bar{v}(F_{\mathsf{dc}})$ oscillates around the value $\bar{v}=0.25$, which corresponds to the position of first harmonic step at $T=0$. In order to calculate $C_{\mathsf{K}}$, we need to transform the values for average velocity into a corresponding binary string. To achieve this we define: $$\label{vN} \langle \bar{v} \rangle=\frac{1}{n}\sum^n_{i=1}\bar{v}_i,$$ where $\bar{v}_i$ corresponds to values of average force taken with step $\delta F_{\mathsf{dc}}$. Then we prescribe values $0$ or $1$ to particular $\bar{v}_i$ according to: $$\label{vi} \bar{v}_i \rightarrow \left\{\begin{array}{rl} 1, & \bar{v}_i \geq \langle \bar{v} \rangle\\ 0, & \bar{v}_i < \langle \bar{v} \rangle. \end{array} \right.$$ Using binary sequences obtained in this way, starting from the mid point of step at $T=0$ in Fig. \[Fig1\](b), we calculate the KC for sequences with increasing number of terms $n$. The output from these calculations are shown in Fig. \[Fig2\] for two different values of temperature. ![\[Fig2\] (Color online) Kolmogorov complexity $C_{\mathsf{K}}$ of the first harmonic and halfinteger step as a function of the length of their corresponding sequences for $K=4$, $\omega=1/2$, $r=0.2$, $F_{\mathsf{ac}}=0.2$, $\nu_0=0.5$ and $T=10^{-5}$, and $10^{-4}$ in (a) and (b) respectively. The dc force was varied with step $\delta F_{\mathsf{dc}}=10^{-5}$.](Fig2.eps){width="8"} As we can see in Fig. \[Fig2\](a), the existence of Shapiro step is obvious, i.e., the Kolmogorov complexity $C_{\mathsf{K}}$ determines the range of parameter $n$ for which $C_{\mathsf{K}}\geq 1$. Therefore, we define a criterion that when $C_{\mathsf{K}}$ becomes less than $1$, the system is no longer on the step, i.e., locked. Following this, we can determine very accurately the step width. If dc force was varied with some step $\delta F_{\mathsf{dc}}$, then the steps width can be determined as $$\label{DF} \Delta F=n_{\mathsf{max}}\, \delta F_{\mathsf{dc}},$$ where $n_{\mathsf{max}}$ is the maximal value of $n$ for which $C_{\mathsf{K}} \geq 1$. The set of parameters in Fig. \[Fig2\] was chosen in such manner that the first halfinteger is larger than the first harmonic step at $T=0$. If the temperature is increased, the steps are melting further as can be seen in Fig. \[Fig2\](b). Here, $C_{K}$ clearly shows not only the size of steps but it also reveals their robustness: harmonic step is more robust under the influence of noise, which is in agreement with [@ACTr]. If we increase $n$ ten times by decreasing the force step to $\delta F_{\mathsf{dc}}=10^{-6}$, we can see in Fig. \[Fig3\] that the results remain unchanged compared to the results in Fig. \[Fig2\](a). ![\[Fig3\] (Color online) Kolmogorov complexity $C_{\mathsf{K}}$ of the first harmonic and halfinteger step as a function of the length of their corresponding sequences for $\delta F_{\mathsf{dc}}=10^{-6}$. The rest of parameters is the same as in Fig. \[Fig2\]. ](Fig3.eps){width="8"} The length of sequence $n$ is essential for the precision of $C_{\mathsf{K}}$, and it is determined by the resolution ($\delta F_{\mathsf{dc}}$) in which plots for $\bar v(F_{\mathsf{dc}})$ are produced. Thus, depending on the size of steps and how precisely we want to measure them, the resolution in which steps are obtained should be adjusted accordingly. In our case, we can see that the resolution, which corresponds to $\delta F_{\mathsf{dc}}=10^{-5}$, provides accurate results for large harmonic or halfintiger steps. Detection and measurement of higher order subharmonic steps, which are typically very small, sometimes requires higher resolution, i.e. larger $n$. *However, we checked that for the force step $\delta F_{\mathsf{dc}}=10^{-3}$, Shapiro step widths agree within $1\%$ with those obtained with high resolution $\delta F_{\mathsf{dc}}=10^{-5}$ (see Appendix). This is due to the fact that decreasing $n$ induce variations of $C_{\mathsf{K}}$ in such manner that complexity is larger for shorter sequences when the system is on the step. Since proposed measure works even for shorter sequences, it could be of relevance in experimental studies.* In our calculation of $C_{\mathsf{K}}$, we started always from the midpoint of the steps as it was shown in Fig. \[Fig1\](b) since they melt symmetrically on the both sides, but in general, any point on the step can be chosen as the starting point, and by the expansion of $n$, the edges of each step can be precisely determined. Finally, we use KC to examine the temperature dependence of Shapiro steps. In Fig. \[Fig4\], the temperature dependence of the first harmonic and halfinteger step is presented. ![\[Fig4\] (Color online) The width of first harmonic $\Delta F_1$ and halfinteger step $\Delta F_{1/2}$ calculated from Kolmogorov complexity for $K=4$, $\omega=1/2$, $r=0.2$, $F_{\mathsf{ac}}=0.2$, $\nu_0=0.5$ and $T=10^{-5}$. The dc force was varied with step $\delta F_{\mathsf{dc}}=10^{-5}$. ](Fig4.eps){width="8"} The results clearly show that though the halfinteger step melts much faster, there is still a region of temperature for which halfinteger step is larger than harmonic one, which is again in very good agreement with the results of temperature dependence obtain in conventional way [@ACTr]. Conclusion {#concl} ========== In this work we applied Kolmogorov complexity on the studies of Shapiro steps in the presence of thermal noise. Conventional analysis of steps from the response function requires some consistent criterion of measure. However, this leads to larger errors in the case of smaller steps or higher temperatures. In contrast, using KC, we were able to detect and measure steps very precisely regardless of the system parameters. Though, we presented results for only harmonic and halfinteger steps, this technique could be also easily applied on the studies of even smallest subharmonic steps. It is important to note that this technique is also model independent, and since it relies only on the output data it could be equally well used for experimental and theoretical investigation of Shapiro steps in any system. For instance, the critical current in irradiated Josephson junctions can be defined as the applied current at which a finite voltage response first appears, and corresponds to a critical depinning force $F_{\mathsf{c}}$, whereas current step $\delta I_{\mathsf{dc}}$ is analogous to the force step $\delta F_{\mathsf{dc}}$ [@Reichard]. Also, widths of Shapiro steps $\Delta V$ in irradiated Josephson junction systems correspond to $\Delta F$ considered in this paper [@ACDS; @EPL]. Shapiro steps have wide applications and Kolmogorov complexity could be a practical tool which can improve measurments of their size in presence of thermal noise. To illustrate that, let us look at the recent studies that attracted great attention: application of Josephson junctions in detection of Majorana fermions [@Jiang; @Houz; @ShukrinovPRB2]. In Ref. [@ShukrinovPRB2], it was shown that the current-voltage characteristics exhibits odd Shapiro steps, where particular sequence of subharmonic steps in the devil’s staircase structure represents a signature of the Majorana states. Thus, detection of Majorana fermions requires very precise observation and measurement of subharmonic Shapiro steps, which in experiments could be a real problem due to noise (all analysis in Ref. [@ShukrinovPRB2] were performed at the zero temperature). We believe that application of KC could help in the solution of this and similar problems. We have been focused here only on Shapiro steps in the presence of thermal effects, but KC could be used for studies of other effects among which certainly interesting is the chaos [@Shukrinov; @Shukrinov; @ACm]. Application of Kolmogorov complexity on the studies of chaotic behavior in systems which exhibit Shapiro steps are part of our future studies and will be published separately. We wish to express our gratitude to Jovan Odavi' c and Gordan Mimi' c for helpful discussions. This work was supported by the Serbian Ministry of Education and Science under Contracts No. OI-171009 and No. III-45010 and by the Provincial Secretariat for High Education and Scientific Research of Vojvodina (Project No. APV 114-451-2201). Complexity of shorter sequences ================================ For the force step $\delta F_{\mathsf{dc}} = 10^{-3}$, we obtain Shapiro step widths within $1\%$ of values calculated with $\delta F_{\mathsf{dc}} = 10^{-5}$ by using relation $\Delta F = n_{\mathsf{max}} \delta F_{\mathsf{dc}}$ (see Fig. \[Fig5\]). ![\[Fig5\] (Color online) Kolmogorov complexity $C_{\mathsf{K}}$ of first harmonic and halfinteger step calculated for $K=4$, $\omega=1/2$, $r=0.2$, $F_{\mathsf{ac}}=0.2$, $\nu_0=0.5$ and two temperatures a) $T=10^{-5}$ and b) $T=10^{-4}$. The dc force was varied with step $\delta F_{\mathsf{dc}}=10^{-3}$. ](Fig5.eps){width="8"} [^1]: \[FN\] According to (\[CK\]), the value of KC for the binary sequence $11010001$ would therefore be $C_{\mathsf{K}}=1.5$. However, the given sequence is too short, $n=8$, and consequently, the obtained result for $C_{\mathsf{K}}$ is not relevant.

--- abstract: 'We discuss localized ground states of the periodic Gross-Pitaevskii equation in the framework of a quantum linear Schrödinger equation with effective potential determined in self-consistent manner. We show that depending on the interaction among the atoms being attractive or repulsive, bound states of the linear self consistent problem are formed in the forbidden zones of the linear spectrum below or above the energy bands. These eigenstates are shown to be exact solitons of the GPE equation. The implications of this bound state interpretation on the existence of a delocalization transition for multidimensional solitons is briefly discussed.' address: | Dipartimento di Fisica “E.R. Caianiello” and Istituto Nazionale di Fisica della Materia (INFM),\ Universitá di Salerno, I-84081 Baronissi (SA), Italy author: - Mario Salerno title: Macroscopic quantum bound states of Bose Einstein condensates in optical lattices --- PACS numbers: 03.75.Fi, 05.30.Jp, 05.45.-a [2]{} One interesting phenomenon occurring in periodic nonlinear systems is the possibility to stabilize localized excitations as a result of the interplay between periodicity and nonlinearity. An example of this is provided by the nonlinear Schrödinger equation (NLS) with periodic potential. It is well known that the defocusing NLS does not admit bright soliton solutions, these being unstable against background decay [@scott]. The presence of a periodic potential, however, allows to stabilize bright solitons against decay, a phenomenon which is presently investigated in connection with Bose Einstein condensates (BEC) in optical lattices (OL). The possibility to form bright solitons in repulsive BEC with OL was analytically and numerically demonstrated, both for a discrete version of the NLS describing BEC arrays in the tight-binding approximation [@smerzi] and for the Gross-Pitaevskii equation (GPE) describing the properties of a continuous BEC in the mean field approximation [@potting; @ks02; @alfimov]. The mechanism underlying soliton formation in periodic structures was identified to be the modulational instability of the Bloch states at the edges of the Brillouin zone [@ks02]. These localized excitations correspond to states with energies inside the gaps of the underlying linear band structure (in nonlinear optics they are called gap solitons) and with an effective mass which depends on the sign of the interaction (for repulsive interactions, bright solitons have negative effective mass, this explaining their existence in BEC with OL [@ks02; @steel]). The usage of linear concepts such as Bloch states, effective mass, etc. [@ks02; @steel; @pethick], makes natural to ask whether nonlinear states could be interpreted in a pure linear (quantum mechanical) context. The aim of the present paper is to address this problem by showing that soliton solutions of the periodic nonlinear Schrödinger equations correspond to bound states of the linear Schrodinger equation with an effective potential which can be determined in self-consistent (SC) manner. This problem will be discussed on the physical example of a Bose Einstein condensate in an optical lattice (OL) described, in mean field approximation, by the following normalized Gross-Pitaevskii equation $$i\psi_t=\left[-\nabla ^{2}+ U_{ol}({\bf{x}}) + \chi|\psi|^{2}\right] \psi \label{gpe}$$ where $\chi$ is the nonlinear parameter, $\bf x$ denotes three dimensional coordinates and $U(\bf x)$ is a periodic potential representing the OL. To discuss bound state features of solitons we restrict to the one dimensional case (the approach however is of general validity and can be applied to NLS type equations in arbitrary dimensions). At the end of the paper we will briefly discuss the implications of the bound state interpretation of localized solutions on the soliton delocalization transition observed in higher dimensions [@flach]. We remark that the properties of solitons of the GPE in optical lattices were studied in [@alfimov] in terms of orbits of a chaotic system. Self-consistent approaches were also used as numerical tools to study discrete breathers of the discrete NLS [@panos] and the stability of gap solitons [@markus]. The physical implications and the full potentiality of the SC approach, however, have not been investigated. Our analysis is based on the simple observation that the stationary localized ground states $\psi_s(x,t) = \psi(x) \exp(-\mu t)$ of the GPE (and more generally of any nonlinear Schrödinger-like equation) can be obtained by solving in a self-consistent manner the following linear Schrödinger problem $$\left[-\nabla ^{2}+ \hat V_{eff}(x) \right] \psi= E \psi \label{schro}$$ with the effective potential $$\hat V_{eff}= \hat U_{ol}(x)+ \hat U_s (x) = A \cos(2 x) + \chi |\hat \psi_s (x)|^2. \label{Veff}$$ Here $\hat U_{ol} \equiv A \cos(2 x) $ is the OL and $\hat U_s$ is the potential associated with a given eigenstate of the quantum problem (\[schro\]). For a self-consistent solution, one starts with a trial wavefunction for $\psi_s$ (typically a gaussian waveform), calculates the effective potential and solves the corresponding eigenvalue problem (\[schro\]). Then, one selects a given eigenstate (for example the ground state but not necessarily) as new trial function and iterates the procedure until convergence is reached. ![ Panel [**(a)**]{} Energy spectrum for the effective potential (\[Veff\]) with $A=3$ and $\chi=0$ (Mathieu equation). Full lines represent exact values of the band edges of the Mathieu equation while dots are the eigenvalues obtained with the above procedure on a lattice of length $L=40 \pi$, with $N=512$ points. Panel [**(b)**]{} Lowest energy band for the effective potential in Eq. (\[Veff\]) with $\psi_s$ taken as the ground state of the system and for $\chi=-1$ (attractive case). Parameters are fixed as in panel (a). Panel [**(c)**]{} The same as in panel (b) but for $A=-3$. Panel [**(d)**]{} Transition from the metastable IS mode to the OS ground state corresponding to the lower level of panel (c). The optical lattice (scaled by a factor 3) is reported as an help to locate the symmetry center of the solutions. Parameters are fixed as in panel (c).[]{data-label="fig1"}](fig1a.eps "fig:"){width="3.8cm" height="3.3cm"} ![ Panel [**(a)**]{} Energy spectrum for the effective potential (\[Veff\]) with $A=3$ and $\chi=0$ (Mathieu equation). Full lines represent exact values of the band edges of the Mathieu equation while dots are the eigenvalues obtained with the above procedure on a lattice of length $L=40 \pi$, with $N=512$ points. Panel [**(b)**]{} Lowest energy band for the effective potential in Eq. (\[Veff\]) with $\psi_s$ taken as the ground state of the system and for $\chi=-1$ (attractive case). Parameters are fixed as in panel (a). Panel [**(c)**]{} The same as in panel (b) but for $A=-3$. Panel [**(d)**]{} Transition from the metastable IS mode to the OS ground state corresponding to the lower level of panel (c). The optical lattice (scaled by a factor 3) is reported as an help to locate the symmetry center of the solutions. Parameters are fixed as in panel (c).[]{data-label="fig1"}](fig1b.eps "fig:"){width="3.8cm" height="3.3cm"} ![ Panel [**(a)**]{} Energy spectrum for the effective potential (\[Veff\]) with $A=3$ and $\chi=0$ (Mathieu equation). Full lines represent exact values of the band edges of the Mathieu equation while dots are the eigenvalues obtained with the above procedure on a lattice of length $L=40 \pi$, with $N=512$ points. Panel [**(b)**]{} Lowest energy band for the effective potential in Eq. (\[Veff\]) with $\psi_s$ taken as the ground state of the system and for $\chi=-1$ (attractive case). Parameters are fixed as in panel (a). Panel [**(c)**]{} The same as in panel (b) but for $A=-3$. Panel [**(d)**]{} Transition from the metastable IS mode to the OS ground state corresponding to the lower level of panel (c). The optical lattice (scaled by a factor 3) is reported as an help to locate the symmetry center of the solutions. Parameters are fixed as in panel (c).[]{data-label="fig1"}](fig1c.eps "fig:"){width="3.8cm" height="3.3cm"} ![ Panel [**(a)**]{} Energy spectrum for the effective potential (\[Veff\]) with $A=3$ and $\chi=0$ (Mathieu equation). Full lines represent exact values of the band edges of the Mathieu equation while dots are the eigenvalues obtained with the above procedure on a lattice of length $L=40 \pi$, with $N=512$ points. Panel [**(b)**]{} Lowest energy band for the effective potential in Eq. (\[Veff\]) with $\psi_s$ taken as the ground state of the system and for $\chi=-1$ (attractive case). Parameters are fixed as in panel (a). Panel [**(c)**]{} The same as in panel (b) but for $A=-3$. Panel [**(d)**]{} Transition from the metastable IS mode to the OS ground state corresponding to the lower level of panel (c). The optical lattice (scaled by a factor 3) is reported as an help to locate the symmetry center of the solutions. Parameters are fixed as in panel (c).[]{data-label="fig1"}](fig1d.eps "fig:"){width="3.8cm" height="3.3cm"} The problem is thus reduced to the diagonalization of the quantum Hamiltonian $$\hat H = \hat H_0 + \hat V_{eff} (x) \label{hamilt}$$ with $\hat H_0\equiv -\nabla ^{2}$ the kinetic energy operator. This can be effectively done by adopting a discrete coordinate space representation $\{x_n=n a\}$, $n=1,...,N$, with $a=L/N$ the discretization constant, $L$ the size of the system and N the total number of points. A basis for this space is simply a basis of $R^N$, i.e. the set of N-component vectors of the type $|n \rangle=(0,...0,1,0,...,0)$, with the $1$ in the position $n$. The effective potential is obviously diagonal in this representation i.e. $\langle n|\hat V_{eff}|n'\rangle = V_{eff} (n a) \delta_{n,n'}$, while $\hat H_0$ is diagonal in the reciprocal representation $|k_n \rangle $, ($k_n= 2\pi/L n $, the two representations being related by the Fourier transform (unitary transformation). The matrix elements of the Hamiltonian $\hat H$ can then be constructed as $$\langle n|\hat H|n'\rangle \equiv H_{n,n'}=\langle n| \hat F^{-1} \hat H_0 \hat F |n'\rangle + V_{eff} (n a) \delta_{n,n'}$$ where $\hat F |n \rangle$ denotes the Fourier transform of the vector $|n \rangle$. ![Wavefunctions and corresponding effective potential for the bound states below the lowest bands of Fig 1a for attractive interaction $\chi=-1$. Panel [**(a)**]{}. OS mode and corresponding effective potential for $E=-1.1667950$ (ground state) and $A=3$. The effective potential was scaled by a factor 6 for graphical convenience. Panel [**(b)**]{}. Same as (a) for the IS mode at $E=-1.0485745$ and $A=-3$. Panel [**(c)**]{}. Same as (a) for the OA mode. $E=-0.999261$. Panel [**(d)**]{}. Same as (b) for the IA mode. $E=-0.9947127$. [**(e)**]{} Energy levels of the OS, IS, OA, IA, nodes inside the gap between the first two bands. The continuous line denotes the lower edge of the second band of Fig. 1a while the dotted lines denote the degenerate levels. Parameters are fixed as for corresponding modes in panels a-d. [**(f)**]{}. Wavefunctions associated to the levels in panel e. For graphical convenience the IS mode was shifted by -1.0 down while the IA and OS modes were shifted up by 0.5 and 1.0, respectively.[]{data-label="fig2"}](fig2a.eps "fig:"){width="3.8cm" height="3.cm"} ![Wavefunctions and corresponding effective potential for the bound states below the lowest bands of Fig 1a for attractive interaction $\chi=-1$. Panel [**(a)**]{}. OS mode and corresponding effective potential for $E=-1.1667950$ (ground state) and $A=3$. The effective potential was scaled by a factor 6 for graphical convenience. Panel [**(b)**]{}. Same as (a) for the IS mode at $E=-1.0485745$ and $A=-3$. Panel [**(c)**]{}. Same as (a) for the OA mode. $E=-0.999261$. Panel [**(d)**]{}. Same as (b) for the IA mode. $E=-0.9947127$. [**(e)**]{} Energy levels of the OS, IS, OA, IA, nodes inside the gap between the first two bands. The continuous line denotes the lower edge of the second band of Fig. 1a while the dotted lines denote the degenerate levels. Parameters are fixed as for corresponding modes in panels a-d. [**(f)**]{}. Wavefunctions associated to the levels in panel e. For graphical convenience the IS mode was shifted by -1.0 down while the IA and OS modes were shifted up by 0.5 and 1.0, respectively.[]{data-label="fig2"}](fig2b.eps "fig:"){width="3.8cm" height="3.cm"} ![Wavefunctions and corresponding effective potential for the bound states below the lowest bands of Fig 1a for attractive interaction $\chi=-1$. Panel [**(a)**]{}. OS mode and corresponding effective potential for $E=-1.1667950$ (ground state) and $A=3$. The effective potential was scaled by a factor 6 for graphical convenience. Panel [**(b)**]{}. Same as (a) for the IS mode at $E=-1.0485745$ and $A=-3$. Panel [**(c)**]{}. Same as (a) for the OA mode. $E=-0.999261$. Panel [**(d)**]{}. Same as (b) for the IA mode. $E=-0.9947127$. [**(e)**]{} Energy levels of the OS, IS, OA, IA, nodes inside the gap between the first two bands. The continuous line denotes the lower edge of the second band of Fig. 1a while the dotted lines denote the degenerate levels. Parameters are fixed as for corresponding modes in panels a-d. [**(f)**]{}. Wavefunctions associated to the levels in panel e. For graphical convenience the IS mode was shifted by -1.0 down while the IA and OS modes were shifted up by 0.5 and 1.0, respectively.[]{data-label="fig2"}](fig2c.eps "fig:"){width="3.8cm" height="3.cm"} ![Wavefunctions and corresponding effective potential for the bound states below the lowest bands of Fig 1a for attractive interaction $\chi=-1$. Panel [**(a)**]{}. OS mode and corresponding effective potential for $E=-1.1667950$ (ground state) and $A=3$. The effective potential was scaled by a factor 6 for graphical convenience. Panel [**(b)**]{}. Same as (a) for the IS mode at $E=-1.0485745$ and $A=-3$. Panel [**(c)**]{}. Same as (a) for the OA mode. $E=-0.999261$. Panel [**(d)**]{}. Same as (b) for the IA mode. $E=-0.9947127$. [**(e)**]{} Energy levels of the OS, IS, OA, IA, nodes inside the gap between the first two bands. The continuous line denotes the lower edge of the second band of Fig. 1a while the dotted lines denote the degenerate levels. Parameters are fixed as for corresponding modes in panels a-d. [**(f)**]{}. Wavefunctions associated to the levels in panel e. For graphical convenience the IS mode was shifted by -1.0 down while the IA and OS modes were shifted up by 0.5 and 1.0, respectively.[]{data-label="fig2"}](fig2d.eps "fig:"){width="3.8cm" height="3.cm"} ![Wavefunctions and corresponding effective potential for the bound states below the lowest bands of Fig 1a for attractive interaction $\chi=-1$. Panel [**(a)**]{}. OS mode and corresponding effective potential for $E=-1.1667950$ (ground state) and $A=3$. The effective potential was scaled by a factor 6 for graphical convenience. Panel [**(b)**]{}. Same as (a) for the IS mode at $E=-1.0485745$ and $A=-3$. Panel [**(c)**]{}. Same as (a) for the OA mode. $E=-0.999261$. Panel [**(d)**]{}. Same as (b) for the IA mode. $E=-0.9947127$. [**(e)**]{} Energy levels of the OS, IS, OA, IA, nodes inside the gap between the first two bands. The continuous line denotes the lower edge of the second band of Fig. 1a while the dotted lines denote the degenerate levels. Parameters are fixed as for corresponding modes in panels a-d. [**(f)**]{}. Wavefunctions associated to the levels in panel e. For graphical convenience the IS mode was shifted by -1.0 down while the IA and OS modes were shifted up by 0.5 and 1.0, respectively.[]{data-label="fig2"}](fig2e.eps "fig:"){width="3.8cm" height="3.cm"} ![Wavefunctions and corresponding effective potential for the bound states below the lowest bands of Fig 1a for attractive interaction $\chi=-1$. Panel [**(a)**]{}. OS mode and corresponding effective potential for $E=-1.1667950$ (ground state) and $A=3$. The effective potential was scaled by a factor 6 for graphical convenience. Panel [**(b)**]{}. Same as (a) for the IS mode at $E=-1.0485745$ and $A=-3$. Panel [**(c)**]{}. Same as (a) for the OA mode. $E=-0.999261$. Panel [**(d)**]{}. Same as (b) for the IA mode. $E=-0.9947127$. [**(e)**]{} Energy levels of the OS, IS, OA, IA, nodes inside the gap between the first two bands. The continuous line denotes the lower edge of the second band of Fig. 1a while the dotted lines denote the degenerate levels. Parameters are fixed as for corresponding modes in panels a-d. [**(f)**]{}. Wavefunctions associated to the levels in panel e. For graphical convenience the IS mode was shifted by -1.0 down while the IA and OS modes were shifted up by 0.5 and 1.0, respectively.[]{data-label="fig2"}](fig2f.eps "fig:"){width="3.8cm" height="3.cm"} -.2cm For an effective construction of these matrix elements one can use the fast Fourier transform while for the computation of the spectrum one can recourse to standard numerical routines for the diagonalization of real symmetric matrices. To check the method we consider first the case of a linear effective potential of the form $V_{eff}=U_{ol}=A \cos(2 x)$ for which the eigenvalue problem reduces to the well known Mathieu equation. In Fig. 1a we depict the lowest part of the spectrum (notice that in this case there is no SC procedure due to the linearity of the problem) from which we see the appearance of a band structure with band edges which exactly coincide with the values obtained for the Mathieu equation (for high energy bands to get good accuracy one needs to increase $N$). In this paper we are mainly interested in the localized states associated with the lowest two bands (i.e., the ones physically most relevant), and for this purpose the choice of $N=256$ will be adequate for most of the calculations. ![Panel [**(a)**]{}. Time evolution of the OS bound state of Fig 2a as obtained from GPE. Panel [**(b)**]{}. Same as in panel (a) for the IS mode of Fig. 2c.[]{data-label="fig3"}](fig3a.eps "fig:"){width="4.cm" height="3.6cm"} ![Panel [**(a)**]{}. Time evolution of the OS bound state of Fig 2a as obtained from GPE. Panel [**(b)**]{}. Same as in panel (a) for the IS mode of Fig. 2c.[]{data-label="fig3"}](fig3b.eps "fig:"){width="4.cm" height="3.6cm"} ![Energy spectrum in correspondence of the localized states above the lowest band of Fig. 1a for repulsive interactions $\chi=1$. Panel [**(a)**]{}. Spectrum associated to the OS mode. Parameters are $A=-3, N=256, L=40 \pi$. Panel [**(b)**]{}. Same as panel (a) but for the IA mode with $A=3$. The continuous lines denote exact band edges of the Mathieu equation. []{data-label="fig4"}](fig4a.eps "fig:"){width="3.8cm" height="3.4cm"} ![Energy spectrum in correspondence of the localized states above the lowest band of Fig. 1a for repulsive interactions $\chi=1$. Panel [**(a)**]{}. Spectrum associated to the OS mode. Parameters are $A=-3, N=256, L=40 \pi$. Panel [**(b)**]{}. Same as panel (a) but for the IA mode with $A=3$. The continuous lines denote exact band edges of the Mathieu equation. []{data-label="fig4"}](fig4b.eps "fig:"){width="3.8cm" height="3.4cm"} In Fig. 1b we show how the lowest band of panel 1a is modified by the nonlinear potential $V_{eff}(x)=A \cos(2 x)+ \chi |\psi_0|^2$, where $\psi_0$ is taken to be the ground state of the system, for the case $\chi<0$ (negative scattering length). A bound state below the band which rapidly converges to a constant value is quite evident. One can see that the state forms from the lower edge of the band and is accompanied by a rearrangement of the extended states inside the band. The corresponding eigenvector is depicted in Fig.2a together with its effective potential. Notice that the potential has an attractive character (potential well) and the bound state is symmetric around a minima of the OL, i.e., it resembles the onsite-symmetric intrinsic localized mode (ILM) of nonlinear lattices (NL) [@sievers]. In the following we shall call it the onsite symmetric (OS) bound state. By shifting the phase of the OL by $\pi$ (i.e. by changing the sign of $A$) one obtains an eigenstate centered on maxima instead than on minima. This bound state is depicted in Fig. 2b and in analogy with NL we shall call it the intersite symmetric (IS) mode. The corresponding spectrum is reported in Fig. 1c. Notice that the IS mode corresponds to the plateau formed just before the decay into the OS mode occurs as shown in panel 1d (also note the appearance of an intersite-symmetric (IA) excited level in Fig. 1c which is absorbed into the band in correspondence of the IS-OS decay). ![Wavefunctions and effective potentials of the bound states levels of Fig. 4 a,b, for the repulsive case $\chi=1$. Panel [**(a)**]{}. OS mode with corresponding effective potential (thin line). Energy is $E=-0.078355$ and $A=-3$. Panel [**(b)**]{}. Same as in Panel (a) for the IA mode. $E=-0.376645$, $A=3$. Panel [**(c)**]{}. Same as in panel (a) for the OA mode. $E=-0.683070$. Panel [**(d)**]{}. Same in panel (b) for the IS mode. $E=-0.691676$. Parameters are fixed as $N=256$, $L=40 \pi$ for panels (a), (b), and $N=512$, $L=40 \pi$ for panels (c), (d). The effective potentials have been reduced by a factor $6$ for graphical convenience. []{data-label="fig5"}](fig5a.eps "fig:"){width="3.8cm" height="3.cm"} ![Wavefunctions and effective potentials of the bound states levels of Fig. 4 a,b, for the repulsive case $\chi=1$. Panel [**(a)**]{}. OS mode with corresponding effective potential (thin line). Energy is $E=-0.078355$ and $A=-3$. Panel [**(b)**]{}. Same as in Panel (a) for the IA mode. $E=-0.376645$, $A=3$. Panel [**(c)**]{}. Same as in panel (a) for the OA mode. $E=-0.683070$. Panel [**(d)**]{}. Same in panel (b) for the IS mode. $E=-0.691676$. Parameters are fixed as $N=256$, $L=40 \pi$ for panels (a), (b), and $N=512$, $L=40 \pi$ for panels (c), (d). The effective potentials have been reduced by a factor $6$ for graphical convenience. []{data-label="fig5"}](fig5b.eps "fig:"){width="3.8cm" height="3.cm"} ![Wavefunctions and effective potentials of the bound states levels of Fig. 4 a,b, for the repulsive case $\chi=1$. Panel [**(a)**]{}. OS mode with corresponding effective potential (thin line). Energy is $E=-0.078355$ and $A=-3$. Panel [**(b)**]{}. Same as in Panel (a) for the IA mode. $E=-0.376645$, $A=3$. Panel [**(c)**]{}. Same as in panel (a) for the OA mode. $E=-0.683070$. Panel [**(d)**]{}. Same in panel (b) for the IS mode. $E=-0.691676$. Parameters are fixed as $N=256$, $L=40 \pi$ for panels (a), (b), and $N=512$, $L=40 \pi$ for panels (c), (d). The effective potentials have been reduced by a factor $6$ for graphical convenience. []{data-label="fig5"}](fig5c.eps "fig:"){width="3.8cm" height="3.cm"} ![Wavefunctions and effective potentials of the bound states levels of Fig. 4 a,b, for the repulsive case $\chi=1$. Panel [**(a)**]{}. OS mode with corresponding effective potential (thin line). Energy is $E=-0.078355$ and $A=-3$. Panel [**(b)**]{}. Same as in Panel (a) for the IA mode. $E=-0.376645$, $A=3$. Panel [**(c)**]{}. Same as in panel (a) for the OA mode. $E=-0.683070$. Panel [**(d)**]{}. Same in panel (b) for the IS mode. $E=-0.691676$. Parameters are fixed as $N=256$, $L=40 \pi$ for panels (a), (b), and $N=512$, $L=40 \pi$ for panels (c), (d). The effective potentials have been reduced by a factor $6$ for graphical convenience. []{data-label="fig5"}](fig5d.eps "fig:"){width="3.8cm" height="3.cm"} We have checked that these bound states coincide with the ones obtained with the approach of Ref. [@alfimov] for the same values of energy. The stability of the OS mode and the decay of the IS mode into the OS state was checked by direct numerical integrations of the GPE (see Fig.3). To obtain the onsite asymmetric (OA) mode one needs to take the first excited state $\psi_1$ as effective potential in the SC procedure. This indeed produces an exact soliton solution of the GPE of type OA as shown in Fig.2c. A shifting of the potential by $\pi$ produce the intersite asymmetric (IA) mode of Fig. 2d. These solutions have the same energies and are more unstable than the IS mode (they, however, do not decay into the ground state but get mixed with the extended states in the band). In general, the effective potentials can be taken of the form $\hat V_{eff}=\hat V_{ol}+ \chi |\hat \psi_n (x)|$ with $\psi_n$ the n-th eigenstate of the eigenvalue problem in (\[schro\]). If the energy of $\psi_n$ lies outside the bands a localized mode of the type described above is produced, while if it lies inside a band, extended states which are nonlinear analogue of the Bloch states [@pethick], are produced. From this we conclude that both localized and extended solutions of the GPE are exact quantum eigenstates of the Schrödinger equation with a suitable effective potential. From the above analysis one expects that below each higher energy bands, eigenstates of the same symmetry type as the ones found for the lowest band should exist. This is precisely what we show in Fig.s 2e, 2f for the energy spectrum and the corresponding eigenstated in the gap below the second band. ![Panel [**(a)**]{}. Two soliton bound state of the repulsive GPE obtained from the SC procedure by using as effective potential $V_{eff}=V_{ol}+|\frac 12 \psi_1-\psi_2]^2$ where $\psi_1, \psi_2$ denote two eigenfunctions at the top of the first band. Panel [**(b)**]{}. Time evolution (modulo square) of the bound state in panel (a) taken as initial condition for the integration of the full GPE. []{data-label="fig6"}](fig6a.eps "fig:"){width="3.8cm" height="3.6cm"} ![Panel [**(a)**]{}. Two soliton bound state of the repulsive GPE obtained from the SC procedure by using as effective potential $V_{eff}=V_{ol}+|\frac 12 \psi_1-\psi_2]^2$ where $\psi_1, \psi_2$ denote two eigenfunctions at the top of the first band. Panel [**(b)**]{}. Time evolution (modulo square) of the bound state in panel (a) taken as initial condition for the integration of the full GPE. []{data-label="fig6"}](fig6b.eps "fig:"){width="3.8cm" height="4.6cm"} Notice that the OA and the IS eigenstates are degenerated (the same occurs also to the OS and IA modes). The OA and IS bound states are both very stable while the energy levels of the OA and IS modes, after establishing a plateau similar to the one in Fig 1c, become unstable (the energy oscillates between this level and the lower edge of the second band). The instability of these modes can be understood as an hybridization of the state (being very close to the band edge) with extended states of the second band and is confirmed by direct numerical integration of the PDE system. Similar localized modes can be found also for repulsive interactions ($\chi>0$), the main difference with the previous case being that now the states appear in the gap above the band edges instead than below. This is shown in Fig. 4 for the lowest energy levels inside the first gap. The corresponding eigenvectors are shown in Fig.5 together with their effective potentials. Notice that the potential has a local repulsive character (it increases in correspondence of the states) so that these bound states could not exist without the OL. We remark that localized modes similar to the ones described in this paper were found also in atomic-molecular BEC using an approach based on Wannier functions [@abdullaev]. Besides localized and extended states, the SC procedure allows to construct full nonlinear bands in reciprocal space (we omit details for brevity). It is worth remarking that more complicated set of solutions of the GPE can be constructed with the SC procedure by taking as effective potentials linear combinations of eigenstates. An example of this is shown in Fig.6 for the case of repulsive interaction. We see that a linear combination of two eigenstates leads to a bound state with two humps which corresponds to a multi-soliton solution of the GPE (see panel b). This is a general property and we conjecture that [*all solutions of the periodic GPE (and more in general of the NLS-like equations with arbitrary potentials) can be obtained with the SC method taking all possible combinations of linear eigenstates as effective potentials*]{}. Before closing this paper we wish to remark that the above bound state interpretation has important consequences on the delocalizing transition [@flach] of localized solutions of the GPE in OL. Since in a 1D potential bound state always exists, it follows from the above analysis that no delocalizing transition of a BEC soliton can occur in this case. On the contrary, for $D \geq 2$ a finite potential depth is required to form a bound state, this implying that a soliton delocalization transition can occur. A detailed investigation of the delocalizing transition of BEC solitons in OL will be reported elsewhere [@bs03]. It is a pleasure to thank Prof. S. De Filippo and Dr. B. Baizakov for interesting discussions. Financial support from a MURST-PRIN-2003 Initiative, and from the European grant LOCNET no. HPRN-CT-1999-00163, is also acknowledged. Alwyn Scott [*Nonlinear Science: Emergence and Dynamics of Coherent Structures,*]{} Oxford Univ. Press, 1999. A. Trombettoni and A. Smerzi, Phys. Rev. Lett. [**86**]{}, 2353 (2001); F. K. Abdullaev, B. B. Baizakov, S. A. Darmanyan, V. V. Konotop, and M. Salerno, Phys. Rev. A [**64**]{}, 043606 (2001). S. Pötting, O. Zobay, P. Meystre, and E. M. Wright, J. Mod. Opt. [**47**]{}, 2653 (2000). V. V. Konotop and M. Salerno, Phys. Rev. A [**65**]{}, 021602 (2002); A. Smerzi, A. Trombettoni, P.G. Kevrekidis, A.R. Bishop, Phys.Rev. Lett. [**89**]{}. 17042 (2002). G. L. Alfimov, V. V. Konotop, and M. Salerno, Europhys. Lett. [**58**]{}, 7 (2002); N.K. Efremidis and D. N. Christodoulides, Phys.Rev.A [**67**]{} 063608 (2003). M. J. Steel and W. Zhang, e-print cond-mat/9810284. D. Diakonov, L.M. Jensen, C.J. Pethick, and H. Smith, Phys.Rev. A [**66**]{} 013604 (2002); P.J. Louis, E.A. Ostrovskaya C.M. Savage, and Y.S. Kivshar, e-print cond-mat/0208516 G. Kalosakas, K. Ø. Rasmussen, and A. R. Bishop, Phys. Rev. Lett. [**89**]{}, 030402 (2002); see also S. Flach, K. Kladko, and R.S. MacKay, Phys. Rev. Lett. [**78**]{}, 1207 (1997). Karen Marie Hilligsœ, Markus K. Oberthaler, and Karl-Peter Marzlin, Phys. Rev. A [**66**]{}, 063605 (2002). P.G. Kevrekidis, K.O Rasmussen, A.R. Bishop, Phys. Rev. E [**61**]{}, 4652 (2000) A.J. Sievers and S. Takeno, Phys.Rev.Lett. [**61**]{}, 970 (1988); O.M. Braun and Yu. S. Kivshar, Phys.Rep. [**[306]{}**]{}, 2 (1998). F.Kh. Abdullaev and V.V. Konotop, Phys. Rev. A [**68**]{} 013605 (2003). B. B. Baizakov, M. Salerno, Phys. Rev. A 2003 (in press).

--- abstract: 'The Receiver Operating Characteristic (ROC) curve is a useful tool that measures the discriminating power of a continuous variable or the accuracy of a pharmaceutical or medical test to distinguish between two conditions or classes. In certain situations, the practitioner may be able to measure some covariates related to the diagnostic variable which can increase the discriminating power of the ROC curve. To protect against the existence of atypical data among the observations, a procedure to obtain robust estimators for the ROC curve in presence of covariates is introduced. The considered proposal focusses on a semiparametric approach which fits a location-scale regression model to the diagnostic variable and considers empirical estimators of the regression residuals distributions. Robust parametric estimators are combined with adaptive weighted empirical distribution estimators to down-weight the influence of outliers. The uniform consistency of the proposal is derived under mild assumptions. A Monte Carlo study is carried out to compare the performance of the robust proposed estimators with the classical ones both, in clean and contaminated samples. A real data set is also analysed.' author: - | Ana M. Bianco$^1$, Graciela Boente$^1$ and Wenceslao González–Manteiga$^2$\ $^1$ Universidad de Buenos Aires and CONICET\ $^2$ Universidad de Santiago de Compostela title: A robust approach for ROC curves with covariates --- [*AMS Subject Classification:*]{} 62F35 Covariates; Robustness, ROC curves; Parametric regression Introduction ============ [\[intro\]]{} The Receiver Operating Characteristic (ROC) curve is a useful tool to size up the capability of a continuous variable or the accuracy of a pharmaceutical or medical test to distinguish between two conditions. ROC curves are a very well known technique in medical studies where a continuous variable or marker (biomarker) is used to diagnose a disease or to evaluate the progression of a disease. The use of ROC curves has become more and more popular in medicine from the early 60’s (see Goncalves *et al.*, 2014, for a historical note and Krzanowsk and Hand, 2009 for further details). ROC curves can also be extended to other general statistical situations such as classification or discrimination, where we typically have a set of individuals or items assigned to one of two classes on the basis of disposable information of that individual. A ROC curve is essentially a plot that represents the diagnostic skill of a binary classifier as the discriminating threshold varies. Assignations are not perfect and may lead to classification errors. In fact, during the assignment procedure some errors may occur, in the sense that an individual or object may be allocated into a wrong class. At this point, ROC curves become an interesting strategy either to evaluate the quality of a given assignment rule or to compare two available procedures. To be more precise, assume that we deal with two populations, henceforth, identified as diseased (*D*) and healthy (*H*) and that a continuous score usually called *biomarker* or *diagnostic variable*, $Y$, is considered for the assignment purpose and whose rule is based on a cut–off value $c$. Thus, according to this assignment rule, an individual is classified as diseased if $Y \ge c$ and as healthy when $Y < c$. Let $F_{D}$ be the distribution of the marker on the diseased population and $F_{H}$ the distribution of $Y$ in the healthy one. From now on, for practical reasons, we denote as $Y_D \sim F_D$ the marker in the diseased population and $Y_{H} \sim F_{H}$ the score in the healthy one. Without loss of generality, we will assume that $Y_D$ is stochastically greater than $Y_{H}$, that is, $\prob(Y_{D} \le c) \le \prob(Y_{H} \le c)$ for all $c$. It is clear that the classification errors depend on the threshold $c$. Therefore, it becomes of interest to study the triplets $\{(c,1-F_{H}(c),1-F_{D}(c)),\; c \in \real \}$, which describes a geometrical object called ROC curve, that reflects the discriminatory capability of the marker. This suggests a different parametrization of this curve in terms of the false positive rate, $1-F_{H}(c)$, leading to $\{(p, 1-F_{D}(F_{H}^{-1}(1-p))), \; p \in (0,1) \}$ and therefore, to $\ROC(p)= 1-F_{D}(F_{H}^{-1}(1-p))), \quad p \in (0,1) $. In this manner, the ROC curve is a complete picture of the performance of the assignment procedure over all the possible threshold values. In practical situations, the discriminatory effectiveness of the biomarker may be improved by several factors. Thus, when for each individual there is additional information contained in measured covariates, it is sensible to include them in the ROC analysis. Through examples Pepe (2003) illustrates how the discriminatory capability of a test is improved by the presence of covariates. For an overview on this topic, we refer to Pardo-Fernández *et al*. (2014). In brief, we may say that the information registered all along the covariates may impact the discrimination capability of the ROC curve. In this situation, in order to have a deeper comprehension of the effect of the covariates, it would be advisable to incorporate this additional covariates information to the ROC analysis instead of considering a *joint* ROC curve, that may lead to oversimplification. This issue can be accomplished in different ways. In the direct methodology, the ROC curve is directly regressed onto the covariates by means of a generalized linear model. Among others, Alonzo and Pepe (2002), Pepe (2003) and Cai (2004) follow this approach. In contrast, in the induced methodology, the markers distribution in each population is modelled separately in terms of the covariates and just after, the induced ROC curve is computed. The papers by Pepe (1998), Faraggi (2003), González-Manteiga *et al.* (2011) and Rodríguez-Álvarez *et al.* (2011a) go in this direction. Besides, Inácio de Carvalho *et al.* (2013) follow a Bayesian nonparametric approach to fit covariate–dependent ROC curves using probability models in each population, while Rodríguez-Álvarez *et al.* (2011b) perform a comparative study of the direct and induced methodologies. In such case, if we denote as $\bX_D$ and $\bX_H$ the covariates for the disease and healthy populations, the conditional ROC curve is defined as $${\ROC}_{\bx}(p) = 1-F_{D}(F_{H}^{-1}(1-p|\bx)|\bx) \,, \label{eq:ROCx}$$ where $F_{j}(\cdot|\bx)$ stands for conditional distribution of $Y_j|\bX_j=\bx$, $j=H, D$. In this paper, we focus on the latter approach through a general regression model. The general methodology to estimate the conditional ROC curve consists in a plug–in procedure where estimators of the regression and of the variance functions together with empirical distribution and quantile function estimators based on the residuals are plugged into the general expression of the conditional ROC curve. Pepe (1997, 1998, 2003), Faraggi (2003), González-Manteiga *et al.* (2011) propose estimators that implement these ideas. Since most of these estimators are based on classical least squares procedures or local averages, they may be very sensitive to anomalous data or small deviations from the model assumptions. The bi–normal model, in which both populations are assumed to be normal, is a very popular choice to fit a ROC curve and one justification for its broad use is its robustness. The term robustness may have different interpretations; in fact, Gonçales *et al.* (2014) discuss the scope of the so–called robustness in the ROC curve scenario. Walsh (1997) performs a simulation study that shows that the bi–normal estimator is sensitive to model misspecifications and to the location of the decision thresholds. In this paper, we focus on robustness, that is, resistance to deviations from the underlying model plus efficiency when this central model holds. During the last decades, robust statistics has pursued the aim of developing procedures that enable reliable inference results, even if small deviations from the model assumptions occur or in the presence of a moderate percentage of outliers. Even when these efforts have been sustained over time across different statistical areas, up to our knowledge, ROC curves have received little attention from this robustness point of view. When no covariates are available, robust estimators of the area under the ROC curve were given in Greco and Ventura (2011) assuming that the distribution functions are known up to a finite–dimensional parameter (see also Farcomeni and Ventura, 2012). In this sense, when covariates are recorded to improve the discrimination power of the biomarker, the main contribution of our paper is to bridge the gap between ROC curves and robustness. We achieve this goal by fitting a location-scale regression model to the diagnostic variable and considering adaptive empirical estimators of the regression residuals distributions. In this respect, our proposal is semiparametric since the errors distribution is not assumed to be known, for example, as in the bi–normal model. Our motivating example consists of the real dataset of a marker for diabetes previously analysed in Faraggi (2003) and Pardo–Fernández *et al*. (2014), in which we add to their analysis a robust perspective focussing on the potential effect of influential data. The observations, that come from a population-based pilot survey of diabetes mellitus in Cairo, Egypt, consist of postprandial blood glucose measurements ($Y$) from a fingerstick in 286 subjects who were divided into healthy (198) and diseased (88) groups according to gold standard criteria of the World Health Organization (1985). It is believed that the aging process may be associated with resistance or relative insulin deficiency among healthy people, therefore postprandial fingerstick glucose levels would be expected to be higher for older persons who do not have diabetes. According to this belief, Smith and Thompson (1996) adjust the ROC curve analysis for covariate information using age ($X$). The obtained ROC curve of the transformed biomarker is given in Figure \[fig:ROC\_diabetes\] together with the ROC curve obtained after removing the 6 outliers detected in the healthy sample through a robust regression fit. Figure \[fig:ROC\_diabetes\] also displays the ROC curve built using the naive approach of using robust regression estimators combined with the usual empirical distribution and quantile function estimators based on the residuals. These plots illustrate that the use of robust regression and variance estimators are not enough to protect the estimation of the ROC curve from the influence of atypical data. This effect may be explained by the fact that large residuals are still present when empirical distribution estimators are computed. This motivates the need of defining appropriate robust estimators of the ROC curve. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- [(a)]{} (b) (c) ![\[fig:ROC\_diabetes\] Estimated ROC surfaces for the Diabetes Data using Age ($X$) as covariate: (a) Classical estimator, (b) Classical estimator without the detected outliers and (c) Naive estimator.](ROC_cl_new.pdf "fig:") ![\[fig:ROC\_diabetes\] Estimated ROC surfaces for the Diabetes Data using Age ($X$) as covariate: (a) Classical estimator, (b) Classical estimator without the detected outliers and (c) Naive estimator.](ROC_cl_so_new.pdf "fig:") ![\[fig:ROC\_diabetes\] Estimated ROC surfaces for the Diabetes Data using Age ($X$) as covariate: (a) Classical estimator, (b) Classical estimator without the detected outliers and (c) Naive estimator.](ROC_hib_new.pdf "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -0.1in In the rest of the paper we will introduce a robust proposal and we will study some of its properties. The paper is organized as follows. Section \[sec:prelim\] reviews some general concepts regarding the conditional ROC curve, while Section \[sec:propuesta\] introduces the robust proposal to estimate the ROC curve focussing in the special situation of a parametric regression model. Section \[sec:consist\] presents some consistency results of the proposed procedure. Finally, in Section \[sec:monte\], a numerical study is conducted to examine the small sample properties of the proposed procedures under a linear and a nonlinear regression model, while the advantages of the proposed methodology are illustrated in Section \[sec:realdata\] on a real data set. All proofs are relegated to the Appendices. Preliminaries ============= [\[sec:prelim\]]{} In this section, we recall the approach considered to model the induced ROC curve when covariates are measured. For that purpose, denote as $Y_D$ and $\bX_{D}$ the biomarker and the covariates measured in the diseased population and as $Y_H$ and $\bX_{H}$ the corresponding ones in the healthy individuals. For the sake of simplicity, we will assume that the covariates of interest are the same in both populations. A general way to include covariates is through a general location–scale regression model which, for simplicity of presentation, we assume homoscedastic, that is, $$\begin{aligned} Y_{D} &=& \mu_{0,D}(\bX_{D})+ \sigma_{0,D} \; \epsilon_{D}\;, \label{modeloD}\\ Y_{H} &=& \mu_{0,H}(\bX_{H})+ \sigma_{0,H} \;\epsilon_{H}\;, \label{modeloH} \end{aligned}$$ where, for $j=D,H$, $\mu_{0,j}$ is the true regression function and $\sigma_{0, j}$ corresponds to the model dispersion, respectively. It is also assumed that the errors $\epsilon_j \sim G_{j}$ are independent of $\bX_j$, for $j=D,H$ and have scale $1$ to properly identify $\sigma_{0,j}$. Furthermore, to identify the regression function avoiding moment conditions, we will assume that $G_j$ has a symmetric distribution. Denote as ${\cal S}$ the common support of $\bX_{D}$ and $\bX_{H}$. It is worth noticing that since the errors and the covariates are independent, for a given $\bx \in \itS$, we have that $$\begin{aligned} F_{Y_{D}}(y|\bx)&=& F_{Y_{D}|X_{D}}(y|\bx)= \prob(Y_{D}\le y|\bX_{D}=\bx) = \prob( \mu_{0,D}(\bX_{D})+ \sigma_{0,D} \;\epsilon_{D} \le y|\bX_{D}=\bx)\\ &=& G_{D}\left(\dfrac{y-\mu_{0,D}(\bx)}{\sigma_{0,D}}\right) \;.\end{aligned}$$ Analogously, we get that the conditional distribution in the healthy distribution satisfies $$F_{Y_{H}}(y|\bx)=F_{Y_{H}|X_{H}}(y|\bx)= G_{H}\left(\dfrac{y-\mu_{0,H}(\bx)}{\sigma_{0,H}}\right) \, .$$ As a consequence, the quantiles of the conditional distributions are related to those of the errors through $F_{j}^{-1}(p|\bx)= \sigma_{0,j}\, G_{j}^{-1}(p) +\mu_{0,j}(\bx)$, for $ j=D,H $, where $G_{j}^{-1}(\cdot)$ denotes the quantile function of the errors $\epsilon_j$. Thus, the conditional ROC curve given $\bx \in \itS$ defined in can be computed as $$\begin{aligned} {\ROC}_{\bx}(p) &=& 1-G_{D}\left(\frac{\mu_{0,H}(\bx) -\mu_{0,D}(\bx)}{\sigma_{0,D}} + \frac{\sigma_{0,H}}{\sigma_{0,D} } \, G_{H}^{-1}(1-p)\right)\,. \label{rocx}\end{aligned}$$ One advantage of this approach is that it enables a very general modelling of the regression functions $\mu_{0,j}$, for $ j=D, H$, since this task can be accomplished from different perspectives. This means that according to the information about the relationship between the biomarker and the covariates and the user’s preferences, the regression functions may be modelled parametrically or either nonparametrically or partly parametrically, even when these last two approaches will be subject of future work. As mentioned in the Introduction, expression of the conditional ROC curve suggests a natural estimation procedure. First, compute estimators of the regression function and the dispersion parameter which allow to obtain the corresponding residuals. Then, estimate $G_{D}$ and $G_{H}^{-1}$ by empirical distribution and quantile function estimators based on the residuals, respectively. Finally, using these estimators in and plugging there–in the obtained estimators of the regression functions and variance parameters, we obtain an estimator of the conditional ROC. Our goal is to introduce a procedure to get reliable and stable $\ROC_{\bx}$ estimators, even when a moderate percentage of outliers arise in one sample or in both of them. Different summary measures of the ROC curve are useful to sum up particular features of the curve. One of the most popular indices is the conditional *area under the curve* (AUC$_{\bx}$), which is computed as $\AUC_{\bx}= \int_0^1 \ROC_{\bx}(p) dp$. Proposal ======== [\[sec:propuesta\]]{} The general procedure --------------------- [\[sec:general\]]{} Suppose that we have a sample from the diseased population, $(y_{D,i},\bx_{D,i})$, $1\le i\le n_D$, that verifies model and one from the healthy population, $(y_{H,i},\bx_{H,i})$, $1\le i\le n_H$, verifying model . Furthermore, assume that the samples are independent from each other. As mentioned above, since the conditional ROC curve is given in equation , an estimation procedure can be obtained following the next steps: i) compute estimators of the regression functions and variance parameters, ii) calculate the corresponding residuals and replace the distribution and quantile functions, $G_D$ and $G_H^{-1}$, by suitable estimators and iii) plug–in estimators of the regression functions and variance parameters in . In order to obtain a final robust estimator of the ROC and AUC curves, it is necessary to consider robust estimators not only in the first step of the described procedure, but also in the second one. In fact, if robust estimators are only considered for the estimation of the regression and variance functions, large residuals would influence the classical empirical distribution and quantile function estimators wasting the efforts made in the first step to get robustness. Taking these ideas into account, we propose the following stepwise procedure: 1. Estimate $\mu_{0,H}(\bx)$, $\sigma_{0,H}$, $\mu_{0,D}(\bx)$, $\sigma_{0,D}$ in a robust fashion from the samples $(y_{H,1},\bx_{H,1}),\dots,$ $(y_{H, n_H},\bx_{H, n_H})$ and $(y_{D,1},\bx_{D,1}),\dots,(y_{D, n_D},\bx_{D, n_D})$, respectively. Denote the resulting estimators by $\wmu_{H}(\bx)$, $\wsigma_{H}$, $\wmu_{D}(\bx)$ and $\wsigma_{D}$. 2. Compute for each sample the standardized regression residuals $$r_{H,i}= \dfrac{y_{H,i}-\wmu_{H}(\bx_{H,i})}{\wsigma_{H}}\quad \mbox{and} \quad r_{D,i}= \dfrac{y_{D,i}-\wmu_{D}(\bx_{D,i})}{\wsigma_{D}}\,.$$ From these residuals, evaluate robust estimators of the distribution and quantile functions, denoted, $\wG_{D}$ and $\wG_{H}^{-1}$, respectively. 3. Plug–in the robust estimators computed in the first two steps into equation to obtain $$\begin{aligned} \widehat{\ROC}_{\bx}(p) = 1-\wG_{D}\left(\frac{\wmu_{H}(\bx) -\wmu_{D}(\bx)}{\wsigma_{D}} + \frac{\wsigma_{H} }{\wsigma_{D} } \, \wG_{H}^{-1}(1-p)\right) \, . \label{rocxhat}\end{aligned}$$ A key point of the above procedure is to provide robust and consistent estimators in the first and second steps. Regarding **Step 1**, the considered regression models and may be either parametric, nonparametric or semiparametric. In each case, suitable robust estimators must be used. In particular, in the parametric case, linear or nonlinear models may be adequate. For instance, when the conditional model is a linear model, the $MM-$estimators introduced in Yohai (1987) are a recommended option, while under a nonlinear one the weighted $MM-$estimators presented in Bianco and Spano (2019) may be used. Beyond the robust estimation of the regression functions and the scales $\sigma_j$, it is necessary to detect outliers in order to obtain a robust version of the empirical distribution and quantile function estimators. Unlike the classical empirical estimators, where all the observations have the same weight, downweighting in the second step atypical points, i.e., those values that lie far away from the bulk of the data, may result in a more resistant procedure. Regarding the estimation of the residual’s distribution ------------------------------------------------------- As in Gervini and Yohai (2002), we consider adaptive weights computed from the empirical distribution of the residuals obtained from a robust fit. To describe the extension of their proposal and to fix ideas, let us consider a general homoscedastic nonlinear regression model. Similar arguments can be consider when the model is fully nonparametric, semiparametric or even heteroscedastic. Assume that we have a random sample $(y_{1},\bx_{1}),\dots,(y_{ n},\bx_{n})$, where $\bx_i$ is a vector of $p$ explanatory variables and $y_i$ is a response variable that satisfies $$y_i= \mu(\bx_i) + u_i= f(\bx_i, \bbe_0) +\sigma_0 \epsilon_i\,, \qquad i=1 \dots n \, , \label{general}$$ with $\bbe \in \real^q$, $\sigma_0$ the scale parameter and $f$ a known function. Note that the dimension of the regression parameter $\bbe$ may be equal or not to that of the covariates. The errors $\epsilon_i$ are independent and identically distributes (i.i.d.) with unknown distribution $G_0 $ and independent of the covariates $\bx_i$. We will assume that $G_0$ is symmetric around 0. Consider robust estimators of regression and scale, let us say $\wmu(\cdot)$ and $\wsigma$, and compute standardized residuals $r_{i}= ({y_{i}-\wmu(\bx_{i})})/{\wsigma} $. In particular, under the nonlinear regression model , $\wmu(\bx_{i})=f(\bx_{i}, \wbbe)$, where for instance, $\wbbe$ is an $S-$ or an $MM-$estimator. On the basis of these residuals, the classical empirical distribution at point $t$ can be computed as $\wG_{n,\mbox{\sc \tiny emp}}(t)= (1/n) \sum_{i=1}^n \indica_{r_i \le t}$. Large values of $|r_i|$ suggest that the corresponding pairs $(y_{i},\bx_{i})$ may be outliers. In that case, under a normal error model, it seems wise to consider as atypical those points whose residuals are larger than a certain cut–off value $t^{\star}$, that is, such that $|r_i|>t^{\star}$. Typically, $t^{\star}$ is chosen as 2.5by taking the standard normal distribution as a benchmark. To take into account these considerations, weighting may be a useful alternative in the computation of the empirical distribution estimator. However, in order to make the cut–off criterion more flexible and more data–driven, adaptive cut–off values could be considered in this process. We compute the adaptive weighted empirical distribution at point $t$ as: $$\wG_n(t)= \frac{1}{\sum_{\ell=1}^n w_{\ell}}\sum_{i=1}^n w_i \indica_{r_i \le t} \, , \label{empiricalw}$$ where the weights $w_i \ge 0$ are based on a weight function $w: \real \to [0,1]$ non-increasing, right continuous, continuous in a neighbourhood of $0$, $w(0)=1$, $w(u)>0$ for $0 < u <1$ and $w(u)=0$ for $ u \ge 1$. The fact that $w(u)=0$ for $ u \ge 1$ ensures that $w_i = 0$ when $|r_i|$ is larger than the selected cut–of value, so, as mentioned in Gervini and Yohai (2002), observations with large residuals are completely eliminated in the weighted estimators. To define the adaptive cut–off values, consider the empirical distribution function of the absolute standardized residuals $r_{i}$ given by $${G}^{+}_n(t)= \frac 1n \sum_{i=1}^n \indica_{|r_i| \le t}\, .$$ and let ${G}^{+}_0(t)$ be the distribution of the absolute errors when $\epsilon_i \sim G_0$. As noted in Gervini and Yohai (2002), if for a large $t$ it happens that ${G}^{+}_n(t) < {G}^{+}_0(t)$, we have that the sample proportion of absolute residuals that exceeds $t$ is greater than the theoretical proportion suggesting that outliers are present among the data. Since in practice the actual distribution of $\epsilon_i$ is unknown, an hypothetical distribution $G$, such as the standardized normal distribution, is assumed. Gervini and Yohai(2002) consider as a measure of the percentage of atypical data $$d_n= \sup_{t \ge 0} \{G^{+}(t)-G^{+}_n(t)\}^+=\sup_{t \ge 0} \{\max\left(G^{+}(t)-G^{+}_n(t),0\right)\} \, ,$$ where $\{\cdot\}^+$ denotes the positive part, $G^{+}$ is the distribution of the random variable $|V|$ when $V \sim G$. Let $|r|_{(1)}\le |r|_{(2)} \le \dots\le |r|_{(n)} $ denote the order statistics of the standardized residuals. As those authors note $$d_n= \max_{1\le i \le n} \left\{ \max\left(G^{+}(|r|_{(i)})-\dfrac{(i-1)}{n},0\right)\right\} \, .$$ Hence, a possible cut–off value may be $$\ot_n= |r|_{i_n}= \min\{t: G^{+}_n(t)\ge 1- d_n\}\,, \label{eq:tnraya}$$ where $i_n=n-[nd_n]$. However, as noted in Gervini and Yohai (2002), the values of $d_n$ may be large for small values of $n$ even when outliers are not present in the sample. Therefore, to combine high efficiency for small samples we define the threshold value as $$t_n= \max(\ot_n, \eta) \,, \label{eq:tn}$$ where $\eta$ is some large quantile of $G^{+}$, that is, $\eta=(G^{+})^{-1}(p)$ for some $p$ close to 1. With this adaptive cut–off value, by means of the weight function $w: \real \to [0,1]$, we define $$w_i= w\left(\dfrac{ r_i }{t_n}\right)\,, \label{eq:pesos}$$ and the adaptive weighted empirical distribution as in , which allows to define also the weighted quantile function. Appendix \[sec:consistG\] provides some uniform consistency results for the adaptive weighted empirical distribution $\wG_n$ defined through and , under mild conditions. Consistency results =================== [\[sec:consist\]]{} The results in this section are based on those concerning the uniform consistency of the weighted distribution function defined in which are given in Appendix \[sec:consistG\]. We will consider a general nonlinear regression model, extensions to other settings, such as nonparametric regression models, can be obtained similarly. Henceforth, $(y_{j,i},\bx_{j,i})$, $1\le i\le n_{\ell}$, for $j=D, H$, stand for independent random samples from the diseased and healthy populations with the same distribution as $(Y_D, \bX_D)\in \real^{p+1}$ and $(Y_H, \bX_H)\in \real^{p+1}$, respectively, where $(Y_D, \bX_D)$ satisfy and $(Y_{H},\bX_{H})$ fulfils . The errors $\epsilon_j \sim G_{j}$ are independent of $\bX_j$, for $j=D,H$. In this situation, using , we get that $${\ROC}_{\bx}(p) = 1-G_{D}\left(\frac{\mu_{0,H}(\bx) -\mu_{0,D}(\bx)}{\sigma_{0,D}} + \frac{\sigma_{0,H}}{\sigma_{0,D}} \, G_{H}^{-1}(1-p)\right)\,.$$ To avoid burden notation, for $j=D, H$, we will denote as $\wG_{j}=\wG_{j,n_j}$ the weighted empirical distribution function defined in using the sample $\left(y_{j,i}, \bx_{j,i}\right)$, $1\le i \le n_j$ and robust consistent estimators $\wmu_j$ and $\wsigma_j$ of $\mu_{0,j}$ and $\sigma^2_{0,j}$, respectively. Then, the estimator of the ROC curve whose uniform consistency we will study is given by $$\begin{aligned} \widehat{\ROC}_{\bx}(p) = 1-\wG_{D}\left(\frac{\wmu_{H}(\bx) -\wmu_{D}(\bx)}{\wsigma_{D} } + \frac{\wsigma_{H} }{\wsigma_{D} } \, \wG_{H}^{-1}(1-p)\right) \, . \label{rocxhatreg}\end{aligned}$$ We will need the following assumptions on the errors distributions and on their estimates: 1. \[ass:A1\] $G_H : \real\to (0, 1)$ has an associated density $g_H$ such that $g_H(y)>0$, for all $y\in \real$. 2. \[ass:A2\] $G_D : \real\to (0, 1)$ is continuous. 3. \[ass:A3\] $\|\wG_{j}-G_j\|_{\infty}\convpp 0$, $j=D,H$. 4. \[ass:A61\] For each fixed $\bx$, $ |\wmu_{j}(\bx)-\mu_{0,j}(\bx)\|\convpp 0$, $j=D,H$. 5. \[ass:A6\] For any compact set $\itK \subset {\cal S}$, $\sup_{\bx\in \itK}|\wmu_{j}(\bx)-\mu_{0,j}(\bx)\|\convpp 0$, $j=D,H$. 6. \[ass:A7\] The regression functions $\mu_{0,j}$ are such that, for any compact set $\itK$ $\sup_{\bx\in \itK} |\mu_{0,j}(\bx)|=A_j<\infty$. \[remark:casonolineal\] If we are dealing with a parametric regression model, i.e., when $\mu_{0,D}(\bx)= f_{D}(\bx, \bbe_{0,D})$ and $\mu_{0,H}(\bx)=f_{H}(\bx, \bbe_{0,H})$ and $\wbbe_j$ and $\wsigma_j$ stand for robust consistent estimators of $\bbe_{0,j}$ and $\sigma^2_{0,j}$, respectively, the estimator of the ROC curve equals $$\widehat{\ROC}_{\bx}(p) = 1-\wG_{D}\left(\frac{f_{H}(\bx, \wbbe_H) -f_{D}(\bx, \wbbe_D)}{\wsigma_{D} } + \frac{\wsigma_{H} }{\wsigma_{D} } \, \wG_{H}^{-1}(1-p)\right) \, .$$ In this framework, conditions under which \[ass:A3\] holds for the linear model $f_{j}(\bx, \bbe_{0,j})=\bx\trasp \bbe_{0,j}$ or more generally, for a nonlinear model are given in the Appendix \[sec:consistG\]. The derivation of conditions that guarantee the validity \[ass:A3\] under nonparametric or semiparametric models are beyond the scope of this paper. On the other hand, \[ass:A61\] to \[ass:A7\] hold if the non–linear regression functions are such that 1. \[ass:A4\] For each fixed $\bx$, the regression functions $f_j(\bx, \bb)$ are continuous in $\bb$. 2. \[ass:A5\] The functions $f_j$ are such that, for any compact set $\itK$ and any sequence $\bbe_n\to \bbe_{0,j}$, we have $\sup_{\bx \in \itK} |f_j(\bx, \bbe_n)- f_j(\bx, \bbe_{0,j})|\to 0$. Further, $\sup_{\bx\in \itK} |f_j(\bx, \bbe_j)|=A_j<\infty$. In particular, these assumptions hold if the regression model is a linear one. \[theo:consist.2\] Let $\left(y_{j,i}, \bx_{j,i}\right)$, $1\le i \le n_j$, $j=D, H$, be independent observations satisfying and , respectively and assume that $\wmu_j$ and $\wsigma_j$ are strongly consistent estimators of $\mu_{0,j}$ and $\sigma_{0,j}$, respectively. Then, under \[ass:A1\] to \[ass:A3\] and \[ass:A61\], 1. $\sup_{0<p<1} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\convpp 0$. 2. If, in addition, \[ass:A6\] holds, $G_D $ has a bounded density $g_D$ and the regression functions $\mu_{0,j}$ satisfy \[ass:A7\], then, for any $\delta>0$ $\sup_{\delta<p<1-\delta} \sup_{\bx\in \itK} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\convpp 0$. 3. Furthermore, assume that $G_D $ has a bounded density $g_D$, the regression functions $\mu_{0,j}$ satisfy \[ass:A7\] and the conditional ROC function is such that, for any $\epsilon>0$, there exists $0<\eta<1$ such that, for any $\bx\in \itK$, ${\ROC}_{\bx}(\eta)<\epsilon$ and $1-{\ROC}_{\bx}(1-\eta)<\epsilon$, then $\sup_{0<p<1 } \sup_{\bx\in \itK} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\convpp 0$. As a consequence of Theorem \[theo:consist.2\], we immediately get the following result. \[theo:consist.1\] Let $\left(y_{j,i}, \bx_{j,i}\right)\sim (Y_j, \bX_j)$, $1\le i \le n_j$, $j=D, H$, be independent observations satisfying $$\begin{aligned} Y_{D} = f_{D}(\bX_{D}, \bbe_{0,D})+ \sigma_{0,D} \epsilon_{D} \qquad \qquad Y_{H}= f_{H}(\bX_{H}, \bbe_{0,H})+ \sigma_{0,H} \epsilon_{H} \label{modeloregDH}\;,\end{aligned}$$ where, for $j=D,H$, the errors $\epsilon_j \sim G_{j}$ are independent of $\bX_j$, for $j=D,H$. Assume that $\wbbe_j$ and $\wsigma_j$ are strongly consistent estimators of $\bbe_{0,j}$ and $\sigma_{0,j}$, respectively. Then, under \[ass:A1\] to \[ass:A4\], 1. $\sup_{0<p<1} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\convpp 0$. 2. If, in addition, $G_D $ has a bounded density $g_D$ and the regression functions $f_j$ satisfy \[ass:A5\], then $\sup_{\bx \in \itK} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)| \convpp 0$. 3. Moreover, for any $\delta>0$ $\sup_{\delta<p<1-\delta} \sup_{\bx\in \itK} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\convpp 0$. 4. Furthermore, assume that $G_D $ has a bounded density $g_D$, the regression functions $f_j$ satisfy \[ass:A5\] and the conditional ROC function is such that, for any $\epsilon>0$, there exists $0<\eta<1$ such that, for any $\bx\in \itK$, ${\ROC}_{\bx}(\eta)<\epsilon$ and $1-{\ROC}_{\bx}(1-\eta)<\epsilon$, then $\sup_{0<p<1 } \sup_{\bx\in \itK} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\convpp 0$. It is worth noticing that the requirement $\sup_{\bx\in \itK}{\ROC}_{\bx}(\eta)<\epsilon$ and $\sup_{\bx\in \itK}1-{\ROC}_{\bx}(1-\eta)<\epsilon$ in (iii) is satisfied when \[ass:A5\] holds and $G_H$ has support on the whole line as stated in \[ass:A1\]. Monte Carlo study ================= [\[sec:monte\]]{} In this section, we summarize the results of a simulation study conducted to study the small sample performance of the proposal given in Section \[sec:propuesta\]. The goal of this numerical experiment is two–fold. On the one hand, we want to illustrate the sensitivity of the classical methods to deviations from the central model. On the other hand, we want to evaluate the performance of our robust proposal under different contamination schemes and to compare it with the classical one. For that purpose, we considered different scenarios and contaminations schemes. In all cases, we generate $Nrep=1000$ datasets of size $n_{D}=n_{H}=n=100$ and $n_{D}=n_{H}=n=200$. To evaluate if the advantages to be observed in the robust procedure depend on linearity, we considered two regression models, a linear and a nonlinear one. Besides, different contaminating schemes are analysed either contaminating one or both populations. To summarize the discrepancy between the estimator and the true ROC surface, we consider two grids of points: $\itG_p=\{p_j\}_{j=1}^{N_p}$ corresponding to equidistant values between $0.01$ and $0.99$ with step $0.01$ and $\itG_x=\{x_i\}_{i=1}^{N_x}$ where the net has step $0.05$ within the interval $[a,b]$ with $a=-1$ and $b=1$ for the linear model, while $a=0$ and $b=1$ for the nonlinear one. The estimators performance is then evaluated using the mean over replications of - the Mean Squared Error ($ MSE $) given by $$MSE=\frac{1}{N_x N_p} \sum_{i=1}^{N_x}\sum_{j=1}^{N_p} \left(\widehat{\ROC}_{x_i}(p_j) -{\ROC}_{x_i}(p_j)\right)^2 \, ,$$ - a measure inspired on the Kolmogorov distance ($ KS $) calculated as $$KS=\sup_{1\le i N_x} \sup_{1\le j \le N_p} \left| \widehat{\ROC}_{x_i}(p_j) - {\ROC}_{x_i}(p_j) \right| \, ,$$ that give a global summary of the mismatch between the estimated $\ROC$ curves and the true ones. Numerical study under a linear model ------------------------------------ [\[sec:linearmodel\]]{} In the first scenario, we consider different homoscedastic linear-mean regression models for the two populations. We considered the same conditions as in Inácio de Carvalho *et al.* (2013), that is, following the linear regression models $$\begin{aligned} y_{D,i} &=& 2+ 4 x_{D,i} + \sigma_{D} \; \epsilon_{D,i} \label{trued}\\ y_{H,i} &=& 0.5+ x_{H,i} + \sigma_{H} \; \epsilon_{H,i} \label{trueh}\; ,\end{aligned}$$ for all $i=1,\dots,n$ $\epsilon_{j,i} \sim N(0,1)$ are independent and independent from $x_{j,i} \sim U(-1,1)$, for $j=D,H$, $ \sigma_D=2 $ and $ \sigma_{H}=1.5$. Besides, the sample from one population was generated independently from the other one. Figure \[fig:surface\_true\] displays the surface corresponding to the true ROC curves generated under the central model given by equations and . ![\[fig:surface\_true\] True ROC surface under the central model given by equations and under the linear model.](ROC_TRUE_C0_3D.pdf "fig:") -0.1in To evaluate the sensitivity of the classical conditional ROC curve and the robust proposal given in Section \[sec:propuesta\], we consider different contamination schemes by varying the sample where we introduce atypical points, the percentage of anomalous data and the size of the outliers. - $C^{H}_{\delta}$: is a contamination in the healthy sample introduced so as to affect the estimation of quantiles of the healthy population. In order to introduce atypical observations, we generate *shift outliers* as follows. The first $m=n \delta$ observations in the healthy dataset were replaced by observations following the model $y_{H,i} = 0.5+ x_{H,i} + S \,\sigma_{H} + \sigma_{H} \epsilon_{H,i} $, where the shift $S\in \itS=\{ 2.5, 5, 7.5, 10, 12.5, 15, 17.5, 20\}$. - $C_{\delta}^D$: corresponds to contaminating the diseased population introduced so as to affect the estimation of the empirical distribution of the diseased population. The atypical observations are introduced in the same fashion as in $C^{H}_{\delta}$, that is, the first $m=n \delta$ observations in the diseased dataset were replaced by observations following the model $y_{D,i} = 0.5+ x_{D,i} + S \,\sigma_{D} + \sigma_{D} \epsilon_{D,i} $, where the shift $S\in \itS$. - $C_{\delta}$: we generate now *shift outliers* in both samples simultaneously. For this end, the first $m=n \delta$ observations in each dataset were replaced by observations generated as follows $$\begin{aligned} y_{D,i} &=& 2+ 4\, x_{D,i} + 20\, \sigma_{D} + \sigma_{D} \epsilon_{D,i} \label{contaminad}\\ y_{H,i} &=& 0.5+ x_{H,i} + 15\, \sigma_{H} + \sigma_{H} \epsilon_{H,i} \label{contaminah}\; ,\end{aligned}$$ We choose two possible contaminating percentages $\delta=0.05$ and $0.10$, that is, a 5% or a 10% of observations are modified, respectively. To avoid burden notation, in all Figures and Tables, $C_0$ stands for the situation of clean samples. To illustrate the behaviour of the ROC curves for clean and contaminated samples, Figure \[fig:surface\] shows the estimated surfaces obtained with the classical and robust estimators from one of the clean samples generated when $n=100$ and when the same sample is corrupted with the shifted outliers generated as in equations and . The estimators of the conditional $\ROC$ curves were computed on the net of points $\itG_x$ and quantiles $\itG_p$, described above. The right panel in Figure \[fig:surface\] illustrates the stability of the proposed method, since the three figures on the right panel are quite similar. On the other hand, the classical estimators are distorted in the presence of outliers, the surface being shifted towards 1 in the central region and flatten towards $0$ specially under $C_{0.10}$. [M GG]{} & Classical Estimators & Robust Estimators\ $C_0$ & ![\[fig:surface\] Estimated surfaces for $n_{D}=n_{H}=100$ under the linear model and for a clean and contaminated sample.](ROC_CL_C0_3D.pdf "fig:") & ![\[fig:surface\] Estimated surfaces for $n_{D}=n_{H}=100$ under the linear model and for a clean and contaminated sample.](ROC_ROB_C0_3D.pdf "fig:")\ $C_{0.05}$ & ![\[fig:surface\] Estimated surfaces for $n_{D}=n_{H}=100$ under the linear model and for a clean and contaminated sample.](ROC_CL_C1_3D.pdf "fig:") & ![\[fig:surface\] Estimated surfaces for $n_{D}=n_{H}=100$ under the linear model and for a clean and contaminated sample.](ROC_ROB_C1_3D.pdf "fig:")\ $C_{0.10}$ & ![\[fig:surface\] Estimated surfaces for $n_{D}=n_{H}=100$ under the linear model and for a clean and contaminated sample.](ROC_CL_C1_10per_3D.pdf "fig:") & ![\[fig:surface\] Estimated surfaces for $n_{D}=n_{H}=100$ under the linear model and for a clean and contaminated sample.](ROC_ROB_C1_10per_3D.pdf "fig:")\ -0.1in To evaluate the effect of the considered contaminations, Tables \[tab:sensi\_5\_100\_H\] to \[tab:sensi\_10\_200\_H\], report the summary measures under $C_{\delta}^H$ and $C_{\delta}^D$ for $\delta=0.05, 0.10$ and $n=100, 200$. It is worth noticing that $MSE$ and $KS$ take values between 0 and 1 and in this range, large deviations correspond to values close to $1$. The reported results show that the classical procedure to estimate the ROC curve is seriously affected by the introduced outliers. It should be taken into account that since the ROC curve varies between $0$ and $1$, the magnitude of the effect is not as evident as in other settings such as in linear regression models. However, when $n=100$, under $C_{0.05}^H$, the $MSE$ is $5.5$ larger when $S=20$ than for clean samples, while the robust procedure remains stable. This effect is more striking in Figures \[fig:MSE\_CH\] and \[fig:MSE\_CD\] which show the plot of the $MSE$ as a function of the level shift $S$ when $n=100$ and $200$ and for the two contamination percentages. The red and blue lines correspond to the classical and robust proposed methods, respectively. Even though a slight influence is observed for the robust procedure under mild outliers ($S=2.5$), which are those more difficult to detect, the whole curve is stable when varying $S$, while the $MSE$ of the classical method quickly increases with the level shift. ------- ----------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Method $C_0$ 2.5 5 7.5 10 12.5 15 17.5 20 $MSE$ Robust 0.0036 0.0040 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 0.0037 Classical 0.0032 0.0049 0.0099 0.0114 0.0122 0.0133 0.0145 0.0160 0.0176 $KS$ Robust 0.1988 0.2156 0.2085 0.2054 0.2056 0.2024 0.2016 0.2016 0.2016 Classical 0.1949 0.3567 0.7172 0.8189 0.8256 0.8256 0.8256 0.8257 0.8258 $MSE$ Robust 0.0036 0.0039 0.0038 0.0038 0.0039 0.0039 0.0039 0.0039 0.0039 Classical 0.0032 0.0038 0.0045 0.0055 0.0067 0.0082 0.0098 0.0117 0.0137 $KS$ Robust 0.1988 0.2041 0.2035 0.2034 0.2037 0.2040 0.2040 0.2040 0.2040 Classical 0.1949 0.2007 0.2130 0.2279 0.2457 0.2640 0.2829 0.3015 0.3196 ------- ----------- -------- -------- -------- -------- -------- -------- -------- -------- -------- : \[tab:sensi\_5\_100\_H\] Sensitivity to the shift size $S$ for $C_{0.05}^H$ and $C_{0.05}^D$ when $n=100$. ------- ----------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Method $C_0$ 2.5 5 7.5 10 12.5 15 17.5 20 $MSE$ Robust 0.0036 0.0058 0.0041 0.0038 0.0038 0.0038 0.0038 0.0038 0.0038 Classical 0.0032 0.0086 0.0228 0.0277 0.0297 0.0317 0.0340 0.0365 0.0393 $KS$ Robust 0.1988 0.2727 0.2411 0.2128 0.2128 0.2128 0.2128 0.2128 0.2128 Classical 0.1949 0.4406 0.7832 0.8834 0.8946 0.8985 0.9007 0.9013 0.9015 $MSE$ Robust 0.0036 0.0048 0.0042 0.0041 0.0041 0.0041 0.0041 0.0041 0.0041 Classical 0.0032 0.0052 0.0064 0.0080 0.0100 0.0123 0.0149 0.0176 0.0204 $KS$ Robust 0.1988 0.2135 0.2085 0.2083 0.2083 0.2083 0.2083 0.2083 0.2083 Classical 0.1949 0.2125 0.2306 0.2522 0.2761 0.2998 0.3229 0.3450 0.3652 ------- ----------- -------- -------- -------- -------- -------- -------- -------- -------- -------- : \[tab:sensi\_10\_100\_H\] Sensitivity to the shift size $S$ for $C_{0.10}^H$ and $C_{0.10}^D$ when $n=100$. ------- ----------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Method $C_0$ 2.5 5 7.5 10 12.5 15 17.5 20 $MSE$ Robust 0.0017 0.0021 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 0.0017 Classical 0.0015 0.0031 0.0083 0.0095 0.0099 0.0104 0.0110 0.0117 0.0125 $KS$ Robust 0.1380 0.1654 0.1463 0.1419 0.1422 0.1413 0.1411 0.1411 0.1411 Classical 0.1363 0.3248 0.7169 0.8207 0.8256 0.8256 0.8256 0.8256 0.8256 $MSE$ Robust 0.0017 0.0019 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 Classical 0.0015 0.0020 0.0023 0.0028 0.0034 0.0042 0.0051 0.0061 0.0073 $KS$ Robust 0.1380 0.1428 0.1407 0.1406 0.1407 0.1408 0.1408 0.1408 0.1408 Classical 0.1363 0.1434 0.1514 0.1627 0.1759 0.1903 0.2049 0.2199 0.2349 ------- ----------- -------- -------- -------- -------- -------- -------- -------- -------- -------- : \[tab:sensi\_5\_200\_H\] Sensitivity to the shift size $S$ for $C_{0.05}^H$ and $C_{0.05}^D$ when $n=200$. ------- ----------- -------- -------- -------- -------- -------- -------- -------- -------- -------- Method $C_0$ 2.5 5 7.5 10 12.5 15 17.5 20 $MSE$ Robust 0.0017 0.0040 0.0020 0.0018 0.0018 0.0018 0.0018 0.0018 0.0018 Classical 0.0015 0.0069 0.0211 0.0253 0.0265 0.0276 0.0288 0.0301 0.0316 $KS$ Robust 0.1380 0.2470 0.1831 0.1465 0.1465 0.1465 0.1465 0.1465 0.1465 Classical 0.1363 0.4269 0.7829 0.8845 0.8937 0.8981 0.9008 0.9012 0.9013 $MSE$ Robust 0.0017 0.0028 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 0.0020 Classical 0.0015 0.0032 0.0040 0.0048 0.0060 0.0073 0.0089 0.0106 0.0124 $KS$ Robust 0.1380 0.1585 0.1461 0.1458 0.1458 0.1458 0.1458 0.1458 0.1458 Classical 0.1363 0.1620 0.1766 0.1930 0.2109 0.2296 0.2482 0.2662 0.2842 ------- ----------- -------- -------- -------- -------- -------- -------- -------- -------- -------- : \[tab:sensi\_10\_200\_H\] Sensitivity to the shift size $S$ for $C_{0.10}^H$ and $C_{0.10}^D$ when $n=200$. Tables \[tab:mse\] summarizes the results obtained when both samples are contaminated. As above, the reported results correspond to the mean of $MSE$ and $KS$ over 1000 replications. As when contaminating only one population, the mean over replications of $MSE$ for the classical procedure is clearly enlarged under $C_\delta$, while those corresponding to the robust procedure are stable for shifted outliers. It should be noticed that, when considering the discrepancy measure $KS$ of the classical procedure, the median over replications under $C_{0.05}$ equals $0.7756$ when the sample size is $n=100$ and the Absolute Median Deviation (<span style="font-variant:small-caps;">mad</span>) is $0$, meaning that for more than half of the samples the obtained global measure equals $0.7756$, which is close to the maximal possible value. This behaviour is also reflected in Figure \[fig:KS\_bxp\] that shows the boxplots of $KS$ for $n=100$ and $n=200$. The boxplot of the classical estimator is completely shifted away under contamination attaining values close to $0.8$. [M GG]{} & $n=100$ & $n=200$\ $C_{0.05}^H$ & ![\[fig:MSE\_CH\] Sensitivity to shift size $S$: $MSE$ under $C_{\delta}^H$ when $n_{D}=n_{H}=100$ and $n_{D}=n_{H}=200$ for $ \delta=0.05, 0.10 $. The red line corresponds to the classical procedure, while the blue one the robust one.](MSE_ROB_CL_n100_cont_3_m_5.pdf "fig:") & ![\[fig:MSE\_CH\] Sensitivity to shift size $S$: $MSE$ under $C_{\delta}^H$ when $n_{D}=n_{H}=100$ and $n_{D}=n_{H}=200$ for $ \delta=0.05, 0.10 $. The red line corresponds to the classical procedure, while the blue one the robust one.](MSE_ROB_CL_n100_cont_3_m_10.pdf "fig:")\ $C_{0.10}^H$ & ![\[fig:MSE\_CH\] Sensitivity to shift size $S$: $MSE$ under $C_{\delta}^H$ when $n_{D}=n_{H}=100$ and $n_{D}=n_{H}=200$ for $ \delta=0.05, 0.10 $. The red line corresponds to the classical procedure, while the blue one the robust one.](MSE_ROB_CL_n200_cont_3_m_10.pdf "fig:") & ![\[fig:MSE\_CH\] Sensitivity to shift size $S$: $MSE$ under $C_{\delta}^H$ when $n_{D}=n_{H}=100$ and $n_{D}=n_{H}=200$ for $ \delta=0.05, 0.10 $. The red line corresponds to the classical procedure, while the blue one the robust one.](MSE_ROB_CL_n200_cont_3_m_20.pdf "fig:") -0.1in [M GG]{} & $n=100$ & $n=200$\ $C_{0.05}^D$ & ![\[fig:MSE\_CD\] Sensitivity to the shift size $S$: $MSE$ under $C_{\delta}^D$ when $n_{D}=n_{H}=100$ and $n_{D}=n_{H}=200$ for $ \delta=0.05, 0.10 $. The red line corresponds to the classical procedure, while the blue one the robust one.](MSE_ROB_CL_n100_cont_4_m_5.pdf "fig:") & ![\[fig:MSE\_CD\] Sensitivity to the shift size $S$: $MSE$ under $C_{\delta}^D$ when $n_{D}=n_{H}=100$ and $n_{D}=n_{H}=200$ for $ \delta=0.05, 0.10 $. The red line corresponds to the classical procedure, while the blue one the robust one.](MSE_ROB_CL_n100_cont_4_m_10.pdf "fig:")\ $C_{0.10}^D$ & ![\[fig:MSE\_CD\] Sensitivity to the shift size $S$: $MSE$ under $C_{\delta}^D$ when $n_{D}=n_{H}=100$ and $n_{D}=n_{H}=200$ for $ \delta=0.05, 0.10 $. The red line corresponds to the classical procedure, while the blue one the robust one.](MSE_ROB_CL_n200_cont_4_m_10.pdf "fig:") & ![\[fig:MSE\_CD\] Sensitivity to the shift size $S$: $MSE$ under $C_{\delta}^D$ when $n_{D}=n_{H}=100$ and $n_{D}=n_{H}=200$ for $ \delta=0.05, 0.10 $. The red line corresponds to the classical procedure, while the blue one the robust one.](MSE_ROB_CL_n200_cont_4_m_20.pdf "fig:")\ -0.1in [G G]{} $n=100$ & $n=200$\ ![\[fig:KS\_bxp\] Boxplots of the measure $KS$ obtained from 1000 replications using the classical and robust estimators under the linear model and .](bx_sup_n100.pdf "fig:") & ![\[fig:KS\_bxp\] Boxplots of the measure $KS$ obtained from 1000 replications using the classical and robust estimators under the linear model and .](bx_sup_n200.pdf "fig:") $n$ ----- ------- ----------- -------- ----------- -------- ----------- -------- Classical Robust Classical Robust Classical Robust 100 $MSE$ 0.0032 0.0036 0.0205 0.0040 0.0349 0.0043 $KS$ 0.1949 0.1988 0.7757 0.2060 0.7993 0.2215 200 $MSE$ 0.0015 0.0017 0.0134 0.0018 0.0269 0.0021 $KS$ 0.1363 0.1380 0.7756 0.1436 0.7997 0.1538 : \[tab:mse\] Mean of $MSE$ and $KS$ over replications under the linear model and , for clean and samples contaminated as in $C_{\delta}$. As mentioned in the Introduction, one of the most popular indices is the *area under the curve*, AUC, which is a summary measure usually considered to evaluate the discriminating effect of the biomarker. When covariates are present, the conditional area under the curve is also used as index of the marker accuracy. It is defined as $\AUC_x= \int_0^1 \ROC_x(p) dp $. Note that in this case, we obtain a single value for each $x$, hence, the function $x\to \widehat{\AUC}_x$ can be plotted for each sample. Taking into account the observed sensitivity of the classical estimators to outliers, it seems natural that this effect will be inherited by the estimators of the conditional area under the curve, ${\AUC}_x$. To evaluate this effect, Figures \[fig:fbx\_sensi\_100\_S5\_C3\] to \[fig:fbx\_sensi\_100\_S20\_C4\] show the functional boxplots of the estimators $\widehat{\AUC}_x$ obtained with the classical and robust procedures, when the sample sizes are $ n_H=n_D=100$, under contaminations $C_{\delta}^H$ and $C_{\delta}^D$ with $\delta=0.05$ and $ 0.10$ and different values of $S$. To facilitate comparisons, in Figure \[fig:fbx\_sensi\_100\_S5\_C3\] we also give the plots corresponding to clean samples. Functional boxplots were introduced by Sun and Genton (2011) and are a useful visualization tool to give a whole picture of the behaviour of a collection of curves. The area in purple represents the 50% inner band of curves, the dotted red lines correspond to outlying curves, the black line indicates the central (deepest) function, while the green line in the plot corresponds to the true $\AUC_x$ curve. As shown in Figure \[fig:fbx\_sensi\_100\_S5\_C3\], when $\delta=0.05$ and the healthy population is contaminated, the shift causes a bias in the classical estimator of $\AUC_x$, so that the central region of the functional boxplot fails to contain the true function for much of its domain. This effect is more striking when $S=15$, where also some outlying curves completely distorted appear (see Figure \[fig:fbx\_sensi\_100\_S15\_C3\]). On the other hand, the effect when contaminating the diseased population is not so devastating for the classical procedure. As shown in Figure \[fig:fbx\_sensi\_100\_S5\_C4\], even though the true curve is not close to the deepest curve it is still within the central region. However, when $S=20$, the 50% inner band is completely enlarged (see Figure \[fig:fbx\_sensi\_100\_S20\_C4\]). As expected, the robust proposal is stable across the considered contaminations. Moreover, by comparing the upper panel in Figure \[fig:fbx\_sensi\_100\_S5\_C3\], we observe that the classical and robust estimators of $\AUC_x$ are quite similar for clean samples when $n=100$ and a similar conclusion holds for $n=200$ (see Figure \[fig:fbx\_200\]). Figures \[fig:fbx\_100\] and \[fig:fbx\_200\] show the functional boxplots of $\widehat{\AUC}_x$ for both the classical and robust estimators when the samples are contaminated according to $C_\delta$, when $n=100$ and $n=200$, respectively. These figures reveal that the effect of outliers on the classical estimator of the ROC curve is inherited by the estimated area under the curve, which is reflected not only by the presence of a great number of outlying curves, but also by the enlargement of the width of the bars of the functional boxplots, as when contaminating only the diseased population. It should be noted that for $n=200$ and for values of $x$ in the range $[0.5,1]$, the true curve is on the limit of the central region. As mentioned above, the robust procedure is stable for the considered contamination. To conclude, these figures make evident the dramatic effect of the introduced outliers on the classical estimates of the area under the ROC curve, while at the same time the robust estimators look very stable. To have a deeper comprehension of the proposal, it is also of interest to see what would happen if in the stepwise procedure described in Section \[sec:general\], robust estimators were considered only in the first step, i.e., only when computing the regression parameters, while the usual empirical distribution and quantile function estimators are used in Step 2. The resulting hybrid procedure is illustrated through the functional boxplots of $\widehat{\AUC}_x$ obtained for $n=100$ and $n=200$ in Figure \[fig:mix\]. These boxplots show that, even when the contamination is less harmful for these estimators than for the classical ones, the true curve lies beyond the functional boxplot 50% inner band of curves when $x\in [0.5,1]$ and $\delta=0.05$ and beyond the limits of the functional boxplot when $\delta=0.10$. [M GG]{} & Classical & Robust\ $C_0$ & ![\[fig:fbx\_sensi\_100\_S5\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and for clean samples and when the samples are contaminated according to $C_{\delta}^H$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_C0.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S5\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and for clean samples and when the samples are contaminated according to $C_{\delta}^H$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_C0.pdf "fig:")\ $C_{0.05}^H$ & ![\[fig:fbx\_sensi\_100\_S5\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and for clean samples and when the samples are contaminated according to $C_{\delta}^H$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_CL_n100_cont_3_m_5_shift_5.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S5\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and for clean samples and when the samples are contaminated according to $C_{\delta}^H$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_ROB_n100_cont_3_m_5_shift_5.pdf "fig:")\ $C_{0.10}^H$ & ![\[fig:fbx\_sensi\_100\_S5\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and for clean samples and when the samples are contaminated according to $C_{\delta}^H$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_CL_n100_cont_3_m_10_shift_5.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S5\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and for clean samples and when the samples are contaminated according to $C_{\delta}^H$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_ROB_n100_cont_3_m_10_shift_5.pdf "fig:")\ -0.1in [M GG]{} & Classical & Robust\ $C_{0.05}^H$ & ![\[fig:fbx\_sensi\_100\_S15\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^H$ for $S=15$ and $ n_H=n_D=100$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_CL_n100_cont_3_m_5_shift_15.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S15\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^H$ for $S=15$ and $ n_H=n_D=100$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_ROB_n100_cont_3_m_5_shift_15.pdf "fig:")\ $C_{0.10}^H$ & ![\[fig:fbx\_sensi\_100\_S15\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^H$ for $S=15$ and $ n_H=n_D=100$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_CL_n100_cont_3_m_10_shift_15.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S15\_C3\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^H$ for $S=15$ and $ n_H=n_D=100$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_ROB_n100_cont_3_m_10_shift_15.pdf "fig:")\ -0.1in [M GG]{} & Classical & Robust\ $C_{0.05}^D$ & ![\[fig:fbx\_sensi\_100\_S5\_C4\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^D$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_CL_n100_cont_4_m_5_shift_5.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S5\_C4\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^D$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_ROB_n100_cont_4_m_5_shift_5.pdf "fig:")\ $C_{0.10}^D$ & ![\[fig:fbx\_sensi\_100\_S5\_C4\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^D$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_CL_n100_cont_4_m_10_shift_5.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S5\_C4\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^D$ for $S=5$ and $ n_H=n_D=100$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_ROB_n100_cont_4_m_10_shift_5.pdf "fig:")\ -0.1in [M GG]{} & Classical & Robust\ $C_{0.05}^D$ & ![\[fig:fbx\_sensi\_100\_S20\_C4\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^D$ for $S=20$ and $ n_H=n_D=100$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_CL_n100_cont_4_m_5_shift_20.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S20\_C4\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^D$ for $S=20$ and $ n_H=n_D=100$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_ROB_n100_cont_4_m_5_shift_20.pdf "fig:")\ $C_{0.10}^D$ & ![\[fig:fbx\_sensi\_100\_S20\_C4\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^D$ for $S=20$ and $ n_H=n_D=100$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_CL_n100_cont_4_m_10_shift_20.pdf "fig:") & ![\[fig:fbx\_sensi\_100\_S20\_C4\] Functional boxplots of $\widehat{\AUC}_x$ for $n=100$ under the linear model and when the samples are contaminated according to $C_{\delta}^D$ for $S=20$ and $ n_H=n_D=100$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbplot_ROB_n100_cont_4_m_10_shift_20.pdf "fig:")\ -0.1in [M GG]{} & Classical Estimators & Robust Estimators\ $C_{0.05}$ & ![\[fig:fbx\_100\]Functional boxplots of $\widehat{\AUC}_x$ for $n=100$, under the linear model and when the samples are contaminated according to $C_{\delta}$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_C1.pdf "fig:") & ![\[fig:fbx\_100\]Functional boxplots of $\widehat{\AUC}_x$ for $n=100$, under the linear model and when the samples are contaminated according to $C_{\delta}$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_C1.pdf "fig:")\ $C_{0.10}$ & ![\[fig:fbx\_100\]Functional boxplots of $\widehat{\AUC}_x$ for $n=100$, under the linear model and when the samples are contaminated according to $C_{\delta}$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_C1_10per_n100.pdf "fig:") & ![\[fig:fbx\_100\]Functional boxplots of $\widehat{\AUC}_x$ for $n=100$, under the linear model and when the samples are contaminated according to $C_{\delta}$. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_C1_10per_n100.pdf "fig:")\ -0.1in [M GG]{} & Classical Estimators & Robust Estimators\ $C_0$ & ![\[fig:fbx\_200\]Functional boxplots of $\widehat{\AUC}_x$ for $n=200$, under the linear model and when the samples are contaminated according to $C_{\delta}$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_C0_n200.pdf "fig:") & ![\[fig:fbx\_200\]Functional boxplots of $\widehat{\AUC}_x$ for $n=200$, under the linear model and when the samples are contaminated according to $C_{\delta}$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_C0_n200.pdf "fig:")\ $C_{0.05}$ & ![\[fig:fbx\_200\]Functional boxplots of $\widehat{\AUC}_x$ for $n=200$, under the linear model and when the samples are contaminated according to $C_{\delta}$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_C1_n200.pdf "fig:") & ![\[fig:fbx\_200\]Functional boxplots of $\widehat{\AUC}_x$ for $n=200$, under the linear model and when the samples are contaminated according to $C_{\delta}$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_C1_n200.pdf "fig:")\ $C_{0.10}$ & ![\[fig:fbx\_200\]Functional boxplots of $\widehat{\AUC}_x$ for $n=200$, under the linear model and when the samples are contaminated according to $C_{\delta}$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_C1_10per_n200.pdf "fig:") & ![\[fig:fbx\_200\]Functional boxplots of $\widehat{\AUC}_x$ for $n=200$, under the linear model and when the samples are contaminated according to $C_{\delta}$ . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_C1_10per_n200.pdf "fig:")\ -0.1in [M GG]{} & $n=100$ & $n=200$\ $C_{0}$ & ![\[fig:mix\]Functional boxplots of $\widehat{\AUC}_x$ obtained with the hybrid estimator for $n=100$ and $n=200$ with clean samples and under 5% and 10% of contamination under the first linear model. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_MIX_C0_n100.pdf "fig:") & ![\[fig:mix\]Functional boxplots of $\widehat{\AUC}_x$ obtained with the hybrid estimator for $n=100$ and $n=200$ with clean samples and under 5% and 10% of contamination under the first linear model. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_MIX_C0_n200.pdf "fig:")\ $C_{0.05}$ & ![\[fig:mix\]Functional boxplots of $\widehat{\AUC}_x$ obtained with the hybrid estimator for $n=100$ and $n=200$ with clean samples and under 5% and 10% of contamination under the first linear model. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_MIX_C1_n100.pdf "fig:") & ![\[fig:mix\]Functional boxplots of $\widehat{\AUC}_x$ obtained with the hybrid estimator for $n=100$ and $n=200$ with clean samples and under 5% and 10% of contamination under the first linear model. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_MIX_C1_n200.pdf "fig:")\ $C_{0.10}$ & ![\[fig:mix\]Functional boxplots of $\widehat{\AUC}_x$ obtained with the hybrid estimator for $n=100$ and $n=200$ with clean samples and under 5% and 10% of contamination under the first linear model. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_MIX_C1_10per_n100.pdf "fig:") & ![\[fig:mix\]Functional boxplots of $\widehat{\AUC}_x$ obtained with the hybrid estimator for $n=100$ and $n=200$ with clean samples and under 5% and 10% of contamination under the first linear model. The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_MIX_C1_10per_n200.pdf "fig:") -0.1in Numerical study under a non–linear regression model --------------------------------------------------- [\[sec:nonlinearmodel\]]{} In this second scenario, we consider an exponential model as in Bianco and Spano (2019), that is, we assume that the observations follow the non–linear regression models $$\begin{aligned} y_{D,i} &=& \beta_{D,1}\; \exp(\beta_{D,2}\, x_{D,i} ) + \epsilon_{D,i} \,,\label{truednl}\\ y_{H,i} &=& \beta_{H,1}\; \exp(\beta_{H,2} \, x_{H,i} ) + \epsilon_{H,i}\; , \label{truehnl}\end{aligned}$$ with $(\beta_{D,1},\beta_{D,2})\trasp=(5,2)$, $(\beta_{H,1},\beta_{H,2})\trasp=(3,1)$ for all $i=1,\dots,n$ $\epsilon_{j,i} \sim N(0,1)$ are independent and independent from $x_{j,i} \sim U(0,1)$, for $j=D,H$. Besides, the sample from one population was generated independently from that of the other one. In this case, in **Step 1**, the robust regression estimators correspond to the weighted $MM-$estimators defined in Bianco and Spano (2019), while the classical ones to the usual least squares estimators for nonlinear regression models. To assess the impact of anomalous data on the estimation of the conditional ROC curve, we introduce *shift outliers* in both populations. To explore the sensitivity of the studied methods to the size of the shift, we vary its magnitude. To this end, the first $m$ observations of each sample were replaced by observations following the models $$\begin{aligned} y_{D,i} &=& \beta_{D,1}\; \exp(\beta_{D,2}\, x_{D,i} ) + z_{D,i} + 0.01 \epsilon_{D,i}\,, \label{contaminadnl}\\ y_{H,i} &=& \beta_{H,1}\; \exp(\beta_{H,2} \, x_{H,i} )+ z_{H,i} + 0.01 \epsilon_{H,i}\; , \label{contaminahnl} \end{aligned}$$ where $x_{j,i}\sim U(0.49,0.5)$ and $\epsilon_{j,i}$ are as above, for $j=D,H$. The shift variables are taken as $ z_{j,i}=S +u_{j,i}$, with $S= 2.5, 5, 7.5, 10, 12.5, 15$, $u_{j,i} \sim N(0,0.01^2)$ for $j=D,H$, $i=1,\dots,m$. We consider similar proportions of anomalous points as in Section \[sec:linearmodel\], that is, we replace $m=n \delta$ points, $\delta=0.05$ and $0.10$, which correspond to a 5% or a 10% of replaced observations. As above, we denote this contamination $C_{\delta}$, while $C_0$ stands for clean samples. Table \[tab:sens\_5\_100\_NL\] summarizes the discrepancy between the true and estimated ROC curves in terms of the mean over replications of the $MSE$. The damage of shift outliers on the conditional ROC curve is striking, since the $MSE$ increases more than 10 times when $n=100$ and more than $20$ times when $n=200$ when $S$ takes the largest values. ----------- ------- ----------- -------- -------- -------- -------- -------- -------- -------- $ \delta$ $n$ $C_0$ 2.5 5 7.5 10 12.5 15 $0.05$ $100$ Robust 0.0023 0.0028 0.0023 0.0023 0.0024 0.0024 0.0024 Classical 0.0019 0.0023 0.0066 0.0127 0.0183 0.0231 0.0269 $0.10$ $100$ Robust 0.0023 0.0032 0.0031 0.0024 0.0024 0.0024 0.0024 Classical 0.0019 0.0029 0.0092 0.0180 0.0240 0.0248 0.0268 $0.05$ $200$ Robust 0.0011 0.0018 0.0011 0.0011 0.0011 0.0011 0.0011 Classical 0.0010 0.0015 0.0062 0.0124 0.0181 0.0229 0.0267 $0.10$ $200$ Robust 0.0011 0.0022 0.0014 0.0011 0.0011 0.0011 0.0011 Classical 0.0010 0.0021 0.0088 0.0178 0.0239 0.0242 0.0249 ----------- ------- ----------- -------- -------- -------- -------- -------- -------- -------- : \[tab:sens\_5\_100\_NL\] Sensitivity of $MSE$ to the shift size $S$ for $C_{\delta}$, when $n=100$ and $200$, under the nonlinear model and . Henceforth, we focus on the particular case of outliers with shift value $S=10$, a mild value among those considered, so as to have a deeper comprehension of the effect of the introduced anomalous points. Table \[tab:mse\_NL\] summarizes the results through the mean of the measures $MSE$ and $KS$. Note that the mean of the summary measures are distorted for the classical procedure. In particular, when considering the measure $KS$ based on the Kolmogorov distance, the mean is enlarged almost 7 times, under $C_{0.05}$ when $n=200$. Figures \[fig:n100-NL\] and \[fig:n200-NL\] show the functionals boxplots of the classical and robust ${\AUC}_x$ obtained for $n=100$ and $n=200$, respectively. Notice that in these boxplots, the estimators of conditional area under the curve were plotted in the range $(-0.5,0.2)$, since for this simulation scheme the ${\AUC}_x$ is almost 1 when the covariate takes values from 0.2 to 0.5. Once again, it becomes evident that the classical estimator suffers from the introduced contamination and that the classical estimator of ${\AUC}_x$ is completely deviated from the true conditional area under the curve, which is plotted in green, while the robust $ {\AUC}_x$ estimator remains very stable. $n$ ----- ------- ----------- -------- ----------- -------- ----------- -------- Classical Robust Classical Robust Classical Robust 100 $MSE$ 0.0019 0.0023 0.0183 0.0024 0.0240 0.0024 $KS$ 0.1881 0.1944 0.9334 0.2001 0.6893 0.2048 200 $MSE$ 0.0010 0.0011 0.0181 0.0011 0.0239 0.0011 $KS$ 0.1352 0.1367 0.9364 0.1395 0.7102 0.1445 : \[tab:mse\_NL\] Mean of $MSE$ and $KS$ over replications for clean and contaminated samples, under the nonlinear model and , for the level shift $S=10$. [M GG]{} & Classical & Robust\ $C_{0}$ & ![\[fig:n100-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=100$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_NL_n100_cont_0_m_0_mod_3.pdf "fig:") & ![\[fig:n100-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=100$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_NL_n100_cont_0_m_0_mod_3.pdf "fig:")\ $C_{0.05}$ & ![\[fig:n100-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=100$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_NL_n100_cont_1_m_5_mod_3.pdf "fig:") & ![\[fig:n100-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=100$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_NL_n100_cont_1_m_5_mod_3.pdf "fig:")\ $C_{0.10}$ & ![\[fig:n100-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=100$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_NL_n100_cont_1_m_10_mod_3.pdf "fig:") & ![\[fig:n100-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=100$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_NL_n100_cont_1_m_10_mod_3.pdf "fig:") -0.1in [M GG]{} & Classical & Robust\ $C_{0}$ & ![\[fig:n200-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=200$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_NL_n200_cont_0_m_0_mod_3.pdf "fig:") & ![\[fig:n200-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=200$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_NL_n200_cont_0_m_0_mod_3.pdf "fig:")\ $C_{0.05}$ & ![\[fig:n200-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=200$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_NL_n200_cont_1_m_10_mod_3.pdf "fig:") & ![\[fig:n200-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=200$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_NL_n200_cont_1_m_10_mod_3.pdf "fig:")\ $C_{0.10}$ & ![\[fig:n200-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=200$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_CL_NL_n200_cont_1_m_20_mod_3.pdf "fig:") & ![\[fig:n200-NL\] Functional boxplots of $\widehat{\AUC}_x$ obtained with the classical and robust estimators for $n=200$ with clean samples and under 5% and 10% of contamination with level shift $S=10$, under the nonlinear model and . The green line corresponds to the true $\AUC_x$ and the dotted red lines to the outlying curves detected by the functional boxplot.](fbx_ROB_NL_n200_cont_1_m_20_mod_3.pdf "fig:") -0.1in Analysis of real data set ========================= [\[sec:realdata\]]{} In this section, we illustrate the benefits of the robust proposed methodology by means of the diabetes real dataset described in the Introduction. Following the analysis given in Faraggi (2003), we transform the marker from both populations using power function $f(t)= -t^{-1/2}$. After this, we assume a linear regression model in each population for the transformed marker $y$, i.e., $$\begin{aligned} y_{D,i} &=& \beta_{D,1} + \beta_{D,2}\, x_{D,i} + \epsilon_{D,i} \,, 1\le i \le 88\,,\label{diseased}\\ y_{H,i} &=& \beta_{H,1} + \beta_{H,2} \, x_{H,i} + \epsilon_{H,i}\; , 1\le i\le 198\,,\label{healthy}\end{aligned}$$ and we compute the classical and robust estimators of the conditional ROC curves, denoted $\widehat{\ROC}_{\bx, \cl}$ and $ \widehat{\ROC}_{\bx }$, respectively. Based on the residuals boxplots of a robust fit, 6 outliers were detected in the healthy sample, labelled as 37, 78, 125, 137, 141 and 150, see the left panel of Figure \[fig:boxplotejemplo\]. The filled red points on the central panel of Figure \[fig:boxplotejemplo\] represent the atypical observations encountered in the healthy sample which correspond to vertical outliers. After removing them, the classical estimator of the conditional ROC curves is recomputed with the remaining points, namely $ \widehat{\ROC}_{\bx, \cl}^{(-6)}$. The upper panel of Figure \[fig:roc\_ejemplo\] displays the estimated surfaces with these three procedures using equidistant grids of points of size 29 and 28 in $p$ and $x$, respectively, between $p=0.01$ and $0.99$ and $x=20$ and $87.5$. In order to facilitate the differences between the estimated surfaces, the middle and lower panel in Figure \[fig:roc\_ejemplo\] show the differences between these estimators, making evident that the robust and classical estimator computed without the outliers are very similar all along the studied range, while the classical estimator computed from the whole sample shows a different pattern, clear in the left panel of Figure \[fig:roc\_ejemplo\] especially for large values of age. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ![\[fig:boxplotejemplo\] The left panel corresponds to the boxplots of the residuals obtained after a robust fit for healthy sample, while the central and right panels to the scatter plots for the healthy and diseased samples.](bxp_residuos_gra.pdf "fig:") ![\[fig:boxplotejemplo\] The left panel corresponds to the boxplots of the residuals obtained after a robust fit for healthy sample, while the central and right panels to the scatter plots for the healthy and diseased samples.](scatter_Glucose_h_gra.pdf "fig:") ![\[fig:boxplotejemplo\] The left panel corresponds to the boxplots of the residuals obtained after a robust fit for healthy sample, while the central and right panels to the scatter plots for the healthy and diseased samples.](scatter_Glucose_d_gra.pdf "fig:") ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -0.1in Final Remarks ============= [\[sec:conclusion\]]{} The ROC curve is a useful graphical tool that measures the discriminating power of a biomarker to distinguish between two conditions or classes. When the practitioner may measure covariates related to the diagnostic variable which can increase the discriminating power, it is sensible to incorporate them in the analysis. To have a deeper comprehension of the effect of the covariates, it would be advisable to incorporate the covariates information to the ROC analysis instead of considering the marginal ROC curve. Conditional ROC curves may be easily estimated using a plug–in procedure. However, the use of classical regression estimators and empirical distribution and quantile functions may lead to estimates which breakdown in the presence of a small amount of atypical data. In this piece of work, we introduce a procedure to robustly estimate the conditional ROC curve. The methodology combines robust regression estimators with a weighted empirical distribution function which downweights the effect of large residuals. We prove that the estimators are uniformly strongly consistent under standard regularity conditions. A simulation study shows that our proposed estimators have good robustness and finite-sample statistical properties. Even though our numerical studies focus on a parametric regression approach, it should be mentioned that our proposal could also be implemented when considering nonparametric or partly parametric regression models, using a robust fit. [ccc]{}\ \ \ $\widehat{\ROC}_{\bx, \cl}$ & $ \widehat{\ROC}_{\bx }$ & $ \widehat{\ROC}_{\bx, \cl}^{(-6)}$\ ![\[fig:roc\_ejemplo\] Diabetes Data: (a) Estimated ROC surfaces and (b) Difference between the estimated ROC surfaces (c) Difference between the estimated ROC surfaces between 0.045 and 0.99.](ROC_cl_gra-other.pdf "fig:") & ![\[fig:roc\_ejemplo\] Diabetes Data: (a) Estimated ROC surfaces and (b) Difference between the estimated ROC surfaces (c) Difference between the estimated ROC surfaces between 0.045 and 0.99.](ROC_rob_gra-other.pdf "fig:") & ![\[fig:roc\_ejemplo\] Diabetes Data: (a) Estimated ROC surfaces and (b) Difference between the estimated ROC surfaces (c) Difference between the estimated ROC surfaces between 0.045 and 0.99.](ROC_so_gra-other.pdf "fig:")\ \ \ \ $\widehat{\ROC}_{\bx }-\widehat{\ROC}_{\bx, \cl}$ & $\widehat{\ROC}_{\bx, \cl}^{(-6)}-\widehat{\ROC}_{\bx, \cl}$ & $\widehat{\ROC}_{\bx}-\widehat{\ROC}_{\bx, \cl}^{(-6)}$\ ![\[fig:roc\_ejemplo\] Diabetes Data: (a) Estimated ROC surfaces and (b) Difference between the estimated ROC surfaces (c) Difference between the estimated ROC surfaces between 0.045 and 0.99.](DIF_rob-cl-gra.pdf "fig:") & ![\[fig:roc\_ejemplo\] Diabetes Data: (a) Estimated ROC surfaces and (b) Difference between the estimated ROC surfaces (c) Difference between the estimated ROC surfaces between 0.045 and 0.99.](DIF_so-cl-gra.pdf "fig:") & ![\[fig:roc\_ejemplo\] Diabetes Data: (a) Estimated ROC surfaces and (b) Difference between the estimated ROC surfaces (c) Difference between the estimated ROC surfaces between 0.045 and 0.99.](DIF_rob-so-gra.pdf "fig:")\ \ \ \ $\widehat{\ROC}_{\bx }-\widehat{\ROC}_{\bx, \cl}$ & $\widehat{\ROC}_{\bx, \cl}^{(-6)}-\widehat{\ROC}_{\bx, \cl}$ &\ ![\[fig:roc\_ejemplo\] Diabetes Data: (a) Estimated ROC surfaces and (b) Difference between the estimated ROC surfaces (c) Difference between the estimated ROC surfaces between 0.045 and 0.99.](DIF_rob-cl_menos1.pdf "fig:") & ![\[fig:roc\_ejemplo\] Diabetes Data: (a) Estimated ROC surfaces and (b) Difference between the estimated ROC surfaces (c) Difference between the estimated ROC surfaces between 0.045 and 0.99.](DIF_so-cl_menos1.pdf "fig:") & -0.1in Appendix A: Proof of Theorem \[theo:consist.2\]. {#sec.appendix} ================================================ We begin by proving (i). Using assumption \[ass:A3\] for $j=H$ and the continuity of the quantile functionals when \[ass:A1\] holds, we get that, for the healthy subjects, $\wG_{H}^{-1}(p) \convpp G_H^{-1}(p)$, for each $0<p<1$. To avoid burden notation denote as $$\begin{aligned} \wDelta(\bx,p)&=& \frac{\wmu_{H}(\bx) -\wmu_{D}(\bx)}{\wsigma_{D} } + \frac{\wsigma_{H} }{\wsigma_{D} } \, \wG_{H}^{-1}(1-p)\,,\\ \Delta(\bx,p)&=& \frac{\mu_{0,H}(\bx) -\mu_{0,D}(\bx )}{\sigma_{0,D}} + \frac{\sigma_{0,H}}{\sigma_{0,D}} \, G_{H}^{-1}(1-p)\,.\end{aligned}$$ Note that the consistency of $\wsigma_j$ and \[ass:A61\] together with the fact that $\wG_{H}^{-1}(p) \convpp G_H^{-1}(p)$, entail that for each fixed $p$ and $\bx$, $\wDelta(\bx,p)\convpp \Delta(\bx,p)$. Therefore, we have that, $$\begin{aligned} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)| &=& \left|\wG_{D}\left(\wDelta(\bx,p)\right)- G_{D}\left(\Delta(\bx,p)\right)\right|\\ &\le & \left|\wG_{D}\left(\wDelta(\bx,p)\right)- G_{D}\left(\wDelta(\bx,p)\right)\right| + \left| G_{D}\left(\wDelta(\bx,p)\right)- G_{D}\left(\Delta(\bx,p)\right)\right|\\ &\le & \left\|\wG_{D}- G_D\right\|_{\infty} + \left| G_{D}\left(\wDelta(\bx,p)\right)- G_{D}\left(\Delta(\bx,p)\right)\right|\end{aligned}$$ which together with the continuity of $G_D$ lead to $\widehat{\ROC}_{\bx}(p) \convpp {\ROC}_{\bx}(p)$, for each fixed $\bx$ and $0<p<1$. Note that for each fixed $\bx$, ${\ROC}_{\bx}(p)$ satisfies the conditions in Lemma S.1.1, so $\sup_{0<p<1} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\convpp 0$. \(ii) Using that $$\begin{aligned} \left|\wDelta(\bx,p)- \Delta(\bx,p)\right|&\le & \frac{1}{\wsigma_{D} } \left\{\left|\wmu_{H}(\bx) -\mu_{0,H}(\bx)\right|+ \left|\wmu_{D}(\bx) -\mu_{0,D}(\bx)\right|\right\} \\ & &+\left| \frac{1}{\wsigma_{D}} - \frac{1}{\sigma_{0,D}} \right|\; \left|\mu_{0,H}(\bx)) -\mu_{0,D}(\bx)\right|\\ && + \frac{\wsigma_{H} }{\wsigma_{D} } \,\left| \wG_{H}^{-1}(1-p)- G_{H}^{-1}(1-p)\right|+ | G_{H}^{-1}(1-p)|\, \left|\frac{\wsigma_{H} }{\wsigma_{D}}-\frac{\sigma_{0,H}}{\sigma_{0,D}} \right|\,,\end{aligned}$$ assumption \[ass:A7\], the consistency of $\wsigma_j$ and the uniform consistency of $\wmu_j$, we get easily that $\sup_{\bx \in \itK} \left|\wDelta(\bx,p)- \Delta(\bx,p)\right|\convpp 0$. Hence, $$\begin{aligned} \sup_{\bx \in \itK} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)| &\le & \left\|\wG_{D}- G_D\right\|_{\infty} + \| g_{D}\|_{\infty} \sup_{\bx \in \itK}\left| \wDelta(\bx,p)- \Delta(\bx,p) \right|\end{aligned}$$ leads to $\sup_{\bx \in \itK} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)| \convpp 0$. Denote as $\wB=\sup_{\delta<p<1-\delta}\sup_{\bx \in \itK} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)|$, then $\wB \le \sum_{\ell=1}^5 \wB_\ell$ where $$\begin{aligned} \wB_1 &=& \left\|\wG_{D}- G_D\right\|_{\infty} \;,\\ \wB_2&=& \| g_{D}\|_{\infty} \left| \frac{1}{\wsigma_{D} } - \frac{1}{\sigma_{0,D} } \right|\; \left(A_H+A_D\right) \;,\\ \wB_3 &=& \| g_{D}\|_{\infty} \frac{1}{\wsigma_{D} } \sup_{\bx \in \itK} \left\{\left|\wmu_{H}(\bx) -\mu_{0,H}(\bx)\right|+ \left|\wmu_{D}(\bx) -\mu_{0,D}(\bx)\right|\right\} \;,\\ \wB_4 &=& \frac{\wsigma_{H} }{\wsigma_{D} } \sup_{\delta<p<1-\delta} \,\left| \wG_{H}^{-1}(1-p)- G_{H}^{-1}(1-p)\right| \;,\\ \wB_5 &=& \sup_{\delta<p<1-\delta}| G_{H}^{-1}(1-p)| \,\left|\frac{\wsigma_{H} }{\wsigma_{D} }-\frac{\sigma_{0,H}}{\sigma_{0,D}} \right| \;.\end{aligned}$$ Assumptions \[ass:A1\] to \[ass:A3\] together with \[ass:A6\] and the consistency of $\wsigma_j$ entail that $\wB_\ell\convpp 0$, for $\ell=1,2,3,5$. Besides, using that $u=\wG_{H}^{-1}(\wG_{H}(u))$ and making the change $\wG_{H}(u)=1-p$, we get that $$\begin{aligned} \sup_{\delta<p<1-\delta} \,\left| \wG_{H}^{-1}(1-p)- G_{H}^{-1}(1-p)\right| &= & \sup_{\wG_{H}^{-1}(\delta)<u<\wG_{H}^{-1}(1-\delta)} \left|G_{H}^{-1}\left(\wG_{H}(u)\right)-u \right|\\ &\le & \sup_{\wG_{H}^{-1}(\delta)<u<\wG_{H}^{-1}(1-\delta)} \frac{1}{g_H(G_H^{-1}(\xi_u))}\left| \wG_{H}(u) -G_{H}(u) \right|\end{aligned}$$ where $\xi_u$ is an intermediate point between $\wG_{H}(u)$ and $G_{H}(u)$. Using that $\wG_{H}^{-1}(1-\delta)\convpp G_{H}^{-1}(1-\delta)$ and $\wG_{H}^{-1}(\delta)\convpp G_{H}^{-1}(\delta)$ and that $G_H^{-1}(\delta/2)<G_{H}^{-1}(\delta)$ and $G_{H}^{-1}(1-\delta) < G_{H}^{-1}(1-\delta/2)$, since $G_H$ has a density, we obtain that for all $\omega \notin \itN$ with $\prob(\itN)=1$, there exists $N_{0,H}$ such that for $n_H\ge N_{0,H}$ , $a(\delta)= G_H^{-1}(\delta/2)<\wG_{H}^{-1}(\delta)$ and $\wG_{H}^{-1}(1-\delta)< G_{H}^{-1}(1-\delta/2)=b(\delta)$. Thus, $\xi_u \in [a(\delta), b(\delta)]=\itI$ so using that $i(\delta)=\inf_{u\in \itI} g_H(u)>0$, we conclude that $$\begin{aligned} \sup_{\delta<p<1-\delta} \,\left| \wG_{H}^{-1}(1-p)- G_{H}^{-1}(1-p)\right| &\le & i(\delta)\, \sup_{a(\delta)<u<b(\delta)} \left| \wG_{H}(u) -G_{H}(u) \right|\le i(\delta) \|\wG_{H} -G_{H}\|_{\infty}\,,\end{aligned}$$ concluding the proof of (ii). We now proceed to derive (iii). Let $\epsilon>0$ be fixed and choose $0<\eta<1$ such that, $\sup_{\bx\in \itK}{\ROC}_{\bx}(\eta)<\epsilon/6$ and $\sup_{\bx\in \itK} (1-{\ROC}_{\bx}(1-\eta))<\epsilon/6$. Denote as $$\begin{aligned} \wB(\eta) & = \sup_{\eta<p<1-\eta}\sup_{\bx \in \itK} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)|\,,\\ \wB_1(\eta)& = \sup_{ p\le \eta}\sup_{\bx \in \itK} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)|\,,\\ \wB_2(\eta)& = \sup_{1-\eta\le p}\sup_{\bx \in \itK} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)|\;.\end{aligned}$$ Hence, $\sup_{0<p<1 } \sup_{\bx\in \itK} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\le \wB(\eta)+\wB_1(\eta)+\wB_2(\eta) $. From (ii), $\wB(\eta)\convpp 0$. Besides, using that ${\ROC}_{\bx}(p)$ is a distribution function and $\widehat{\ROC}_{\bx}(p)$ is non-decreasing in $p$, we get that for any $p\le \eta$, $\bx \in \itK$, $$|\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)| \le \max\left\{\widehat{\ROC}_{\bx}(\eta), {\ROC}_{\bx}(\eta)\right\}\,,$$ so $\wB_1(\eta)\le \sup_{\bx\in \itK} \max\left\{\widehat{\ROC}_{\bx}(\eta), {\ROC}_{\bx}(\eta)\right\}=\wC_1(\eta)$ . Similarly, we obtain that $\wB_2(\eta)\le \sup_{\bx\in \itK} \max\left\{1-\widehat{\ROC}_{\bx}(1-\eta), 1-{\ROC}_{\bx}(1-\eta)\right\}=\wC_2(\eta)$. Using that $\sup_{\bx\in \itK}{\ROC}_{\bx}(\eta)<\epsilon$ and $\sup_{\bx\in \itK} (1-{\ROC}_{\bx}(1-\eta))<\epsilon$ and that for any fixed $0<p<1$, $\sup_{\bx \in \itK} |\widehat{\ROC}_{\bx}(p) - {\ROC}_{\bx}(p)| \convpp 0$, we conclude that there exists $\itN$ such that $\prob(\itN)=0$ and for $\omega\notin \itN$, $ \wB(\eta) \to 0$, $\wC_1(\eta) \to \sup_{\bx\in \itK} {\ROC}_{\bx}(\eta)< \epsilon/{6}$ and $\wC_2(\eta) \to \sup_{\bx\in \itK} 1- {\ROC}_{\bx}(1-\eta)< {\epsilon}/{6}$. Hence, for $n_H$ and $n_D$ large enough, we obtain that $ \wB(\eta)<\epsilon/3$, $\wC_\ell(\eta)<\epsilon/3$, for $\ell=1,2$ which leads to $\sup_{0<p<1 } \sup_{\bx\in \itK} |\widehat{\ROC}_{\bx}(p)- {\ROC}_{\bx}(p)|\le \epsilon$, concluding the proof. Appendix B {#sec:consistG} ========== In this section, we investigate the validity of assumption \[ass:A3\]. For that purpose, we will derive the uniform strong consistency of $\wG_n(t)$ defined in in two situations, under a linear model or a non–linear one, since for the former we can also include a hard rejection weight function to define the weights $w_i$. It is worth noticing that our results generalize those given in Gervini and Yohai (2002) in two directions: we extend their results beyond the linear model to a non–linear one and we obtain almost surely uniform consistency instead of results in probability. From now on, for any measure $Q$, we denote as $N(\epsilon, \itF, L_s(Q))$ and $N_{[\;]}(\epsilon, \itF, L_s(Q))$ the covering and bracketing numbers of the class $\itF$ with respect to the distance in $ L_s(Q)$, as defined, for instance, in van der Vaart and Wellner (1996). Linear Model ------------ [\[sec:appendix1\]]{} Throughout this section, we will assume that we have a random sample $(y_{1},\bx_{1}),\dots,(y_{ n},\bx_{n})$, where $\bx_i$ is a vector of $p$ explanatory variables and $y_i$ is a response variable that satisfy the linear regression model $$y_i= \bx_i \trasp \bbe_0 + u_i= \bx_i \trasp \bbe_0 +\sigma_0 \epsilon_i, i=1 \dots n \, ,$$ with $\bbe_0 \in \real^p$ and the errors $\epsilon_i$ are i.i.d. and independent of $\bx_i$ with unknown distribution $G_0(\cdot)$ and $\sigma_0$ is the scale parameter. As above, we will assume that $G_0$ is symmetric around 0. From now on, $\wbbe$ and $\wsigma$ stand for robust consistent estimators of $\bbe_0$ and $\sigma_0$, so the standardized residuals are given by $$r_{i}= \dfrac{y_{i}-\bx_{i}\trasp \wbbe }{\wsigma} \, .$$ Based on the residuals the adaptive weighted empirical distribution given in is defined using the weights $w_i=w(r_i/t_n)$ defined in with $t_n$ given in . To derive uniform consistency results, we will need the following set of assumptions: 1. \[ass:peso\] The weight function $w: \real \to [0,1]$ is even, non-increasing on $[0, +\infty)$, continuous, $w(0)=1$, $w(u)>0$ for $0 < u <1$ and $w(u)=0$ for $ |u| \ge 1$. 2. \[ass:G0\] $G_0$ is a continuous distribution function, symmetric around 0. 3. \[ass:consis\] The estimators $\wbbe$ and $\wsigma$ are such that $\wbbe\convpp \bbe_0$ and $\wsigma\convpp \sigma_0$. Define the values $$\begin{aligned} d_0 &=& \sup_{t \ge 0} \{G^{+}(t)-G^{+}_0(t)\}^+=\sup_{t \ge 0} \{\max\left(G^{+}(t)-G^{+}_0(t),0\right)\} \label{eq:d0}\\ \ot_0 &=& (G_0^{+})^{-1}(1-d_0)=G_0^{-1}\left(1-\frac{d_0}2\right) \label{eq:ot0}\\ t_0 &=& \max(\ot_0, \eta) \label{eq:t0} \end{aligned}$$ As mentioned in Gervini and Yohai (2002), when $G^{+}$ is stochastically larger or equal than $G_0^{+}$, we have that $t_0=\infty$, so $\wG_n$ defined in will converge to $G_0$. Furthermore, consider the functions $$\begin{aligned} h_{\infty}(t) &=& \esp_{G_0} w\left(\frac{\epsilon_1}{ \, t}\right) \label{eq:hinfty}\\ h_0(t,s) &=& \esp_{G_0} w\left(\frac{\epsilon_1}{ \, t}\right)\indica_{\epsilon_1\le s } \label{eq:h0} \end{aligned}$$ The following lemma is a well known result regarding continuous distributions, whose proof we include for completeness. 0.1in \[lema:appendix1.1\] Let $F_n:\real \to [0,1]$ and $F:\real \to [0,1]$ be non–decreasing functions such that $F$ is continuous, $\lim_{t\to +\infty}F(t)=1$ and $\lim_{t\to -\infty}F(t)=0$. Then, if $F_n(t)\convpp F(t)$, for any $t\in \real$, we also have that $\|F_n-F\|_{\infty} \convpp 0 $. <span style="font-variant:small-caps;">Proof.</span> Given $\epsilon>0$, let $a$ and $b$ be such that $F(a)<\epsilon$ and $F(b)>1-\epsilon$. Furthermore, using that $F$ is uniformly continuous on $[a,b]$, we get that there exists $\delta$ such that $$|t-s|<\delta, t,s \in [a,b] \Rightarrow |F(t)-F(s)|<\epsilon$$ Let $a=a_0<a_2<\dots<a_k=b$, be a grid such that $a_j-a_{j-1}<\delta$, $1\le j\le k$. Then, we have that for any $t<a$, $F_n(t)-F(t)\le F_n(a)\le F_n(a)-F(a)+ F(a)\le |F_n(a)-F(a)|+\epsilon$, while $F(t)-F_n(t)\le F(a)<\epsilon$, so $$\label{eq:Fna} \sup_{t<a}|F_n(t)-F(t)|\le |F_n(a)-F(a)|+\epsilon\,.$$ Similarly, $$\label{eq:Fnb} \sup_{t>b}|F_n(t)-F(t)|\le |F_n(b)-F(b)|+\epsilon\,.$$ Finally, given $t\in [a,b]$, there exists $1\le j\le k$ such that $t\in [a_{j-1}, a_j]$, so that $$\begin{aligned} F_n(t)-F(t) & \le & F_n(a_j)-F(a_{j-1})\le F_n(a_j)-F(a_{j})+ F(a_{j})-F(a_{j-1})\\ & \le & \epsilon+\max_{1\le j\le k}|F_n(a_j)-F(a_{j})| \,. \end{aligned}$$ Similarly, $$\begin{aligned} F (t)-F_n(t) & \le & F(a_j)-F_n(a_{j-1})\le F(a_j)-F(a_{j-1})+ F(a_{j-1})-F_n(a_{j-1})\\ & \le & \epsilon+\max_{1\le j\le k}|F_n(a_j)-F(a_{j})|\,, \end{aligned}$$ so $$\label{eq:Fnab} \sup_{a\le t\le b}|F_n(t)-F(t)|\le \epsilon+\max_{1\le j\le k}|F_n(a_j)-F(a_{j})|\,.$$ Let $\itN$ be such that, for $\omega\notin \itN$, $F_n(a_j)\to F(a_j)$ and $\prob(\itN)=0$. Then, using , and we get that $$\prob(\limsup \|F_n-F\|_{\infty}<\epsilon)=1\,,$$ for any $\epsilon>0$, concluding the proof. 0.1in \[lema:appendix1.2\] Under \[ass:G0\] and \[ass:consis\], we have that 1. $\|G_n^{+}-G_0^{+}\|_{\infty}\convpp 0$. 2. $d_n\convpp d_0$. 3. $\ot_n\convpp \ot_0$. <span style="font-variant:small-caps;">Proof.</span> a) Let us consider the family of functions $$\itF=\{f_{\bthech,\; \kappa}(u,\bx)=\indica_{|u-\bx\trasp\bthech|\le \kappa\; t} \mbox{ for } (\bthe, t,\kappa)\in \real^p\times \real_{\ge 0}\times \real_{> 0}\} \,.$$ First, note that $$f_{\bthech,\; \kappa}(u,\bx)=\indica_{|u-\bx\trasp\bthech|\le \kappa \; t}=\indica_{C_{(\kappa^{-1}, \bthech\; \kappa^{-1}, \; t)}} \,,$$ where the set $C_{(s, \bthech,\; t)}= A_{(s, \bthech,\; t)}\cap B_{(s, \bthech,\; t)}$, with $ A_{(s, \bthech,\; t)}=\{(u,\bx)\in \real^{p+1}: su-\bx\trasp\bthe -t \le 0\}$ and $ B_{(s, \bthech,\; t)}= \{(u,\bx)\in \real^{p+1}: 0\le su-\bx\trasp\bthe +t \} $. Define the classes of sets $$\begin{aligned} \itA &=& \{ A_{(s, \bthech,\; t))}: (\bthe, s,\; t))\in \real^p\times\real_{\ge 0}\times \real_{>0}\}\\ \itB &=& \{ B_{(s, \bthech,\; t))}: (\bthe, s,\; t))\in \real^p\times\real_{\ge 0}\times \real_{>0}\}\,. \end{aligned}$$ Taking into account that $\{g(u,\bx)=su-\bx\trasp\bthe -t ; (\bthe, s,\; t))\in \real^p\times\real_{\ge 0}\times \real_{>0}\}$ is a finite–dimensional space of functions with dimension $p+2$, from Lemmas 9.6, 9.8 and 9.9 in Kosorok (2008) we get that $\itA$ and $\itB$ are VC-classes with index at most $p+4$. Furthermore, $\itC=\itA\cap\itB$ is also a VC-class with index smaller or equal than $2p+7$. Taking into account that $C_{(s, \bthech)}\in \itC$, applying again Lemma 9.8 in Kosorok (2008), we get that the class of functions $\itF$ is a VC-class with index $V(\itF)$ smaller or equal than $2p+7$. Note that the envelope of $\itF$ equals $F\equiv 1$. Hence, Theorem 2.6.7 in van der Vaart and Wellner (1996) entails that, there exists a universal constant $K$ such that, for any measure $Q$ $$N(\epsilon, \itF, L_1(Q)) \le K \; V(\itF) \left(16 e\right)^{V(\itF)} \left(\frac{1}{\epsilon}\right)^{V(\itF)-1}\, ,$$ which together with Theorem 2.4.3 in van der Vaart and Wellner (1996) or Theorem 2.4 in Kosorok (2008), leads to $$\label{eq:glivenko} \sup_{f\in \itF} | P_n f- P f| \convpp 0\,,$$ where we have used the standard notation in empirical processes, i.e., $P f=\esp f(u,\bx)$ and $P_n f=(1/n) \sum_{i=1}^n f(u_i, \bx_i)$. Note that $G_n^{+}$ can be written as $$G_n^{+}(t) =P_n f_{\wbthech, \;\wsigma\,t}(u_i,\bx_i)$$ with $\wbthe=\wbbe-\bbe_0$. Denote as $M(\bthe, \kappa)= P f_{\bthech,\; \kappa}(u,\bx)$. Then, using , we conclude that $$\sup_{t\ge 0}\left|G_n^{+}(t) - M( \wbthech, \;\wsigma\,t)\right|\le \sup_{f\in \itF} | P_n f- P f| \convpp 0\,.$$ It remains to show that $$\sup_{t\ge 0}\left| M( \wbthe, \;\wsigma\,t)- G_0^{+}(t)\right| \convpp 0\,.$$ Note that $$M(\bthe, \kappa)= P f_{\bthech,\; \kappa}(u,\bx)= \prob(|u-\bx\trasp\bthe|\le \kappa) \,,$$ hence $$M(0, \sigma_0\, t)= \prob( |u|\le \sigma_0 \, t)=G_0^{+}(t)\,.$$ Therefore, we have to show that $$\sup_{t\ge 0}\left| M( \wbthe, \;\wsigma\,t)- M(0, \sigma_0\, t)\right| \convpp 0\,.$$ First observe that $$\begin{aligned} M(\bthe, \kappa)&=& \prob \left( -\kappa + \bx\trasp\bthe\le u \le \kappa+\bx\trasp\bthe\right)\\ &=& \esp \left\{G_0(\bx\trasp\bthe+\kappa)- G_0(\bx\trasp\bthe- \kappa)\right\}\, . \end{aligned}$$ The continuity of $G_0$ and the Dominated Convergence Theorem entail that $M(\bthe, \kappa)$ is a continuous function of its arguments, which together with \[ass:consis\], entails that $M( \wbthe, \;\wsigma\,t)- M(0, \sigma_0\, t) \convpp 0$, for each fixed $t$. Now $$\wM(t)=M( \wbthe, \;\wsigma\,t)=\prob\left(\frac{|u-\bx\trasp\wbthe|}{\wsigma}\le t\right)$$ is a bounded monotone function of $t$, while $M(0, \sigma_0\, t)=G_0^{+}(t)$ is also bounded, monotone and continuous, thus, from Lemma \[lema:appendix1.1\] we obtain that the convergence is indeed uniform, that is, $\sup_{t\ge 0}\left| M( \wbthe, \;\wsigma\,t)- M(0, \sigma_0\, t)\right| \convpp 0 $, concluding the proof of a). b\) As in Gervini and Yohai (2002), $|d_n- d_0|\le \|G_n^{+}-G_0^{+}\|_{\infty}$ and the result follows. c\) To show that $\ot_n\convpp \ot_0$, it is enough to show that $G_0^{+}(\ot_n)\convpp G_0^{+}(\ot_0)=1-d_0$ which follows from Lemma 3.1 in Gervini and Yohai (2002) distinguishing the cases $\ot_0<\infty$ and $\ot_0=\infty$. 0.2in \[lema:appendix1.3\] Assume that either $w(t)= \indica_{[-1,1]}(t)$ or $w$ satisfies \[ass:peso\]. Then, we have that $\sup_{f\in \itF} | P_n f- P f| \convpp 0$, where $$\itF=\{f_{\bthech,\; \kappa, \nu}(u,\bx)=w\left(\nu({u -\bx\trasp\bthe})\right)\, \indica_{u-\bx\trasp\bthech\le \kappa\, s} \mbox{ for } (\bthe, \kappa, \nu)\in \real^p\times \real_{\ge 0}\times \real_{\ge 0}\} \,.$$ <span style="font-variant:small-caps;">Proof.</span> Assume that $w$ satisfies \[ass:peso\] and note that $\itF\subset \itF_1 \cdot \itF_2$ where $$\begin{aligned} \itF_1 &=& \{f_{\bthech,\, \nu}(u,\bx)=w\left(\nu({u -\bx\trasp\bthe})\right)\, \mbox{ for } (\bthe, \nu)\in \real^p\times \real_{\ge 0}\}\\ \itF_2 &=& \{f_{\bthech,\; \kappa }(u,\bx)= \indica_{u-\bx\trasp\bthech\le \kappa\, s} \mbox{ for } (\bthe, \kappa )\in \real^p\times \real_{\ge 0}\}\,. \end{aligned}$$ The classes $\itF$, $\itF_1$ and $\itF_2$ have envelope 1, hence we have easily that, for any measure $Q$, $$N(2\,\epsilon, \itF, L_1(Q))\le N(\epsilon, \itF_1, L_1(Q)) N(\epsilon, \itF_2, L_1(Q))\,,$$ so that to show $\sup_{f\in \itF} | P_n f- P f| \convpp 0$, it will be enough to prove that, for $j=1,2$, $$\label{eq:NPn} \frac{1}{n} \log N(\epsilon, \itF_j, L_1(P_n)) \convprob 0\,.$$ As in the proof of Lemma \[lema:appendix1.2\], it is easy to see that $\itF_2$ is a VC-class with index $V_2=p+3$, so $$N(\epsilon, \itF_2, L_1(Q)) \le K \; V_2 \left(16 e\right)^{V_2} \left(\frac{1}{\epsilon}\right)^{V_2-1}\, ,$$ leading to , when $j=2$. On the other hand, the family $$\itR = \left\{\nu({u -\bx\trasp\bthe})\,:\; \bthe \in \real^p, \nu \in \real_{\ge 0}\right\}$$ is a subset of the vector space of all linear functions in $p+1$ variables. It follows from Lemma 2.6.15 of van der Vaart and Wellner (1996) that $\itR$ has VC-index at most $p+3$. Note that $w$ is an even function, non-increasing on $[0,+\infty)$, hence it can be written as $w = w^{(1)}+w^{(2)}$, where $w^{(1)}(x)=w(x)\indica_{[0,+\infty)}(x)$ is non–increasing and $w^{(2)}(x)=w(x)\indica_{(-\infty,0)}(x)$ is non–decreasing. Using the permanence property for VC-classes, see Lemma 9.9 in Kosorok (2008), we obtain that the classes of functions $\itR_{w^{(1)}}=w^{(1)} \circ \itR$ and $\itR_{w^{(2)}}=w^{(2)} \circ \itR$ are VC–classes with VC–index at most $p+ 3$. Furthermore, the classes $\itR_{w^{(j)}}$, $j=1,2$, have envelope 1. Then, Theorem 2.6.7 of Van der Vaart and Wellner (1996) implies that there exists a universal constant $K$ such that, for any probability measure $Q$ on $\real^{p+1}$ and any $0<\epsilon<1$, we have that $$N(\epsilon , \itR_{w^{(j)}}, L_1(Q)) \leq K (p+3)\; (16e)^{(p+3)}\left(\frac 1\epsilon\right)^{p+2}\,.$$ Note that $\itR_{w^{(1)}} + \itR_{w^{(2)}}$ has also constant envelope equal to $2 $. Therefore, $$\begin{aligned} N(2 \epsilon , \itR_{w^{(1)}} + \itR_{w^{(2)}}, L_1(Q)) &\leq N( \epsilon ,\itR_{w^{(1)}} , L_1(Q)) \times N( \epsilon ,\itR_{w^{(2)}} , L_1(Q)) \\ &\leq \left[K (p+3)\; (16e)^{(p+3)}\left(\frac 1\epsilon\right)^{p+2}\right]^2\,.\end{aligned}$$ Finally noting that $\itF_1 $ has constant envelope equal to 1 and $\itF_1 \subset \itR_{w^{(1)}} + \itR_{w^{(2)}}$, we get that $$N(2 \epsilon , \itF_1 , L_1(P_n)) \le \left[K (p+3)\; (16e)^{(p+3)}\left(\frac 1\epsilon\right)^{p+2}\right]^2\,,$$ concluding the proof. When $w(t)= \indica_{[-1,1]}(t)$ the result is straightforward using that $$\begin{aligned} \itF_1 &=& \{f_{\bthech,\, \nu}(u,\bx)=\indica_{\nu({u -\bx\trasp\bthe})\le 1}\indica_{\,-\,\nu({u -\bx\trasp\bthe})\le 1}\, \mbox{ for } (\bthe, \nu)\in \real^p\times \real_{\ge 0}\} \end{aligned}$$ and similar arguments to those consider above. 0.2in \[prop:appendix1.1\] Assume that $w(t)= \indica_{[-1,1]}(t)$ or $w$ satisfies \[ass:peso\]. Under \[ass:G0\] to \[ass:consis\], we have that 1. if $t_0<\infty$, $$\sup_{s\in \real} \left|\wG_n(s)- \frac{h_0(t_0,s)}{h_{\infty}(t_0)}\right|\convpp 0\,,$$ with $h_\infty(t_0)$ and $h_0(t_0,s)$ defined in and , respectively. 2. if $t_0=\infty$, $\|\wG_n-G_0\|_{\infty}\convpp 0$. <span style="font-variant:small-caps;">Proof.</span> When $t_0=\infty$, using that $G_0$ is a bounded, monotone and continuous function and that $\wG_n$ is monotone, it will be enough to show that for each $s\in \real$, $\wG_n(s)\convpp G(s)$. On the other hand, when $t_0<\infty$, standard arguments allow to show that $F(s)= {h_1(t_0,s)}/{h_{\infty}(t_0)}$ is a bounded, monotone and continuous function of $s$ and the uniform convergence also follows from the pointwise one. Denote as $\wnu_n= 1/(t_n\, \wsigma_n )$, $\nu_0= 1/(t_0\, \sigma_0 )$, where we understand that if $t_0=+\infty$, $\nu_0=0$. Then $\wnu_n\convpp \nu_0$. We will begin by showing that $$\label{eq:aprobar1} \frac{1}{n} \sum_{i=1}^n w_i I(r_i \le s) \convpp h_0(t_0,s) =\left\{\begin{array}{lr} \esp_{G_0} w\left(\dfrac{\epsilon_1}{ \, t_0}\right)\indica_{\epsilon_1\le s } & \mbox{ if } t_0<\infty\\ \esp_{G_0} \indica_{\epsilon_1\le s }=G_0(s) & \mbox{ if } t_0=\infty \, . \end{array} \right.$$ For that purpose and noting that $r_i=(u_i-\bx_i\trasp \wbthe)/\wsigma$ with $\wbthe=\wbbe-\bbe_0$, define the class of functions $$\itF=\{f_{\bthech,\; \kappa, \nu}(u,\bx)=w\left(\nu({u -\bx\trasp\bthe})\right)\, \indica_{u-\bx\trasp\bthech\le \kappa\, s} \mbox{ for } (\bthe, \kappa, \nu)\in \real^p\times \real_{\ge 0}\times \real_{\ge 0}\} \,.$$ Lemma \[lema:appendix1.3\] entails that $$\sup_{f\in \itF} | P_n f- P f| \convpp 0\,,$$ then, using that $$\frac{1}{n} \sum_{i=1}^n w_i I(r_i \le s)= P_n f_{\wbthech, \;\wsigma,\; \wnu_n}\, ,$$ we obtain that $$\frac{1}{n} \sum_{i=1}^n w_i I(r_i \le s) - P f_{\wbthech , \;\wsigma,\; \wnu_n} \convpp 0\,.$$ It remains to show that $$P f_{\wbthech , \;\wsigma,\; \wnu_n }\convpp P f_{0,\, \sigma_0,\; \nu_0} = h_0(t_0,s)\,,$$ which will follow if we derive that $$\begin{aligned} A_n&=&P f_{\wbthech, \;\wsigma,\; \wnu_n }-\esp w\left(\nu_0 u \right)\, \indica_{u-\bx\trasp\wbthech\le \wsigma \, s} \convpp 0 \label{eq:An}\\ B_n&=& \esp w\left(\nu_0 u \right)\, \indica_{u-\bx\trasp\wbthech\le \wsigma \, s} -h_0(t_0,s) \convpp 0 \, . \label{eq:Bn} \end{aligned}$$ We begin by considering the situation where $w$ satisfies \[ass:peso\]. Noting that $$\begin{aligned} |A_n| &=& \left|\esp \left\{ w\left(\nu_n(u-\bx\trasp\wbthe)\right) - w\left(\nu_0\, u \right)\right\} \indica_{u-\bx\trasp\wbthech\le \wsigma \, s} \right|\\ &\le & \esp \left| w\left(\nu_n(u-\bx\trasp\wbthe)\right) - w\left(\nu_0\, u \right)\right|\, , \end{aligned}$$ using the Dominated Convergence Theorem, the continuity of $w$ and the fact that $\wnu_n\convpp \nu_0$ and $\wbthe\convpp 0$, we obtain that $A_n\convpp 0$, concluding the proof of . When $w= \indica_{[-1,1]}$, we have that $$\begin{aligned} w\left(\nu_n(u-\bx\trasp\wbthe)\right) - w\left(\nu_0\, u \right)&=&\indica_{\nu_n(u-\bx\trasp\wbthech)\le 1}\;\indica_{-1\le \nu_n(u-\bx\trasp\wbthech)}- \indica_{\nu_0\, u \le 1}\; \indica_{-1\le \nu_0\, u }\\ &=& \indica_{\nu_n(u-\bx\trasp\wbthech)\le 1}\;\left\{\indica_{-1\le \nu_n(u-\bx\trasp\wbthech)}- \indica_{-1\le \nu_0\, u }\right\}\\ &&+ \left\{\indica_{\nu_0\, u \le 1}-\indica_{\nu_n(u-\bx\trasp\wbthech)\le 1}\right\}\; \indica_{-1\le \nu_0\, u }\;,\end{aligned}$$ so $$\begin{aligned} |A_n| &\le & \esp \left| w\left(\nu_n(u-\bx\trasp\wbthe)\right) - w\left(\nu_0\, u \right)\right| \\ & \le & \esp \left| \indica_{-1\le \nu_n(u-\bx\trasp\wbthech)}- \indica_{-1\le \nu_0\, u }\right| +\esp \left| \indica_{\nu_0\, u \le 1}-\indica_{\nu_n(u-\bx\trasp\wbthech)\le 1}\right|\\ & \le & \esp \left| \indica_{-\frac{1}{\nu_n} + \bx\trasp\wbthech \le u}- \indica_{-\frac{1}{\nu_0}\le \, u }\right| +\esp \left| \indica_{ u \le \frac{1}{\nu_0}}-\indica_{ u\le \frac{1}{\nu_n}+ \bx\trasp\wbthech}\right|\\ & \le & \esp \left| \indica_{ u < -\frac{1}{\nu_n} + \bx\trasp\wbthech}- \indica_{ u < -\frac{1}{\nu_0}}\right| +\esp \left| \indica_{ u \le \frac{1}{\nu_0}}-\indica_{ u\le \frac{1}{\nu_n}+ \bx\trasp\wbthech}\right|\\ &\le & \esp \left|G_0\left(\frac{1}{\sigma_0}\left[ -\frac{1}{\nu_n} + \bx\trasp\wbthe \right] \right) - G_0\left(-\frac{1}{\sigma_0\;\nu_0} \right) \right| + \esp \left|G_0\left( \frac{1}{\sigma_0}\left[ \frac{1}{\nu_n} + \bx\trasp\wbthe \right] \right) - G_0\left( \frac{1}{\sigma_0\;\nu_0} \right) \right| \, , \end{aligned}$$ where we understand that $\indica_{ u < -1/{\nu_0}}=0$, $G_0\left(- {1}/{\sigma_0\;\nu_0}\right) =0$, $\indica_{ u < 1/{\nu_0}}=1$ and $G_0\left( {1}/{\sigma_0\;\nu_0}\right) =1$ if $\nu_0=0$. Now the proof follows from the continuity of $G_0$ is $t_0<\infty$ and from the fact that $\lim_{u\to -\infty} G_0(u)=0$ while $\lim_{u\to +\infty} G_0(u)=1$. To derive , note that $$\begin{aligned} | B_n|&=& \left|\esp w\left(\nu_0 u \right)\, \indica_{u-\bx\trasp\wbthech\le \wsigma \, s} - \esp w\left(\nu_0\, u\right)\indica_{u\le \sigma_0 \, s }\right|\\ &\le & \esp \left|\indica_{u\le \bx\trasp\wbthech+ \wsigma \, s} -\indica_{u\le \sigma_0 \, s }\right| \, . \end{aligned}$$ If $\bx\trasp\wbthe+ \wsigma \, s\le \sigma_0 \, s $, then $\indica_{u\le \bx\trasp\wbthech+ \wsigma \, s} =1$ implies that $\indica_{u\le \sigma_0 \, s} =1$, so that $\Delta(u)=\indica_{u\le \bx\trasp\wbthech+ \wsigma \, s} -\indica_{u\le \sigma_0 \, s }=0$. Similarly, if $\indica_{u\le \sigma_0 \, s} =0$, then $\indica_{u\le \bx\trasp\wbthech+ \wsigma \, s} =0$ and $\Delta(u)=0$. Therefore, when $\bx\trasp\wbthe + \wsigma \, s\le \sigma_0 \, s $, $\Delta(u)=1$ if and only if $ \bx\trasp\wbthe + \wsigma \, s< u\le \sigma_0 \, s$. On the other hand, if $\bx\trasp\wbthe + \wsigma \, s\ge \sigma_0 \, s $, then $\Delta(u)=1$ if and only if $ \sigma_0 \, s < u\le \bx\trasp\wbthe + \wsigma \, s$. Note that the fact that $\wbthe\convpp 0$ and $\wsigma\convpp \sigma_0$ entails that $\bx\trasp\wbthe+ \wsigma \, s\convpp \sigma_0 \, s $, for each $\bx$. Let $\itC=\{\bx: \bx\trasp\wbthe+ \wsigma \, s\le \sigma_0 \, s\}$ and $\overline{\itC}$ its complement, then $$\begin{aligned} | B_n| &\le & \esp \;\indica_{\itC} \;\indica_{ \bx\trasp\wbthech+ \wsigma \, s<u\le \sigma_0 \, s } + \esp \;\indica_{\overline{\itC}} \;\indica_{ \sigma_0 \, s <u\le \bx\trasp\wbthech+ \wsigma \, s }\\ &\le & \esp \;\indica_{\itC} \;\left\{G_0( \, s )- G_0\left(\frac{\bx\trasp\wbthe + \wsigma \, s}{\sigma_0} \right)\right\} + \esp \;\indica_{\overline{\itC}} \;\left\{ G_0\left(\frac{\bx\trasp\wbthe + \wsigma \, s}{\sigma_0} \right)- G_0( \, s ) \right\} \\ &\le & \esp \left | G_0\left(\frac{\bx\trasp\wbthe + \wsigma \, s}{\sigma_0} \right)- G_0( \, s ) \right| \end{aligned}$$ and follows immediately from the continuity of $G_0$ and \[ass:consis\], concluding the proof of . Similar arguments allow to show that $$\label{eq:aprobar2} \frac{1}{n} \sum_{i=1}^n w_i \convpp h_\infty(t_0) =\left\{\begin{array}{lr} \esp_{G_0} w\left(\dfrac{\epsilon_1}{ \, t_0}\right) & \mbox{ if } t_0<\infty\\ 1 & \mbox{ if } t_0=\infty \end{array} \right.$$ and the desired result follows now easily combining and . Non–linear Model ---------------- [\[sec:appendix2\]]{} In this section, we assume that we have a random sample $(y_{1},\bx_{1}),\dots,(y_{ n},\bx_{n})$, where $\bx_i$ is a vector of $p$ explanatory variables and $y_i$ is a response variable that satisfy $$y_i= f(\bx_i , \bbe_0) + u_i= f(\bx_i , \bbe_0) +\sigma_0 \epsilon_i, i=1 \dots n \, ,$$ with $\bbe_0 \in \real^q$ and the errors $\epsilon_i$ are i.i.d. and independent of $\bx_i$ with unknown distribution $G_0(\cdot)$ and $\sigma_0$ is the scale parameter. As above, the residuals are defined using robust strongly consistent estimators of $\bbe_0$ and $\sigma_0$, let us say $\wbbe$ and $\wsigma$ as $$r_{i}= \dfrac{y_{i}-f(\bx_{i}, \wbbe) }{\wsigma} = \dfrac{u_{i}-\left[f(\bx_{i}, \wbbe)-f(\bx_i , \bbe_0)\right] }{\wsigma} \, .$$ We compute the adaptive weighted empirical distribution at point $t$ as in with $$w_i= w\left(\dfrac{ r_i }{t_n}\right)\,,$$ where as in Section \[sec:appendix1\], the adaptive cut–off values are defined through . The following additional assumptions are required to provide a general framework to deal with non–linear models. 1. \[ass:clasef\] The class of functions $$\itF=\{ f(\bx , \bbe)\,, \|\bbe-\bbe_0\|\le 1\}$$ with enveloppe $F\in L^1(P_{\bx})$ is such that $N_{[\;]}(\epsilon, \itF, L_1(P_{\bx}))<\infty$, where $P_{\bx}$ is the probability measure of $\bx$. 2. \[ass:denG0\] $G_0$ has a bounded density $g_0$. 3. \[ass:fcont\] $f(\bx,\bbe)$ is a continuous function of $\bbe$ for each $\bx$ and $F(\bx)=\sup_{\|\bbe-\bbe_0\|\le 1} f(\bx,\bbe)\in L^1(P_{\bx})$. It is worth noticing that Lemma 3.10 in van der Geer (2000) entails that \[ass:clasef\] holds if \[ass:fcont\] holds. Lemma \[lema:appendix2.1\] below is an intermediate result needed to derive Lemma \[lema:appendix2.2\] which is the non–linear counterpart of Lemma \[lema:appendix1.2\]. \[lema:appendix2.1\] Assume that \[ass:G0\], \[ass:clasef\] and \[ass:denG0\] hold. Denote $\itV_0=\{\bbe: \|\bbe-\bbe_0\|\le 1\}$ and $\itI_0=[\sigma_0/2, 2\,\sigma_0]$ and for any fixed $t\ge 0$ consider the family of functions $$\itH=\{h_{\bbech,\; \sigma}(y,\bx)=\indica_{|y-f(\bx , \bbech)|\le \sigma\, t} \mbox{ for } (\bbe, \sigma)\in \itV_0\times \itI_0\}\,.$$ Then, $ \sup_{h\in \itH} | P_n h- P h| \convpp 0$. <span style="font-variant:small-caps;">Proof.</span> First, note that $$h_{\bbech,\; \sigma}(y,\bx)=\indica_{|y-f(\bx , \bbech)|\le \sigma \; t}= h^{(1)}_{\bbech,\; \sigma}(y,\bx)\,h^{(2)}_{\bbech,\; \sigma}(y,\bx)\,,$$ where $$\begin{aligned} h^{(1)}_{\bbech,\; \sigma}(y,\bx) &=& \indica_{ y-f(\bx , \bbech) - \sigma \; t\le 0}\\ h^{(2)}_{\bbech,\; \sigma}(y,\bx) &=& \indica_{ 0\le y-f(\bx , \bbech) +\sigma \; t }\, . \end{aligned}$$ Denote as $\itH^{(j)}=\{h^{(j)}_{\bbech,\; \sigma}(y,\bx)\,, (\bbe, \sigma)\in \itV_0\times \itI_0\} $. Taking into account that $\itH\subset \itH^{(1)}\cdot \itH^{(2)}$ and that the functions $h^{(j)}_{\bbech,\; \sigma}$ are non–negative and bounded by 1, to show that $$\sup_{h\in \itH} | P_n h- P h| \convpp 0\,,$$ it will be enough to show that $N_{[\;]}(\epsilon, \itH^{(j)}, L_1(P ))<\infty$, for $j=1,2$, where $P$ is the probability measure of $(y,\bx)$. We will derive the result for $\itH^{(1)}$, the proof for $\itH^{(2)}$ been analogous. Let $\epsilon>0$ and denote $\delta=\sigma_0\epsilon/(2\,\|g_0\|_{\infty})$. Then, the fact that $\itI_0$ is compact entails that there exist $k\le 2\, t\, \sigma_0/\delta$ $\sigma_0/2=\sigma_1\le \dots\le\sigma_k=2\, \sigma_0$ such that $\sigma_j-\sigma_{j-1}\le \delta/ t $. Denote $M=N_{[\;]}(\delta, \itF, L_1(P_{\bx}))$, then there exists $\{(f_{j,L}, f_{j,U})\}_{1\le j\le M}$ such that, for any $f\in \itF$ there exists $j$ such that $f_{j,L} \le f\le f_{j,U} $ and $\esp f_{j,U}-f_{j,L} \le \delta$. Fix $\bbe\in \itV_0$ and $\sigma\in \itI_0$ and let $1\le j\le M$ and $1\le \ell \le k-1$, be such that $\sigma\in [\sigma_\ell, \sigma_{\ell+1}]$ and $f_{j,L}(\bx) \le f(\bx)\le f_{j,U}(\bx) $, for all $\bx$. Then, using that $t\ge 0$ we obtain that $$g_{\ell, j,L}(y,\bx)=y- f_{j,U}(\bx) - \sigma_{\ell+1} \; t \le y-f(\bx , \bbe) - \sigma \; t\le y-f_{j,L}(\bx) - \sigma_{\ell} \; t=g_{\ell, j, U}(y,\bx)\, ,$$ so that $$\indica_{g_{\ell, j,U}(y,\bx)\le 0}\le h^{(1)}_{\bbech,\; \sigma}(y,\bx)\le \indica_{g_{\ell, j,L}(y,\bx)\le 0}\,.$$ Denote $h_{\ell,j,L}=\indica_{g_{\ell, j,U}(y,\bx)\le 0}$ and $h_{\ell,j,U}=\indica_{g_{\ell, j,L}(y,\bx)\le 0}$. We will show that $\esp |h_{\ell,j,U}- h_{\ell,j,L}|<\epsilon$, that is, $\{(h_{\ell,j,L}, h_{\ell,j,U}\}_{1\le \ell \le M, 1\le j\le k}$ is an $\epsilon-$bracket for $\itH^{(1)}$, so $N_{[\;]}(\epsilon, \itH^{(j)}, L_1(P ))\le k M<\infty$. Using that $h_{\ell,j,L}\le h_{\ell,j,U}$, $g_{\ell, j,L}(y,\bx)\le g_{\ell, j,U}(y,\bx)$ and that $$g_{\ell, j,L}(y,\bx)=u+ f(\bx, \bbe_0)- f_{j,U}(\bx) - \sigma_{\ell+1} \; t\quad g_{\ell, j,U}(y,\bx)=u+ f(\bx, \bbe_0)- f_{j,L}(\bx) - \sigma_{\ell} \; t$$ we get that $$\begin{aligned} \esp |h_{\ell,j,U}- h_{\ell,j,L}| &=& \esp \indica_{g_{\ell, j,L}(y,\bx)\le 0}- \indica_{g_{\ell, j,U}(y,\bx)\le 0} = \prob \left(g_{\ell, j,L}(y,\bx)\le 0\right) -\prob\left( g_{\ell, j,U}(y,\bx)\le 0 \right)\\ &=& \prob \left(u\le f_{j,U}(\bx) + \sigma_{\ell+1} \; t- f(\bx, \bbe_0)\right)- \prob\left(u\le f_{j,L}(\bx) + \sigma_{\ell} \; t- f(\bx, \bbe_0)\right)\\ &=& \esp\left\{G_0\left(\frac{ f_{j,U}(\bx) + \sigma_{\ell+1} \; t- f(\bx, \bbe_0)}{\sigma_0}\right)-G_0\left(\frac{f_{j,L}(\bx) + \sigma_{\ell} \; t- f(\bx, \bbe_0)}{\sigma_0}\right)\right\} \end{aligned}$$ Thus, using that $G_0$ has a bounded density $g_0$, we obtain that $$\begin{aligned} \esp |h_{\ell,j,U}- h_{\ell,j,L}| &\le & \|g_0\|_{\infty}\esp\left\{ \left|\frac{ f_{j,U}(\bx) + \sigma_{\ell+1} \; t- f(\bx, \bbe_0)}{\sigma_0} -\frac{f_{j,L}(\bx) + \sigma_{\ell} \; t- f(\bx, \bbe_0)}{\sigma_0}\right|\right\}\\ &\le & \frac{\|g_0\|_{\infty}}{\sigma_0}\left\{ \left(\sigma_{\ell+1}- \sigma_{\ell} \right)\; t+ \esp \left| f_{j,U}(\bx) -f_{j,L}(\bx) \right|\right\}\le 2\delta \; \frac{\|g_0\|_{\infty}}{\sigma_0} =\epsilon\,, \end{aligned}$$ concluding the proof. 0.2in \[lema:appendix2.2\] Assume that \[ass:G0\], \[ass:consis\], \[ass:denG0\] and \[ass:fcont\] hold. Then, we have that 1. $\|G_n^{+}-G_0^{+}\|_{\infty}\convpp 0$, where $$G^{+}_n(t)= \frac 1n \sum_{i=1}^n I(|r_i| \le t) \quad \quad r_i= \dfrac{y_{i}-f(\bx_{i}, \wbbe) }{\wsigma}$$ and $G^{+}_0(t)$ is the distribution of the absolute errors when $\epsilon_i \sim G_0$ 2. $d_n\convpp d_0$. 3. $\ot_n\convpp \ot_0$. <span style="font-variant:small-caps;">Proof.</span> a) Using Lemma \[lema:appendix1.1\], it will be enough to show that for each fixed $t$ $$\label{eq:Gn+toG+} G_n^{+}(t)-G_0^{+}(t)\convpp 0\,.$$ Denote $\itV_0=\{\bbe: \|\bbe-\bbe_0\|\le 1\}$ and $\itI_0=[\sigma_0/2, 2\,\sigma_0]$. Let us consider the family of functions $$\itH=\{h_{\bbech,\; \sigma}(y,\bx)=\indica_{|y-f(\bx , \bbech)|\le \sigma\, t} \mbox{ for } (\bbe, \sigma)\in \itV_0\times \itI_0\}\,.$$ Using that \[ass:fcont\] implies \[ass:clasef\], Lemma \[lema:appendix2.1\] entails that $$\label{eq:glivenko2} \sup_{h\in \itH} | P_n h- P h| \convpp 0\,.$$ On the other hand, $G_n^{+}$ can be written as $$G_n^{+}(t) =P_n h_{\wbbech, \;\wsigma }(y_i,\bx_i)\,.$$ Hence, if we denote as $M(\bbe, \sigma)= P h_{\bbech,\; \sigma}$, using and the fact that \[ass:consis\] entails that with probability 1, for $n$ large enough, $(\wbbe,\wsigma)\in \itV_0\times \itI_0$, we conclude that $$\left|G_n^{+}(t) - M( \wbthech, \;\wsigma )\right| \convpp 0\,.$$ It remains to show that $$M( \wbthe, \;\wsigma )- G_0^{+}(t) \convpp 0\,.$$ Note that $$M(\bbe, \sigma)= P h_{\bbech,\; \sigma}= \prob(|y-f(\bx , \bbech)|\le \sigma\, t) \, ,$$ hence $$M(\bbe_0, \sigma_0 )= \prob( |u|\le \sigma_0 \, t)=G_0^{+}(t)\,.$$ Therefore, we have to show that $$M( \wbthe, \;\wsigma\,t)\convpp M(\bbe_0, \sigma_0)\,.$$ First observe that $$\begin{aligned} M(\bbe, \sigma)&=& \prob \left( -\sigma\, t + f(\bx,\bbe)- f(\bx,\bbe_0) \le u \le \sigma\, t+ f(\bx,\bbe)- f(\bx,\bbe_0) \right)\\ &=& \esp \left\{G_0(\sigma\, t+ f(\bx,\bbe)- f(\bx,\bbe_0))- G_0(-\sigma\, t + f(\bx,\bbe)- f(\bx,\bbe_0))\right\} \end{aligned}$$ The continuity of $G_0$ and $f(\bx, \bbe)$ and the Dominated Convergence Theorem entail that $M(\bbe, \sigma)$ is a continuous function of its arguments, which together with \[ass:consis\], entails that $M( \wbthe, \;\wsigma )- M(\bbe_0, \sigma_0 ) \convpp 0$, for each fixed $t$, concluding the proof of a). b\) and c) follow as in Lemma Lemma \[lema:appendix1.2\]. 0.2in As in Section \[sec:appendix1\], denote $\wnu_n= 1/(t_n\, \wsigma_n )$, $\nu_0= 1/(t_0\, \sigma_0 )$, where we understand that if $t_0=\infty$, $\nu_0=0$. Furthermore, let $\itJ_0$ be a compact interval with non–empty interior, such that $\nu_0\in \itJ_0$. Lemma \[lema:appendix2.3\] is the non–linear counterpart of Lemma \[lema:appendix1.3\]. Note that a bounded density is needed when a general non–linear model is considered, as well as a continuous weight function. \[lema:appendix2.3\] Under \[ass:peso\], \[ass:G0\], \[ass:denG0\] and \[ass:fcont\], we have that $\sup_{g\in \itG} | P_n g- P g| \convpp 0$, where $$\itG=\{g_{\bbech,\; \sigma,\; \nu}(y,\bx)=w\left(\nu({y -f(\bx,\bbe)})\right)\, \indica_{y-f(\bx , \bbech)\le\sigma\, t} \mbox{ for } (\bbe, \sigma, \nu)\in \itV_0\times \itI_0 \times \itJ_0\} \,.$$ <span style="font-variant:small-caps;">Proof.</span> Note that $\itG\subset \itG_1 \cdot \itG_2$ where $$\begin{aligned} \itG_1 &=& \{g_{\bbech,\, \nu}(y,\bx)=w\left(\nu(y -f(\bx,\bbe))\right)\, \mbox{ for } (\bbe, \nu)\in \itV_0\times \itJ_0\}\\ \itG_2 &=& \{g_{\bbech,\; \sigma }(y,\bx)= \indica_{y-f(\bx , \bbech)\le\sigma\, t} \mbox{ for } (\bbe, \sigma )\in \itV_0\times \itI_0 \}\,. \end{aligned}$$ The classes $\itG$, $\itG_1$ and $\itG_2$ have envelope 1 and are classes of non–negative functions, hence we have easily that, $$N_{[\;]}(2\,\epsilon, \itG, L_1(P))\le N_{[\;]}(\epsilon, \itG_1, L_1(P)) N_{[\;]}(\epsilon, \itG_2, L_1(P))\,,$$ so that to show $\sup_{g\in \itG} | P_n g- P g| \convpp 0$, it will be enough to prove that, for $j=1,2$, $$\label{eq:NP2} N_{[\;]}(\epsilon, \itG_j, L_1(P))<\infty\,.$$ Note that, when $j=2$, follows from the proof of Lemma \[lema:appendix2.1\]. On the other hand, the continuity of $w$ and \[ass:fcont\] entail that $w\left(\nu(y -f(\bx,\bbe))\right)$ is a continuous function of $(\nu, \bbe)$ for each $(y,\bx)$. Then, Lemma 3.10 in van der Geer (2000) entails that $ N_{[\;]}(\epsilon, \itG_1, L_1(P))<\infty$, concluding the proof. 0.2in \[prop:appendix2.1\] Under \[ass:peso\] to \[ass:consis\] and \[ass:denG0\] and \[ass:fcont\], we have that 1. if $t_0<\infty$, $$\sup_{s\in \real} \left|\wG_n(s)- \frac{h_0(t_0,s)}{h_{\infty}(t_0)}\right|\convpp 0\,,$$ with $h_\infty(t_0)$ and $h_0(t_0,s)$ defined in and , respectively. 2. if $t_0=\infty$, $\|\wG_n-G_0\|_{\infty}\convpp 0$. <span style="font-variant:small-caps;">Proof.</span> When $t_0=\infty$, using that $G_0$ is a bounded, monotone and continuous function and that $\wG_n$ is monotone, from Lemma \[lema:appendix1.1\], it will be enough to show that for each $s\in \real$, $\wG_n(s)\convpp G(s)$. On the other hand, when $t_0<\infty$, standard arguments allow to show that $F(s)= {h_1(t_0,s)}/{h_{\infty}(t_0)}$ is a bounded, monotone and continuous function of $s$ and the uniform convergence also follows from the pointwise one. Taking into account that $\wnu_n\convpp \nu_0$, we have that with probability 1, for $n$ large enough $\wnu_n\in \itJ_0$. As in the proof of Proposition \[prop:appendix1.1\], we will begin by showing that $$\label{eq:aprobar12} \frac{1}{n} \sum_{i=1}^n w_i I(r_i \le s) \convpp h_0(t_0,s) =\left\{\begin{array}{lr} \esp_{G_0} w\left(\dfrac{\epsilon_1}{ \, t_0}\right)\indica_{\epsilon_1\le s } & \mbox{ if } t_0<\infty\;,\\ \esp_{G_0} \indica_{\epsilon_1\le s }=G_0(s) & \mbox{ if } t_0=\infty \;. \end{array} \right.$$ For that purpose and noting that $r_i=(y_i-f(\bx, \wbbe))/\wsigma$, define the class of functions $$\itG=\{g_{\bbech,\; \sigma,\; \nu}(y,\bx)=w\left(\nu( y -f(\bx,\bbe))\right)\, \indica_{y-f(\bx , \bbech)\le\sigma\, s} \mbox{ for } (\bbe, \sigma, \nu)\in \itV_0\times \itI_0 \times \itJ_0\} \,.$$ Lemma \[lema:appendix2.3\] entails that $$\sup_{g\in \itG} | P_n g- P g| \convpp 0\,,$$ then, using that $$\frac{1}{n} \sum_{i=1}^n w_i I(r_i \le s)= P_n g_{\wbbech, \;\wsigma, \; \wnu_n}\;,$$ we obtain that $$\frac{1}{n} \sum_{i=1}^n w_i I(r_i \le s) - P g_{\wbbech, \;\wsigma, \; \wnu_n} \convpp 0\,.$$ It remains to show that $$P g_{\wbbech, \;\wsigma ,\;\wnu_n}\convpp P g_{\bbech_0,\, \sigma_0,\; \nu_0} = h_0(t_0,s)\,.$$ which will follow if we derive that $$\begin{aligned} A_n&=&P g_{\wbbech , \;\wsigma,\;\wnu_n }-\esp w\left(\nu_0 u \right)\, \indica_{y-f(\bx,\wbbech)\le \wsigma \, s} \convpp 0 \label{eq:An2}\\ B_n&=& \esp w\left(\nu_0 u \right)\, \indica_{y-f(\bx,\wbbech)\le \wsigma \, s} -h_0(t_0,s) \convpp 0 \;. \label{eq:Bn2} \end{aligned}$$ Noting that $$\begin{aligned} |A_n| &=& \left|\esp \left\{ w\left(\nu_n\left[y-f(\bx,\wbbe)\right]\right) - w\left(\nu_0\, u \right)\right\} \indica_{y-f(\bx,\wbbech)\le \wsigma \, s} \right|\\ &\le & \esp \left| w\left(\nu_n\left[y-f(\bx,\wbbe)\right]\right) - w\left(\nu_0\, u \right)\right| \;, \end{aligned}$$ using the Dominated Convergence Theorem, the continuity of $w$ and the fact that $\wnu_n\convpp \nu_0$ and $\wbthe\convpp 0$, we obtain that $A_n\convpp 0$, concluding the proof of . To derive , using that $0\le w(x)\le 1$, we get that $$\begin{aligned} | B_n|&=& \left|\esp w\left(\nu_0 u \right)\, \indica_{y-f(\bx,\wbbech)\le \wsigma \, s} - \esp w\left(\nu_0\, u\right)\indica_{u\le \sigma_0 \, s }\right|\\ &\le & \esp \left|\indica_{u\le f(\bx,\wbbech)-f(\bx,\bbech_0)+ \wsigma \, s} -\indica_{u\le \sigma_0 \, s }\right|\;. \end{aligned}$$ As in the proof of Proposition \[prop:appendix1.1\], we have that, if $f(\bx,\wbbe )-f(\bx,\bbe_0)+ \wsigma \, s \le \sigma_0 \, s $, then $\Delta(u)=\indica_{u\le \bx\trasp\wbthech+ \wsigma \, s} -\indica_{u\le \sigma_0 \, s }= 1$ if and only if $ f(\bx,\wbbe )-f(\bx,\bbe_0)+ \wsigma \, s < u\le \sigma_0 \, s$. On the other hand, if $f(\bx,\wbbe )-f(\bx,\bbe_0)+ \wsigma \, s \ge \sigma_0 \, s $, then $\Delta(u)=1$ if and only if $ \sigma_0 \, s < u\le f(\bx,\wbbe )-f(\bx,\bbe_0)+ \wsigma \, s$. Note that the fact that $\wbbe\convpp 0$ and $\wsigma\convpp \sigma_0$ together with the continuity of $f(\bx, \bbe)$ entails that $f(\bx,\wbbe )-f(\bx,\bbe_0)+ \wsigma \, s\convpp \sigma_0 \, s $, for each $\bx$. Let $\itC=\{\bx: f(\bx,\wbbe )-f(\bx,\bbe_0)+ \wsigma \, s \le \sigma_0 \, s\}$ and $\overline{\itC}$ its complement, then $$\begin{aligned} | B_n| &\le & \esp \;\indica_{\itC} \;\indica_{f(\bx,\wbbech)-f(\bx,\bbech_0)+ \wsigma \, s <u\le \sigma_0 \, s } + \esp \;\indica_{\overline{\itC}} \;\indica_{ \sigma_0 \, s <u\le f(\bx,\wbbech)-f(\bx,\bbech_0)+ \wsigma \, s }\\ &\le & \esp \left | G_0\left(\frac{f(\bx,\wbbe )-f(\bx,\bbe_0)+ \wsigma \, s}{\sigma_0} \right)- G_0( \, s ) \right| \end{aligned}$$ and follows immediately from the continuity of $G_0$ and \[ass:consis\], concluding the proof of . Similar arguments allow to show that $$\label{eq:aprobar22} \frac{1}{n} \sum_{i=1}^n w_i \convpp h_\infty(t_0) =\left\{\begin{array}{lr} \esp_{G_0} w\left(\dfrac{\epsilon_1}{ \, t_0}\right) & \mbox{ if } t_0<\infty\\ 1 & \mbox{ if } t_0=\infty \end{array} \right.$$ and the desired result follows now easily combining and . References {#references .unnumbered} ========== Alonzo, T. A. and Pepe, M. S. (2002). Distribution-free ROC analysis using binary regression techniques. *Biostatistics*, **3**, 421-432. Bianco, A. M. and Spano, P. (2019). Robust inference for nonlinear regression models. *Test*, **28**, 369-398. Cai, T. (2004). Semiparametric ROC regression analysis with placement values. *Bio statistics*, **5**, 45-60. Inácio de Carvalho, V., Jara, A., Hanson, T. E. and de Carvalho, M. (2013). Bayesian nonparametric ROC regression modeling, *Bayesian Analysis*, **8**, 623-646. Faraggi, D. (2003). Adjusting receiver operating characteristic curves and related indices for covariates. *Journal of the Royal Statistical Society*, Ser. D, **52**, 1152-1174. Farcomeni, A. and Ventura, L. (2012). An overview of robust methods in medical research. *Statistical Methods in Medical Research*, **21**, 111-133. Gervini, D. and Yohai, V. J. (2002). A class of robust and fully efficient regression estimators. *Annals of Statistics*, **30**, 583-616. Goncalves, L., Subtil, A., Oliveira, M. R. and Bermudez, P. (2014) Roc Curve Estimation: An Overview. *REVSTAT-Statistical Journal*, **12**, 1-20. González-Manteiga, W., Pardo-Fernández, J. C., and Van Keilegom, I. (2011). ROC curves in non-parametric location-scale regression models.*Scandinavian Journal of Statistics*, **38**, 169-184. Greco, L. and Ventura, L. (2011). Robust inference for the stress-strength reliability. *Statistical Papers*, **52**, 773-788. Kosorok, M. (2008). *Introduction to Empirical Processes and Semiparametric Inference*. Springer–Verlag, New York. Krzanowski, W. J. and Hand, D. J. (2009). *ROC curves for continuous data*. Chapman and Hall/CRC, Boca Raton. Pardo-Fernández, J. C., Rodríguez-Alvarez, M. X. and Van Keilegom, I. (2014). A review on ROC curves in the presence of covariates. *REVSTAT Statistical Journal*, **12**, 21-41. Pepe, M. S. (1997). A regression modelling framework for receiver operating characteristic curves in medical diagnostic testing. *Biometrika*, **84**, 595-608. Pepe, M. S. (1998). Three approaches to regression analysis of receiver operating characteristic curves for continuous test results. *Biometrics*, **54**, 124-135. Pepe, M. S. (2003). *The Statistical Evaluation of Medical Tests for Classification and Prediction*, Oxford University Press, New York. Rodríguez-Alvarez, M. X., Roca-Pardiñas, J., and Cadarso-Suárez, C. (2011a). ROC curve and covariates: extending the induced methodology to the non-parametric framework. *Statistics and Computing*, **21**, 483-495. Rodríguez-Alvarez, M. X., Tahoces, P. C., Cadarso-Suárez, C., and Lado, M. J. (2011b). Comparative study of ROC regression techniques—applications for the computer-aided diagnostic system in breast cancer detection. *Computational Statistics and Data Analysis*, **55**, 888-902. Sun, Y. and Genton, M. G. (2011). Functional boxplots. *Journal of Computational and Graphical Statistics*, **20**, 316-334. Van de Geer, S. (2000). *Empirical Processes in M–Estimation*, Cambridge University Press, United States of America. van der Vaart, A. and Wellner, J. (1996). *Weak Convergence and Empirical Processes. With Applications to Statistics*. Springer–Verlag, New York. Walsh, S. J. (1997). Limitations to the robustness of binormal ROC curves: effects of model misspecification and location of decision thresholds on bias, precision, size and power, *Statistics in Medicine*, **16**, 669-679. Yohai, V. J. (1987). High Breakdown-Point and High Efficiency Robust Estimates for Regression. *Annals of Statistics*, **15**, 642-656.

--- abstract: 'We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields.' address: ' [ American Institute of Mathematicsfarmer@aimath.orgDepartment of MathematicsYonsei Universityhaseo@yonsei.ac.kr]{} ' author: - 'David W. Farmer and Haseo Ki' title: 'Landau-Siegel zeros and zeros of the derivative of the Riemann zeta function' --- [^1] Introduction ============ The spacing between zeros of the Riemann zeta-function and the location of zeros of the derivative of the zeta-function are closely related problems which have connections to other topics in number theory. For example, if the zeta-function had a large number of pairs of zeros that were separated by less than half their average spacing, one would obtain an effective lower bound on the class numbers of imaginary quadratic fields [@M; @CI]. Also, Speiser proved that the Riemann hypothesis is equivalent to the assertion that the nontrivial zeros of the derivative of the zeta-function, $\zeta'$, are to the right of the critical line [@Sp]. There is a quantitative version of Speiser’s theorem [@LM] which is the basis for Levinson’s method [@L]. In Levinson’s method there is a loss caused by the zeros of $\zeta'$ which are close to the critical line, so it would be helpful to understand the horizontal distribution of zeros of $\zeta'$. The intuition is that the spacing of zeros of the zeta-function should determine the horizontal distribution of zeros of the derivative. Specifically, a pair of closely spaced zeros of $\zeta(s)$ gives rise to a zero of $\zeta'(s)$ close to the critical line. Our main result is a partial converse, showing that *sufficiently many* zeros of $\zeta'(s)$ close to the $\tfrac12$-line implies the existence of many closely spaced zeros of $\zeta(s)$. See Theorem \[t:la\]. We assume the Riemann hypothesis and write the zeros of $\zeta$ as $\rho_j=\tfrac12+i\gamma_j$ and the zeros of $\zeta'$ as $\beta_j'+i\gamma_j'$, where in both cases we list the zeros by increasing imaginary part. We consider the normalized gaps between zeros of $\zeta$ and the normalized distance of $\rho_j'$ to the right of the critical line, given by $$\begin{aligned} \label{def:lambdas} \lambda_j=\mathstrut &(\gamma_{j+1}-\gamma_j)\log\gamma_j \cr \lambda_j'=\mathstrut & (\beta_j'-\tfrac12) \log\gamma_j'.\end{aligned}$$ We are interested in how small the normalized gaps can be, and how small the normalized distance to the critical line can be, so we set $$\begin{aligned} \lambda=\mathstrut &\liminf_{j\to\infty} \lambda_j\\ \lambda'=\mathstrut &\liminf_{j\to\infty}\lambda_j' .\end{aligned}$$ We also consider the cumulative densities of $\lambda_j$ and $\lambda_j'$, given by $$\begin{aligned} m(\nu) =\mathstrut & \liminf_{J\to\infty} \frac{1}{J} \, \#\{j\le J\ :\ \lambda_j \le \nu\}\cr m'(\nu) =\mathstrut & \liminf_{J\to\infty} \frac{1}{J}\, \#\{j\le J\ :\ \lambda_j' \le \nu\}.\end{aligned}$$ Soundararajan’s [@S] Conjecture B states that $\lambda=0$ if and only if $\lambda'=0$. This amounts to conjecturing that zeros of $\zeta'(s)$ close to the $\tfrac12$-line can only arise from a pair of closely spaced zeros of $\zeta(s)$. Zhang [@Z] showed that (on RH) $\lambda=0$ implies $\lambda'=0$. Thus, Soundararajan’s conjecture is almost certainly true because $\lambda=0$ follows from standard conjectures on the zeros of the zeta-function, based on random matrix theory. However, the second author[@K] showed that $\lambda=0$ and $\lambda'=0$ are not logically equivalent. Specifically, Ki[@K] proved \[thm:ki\] (Haseo Ki [@K]) Assuming RH, $\lambda' >0$ is equivalent to $$\label{eqn:zetacondition} M(\gamma_j):= \sum_{0<|\gamma_j-\gamma_n|<1} \frac{1}{\gamma_j-\gamma_n} =O(\log \gamma_j) .$$ Note that the theorem implies Zhang’s result (that $\lambda=0$ implies $\lambda'=0$), because if $\lambda=0$ then for some $j$ the sum in will be large because an individual term in the sum is large. But that is not the only way for $M(\gamma_j)$ to be large. It is possible that there could be an imbalance in the distribution of zeros, such as a very large gap between neighboring zeros, which makes the sum large because many small terms have the same sign. For example, suppose there were consecutive zeros of the zeta function with a gap of size 1, followed by $c \log T$ zeros equally spaced (this cannot happen, but we are illustrating a point). Then $M(\gamma)$ would be $\gg \log T \log\log T$. That possibility is the reason attempts to prove $\lambda'=0$ implies $\lambda=0$ have been unsuccessful. For example, Garaev and Y[i]{}ld[i]{}r[i]{}m [@GY] required the stronger assumption $\lambda_J'(\log\log \gamma_J')^2=o(1)$ in order to conclude $\lambda_J=o(1)$. The discussion in the previous paragraph shows that, without detailed knowledge of the distribution of zero spacings, one requires $ M(\gamma)\ge C \log T \log\log T$ for any $C>0$ in order to conclude $\lambda =0$. It is possible that this could be improved by proving results about the rigidity of the spacing between zeros of the zeta function. Random matrix theory could give a clue about the limits of this approach. This would involve finding the expected maximum of the random matrix analogue of the sum $$\label{eqn:zetairregular} \sum_{\frac{1}{\log \gamma_j}<|\gamma_j-\gamma_n|<1} \frac{1}{\gamma_j-\gamma_n}.$$ Unfortunately, the necessary random matrix calculation may be quite difficult because a lower bound on $|\gamma_j-\gamma_n|$ requires the exclusion of a varying number of intervening zeros, so the combinatorics of the random matrix calculation may be intricate. In this paper we consider not $\lambda$ and $\lambda'$, but the density functions $m(\nu)$ and $m'(\nu)$. In the next section we illustrate this with the example described above, and then we state our main result. Examples with equally spaced zeros {#sec:pictures} ---------------------------------- We illustrate Theorem \[thm:ki\] with examples which can help build intuition for why $\lambda'=0$ does not imply $\lambda=0$. Our example involves degree $N$ polynomials with all zeros on the unit circle. In other words, characteristic polynomials of matrices in the unitary group $U(N)$. In these examples. $\lambda>0$ but $\lambda'=0$, where $\lambda$ and $\lambda'$ refer respectively to the large $N$ limits of the normalized gap between zeros, and the rescaled distance between zeros of the derivative and the unit circle. This is the random matrix analogue of $\lambda$ and $\lambda'$ for the zeta function. Figure \[fig:1a\] illustrates the case of 16 zeros in the interval$\{e^{i\theta}\ :\ 0\le \theta\le \pi/2\}$. The plot on the left shows the zeros of the polynomial and its derivative. The figure on the right is the same plot “unrolled”: the horizontal axis is the argument, and the vertical axis is the distance from the unit circle, rescaled by a constant factor. \[0.7\][![On the left, the zeros and the zeros of the derivative of a degree 16 polynomial having all zeros in $\frac14$ of the unit circle. On the right, the image of those zeros under the mapping $r e^{i \theta} \mapsto (\theta, 2 \pi \cdot 16(1-r))$. Zeros of the function are shown as small squares, and zeros of the derivative as small dots. []{data-label="fig:1a"}](plot1a.eps "fig:")]{} 0.5in \[0.7\][![On the left, the zeros and the zeros of the derivative of a degree 16 polynomial having all zeros in $\frac14$ of the unit circle. On the right, the image of those zeros under the mapping $r e^{i \theta} \mapsto (\theta, 2 \pi \cdot 16(1-r))$. Zeros of the function are shown as small squares, and zeros of the derivative as small dots. []{data-label="fig:1a"}](plot1aT.eps "fig:")]{} Figure \[fig:1b1c\] is the analogue of the plot on the right side of Figure \[fig:1a\], for 101 zeros and 501 zeros. Note that in these examples $\lambda\sim \pi/2$. \[0.7\][![Unrolled and rescaled zeros of the derivative of a polynomial with zeros equally spaced along the arc $\{e^{i\theta}\ :\ 0\le \theta\le \pi/2\}$. The polynomial has degree 101 (left) and 501 (right). []{data-label="fig:1b1c"}](plot1bT.eps "fig:")]{} 0.5in \[0.7\][![Unrolled and rescaled zeros of the derivative of a polynomial with zeros equally spaced along the arc $\{e^{i\theta}\ :\ 0\le \theta\le \pi/2\}$. The polynomial has degree 101 (left) and 501 (right). []{data-label="fig:1b1c"}](plot1cT.eps "fig:")]{} In Figure \[fig:1b1c\] the vertical scales are stretched by a factor of $2 \pi N (1-r)$ where $N=101$ and $501$, respectively. Figures \[fig:1a\] and \[fig:1b1c\] illustrate that, with this unrolling and rescaling, the zeros of the derivative approach a circle. We see that even though $\lambda>0$ we have $\lambda'=0$, but furthermore, since the zeros lie on a (rescaled) circle, we have $m'(\nu)\gg \nu^2$ as $\nu\to 0$. Thus, we can have $m'(\nu)>0$ for all $\nu>0$, yet $m(\nu)=0$ for $\nu$ sufficiently small. We believe that the above example is the limit of this behavior, and we make the following conjecture, which we view as a refinement of Soundararajan’s conjecture. \[conj:conjecture1\] If $m'(\nu) \gg \nu^\alpha$ for some $\alpha<2$, then $m(\nu)>0$ for all $\nu>0$. We intend this as a general conjecture, applying to the Riemann zeta function but also to other cases such as a sequence of polynomials with all zeros on the unit circle. For applications to lower bounds of class numbers [@M; @CI] one does not actually need $m(\nu)>0$ for $\nu < \pi $; it is sufficient to show that a relatively small number of gaps between zeros of the zeta function are small. Our main result, Theorem \[t:la\], obtains such bounds from estimates on the zeros of the derivative of the zeta function. Denote $ \log_{(2)}t=\log\log t $. \[t:la\] Assume RH. Suppose that for all $\nu>0$, $$\#\{0<\gamma'<T:\left(\beta'-\tfrac{1}{2}\right)\log\gamma'\le\nu\}\geqslant e^{-C(\nu)}T\log T\qquad(\nu>0,\,T\to\infty),$$ as $T\to \infty$, where $C(\nu)>0$ for $\nu>0$ with $\lim_{\nu\to0^+}\sqrt{\nu}C(\nu)=0$ and $\lim_{\nu\to0^+}C(\nu)=\infty$. Then $$\label{eqn:mainresult} \liminf_{T\to\infty}\frac{\#\{\gamma_n\le T:(\gamma_{n+1}-\gamma_n)\log\gamma_n\le\nu\}}{T\log T/\logg T}>0 $$ for all $\nu>0$. The conclusion of the theorem is weaker than $m(\nu)>0$ for $\nu>0$, but only by a factor of $\logg T$. Thus, it is more than sufficient to apply the results of Conrey and Iwaniec [@CI]. In particular, Theorem \[t:la\] shows that it is possible to obtain lower bounds for class numbers of imaginary quadratic fields from knowledge of the density of zeros of the derivative of the Riemann zeta function. There is an apparent discrepancy between Conjecture \[conj:conjecture1\] and Theorem \[t:la\] which we wish to clarify. In Theorem \[t:la\] we allow exponential decrease of $m'(\nu)$ as $\nu\to 0$. While the conclusion of the theorem is weaker than $m(\nu)>0$ by a factor of $\logg T$, it may seem curious that the condition in Conjecture \[conj:conjecture1\] requires $m'(\nu)$ to be relatively large as $\nu\to 0$. Indeed, the examples in Section \[sec:pictures\] show that the condition in Conjecture \[conj:conjecture1\] cannot be improved for general functions. The reason for the apparent inconsistency is that, as described in Section \[sec:boundMgammac\], our method relies on a bound on the moments of the logarithmic derivative. For the Riemann zeta function one expects $$\label{eqn:zpzbound} \int_{T}^{2T} \left| \frac{\zeta'}{\zeta}\left(\frac12 + \frac{1}{\log T} + i t\right) \right|^{2k} dt \ll_k T\log^{2k}T.$$ The bound should follow by the method of Selberg [@Sel], although we give a conditional proof that allows us to explicitly determine the implied constant. Such a bound, for one fixed $k$, would establish a weaker version of Theorem \[t:la\] that required $m'(\nu)\gg \nu^{-2 k}$. However, more general functions like the polynomials in Section \[sec:pictures\] do not satisfy an analogous bound to . In fact, they are very large on the unit circle and do not satisfy the analogue of the Lindelöf hypothesis. Conjecture \[conj:conjecture1\] is intended to cover those more general cases, while stronger statements should be true for the zeta function. It is interesting to speculate on the precise nature of the function $m'(\nu)$ for the Riemann zeta function. Dueñez [*et. al.*]{} [@Due] give a detailed analysis of the relationship between small gaps between zeros of the zeta function (and analogously for zeros of the characteristic polynomial of a random unitary matrix) and the zeros of the derivative which arise from the small gaps. For the case of the Riemann zeta function they indicate that the random matrix conjectures for the zeros of the zeta function should imply $$m_\zeta'(\nu) \sim \frac{8}{9\pi} \nu^{\frac{3}{2}},$$ as conjectured by Mezzadri [@Mez03]. That calculation is based on a more general result which suggests that if $m(\nu)\sim \kappa \nu^\beta$ then $m'(\nu) \sim \kappa' \nu^{\beta/2}$ where $$\kappa'=2\pi \frac{\kappa}{\beta} \left(\frac{2}{\pi}\right)^\beta.$$ The factor of $2\pi$ comes from a different normalization used in [@Due] and here we work with the cumulative distribution functions $m$ and $m'$, while in [@Due] they use density functions. That derivation assumed that zeros of $\zeta'$ close to the $\tfrac12$-line only arise from closely spaced zeros of the zeta-function. The discussion above shows that, without further knowledge of the zeros, this is not a valid assumption. But, as indicated in our Conjecture \[conj:conjecture1\], if $\beta<4$ then we believe that the almost all zeros close to the $\tfrac12$-line do arise in such a manner. The random matrix prediction for the neighbor spacing of zeros of the zeta-function has $\kappa=\pi/6$ and $\beta=3$, which is covered by Conjecture \[conj:conjecture1\]. So our results support the analysis of Dueñez [*et. al.*]{} [@Due]. The remainder of this paper is devoted to the proof of Theorem \[t:la\]. Proof of Theorem \[t:la\] ========================= Theorem \[t:la\] says that sufficiently many zeros of $\zeta'$ close to the $\frac12$-line can only arise from closely spaced zeros of the zeta-function. If $\rho'=\beta'+i\gamma'$ is a zero of $\zeta'$, then we denote by $\rho_c=\frac12+i\gamma_c$ the zero of the zeta-function which is closest to $\rho'$. Thus, we must show that if there are many $\beta'$ very close to $\tfrac12$, then often there is another zero of the zeta-function close to $\gamma_c$. Our approach involves a study of the quantity $$M_{\gamma_c}=\sum_{0<|\gamma-\gamma_c|\le X(\gamma_c)} \frac{1}{\gamma-\gamma'},$$ where the range in the sum, $X(\gamma_c)$, turns out to be a limiting factor in our method. By analogy to a similar quantity studied in [@K], we expect that $M_{\gamma_c}$ should be large if and only if $\beta'-\frac12$ is small. And just like in [@K], there are two ways that $M_{\gamma_c}$ can be large. There could be an individual term which is large. That would happen if $\gamma'$ was near two $\gamma$s that are very close together. Or there could be a large imbalance in the the distribution of the $\gamma$s, for example if there was an unusually large gap between $\gamma_c$ and one of the adjacent zeros. We must show that the second possibility cannot occur too often. This is accomplished by showing that an imbalance in the distribution of zeros causes the zeta function to be large, and bounds on moments of the zeta function show that this cannot happen too often. The proof involves two steps. Assume the zeros of the zeta function rarely get close together. First we show that if $\beta'-\frac12$ is small then $M_{\gamma_c}$ is large. Second, we show that if $M_{\gamma_c}$ is large then usually $\frac{\zeta'}{\zeta}(s)$ is large near $\tfrac12+i\gamma'$, subject to our assumption that the zeros of the zeta function rarely get close together. Standard bounds for the moments of $\frac{\zeta'}{\zeta}(\sigma+i t)$ let us conclude that $\beta'-\frac12$ cannot be small too often, which is what we wanted to prove. The relationship between $M_{\gamma_c}$ and $\zeta'/\zeta$ relies on an estimate for $\zeta'/\zeta$ in terms of a short sum over zeros. Suppose we have $$\label{eqn:logderiv1} \frac{\zeta'}{\zeta}(s) = \sum_{|\gamma-t|< X(T)} \frac{1}{s-\rho} + O(\log T).$$ On RH, with $X(t)=1/\logg T$ the above holds for all $t$ [@Ti]. Using this, instead of our  below, leads to a weaker version of Theorem \[t:la\], where the $\logg T$ in the denominator of is replaced by $\log T$. We prove the following strengthening of , but only near almost all $\gamma$. \[p:logdea\] Assume RH. Let $m_0$ be a positive integer. If $C^*>1$ is sufficiently large, then the number of $\gamma_n<T$ such that $$\label{e:logdea} \frac{\zeta'}{\zeta}(s)= \sum_{|\gamma-t|\le \frac{C^*\logg\gamma}{\log\gamma}}\frac{1}{s-\rho} + O(\log\gamma_n)$$ for $s=1/2+1/\log\gamma_n+it$ with $t\geqslant10$ and $|\gamma_n-t|\le A/\log\gamma_n$ is $$\frac{T}{2\pi}\log T+O\left(\frac{T}{(\log T)^{m_0}}\right)$$ as $T\to\infty$. The proof of Proposition \[p:logdea\] is in Section \[sec:logdea\]. Restricting to zeros with special properties -------------------------------------------- We begin the proof of Theorem \[t:la\]. The lemmas in this section show that, in the context of the proof of Theorem \[t:la\], we only have to deal with zeros that are well spaced. Suppose, for the purposes of contradiction, that there exists $\epsilon>0$ so that $$\label{eqn:nosmallgaps} \liminf_{T\to\infty}\frac{\#\{\gamma_n\le T:\gamma_{n+1}-\gamma_n\le\epsilon/\log\gamma_n\}}{T\log T/\logg T}=0.$$ Then, we can find a sequence $\langle T_l\rangle$ such that $T_1$ is sufficiently large, $T_l\to\infty$ and $$\#\{\gamma_n\le T_l:\gamma_{n+1}-\gamma_n\le\epsilon/\log\gamma_n\}=o\left(T_l\log T_l/\logg T_l\right)$$ as $l\to\infty$. We set $$T=T_l. $$ The following lemma shows that we can restrict our attention to those zeros whose immediate neighbors are well spaced. \[lem:density\] Let $K=4C^*\left[\logg T\right]$. Under assumption  we have $$\label{e:density} \# \{\gamma_n<T:0<|m|\le K,\,|\gamma_{n+m}-\gamma_{n+m-1}|\geqslant\frac{\epsilon}{2\log\gamma_n}\}=\frac{T}{2\pi}\log T(1+o(1)).$$ For each $m=\pm1,\pm2,\ldots$, let $$A_m=\{\gamma_n<T:|\gamma_{n+m}-\gamma_{n+m-1}|\geqslant\frac{\epsilon}{2\log\gamma_n}\}.$$ Here, we exclude the case $n+m\le1$. By assumption  have $$\#(A_m)=\frac{T}{2\pi}\log T+o\left(\frac{T\log T}{\logg T}\right)$$ for $0<|m|\le\log T$. We see that $$\aligned \#\left(\bigcap_{0<|m|\le K}A_m\right)=&\sum_{0<|m|\le K}\#(A_m)-\sum_{\substack{-K\le m<K\\ m\not=0}}\#\left(A_m\cup\bigcap_{\substack{m<l\le K\\ l\not=0}}A_l\right)\\ \geqslant&2K\frac{T}{2\pi}\log T+o\left(\frac{KT\log T}{\logg T}\right)-(2K-1)\frac{T}{2\pi}\log T+O(K\log T)\\ =&\frac{T}{2\pi}\log T+o(T\log T). \endaligned$$ The next Proposition shows that we can restrict to intervals where the number of zeros is close to its average. Fix $C^*>1$, let $l_1$ and $l_2$ be integers, and for $\tfrac12+i\gamma$ a zero of the zeta function set $$N(\gamma,l_1,l_2)= N\left(\gamma+\frac{l_2C^*\logg\gamma}{\log\gamma}\right) -N\left(\gamma+\frac{l_1C^*\logg\gamma}{\log\gamma}\right) -\frac{(l_2-l_1)C^*\logg\gamma}{2\pi}$$ Using an argument in [@Iv], we get the following. \[p:dnumber\] Let $m_0>0$. There exists $C>0$ such that the number of $\gamma_n<T$ with $$N(\gamma_n,l_1,l_2)\le C\logg T$$ is $$\frac{T}{2\pi}\log T+O\left(\frac{T}{(\log T)^{m_0}}\right)\qquad(T\to\infty),$$ provided that $|l_1|,|l_2|\le\log T/(C^*\logg T)$ and $0<l_2-l_1\le2\log T/(C^*\logg T)$. The proof of Proposition \[p:dnumber\] is in Section \[sec:dnumber\]. Lower bound for $M_{\gamma_c}$ ------------------------------ Let $\beta'+i\gamma'$ be a zero of $\zeta'$, and (assuming RH) let $\tfrac12+i\gamma_c$ be the zero of the zeta function which is closest to $i\gamma'$. If there are two closest zeros, choose the one nearer to the origin. We will use the above lemmas to give a lower bound for $M_{\gamma_c}$, assuming $\beta'-\tfrac12$ is small. Let $Z(T)$ be the set of $\gamma_c < T$ which satisfy the following three conditions: $$\begin{aligned} \gamma_c\in\mathstrut & \{\gamma_n<T : 0<|m|\le K,\, |\gamma_{n+m}-\gamma_{n+m-1}|\geqslant \frac{\epsilon}{2\log\gamma_n}\}; \label{eqn:gammacset}\\ N(\gamma_c,l_1,l_2)\le\mathstrut & C\logg T\qquad \left(-\frac{\log T}{C^*\logg T}\le l_1<l_2\le\frac{\log T}{C^*\logg T}\right); \label{eqn:gammacN}\\ \frac{\zeta'}{\zeta}(s)=\mathstrut & \sum_{|\gamma-t|\le\frac{\logg\gamma}{\log\gamma}} \frac{1}{s-\rho} \label{eqn:gammaczpz} + O(\log \gamma_c),\end{aligned}$$ where $s=1/2+1/\log\gamma_c+it$ and $|\gamma_c-t|\le A/\log\gamma_c$. By the lemmas in the previous section, as $T\to\infty$ the set $Z(T)$ contains $\sim \frac{1}{2\pi}T\log T$ elements. For the remainder of the proof we will assume $\gamma_c\in Z(T)$. Recall Titchmarsh [@Ti], Theorem 9.6(A): $$\label{eqn:zpzeasy} \frac{\zeta'}{\zeta}(s)=-\frac{1}{2}\log t+O(1)+\sum_{\rho}\left(\frac{1}{s-\rho}-\frac{1}{\rho}\right),$$ uniformly for $t\geqslant10$ and $-1\le\sigma\le2$. Let $\beta'+i\gamma'$ be a zero of $\zeta'(s)$ where $0<\gamma'<T$ is sufficiently large. Taking the real part we have $$\label{e:gprime} \frac{1}{2}\log\gamma'+O(1) = \frac{\beta'-\frac{1}{2}} {\left(\beta'-\frac{1}{2}\right)^2+(\gamma'-\gamma_c)^2} + \sum_{\gamma\not=\gamma_c} \frac{\beta'-\frac{1}{2}} {\left(\beta'-\frac{1}{2}\right)^2+(\gamma'-\gamma)^2}.$$ There are three cases to consider. [**Case 1.**]{} $\beta'-1/2>|\gamma'-\gamma_c|$. Then, by (\[e:gprime\]), we get $$\frac{1}{2}\log\gamma'\geqslant\frac{1}{2\left(\beta'-\frac{1}{2}\right)}.$$ Thus, we have $\beta'-1/2 \gg 1/\log\gamma'$. [**Case 2.**]{} $\beta'-1/2\le|\gamma'-\gamma_c|$ and $|\gamma'-\gamma_c|>\delta(\epsilon)/\log\gamma'$, where $\delta(\epsilon)=8/\epsilon^2$. By , , and , we have $$\aligned \frac{1}{2}\log\gamma'\ll&\left(\beta'-\frac{1}{2}\right)\log^2\gamma'+\sum_{m=1}^{\infty} \frac{\beta'-\frac{1}{2}}{\left(\frac{m\epsilon}{\log\gamma'}\right)^2}+\sum_{m=0}^{\infty} \frac{\left(\beta'-\frac{1}{2}\right)\logg\gamma'} {\left(\frac{\logg\gamma'}{\log\gamma'}\right)^2+\left(\frac{m\logg\gamma'}{\log\gamma'}\right)^2}\\ \ll&\left(\beta'-\frac{1}{2}\right)\log^2\gamma' \endaligned$$ and so again we have $$\beta'-\frac{1}{2}\gg\frac{1}{\log\gamma'}.$$ Here the implied constants depend only on $\epsilon$. [**Case 3.**]{} $\beta'-1/2\le|\gamma'-\gamma_c|$ and $|\gamma'-\gamma_c|\le\delta(\epsilon)/\log\gamma'$. Using , , and , as in Case 2, we get $$\frac{1}{2}\log\gamma'\geqslant\frac{\beta'-\frac{1}{2}}{2(\gamma'-\gamma_c)^2}$$ $$\frac{1}{2}\log\gamma'\ll\frac{\beta'-\frac{1}{2}}{(\gamma'-\gamma_c)^2}+\left(\beta'-\frac{1}{2}\right)\log^2\gamma'\\ \ll\frac{\beta'-\frac{1}{2}}{(\gamma'-\gamma_c)^2}.$$ Thus we have $$\label{eqn:twosided} (\gamma'-\gamma_c)^2\log\gamma'\ll \beta'-\frac{1}{2}\ll(\gamma'-\gamma_c)^2\log\gamma'.$$ Here the implied constants depend only on $\epsilon$. By  and the conditions of Case 3 we have $$\label{eqn:nosum} \frac{\gamma_c-\gamma'}{\left(\beta'-\frac{1}{2}\right)^2+(\gamma'-\gamma_c)^2} -\frac{1}{\gamma_c-\gamma'}=O(\log\gamma').$$ Now take the imaginary part of  to get $$\label{eqn:sum1} \sum_{0<|\gamma-\gamma_c| \le \frac{C^*\logg\gamma_c}{\log\gamma_c}} \frac{\gamma-\gamma'}{\left(\beta'-\frac{1}{2}\right)^2+(\gamma'-\gamma)^2} +\frac{\gamma_c-\gamma'} {\left(\beta'-\frac{1}{2}\right)^2+(\gamma'-\gamma_c)^2} = O\left(\log\gamma'\right).$$ Finally, by we have $$\begin{aligned} \label{eqn:Mcsum} \sum_{0<|\gamma-\gamma_c|\le \frac{C^*\logg\gamma_c}{\log\gamma_c}} \frac{\gamma-\gamma'}{\left(\beta'-\frac{1}{2}\right)^2 +(\gamma'-\gamma)^2}-M_{\gamma_c} =\mathstrut& \sum_{k=1}^{\infty}\frac{\left(\beta'-\frac{1}{2}\right)^2}{\left(\frac{\epsilon k}{\log\gamma'}\right)^3}\\ =\mathstrut&O(\log\gamma'),\end{aligned}$$ where $$M_{\gamma_c}=\sum_{0<|\gamma-\gamma_c|\le \frac{C^*\logg\gamma_c}{\log\gamma_c}}\frac{1}{\gamma-\gamma'}.$$ By combining , , , and , we have $$O(\log\gamma')=M_{\gamma_c}+\frac{1}{\gamma_c-\gamma'}=M_{\gamma_c} + A_{\gamma_c} \sqrt{\frac{\log\gamma'}{\beta'-\frac{1}{2}}},$$ where $1\ll A_{\gamma_c}\ll 1 $, with the implied constants depending only on $\epsilon$. Let $\nu$ be a positive number. Suppose that $$\left(\beta'-\frac{1}{2}\right)\log\gamma'\le\nu.$$ Then, for sufficiently small $\nu$, we see that only Case 3 is possible for sufficiently large $\gamma'$, namely we have $$M_{\gamma_c}+ A_{\gamma_c} \sqrt{\frac{\log\gamma'}{\beta'-\frac{1}{2}}} =O(\log \gamma').$$ By this, the assumption in Theorem \[t:la\], and the fact that $\#Z(T)\sim \frac{1}{2\pi} T\log T$, we have $$e^{-C(\nu)}\le\frac{1}{\frac{T}{2\pi}\log T}\#\{0<\gamma'<T: \gamma_c\in Z(T) \text{ and } |M_{\gamma_c}|\gg \frac{\log\gamma'}{\sqrt{\nu}}(1+O(\sqrt{\nu}))\}.$$ By the last inequality we have $$\label{e:nui} \frac{e^{-C(\nu)}\log^{2k}T}{\nu^{k}}(1+O(\sqrt{\nu}))^{2k}\frac{T}{2\pi}\log T\ll\sum_{\substack{\frac{T}{\log T}\le\gamma'\le T\\ \left(\beta'-\frac{1}{2}\right)\log\gamma'\le\nu\\ \gamma_c\in Z(T)}} |M_{\gamma_c}|^{2k}.$$ In the next section we describe upper bounds for the moments of $M_{\gamma_c}$. This will contradict  and complete the proof of Theorem \[t:la\]. Bounding the moments of $M_{\gamma_c}$ {#sec:boundMgammac} -------------------------------------- We obtain an upper bound on $M_{\gamma_c}$ from a bound on moments of the logarithmic derivative of the zeta function. This makes use of that fact that, assuming the zeros of the zeta function do not get close together, the logarithmic derivative can be approximated either by a short sum over zeros, or by a short Dirichlet series. \[l:moment\] Assume RH. Let $\gamma'<T$ such that $|\gamma'-\gamma_c|\le\delta(\epsilon)/\log\gamma'$ and assume – . Then $$\frac{M_{\gamma_c}}{i}+\frac{\zeta'}{\zeta}\left(\frac{1}{2}+\frac{1}{\log T}+it\right)=O_\epsilon (\log T),$$ for $|t-\gamma'|\le A/\log\gamma'$. By the assumptions we have $$\begin{aligned} \frac{M_{\gamma_c}}{i}+\frac{\zeta'}{\zeta}\left(\frac{1}{2}+\frac{1}{\log T}+it\right) =\mathstrut&\frac{M_{\gamma_c}}{i}+\sum_{0<|\gamma-\gamma_c|\le \frac{C^*\logg\gamma_c}{\log\gamma_c}}\frac{1}{\frac{1}{\log T}+i(t-\gamma)} +O(\log T)\\ =\mathstrut&\sum_{0<|\gamma-\gamma_c|\le \frac{C^*\logg\gamma_c}{\log\gamma_c}} \frac{(\frac{1}{\log T}+i (t-\gamma))} {(\gamma-\gamma')(\frac{1}{\log T}+i(t-\gamma))} +O(\log T)\\ \ll\mathstrut &\sum_{m=1}^{\infty} \frac{\frac{1+\delta(\epsilon)}{\log\gamma'}} {\left(\frac{m\epsilon}{\log\gamma'}\right)^2}+O(\log T)\\ =\mathstrut &O_\epsilon (\log T).\end{aligned}$$ \[lem:zpzapprox\] Assume RH and – . Let $s=\frac12+\frac{1}{\log T}+i t$ with $|t|\le T$, and let $x=T^{1/100 k}$. Then if $|\gamma'-\gamma_c|\le\delta(\epsilon)/\log\gamma'$ and $|t-\gamma'|\le \epsilon /\log\gamma'$, we have $$\frac{\zeta'}{\zeta}(s)=-\sum_{n<x^2}\frac{\Lambda_x(n)}{n^s}+O_\epsilon(k\log T),$$ where $$\Lambda_x(n)= \begin{cases} \Lambda(n) & 1\le n\le x \cr \Lambda(n)\frac{\log(\frac{x^2}{n})}{\log x} & x\le n\le x^2 \end{cases} .$$ By [@Ti], Theorem 14.20, $$\begin{aligned} \frac{\zeta'}{\zeta}(s)=\mathstrut &-\sum_{n<x^2}\frac{\Lambda_x(n)}{n^s}+\frac{x^{2(1-s)}-x^{1-s}}{(1-s)^2\log x}\cr &+ \frac{1}{\log x}\sum_{q=1}^{\infty}\frac{x^{-2q-s}-x^{-2(2q+s)}}{(2q+s)^2}+\frac{1}{\log x}\sum_{\rho}\frac{x^{\rho-s}-x^{2(\rho-s)}}{(s-\rho)^2}.\end{aligned}$$ The assumptions on the zero spacings give the claimed bound on the terms involving the zeros. \[lem:sound\] (Soundararajan, Lemma 3 of [@So1]) Let $T$ be large, and let $2\le x\le T$. Let $k$ be a natural number such that $x^k\le T/\log T$. For any complex numbers $a(p)$ we have $$\int_T^{2T}\left|\sum_{p\le x}\frac{a(p)}{p^{\frac{1}{2}+it}}\right|^{2k}dt\ll k!\, T \left(\sum_{p\le x} \frac{|a(p)|^2}{p}\right)^{k} ,$$ where the sum is over the primes. We assemble the above lemmas to bound the moments of $M_{\gamma_c}$. By Lemma \[l:moment\] and Lemma \[lem:zpzapprox\], with $A=A_\epsilon$ a constant depending only on $\epsilon$, which may be different in each inequality, we have $$\begin{aligned} \label{eqn:Mgammacbound} |M_{\gamma_c}|^{2k} \ll \mathstrut & A^{2k} \log^{2k}T + 2^{2k} \left| \frac{\zeta'}{\zeta}\left(\frac12 + \frac{1}{\log T} + i t\right) \right|^{2k} \cr \ll \mathstrut & A^{2k} k^{2k} \log^{2k}T + 2^{2k} \left|\sum_{n<x^2} \frac{\Lambda_x(n)}{n^{\frac12 + \frac{1}{\log T} + i t}}\right|^{2k}\cr \ll \mathstrut & A^{2k} k^{2k} \log^{2k}T + 2^{2k} \left|\sum_{p<x^2} \frac{\Lambda_x(p)}{p^{\frac12 + \frac{1}{\log T} + i t}}\right|^{2k},\end{aligned}$$ where $x=T^{1/100k}$, for $|t-\gamma'|\le \delta(\epsilon)/\log\gamma'$, provided $\gamma_c$ satisfies – . That is, provided $\gamma_c\in Z(T)$. Integrating inequality  over the set $$\left\{T/{\log T} <t < T\ :\ |t-\gamma_c|< \delta(\epsilon)/\log T \text{ for some } \gamma_c\in Z(T) \right\}$$ and then using Lemma \[lem:sound\] we get $$\begin{aligned} \frac{\delta(\epsilon)}{\log T} \sum_{\substack{\frac{T}{\log T}\le\gamma'\le T\\ \left(\beta'-\frac{1}{2}\right)\log\gamma'\le\nu \\ \gamma_c\in Z(T)}} |M_{\gamma_c}|^{2k} \ll \mathstrut & A^{2k} k^{2k} T\log^{2k}T + 2^{2k} \int_{\frac{T}{\log T}}^T \left| \sum_{p<x^2} \frac{\Lambda_x(p)}{p^{\frac12 + \frac{1}{\log T} + i t}} \right|^{2k}dt\cr \ll \mathstrut& A^{2k} k^{2k} T\log^{2k}T + 2^{2 k} k!\, T \left( \sum_{p<x^2} \frac{\Lambda_x(p)^2}{p^{1 + \frac{2}{\log T} }} \right)^{k} \cr \ll \mathstrut& A^{2k} k^{2k} T\log^{2k}T.\end{aligned}$$ The last step used $\Lambda_x(p)\le \Lambda(p)$ and the fact that $$\sum_{p\le x}\frac{\Lambda(p)^2}{p} \ll \log^2 x,$$ which is a weak form of the prime number theorem. Rearranging the above inequality and combining with (\[e:nui\]), we have $$\frac{e^{-C(\nu)}}{\nu^k}(1+O(\sqrt{\mathstrut\nu}))^{2k}T\log^{2k+1}T \ll A^{2k} k^{2k} T\log^{2k+1}T,$$ which rearranges to give $$\left(1+O(\sqrt{\mathstrut\nu})\right)^{2k} \ll A^{2k} k^{2k} \nu^k e^{C(\nu)}.$$ Letting $k=[1/\sqrt{A^2 e\nu}]$, we have a contradiction if $\sqrt{\mathstrut\nu}C(\nu) \to 0$ as as $\nu\to 0$. This completes the proof of Theorem \[t:la\]. Proofs of technical results =========================== In this section we provide the proofs of Proposition \[p:logdea\] and Proposition \[p:dnumber\]. Proof of Proposition \[p:dnumber\] {#sec:dnumber} ---------------------------------- A special case of the Proposition is the following: There exists $C_1>0$ such that the number of $\gamma_n<T$ satisfying $$N\left(\gamma_n+\frac{lC^*\logg T}{\log T}\right)-N\left(\gamma_n+\frac{(lC^*-1)\logg T}{\log T}\right)\le C_1\logg T,$$ for all $|l|\le\log T/(C^*\logg T)$, is $$\frac{T}{2\pi}\log T+O\left(\frac{T}{(\log T)^{m_0}}\right).$$ Here $C_1$ is not depending on $C^*$. The proof of Claim follows easily from the same method below. Thus, we omit the proof of it. From now on, we are assuming that $\gamma_n$ satisfies $T/(\log T)^{m_0+1}<\gamma_n<T$ and Claim. We recall $$\begin{aligned} \int_T^{T+H}|S(t+h)-S(t)|^{2k}dt=\mathstrut&\frac{H(2k)!}{(2\pi^2)^kk!}\log^k(2+h\log T)\\ &+O\left(H(ck)^k\left(k+\log^{k-1/2}(2+h\log T)\right)\right)\end{aligned}$$ uniformly for $T^a<H\le T$, $a>1/2$, $0<h<1$ and any positive integer $k$, where $c$ is a positive constant and $S(t)=\frac{1}{\pi}\arg\zeta(1/2+it)$. For this, see [@Ts Theorem 4]. Thus we have $$\label{e:tsang} \int_0^T|S(t+h)-S(t)|^{2k}dt\ll T\left(Ak\right)^{2k},$$ where $\log(2+h\log T)\ll k$. We note that $$\label{e:ost} S(t+h)-S(t)=N(t+h)-N(t)-\frac{h}{2\pi}\log t+O\left(\frac{h^2+1}{t}\right),$$ where $N(t)$ is the number of zeros of $\zeta(s)$ in $0<\Im s<t$. By this, we have $$\widetilde{S}(t,l_1,l_2)=N\left(t+\frac{(l_2-l_1)C^*\logg T}{\log T}\right)-N(t)-\frac{(l_2-l_1)C^*\logg T}{2\pi}+O\left(\frac{1}{t}\right),$$ where $$\widetilde{S}(t,l_1,l_2)=S\left(t+\frac{(l_2-l_1)C^*\logg T}{\log T}\right)-S(t)$$ Using Claim, the last formula and (\[e:ost\]), we have $$\begin{aligned} N(n,l_1,l_2)\le\mathstrut &\left|\widetilde{S}(t,l_1,l_2)\right|+\logg T\\ \le\mathstrut &\left|\widetilde{S}(t-h,l_1,l_2)\right|+3\logg T\\ &\mathstrut +\sum_{j=1}^{2}N\left(\gamma_n+\frac{l_jC^*\logg T}{\log T}\right)-N\left(\gamma_n+\frac{(l_jC^*-1)\logg T}{\log T}\right)\\ \le\mathstrut &C_2\logg T+\left|\widetilde{S}(t-h,l_1,l_2)\right|\end{aligned}$$ for $t=\gamma_n+l_1C^*\logg T/\log T$ and $0\le h\le\logg T/\log T$, where $C_2=\max\{2C_1+3,A\}$. Using this, we have $$\begin{aligned} \sum_{\substack{\frac{T}{(\log T)^{m_0+1}}<\gamma_n<T\\ N(n,l_1,l_2)\geqslant C\logg T}}(C\logg T)^{2k} \mathstrut & \mathstrut \frac{\logg T}{\log T}\\ \ll \mathstrut &T\log T(2C_2\logg T)^{2k}+\sum_{\gamma_n<T}\int_{\gamma_n+\frac{(l_1C^*-1)\logg T}{\log T}}^{\gamma_n+\frac{l_1C^*\logg T}{\log T}}\left|2\widetilde{S}(t,l_1,l_2)\right|^{2k}dt\\ \ll \mathstrut & T\log T(2C_2\logg T)^{2k} +\log T\int_0^T\left|2\widetilde{S}(t,l_1,l_2)\right|^{2k}dt\\ \ll \mathstrut &T\log T\left((2C_2\logg T)^{2k}+(2C_2k)^{2k}\right)\end{aligned}$$ for any sufficiently large $T$ and any $|l_1|,|l_2|\le\log T/(C^*\logg T)$ with $0<l_2-l_1\le 2\log T/(C^*\logg T)$. We put $$k=[\logg T]\qquad\text{and}\qquad C=e^{m_0+2}(2C_2+1).$$ By these and the last inequality, we have $$\sum_{\substack{|l_1|,|l_2|\le\frac{\log T}{C^*\logg T}\\0<l_2-l_1\le\frac{2\log T}{C^*\logg T}}} \sum_{\substack{\frac{T}{(\log T)^{m_0+1}}<\gamma_n<T\\ N(n,l_1,l_2)\geqslant C\logg T}}1\ll\frac{T(\log T)^4(2C_2\logg T)^{2k}}{\left(C\logg T\right)^{2k}}\ll\frac{T}{(\log T)^{m_0}}.$$ We complete the proof of Proposition \[p:dnumber\]. Proof of Proposition \[p:logdea\] {#sec:logdea} --------------------------------- We recall $$\frac{\zeta'}{\zeta}(s)=O(\log t)+\sum_{|\gamma-t|\le1}\frac{1}{s-\rho}$$ holds uniformly for $t>1$ and $-2\le{\mathbb R}s\le1$. For this, see [@Ti Theorem 9.6 (A)]. Using the last formula, it suffices to show that the number of $\gamma_n$ in $T/(\log T)^{m_0+1}\le\gamma_n<T$ such that $\gamma_n$ satisfies the condition in Proposition \[p:dnumber\] and $$\sum_{\frac{C^*\logg T}{\log T}<|\gamma_n-\gamma_m|\le1}\frac{1}{\gamma_n-\gamma_m}=O(\log T)$$ is $$\frac{T}{2\pi}\log T+O\left(\frac{T}{(\log T)^{m_0}}\right)\qquad(T\to\infty),$$ because for $s=1/2+1/\log T+it$ and $|\gamma_n-t|\le A/\log T$, we have $$\sum_{\frac{C^*\logg T}{\log T}<|\gamma_n-\gamma_m|\le1}\frac{1}{s-\rho}-\frac{1}{i(\gamma_n-\gamma_m)}=O\left(\frac{1}{\log T}\sum_{m=1}^{\infty}\frac{\logg T}{\left(\frac{m\logg T}{\log T}\right)^2}\right)=O(\log T).$$ We recall that Proposition \[p:dnumber\] implies $$N\left(\gamma_n+\frac{lC^*\logg T}{\log T}\right) =N(\gamma_n)+\frac{lC^*\logg T}{2\pi}+O\left(\logg T\right)$$ for any integer $l$ with $|l|\le\log T/(C^*\logg T)$. This immediately implies that for a sufficiently large $C^*>1$, we have $$\max_{0\le k\le N}|2\gamma_n-\gamma_{m_2+k}-\gamma_{m_1-k}|\frac{\log T}{2\pi}=O(\logg T),$$ where $\gamma_{m_1}$ is the greatest one in $[\gamma_n-1,\gamma_n-C^*\logg T/\log T)$, $\gamma_{m_1}$ the least one in $(\gamma_n+C^*\logg T/\log T,\gamma_n+1]$ and $N$ the largest positive integer such that $\gamma_{m_1-N}$ and $\gamma_{m_2+N}$ belong to $[\gamma_n-C^*\logg T/\log T,\gamma_n+C^*\logg T/\log T]$. By this and putting $ M(n)=\max_{0\le k\le N}|2\gamma_n-\gamma_{m_2+k}-\gamma_{m_1-k}|,$ we have $$M(n)\ll\frac{\logg T}{\log T}$$ Using this and Proposition \[p:dnumber\] and the fact [@Ti Theorems 9.3 and 14.13] that the number of zeros between $t$ and $t+1$ is $$\frac{\log t}{2\pi}+O\left(\frac{\log t}{\logg t}\right)\qquad(\text{as } t\to\infty),$$ we have $$\begin{aligned} \sum_{\frac{C^*\logg T}{\log T}<|\gamma_n-\gamma_m|\le1}\frac{1}{\gamma_n-\gamma_m}=\mathstrut&\sum_{0\le k\le N}\frac{2\gamma_n-\gamma_{m_2+k}-\gamma_{m_1-k}}{(\gamma_n-\gamma_{m_2+k})(\gamma_n-\gamma_{m_1-k})}+O(\log T)\\ =\mathstrut&O\left(M(n)\sum_{k=1}^{\infty}\frac{\logg T}{\left(\frac{k\logg T}{\log T}\right)^2}\right)+O(\log T)\\ =\mathstrut&O(\log T).\end{aligned}$$ Thus, we complete the proof of Proposition \[p:logdea\]. [99]{} J.B. Conrey and H. Iwaniec, *Spacing of zeros of Hecke $L$-functions and the class number problem*, Acta Arith. 103 (2002) no. 3, 259-312. E. Dueñez, D.W. Farmer, S. Froehlich, C. P. Hughes, F. Mezzadri, and T. Phan, *Roots of the derivative of the Riemann zeta function and of characteristic polynomials*, preprint. D.W. Farmer and R. Rhoades, *Differentiation evens out zero spacing*, Trans. Amer. Math. Soc., Vol 37, No. 9, 2005, p3789-3811. arXiv:math.NT/0310252 M.Z. Garaev and C.Y. Y[i]{}ld[i]{}r[i]{}m, *On small distances between ordinates of zeros of $\zeta(s)$ and $\zeta'(s)$*, Int. Math. Res. Notices [**2007**]{} (2007) Art. ID rnm091, 14 pp. A. Iv[i]{}ć, *On small values of the Riemann zeta-function on the critical line and gaps between zeros*, Lietuvos Mat. Rinkinys [**42**]{} (2002), 31-45 H. Ki, *The zeros of the derivative of the Riemann zeta function near the critical line*, Int Math Res Notices (2008) Vol. 2008, Art. ID rnn064, 23 pp. arXiv:0701726 Levinson N. *More than one third of zeros of Riemann’s zeta function are on $\sigma = 1/2$*, Adv. Math. [**13**]{}, 383–436 (1974). N. Levinson and H. Montgomery, *Zeros of the derivatives of the Riemann zeta-function*, Acta Math. 133 (1974), 49-65. Mezzadri F. Random matrix theory and the zeros of $\zeta'(s)$. *J. Phys. A: Math. Gen.* **36**, 2945–2962 (2003). H. Montgomery, *The pair correlation of zeros of the zeta function*, Proc. Sympos. Pure Math. 24, Amer. Math. Soc., Providence, R. I. (1973), 181–193. A. Selberg, On the normal density of primes in small intervals,and the difference between consecutive primes, Arch. Math. Naturvid. 47, No. 6 (1943), 87–105. Soundararajan, *The horizontal distribution of zeros of $\zeta'(s)$*, Duke Math. J. (1998) [**91**]{}, no 1, 33-59. Soundararajan, *Moments of the Riemann zeta-function*, 11 pp., to appear in Annals of Math. arXiv:0612106. Speiser A. *Geometrisches zur Riemannschen Zetafunktion*, Math. Ann. [**110**]{} 514–21 (1934). E. C. Titchmarsh, *The Theory of the Riemann Zeta-function*, 2nd ed., revised by D. R. Heath-Brown, Oxford University Press, Oxford, 1986 K.-M. Tsang, *Some $\Omega$-theorems for the Riemann zeta-function*, Acta Arith. [**46**]{} (1986), 369–395 Y. Zhang, *On the zeros of $\zeta'(s)$ near the critical line*, Duke Math. J. (2001) [**110**]{}, No. 3 555-572. [^1]: Research of the first author supported by the American Institute of Mathematics, and the National Science Foundation. Research of the second author supported by the Korea Research Foundation Grant funded by the Korean Government(MOEHRD, Basic Research Promotion Fund)(KRF-2008-313-C00009).

--- abstract: 'We present new mid-infrared $N$-band spectroscopy and $Q$-band photometry of the local luminous infrared galaxy NGC 1614, one of the most extreme nearby starbursts. We analyze the mid-IR properties of the nucleus (central 150pc) and four regions of the bright circumnuclear (diameter$\sim 600$pc) star-forming (SF) ring of this object. The nucleus differs from the circumnuclear SF ring by having a [strong 8–12 continuum]{} (low 11.3 PAH equivalent width). These characteristics, together with the nuclear X-ray and sub-mm properties, can be explained by an X-ray weak active galactic nucleus (AGN), or by peculiar SF with a short molecular gas depletion time and producing an enhanced radiation field density. In either case, the nuclear luminosity ($L_{\rm IR}<$6$\times$10$^{43}$ergs$^{-1}$) is only $<$5% of the total bolometric luminosity of NGC 1614. So this possible AGN does not dominate the energy output in this object. We also compare three star-formation rate (SFR) tracers (Pa$\alpha$, 11.3 PAH, and 24 emissions) at 150pc scales [in the circumnuclear ring]{}. In general, we find that the SFR is underestimated (overestimated) by a factor of 2–4 (2–3) using the 11.3 PAH (24) emission with respect to the extinction corrected Pa$\alpha$ SFR. The former can be explained because we do not include diffuse PAH emission in our measurements, while the latter might indicate that the dust temperature is particularly warmer in the central regions of NGC 1614.' author: - | \ $^{1}$Centro de Astrobiología (CSIC/INTA), Ctra de Torrejón a Ajalvir, km 4, 28850, Torrejón de Ardoz, Madrid, Spain\ $^{2}$ASTRO-UAM, UAM, Unidad Asociada CSIC\ $^{3}$Instituto de Física de Cantabria, CSIC-Universidad de Cantabria, 39005 Santander, Spain\ $^{4}$Observatorio Astronómico Nacional (OAN-IGN)-Observatorio de Madrid, Alfonso XII, 3, 28014, Madrid, Spain\ $^{5}$Núcleo de Astronomía de la Facultad de Ingeniería, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago, Chile\ $^{6}$Instituto de Astrofísica de Andalucía, Glorieta de las Astronomía, s/n, 18008 Granada, Spain\ $^{7}$Centro de Radioastronomía y Astrofísica (CRyA-UNAM), 3-72 (Xangari), 8701, Morelia, Mexico\ $^{8}$Subaru Telescope, 650 North A’ohoku Place, Hilo, Hawaii, 96720, U.S.A.\ $^{9}$Gemini Observatory, Casilla 603, La Serena, Chile\ $^{10}$Centro de Estudios de la Física del Cosmos de Aragón, 44001 Teruel, Spain\ $^{11}$Instituto de Astrofísica de Canarias, Vía Láctea s/n, 38205 La Laguna, Tenerife, Spain title: 'Sub-arcsec mid-IR observations of NGC 1614: Nuclear star-formation or an intrinsically X-ray weak AGN?' --- \[firstpage\] galaxies: active – galaxies: nuclei – galaxies: starburst – galaxies: individual: NGC 1614 – infrared: galaxies Introduction {#s:intro} ============ Ultra-luminous and luminous infrared galaxies (U/LIRGs) are objects with infrared (IR) luminosities ($ L_{\rm IR}$) between 10$^{11}$ and 10$^{12}$ (LIRGs) and $>$10$^{12}$  (ULIRGs). Locally, objects with such high IR luminosities are unusual. However, between $z\sim 1$ and 2, galaxies in the LIRG and ULIRG luminosity ranges dominate the star-formation rate (SFR) density of the Universe [@PerezGonzalez2005; @LeFloch2005; @Caputi2007; @Magnelli2011]. Therefore, the study at high-angular resolution of local LIRGs provides a unique insight into extreme SF environments similar to those of high-$z$ galaxies near the SFR density peak of the Universe [@Madau2014]. NGC 1614 (Mrk 617) is the second most luminous galaxy within 75Mpc ($\log L_{\rm IR}=11.6$; @SandersRBGS) and according to optical spectroscopy its nuclear activity is classified as composite [@Yuan2010]. It is an advanced minor merger (3:1–5:1 mass ratio; @Vaisanen2012) located at 64Mpc (310pcarcsec$^{-1}$) with long tidal tails. Its bolometric luminosity is dominated by a strong starburst in the central kpc [@AAH01; @Imanishi2010], and, so far, there is no clear evidence of an active galactic nucleus (AGN) in NGC 1614 [@Herrero-Illana2014]. The central kpc of NGC 1614 contains a compact nucleus (45-80pc), which dominates the near-IR continuum emission, and a bright circumnuclear SF ring (diameter$\sim600$pc), which is predominant in Pa$\alpha$ [@AAH01] and other SF indicators like the polycyclic aromatic hydrocarbon (PAH) emission [@DiazSantos2008; @Vaisanen2012], cold molecular gas [@Konig2013; @Sliwa2014; @Xu2015], and radio continuum [@Olsson2010; @Herrero-Illana2014]. In addition, @GarciaBurillo2015 found a massive cold molecular gas outflow (3$\times$10$^7$$M_\odot$; $\dot{M}_{\rm out}\sim$40$M_{\odot}$yr$^{-1}$) which can be powered by the SF in the ring. A bright obscured AGN is discarded by X-ray observations [@Pereira2011; @Herrero-Illana2014]. However, previous mid-IR $N$-band imaging of NGC 1614 showed that the compact nucleus has a relatively high surface brightness [@Soifer2001; @DiazSantos2008; @Siebenmorgen2008]. Therefore, these observations suggest an enhanced mid-IR luminosity to SFR (as inferred from the observed Pa$\alpha$ luminosity) ratio in the nucleus [@DiazSantos2008], which might indicate the presence of an active nucleus. However, without high angular resolution spectroscopy no detailed studies were possible. {width="\textwidth"} In this paper we present the first high-angular resolution ($\sim$05 ) $N$-band (7.5–13) spectroscopy of the nucleus and surrounding star-forming ring of NGC 1614, as well as $Q$-band 24.5 imaging using CanariCam on the 10.4m Gran Telescopio CANARIAS (GTC). First, we describe the new observations in Section \[s:data\]. The extraction of the spectra and photometry, and a simple two component modeling are presented in Section \[s:analysis\]. We explore the AGN or SF nature of the nucleus in Section \[ss:agn\_vs\_sf\], and, in Section \[ss:sfr\_tracers\], the reliability of several SFR tracers at 150pc scales is discussed. The main conclusions are presented in Section \[s:conclusions\]. Throughout this paper we assume the following cosmology $H_{\rm 0} = 70$kms$^{-1}$Mpc$^{-1}$, $\Omega_{\rm m}=0.3$, and $\Omega_{\rm \Lambda}=0.7$ and the @Kroupa2001 IMF. Observations and Data Reduction {#s:data} =============================== Mid-IR Imaging -------------- We obtained $Q$-band diffraction limited (05) images of NGC 1614 using the Q8 filter ($\lambda_{\rm c}=24.5$, width at 50% cut-on/off of $\Delta\lambda = 0.8$) of CanariCam (CC; @Telesco2003CC) on the 10.4m GTC during December 2nd 2014. These observations are part of the ESO/GTC large program 182.B-2005 (PI Alonso-Herrero). The plate scale of CC is 008/pixel and its field of view is 26$\times$19, so it covers the central 6kpc of NGC 1614. Three exposures were taken with an on-source integration of 400s each. To reduce the data we used the <span style="font-variant:small-caps;">redcan</span> pipeline [@GonzalezMartin2013RedCan]. It performs the flat-fielding, stacking, and flux calibration of the individual exposures. The three reduced images were then combined after correcting the different background levels (right panel of Figure \[fig:maps\]). For the flux calibration the standard star HD 28749 was observed. It is relatively weak at 24.5 (1.2Jy; @Cohen1999) so the absolute calibration error of our $Q$-band observations is $\sim$20%. To check the flux calibration we also compared the integrated flux of NGC 1614 in our 24.5 image (6.0$\pm$0.9Jy) with the [*Spitzer*]{}/MIPS 24 flux (5.7$\pm$0.3Jy; @Pereira2015not). Both values are in good agreement. In addition, $N$-band imaging of this galaxy was previously obtained using Gemini/T-ReCS in the Si2 filter ($\lambda_{\rm c}=8.7$, $\Delta\lambda = 0.8$). This image was published by @DiazSantos2008 and it is shown in the middle panel of Figure \[fig:maps\]. The angular resolution of this observation estimated from the calibration star image is 04. Mid-IR Spectroscopy ------------------- We obtained $N$-band spectroscopy (7.5–13) of NGC 1614 with GTC/CanariCam on September 8th 2013 and January 5th 2014. The low spectral resolution ($R\sim175$) grating was used. These observations are also part of the ESO/GTC large program 182.B-2005. The nucleus of NGC 1614 was observed with a slit of 052 width using two perpendicular orientations (position angles 0 and 90[$^\circ$]{}). The approximate location of the slits is overplotted in the middle panel of Figure \[fig:maps\]. The on-source integration time for each of the slit orientations was 1200s. The standard star HD 28749 was observed in spectroscopy mode to provide the absolute flux calibration and telluric correction. From the two-dimensional spectrum of the standard star we derive that the angular resolution, $\sim$05, is approximately constant with the wavelength both nights. That is, the spectroscopy was not obtained in diffraction limited conditions. The data were reduced using the <span style="font-variant:small-caps;">redcan</span> pipeline. Flat-fielding, stacking, wavelength calibration, and flux calibration of the exposures are performed by this software. The spectra were extracted using a custom procedure (see Section \[ss:image\_modeling\]) instead of the default <span style="font-variant:small-caps;">redcan</span> extraction. Analysis and Results {#s:analysis} ==================== Image modeling and spectral extraction {#ss:image_modeling} -------------------------------------- The *HST*/NICMOS Pa$\alpha$ image of NGC 1614 (@AAH01, see Fig 1) revealed that the angular separation between the nucleus and the star-forming ring is 05–07, which is comparable to the angular resolution of the CC Q-band imaging and N-band spectroscopy. Therefore, to disentangle the emission produced by the different regions we modeled the mid-IR image with the highest resolution (i.e., the 8.7 T-ReCS image) with <span style="font-variant:small-caps;">galfit</span> [@Peng2010GALFIT]. @Imanishi2011 published $Q$-band imaging of NGC 1614 at 17.7. In this image, the emissions from the SF ring and the nucleus are not as clearly separated as in the CC 24.5 image, probably due to the slightly worse angular resolution (07; @Asmus2014). We used six Gaussian spatial components (nucleus, north, south, east, west, and diffuse) convolved with the PSF, to reproduce the 8.7 image (see Figure \[fig:galfit\]). These components are motivated by the Pa$\alpha$ morphology (Figure \[fig:maps\]) and is the minimum number of components needed to reproduce the mid-IR images. The position and full width half maximum (FWHM) of these components are listed in Table \[tab:components\]. According to this decomposition, the ring is located $\sim$06 away from the nucleus (190pc) and has a FWHM of $\sim$05–07 (160–220pc). The residuals of the model are less than 20% (Figure \[fig:galfit\]). To extract the fluxes from the CC Q-band image, we used <span style="font-variant:small-caps;">galfit</span> fixing the relative positions and widths of these components, but allowing their intensities to vary (see Figure \[fig:galfit\] and Table \[tab:fluxes\]). ![<span style="font-variant:small-caps;">galfit</span> models of the T-ReCS 8.7 (top panels), and CanariCam 24.5 (bottom panels) observations of the nuclear regions of NGC 1614. The observed image, the best model and the residuals are shown in the left, middle, and right panels, respectively. The color scale is in Jyarcsec$^{-2}$ units. \[fig:galfit\]](NGC1614_galfit.pdf){width="50.00000%"} --------- ------------------ ---------- -- -- -- -- -- -- -- -- Region $d^a$ FWHM$^b$ (arcsec) (arcsec) Nucleus [  $\cdots$  ]{} 0.21 Diffuse [  $\cdots$  ]{} 2.4 N 0.52 0.73 S 0.59 0.72 E 0.60 0.54 W 0.61 0.45 --------- ------------------ ---------- -- -- -- -- -- -- -- -- : Spatial decomposition nuclear region and circumnuclear ring of star formation of NGC 1614\[tab:components\] **Notes:** $^{(a)}$ Angular distance between the nucleus and the component. $^{(b)}$ Deconvolved FWHM. Similarly, we used this information to extract the CC N-band spectra. For each wavelength we generated a synthetic image taking into account the CC N-band PSF, and then we simulated the two slit orientations (P.A. 0 and 90[$^\circ$]{}) to obtain the one-dimensional spatial profiles. We varied the intensities of the different regions to reproduce simultaneously the observed N-S and E-W profiles at each wavelength. The resulting spectra are plotted in Figure \[fig:cc\_spec\]. The fluxes at 10 and 12, and the 11.3 PAH flux and equivalent width (EW) are listed in Table \[tab:fluxes\]. ![Mid-IR CanariCam spectra of the nucleus and different regions in the star-forming ring. The nuclear (orange), north (green), south (dark blue), and east (purple) spectra are shifted by 1.3, 0.8, 0.55, and 0.3Jy, respectively. The vertical lines mark the wavelength of the 7.7, 8.6, and 11.3 PAH features (dashed line). The shaded gray regions mark low-atmospheric transmission spectral ranges. \[fig:cc\_spec\]](fig_spec_cc.pdf){width="46.00000%"} --------- ----------------- ----------------- ------------------- ---------------------------------- ---------------------------- ------------------------- -- -- -- -- Region f$_\nu$(10)$^a$ f$_\nu$(12)$^a$ f$_\nu$(24.1)$^b$ 11.3 PAH$^c$ ${EW_{\rm 11.3\micron}}^d$ f$_\nu$(Pa$\alpha$)$^e$ (mJy) (mJy) (Jy) (10$^{-13}$ergcm$^{-2}$s$^{-1}$) (10$^{-3}$) (mJy) Nucleus 120 $\pm$ 8 210 $\pm$ 20 $<$0.5 3.2 $\pm$ 0.6 79 $\pm$ 8 2.2 N 91 $\pm$ 10 390 $\pm$ 30 1.7 $\pm$ 0.6 12.7 $\pm$ 0.2 220 $\pm$ 5 5.2 S 81 $\pm$ 10 390 $\pm$ 50 1.6 $\pm$ 0.4 9.3 $\pm$ 0.5 170 $\pm$ 3 5.0 E 61 $\pm$ 6 130 $\pm$ 10 2.0 $\pm$ 0.7 13.2 $\pm$ 0.3 500 $\pm$ 20 5.5 W 73 $\pm$ 20 280 $\pm$ 30 0.6 $\pm$ 0.3 14.2 $\pm$ 0.4 350 $\pm$ 10 4.6 --------- ----------------- ----------------- ------------------- ---------------------------------- ---------------------------- ------------------------- -- -- -- -- **Notes:** 3$\sigma$ upper limits are indicated for non-detections. The uncertainties do not include the $\sim$10–15% absolute calibration error. All the wavelengths are rest-frame. $^{(a)}$ The monochromatic 10 and 12 fluxes are measured in the CC spectra of each region (see Section \[ss:image\_modeling\]). $^{(b)}$ 24.1 fluxes derived from the CC $Q$-band imaging (see Section \[ss:image\_modeling\]). $^{(c)}$ Flux of the 11.3 PAH feature. $^{(d)}$ EW of the 11.3 PAH. $^{(e)}$ Pa$\alpha$ flux measured in the continuum subtracted F190N NICMOS images [@AAH06s]. To convert to flux units, these values should be multiplied by 1.56$\times$10$^{-14}$ ergcm$^{-2}$s$^{-1}$mJy$^{-1}$. Spectral Modeling {#ss:modeling} ----------------- For the five selected regions, we decomposed the $N$-band spectra together with the 24 photometry using a two component model consisting of a modified black-body with $\beta=2$ and a PAH emission template. The latter is derived from the [*Spitzer*]{}/IRS starburst template presented by @Smith07 after removing the dust continuum emission (see @Pereira2015not for details). We excluded the low-atmospheric transmission spectral ranges marked in Figure \[fig:cc\_spec\] for the fitting. The black-body temperature and the intensities of the black-body and the PAH template are free parameters of the model. In addition, we let the relative strength of the 11.3 PAH feature free during the fit since the strength of the different PAH features varies both in starbursts [@Smith07] and Seyfert galaxies [@Diamond2010]. We calculated the warm dust mass using the following relation $$M_{\rm dust} = \frac{D^2 f_{\nu}}{\kappa_{\nu} B_{\nu} (T_{\rm dust})}$$ where $D$ is the distance, $f_{\nu}$ the observed flux, $\kappa_{\nu}$ the absorption opacity coefficient, and $B_{\nu} (T_{\rm dust})$ the Planck’s blackbody law, all of them evaluated at 10. We assumed $\kappa_{10\mu m}=1920$cm$^{2}$g$^{-1}$ [@Li2001]. The results of the fits are shown in Table \[tab:models\] and Figure \[fig:models\]. The mid-IR emission of the SF ring regions are well fitted by a combination of a PAH component, which dominates the emission below 9, and a warm ($T\sim$110K) dust continuum component which dominates the emission at longer wavelengths. By contrast, the nuclear 8–13 spectrum is completely dominated by a warmer ($T\sim$160K) dust component. ![Best-fit of the modified blackbody+PAH template models to the CC spectroscopy (solid black line) and 24.5 photometry (white circle) of the different regions of NGC 1614. The 1$\sigma$ range of the best-fit model is indicated by the red shaded area. The solid green line and the dashed blue line represent the PAH template and the modified blackbody continuum, respectively. \[fig:models\]](fig_modelfit.pdf){width="46.00000%"} --------- -------------------------------------------------------- ------------------------------- ------------------------------- -- -- -- -- -- -- -- Region $\frac{\rm 7.7\micron\ PAH}{\rm 11.3\micron\ PAH}$$^a$ $T^{\rm warm}_{\rm dust}$$^b$ $M^{\rm warm}_{\rm dust}$$^c$ (K) (10$^3$) Nucleus 1.7 $\pm$ 0.2 160 $\pm$ 8 0.2$^{+0.3}_{-0.05}$ N 3.6 $\pm$ 0.5 110 $\pm$ 3 7.6$^{+6.7}_{-3.4}$ S 3.1 $\pm$ 0.4 109 $\pm$ 4 8.9$^{+3.2}_{-2.3}$ E 2.9 $\pm$ 0.5 108 $\pm$ 6 6.3$^{+1.8}_{-4.3}$ W 3.3 $\pm$ 0.6 115 $\pm$ 3 3.9$^{+2.2}_{-1.3}$ --------- -------------------------------------------------------- ------------------------------- ------------------------------- -- -- -- -- -- -- -- : Results from the modeling of the CC data\[tab:models\] **Notes:** $^{(a)}$ Ratio between the intensities of the modeled 7.7 and 11.3 PAH features (Section \[ss:modeling\]). $^{(b)}$ Temperature of the warm dust component detected in the mid-IR. $^{(c)}$ Mass of the warm dust (see Section \[ss:modeling\] for details). The AGN or SF Nature of the Nucleus {#ss:agn_vs_sf} =================================== The nature of the nucleus (central 150pc) of NGC 1614 is not well established. In part, this is because it is surrounded by a circumnuclear ring with strong star-formation (ring SFR$\sim$40yr$^{-1}$; @AAH01), which masks the relatively weak nuclear emission when observed at lower angular resolutions. In the high-angular resolution (011) *HST*/NICMOS images, the nucleus is slightly resolved and shows near-IR colors compatible with stellar emission, although the CO index is inconsistent with an old stellar population [@AAH01]. To explain this, @AAH01 suggested that the nuclear SF is more evolved than that of the star-forming ring. Based on *ASCA* X-ray observations, @Risaliti2000 suggested that NGC 1614 may host a Compton-thick AGN. However, the FeK 6.4keV line, which usually has a high EW in Compton-thick AGNs (although it depends on the obscuring matter geometry; see e.g., @Fabian2002), is not detected in more sensitive *XMM-Newton* observations [@Pereira2011]. More recently, the non-detection of CO(6–5) emission and 435 continuum in the nucleus in high-resolution ALMA observations implies that the amount of dust and molecular gas is much lower than that expected for a Compton-thick AGN [@Xu2015]. Similarly, interferometric radio continuum observations reveal that the nuclear emission is mostly thermal and relatively weak, which also supports the non-AGN nuclear activity [@Herrero-Illana2014]. Consequently, SF, as traced by the nuclear Pa$\alpha$ emission, would be the dominant energy source of the nucleus of NGC 1614. Our new mid-IR data challenge these previous results. The nuclear spectrum shows a strong 12 mid-IR (and a low EW of the 11.3 PAH feature), and the nuclear 24 continuum is weak in comparison with the SF ring emission. Therefore, the nucleus of NGC 1614 presents some characteristics (weak X-ray and far-IR emissions, lacking molecular gas, strong 12 mid-IR continuum, and Pa$\alpha$ emission) that cannot be explained in a standard AGN or SF context. In the following, we discuss possible modifications to the AGN and SF scenarios to explain the observations available so far. X-ray weak AGN? --------------- ![Comparison of the nuclear NGC 1614 mid-IR spectrum (black) and the average spectra of Type 1 (red) and Type 2 (blue) Seyfert galaxies from @AAH2014 normalized at 12. The shaded regions represent the 1$\sigma$ dispersion of the averaged spectra. \[fig:agn\_comparison\]](compara_agn.png){width="45.00000%"} ### Mid-IR AGN evidences The CC spectrum of the nucleus is remarkably different from the spectra of the star-forming regions in the ring of SF. It shows a strong mid-IR continuum relative to the PAH emission, a dust temperature higher than in the SF regions of the ring (Table \[tab:models\]), and a continuum peak at around $\sim$20 (Figure \[fig:models\]). Differences in the dust continuum emission are also evident if we consider the 24 to 10 flux ratio which is $\sim$10-30 in the ring and $<$5 in the nucleus (see Table \[tab:fluxes\] and Figure \[fig:maps\]). Low 24/10 ratios are predicted by AGN torus models because of the high dust temperatures reached in the torus (150–1500K; @Nenkova2008). Moreover, mid-IR [*Spitzer*]{}/IRS spectroscopy of active galaxies shows that for $\sim$30% of them (including Type 1 and 2 Seyfert objects) the mid-IR spectra peaks at $\sim$20 indicating that a warm dust component ($T\sim150-170$K) dominates the mid-IR emission [@Buchanan2006; @Wu2009]. This trend is also observed in ground-based sub-arcsecond mid-IR spectroscopic surveys of Seyfert galaxies (e.g., @AAH2011torus [@RamosAlmeida2009; @RamosAlmeida2011]). In Figure \[fig:agn\_comparison\] we compare the average Type 1 and Type 2 AGN mid-IR spectra obtained by @AAH2014 for nearby Seyfert galaxies. It shows that they all have similar continuum slopes. This suggests that the warm dust conditions in the nucleus of NGC 1614 are similar to those found in Seyfert galaxies. [Although, the emission of hotter dust (at $\sim$8) is weaker in NGC 1614.]{} The minimum 11.3 PAH EW is located at the nucleus (79 $\pm$ 8)$\times10^{-3}$ (Table \[tab:fluxes\]). This behavior is also observed in local Seyfert galaxies, and it is explained in these objects by the increased AGN continuum contribution in the nucleus [@AAH2014; @Esquej2014; @RamosAlmeida2014; @GarciaBernete2015]. In addition, in the nucleus of NGC 1614, the 11.3 PAH feature is enhanced by a factor of $\sim$2 with respect to the 7.7 PAH feature (Table \[tab:models\]). Similar enhancements of the 11.3 PAH feature are observed in active galaxies although on kpc scales [@Diamond2010]. ### Weak X-ray Emission {#ss:xrayweak} A correlation between the 12 and the 2–10keV luminosities is observed for Seyfert galaxies [@Horst2008; @Levenson2009; @Gandhi2009; @Asmus2011]. For the nuclear 12 luminosity measured from the spectrum of NGC 1614 ($\nu L_{\nu}=$2.6$\times$10$^{43}$ergs$^{-1}$) the expected hard X-ray luminosity would be 1.6$\times$10$^{43}$ergs$^{-1}$ according to the @Gandhi2009 relation. Threfore, both the nuclear 12 and expected 2–10keV luminosities are comparable to that of an average local Seyfert galaxy (see Figure 1 of @Gandhi2009). However, the observed integrated hard X-ray luminosity of this galaxy is just 1.4$\times$10$^{41}$ergs$^{-1}$, almost a factor of 200 lower than expected for an AGN, and most of it can be explained by the hard X-ray emission from star-formation (i.e., high-mass X-ray binaries; @Pereira2011). Similarly, the soft X-ray emission is also better explained by star-formation [@Pereira2011; @Herrero-Illana2014]. If an AGN is present in the nucleus of NGC 1614, three possibilities may explain the weakness of the X-ray emission: it may be a strongly variable source observed during its low state; it may be a Compton-thick AGN so the 2–10keV emission is absorbed; or it may be an intrinsically X-ray weak AGN. There are three hard X-ray observations of NGC 1614 during 18yr (Table \[tab:xray\]) which show that the variability is less than a factor of 2. So it is not likely that X-ray variability is the reason for the X-ray weakness. The Compton-thick AGN possibility was rejected by @Xu2015 based on the low amount of molecular gas and cold dust in the nucleus. Moreover, NGC 1614 is not detected in the 14–195keV [*Swift*]{}/BAT 70-Month Hard X-ray Survey [@Baumgartner2013]. If NGC 1614 would be a Compton-thick AGN with an intrinsic 2–10keV luminosity of 1.6$\times$10$^{43}$ergs$^{-1}$ (see above), its 14–195keV flux would be[^1] 6$\times$10$^{-11}$ergcm$^{-2}$s$^{-1}$, which is $\sim$4 times the 5$\sigma$ sensitivity of the [*Swift*]{}/BAT survey. Finally, it is also possible that the X-ray emission of the NGC 1614 AGN is intrinsically weak. The ultra-luminous IR galaxy Mrk 231 [@Teng2014], as well as several quasars [@Leighly2007; @Miniutti2012; @Luo2014], have X-ray luminosities 30–100 times weaker than those predicted by the $\alpha_{\rm OX}$[^2] vs. $L_{\rm 2500A})$ correlation, probably due to a distortion of the accretion disk corona [@Miniutti2012; @Luo2013]. In the case of NGC 1614, the nuclear UV emission is completely obscured (see @Petty2014), so a direct comparison with the results for these X-ray weak AGNs is not possible. However, using the 12 emission we obtain that the [observed]{} 2–10keV emission is more than two orders of magnitude lower than the expected value, similar to the X-ray weakness observed on those objects. ------------ ------------------ ---------------------------------- ------ -- -- -- -- -- -- -- Date Telescope Flux Ref. (10$^{-13}$ergcm$^{-2}$s$^{-1}$) 1994-02-16 [*ASCA*]{} 5.6 1 2003-02-13 [*XMM-Newton*]{} 2.7$\pm$0.4 2 2012-04-10 [*Swift*]{} 2.5$\pm$0.4 3 ------------ ------------------ ---------------------------------- ------ -- -- -- -- -- -- -- : 2–10keV X-ray observations of NGC 1614\[tab:xray\] **References:** (1) @Risaliti2000; (2) @Pereira2011; (3) @Evans2014. or Nuclear Star-formation? -------------------------- Alternatively, it is possible to explain the nuclear observations assuming only star-formation (SF). However, the nuclear SF and the SF taking place in the ring surrounding the nucleus must have very different characteristics. In particular, the nuclear mid-IR spectrum shows a strong 8–12 continuum that is not present in the ring spectra (Figure \[fig:models\]), and the nucleus remains undetected in the 435 far-IR continuum and CO(3–2) maps [@Xu2015; @Usero2015] while the ring is clearly detected. In our nuclear mid-IR spectrum, we detect the 11.3 PAH feature which is usually associated with SF (mostly B stars, see @Peeters2004). Using the $L_{\rm 11.3\mu m\,PAH}$ SFR calibration of @Diamond-Stanic2012, we estimate a nuclear SFR of $\sim$0.9yr$^{-1}$ (Table \[tab:sfr\] and see Section \[ss:sfr\_tracers\]). We also used the nuclear Pa$\alpha$ flux [@DiazSantos2008] to derive a SFR $\sim$1.5yr$^{-1}$ (assuming $A_{\rm k}=$0.3mag; @AAH01), so both SFR tracers are in agreement within a factor of 2. Finally, we used the IR continuum upper limits at 24 and 432 to derive an upper limit for the nuclear IR (4-1000) luminosity of $<$6$\times$10$^{43}$ergs$^{-1}$. This upper limit is compatible with the expected IR luminosity for a SFR$\sim$1.5yr$^{-1}$ ($\sim$4$\times$10$^{43}$ergs$^{-1}$; @Kennicutt2012). Therefore, all these IR SFR tracers are compatible and they indicate that the nuclear SFR is $\leq$1.5yr$^{-1}$, that is, less than $<$2% of the total SFR of NGC 1614 ($\sim$100yr$^{-1}$; @Pereira2015not). However, the nuclear and the integrated IR (8–500) spectral energy distributions are very different. The ring is detected at 435 [@Xu2015] and 24 (Figure \[fig:maps\]),but the nucleus is not. Therefore, this implies that the dust temperature is much higher in the nucleus, as already suggested by our mid-IR data. This higher nuclear dust temperature (Table \[tab:models\]) can be explained by the enhanced radiation field density, which is expected to increase the dust temperature (see @Draine07), due to an increased density of young stars in the nucleus (or an AGN, see Section \[ss:xrayweak\]). Molecular gas is not detected in the nucleus of NGC 1614. From the 05 resolution CO(3-2) ALMA observations of NGC 1614, @Usero2015 estimate an upper limit to the nuclear molecular gas mass of 3$\times$10$^{6}$[^3]. This low molecular gas mass puts the nucleus of NGC 1614 well above the Kennicutt-Schmidt relation (see Figure 8 of @Xu2015). Consequently, the molecular gas depletion time is $<$3Myr, much lower than in normal galaxies at 100pc scales (1–3Gyr; e.g., @Leroy2013), and also lower than in local ULIRGs (70–100Myr; e.g., @Combes2013). A short depletion time might indicate that the ignition of the nuclear SF occurred earlier than in the ring (see @AAH01). [Therefore, the nuclear starburst would have consumed a larger fraction of the original cold molecular gas than the younger starburst of the ring. Actually, the evolutionary state of the SF regions is commonly used to explain the dispersion of individual SF regions in the Kennicutt-Schmidt relation (e.g., @Onodera2010 [@Schruba2010; @Kruijssen2014]).]{} However, the integrated (including nucleus and SF ring) dense molecular gas depletion time in NGC 1614 is also shorter ($\sim$10Myr) than in other LIRGs ($\sim$50Myr; @GarciaBurillo2012), so it is not obvious to associate the particularly short nuclear depletion time with older SF. Alternatively, a massive molecular outflow, produced by an AGN or SN explosions (see @GarciaBurillo2015), could have swept most the molecular gas away from the nucleus. On the other hand, the hard X-ray luminosity of this object is also compatible with a SF origin [@Pereira2011], although most of the emission would be produced in the ring. Unfortunately, the angular resolution of the *Chandra* X-ray data is not sufficient to separate the nucleus and the ring [@Herrero-Illana2014]. Note that, in principle, a combination of SF and a normal AGN would be also possible. However, this assumption suffers the same problems explaining the observations than the SF and AGN individually. For these reason, we do not discuss this AGN$+$SF composite possibility. ------------------- ------------------- ------------------ ---------------- ---------- -- -- -- -- -- -- \[-1.5ex\] \[-2.5ex\] Region $A_{\rm k}$$^{a}$ Pa$\alpha$$^{b}$ 11.3 PAH$^{c}$ 24$^{d}$ Nucleus$^\star$ 0.3 1.5 0.9 $<$6 N 0.7 9.3 4.1 22 S 0.8 11.2 3.1 21 E 0.6 7.8 4.1 26 W 1.0 16.4 5.2 7 ------------------- ------------------- ------------------ ---------------- ---------- -- -- -- -- -- -- : SFR from different IR tracers\[tab:sfr\] **Notes:** $^{(a)}$ $K$-band extinction in magnitudes derived from the stellar colors [@AAH01]. $^{(b)}$ Extinction corrected Pa$\alpha$ SFR using the @Kennicutt2012 calibration assuming H$\alpha$/Pa$\alpha$ = 8.51. $^{(c)}$ SFR obtained from the 11.3 PAH luminosities (Table \[tab:fluxes\]) based on the @Diamond-Stanic2012 calibration. We multiplied by 2 our 11.3 PAH luminosities to account for the different method used to measure the PAH features (local continuum vs. full decomposition, see @Smith07). $^{(d)}$ SFR derived from the monochromatic 24 luminosities (Table \[tab:fluxes\]) using the @Rieke2009 calibration. $^{(\star)}$ Nuclear SFR derived assuming that all the nuclear emission is produced by SFR (i.e., no AGN). SFR Tracers at $\sim$150 Scales {#ss:sfr_tracers} =============================== Using the new CC mid-IR data (11.3 PAH and 24 continuum) in combination with the NICMOS Pa$\alpha$ image, we can test several SFR calibrations at 150pc scales in this galaxy. In Table \[tab:sfr\] we show a summary of the SFR derived using these tracers for the five regions we defined in NGC 1614. We used the calibrations of @Kennicutt2012, @Diamond-Stanic2012, and @Rieke2009 for the Pa$\alpha$, 11.3 PAH, and 24 tracers, respectively. The Pa$\alpha$ emission was corrected for extinction using the near-IR continuum colors (see @AAH01). Since the extinction corrected Pa$\alpha$ calibration is a direct measurement of the number of ionizing photons produced by young stars, we consider it as the reference SFR tracer. The 24 luminosity gives the highest SFR values (2–3 and 5–7 times higher than those derived from the Pa$\alpha$ and 11.3 PAH luminosities, respectively), except in the W region of the ring. The modeling of the radio emission of the W region indicates the presence of supernovae (SN; @Herrero-Illana2014), so it could be more evolved than the rest of the ring. Therefore, a lower amount of young stars would be dust embedded in this region reducing the warm dust emission. The disagreement between the extinction corrected Pa$\alpha$ and the 24 SFR values is $\sim$0.4dex, which is higher than the calibration uncertainty (0.2dex). Although, in principle, both tracers should produce similar SFR estimates (see Equations 5 and 8 of @Rieke2009). There are two possibilities to explain this. First, it is possible that even the extinction corrected Pa$\alpha$ emission underestimates the SFR. In extremely obscured regions (e.g., $A_{\rm v}>$15–20mag), dust might absorb the Pa$\alpha$ emission completely, as well as part of the ionizing photons, and therefore, rendering any extinction correction ineffective. Alternatively, an increase of the dust temperature at high SFR densities, like in the SF ring of NGC 1614, can produce enhanced 24 emission that might not be taken into account by the 24 SFR calibration which is valid for integrated emission of galaxies (e.g., @Calzetti2010). The stellar $A_{\rm k}$ measured in the SF ring of NGC 1614 is 0.6–1.0mag ($A_{\rm v}=$5–10mag; @AAH01), so the obscuration level is not as extreme as observed in some ULIRGs ($A_{\rm v}=$8–80mag; @Armus07). In addition, the 9.7 silicate absorption in the SF ring spectra is not very deep (Figure \[fig:models\]). Therefore, this favors the second possibility. That is, an increased 24 emission in the SF ring of NGC 1614 due to a warmer dust emission. According to Table \[tab:sfr\], the SFR derived from the 11.3 PAH luminosity is 2–4 times lower than that derived from Pa$\alpha$. The 11.3 PAH SFR calibration is based on $\sim$kpc integrated measurements [@Diamond-Stanic2012]. However, it is known that the PAH emission, and in particular the 11.3 PAH emission, is more extended than the warm dust continuum and other ionized gas tracers (e.g., \[\]12.81; @DiazSantos2011). Actually, $\sim$30–40% of the total PAH emission is not related to recent SF [@Crocker2013]. Therefore, this SFR calibration possibly includes a considerable amount of PAH emission not produced by young stars. [In addition, using templates of SF galaxies, @Rieke2009 showed that the 11.3 PAH contribution to the total IR luminosity drops by a factor of $\sim$2.5 for galaxies with $L_{\rm IR}>10^{11}$. A similar result was found by @AAH2013 for a sample of local LIRGs.]{} [A combination of these reasons might]{} explain why we obtain these relatively low SFR estimates from the 11.3 PAH luminosities for the $\sim$150pc SF regions in the ring of NGC 1614. Conclusions {#s:conclusions} =========== We analyzed new GTC/CC high-angular resolution ($\sim$05) mid-IR observations of the local LIRG NGC 1614. The new $N$-band spectroscopy and $Q$-band imaging are combined with existing *HST*/NICMOS Pa$\alpha$ and T-ReCS 8.7 images to study the properties of the bright circumnuclear SF ring and the nucleus of this object. The main results are the following: 1. We extracted mid-IR spectra from four different regions in the circumnuclear SF ring and from the nuclear region (central 05$\sim$150pc). The spectra from the SF ring are typical of a SF region with strong PAH emission and a shallow 9.7 silicate absorption. By contrast, the nuclear spectrum has a strong mid-IR continuum, which dominates its mid-IR emission, and weak PAH emission (EW$_{\rm 11.3\micron}$=80$\times$10$^{-3}$). Similarly, the SF ring is clearly detected in the 24.5 image, as expected for a SF region, while the nucleus is [weaker]{} at this wavelength. 2. A two component model, consisting of a modified black-body with $\beta=2$ and a PAH emission template, reproduces the observed $N$ spectra and $Q$ photometry well. The main differences between the nuclear and the SF ring observations are: the higher dust temperature in the nucleus (160K in the nucleus vs. $\sim$110K in the ring); the lower PAH EW; and the lower nuclear 7.7/11.3 PAH ratio. 3. The above results based on the mid-IR data, suggest that an AGN might be present in the nucleus. However, this is at odds with the low X-ray luminosity of NGC 1614 ($\sim$200 times lower than that expected for an AGN with the observed 12 continuum luminosity). Since the hard (2–10keV) X-ray emission shows no variability, and likely it is not a Compton-thick AGN, if an AGN is present in NGC 1614, it must be an intrinsically X-ray weak AGN. We also calculated an upper limit to the IR luminosity of the nucleus, $<$6$\times$10$^{43}$ergs$^{-1}$. 4. Alternatively, SF can explain the observations of the nucleus too. However, we need to invoke extremely short molecular gas depletion times ($<$3Myr [for a nuclear SFR of $\sim$1–1.5yr$^{-1}$]{}), and an increased radiation field density to explain the observed hot dust in the nucleus. 5. Finally, we compared three SFR tracers at 150pc scales [in the circumnuclear ring]{}: extinction corrected Pa$\alpha$, 11.3 PAH, and 24 continuum. Since the extinction is not extremely high ($A_{\rm v}<10$mag), we take as reference the Pa$\alpha$ derived SFR. In general, the 24 SFR overestimates the SFR by a factor of 2–3, while the 11.3 PAH underestimates the SFR by a factor of 2–4. The former might be explained if the dust temperature is higher in the SF regions of NGC 1614, while the latter [could be because we do not include diffuse PAH emission in our measurements as well as because the PAH contribution to the total IR luminosity might be reduced in LIRGs.]{} 6. In the West region of the ring, the 24 emission is $\sim$5 times weaker than expected based on the observed Pa$\alpha$/24 ratio in this galaxy. We propose that this is because this is a more evolved SF region (SN are present; @Herrero-Illana2014) where a larger fraction of the young stars are not dust embedded. In summary, our mid-IR data suggest that an intrinsically X-ray weak AGN ($L^{\rm AGN}_{\rm bol}\sim$10$^{43}$ergs$^{-1}$, $<$5% of the NGC 1614 bolometric luminosity) might be present in the nucleus of NGC 1614. However, SF with a short molecular gas depletion time and increased dust temperatures can explain the observations as well. In order to further investigate the nature of the nucleus of this galaxy, IR and sub-mm high-angular resolution observations are needed. Acknowledgments {#acknowledgments .unnumbered} =============== We thank the anonymous referee for useful comments and suggestions. We thank the GTC staff for their continued support on the CanariCam observations. We acknowledge support from the Spanish Plan Nacional de Astronomía y Astrofísica through grants AYA2010-21161-C02-01, and AYA2012-32295. AAH and AA acknowledges funding from the Spanish Ministry of Economy and Competitiveness under grants AYA2012-31447 and AYA2012-38491-CO2-02, which are party funded by the FEDER program. MAPT acknowledges support from the Spanish MICINN through grant AYA2012-38491-C02-02. CRA acknowledges support from a Marie Curie Intra European Fellowship within the 7th European Community Framework Programme (PIEF-GA-2012-327934). Based on observations made with the Gran Telescopio Canarias (GTC), installed in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias, in the island of La Palma. Partially based on observations obtained at the Gemini Observatory (program GS-2006B-Q-9), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). [76]{} natexlab\#1[\#1]{} , A., [Engelbracht]{}, C. W., [Rieke]{}, M. J., [Rieke]{}, G. H., & [Quillen]{}, A. C. 2001, , 546, 952 , A., [Pereira-Santaella]{}, M., [Rieke]{}, G. H., [et al.]{} 2013, , 765, 78 , A., [Ramos Almeida]{}, C., [Esquej]{}, P., [et al.]{} 2014, , 443, 2766 , A., [Ramos Almeida]{}, C., [Mason]{}, R., [et al.]{} 2011, , 736, 82 , A., [Rieke]{}, G. H., [Rieke]{}, M. J., [et al.]{} 2006, , 650, 835 , L., [Charmandaris]{}, V., [Bernard-Salas]{}, J., [et al.]{} 2007, , 656, 148 , K. A. 1996, in Astronomical Society of the Pacific Conference Series, Vol. 101, Astronomical Data Analysis Software and Systems V, ed. G. H. [Jacoby]{} & J. [Barnes]{}, 17 , D., [Gandhi]{}, P., [Smette]{}, A., [H[ö]{}nig]{}, S. F., & [Duschl]{}, W. J. 2011, , 536, A36 , D., [H[ö]{}nig]{}, S. F., [Gandhi]{}, P., [Smette]{}, A., & [Duschl]{}, W. J. 2014, , 439, 1648 , W. H., [Tueller]{}, J., [Markwardt]{}, C. B., [et al.]{} 2013, , 207, 19 , A. D., [Wolfire]{}, M., & [Leroy]{}, A. K. 2013, , 51, 207 , C. L., [Gallimore]{}, J. F., [O’Dea]{}, C. P., [et al.]{} 2006, , 132, 401 , D., [Wu]{}, S.-Y., [Hong]{}, S., [et al.]{} 2010, , 714, 1256 , K. I., [Lagache]{}, G., [Yan]{}, L., [et al.]{} 2007, , 660, 97 , M., [Walker]{}, R. G., [Carter]{}, B., [et al.]{} 1999, , 117, 1864 , F., [Garc[í]{}a-Burillo]{}, S., [Braine]{}, J., [et al.]{} 2013, , 550, A41 , A. F., [Calzetti]{}, D., [Thilker]{}, D. A., [et al.]{} 2013, , 762, 79 , A. M. & [Rieke]{}, G. H. 2010, , 724, 140 , A. M. & [Rieke]{}, G. H. 2012, , 746, 168 , T., [Alonso-Herrero]{}, A., [Colina]{}, L., [et al.]{} 2008, , 685, 211 , T., [Charmandaris]{}, V., [Armus]{}, L., [et al.]{} 2011, , 741, 32 , B. T. & [Li]{}, A. 2007, , 657, 810 , P., [Alonso-Herrero]{}, A., [Gonz[á]{}lez-Mart[í]{}n]{}, O., [et al.]{} 2014, , 780, 86 , P. A., [Osborne]{}, J. P., [Beardmore]{}, A. P., [et al.]{} 2014, , 210, 8 , A. C., [Wilman]{}, R. J., & [Crawford]{}, C. S. 2002, , 329, L18 , P., [Horst]{}, H., [Smette]{}, A., [et al.]{} 2009, , 502, 457 , I., [Ramos Almeida]{}, C., [Acosta-Pulido]{}, J. A., [et al.]{} 2015, , 449, 1309 , S., [Combes]{}, F., [Usero]{}, A., [et al.]{} 2015, , 580, A35 , S., [Usero]{}, A., [Alonso-Herrero]{}, A., [et al.]{} 2012, , 539, A8 , O., [Rodr[í]{}guez-Espinosa]{}, J. M., [D[í]{}az-Santos]{}, T., [et al.]{} 2013, , 553, A35 , R., [P[é]{}rez-Torres]{}, M. [Á]{}., [Alonso-Herrero]{}, A., [et al.]{} 2014, , 786, 156 , H., [Gandhi]{}, P., [Smette]{}, A., & [Duschl]{}, W. J. 2008, , 479, 389 , M., [Imase]{}, K., [Oi]{}, N., & [Ichikawa]{}, K. 2011, , 141, 156 , M., [Nakagawa]{}, T., [Shirahata]{}, M., [Ohyama]{}, Y., & [Onaka]{}, T. 2010, , 721, 1233 , R. C. & [Evans]{}, N. J. 2012, , 50, 531 , S., [Aalto]{}, S., [Muller]{}, S., [Beswick]{}, R. J., & [Gallagher]{}, J. S. 2013, , 553, A72 , P. 2001, , 322, 231 , J. M. D. & [Longmore]{}, S. N. 2014, , 439, 3239 , E., [Papovich]{}, C., [Dole]{}, H., [et al.]{} 2005, , 632, 169 , K. M., [Halpern]{}, J. P., [Jenkins]{}, E. B., [et al.]{} 2007, , 663, 103 , A. K., [Walter]{}, F., [Sandstrom]{}, K., [et al.]{} 2013, , 146, 19 , N. A., [Radomski]{}, J. T., [Packham]{}, C., [et al.]{} 2009, , 703, 390 , A. & [Draine]{}, B. T. 2001, , 554, 778 , B., [Brandt]{}, W. N., [Alexander]{}, D. M., [et al.]{} 2013, , 772, 153 , B., [Brandt]{}, W. N., [Alexander]{}, D. M., [et al.]{} 2014, , 794, 70 , P. & [Dickinson]{}, M. 2014, , 52, 415 , B., [Elbaz]{}, D., [Chary]{}, R. R., [et al.]{} 2011, , 528, A35 , A., [Risaliti]{}, G., [Gilli]{}, R., [et al.]{} 2004, , 351, 169 , G., [Brandt]{}, W. N., [Schneider]{}, D. P., [et al.]{} 2012, , 425, 1718 , M., [Sirocky]{}, M. M., [Nikutta]{}, R., [Ivezi[ć]{}]{}, [Ž]{}., & [Elitzur]{}, M. 2008, , 685, 160 , E., [Aalto]{}, S., [Thomasson]{}, M., & [Beswick]{}, R. 2010, , 513, A11 , S., [Kuno]{}, N., [Tosaki]{}, T., [et al.]{} 2010, , 722, L127 , E., [Spoon]{}, H. W. W., & [Tielens]{}, A. G. G. M. 2004, , 613, 986 , C. Y., [Ho]{}, L. C., [Impey]{}, C. D., & [Rix]{}, H.-W. 2010, , 139, 2097 , M., [Alonso-Herrero]{}, A., [Colina]{}, L., [et al.]{} 2015, , 577, A78 , M., [Alonso-Herrero]{}, A., [Santos-Lleo]{}, M., [et al.]{} 2011, , 535, A93 , P. G., [Rieke]{}, G. H., [Egami]{}, E., [et al.]{} 2005, , 630, 82 , S. M., [Armus]{}, L., [Charmandaris]{}, V., [et al.]{} 2014, , 148, 111 , C., [Alonso-Herrero]{}, A., [Esquej]{}, P., [et al.]{} 2014, , 445, 1130 , C., [Levenson]{}, N. A., [Alonso-Herrero]{}, A., [et al.]{} 2011, , 731, 92 , C., [Levenson]{}, N. A., [Rodr[í]{}guez Espinosa]{}, J. M., [et al.]{} 2009, , 702, 1127 , G. H., [Alonso-Herrero]{}, A., [Weiner]{}, B. J., [et al.]{} 2009, , 692, 556 , G., [Gilli]{}, R., [Maiolino]{}, R., & [Salvati]{}, M. 2000, , 357, 13 , D. B., [Mazzarella]{}, J. M., [Kim]{}, D.-C., [Surace]{}, J. A., & [Soifer]{}, B. T. 2003, , 126, 1607 , A., [Leroy]{}, A. K., [Walter]{}, F., [Sandstrom]{}, K., & [Rosolowsky]{}, E. 2010, , 722, 1699 , R., [Haas]{}, M., [Pantin]{}, E., [et al.]{} 2008, , 488, 83 , K., [Wilson]{}, C. D., [Iono]{}, D., [Peck]{}, A., & [Matsushita]{}, S. 2014, , 796, L15 , J. D. T., [Draine]{}, B. T., [Dale]{}, D. A., [et al.]{} 2007, , 656, 770 , B. T., [Neugebauer]{}, G., [Matthews]{}, K., [et al.]{} 2001, , 122, 1213 , C. M., [Ciardi]{}, D., [French]{}, J., [et al.]{} 2003, Proc. SPIE, 4841, 913 , S. H., [Brandt]{}, W. N., [Harrison]{}, F. A., [et al.]{} 2014, , 785, 19 , A. 2015 in prep. , P., [Rajpaul]{}, V., [Zijlstra]{}, A. A., [Reunanen]{}, J., & [Kotilainen]{}, J. 2012, , 420, 2209 , Y., [Charmandaris]{}, V., [Huang]{}, J., [Spinoglio]{}, L., & [Tommasin]{}, S. 2009, , 701, 658 , C. K., [Cao]{}, C., [Lu]{}, N., [et al.]{} 2015, , 799, 11 , T.-T., [Kewley]{}, L. J., & [Sanders]{}, D. B. 2010, , 709, 884 \[lastpage\] [^1]: Using <span style="font-variant:small-caps;">XSPEC</span> [@Arnaud1996] and assuming a power-law spectrum with $\Gamma=1.9$ [@Marconi2004] and $N_{\rm H}=$3$\times$10$^{24}$cm$^{-2}$. Increasing the $N_{\rm H}$ up to 10$^{26}$cm$^{-2}$ the flux would be reduced by factor of two. [^2]: $\alpha_{\rm OX}=-0.384\log (L(2{\rm \,keV}) \slash L(2500\AA) $ [^3]: Assuming a CO(3-2) to CO(1-0) ratio of $\sim$1 and the Galactic CO-to-H$_2$ conversion factor [@Bolatto2013]. Using the conversion factor for ULIRGs it would be a factor of $\sim$4 lower.

--- abstract: | It is shown that two gravitating scalar fields may form a thick brane in 5D spacetime. The necessary condition for the existence of such a regular solution is that the scalar fields potential must have local and global minima. Key words: thick brane, scalar fields author: - 'Vladimir Dzhunushaliev [^1]' title: Thick brane solution in the presence of two interacting scalar fields --- Introduction ============ In recent years there has been a revived interest in theories having a greater number of spatial dimensions than the three that are observed. In contrast to the original Kaluza-Klein theories of extra dimensions, the recent incarnations of extra dimensional theories allow the extra dimensions to be large and even infinite in size (in the original Kaluza-Klein theories the extra dimensions were curled up or compactified to the experimentally unobservable small size of the Planck length: $10^{-33}$cm). These new extra dimensional theories have opened up new avenues to explaining some of the open questions in particle physics (the hierarchy problem, nature of the electro-weak symmetry breaking, explanation of the family structure) and astrophysics (the nature of dark matter, the nature of dark energy) [@arkani] - [@gogberashvili]. In addition they predict new experimentally measurable phenomenon in high precision gravity experiments, particle accelerators, and in astronomical observations. Most of the brane world models use infinitely thin branes with delta-like localization of matter. However these models are generally regarded as an approximation since any fundamental underlying theory, such as quantum gravity or string theory, must contain a fundamental length beyond which a classical space-time description is impossible. It is therefore necessary to justify the infinitely thin brane approximation as a well-defined limit of a smooth structure – a thick brane – obtainable as a solution to coupled gravitational and matter field equations. One early example of a thick brane comes from the 5 dimensional model considered in [@rubakov] where one had a topologically non-trivial field configuration for the scalar field. In Ref. [@akama], the picture is presented that our universe is a dynamically localized 3-brane in a higher dimensional space (”brane world“ ). As an example, the dynamics of the Nielsen-Olesen vortex type in six dimensional spacetime is adopted to localize our space-time within a 3-brane. At low energies, everything is trapped in the 3-brane, and the Einstein gravity is induced through the fluctuations of the 3-brane. It is therefore of great interest to formulate some general requirements on the brane world design leading to the appearance of stable, thick branes having a well-defined zero thickness limit and able to trap ordinary matter. Taking a physically reasonable stress-energy tensor it was shown that in 6 dimensions [@gogberashvili2] and also higher extra dimensions [@singleton] one can trap all the Standard Model fields using gravity alone. In Ref’s [@bronnikov] thick brane world models are studied as $\mathbb Z_2$-symmetric domain walls supported by a scalar field with an arbitrary potential $V(\phi)$ in 5D general relativity and it was shown that in the framework of 5D gravity, a globally regular thick brane always has an anti-de Sitter asymptotic and is only possible if the scalar field potential $V(\phi)$ has an alternating sign. In Ref’s [@Gremm1] - [@barcelo] some properties of brane models was investigated: localization of gravity, graviton ground state, stability and so on. In Ref. [@Barbosa-Cendejas:2005kn] a comparative analysis of localization of 4D gravity on a non $Z_2$-symmetric scalar thick brane in both 5-dimensional Riemannian space time and pure geometric Weyl integrable manifold is presented. Multidimensional space-times with large extra dimensions turned out to be very useful when addressing several problems of the recent non–supersymmetric string model realization of the Standard Model at low energy with no extra massless matter fields [@kokorelis]. In Ref. [@Dzhunushaliev:2003sq] it is shown that two interacting non-gravitating scalar fields with a non-trivial potential may have a regular spherically symmetric solution. This solution shows that one can avoid the Derrick’s theorem [@derrick] forbidding the existence of regular static solution in the spacetime with the dimension greater 2 for scalar fields if the potential has a local minimum besides global one. This result allows us to assume that the inclusion of gravitation may not destroy the regularity of similar solutions in 5D spacetime. In Ref. [@Bronnikovc] one example of spherically symmetric solution with a gravitating scalar field is given but in contrast with the solution that will be presented here the potential of the scalar field in Ref. [@Bronnikovc] is negative. The goal of this investigation is to show that there exists a new kind of thick brane solutions that is different with thick brane solutions found in Ref’s [@DeWolfe:1999cp] [@Bronnikov:2005bg]. We will show that the asymptotical behavior of one scalar field allow us to offer trapping of Maxwell electrodynamics and spinor fields on the brane. Especially it is necessary to note that the consideration of two scalar fields allow us to obtain the regular thick brane solution with the potential bounded from below. Initial equations ================= We consider 5D gravity + two interacting fields. The key for the existence of a regular solution here is that the scalar fields potential have to have *local* and *global* minima, and at the infinity the scalar fields tend to a local but *not* to global minimum. The 5D metric is $$ds^2 = a(y) \eta_{\mu \nu} dx^\mu dx^\nu - dy^2, \label{sec2-10}$$ where $\mu ,\nu = 0,1,2,3$; $y$ is the $5^{th}$ coordinate; $\eta_{\mu \nu} = \left\{ +1, -1, -1, -1 \right\}$ is the 4D Minkowski metric. The Lagrangian for scalar fields $\phi$ and $\chi$ is $$\mathcal L = \frac{1}{2} \nabla_A \phi \nabla^A \phi + \frac{1}{2} \nabla_A \chi \nabla^A \chi - V(\phi, \chi) , \label{sec2-20}$$ where $A= 0,1,2,3,5$. The potential $V(\phi, \chi)$ is $$V(\phi, \chi) = \frac{\lambda_1}{4} \left( \phi^2 - m_1^2 \right)^2 + \frac{\lambda_2}{4} \left( \chi^2 - m_2^2 \right)^2 + \phi^2 \chi^2 - V_0 , \label{sec2-30}$$ where $V_0$ is a constant which can be considered as a 5D cosmological constant $\Lambda$. We consider the case when the functions $\phi, \chi$ are $\phi(y), \chi(y)$. The 5D Einstein and scalar field equations are $$\begin{aligned} R^A_B - \frac{1}{2} \delta^A_B R &=& \varkappa T^A_B , \label{sec2-40}\\ \frac{1}{\sqrt{G}} \nabla_A \left( \sqrt{G} G^{AB} \nabla_B \phi \right) &=& - \frac{\partial V\left( \phi, \chi \right)}{\partial \phi} , \label{sec2-50}\\ \frac{1}{\sqrt{G}} \nabla_A \left( \sqrt{G} G^{AB} \nabla_B \chi \right) &=& - \frac{\partial V\left( \phi, \chi \right)}{\partial \chi} , \label{sec2-60}\end{aligned}$$ where $\varkappa$ is the 5D gravitational constant; $G_{AB}$ is the 5D metric and $G$ is the corresponding determinant. After substituting metric into Eq’s - we have the following equations $$\begin{aligned} -3 \frac{a''}{a} - 3 \frac{a'^2}{a^2} &=& \frac{\varkappa}{4} \left[ \phi'^2 + \chi'^2 + \frac{\lambda_1}{2} \left( \phi^2 - m_1^2 \right)^2 + \frac{\lambda_2}{2} \left( \chi^2 - m_2^2 \right)^2 + 2 \phi^2 \chi^2 - 2 V_0 \right] , \label{sec2-70}\\ - 6 \frac{a'^2}{a^2} &=& \frac{\varkappa}{4} \left[ - \phi'^2 - \chi'^2 + \frac{\lambda_1}{2} \left( \phi^2 - m_1^2 \right)^2 + \frac{\lambda_2}{2} \left( \chi^2 - m_2^2 \right)^2 + 2 \phi^2 \chi^2 - 2 V_0 \right] , \label{sec2-80}\\ \phi'' + 4 \frac{a'}{a} \phi' &=& \phi \left[ 2 \chi^2 + \lambda_1 \left( \phi^2 - m_1^2 \right) \right] , \label{sec2-90}\\ \chi'' + 4 \frac{a'}{a} \chi' &=& \chi \left[ 2 \phi^2 + \lambda_2 \left( \chi^2 - m_2^2 \right) \right] , \label{sec2-100}\end{aligned}$$ where $\frac{d (\cdots)}{ dy} = (\cdots)'$. Let us introduce the following dimensionless functions $a/\sqrt{\varkappa/6} \rightarrow a$, $\phi \sqrt{\varkappa/3} \rightarrow \phi$, $ \chi\sqrt{\varkappa/3} \rightarrow \chi$, $2\left( \varkappa/6 \right)^2 V_0 \rightarrow V_0$, $m_{1,2}\sqrt{\varkappa/3} \rightarrow m_{1,2}$, $\lambda_{1,2}/2 \rightarrow \lambda_{1,2}$ and the dimensionless variable $y/\sqrt{\varkappa/6} \rightarrow y$. After algebraical transformations Eq’s - have the following form $$\begin{aligned} \frac{a''}{a} - \frac{a'^2}{a^2} &=& - \frac{1}{2}\left( \phi'^2 + \chi'^2 \right), \label{sec2-75}\\ \frac{a'^2}{a^2} &=& \frac{1}{8} \left[ \phi'^2 + \chi'^2 - \frac{\lambda_1}{2} \left( \phi^2 - m_1^2 \right)^2 - \frac{\lambda_2}{2} \left( \chi^2 - m_2^2 \right)^2 - \phi^2 \chi^2 + 2 V_0 \right] , \label{sec2-85}\\ \phi'' + 4 \frac{a'}{a} \phi' &=& \phi \left[ \chi^2 + \lambda_1 \left( \phi^2 - m_1^2 \right) \right] , \label{sec2-95}\\ \chi'' + 4 \frac{a'}{a} \chi' &=& \chi \left[ \phi^2 + \lambda_2 \left( \chi^2 - m_2^2 \right) \right] . \label{sec2-105}\end{aligned}$$ It is easy to see that Eq. is the consequence of Eq. : if we take a derivative from the LHS and RHS of Eq. then we shall receive Eq. . The boundary conditions are $$\begin{aligned} a(0) &=& a_0 , \label{sec2-110}\\ a'(0) &=& 0 , \label{sec2-120}\\ \phi(0) &=& \phi_0 , \quad \phi'(0) = 0 , \label{sec2-130}\\ \chi(0) &=& \chi_0 , \quad \chi'(0) = 0 . \label{sec2-140}\end{aligned}$$ The boundary condition - and Eq. give us the following constraint $$V_0 = \frac{\lambda_1}{4} \left( \phi^2_0 - m_1^2 \right)^2 + \frac{\lambda_2}{4} \left( \chi^2_0 - m_2^2 \right)^2 + \frac{1}{2} \phi^2_0 \chi^2_0, \label{sec2-150}$$ Numerical investigation {#num} ======================= For the numerical calculations we choose the following parameters values $$a_0 = \phi_0 = 1, \quad \chi_0 = \sqrt{0.6}, \quad \lambda_1 = 0.1, \quad \lambda_2 = 1.0 . \label{sec3-10}$$ We apply the methods of step by step approximation for finding of numerical solutions using the MATHEMATICA package (the details of similar calculations can be found in Ref. [@Dzhunushaliev:2003sq], the corresponding MATHEMATICA program can be found in \*.tar.gz file of the archived version of this paper [@Dzhunushaliev:2006vv]). **Step 1**. On the first step we solve Eq. (having zero approximations $a_0(y) = a_0, \chi_0(y) = m_1 \tanh y$). The regular solution exists for a special value $m^*_{1,i}$ only. For $m_1 < m^*_{1,i}$ the function $\chi_i(y) \rightarrow +\infty$ and for $m_1 > m^*_{1,i}$ the function $\chi_i(y) \rightarrow -\infty$ (here the index $i$ is the approximation number). One can say that in this case we solve *a non-linear eiqenvalue problem*: $\chi_i^*(y)$ is the eigenstate and $m_{1,i}^*$ is the eigenvalue on this Step. **Step 2**. On the second step we solve Eq. using zero approximation $a_0(y)$ for the function $a(y)$ and the first approximation $\chi_1^*(y)$ for the function $\chi(y)$ from the Step 1. For $m_2 < m^*_{2,1}$ the function $\phi_1(y) \rightarrow +\infty$ and for $m_2 > m^*_{2,1}$ the function $\phi_1(y) \rightarrow -\infty$. Again we have *a non-linear eiqenvalue problem* for the function $\phi_1(y)$ and $m^*_{2,1}$. **Step 3**. On the third step we repeat the first two steps that to have the good convergent sequence $\phi_i^*(y), \chi_i^*(y)$. Practically we have made three approximations. **Step 4**. On the next step we solve Eq. which gives us the function $a_1(y)$. **Step 5**. On this step we repeat Steps 1-4 necessary number of times that to have the necessary accuracy of definition of the functions $a^*(y), \phi^*(y), \chi^*(y)$. After Step 5 we have the solution presented on Fig. \[fig1\]. These numerical calculations give us the eigenvalues $m_1^* \approx 2.122645756$, $m_2^* \approx 1.3721439906788$ and eigenstates $a^*(y), \phi^*(y), \chi^*(y)$. The derived solution was verified by using the standard numerical method of solving the differential equations in the MATHEMATICA package (the corresponding MATHEMATICA program can be found in \*.tar.gz file of the archived version of this paper [@Dzhunushaliev:2006vv]). It easy to see that the asymptotical behavior of the solution is $$\begin{aligned} a(y) &\approx& a_\infty e^{-k_a y} , \quad k_a^2 = \frac{1}{4} \left( V_0 - \frac{\lambda_2}{4} m_2^4 \right) , \label{sec3-20}\\ \phi(y) &\approx& m_1 + \phi_\infty e^{-k_\phi y} ,\quad k_\phi = 2k_a + \sqrt{4 k_a^2 + 2 \lambda_1 m_1^2} , \label{sec3-30}\\ \chi(y) &\approx& \chi_\infty e^{-k_\chi y} ,\quad k_\chi = 2k_a + \sqrt{4 k_a^2 + m_1^2 - \lambda_2 m_2^2} , \label{sec3-40}\end{aligned}$$ where $a_\infty, \phi_\infty, \chi_\infty$ are constants. The dimensionless energy density is $$e(y) = 2 \left( \frac{\varkappa}{3} \right)^2 \varepsilon(y) = \frac{1}{4} \left[ \phi'^2 + \chi'^2 + \frac{\lambda_1}{2} \left( \phi^2 - m_1^2 \right)^2 + \frac{\lambda_2}{2} \left( \chi^2 - m_2^2 \right)^2 + \phi^2 \chi^2 - 2 V_0 \right] \label{sec3-50}$$ and it is presented in Fig. \[fig2\]. Taking into account that the quantity $V\left( \phi(\infty), \chi(\infty) \right)$ is absolutely similar to a 5D cosmological constant, we can introduce a dimensionless brane tension $$\sigma = 2 \int \limits_0^\infty \biggl[ e(y) - V\Bigl( \phi(\infty), \chi(\infty) \Bigl) \biggl] dy \approx 0.74 . \label{sec3-55}$$ According to Eq. one can define the thickness $\Delta$ of the presented thick brane as $$\Delta \approx \max \left\{ k_\phi, k_\chi \right\}. \label{sec3-60}$$ The key role for understanding why such regular solution may exist belongs to the fact that the potential has the local and global minima. The profile of the potential $V(\phi, \chi)$ is presented in Fig. \[fig3\]. ![The profile of the potential $V(\phi, \chi)$.[]{data-label="fig3"}](potential){height="9cm" width="9cm"} Trapping of the matter ====================== Now we would like to consider trapping of the electromagnetic and spinor fields on the above derived thick brane. The Lagrangian of interacting electromagnetic and scalar fields is taken from : $$\label{Lagr_int} L_{eff}=-\frac{1}{4} \tilde{F}_{BC} \tilde{F}^{BC} + \alpha \phi^2 \tilde{A}_B \tilde{A}^B - m^2 \tilde{A}_B \tilde{A}^B ,$$ where $\tilde{F}_{BC} = \tilde{A}_{B,C} - \tilde{A}_{C,B}$ is the 5D electromagnetic tensor with 5-dimensional vector potential $\tilde{A}_B(x^{D})$ and scalar field $\varphi(y)$ depending only on the extra coordinate $y$; $\alpha$ - an arbitrary constant, $m$ is the mass of vector field $\tilde{A}_B$. The 5D Maxwell equations will be: $$\label{max} D_C \tilde{F}^{BC}=\tilde{A}^B(x^{D}) \left[ \alpha \phi^2 - m^2 \right].$$ Let us rewrite Eq. (\[max\]) as follows: $$\label{max1} D_{\nu} \tilde{F}^{B \nu}+D_5 \tilde{F}^{B 5}= \tilde{A}^B(x^{D}) \left[ \alpha \phi^2 - m^2 \right].$$ We will use the gauge $\tilde{A}_5=0$ and search for a solution of (\[max1\]) in the form: $$\begin{aligned} D_\nu \tilde{F}^{\mu \nu} &=& 0, \label{max2}\\ D_5 \tilde{F}^{B 5} &=& \tilde{A}^B(x^{D}) \left[ \alpha \phi^2 - m^2 \right] \label{max3}\end{aligned}$$ For the solution we will use the following ansatz $$\tilde{A}^B(x^{D}) = A^B(x^{\mu}) f(y),$$ where $A^B(x^{\mu})$ is the 4D electromagnatic potential function only on 4D coordinates. Then from Eq’s we will have $$\begin{aligned} D_\nu F^{\mu \nu} &=& 0, \label{sec4-10}\\ f^{\prime \prime} + 4\frac{a^{\prime}}{a} f^{\prime} &=& \frac{1}{a^4} \frac{d}{dy} \left( a^4 \frac{df}{dy} \right) = f \left( \alpha \phi^2 - m^2 \right) \label{sec4-20}\end{aligned}$$ where the first equation is the usual 4D Maxwell equations on the brane. The solution of the second equation on the background of the thick brane is presented in Fig.\[EM\]. Here it is necessary to note that again the regular solution $f(y)$ exists for an exceptional value of the parameter $m$ only. It is easy to see from Eq. : this equation is exactly Schrodinger equation with the potential $\alpha \phi^2$ (which is a hole). Eq.  has a regular solution describing a particle in a hole for an exceptional value of $m$ that is an eigenvalue of the Schrodinger equation . As one can see, the EM field is trapped on the 4D brane. In this case the electromagnetic fields in the bulk are $$\tilde{A}(x^B) = A(x^\mu) f(y)$$ where $f(y)$ is the exponentially decreasing function. Let us consider further the question about trapping of fermion fields on the brane. In the simplest case such a possibility was pointed out in Ref. [@Rub] at consideration of the brane model as the model of domain wall. In this work the model of one real scalar field $\phi$ with two degenerated minima was introduced for description of the domain wall in 5D spacetime. In this case existing kink solution has its asymptotes in these minima with constant values of the field $\phi$. In our case similar situation occurs: two scalar fields $\phi, \chi$ create the system with two local minima, and the solutions tend asymptotically to one of these minima where the field $\chi$ tends to zero and $\phi$ to the constant values as in the case from Ref. [@Rub]. It allows us to investigate trapping of fermions on the brane for our case by analogy with Ref. [@Rub]. The curved space 5D gamma matrices are $$\Gamma^{\mu} = \frac{1}{\sqrt{a}} \gamma^{\mu},\qquad \Gamma^{r}=-i \gamma^{5},$$ where $\gamma^{\mu}$ and $\gamma^{5}$ are the usual Dirac matrices in 4D theory $$\gamma^\mu = \left\{ \left( \begin{array}{cc} 0 & I \\ I & 0 \end{array} \right), \left( \begin{array}{cc} 0 & \vec \sigma \\ -\vec \sigma & 0 \end{array} \right) \right\}, \quad \gamma^5 = \left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \end{array} \right) \label{1.3}$$ where $\sigma^i$ are usual Pauli matrices in flat spacetime. Then using the action for interacting scalar $\varphi$ and fermion $\Psi$ fields we have $$\label{action_int} S_{\Psi}=\int d^4x\, dr \left( i \overline{\Psi} \Gamma^A D_A \Psi - h \phi \overline{\Psi} \Psi \right)$$ here $\phi$ is the scalar field from the Lagrangian and $h$ is a constant. The Dirac equation can be written in the form $$\label{Dirac_eq} i \Gamma^A D_A \Psi-h \phi(y) \Psi =0.$$ Here $D_A=\partial_A+\Upsilon_A$, where pseudo-connection $\Upsilon_A$ can be defined as follows [@Ying] $$\Upsilon_A=\frac{1}{2} e_{\bar{M}}^N \left(\partial_A e_N^{\bar{M}}-\partial_N e_A^{\bar{M}}\right),$$ where the vielbein $e_A^{\bar{M}}$ is defined via $g_{A B}= e_A^{\bar{M}} e_B^{\bar{N}} \eta_{\bar{M} \bar{N}}$, and the inverse vielbein $e_{\bar{M}}^A$ via $g^{A B}= e_{\bar{M}}^A e_{\bar{N}}^B \eta^{\bar{M} \bar{N}}$. For our case $\Upsilon_A=\left(0, 0, 0, 0, a^{\prime}/a\right)$. Let us consider ansatz $$\label{ansatz1} \Psi (x^B) = \psi(x^\mu) \Psi_0(y)$$ If we are interested in localization of zero modes, then, as it was shown in Ref. [@Jack], there are the solutions of Eq. (\[Dirac\_eq\]) with 4D mass $m=0$. For the zero mode $\gamma^{\mu}D_{\mu}\psi=0$, and the Dirac equation (\[Dirac\_eq\]) turns out in the equation: $$\label{Dirac_eq_1} \gamma^5 \left( \partial_r +\frac{a^{\prime}}{a}\right) \Psi_0 = h \phi(y) \Psi_0,$$ where $^\prime$ means the derivative with respect to $r$. Eq. with account of has the following solution: $$\label{Dirac_sol} \Psi = \exp{\left[ -\int_{0}^{r} dr^{\prime}\left(\frac{a^{\prime}}{a}+ h \phi(r^{\prime})\right) \right]} \psi (x^\mu),$$ where $\psi (x^\mu)$ is the usual solution of 4D Weyl equation, and the condition $\gamma_5 \Psi_0=-\Psi_0$ is taken into account. As it was shown in Section \[num\], the sum $\left(a^{\prime}/a+ h \phi\right)$ tends asymptotically to some constant. So the zero mode (\[Dirac\_sol\]) is localized near $r=0$, i.e. on the brane, and decreases exponentially at large $r$: $\Psi_0 \propto \exp{(-m_5 \left| y \right|)}$. Let us note that one can include the function $\chi$ in Eq’s and by the following way:\ $\alpha \phi^2 \tilde{A}_B \tilde{A}^B \stackrel{\text{change}}{\longrightarrow} \alpha \left( \phi^2 + \chi^2 \right) \tilde{A}_B \tilde{A}^B$ and $h \phi \overline{\Psi} \Psi \stackrel{\text{change}}{\longrightarrow} h \left( \phi + \chi \right) \overline{\Psi} \Psi$ but it does not matter because the asymptotical behavior of the function $\phi \stackrel{r \rightarrow \infty}{\longrightarrow} m_1$ is important only (as $\chi \stackrel{r \rightarrow \infty}{\longrightarrow} 0$). Let us note that we do not consider trapping of scalar fields on this brane. The reason is very simple: we have shown *exactly* that two scalar fields with Lagrangian are confined on the brane. The situation is even better: these scalar fields create the brane ! It is necessary note that in the process of numerical calculation we have obtained a domain wall solution without gravity, i.e. two scalar fields can create the solution with the planar symmetry and switching on the gravity does not destroy this solution. Discussion and conclusions ========================== Now we would like to list the essential specialities of the presented solution: 1. The existence of the solution crucially depends on the number of interacting scalar fields $(n > 1)$ and the presence of the non-trivial potential $V(\phi, \chi)$ which has local and global minima. At the infinity the scalar fields tend to local minimum and the potential has alternating sign when $r \in [0, \infty]$ that to the existence of the presented solution. The numerical investigation shows that in the presence of one scalar field the similar solution does not exist. 2. The advantage of the presented solution is that the asymptotical behavior of the scalar field $\phi$ allow us to obtain trapping of electromagnetic and spinor fields on the brane. 3. Let us note that the thick brane solution presented here differs from the thick brane solutions presented in Ref’s [@DeWolfe:1999cp] and [@Bronnikov:2005bg] that: 1. thick brane solution from Ref. [@DeWolfe:1999cp] is obtained for the scalar field with the potential unbounded from below that in contrast with our potential which is bounded from below. 2. in Ref. [@Bronnikov:2005bg] the thick brane solution is obtained for scalar fields having non-trivial asymptotical topological structure in contrast with our solution. 4. The solution is topologically trivial. It means that at the infinity two scalar fields do not form a hedgehog configuration in contrast with the thick brane solutions presented in Ref. [@Bronnikov:2005bg]. 5. The quantity $V(\phi(\infty), \chi(\infty))$ can be considered as a 5D cosmological constant $\Lambda$. 6. \[4\] In Ref. [@Dzhunushaliev:2003sq] it is shown that after some simplification and assumtions the SU(3) gauge Lagrangian can be reduced to the Lagrangian describing interacting scalar fields $\phi$ and $\chi$. This remark allows us to assume that a real thick brane can be formed by a 5D gauge condensate which is described by interacting scalar fields. 7. According to the previous item (\[4\]) the 5D mechanism of trapping the matter on a thick brane may be similar to the confinement mechanism in 4D quantum chromodynamics. In this case trapping of the corresponding quantum gauge fields on the thick brane is non-perturbative and can not be investigated using Feynman diagram technique. 8. If the thick brane is formed with the help of a gauge condensate then the problem of the stability of the thick brane becomes very non-trivial. It occurs because a non-static condensate has to be described in much more complicated manner than static condensate in Ref. [@Dzhunushaliev:2003sq]. It is connected to that fact that the change in time of quantum object is connected not only to the change of this quantity but also to the change of a wave function as well. 9. From the mathematical point of view the presented solution is an eigenstate for a nonlinear eigenvalue problem. [99]{} N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B 429, 263 (1998);\ I. Antoniadis, S. Dimopoulos and G. Dvali, Nucl. Phys. B 516, 70 (1998). L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370 (1999);\ L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 4690 (1999). M. Gogberashvili, Int. J. Mod. Phys. D. 11, 1635 (2002);\ Mod. Phys. Lett. A, 14, 2025 (1999). V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett., **B125**, 136 (1983). K. Akama, ”Pregeometry“ in Lecture Notes in Physics, 176, Gauge Theory and Gravitation, Proceedings, Nara, 1982, edited by K. Kikkawa, N. Nakanishi and H. Nariai, 267-271 (Springer-Verlag,1983), (A TeX-typeset version is also available in e-print hep-th/0001113. Merab Gogberashvili, Douglas Singleton, Phys.Lett., **B582**, 95-101(2004), hep-th/0310048;\ Merab Gogberashvili, Douglas Singleton, Phys.Lett., **B582**, 95-101(2004), hep-th/0310048. Douglas Singleton, Phys.Rev., **D70**, 065013(2004), hep-th/0403273. K.A. Bronnikov and B.E. Meierovich, Grav. Cosmol. 9, 313-318 (2003),\ Sergei T. Abdyrakhmanov, Kirill A. Bronnikov, Boris E. Meierovich, Grav.Cosmol., **11**, 82-86 (2005); gr-qc/0503055. M. Gremm, Phys. Lett., **B478**, 434 (2000). M. Gremm, Phys. Rev., **D62**, 044017 (2000);\ A. Davidson and P.D. Mannheim, “Dynamical localizations of gravity”, hep-th/0009064;\ C. Csaki, J. Erlich, T.J. Hollowood and Y. Shirman, Nucl. Phys., **B581**, 309 (2000);\ A. Wang, Phys. Rev., **D66** 024024 (2002);\ D. Bazeia, F.A. Brito and J.R. Nascimento, Phys. Rev., **D68** 085007 (2003);\ D. Bazeia, C. Furtado and A. R. Gomes, JCAP [**0402**]{}, 002 (2004); hep-th/0308034. S. Kobayashi, K. Koyama and J. Soda, Phys. Rev., **D65** 064014 (2002). A. Melfo, N. Pantoja and A. Skirzewski, Phys. Rev., **D67** 105003 (2003). R. Ghoroku and M. Yahiro, “Instability of thick brane worlds”, hep-th/0305150. C. Barceló, C. Germani and C.F. Sopuerta, Phys. Rev., **D68** 104007 (2003). N. Barbosa-Cendejas and A. Herrera-Aguilar, JHEP [**0510**]{}, 101 (2005), hep-th/0511050. C. Kokorelis, Nucl. Phys. B[**677**]{} 115 (2004), hep-th/0207234. V. Dzhunushaliev, Hadronic J. Suppl.  [**19**]{}, 185 (2004), hep-ph/0312289. G.H. Derrick, J. Math Phys. [**5**]{} 1252 (1964). K.A. Bronnikov and G.N. Shikin, Grav. Cosmol. 8, 107 (2002), gr-qc/0109027. O. DeWolfe, D. Z. Freedman, S. S. Gubser and A. Karch, Phys. Rev. D [**62**]{}, 046008 (2000), hep-th/9909134. K. A. Bronnikov and B. E. Meierovich, J. Exp. Theor. Phys.  [**101**]{}, 1036 (2005), \[Zh. Eksp. Teor. Fiz.  [**101**]{}, 1184 (2005)\], gr-qc/0507032. V. Dzhunushaliev, “Thick brane solution in the presence of two interacting scalar fields,” gr-qc/0603020. V. Rubakov, M. Shaposhnikov, [*Phys. Lett.*]{} [**B125**]{}, 136 (1983);\ V. Rubakov, [*Physics-Uspekhi*]{} [**44**]{}, 871 (2001). Ying-Qiu Gu, “Simplification of the covariant derivatives of spinors”, gr-qc/0610001. R. Jackiw, C. Rebbi, [*Phys. Rev.*]{} [**D13**]{}, 3398 (1976). [^1]: Senior Associate of the Abdus Salam ICTP

--- abstract: 'We construct the regularized Wheeler–De Witt operator demanding that the algebra of constraints of quantum gravity is anomaly free. We find that for only a small subset of all wavefunctions being integrals of scalar densities this condition can be satisfied. It turns out that the resulting operator is much simpler than the one used in [@JK] to find exact solutions of Wheeler–De Witt equation. We proceed to finding exact solutions of quantum gravity and we discuss their interpretation making use of the quantum potential approach to quantum theory.' author: - | A. Błaut[^1] and J. Kowalski–Glikman[^2]\ Institute for Theoretical Physics\ University of Wrocław\ Pl. Maxa Borna 9\ Pl–50-204 Wrocław, Poland title: Algebra of Constraints and Solutions of Quantum Gravity --- Introduction ============ One of the most outstanding problems of modern theoretical physics is the construction of quantum theory of gravity [@reviews]. Indeed, it have been claimed many times that various unsolved problems like the cosmological constant problem, the problem of origin of the universe, the problem of black holes radiation will find their ultimate solution once this theory is finally constructed and properly understood. Some [@Emperor], claim that the theory of quantum gravity will also shed some light on the fundamental problems of quantum mechanics and even on the origin of mind. These all prospects are very exciting indeed, however the sad fact remain that the shapes of the future theory are still very obscured. Nowadays there are two major ways of approaching the problem of quantum gravity. The first one is associated with the broad term “superstrings”. In this approach the starting point is a two-dimensional quantum field theory which yields quantum gravity as one of resulting low-energy effective theories. It is clear that in superstrings, as in other, less developed approaches in whose gravity appears as an effective theory, it does not make sense to try to “quantise” classical gravity. In the canonical approach one does something opposite: the idea is to pick up some structures which appear already at the classical level and then promote them to define the quantum theory. In both the standard canonical approach in metric representation, which we will follow here, and in the approach based on loop variables [@loop], these fundamental structures are constraints of the classical canonical formalism reflecting the symmetries of the theory and their algebra. There are good reasons for such an approach. The equivalence principle is the main physical principle behind the classical theory of gravity; this principle leads to the general coordinate invariance and selects the Einstein–Hilbert action as the simplest possible one. Another building block of the quantum theory is the quantisation procedure. Here one encounters the problem as to if a generalization of the standard Dirac procedure of quantisation of gauge theories is necessary. This would be the case if one shows that the standard approach is not capable of producing any interesting results. It is not excluded that this may be eventually the result of possible failure of investigations using standard techniques, however, in our opinion, at the moment there is no reason to modify the basic principles of quantum theory. Our starting point consists therefore of - The classical constraints of Einstein’s gravity: the diffeomorphism constraint generating diffeomorphism of the spatial three-surface “of constant time” $${\cal D}_a=\nabla_b\, \pi^{ab}$$ and the hamiltonian constraint generating “pushes in time direction”: $${\cal H} = \kappa^2 G_{abcd}\pi^{ab}\pi^{cd} - \frac1{\kappa^2}\sqrt h (R +2\Lambda)$$ In the formulas above $\pi^{ab}$ are momenta associated with the three-metric $h_{ab}$, $$G_{abcd}=\frac1{2\sqrt h}\left(h_{ac}h_{bd}+h_{ad}h_{bc}-h_{ab}h_{cd}\right)$$ is the Wheeler–De Witt metric, $R$ is the three-dimensional curvature scalar, $\kappa$ is the gravitational constant, and $\Lambda$ the cosmological constant. The constraints satisfy the following algebra $$[{\cal D}, {\cal D}] \sim {\cal D},\label{difdif}$$ $$[{\cal D}, {\cal H}] \sim {\cal H},\label{difham}$$ $$[{\cal H}, {\cal H}] \sim {\cal D}. \label{hamham}$$ - The rules of quantisation given by the metric representation of the canonical commutational relations $$\left[\pi^{ab}(x),h_{cd}(y)\right]=- i\delta^{(a}_c\delta^{b)}_d\delta(x,y),$$ $$\pi^{ab}(x)=-i\frac{\delta}{\delta h_{ab}(x)}.$$ Sadly, in the canonical approach, the points (i) and (ii) above encompass the whole of the input in our disposal in construction of the quantum theory. In particular, we do not know what is the correct physical inner product, and thus we do not know if the relevant operators are hermitean or not. Besides, we do not even know if we should demand these operators to be hermitean: the hamiltonian annihilates the physical states (the famous time problem [@ishamtime]) and thus unitary evolution does not play the privileged role anymore. It follows that we cannot distinguish “relevant” wave functions by demanding that they are normalizable, as in the case of quantum mechanics, in fact, since the probabilistic interpretation of the “wavefunction of the universe” is doubtful, it is not clear if the norm of this wavefunction is to be 1. In the recent paper [@JK] a class of exact solutions of the Wheeler–De Witt equation was found. In that paper we used the heat kernel to regularize the hamiltonian operator and inserted the particular ordering. The question arises what is the level of arbitrariness in this construction. In other words, could we construct other (possibly simpler) regularized hamiltonian operators and what would be their properties? This question is the subject of the present paper. It is clear from the discussion above that the only principle, we can base our construction on is the principle that the algebra of constraints is to be anomaly–free, that is, the corresponding algebra of commutators of quantum constraints is weakly identical with the classical one. This means that the structure of the Poisson bracket algebra (\[difdif\]–\[hamham\]) is to be preserved, in the sense which will be explained below, on the quantum level. The following section is devoted to the analysis of this problem. In section 3 we investigate solutions of the resulting equations, and in section 4 we seek interpretation of the wavefunctions making use of the quantum potential approach to quantum mechanics. In the final section we draw our conclusions and describe the open problems. The commutator algebra and construction of regularized operators ================================================================ As we explained in Introduction, our starting point in construction of the quantum hamiltonian operator (the Wheeler–De Witt operator) is the algebra (\[difdif\]–\[hamham\]) and we demand that the same algebra holds on the quantum level. At this point we encounter immediately the problem, well known from the investigations of anomalies in quantum field theories, that the sole algebra of regularized operators is meaningless unless the space of states on which these operators act is defined [*a priori*]{}[^3]. This follows from the fact that the transition from regularised to renormalised action of an operator depends crucially on what particular state this operator acts (see below.) We will chose our starting space of states to be the space of integrals over compact three-space $M$ of scalar densities like ${\cal V}=\int_M\sqrt h$, ${\cal R}=\int_M\sqrt h R$, etc.; $$\Psi = \Psi({\cal V}, {\cal R}, \ldots).$$ We choose the following representation of the diffeomorphism constraint $${\cal D}_a(x)=-i\nabla_b^{x} \frac{\delta}{\delta h_{ab}(x)},$$ where we employed the notation $\nabla_b^{x}$ meaning that the covariant derivative acts at the point $x$. Then we see that diffeomorphism constraint annihilates all the states and the commutator relation (\[difdif\]) is identically satisfied. Moreover we see that the relation (\[difham\]) reduces to the formal relation $${\cal D}({\cal H}\Psi)\sim {\cal H}\Psi.\label{difham1}$$ Now we must turn to the heart of the problem, the construction of the Wheeler–De Witt operator. It is well known that second functional derivative acting at the same point on a local functional produces divergent result. We deal with this problem by making the point split in the kinetic term, to wit $$G_{abcd}(x)\pi^{ab}(x)\pi^{cd}(x) \Longrightarrow \int\, dx'\, K_{abcd}(x,x';t) \frac{\delta}{\delta h_{ab}(x)}\frac{\delta}{\delta h_{cd}(x')},$$ where $K_{abcd}(x,x';t)$ satisfies $$\lim_{t\rightarrow 0^+}K_{abcd}(x,x';t)=\delta(x,x').$$ By virtue of the correspondence principle, we take $$K_{abcd}(x,x';t)=G_{abcd}(x')\triangle(x,x';t)\left(1 + K(x,t)\right),$$ where $$\triangle(x,x';t)=\frac{\exp\left(-\frac1{4t} h_{ab}(x-x')^a(x-x')^b\right)}{4\pi t^{3/2}}$$ and $K(x,t)$ is analytic in $t$. Next we must resolve the ordering ambiguity in the operator ${\cal H}$. To this end we add the new term $L_{ab}(x)\frac{\delta}{\delta h_{ab}(x)}$, where $L_{ab}$ is a tensor to be derived along with $K(x,t)$. Thus the final form of the Wheeler–De Witt operator is[^4] $${\cal H}(x)=\kappa^2\int\, dx'\, K_{abcd}(x,x';t) \frac{\delta}{\delta h_{ab}(x)}\frac{\delta}{\delta h_{cd}(x')} +$$ $$+ L_{ab}(x)\frac{\delta}{\delta h_{ab}(x)} + \frac1{\kappa^2}\sqrt h (R +2\Lambda).\label{qham}$$ To set the stage, we still need to define the action of operators on states. To this end we must discuss the issue of regularization and renormalization. The operator (\[qham\]) acting on a state (defined as an integral of a scalar density) produces, in general, terms with arbitrary (positive and negative) powers of $t$. This provides the regularized version of the operator since all the terms are finite, and singularities of the form $\delta(0)$ are traded for terms which are singular for $t\rightarrow0$. Observe that, as noted above, the singular part of the action of the operator on a state depends on this state. To renormalize, we follow the procedure proposed by Mansfield [@Mansfield] which result in the following: the terms with positive powers of $t$ are dropped, and the singular terms of the form $t^{-k/2}$ are replaced by the renormalization coefficients $\rho^k$. This procedure provides us with the finite action of the Wheeler–De Witt operator on any state. Now we can turn to the interpretation of equation (\[difham1\]). We understand it in the following way. An operator acts on a state and after renormalization gives another state depending on renormalization constants. On this resulting state the second operator acts. Thus the formal relation (7) is defined to mean $${\cal D}\,({\cal H}\Psi)_{ren}\sim ({\cal H}\Psi)_{ren},\label{difhamf}$$ and, similarly, for the hamiltonian– hamiltonian commutator $$\left({\cal H}[N]\,\,({\cal H}[M]\Psi)_{ren}\right)_{ren} - \left({\cal H}[M]\,\,({\cal H}[N]\Psi)_{ren}\right)_{ren}=0\label{hamhamf}$$ since $\Psi$ is diffeomorphism invariant. In the formula above we used the smeared form of the Wheeler–De Witt operator $${\cal H}[M]=\int dx\, M(x){\cal H}(x).$$ Let us turn back to equation (\[difhamf\]). Since the action of diffeomorphism is standard, it suffices to check that $({\cal H}\Psi)_{ren}$ is a scalar density. But this is clearly the case: the first functional derivative acting on a state produces a tensor density ${\sf T}^{ab}(x')$. After acting by the second derivative and contracting indices, we obtain the terms of the form $${\sf T}_0(x')\delta(x',x) + {\sf T}_1(x')\circ \nabla^{x'}\circ \nabla^{x'} \delta(x',x) + {\sf T}_2(x')\circ \nabla^{x'}\circ \nabla^{x'}\circ \nabla^{x'}\circ \nabla^{x'}\delta(x',x) + \ldots$$ where $\circ$ denotes various indices contractions, and ${\sf T}_n$ are tensor densities. These terms are multiplied by $\triangle(x,x';t)$ and integrated over $x'$. Now we integrate by parts which results in replacing covariant derivatives acting on $K$ with appropriate powers of $t^{- 1}$ multiplied by some coefficients. After renormalization we obtain a scalar density as required. The action of the $L$ term clearly gives the same result. Thus [*For the states being integrals of scalar densities there is no anomaly in the diffeomorphism — hamiltonian commutator*]{} This result is quite important because the anomaly in the string theory appears in the diffeomorphism — hamiltonian commutator. Now we turn to the most complicated problem, the hamiltonian — hamiltonian commutator (\[hamhamf\]). Our goal will be to find the maximal space of states together with conditions defining coefficients $K$ and $L_{ab}$ of the Wheeler–De Witt operator. Our approach is based on the following [Claim]{}. [*If $({\cal H}\Psi)_{ren}$ contains terms which contain four or more derivatives of the metric like $R^2$, $R_{ab}R^{ab}$ etc., then (\[hamhamf\]) cannot be satisfied.*]{} We have checked this claim for terms proportional to square of three-curvature; it is clear from this computation that the claim holds for higher order terms as well unless there are some miraculous cancellations. We leave it as an open problem to check if the claim above is generally valid. Let us start with the simplest state $\Psi=1$. Then the action of the first smeared operator gives $$({\cal H}[M]\Psi)_{ren} = \int dx\, \sqrt h(x) M(x) (R(x) +2\Lambda).$$ After rather tedious computation one finds in the commutator the term proportional to $N\nabla^a M-M\nabla^a M$ which must vanish, to wit $$\rho^{(1)}\frac32\nabla_a K - \frac1{\kappa^2}\left(\nabla_a L + \nabla_b L_a^b\right) =0,\label{coeff}$$ where $L= h_{ab}L^{ab}$. Turning to the states depending of ${\cal V}=\int_M d^3x\, \sqrt h$, we find, taking the Claim above into account that $K$ and $L_{ab}$ can only contain terms at most linear in Ricci tensor. Given that, there is no anomaly. Thus we take $$L_{ab}=\frac1{\kappa}\alpha h_{ab} + \kappa(\beta h_{ab} R +\gamma R_{ab})$$ where in the first term we included the gravitational constant for dimensional resons. All states depending on integrals of scalars constructed from powers of curvature tensor will necessarily lead to terms excluded by virtue of the Claim. This means that not all states of this form will lead to the anomaly-free algebra: the wavefunction will have to satisfy equations guaranteeing that such terms are absent. These equations, for the case of states depending on ${\cal R}=\int_M d^3x\, \sqrt h R$ will further restrict the form of the regularized Wheeler–De Witt operator. Now we turn to the wavefunction $\Psi=\Psi({\cal R})$. As we argued above, in the action of the Wheeler–De Witt operator on this state all terms with four derivatives must vanish. These terms are $$\kappa^2\frac{\partial^2 \Psi}{\partial{\cal R}^2}\left( R_{ab}R^{ab} -\frac38 R^2\right) + \kappa\frac{\partial \Psi}{\partial{\cal R}}\left(-\gamma R_{ab}R^{ab} + \frac12(\gamma + \beta) R^2\right) =0,\label{R2}$$ from which we obtain conditions on the coefficients $$\beta = -\frac14\gamma,\label{solcoeff}$$ and from (\[R2\]) we find that $\Psi({\cal R})$ must be of the form $$\Psi({\cal R})= \exp\left(\frac{\gamma}{\kappa}{\cal R}\right).$$ Thus $L_{ab} = \frac{\kappa}\alpha h_{ab} + \kappa\gamma(R_{ab}- \frac14 h_{ab} R)$. But then it follows from (\[coeff\]) that $K(x,t)$ must be constant. It is possible, in principle, to construct $K$ from global integrals like ${\cal V}$ and/or ${\cal R}$, for example $K=t\frac{{\cal R}}{{\cal V}}$, however we will not pursue this (interesting) possiblity here. Thus the final form of the regularised Wheeler–De Witt operator is $${\cal H}(x)=\kappa^2\int\, dx'\, G_{abcd}(x')\triangle(x,x';t) \frac{\delta}{\delta h_{ab}(x)}\frac{\delta}{\delta h_{cd}(x')} +$$ $$+ \left(\frac1{\kappa}\alpha h_{ab}+ \kappa\gamma(-\frac14h_{ab}R + R_{ab})\right)(x) \frac{\delta}{\delta h_{ab}(x)}+ \frac1{\kappa^2}\sqrt h (R +2\Lambda).\label{qhamf}$$ The formula (\[qhamf\]) completes our construction of the Wheeler–De Witt operator. As compared to the choice made in the paper [@JK], where we used the heat kernel and $L_{ab}$ was its functional derivative, here we gained much more freedom in the form of two independent constants. In particular, we can make the constants $\alpha$ and $\gamma$ complex. This fact is very important in view of the quantum potential interpretation of our results (see Section 4), where it turns out that only complex wavefunctions lead to time-evolving universes. Solutions ========= From the previous section we know that the most general form of the Wheeler- De Witt operator satisfying our criteria is given by equation (\[qhamf\]), with coefficients $\alpha$ and $\gamma$ being still not fixed. Now, employing this operator, we will try to find a class of solutions of the Wheeler–De Witt equation. It should be stressed at this point that we regard the existence of a maximal possible space of solutions as an ultimate condition fixing the operator completely. Thus our goal is twofold: to find solutions and to fix the operator as to allow for the maximal possible number of them. We will consider only the states of the form $\Psi = \Psi({\cal V}, {\cal R})$. Since we have already taken care of the terms proportional to squares of Ricci tensor in (\[solcoeff\]), we have two equations for multipliers of $\sqrt h R(x)$ and $\sqrt h$. They read, for $\sqrt h R$: $$-\frac14\kappa^2\frac{\partial^2\Psi}{\partial{\cal V}\partial{\cal R}}+ \frac78\kappa^2\rho^{(3)}\frac{\partial\Psi}{\partial{\cal R}} + \frac18\kappa\gamma\frac{\partial\Psi}{\partial{\cal V}} + \frac1{2\kappa}\alpha\frac{\partial\Psi}{\partial{\cal R}} + \frac1{\kappa^2}\Psi=0,\label{R}$$ and for $\sqrt h$: $$-\frac38\kappa^2\frac{\partial^2\Psi}{\partial{\cal V}^2} -\frac{21}{8}\kappa^2\rho^{(3)}\frac{\partial\Psi}{\partial{\cal V}} - \frac34\kappa^2\rho^{(5)}\frac{\partial\Psi}{\partial{\cal R}} + \frac3{2\kappa}\alpha\frac{\partial\Psi}{\partial{\cal V}} + \frac2{\kappa^2}\Lambda\Psi=0.\label{L}$$ Now we must consider two cases: 1. $\Psi$ does not depend on ${\cal R}$. Then from equations above we find $$\frac18\kappa\gamma\frac{\partial\Psi}{\partial{\cal V}} + \frac1{\kappa^2}\Psi=0,\label{R1}$$ $$-\frac38\kappa^2\frac{\partial^2\Psi}{\partial{\cal V}^2} -\left(\frac{21}{8}\kappa^2\rho^{(3)} - \frac3{2\kappa}\alpha\right)\frac{\partial\Psi}{\partial{\cal V}} + \frac2{\kappa^2}\Lambda\Psi=0.\label{L1}$$ To make equations (\[R1\]) and (\[L1\]) consistent with each other, the coefficients must satisfy the relation $$\frac{2\Lambda}{\kappa^2}\gamma^2 + \left(\frac{21\rho^{(3)}} {\kappa}- \frac{3\alpha}{16\kappa^4} \right)\gamma-24\frac{1}{\kappa^4}=0 \label{cond1}$$ and the solution is $$\Psi({\cal V})=\exp\left(-\frac{8}{\kappa^3\gamma}{\cal V}\right). \label{sol1}$$ 2. In the case when $\Psi$ depends on both ${\cal V}$ and ${\cal R}$, we must take into account the fact that $\Psi=\tilde\Psi({\cal V})\exp\left(\frac{\gamma}{\kappa}{\cal R}\right)$. Then we find the following equations $$-\frac18\kappa\gamma\frac{\partial\tilde\Psi}{\partial{\cal V}}+ \left(\frac78\kappa\gamma\rho^{(3)} + \frac1{2\kappa^2}\alpha\gamma + \frac1{\kappa^2}\right)\tilde\Psi=0,\label{R3}$$ $$-\frac38\kappa^2\frac{\partial^2\tilde\Psi}{\partial{\cal V}^2} -\left(\frac{21}{8}\kappa^2\rho^{(3)}- \frac3{2\kappa}\alpha\right)\frac{\partial\tilde\Psi}{\partial {\cal V}} - \left(\frac34\rho^{(5)}\frac{\gamma}{\kappa}- \frac2{\kappa^2}\Lambda\right)\tilde\Psi=0.\label{L3}$$ From equation (\[R3\]) we find the solution for $\tilde\Psi$; thus $$\Psi({\cal V},{\cal R})=\exp\left\{\left( 7\rho^{(3)}+\frac{4\alpha}{\kappa^3}+\frac{8}{\kappa^3\gamma}\right) {\cal V}\right\} \exp\left\{\frac{\gamma}{\kappa}{\cal R}\right\}.\label{sol2}$$ Substituting (\[sol2\]) into (\[L3\]) we find another condition on the coefficients $\gamma$ and $\alpha$. Together with (\[cond1\]) it forms a system of equations which turns into a sixth order algebraic equation for $\gamma$. Each of the solutions of the latter defines unambigously the operator and the set of its solutions. Quantum potential interpretation ================================ In the previous section we found a class of solutions of the Wheeler - De Witt equation. However the physical interpretation of these “wavefunctions of the universe” is quite obscured. It turns out that there exists a nice device which makes it possible to interpret a wavefunction in terms of modified particle or field dynamics. This approach is an extension of works of David Bohm on interpretation of quantum mechanics [@Bohm] and was presented in [@qp] (see also [@shtanov].) It should be stressed at this point that we use here the quantum potential approach solely as a technical device to picture the wavefunction and we do not attempt to discuss the issue of interpretation of quantum theory. As compared with the work [@qp] here we have to do with one important modification resulting from the presence of the $L$ term in our Wheeler - De Witt operator. Therefore we repeat firs the most important steps, referring the reader to the original paper [@qp] for more details. Assume that the wavefunction of the universe is of the form $$\Psi=e^{\Gamma}e^{i\Sigma}.$$ The idea is to substitute this wavefunction to the Wheeler - De Witt equation and consider only the real part of the resulting equation. We obtain $$-\kappa^2G_{abcd}(x)\frac{\delta\Sigma}{\delta h_{ab}(x)}\frac{\delta\Sigma}{\delta h_{cd}(x)}+\frac1{\kappa}\sqrt{h(x)} (R(x)+2\Lambda) +\Re(L)_{ab}(x)\frac{\delta\Gamma}{\delta h_{ab}(x)}$$ $$-\Im(L)_{ab}(x)\frac{\delta\Sigma}{\delta h_{ab}(x)} +e^{-\Gamma} \kappa^2\left(\frac{\delta^{2}e^{\Gamma}}{\delta h^{2}}\right)_{ren}(x)=0, \label{HJ}$$ where $\Re(L)_{ab}$ and $\Im(L)_{ab}$ denote the real and imaginary part of $L_{ab}$, respectively In the last term we used the abbreviated notation to indicate that the action of the second functional derivative is renormalised. Now one identifies momenta with the (functional) gradient of $\Sigma$, to wit $$p^{ab}(x) = \frac{\delta\Sigma}{\delta h_{ab}(x)}.\label{p}$$ Then the first two terms in (\[HJ\]) are identical with the hamiltonian constraint of classical general relativity. The remaining terms are understood as quantum corrections (if we reintroduced $\hbar$ all these terms would become multiplied by $\hbar^2$.) The wave function is subject to the second set of equations, namely the ones enforcing the three dimensional diffeomorphism invariance. These equations read (for imaginary part) $$\nabla^a\frac{\delta\Sigma}{\delta h_{ab}(x)} = \nabla^a\, p_{ab} =0\label{3diff}$$ Thus our theory is defined by two equations (\[HJ\]) with functional derivatives of $\Sigma$ replaced by $p^{ab}$ as in (\[p\]), and (\[3diff\]). Now we can follow without any alternations the derivation of Gerlach [@gerlach] to obtain the full set of ten equations governing the quantum gravity theory in quantum potential approach $$\begin{aligned} 0&=& {\cal H}^a = \nabla_a\, p^{ab} ,\label{diff1}\\ 0&=& {\cal H}_{\bot} = -\kappa^2G_{abcd}(x)p^{ab}p^{cd}+\frac1{\kappa^2}\sqrt{h(x)} (R(x)+2\Lambda)\nonumber \\ &+&\Re(L)_{ab}(x)\frac{\delta\Gamma}{\delta h_{ab}(x)} -\Im(L)_{ab}(x)p^{ab} +\kappa^2e^{-\Gamma} \left(\frac{\delta^{2}\,e^{\Gamma}}{\delta h^{2}}\right)_{ren}(x), \label{HQ}\\ &&\dot{h}_{ab}(x,t) = \left\{ h_{ab}(x,t),\, {\cal H}[N,\vec{N}]\right\},\label{1}\\ &&\dot{p}^{ab}(x,t) = \left\{ p^{ab}(x,t),\, {\cal H}[N,\vec{N}]\right\}.\label{2}\end{aligned}$$ In equations above, $\{\star,\, \star\}$ is the usual Poisson bracket, and $${\cal H}[N,\vec{N}]=\int\, d^3x \left( N(x){\cal H}_{\bot}(x) + N^a(x){\cal H}_a(x)\right)$$ is the total hamiltonian (which is a combination of constraints). It might seem puzzling at the first sight why to a single wavefunction there corresponds a set of equations with, clearly, many solutions. The resolution of this problem is that the wavefunction, as a rule, is sensitive only to some aspects of the configuration. For example, the wavefunction (\[sol1\]) depends on ${\cal V}$ only, and thus any configuration with given volume leads to the same numerical value of it. The above dynamical equations provide us with much more detailed information concerning the dynamics of the system than the wavefunction alone. Now we apply this formalism to the case of the wavefunction depending only on ${\cal V}$, (\[sol1\]) Then $\Gamma = -\frac{8}{\kappa^3\gamma\gamma^*}{\Re(\gamma)\cal V}$. Let us inspect equation (\[HQ\]). The term $$\left(\frac{\delta^{2}e^{\Gamma}}{\delta h^{2}}\right)_{ren}(x)$$ provides us only with modification of the cosmological constant. The term $$\Re(L)_{ab}(x)\frac{\delta\Gamma}{\delta h_{ab}(x)}$$ modifies both the cosmological constant [*and*]{} the coefficient of the term $\sqrt h R$. Taken together these modification can be written as $$\frac1{\kappa^2}\sqrt h(R+2\Lambda)\; \Longrightarrow\; \frac1{\tilde\kappa^2}\sqrt h(R+2\tilde\Lambda),$$ where $\tilde\kappa$ and $\tilde\Lambda$ are modified gravitational and cosmological constants, respectively. Our final, modified, hamiltonian constraint reads therefore $${\cal H}_\bot(x) = - \kappa^2G_{abcd}(x)p^{ab}p^{cd}+\frac{1}{\tilde\kappa^2}\sqrt{h(x)} (R(x)+2\tilde\Lambda)$$ $$-\frac{\Im(\alpha)}{\kappa}h_{ab}p^{ab} - \kappa\Im(\gamma)\left(R_{ab}p^{ab}- \frac14 h_{ab}p^{ab}R\right).$$ Now it is clear that the modified hamiltonian above is [*not*]{} a first class constraint. This follows from the presence of the terms linear in $p$. Thus the effective theory has less symmetries than the classical general relativity. It follows then that the parameter $N$ in the definition of total hamiltonian is not free anymore, rather it should be fixed by the requirement that the hamiltonian is time independent, to wit, $$\left\{{\cal H}_\bot(x), {\cal H}[N, \vec{N}] \right\} =0 \;\;\; \mbox{(weakly).}$$ It should also be noted that even if the terms linear in $p$ were not present (if the Poisson bracket constraint algebra would close), we still would not be able to recover the standard four dimensional action $\int d^4x\sqrt g({}^{(4)}R+2\Lambda)$ from the hamiltonian action $$\int d^3xdt\left(p^{ab}\dot h_{ab} - {\cal H}[N,\vec{N}]\right).$$ The reason is that the coefficients $\kappa$ and $\tilde\kappa$ are not identical and therefore the three curvature and external curvature components of the four dimensional curvature scalar would be multiplied by different coefficients. It should be stressed that the situation described for the case of this particular solution is quite generic. The conclusion, we draw from these computation is that the four dimensional general coordinate invariance seems to be broken by quantum corrections (besides, this provides the ultimate solution of the celebrated problem of time.) This symmetry is restored when quantum effects are neglegible (because the ordering problems leading to introducing the $L$-term disappears in the semiclassical limit.) Conclusions =========== The main problem, we address in this paper was to find a set of conditions which would make it possible to construct the regularized Wheeler–De Witt operator and a class of states for which the algebra of quantum constraints is anomaly free. It turned out that this space of states is quite modest but we were able to find some physical states (solutions of quantum gravity.) We then tried to find an interpretation of one of the solutions employing the method of quantum potential. We found that the resulting modified 3+1 theory does not possess the symmetry of time translation anymore. It is hard to say at this moment if this is just an artefact of employing the canonical quantisation method, where, by construction of the formalism, the time translation symmetry is very volnurable from the very beginning or if it signifies some real physical effect at the Planck scale. It may also happen to be an artefact of the quantum potential interpretation of the wavefunction. These questions should be certainly further investigated. Another direction of research is to include the coupling of gravity to matter fields like the scalar field or the supergravity. We are now in position to construct relevant regularised operators in both cases, but it turns out that to find solutions in these cases is (surprisingly?) hard. These open problems are subject of intensive investigations and we hope to be able to present the results soon. [99]{} C. Isham, [*Conceptual and Geometrical Problems in Quantum Gravity*]{}, Recent Aspects of Quantum Fields, H. Mitter and H. Gausterer eds., Springer Verlag 1992; C. Isham, [*Prima Facie Questions in Quantum Gravity*]{} in Canonical Gravity: From Classical to Quantum, J. Ehlers and H. Friedrich eds., Springer Verlag, 1994; R. Penrose, [*The Emperors New Mind*]{}, Oxford University Press, 1990. A. Ashtekar, [*Lectures on Non-Perturbative Quantum Gravity*]{}, World Scientific, 1991; Lee Smolin, [Recent Developments in Nonperturbative Quantum Gravity]{}, Lectures given at 1992 International GIFT Seminar on Theoretical Physics; A. Ashtekar, [*Polymer Geometry at Planck Scale and Quantum Einstein Equation*]{}, hep-th 9601054 C.J. Isham, [*Canonical Quantum Gravity and the Problem of Time*]{} in Integral Systems, Quantum Groups, and Quantum Field Theories, L.A. Ibort and M.A. Rodriguez eds., Kluwer Academic Publishers, 1993. T. Horiguchi, K. Maeda and M. Sakamoto, Phys.  Lett., 105 (1995); K. Maeda and M. Sakamoto, Phys. Rev. [**D**]{}, in press. J. Kowalski-Glikman and K. Meissner, hep-th 9601062, Phys. Lett. [**B**]{}, in press. Paul Mansfield, Nucl. Phys. [**B**]{}418, (1994), 113. D. Bohm, Phys. Rev. [**85**]{}, [166]{} (1952), Phys. Rev. [**85**]{}, 180 (1952); D. Bohm and B.J. Hiley, Phys. Rep. [**144**]{}, 323 (1987); D. Bohm, B.J. Hiley, and P.N. Koloyerou Phys. Rep. [**144**]{}, 349 (1987); D. Bohm, B.J. Hiley, [*The Undivided Universe: An Ontological Interpretation of Quantum Theory*]{}, Routledge & Kegan Paul, London, 1993; J. S. Bell [*Speakable and Unspeakable in Quantum Mechanics*]{}, Cambridge University Press, 1987. J. Kowalski-Glikman; [*Quantum Potential and Quantum Gravity*]{} gr-qc 9511014, to appear in the volume dedicated to J. Lukierski, World Scientific 1996. Yu. Shtanov [*Pilot Wave Quantum Gravity*]{}, gr-qc 9503005. Ulrich H. Gerlach, Phys. Rev. [**177**]{}, 1929 (1969). [^1]: e-mail address ablaut@ift.uni.wroc.pl [^2]: e-mail address jurekk@ift.uni.wroc.pl [^3]: Here our approach differs from the one used in, for example [@Maeda], where the authors choose to analyze the algebra of quantum constraints without defining the space of states. [^4]: In the paper [@JK] we took $\tilde K_{abcd}(x,x')=G_{abcd}(x) \tilde K(x,x')$, where $\tilde K$ is a heat kernel, and $L_{ab}$ was taken to be the functional derivative of $\tilde K_{abcd}$ with respect to $h_{cd}$.

--- abstract: | A tiling of ${{\mathbb R}}^d$ is [*repulsive*]{} if no $r$-patch can repeat arbitrarily close to itself, relative to $r$. This is a characteristic property of aperiodic order, for a non repulsive tiling has arbitrarily large local periodic patterns. We consider an aperiodic, repetitive tiling $T$ of ${{\mathbb R}}^d$, with finite local complexity. From a spectral triple built on the discrete hull $\Xi$ of $T$, and its Connes distance, we derive two metrics ${d_{\text{\rm sup}}}$ and ${d_{\text{\rm inf}}}$ on $\Xi$. We show that $T$ is repulsive if and only if ${d_{\text{\rm sup}}}$ and ${d_{\text{\rm inf}}}$ are Lipschitz equivalent. This generalises previous works for subshifts by J. Kellendonk, D. Lenz, and the author. author: - | J. Savinien$^{1,2}$\ [$^{1}$ UniversitŽ de Lorraine, Institut Elie Cartan de Lorraine, UMR 7502, Metz, F-57045, France.]{}\ [$^{2}$ CNRS, Institut Elie Cartan de Lorraine, UMR 7502, Metz, F-57045, France.]{} title: A metric characterisation of repulsive tilings --- Introduction ============ In two recent articles in collaboration with J. Kellendonk and D. Lenz [@KS10; @KLS11], we used constructions of non commutative geometry [@Co94] to derive a new characterisation of aperiodically ordered $1d$-subshifts. We showed that a minimal and aperiodic subshift $X$ has bounded powers if and only if two metrics derived from the Connes distance of a spectral triple over $X$ are Lipschitz equivalent. An essential ingredient to obtain this result is the notion of [*privileged words*]{} [@KLS11]. In this paper, we generalise this formalism and this results to tilings of ${{\mathbb R}}^d$. The essential ingredient here is the notion of [*privileged patches*]{} of a tiling. A $1d$-subshift has bounded powers if its language does not contain arbitrarily large powers, [*i.e.*]{} there is an integer $p$ such that $n$-fold repetitions $w^n=w\cdots w$ of a word $w$ cannot occur for $n>p$. Linearly recurrent subshifts, which are usually considered highly ordered, share this property [@LP03; @Dur00; @Dur03]. Loosely speaking, bounded powers means that no factor can repeat too close, or overlap too much, along a sequence in the subshift. Bounded powers is equivalent to the property that a complete first return $u'$ of a word $u$ must be longer than a uniform constant times the length of $u$: $|u'|> C |u|$. The corresponding notion for tilings is [*repulsiveness*]{}: no patch can repeat arbitrarily close to itself relative to its size, see equation . A non repulsive tiling has arbitrarily large local periodic patterns – the analogue of arbitrarily large powers. As for subshifts, linearly repetitive tilings are repulsive [@Pat98; @Len04]. The property of bounded (or unbounded) powers in a subshift is measured by privileged words. Privileged words are iterated complete first returns to letters of the alphabet. Privileged words were introduced in [@KLS11], and have recently encountered a lot of interest in the combinatorics of words [@Pelto13; @FPZ13; @Pelto; @FJS13]. For rich subshifts [@GJWZ09] privileged words coincide exactly with palindromes (see [@KLS11] Section 2.2 for further details). We generalise this notion to tilings. We define privileged patches: a notion of iterated complete first returns to the prototiles, see Section \[sec-priv\]. For $1d$ subshifts, a privileged patch is a generalisation of a privileged word obtained with bilateral versions of complete first returns. Because of the geometry in ${{\mathbb R}}^d$, the combinatorics of patches is much more involved than that of words. We need a few technical lemmas to deal with this. But the crucial point is the generalisation of privileged words to the tilings setting. Once the right definition of privileged patch is at hand, our formalism for subshifts essentially goes through for tilings of ${{\mathbb R}}^d$. The spectral triple we used in [@KLS11] for subshift is build from the tree of privileged words of the subshift. The spectral triple we use here is the same one built on the tree of privileged patches of the tiling. This allows us to characterise repulsive tilings by Lipschitz equivalence of two metrics derived from the Connes distance of the spectral triple, in complete analogy with the case of subshifts treated in [@KLS11]. Our initial motivation in studying properties of aperiodically ordered subshifts and tilings, came from non commutative geometry (NCG) [@Co94]. Namely we were interested in the construction of non commutative Riemanian structures, [*i.e.*]{} spectral triples, over totally disconnected spaces defined by tilings and subshifts. As it turns out, and as in [@KLS11], the criterium for aperiodic order we derive here can be explained in a rather combinatorial way, without introducing the framework of NCG and giving the details of the construction of the spectral triple. So we follow this line in the paper: we give the criterium [*ad hoc*]{} to state and prove our result. And in the last section we describe briefly the underlying spectral triple. The paper is organised as follows. In Section \[sec-basics\] we remind the reader of the basic definitions for tilings of ${{\mathbb R}}^d$, and the classical results we need. We introduce privileged patches in Section \[sec-priv\], and state some combinatorial properties, including technical lemmas which allows us to adapt our formalism for subshifts to tilings of ${{\mathbb R}}^d$. In Section \[sec-tree\] we explain the construction of the tree of privileged patches, from which we define the two Connes metrics. In Section \[sec-charact\] we state and prove our main result, namely that a tiling is repulsive if and only if the Connes metrics are Lipschitz equivalent. The construction of the spectral triple, from which the Connes metrics are derived, is given briefly in Section \[sec-ST\]. [**Aknowlegements.**]{} The author would like to thank J. Kellendonk and D. Lenz for useful discussions, and encouragements to publish this work. Basic definitions {#sec-basics} ================= A [*tile*]{} of ${{\mathbb R}}^d$ is a subset $t \subset {{\mathbb R}}^d$ which is homeomorphic to a closed ball. A [*tiling*]{} of ${{\mathbb R}}^d$, is a countable family of tiles, $T=\{ t_i\}_{i\in{{\mathbb N}}}$, which have pairwise disjoint interiors and whose union covers ${{\mathbb R}}^d$. Given a tiling $T$, we specify a [*marker*]{} [^1] in each of its tile $t$: a point $x(t)\in {{\mathbb R}}^d$ in its interior. A [*translate*]{} of a family $F=\{ t_j\}_{j\in J}$ of tiles of $T$, is a family $F+a=\{t_j + a\}_{j\in J}$, for some $a\in{{\mathbb R}}^d$. Let $x$ be the marker of a tile of $T$, and $r>0$. We call an [*$r$-patch*]{}, or a [*patch*]{} of radius $r$, the finite family of tiles of $T-x$ all of whose markers lie inside the open ball $B(0,r)$. In addition, $r$ is maximal with respect to the family of tiles defining the patch. As a consequence, the only $0$-patch is the empty patch. The patches made of a single tile (containing the marker of a single tile), are called [*prototiles*]{}. Consider an $r$-patch $p$ of $T$. Given a family $F=\{ t_j\}_{j\in J}$ of tiles of $T$, we say that [*$p$ occurs in $F$*]{}, if there is a translate of $p$ which is a subset of $F$: $p+a\subset F$ for some $a \in {{\mathbb R}}^d$. The translate $p+a$ is called an [*occurrence*]{} of $p$ in $F$. Given a subset $U$ of ${{\mathbb R}}^d$, we say that [*$p$ occurs in $U$*]{}, if there is an occurrence of $p$ in $T$, the union of all of whose tiles is a subset of $U$. We mean that a patch $p$ is marked at the origin: $x(p)=0$. And that an occurrence of $p$ in $T$, in a family of tiles $F$, or in a subset of ${{\mathbb R}}^d$, is some translated copy $p+a$ marked at $a$: $x(p+a)=a$. We will consider tilings satisfying the following three properties. \[def-hypT\] A tiling $T$ of ${{\mathbb R}}^d$ is called (i) [*aperiodic*]{} if $T+a=T$ implies $a=0$; (ii) [*repetitive*]{} if for any $r>0$, and any $r$-patch $p$ of $T$, there exists $R>0$ such that $p$ occurs in any ball of radius $R$; (iii) [*FLC*]{}, or has [*Finite Local Complexity*]{}, if for any $r>0$ there are finitely many $r$-patches. Let $T$ be a repetitive and FLC tiling of ${{\mathbb R}}^d$, and $p$ an $r$-patch of $T$. The [*Delone set of occurrences of $p$ in $T$*]{} is the set $L_p$ of markers of all occurrences of $p$ in $T$. This is a Delone set as the distance between nearest points of $L_p$ is uniformly bounded. We let ${r_{\text{\rm pack}}}(L_p)$ (resp. ${r_{\text{\rm cov}}}(L_p)$) be one half of that uniform minimal distance (resp. maximal distance). It is called the [*packing radius*]{} of $L_p$ (resp. [*covering radius*]{}): any ball of radius ${r_{\text{\rm pack}}}(L_p)$ (resp. ${r_{\text{\rm cov}}}(L_p)$) contains at most (resp. at least) one point of $L_p$. A tiling $T$ is said to be [*repulsive*]{} if $$\label{eq-rep} \ell = \inf \Bigl\{ \frac{{r_{\text{\rm pack}}}(L_p)}{r} \, : \, \text{\rm $p$ an $r$-patch of $T$} \Bigr\} > 0.$$ Informally, $\ell>0$ means that patches cannot overlap too much. On the contrary, in a non-repulsive tiling, there are arbitrarily large $r$-patches with arbitrarily close occurrences relative to $r$. Such occurrences overlap over an arbitrarily large proportion of their tiles. This implies that a non-repulsive tiling has arbitrarily large local periodic patterns, see Figure \[fig-nrep\]. We now fix an aperiodic, repetitive, and FLC tiling $T$ of ${{\mathbb R}}^d$ and assume that there is a tile whose marker lies at the origin. We endow the family of all of its translates, $T+{{\mathbb R}}^d$, with the following topology. A base of open sets is given by the acceptance domains of patches: for $p$ an $r$-patch of $T$ $$[p] = \bigl\{ T'\in \ T+{{\mathbb R}}^d \, : \, p \ \text{\rm occurs at the origin in } T' \bigr\}$$ If $q$ is a patch contained in $p$, which we write $q\subseteq p$, then $[p]\subset [q]$. Hence two tilings are close for this topology, if they agree on a large patch around the origin. The [*discrete hull*]{} of a $T$ is the closure of its translates in this topology: $$\Xi = \overline{ \bigl\{ T + a \, : \, a \in {{\mathbb R}}^d, \, T + a \, \text{\rm has a marker at the origin} \bigr\}}\,.$$ As a consequence of the hypothesis in Definition \[def-hypT\] the following classical results hold: 1. $\Xi$ is a Cantor set (compact, totally disconnected, with no isolated point); 2. the family of acceptance domains $[p]$ is a countable base of clopen sets[^2] for $\Xi$; 3. any $T'\in \Xi$ satisfies the hypothesis of Definition \[def-hypT\], and the closure of $T'+{{\mathbb R}}^d$ is $\Xi$. The discrete hull is metrizable. Any function $\delta: [0,+\infty) \rightarrow (0,1]$, which decreases and has limit $0$ at $+\infty$, defines a ultra-metric on $\Xi$ as follows: $$\label{eq-ultrametric} d_\delta (T_1,T_2) = \inf \bigl\{ \delta(r) : \text{\rm there is an $r$-patch $p$ occurring in both $T_1$ and $T_2$ at the origin} \bigr\}.$$ Privileged patches {#sec-priv} ================== Given an $r$-patch $p$, we say that an $r'$-patch $p'$ is [*derived*]{} from $p$ if (i) $p$ is contained in $p'$; (ii) $p$ occurs at least twice in $p'$; (iii) for any ${\widetilde}{r}<r'$, and any ${\widetilde}{r}$-patch $q$ contained in $p'$, $p$ occurs at most once in $q$. See Figure \[fig-derpatch\] for an illustration. Condition (ii) means that $p'$ contains $p$ as a subpatch (hence with marker at the origin), plus another translate $p+a$ for some $a\neq 0$. Conditions (ii) and (iii) mean that $p'$ is a minimal extension of $p$ containing two occurrences of $p$. ![A patch $p'$ derived from $p$.[]{data-label="fig-derpatch"}](derpatch-0.pdf) We define [*privileged patches*]{} inductively, as follows: 1. the empty patch is the only privileged patch of order $0$; 2. the prototiles of $T$, are the privileged patches of order $1$; 3. for $n>1$ a privileged patch of order $n$ is an $n$-th iterated derived patch from the empty patch. For $1d$-subshifts, [*i.e.*]{} symbolic one-dimensional tilings, this is a two-sided version of [*privileged words*]{} introduced in [@KLS11]. Let us state some elementary properties of derived patches. The first two Lemmas are needed to build the tree of privileged patches in the next Section. \[lem-der\] Let $q$ be a patch derived from some patch, then (i) there exists a unique patch $p$ such that $p'=q$; (ii) if $q$ is privileged, then there exists a unique privileged patch $p$ such that $p'=q$; (iii) if $q$ is privileged, and $p$ is a privileged patch contained in $q$, then there exists $i\ge 0$ such that $q$ is an $i$-th iterated derived patch from $p$, which we write $p^{(i)}=q$. \(i) Assume that $q=p'_1=p'_2$, for two distincts patches $p_i$ of radius $r_i$, $i=1,2$. We may assume $r_2<r_1$, but then $p_2 \subsetneq p_1$, and this implies $p'_2 \subsetneq p'_1$ a contradiction. \(ii) If $q$ is privileged, by definition there exists a privileged patch $p$ such that $p'=q$, and by (i) $p$ is unique. \(iii) We prove this by induction on the radius of $q$. Let $(r_n)_{{\mathbb N}}$ be the non-decreasing sequence of radii of privileged patches of $T$ (which exists by FLC). The property is obvious for privileged patches of radius $r_1$: $q$ is a prototile with smallest radius and is derived for the empty patch. Assume the property holds for all privileged patches of radii less than or equal to $r_{n}$, for some $n>1$. Consider a privileged patch $q$ of radius $r_{n+1}$, and a privileged patch $p\subseteq q$. The case $p=q=p^{(0)}$ is trivial, so assume $p\subsetneq q$. By (ii) there exists a unique privileged patch ${\widetilde}p\subsetneq q$ with ${\widetilde}p^{(1)}=q$. Case $p\subseteq{\widetilde}p$ : by induction ${\widetilde}p = p^{(j)}$ for some $j\ge 0$, and so $q=p^{(j+1)}$. Case ${\widetilde}p\subsetneq p$ : by induction there is a $j>0$ such that $p={\widetilde}p^{(j)} = q^{(j-1)}$, which implies $q\subseteq p$ a contradiction. \[lem-rder\] (i) Let $p$ be an $r$-patch, and $p'$ an $r'$-patch derived from $p$, then $$2 {r_{\text{\rm pack}}}( L_p) + r \le r' \le 2 {r_{\text{\rm cov}}}( L_p) + r.$$ (ii) Let $(p_n)_{n\ge1}$ be a sequence of $r_n$-patches, such that $p_{n+1}$ is derived from $p_n$ for all $n$. Then $ r_{n+1} \ge 2n \,{r_{\text{min}}}$, where ${r_{\text{min}}}$ is the radius of the smallest prototile. If in addition $T$ is repulsive, then $r_{n+1}\ge (2\ell + 1)^n {r_{\text{min}}}$. The first claim follows at once from the definition. We use the first inequality in (i) inductively to get $$r_{n+1} \ge 2 {r_{\text{\rm pack}}}( L_{p_{n}}) + r_{n} \ge 2 \, {r_{\text{min}}}+ r_{n} \ge \cdots \ge 2n \, {r_{\text{min}}}.$$ If $T$ is repulsive, then ${r_{\text{\rm pack}}}( L_{p_{j}}) \ge \ell r_j$ for all $j$, see equation , so one gets $$r_{n+1} \ge 2 {r_{\text{\rm pack}}}( L_{p_{n}}) + r_{n} \ge (2 \ell +1 ) r_{n} \ge \cdots \ge (2\ell +1)^n {r_{\text{min}}}.$$ The following technical lemma is analogous to Lemma 3.8 in [@KLS11]. It states that if a tiling is not repulsive, then one can find arbitrary long sequences of derived privileged patches whose radii grow “slowly”. \[lem-nrep\] If $T$ is not repulsive, then for all $m\in{{\mathbb N}}$, there exists privileged patches $p_0, p_1,\ldots p_m$, of radii $r_0, r_1, \ldots r_m$ respectively, such that (i) $p_{j}=p_0^{(j)}$, for all $j=1, \ldots, m$, (ii) $r_m \le 2 r_1$. Consider a non repulsive tiling $T$, and fix an integer $m>1$. Since the infimum in equation  is zero, for any $0<\epsilon <1/(8m)$, there is an $r_q$-patch $q$ of $T$ for which ${r_{\text{\rm pack}}}(L_q) / r_q < \epsilon$. By FLC there are two copies of $q$ which occur at some markers $x$ and $y$ of tiles in $T$, satisfying $|x-y|=2{r_{\text{\rm pack}}}(L_q)$. Set $a={r_{\text{\rm pack}}}(L_q)$. Consider the largest privileged patch $p$ contained in $q$, with same marker, and of radius $r\le r_q$. We must have $r\ge r_q/2$ for otherwise, as $p$ occurs both at $x$ and $y$, then one of its derived patches $p'$ would have radius $r' \le r+ 2a < r_q/2 + 2\epsilon r_q < r_q$. Hence $p'$ would be contained in $q$, which contradicts maximality of $p$ in $q$. Consider the largest privileged patch $p_0\subset p$, with same marker, and radius $r_0 < r/2$. Since $p_0$ occurs both at $x$ and $y$, it has a derived patch $p_1\subset p$, with radius $r_1 \le r_0 + 2a < r/2 + \epsilon r_q<r$, which is thus a proper sub-patch of $p$. By Lemma \[lem-der\] (iii), we have $p=p_1^{(n-1)}=p_0^{(n)}$ for some $n>1$. Again $p_1$ occurs both at $x$ and $y$, so it has a derived patch $p_2=p_0^{(2)} \subsetneq p$ of radius $r_2\le r_1+2a \le r_0 + 4a$. See Figure \[fig-nrep\] for an illustration. ![The dotted circles, in order of decreasing radii, are the translates $p+y$, $p_1+y$, and $p_0+y$. The small circles along the horizontal axis are occurrences of the same patch, and illustrate the local periodic pattern generated by the overlaping of $p+x$ and $p+y$.[]{data-label="fig-nrep"}](nrep-0.pdf) We iterate this argument to obtain that there is a patch $p_j=p_0^{(j)}$ with radius $r_j \le r_0 + 2ja$, for all $j\le n$. For $j=n$ this last inequality implies $n \ge r/(4a) \ge r_q /(8a) > 1/ (8\epsilon) > m$. We have thus build a sequence of privileged patches $p_0, p_1, \cdots p_m, \cdots p_n=p$, whose first $m+1$ terms give the sequence in (i). But $p_0$ is the largest privileged patch in $p$ (with same marker and) of radius $r_0< r/2$, hence $p_1=p_0'$ has radius $r_1\ge r/2= r_n/2 \ge r_m/2$, which proves (ii). The tree of privileged patches and the Connes metrics {#sec-tree} ===================================================== We build the tree ${{\mathcal T}}$ of privileged patches of $T$ inductively as follows: 1. [(0)]{} the root of ${{\mathcal T}}$ stands for the empty patch; 2. [(1)]{} vertices of order $1$ stand for privileged patches of order 1 (prototiles), each of which is linked by one edge to the root; 3. [(n)]{} vertices of order $n>1$ stand for privileged patches of order $n$, each of which is linked by one edge to the vertex of order $n-1$ corresponding to the patch it is derived from. The tree ${{\mathcal T}}$ is well-defined by Lemma \[lem-der\] (ii): each vertex of level $n+1$ is linked to a unique vertex of level $n$, for all $n$. We let $\partial {{\mathcal T}}$ be the set of infinite rooted path in ${{\mathcal T}}$: $\xi =(\xi_n)_{n\ge 0} \in \partial {{\mathcal T}}$ is a sequence of privileged patches, with $\xi_{n+1}$ derived from $\xi_n$ for all $n$. Given a vertex $v\in {{\mathcal T}}$, we let $[v] \subset \partial {{\mathcal T}}$ be the [*cylinder*]{} of $v$, namely the set of all infinite paths through $v$. \[prop-tree\] The set $\partial {{\mathcal T}}$ of infinite paths in ${{\mathcal T}}$, endowed with the topology of cylinders, is homeomorphic to the discrete hull $\Xi$. The sets $\partial {{\mathcal T}}$ and $\Xi$ are easily seen to be isomorphic. Given a tiling $T$ in $\Xi$, let $\xi_0$ be the empty patch and $\xi_1$ the prototile occurring in $T$ at the origin. Since $T$ is repetitive, there is a (unique) privileged patch $\xi_2$ derived from $\xi_1$ which occurs in $T$ at the origin. We construct inductively a (unique) sequence of privileged patches occurring at the origin of $T$, which defines an infinite path in ${{\mathcal T}}$. Conversely, by Lemma \[lem-rder\] (ii), a sequence of privileged patches in $\partial {{\mathcal T}}$ defines a unique tiling in $\Xi$. A basis for the topology of $\Xi$ is given by the acceptance domains $[p]$ of patches. While cylinders correspond to acceptance domains of privileged patches, hence yield a coarser topology on $\Xi$. Given a patch $p$, let $p_0$ is the greatest privileged patch contained in $p$, and $p_1, \ldots p_k$ the patches derived from $p_0$. Then $[p] \subset [p_1] \cup \ldots \cup [p_k]$, so both topologies agree. A [*weight function*]{} is any function $\delta: [0,+\infty) \rightarrow (0,1]$, which decreases and has limit $0$ at $+\infty$. A weight function allows us to defined a ultra metric on $\Xi$, as in equation , and to build a spectral triple on $C(\Xi)$ as explained in Section \[sec-ST\]. The Connes distance of that spectral triple yields two pseudo-metrics on $\partial {{\mathcal T}}\simeq \Xi$, which we now define. Given $\xi, \xi'\in \partial {{\mathcal T}}$, we let $\xi {\, {\widetilde}\wedge \,}\xi'$ denote the vertex at which the paths branch in ${{\mathcal T}}$, and ${{\mathcal O}}(\xi {\, {\widetilde}\wedge \,}\xi')$ the order of that vertex. If we identify $\xi,\xi'$ with tilings $T,T'\in\Xi$ by Proposition \[prop-tree\], then $\xi{\, {\widetilde}\wedge \,}\xi'$ is the greatest common privileged patch which occurs in both $T$ and $T'$ at the origin. The following define two metrics[^3] on $\partial {{\mathcal T}}$: $$\label{eq-dinf} {d_{\text{\rm inf}}}(\xi, \xi') = \left\{ \begin{array}{ll} \delta\bigl( r_{{{\mathcal O}}(\xi {\, {\widetilde}\wedge \,}\xi')} \bigr)& \text{\rm if } \xi \neq \xi' \\ 0 & \text{\rm if } \xi = \xi' \end{array}\right.,$$ and $$\label{eq-dsup} {d_{\text{\rm sup}}}(\xi,\xi') = {d_{\text{\rm inf}}}(\xi,\xi') + \sum_{n> {{\mathcal O}}(\xi {\, {\widetilde}\wedge \,}\xi')} \delta(r_n) + \delta(r'_n) \ , $$ where $r^{(')}_n$ is the radius of the patch $\xi^{(')}_n$ (so one has $r_n=r'_n$ for all $n\le {{\mathcal O}}(\xi {\, {\widetilde}\wedge \,}\xi')$). Clearly ${d_{\text{\rm inf}}}\le {d_{\text{\rm sup}}}$, and one easily sees that ${d_{\text{\rm inf}}}$ and ${d_{\text{\rm sup}}}$ are Lipschitz equivalent if and only if $$\label{eq-Lip} \exists C>0, \, \forall \xi \in \partial {{\mathcal T}}, \, \forall m\in {{\mathbb N}}, \qquad \delta(r_m)^{-1} \sum_{k \ge 1} \delta(r_{m+k}) \le C \,.$$ Characterisation of repulsive tilings {#sec-charact} ===================================== We state our main result. Let $T$ be an aperiodic, repetitive, and FLC tiling of ${{\mathbb R}}^d$, as in Definition \[def-hypT\]. Consider the tree ${{\mathcal T}}$ of privileged patches of $T$ as in the previous section. Let $\delta: [0,+\infty) \rightarrow (0,1]$ be a weight function as in the previous section (decreasing with limit $0$ at infinity), and assume that there exists $c_1, c_2>0$ such that $$\label{eq-delta} \delta(ab) \le c_1 \delta(a) \delta(b), \qquad \delta(2a)\ge c_2 \delta(a), \ \forall a,b \ge 0.$$ Consider the metrics ${d_{\text{\rm inf}}}$ and ${d_{\text{\rm sup}}}$ on $\partial {{\mathcal T}}$ given in equations  and . \[thm-main\] The following are equivalent: (i) $T$ is repulsive, (ii) ${d_{\text{\rm inf}}}$ and ${d_{\text{\rm sup}}}$ are Lipschitz equivalent. Assume $T$ is repulsive, so $\ell>0$ in equation . Upon rescalling $\delta$, we may assume that $c_1=1$ in equation , and $\delta(2\ell +1)<1$. Pick $m\in{{\mathbb N}}$ and $\xi\in\partial {{\mathcal T}}$. Let $r_n$ be the radius of the patch $\xi_n$, $n\ge 0$. By Lemma \[lem-rder\] (ii), for any $k\ge 1$ we have $r_{m+k}\ge (2\ell +1)^k r_m$. Hence $$\delta(r_{m})^{-1} \sum_{k\ge 1 } \delta(r_{m+k}) \le \delta(r_{m})^{-1} \sum_{k\ge 1 } \delta((2\ell+1)^k) \delta(r_{m}) \le \sum_{k\ge 1 } \delta(2\ell +1)^k\,,$$ where the last two inequalities follows from equation . The converging geometric series on the right hand side gives a uniform bound in equation , which proves that ${d_{\text{\rm inf}}}$ and ${d_{\text{\rm sup}}}$ are Lipschitz equivalent. Assume that $T$ is not repulsive. Fix an integer $N$ (large), and consider a sequence of privileged patches $p_0, p_1, \ldots p_{m}$, $m>N$, as in Lemma \[lem-nrep\]. Choose an infinite path $\xi\in\partial {{\mathcal T}}$ going through the vertices associated with $p_0, p_1 \ldots p_{m}$. Upon a change of index, we may assume that $\xi_j$ corresponds to $p_j$, for $j=1, \ldots m$. Then $$\delta(r_{1})^{-1} \sum_{k\ge 1} \delta(r_{1+k}) \ge \frac{1}{\delta(r_{1})} \sum_{j=2}^{m} \delta(r_j) \ge \frac{m-1}{\delta(r_{1})}\delta(r_m) \ge \frac{N}{\delta(r_1)} \delta(2r_1)\ge Nc_2\,,$$ where we used that $\delta$ decreases, and equation . Since $N$ was arbitrary, one cannot bound the series on the left hand side. So there exists no uniform bound in equation , and thus ${d_{\text{\rm inf}}}$ and ${d_{\text{\rm sup}}}$ are not Lipschitz equivalent. The spectral triple {#sec-ST} =================== For the sake of completeness, we remind the reader of the spectral triple on $C(\Xi)\cong C(\partial {{\mathcal T}})$ whose Connes distance yields ${d_{\text{\rm inf}}}$ and ${d_{\text{\rm sup}}}$. The construction in [@KS10] is given for any tree, and in [@KLS11] for the tree of privileged words of a $1d$-subshift, which we rewrite here for the tree of privileged patches defined in Section \[sec-tree\]. These constructions are related to other spectral triples build for metric spaces [@Ri99; @Ri04; @CI07] or more specifically fractals [@GI03; @GI05; @CIL08] and ultrametric Cantor sets [@PB09]. We refer the reader to [@KS10] and [@KLS11] for details and proofs. We consider the tree ${{\mathcal T}}=({{\mathcal T}}^0,{{\mathcal T}}^1)$ of privileged patches, and a weight $\delta$ as in Section \[sec-tree\]. We add [*horizontal edges*]{} ${{\mathcal H}}$ to the graph ${{\mathcal T}}$: ${{\mathcal H}}=\cup_{n\ge 1} {{\mathcal H}}_n$, and ${{\mathcal H}}_n$ is a set of oriented edges between vertices of level $n$ in ${{\mathcal T}}$ defined as follows. If $u_1,u_2\in {{\mathcal T}}$, then there is one horizontal edge $h\in {{\mathcal H}}_n$ with source $s(h)=u_1$ and range $r(h)=u_2$, if and only if $u_1$ and $u_2$ stand for two distinct privileged patches of order $n$ both of which are derived from the same privileged patch of order $n-1$. Given any such $h$, there is then an edge $h^{\text{\rm op}}\in {{\mathcal H}}$ with reverse orientation: $r(h^{\text{\rm op}})=s(h)$ and $s(h^{\text{\rm op}})=r(h)$, and $(h^{\text{\rm op}})^{\text{\rm op}}=h$. We fix an orientation of ${{\mathcal H}}$, and write the decomposition into positively and negatively oriented edges ${{\mathcal H}}={{\mathcal H}}_+ \cup {{\mathcal H}}_- $. A [*choice*]{} is a map $\tau: {{\mathcal T}}^{0} \rightarrow \partial {{\mathcal T}}$ such that $\tau(v)$ is an infinite path through vertex $v$. The [*approximation graph*]{} $G_\tau=(V,E)$ is defined by $$V= \tau({{\mathcal T}}^0), \qquad E= \tau\times \tau ({{\mathcal H}}).$$ The orientation on ${{\mathcal H}}$ is carried over to $E=E_+ \cup E_-$. We endow $G_\tau$ with the graph metric induced by the weight $\delta$: for $e=\tau\times\tau(h) \in E$ we set the length of $e$ to be $\ell(e) = \delta(r_h)$, where $r_h$ is the radius of the privileged patch from which $s(h)$ and $r(h)$ are derived. The set of vertices $V$ is dense in $\partial {{\mathcal T}}$, and the set of edges $E$ encodes adjacencies according to the choice $\tau$. We consider the spectral triple associated with the approximation graph $G_\tau=(V,E)$: $\bigl( C(\partial{{\mathcal T}}), \ell^2(E), \pi_\tau, D\bigr)$. The C$^\ast$-algebra $C(\partial {{\mathcal T}})$ of continuous functions over $\partial{{\mathcal T}}$ is faithfully represented on the Hilbert space $\ell^2(E)$ by $$\pi_\tau (f) {\varphi}(e) = f\bigl( s(e) \bigr) {\varphi}(e).$$ The Dirac $D$ is the unbounded operator on $\ell^2(E)$, with compact resolvent, given by $$D{\varphi}(e) = \frac{1}{\ell(e)} {\varphi}( e^{\text{\rm op}}).$$ The “non commutative derivation” is the finite difference operator $$\bigl[ D,\pi_\tau(f) \bigr] {\varphi}(e) = \frac{f(s(e)) - f(r(e))}{\ell(e)} {\varphi}(e^{\text{\rm op}}),$$ which is bounded over the pre-C$\ast$-algebra $C_{Lip}(\partial {{\mathcal T}})$ of Lipschitz continuous functions over $\partial {{\mathcal T}}$. The Connes distance of the spectral triple is a pseudo-metric over $\partial {{\mathcal T}}$ which reads $$\begin{aligned} d_\tau (\xi, \xi') &=& \sup_{f\in C(\partial {{\mathcal T}})} \Bigl\{ \bigl| f(\xi) - f(\xi')\bigr| : \| [D, \pi_\tau(f) \| \le 1 \Bigr\} \\ & = & \sup_{f\in C(\partial {{\mathcal T}})} \Bigl\{ \bigl| f(\xi) - f(\xi') \bigr| : |f(s(e)) - f(r(e))| \le \ell(e), \, \forall e\in E \Bigr\}\end{aligned}$$ where the norm is the operator norm on $\ell^2(E)$. It is an extension of the graph metric of $G_\tau$ to $\partial {{\mathcal T}}$. If it is continuous for the topology of ${{\mathcal T}}$, it reads explicitly: $$d_\tau(\xi, \xi') = {d_{\text{\rm inf}}}(\xi, \xi') + \sum_{n> {{\mathcal O}}(\xi{\, {\widetilde}\wedge \,}\xi')} b_\tau(\xi_n) \delta(r_n) + b_\tau(\xi'_n) \delta(r'_n)\,,$$ where $\xi^{(')}=(\xi^{(')}_n)$, $r^{(')}_n$ is the radius of the patch $\xi^{(')}_n$, and for $\eta\in\partial{{\mathcal T}}$, $b_\tau(\eta_n) = 1$ if $\eta$ and $\tau(\eta_n)$ branch at $\eta_n$, and $b_\tau(\eta_n) = 0$ otherwise. The infimum and supremum of $d_\tau$ over choices $\tau$ yield the metrics ${d_{\text{\rm inf}}}$ of equation  and ${d_{\text{\rm sup}}}$ and  respectively. [BEL]{} A. Connes. [*Noncommutative geometry*]{}. Academic Press Inc., San Diego, CA, 1994. E. Christensen, C. Ivan. “Sums of two-dimensional spectral triples”. [*Math. Scand.*]{} [**100**]{} (2007) 35–60. E. Christensen, C. Ivan, M.L. Lapidus. “Dirac operators and spectral triples for some fractal sets built on curves”. [*Adv. Math.*]{} [**217**]{} (2008), no. 1, 42–78. F. Durand. “Linearly recurrent subshifts have a finite number of non-periodic subshift factors”. [*Ergodic Theory Dynam. Systems*]{} [**20**]{} (2000) 1061–1078. F. Durand. Corrigendum and addendum to [@Dur00]. [*Ergodic Theory Dynam. Systems*]{} [**23**]{} (2003) 663–669. A. Glen, J. Justin, S. Widmer, L.Q. Zamboni. “Palindromic richness”. [*European J. Combin.*]{} [**30**]{} (2009) 510–531. D. Guido, T. Isola. “Dimensions and singular traces for spectral triples, with applications for fractals”. [*J. Func. Anal.*]{} [**203**]{} (2003) 362–400. D. Guido, T. Isola. “Dimension and spectral triples for fractals in ${{\mathbb R}}^n$”. In Advances in Operator Algebras and Mathematical Physics, [*Theta Ser. Adv. Math.*]{} [**5**]{}, Theta, Bucarest (2005), 89–108. M. Forsyth, A. Jayakumar, J. Shallit. “Remarks on Privileged Words”. [*Eprint*]{} arXiv:1306.6768 (math.CO). A.E. Frid, S. Puzynina, L. Zamboni. “On palindromic factorization of words”. [*Adv. in Appl. Math.*]{} [**50**]{} (2013) 737-748. J. Kellendonk, J. Savinien. “Spectral triples and characterization of aperiodic order”. [*Proc. London Math. Soc.*]{} [**104**]{} (2012) 123-157. J. Kellendonk, D. Lenz, J. Savinien. “A characterisation of subshifts with bounded powers”. [*Discrete Math.*]{} [**313**]{} (2013) 2881-2894. J. Lagarias, P. Pleasants. “Repetitive Delone sets and quasicrystals”. [*Ergodic Theory Dynam. Systems*]{} [**23**]{} (2003) 831–867. D. Lenz. “Aperiodic linearly repetitive delone sets are densely repetitive”. [*Discrete Comput. Geom.*]{} [**31**]{} (2004) 323-326. J. Patera (ed.): [*Quasicrystals and Discrete Geometry*]{}, Toronto, ON, 1995. Fields Inst. Monogr., vol. 10. Am. Math. Soc., Providence (1998) J. Pearson, J. Bellissard. “Noncommutative Riemannian geometry and diffusion on ultrametric Cantor sets”. [*J. Noncommut. Geom.*]{} [**3**]{} (2009) 447–481. J. Peltomäki. “Introducing Privileged Words: Privileged Complexity of Sturmian Words”. [*Theoret. Comput. Sci.*]{} [**500**]{} (2013) 57-67. J. Peltomäki. “Privileged Factors in the Thue-Morse Word - A Comparison of Privileged Words and Palindromes”. [*Eprint*]{} arXiv:1306.6768 (math.CO). M. Rieffel. “Metrics on state spaces”. [*Doc. Math.*]{} [**4**]{} (1999) 559–600. M. Rieffel. “Compact Quantum Metric Spaces”. [*Operator algebras, quantization, and noncommutative geometry*]{} 315–330, Contemp. Math. [**365**]{}, Amer. Math. Soc., Providence (2004). [^1]: sometimes also called a [*puncture*]{}, so that one talks about [*punctured tilings*]{}. [^2]: closed and open sets [^3]: ${d_{\text{\rm inf}}}$ is a ultra-metric, ${d_{\text{\rm sup}}}$ is valued in $[0,+\infty]$.

--- abstract: | The recent TRIUMF experiment for $\mu^- p \rightarrow n \nu_{\mu} \gamma$ gave a surprising result that the induced pseudoscalar coupling constant $g_P$ was larger than the value obtained from $\mu^- p \rightarrow n \nu_{\mu}$ experiment as much as 44 %. Reexamining contribution of the axial vector current in electromagnetic interaction, we found an additional term to the matrix element which was used to extract the $g_P$ value from the measured photon energy spectrum. This additional term, which plays a key role to restore the reliability of $g_P ( - 0.88 m_{\mu}^2 ) = 6.77 g_A (0)$, is shown to affect the $g_P$ quenching problems in nucleus. address: | Department of Physics, Yonsei University, Seoul, 120-749, Korea\ (10 November, 1998) author: - 'Myung Ki Cheoun [^1], K.S.Kim, B.S.Han and Il-Tong Cheon' title: Radiative Muon Capture and Induced Pseudoscalar Coupling Constant in Nuclear Matter --- Introduction ============ The matrix element of vector and axial vector currents are generally given as $$\begin{aligned} \langle N (p^{'}) \vert V_a^{\mu} (0) \vert N (p) \rangle & = {\bar u} ( p^{'}) [ G_V ( q^2) \gamma^{\mu} + {{G_S ( q^2 )} \over { 2 m}} q^{\mu} + G_M ( q^2) \sigma^{\mu \nu} q_{\nu} ] {\tau_a \over 2} u(p) \nonumber \\ \langle N (p^{'}) \vert A_a^{\mu} (0) \vert N (p) \rangle &= {\bar u} ( p^{'}) [ G_A ( q^2) \gamma^{\mu} + {{G_P ( q^2 )} \over { 2 m}} q^{\mu} + G_T ( q^2) \sigma^{\mu \nu} q_{\nu} ] \gamma_5 {\tau_a \over 2} u(p)~, \nonumber \\\end{aligned}$$ where $G_A (0) = g_A (0),~ G_M(0) = g_M(0),~ G_V(0) = g_V(0)$ and $~G_P ( q^2) = ( {{ 2 m } \over { m_{\mu}}} ) g_P ( q^2) $ with the nucleon and muon masses, $m $ and $ m_{\mu}$. $\tau_a$ is the isospin operator. $G_S$ and $G_T$ belong to the second class current which has a different G-parity from the first class current, and they are assumed to be absent from the muon capture to be discussed in this paper. On the basis of PCAC (Partially Conserved Axial Current), the induced pseudoscalar coupling constant is calculated as $$g_P ( -0.88 m_{\mu}^2 ) = { { 2 m ~ m_{\mu} } \over { m_{\pi}^2 + 0.88 m_{\mu}^2}} g_A (0) = 6.77 g_A(0 ).$$ This value is confirmed by an experiment of the ordinary muon capture (OMC) on a proton, ${\mu}^- p \rightarrow n \nu_{\mu}$ [@Ba81]. However, in order to obtain more precise data, the TRIUMF group measured recently the photon energy spectrum of the radiative muon capture (RMC) on a proton, $\mu^- p \rightarrow n \nu_{\mu} \gamma$ and extracted a surprising result [@Jo96] $${\hat g_P} \equiv g_P ( - 0.88 m_{\mu}^2 ) / g_A (0) = 9.8 \pm 0.7 \pm 0.3~.$$ It exceeds the value obtained from OMC as much as 44%. This discrepancy is serious because the theoretical value of $g_P$ is predicted in a fundamental manner based on PCAC and agrees with the OMC value. As long as PCAC is assumed to be creditable, a doubt may be cast on the result of TRIUMF experiment. Recent calculations [@Me97; @An97] by chiral perturbation also says such a doubt. However, in order to solve this puzzle, one has to reexamine carefully the Beder-Fearing formula [@Fe80; @Be87], which is a phenomenological model, used to extract the $g_P$ value from the measured RMC spectrum. In finite nuclei, through the theoretical analyses of OMC experimental results, it is already reported [@Ha96] that the ${\hat g}_P$ value is quenched in medium-heavy and heavy nuclei while it is enhanced in light nuclei. Since these analyses are carried out before the recent TRIUMF experiment one needs to reconsider those analyses from another viewpoint. In this paper we present more successful analysis for the recent TRIUMF data and show some progressive results for ${\hat g}_P$ quenching problems in nucleus by applying our results on proton to nuclear matter. Basic Formulae ============== We start from the ordinary linear-$\sigma$ model ; $${\cal L}_0 = {\bar \Psi} [ i \gamma^{\mu} \partial_{\mu} - g ( \sigma + i {\vec \tau} \cdot {\vec \pi} \gamma_5 )] \Psi + { 1 \over 2} [ {( \partial_{\mu} {\vec \pi} )}^2 + {( \partial_{\mu}{\sigma} )}^2 ] + { 1 \over 2} {\mu}^2 ( {\vec \pi}^2 + {\sigma}^2 ) - { {\lambda}^2 \over 4} {( {\vec \pi}^2 + {\sigma}^2 )}^2$$ , which gives the following axial current $$A_{\mu}^a = {\bar \Psi} \gamma_{\mu} \gamma_5 { {\tau_a} \over 2} \Psi + {\pi}^{a} {\partial}_{\mu} \sigma - \sigma {\partial}_{\mu} {\pi}^a ~.$$ By the spontaneous breakdown of chiral symmetry, $\sigma$ field is shifted to ${\sigma}^{'} = \sigma - {\sigma}_0$ with ${\sigma}_0 = f_{\pi}$. Consequently, the pion appears as Nambu-Goldstone boson. The PCAC can be satisfied by the additional inclusion of the explicit chiral symmetry breaking term as well known. But the axial current $$A_{\mu}^a = {\bar \Psi} \gamma_{\mu} \gamma_5 { {\tau_a} \over 2} \Psi - f_{\pi} {\partial}_{\mu} {\pi}^a ~$$ gives $g_A = 1$ in the tree approximation. Following the recipe of Akhmedov [@Akh89] to cure this problem, we add chiral invariant lagrangian ${\cal L}_1$ to ${\cal L}_0$, $${\cal L}_1 = C[ {\bar \Psi} {\gamma}_{\mu} {{\vec \tau} \over 2} \Psi ( {\vec \pi} \times {\partial}_{\mu} {\vec \pi}) + {\bar \Psi} {\gamma}_{\mu} \gamma_5 {{\vec \tau} \over 2} \Psi ( {\vec \pi} {\partial}_{\mu} {\sigma} - \sigma {\partial}_{\mu} {\vec \pi}) ]~,$$ where arbitrary parameter $C$ is determined so that the axial current pertinent to nucleons in ${\cal L} = {\cal L}_0 + {\cal L}_1$ $${}^{(N)} {A_{\mu}^{a}} = {\bar \Psi} {\gamma}_{\mu} {\gamma_5} {{{\tau}_a } \over 2} \Psi [ 1 + C^2 ( {\vec \pi}^2 + {\sigma}^2 ) ]$$ should satisfy ${}^{(N)}{A_{\mu}^{a}} = g_A {\bar \Psi} {\gamma}_{\mu} {\gamma}_5 {{{\tau}_a} \over 2} \Psi $ with $g_A = 1.26$. The Goldberger-Treiman relation then is satisfied exactly. As a consequence, ${}^{(N)} A_{\mu}$ includes the contribution not only from the nucleon but also from the $\pi - N$ interactions. Now, the axial vector current consists of the nucleon and pion sectors as $$\begin{aligned} A^{\mu}_a ( x) = {}^{(N)}\! A_a^{\mu} ( x) + {}^{(\pi)} A_a^{\mu} (x) \\ \nonumber = {}^{(N)} A_a^{\mu} ( x) + f_{\pi} \partial^{\mu} {\phi}_a (x)~, \end{aligned}$$ where $f_{\pi}$ is the pion decay constant. $\phi_a (x) $ is the pion field. To describe RMC, we need a radiative axial current, which is used to obtain the transition amplitude of RMC by coupling to the weak current of lepton line. Three different methods are considered in order to construct a radiative axial current. The 1st method [@Ch98] is to start from the above lagrangians ${\cal L} = {\cal L}_0 + {\cal L}_1$ using covariant derivative, by which we introduce a photon field in U(1) gauge invariant way. The outcoming lagrangian gives a radiative axial current, which characteristic is its non-conservation through the explicit chiral symmetry breaking due to the electromagnetic interaction. The second method [@Ch97] is to use the extended Euler equation [@Ad65] for the lagrangian ${\cal L} = {\cal L}_0 + {\cal L}_1$. The third is to make it directly from the above axial currents, eq.(9), by exploiting it minimal coupling scheme to the momenta of relevant particles. Here we follow the third one. Of course the final radiative axial currents from these three different methods turned out to be equivalent. Let us begin from the divergences of $^{(N)} A_a^{\mu} (x)$ and $^{(\pi)} A_a^{\mu} (x)$, $$\partial_{\mu} {}^{(N)} A_a^{\mu} (x) = {\partial}_{\mu} [ g_A {\bar \Psi} (x) {\gamma}^{\mu} \gamma_5 { \tau_a \over 2} \Psi (x) ] \equiv f_{\pi} J_a^{N}~,$$ $$\partial_{\mu} {}^{(\pi)} A_a^{\mu} (x) = f_{\pi} \partial^2 \phi_a (x) = - f_{\pi} [ m_{\pi}^2 \phi_a (x) + J_a^N ]~,$$ where we used pion field equation from the above lagrangians $$(\partial^2 + m_{\pi}^2 ) \phi_a = -J_a^N~.$$ The quantity $J_a^N$ denotes the pion source term. Therefore, the divergence of total axial currents is given in the following way $$\begin{aligned} \partial_{\mu} A_a^{\mu} (x) & = & \partial_{\mu} {}^{(N)} A_a^{\mu} (x) + \partial_{\mu} {}^{(\pi)} A_a^{\mu} (x) \\ \nonumber & =& - f_{\pi} [ m_{\pi}^2 \phi_a (x) + J_a^N ] + f_{\pi} J_a^N \\ \nonumber & =& - f_{\pi} m_{\pi}^2 \phi_a (x) ~.\end{aligned}$$ This is PCAC. By eqs. (10) and (12), one can obtain $$\phi_a (x) = -{ 1 \over { f_{\pi} ( \partial^2 + m_{\pi}^2 )}} \partial_{\mu} {}^{(N)}\!A_a^{\mu} (x).$$ Substitution of eq.(14) into eq.(9) yields $$A_a^{\mu} (x) = {}^{(N)}\!A_a^{\mu} (x) - {i \over { \partial^2 + m_{\pi}^2} } (i\partial)^{\mu} [ \partial_{\mu} {}^{(N)}A_a^{\mu} (x) ].$$ In order to clarify a role of the axial current in description of the radiation process, we adopt the minimal coupling prescription, i.e. $\partial_{\lambda} \rightarrow \partial_{\lambda} - ie {\cal A}_{\lambda}$ and $q^{\mu} \rightarrow q^{\mu} - e {\cal A}^{\mu}$. This procedure leads to $$\begin{aligned} A_a^{\mu} (x) & = & {}^{(N)}\! A_a^{\mu} (x) - { i \over {\partial^2 + m_{\pi}^2 }} (i\partial)^{\mu} [ \partial_{\lambda} {}^{(N)} A_a^{\lambda} (x)] + { i e \over {\partial^2 + m_{\pi}^2 }} \epsilon^{\mu} [ \partial_{\lambda} {}^{(N)} A_a^{\lambda} (x)] \nonumber \\ &&- { e \over {\partial^2 + m_{\pi}^2 }} (i\partial)^{\mu} [ \epsilon_{\lambda} {}^{(N)} A_a^{\lambda} (x)] + { e^2 \over {\partial^2 + m_{\pi}^2 }} [ \epsilon^{\mu} \partial_{\lambda} {}^{(N)} A_a^{\lambda} (x)]~,\end{aligned}$$ where the potential ${\cal A}_{\lambda}$ is replaced by the photon polarization vector $\epsilon_{\lambda}$. Notice here that $\partial_{\lambda} {}^{(N)}A_a^{\lambda} (x) = g_A m {\bar \Psi} (x) i \gamma_5 \tau_a \Psi (x)$. The last term in eq.(16) can be neglected because it appears at $O(e^2)$ order. Thus we can express the axial current in the radiative processes in the following realistic form $$\begin{aligned} A_a^{\mu} (x) & = & {\bar \Psi} (x) [ g_A \gamma^{\mu} \gamma_5 + { {g_P (q^2) } \over { m_{\mu}}} q^{\mu} \gamma_5 - { {e g_P ( q^2) } \over { m_{\mu}}} \epsilon^{\mu} \gamma_5 ] {\tau_a \over 2} \Psi (x) \\ \nonumber & & - { {e g_P (q^2) } \over { 2 m m_{\mu}}} q^{\mu} [ {\bar \Psi} (x) \epsilon_{\alpha} \gamma^{\alpha} \gamma_5 {\tau_a \over 2} \Psi (x) ]~.\end{aligned}$$ The fourth term, which corresponds to “Seagull term”, is missing [@Ch97] in the previous calculations [@Fe80; @Be87]. Following the Fearing’s formulation and notation [@Fe80] for the diagrams given in ref. [@Fe80], one can evaluate the relativistic amplitude of RMC on a proton as $$M_{ f i} = M_a + M_b + M_c + M_d + M_e + \Delta M_e$$ with $$\begin{aligned} M_a &=& - \epsilon_{\alpha} {\bar u}_n \Gamma^{\delta} ( Q) u_p \cdot {\bar u}_{\nu} \gamma_{\delta} ( 1 - \gamma_5) { { \mu\!\!\!/ - {k \!\!\!/} + m_{\mu} } \over { - 2 k \cdot \mu}} \gamma^{\alpha} u_{\mu} ~,\\ \nonumber M_b &=& \epsilon_{\alpha} L_{\delta} {\bar u}_n \Gamma^{\delta} ( K) { {{p \!\!\!/}-{ k \!\!\!/}+ m_p} \over {- 2 k \cdot p}} (\gamma^{\alpha} - i \kappa_p { \sigma^{\alpha \beta} \over { 2 m_p}} k_{\beta} ) u_p ~, \\ \nonumber M_c &=& \epsilon_{\alpha} L_{\delta} {\bar u}_n ( - i \kappa_n { \sigma^{\alpha \beta} \over {2 m_n }} k_{\beta} ) { {{n \!\!\!/}+{ k \!\!\!/}+ m_n} \over { 2 k \cdot n}} \Gamma^{\delta} ( K) u_p ~,\\ \nonumber M_d &=& - \epsilon_{\alpha} L_{\delta} {\bar u}_n ( { {2 Q^{\alpha} + k^{\alpha}} \over { Q^2 - m_{\pi}^2 }} { g_P ( K^2) \over m_{\mu} } K^{\delta} \gamma_5 ) u_p ~,\\ \nonumber M_e &=& \epsilon_{\alpha} L_{\delta} {\bar u}_n ( { {i g_M } \over {2 m} } \sigma^{\delta \alpha} + {{g_P ( Q^2)} \over{m_{\mu}} } \gamma_5 g^{\delta \alpha} ) u_p ~,\\ \nonumber \Delta M_e & =& - \epsilon_{\alpha} L_{\delta} {\bar u}_n ( { { g_P (K^2)} \over {2 m m_{\mu}} } Q^{\delta} \gamma_5 \gamma^{\alpha} ) u_p ~,\end{aligned}$$ where $$\Gamma^{\delta} ( q ) = g_V \gamma^{\delta} + { { i g_M } \over { 2 m}} \sigma^{\delta \beta } q_{\beta} + g_A \gamma^{\delta} \gamma_5 + { {g_P (q^2)} \over m_{\mu}} q^{\delta} \gamma_5 ~,$$ $L_{\delta} = {\bar u}_{\nu} \gamma_{\delta} ( 1 - \gamma_5) u_{\mu}, K = n - p + k $ and $ Q = n - p$ with momenta of neutron, proton and photon, $ n, p $ and $ k$, respectively. And $m \sim m_p \sim m_n$. Other constants are taken as $g_V = 1.0, g_A = - 1.25, g_M = 3.71, \kappa_p = 1.79$ and $\kappa_n = - 1.91$ [@Fe80]. $M_e$ term is originated from the third term in eq.(17) and $\Delta M_e$ term comes from the fourth term. But the latter, $\Delta M_e$, is missing in the paper by Fearing [@Fe80; @Be87]. Accordingly, this term was not included in the previous procedure of extracting $g_P$ value from the experimental RMC photon energy spectrum [@Jo96]. The above transition amplitude can be also understood in terms of pseudo vector (PV) coupling scheme between nucleons and virtual pion, through which external axial current interacts with nucleons. Moreover in nuclear matter, this PV coupling type is preferred rather than PS coupling type because the former is consistent with PCAC while the latter contradicts to PCAC in nuclear matter [@Akh89]. The RMC transition rate is given by $${ {d \Gamma_{RMC}} \over {d k}} = { {\alpha G^2 \vert \phi_{\mu} \vert^2 m_N } \over { {( 2 \pi )}^2 }} \int_{-1}^{1} dy { { k E_{\nu}^2 } \over { W_0 - k( 1 - y) }} { 1 \over 4} \sum_{spins} \vert M_{f i } {\vert}^{2} ~,$$ where $\alpha$ is the fine structure constant, $G$ is the standard weak coupling constant, $ y = {\hat k } \cdot {\hat \nu},~ k_{max} = ( W_0^2 - m_n^2 ) / 2 W_0,~ E_{\nu} = W_0 ( k_{max} - k ) / [ W_0 - k( 1 - \nu)],~ W_0 = m_p + m_n $ - (muon binding energy) and $\vert \phi_{\mu} \vert^2$ is the absolute square of muon wave function averaged over the proton which is taken as a point Coulomb. In order to compare to the experimental results, we take the following steps. For liquid hydrogen target, muon capture is dominated through the ortho and para $p \mu p$ molecular states [@Jo96; @Ba82]. Since these molecular states can be attributed to the combinations of hyperfine states of $\mu p$ atomic states [@Ba82] i.e. single and triplet states, we decompose the statistical spin mixture ${1 \over 4} \sum_{spins} \vert M_{ f i } {\vert}^2$ into such hyperfine states by reducing $4 \times 4 $ matrix elements to $2 \times 2$ spin matrix elements. At this step, we confirmed that when the $\Delta M_e$ term was not included, eq.(21) reproduced the curves given in ref. [@Be87]. For the description of RMC in nuclear matter, we follow the Fearing’s paper [@Fe89], i.e. we adopted the relativistic mean field theory [@Se86] where the nucleons are treated as free Dirac particles with effective mass due to the scalar and vector potential. Then the nucleus are the Fermi gas. Our RMC capture rate in nuclear matter is given in the following way $$\begin{aligned} \Gamma_{RMC}^{NM}& = &{{\alpha G^2 } \over { 4 {\pi}^2}} { { 3 m_P m_N } \over { 4 \pi k_F^3}} \vert \phi_{\mu} {\vert}^2 \int_{k_{min}}^{k_{max}} dk \int_{{(cos {\theta}_k )}_{min}} ^{{(cos {\theta}_k )}_{min}} d cos {\theta}_k \int_{p_{min}}^{k_F} dp \int_{k_F}^{n_{max}} dn \nonumber \\ & & \int_{0}^{ 2 \pi} d \phi_n { { E_{\nu} n p^2 k } \over { E_p E_n \vert {\vec n} + {\vec p} \vert}} {1 \over 4} \sum_{spins} \vert M_{f i } {\vert}^{2} ~,\end{aligned}$$ where the integration intervals of nucleon momenta, which comes from the Pauli blocking in nuclear matter, are calculated in detailed kinematics and $k_F$ is the Fermi momentum. For finite nuclei, we need to know the corresponding muon wave functions and $k_F$ values, but which depends on the model. We already suggested a model [@Ch97] for this purpose, but will be skipped here and concentrate on the case of nuclear matter. Results and Discussions ======================= Our results for RMC on proton are shown in Fig.1. The solid curve is the spectrum obtained in ref. 2., i.e. the result without $\Delta M_e$ term for ${\hat g}_P = 9.8$. On the other hand, the dotted curve is calculated without $\Delta M_e$ term for ${\hat g}_P$ = 6.77. This curve is obviously much lower than the solid curve. When $\Delta M_e$ term is taken into account for ${\hat g}_P$ = 6.77, we obtain the dashed curve which is very close to the solid curve for the energy spectrum on $k \ge 60 MeV$. The minor discrepancy may be due to the neglect of higher order contributions and other degree of freedom such as $\Delta$. Our result shows that $\Delta M_e$ term restores the credit of ${\hat g}_P = 6.77$. The number of RMC photons observed for $k \ge 60 MeV $ is 279 $\pm$ 26 and the number of those from the solid curve is 299, while our result obtained by integrating the dotted curve spectrum is 273. Since the contribution of $\Delta$ degree of freedom is known to be a few percent [@Be87], it is not included in the present calculation. Vector mesons such as $\rho$ and $\omega$ are also turned out to have very small contributions in this calculation. Higher order terms are pointed out to be insignificant [@Fe80]. It is confirmed that the matrix elements upto the $M_e$ term in eq.(19) satisfy gauge invariance. However, it is broken when the $\Delta M_e$ term and $\Delta$ degree of freedom are included. As far as diagram method is adopted, the gauge invariance is more or less broken because a series of diagrams will be cut somewhere. In order to estimate the rate of gauge invariance broken by the $\Delta M_e$ term, we evaluated the spectrum with $\Delta M_e$ alone. The result is shown by a dot-dashed curve in Fig.1. The gauge invariance breaking is not so large as expected. However, in the limit $m_{\mu} \rightarrow 0$, the gauge invariance is restored. In spite of such a burden of gauge invariance, our present calculation shows that ${\hat g}_P$ = 6.77 is reasonable for both OMC and RMC on a proton. Figure 2 shows the photon energy spectrum in nuclear matter, which is presented as the ratio of RMC to OMC in order to reduce the uncertainty from the nuclear structure. We compared our amplitude to Fearing’s analysis, which is PS coupling and has been used to extract the ${\hat g}_P$ from the experimental data. At the same ${\hat g}_P$, our ratio R is higher than the PS coupling scheme. It means that ${\hat g}_P$ value to fit some experimental data becomes lower in our PV scheme. Therefore ${\hat g}_P$ quenching rate is smaller than the usual PS coupling scheme. These behaviour are nearly independent of effective nucleon mass in nuclear matter (see the solid, dotted and dashed lines). Figure 3 shows another interesting results for ${\hat g}_P$ quenching in finite nuclei. The larger $k_F$, which may mean the heavier nuclei, the smaller becomes the ratio R. As a result, ${\hat g}_P$ quenching may be larger in heavier nuclei. But in relatively lower $k_F$ region just reversed results are shown. Consequently ${\hat g}_P$ may be enhanced in lower $k_F$ region. Since we have to integrate the nucleon’s Fermi motion from the possible lowest momentum up to the Fermi momentum, the mechanism in lower $k_F$ region could play a role of compensating ${\hat g}_P$ quenching in larger $k_F$ region. Therefore this could be an indication for the ${\hat g}_P$ enhancement in light nuclei. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the KOSEF and the Korean Ministry of Education (BSRI-97-2425). G.Bardin [*et al.*]{}, Phys. Lett. [**104B**]{}, 320 (1981) and T.P.Gorringe [*et al.*]{}, Phys. Rev. Lett. [**72**]{}, 3472 (1994). G.Jonkmans [*et al.*]{}, Phys. Rev. Lett. [**77**]{}, 4512 (1996). M.Hasinoff, Non-Nucleonic Degrees of Freedom Symposium, Osaka, RCNP, Japan,(1996). T.Meissner, F.Myhrer and K.Kubodera, preprint, nucl-th/9707019, (1997). S.Ando and D.P.Min, preprint, hep-th/9707504, (1997). H.W.Fearing, Phys. Rev. [**21**]{}, 1951(1980). D.S.Beder and H.W.Fearing, Phys. Rev. [**D35**]{}, 2130 (1987). Myung Ki Cheoun, K.S.Kim and Il-Tong Cheon, preprint, nuch-th/9811005 (1998). Il-Tong Cheon and Myung Ki Cheoun, Proceedings of ISR ’97, Tashkent, (1997) and preprint, nucl-th/9811009, (1998). S.Adler, Phys. Rev., [**139**]{}, B1638 (1965). Il-T.Cheon and M.T.Jeong, J.Phys.Soc.Jpn. [**61**]{}, 2726 (1992). E.Kh.Akhmedov, Nucl. Phys., [**A500**]{}, 596 (1989). D.D.Bakalov, M.P.Faifman, L.I.Ponomarev and S.I.Vinitsky, Nucl. Phys. [**A384**]{}, 302(1982). Harold W. Fearing and G.E.Walker, Phys. Rev., [**C39**]{}, 2349(1989). D.Serot and J.D.Walecka, in Advances in Nuclear Physics, edited by J.W.Negles and E.Vogt, Vol. 16, Plenum, New York, (1986). Figure Captions Figure 1. Photon energy spectrum for triplet states in RMC on proton. The solid curve is deduced without $\Delta M_e$ term for ${\hat g}_P$ = 9.8, whose result corresponds to the experimental results in ref.2. The dotted curve is obtained without $\Delta M_e$ term for ${\hat g}_P = 6.77$. The dashed curve is with $\Delta M_e$ for ${\hat g}_P$ = 6.77. The dot-dashed curve is calculated with $\Delta M_e$ term alone for ${\hat g}_P$ = 6.77. Figure 2. The photon energy spectrum for the ratio of RMC and OMC in nuclear matter. The thick curves are results from our transition amplitudes, but thin curves are Fearing’s amplitudes. The solid curves are for $ M^* = M $, the dotted curves are for $M^* = 0.57 M$ and the dashed curves are for $M^* = 0.7 M$. Figure 3. The ratio of RMC and OMC versus fermi momentum $k_F$. The dot-dashed curve ($k_{\gamma}$ = 60 MeV) and dotted curve ($k_{\gamma}$ = 80 MeV) come from our amplitude, while the solid and long dashed curves are from Fearing’s, respectively from $k_{\gamma}$ = 60 and 80 MeV. [^1]: e.mail : cheoun@phya.yonsei.ac.kr

--- abstract: 'We consider serious conceptual problems with the application of standard perturbation theory, in its zero temperature version, to the computation of the dressed [Fermi surface]{}for an interacting electronic system. In order to overcome these difficulties, we set up a variational approach which is shown to be equivalent to the renormalized perturbation theory where the dressed [Fermi surface]{}is fixed by recursively computed counterterms. The physical picture that emerges is that couplings that are irrelevant tend to deform the [Fermi surface]{}in order to become more relevant (irrelevant couplings being those that do not exist at vanishing excitation energy because of kinematical constraints attached to the [Fermi surface]{}). These insights are incorporated in a renormalization group approach, which allows for a simple approximate computation of [Fermi surface]{}deformation in quasi one-dimensional electronic conductors. We also analyze flow equations for the effective couplings and quasiparticle weights. For systems away from half-filling, the flows show three regimes corresponding to a Luttinger liquid at high energies, a Fermi liquid, and a low-energy incommensurate spin-density wave. At half-filling Umklapp processes allow for a Mott insulator regime where the dressed [Fermi surface]{}is flat, implying a confined phase with vanishing effective transverse single-particle coherence. The boundary between the confined and Fermi liquid phases is found to occur for a bare transverse hopping amplitude of the order of the Mott charge gap of a single chain.' address: | $^1$ Laboratoire de Physique Théorique et Hautes Energies, CNRS UMR 7589, Université Paris VII-Denis Diderot,\ 4, Place Jussieu, 75252 Paris Cedex 05, France. author: - 'Sébastien Dusuel$^{1}$, Benoît Douçot$^{1}$' title: 'Interaction-induced Fermi surface deformations in quasi one-dimensional electronic systems' --- Introduction {#sec:intro} ============ One of the striking results obtained in the last decade on strongly correlated electronic systems is the coexistence of a notion of [Fermi surface]{}and of strong deviations from the predictions of Fermi liquid theory for many low-energy properties. This has been extensively studied experimentally for high-temperature superconducting cuprates, where angular resolved photoemission spectroscopy (ARPES) has revealed the presence of [Fermi surface]{}arcs, even in the underdoped regime which is characterized by the pseudo-gap seen with most low-energy probes.[@Timusk99] Although these systems exhibit intermediate or even strong electron interactions, they have triggered many theoretical works using perturbative tools.[@Zanchi96; @Halboth00; @Honerkamp01] At the beginning of any perturbative analysis, the shape of the [Fermi surface]{}is crucial in determining which couplings survive in an effective low-energy description.[@Shankar94] For most crystalline materials the absence of continuous rotational invariance allows for a deformation of the [Fermi surface]{}away from the bare free electron [Fermi surface]{}, as interactions are switched on. In many metallic systems this effect is not expected to play much role beyond usual renormalizations of effective parameters of band theory. But in some situations, like the vicinity of a Van-Hove singularity, the presence of a nesting vector, or for strongly anisotropic conductors, it seems essential to understand how to compute the dressed [Fermi surface]{}, since it is the relevant object for the construction of an effective low-energy theory. In the case of quasi one-dimensional (quasi 1D) systems, this [Fermi surface]{}deformation is intimately connected to the widely studied notion of transverse coherence. Experimental and theoretical investigations converge towards a description in terms of almost uncoupled Luttinger liquids along the chains, at high enough energies.[@Jerome82_dans_articles; @Bourbonnais91] At low energies, optical conductivity measurements[@Vescoli98] have shown the existence of two types of behaviors: either the system remains confined in a Mott-insulator phase (in the TMTTF compounds) or the transverse hopping of electrons takes over and establishes a long-ranged transverse phase coherence, leading to a two-dimensional (2D) Fermi liquid phase (for the TMTSF). In the latter case the dressed [Fermi surface]{}remains warped while in the former it becomes completely flat under the effect of sufficiently strong interactions.[@Prigodin79; @Bourbonnais85] Because of their difficulty, precise computations of [Fermi surface]{}deformations for model systems have been undertaken only recently. A direct numerical evaluation of the electron propagator to second order in interaction has been performed for the 2D Hubbard model.[@Zlatic95; @Halboth97] Similar studies have also been carried for more phenomenological models where electrons are scattered by dynamical spin fluctuations.[@Yanase99; @Morita00] Although these computations yield valuable physical understanding of the processes involved in the [Fermi surface]{}deformation, they suffer from at least two serious problems. First, they identify the dressed [Fermi surface]{}with the locus of points in $k$-space for which the dressed quasiparticle energy is equal to the (interacting) chemical potential, which is of course correct. But this does not imply that the imaginary part of the [self-energy]{}vanishes on this surface and for frequencies equal to the chemical potential. Therefore this procedure does not lead to a picture of asymptotically stable quasiparticles at low energies. This remark is valid in the zero temperature approach, which is the only one we are using in this paper, because of its conceptual simplicity. Second, this problem is not cured while going to higher orders in perturbation theory. Furthermore, some new problems arise (namely infrared divergences) at these higher orders for both zero and finite temperature formalisms. The underlying assumption of the standard perturbation scheme as used above is that one can generate the interacting ground-state by adiabatically switching on the interactions, starting from the non-interacting ground-state. This has to be questioned for large systems for which the ground-state lies at the edge of an energy continuum. Because of this, the perturbation algorithm acting on various excited states of the original systems, associated to different shapes of the [Fermi surface]{}, has the possibility to generate energy levels’ crossings. This implies that the seed state to be used in perturbation theory is not known a priori, when interactions do deform the [Fermi surface]{}. This difficulty has been pointed out in the sixties by Kohn and Luttinger,[@Kohn60] and also Nozières.[@Nozieres_anglais] These ideas have been revived recently in a mathematically rigorous framework.[@Feldman96] The conclusion of all these works is that a sound formalism is obtained when one works with a bare propagator which singularities are pinned to the [*dressed*]{} [Fermi surface]{}. This is achieved in practice by the introduction of counterterms, which have to be computed order by order in perturbation theory. The main difficulty in practical implementations of this philosophy (which may be called renormalized perturbation theory) is that it provides only an implicit determination of the dressed [Fermi surface]{}, since this algorithm expresses the bare [Fermi surface]{}as a function of the dressed one. Although formally this connection has been proved to be invertible,[@Feldman98] this remains a formidable task which has never been, to our knowledge, practically undertaken. Note that the necessity to use these counterterms is not a pathology of the zero temperature approach. It also appears in the Matsubara formalism at finite temperature which is the one used in the rigorous works just described. As a first step towards the realization of this program, several groups have performed self-consistent computations. Their basic principle is to start with a trial [Fermi surface]{}, which is adjusted so that it matches with the calculated [Fermi surface]{}. A first example follows directly the standard Hartree-Fock method.[@Valenzuela01] It has been applied to the 2D Hubbard model in the presence of second-neighbor hopping and nearest neighbor interaction, and the possibility of a change in [Fermi surface]{}topology (from hole-like to electron-like) has been observed. A rather sophisticated scheme has also been developed by Nojiri,[@Nojiri99] in which the [self-energy]{}is self-consistently computed from the corresponding second order Feynman diagram. This work addressed the simplest 2D Hubbard model with on-site interaction for which the [Fermi surface]{}deformation was found to be very small and to preserve the [Fermi surface]{}topology. Note that the quantitative difference between this self-consistent scheme and a standard perturbation theory[@Zlatic95; @Halboth97] appears to be small. In spite of their merits, these approaches lack the ability to keep track of the growth of some effective couplings, as the typical energy scale is lowered. These effects play a crucial role for the 2D Hubbard model near half-filling, or for quasi 1D conductors. A natural way of handling these trends is to use a renormalization group (RG) approach. Several groups have incorporated the RG methodology in the computation of the dressed [Fermi surface]{}.[@Prigodin79; @Bourbonnais85; @Kishine98; @Honerkamp01] Similar studies have also been carried for two coupled chains where the [Fermi surface]{}reduces to four Fermi points.[@Fabrizio93; @Tsuchiizu99; @LeHur01] Our understanding of these works is that they always begin with a known bare [Fermi surface]{}and compute the evolution of the effective [Fermi surface]{}, as the high-energy cut-off is gradually decreased. Although this is very reasonable on physical grounds, we may wonder whether this fits with the general rigorous analysis described in the last but one paragraph. We believe there are two ways to combine the corresponding requirements with a RG approach. The first one uses the renormalized perturbation theory described above, with a running energy cut-off. After the usual mode integration in a small energy shell, the kinetic term in the effective action is corrected to preserve the shape of the dressed [Fermi surface]{}. In the process of integrating the RG flow, one has to keep track of and sum all these counterterms to obtain the bare [Fermi surface]{}as a function of the dressed one. Alternatively, one would fix the bare high-energy theory, and perform the mode integration is such a way that modes being integrated out always remain at a finite distance from the flowing [Fermi surface]{}. But then one has to ensure that all modes are integrated over exactly once with a uniform weight. This is indeed possible but requires some slight modifications of the Wilson-Polchinski usual RG equations.[@Drazen] We believe the practical implementation of either approach remains to be attempted. The bulk of this paper is composed of three sections. Sec. \[sec:csfs\] begins with a general discussion of some difficulties with the standard perturbation theory. We then develop a physical understanding of the driving force that deforms the [Fermi surface]{}on the basis of a simple variational calculation for a system of two spinless chains. The main insight gained here is that the couplings which tend to deform the [Fermi surface]{}are those for which external momenta of in and out going particles can not be simultaneously taken on the [Fermi surface]{}, because of momentum conservation. In the RG language, these interactions are usually called irrelevant. We finally establish the equivalence between this procedure and a standard renormalized perturbation theory where the dressed [Fermi surface]{}is fixed by counterterms. The reader interested in more technical aspects is referred to Appendices \[app:equi\], \[app:diff\_re\] and \[app:diff\_im\] (the first two begin with some simple first order calculations on the system of two spinless chains, whose results can be compared to the ones obtained in Sec. \[sec:csfs\]). In Secs. \[sec:RG\_formalism\] and \[sec:RG\_numerical\] we show how the RG can be implemented in the study of quasi 1D systems. We want to emphasise that we have not made use of a single RG scheme, but of two coupled RG schemes. We describe our motivations for performing such a study in Secs. \[sec:sub:sub:gen\_th\_an\] and \[sec:sub:motiv\_use\_two\_RG\], but let us very briefly explain what they are, before coming to a more detailed description of Secs. \[sec:RG\_formalism\] and \[sec:RG\_numerical\]. The field-theoretical RG in the spirit of Gell-Mann and Low[@Gell-Mann54] is a simple but powerful way of computing low-energy properties of systems described by a renormalizable field-theory. This is why we adopted it for this purpose (this method is discussed in detail in Appendix \[app:field\_th\]). However, it cannot be used to compute the dressed [Fermi surface]{}, for the simple reason that the [Fermi surface]{}is defined as the locus of the zeros, in $k$-space of the inverse propagator evaluated at [*zero frequency*]{}. There is thus no low-energy scale $\nu$ that can be varied to get RG equations as is done for example for the low-energy vertices, when relating the values of these vertices at two different scales $\nu$ and $\nu'$. However, one can use the approach known under the name cut-off scaling, and developed by Sólyom.[@Solyom79_dans_articles] This RG does not suffer from the limitation just described, because it is the high-energy cut-off and not the low-energy scale that is varied, and we have used it for the computation of the dressed [Fermi surface]{}. The high-energy part of the flows, in which the [Fermi surface]{} deformation takes place, is thus described by the cut-off scaling. The dressed [Fermi surface]{}that one obtains in this way then serves as an input parameter for the field-theoretical RG which governs the low-energy part of the flows. Let us say that RG flow equations appear neither in Sec. \[sec:RG\_formalism\] nor in Sec. \[sec:RG\_numerical\], but they all have been gathered in Appendix \[app:flow\_eq\]. In Sec. \[sec:RG\_formalism\] we set up the cut-off scaling approach for the study of [Fermi surface]{}deformations in a quasi 1D system of weakly coupled electronic chains. In order to make the ideas more concrete, this method is then applied to the simplest possible example, and we end the section with a comparison to other methods that can be found in the literature. We then turn to numerical investigations, that are presented in Sec. \[sec:RG\_numerical\], for short range, Hubbard-like, repulsive electron interactions. Sec. \[sec:sub:first\_num\_std\] deals with considerations about systems away from half-filling which exhibit an incommensurate nesting vector for their [Fermi surface]{}. The flow pattern involves a high-energy Luttinger liquid regime, followed by a Fermi liquid at intermediate energy, and finally a long-range ordered spin-density wave (SDW) phase is the stable low-energy attractor. Special attention has been given to the scale and transverse size dependence of the quasiparticle weight. We then focus on the half-filled (and nearly half-filled) case in Sec. \[sec:sub:umklapps\_limitations\], where Umklapp processes may drive the system into a confined low-energy phase and pin the SDW on the crystal lattice. In particular we study the cross-over between the confined and the Fermi liquid regimes. It is shown to occur for bare values of the inter-chain hopping of the order of the 1D Mott charge gap. Computing the shape of the [Fermi surface]{}: various difficulties and their resolution {#sec:csfs} ======================================================================================= General considerations {#sec:sub:gen_cons} ---------------------- As emphasized in the Introduction, the computation of the dressed Fermi surface in an interacting metallic state encounters some obstacles because of the presence of a continuum of low-lying energy states in the immediate vicinity of the non-interacting ground-state. This has been already discussed in a very inspiring paper by Kohn and Luttinger.[@Kohn60] There, they have shown that the standard Brueckner-Goldstone perturbation theory for the ground-state energy is not consistent with a careful procedure of taking the zero-temperature limit of the total energy computed in the grand-canonical ensemble. They interpret this failure in terms of the pattern of energy levels of an interacting Fermi system as a function of the interaction strength. When the shape of the Fermi surface changes, a deep reshuffling of the spectrum takes place, leading to a huge number of level crossings. A simple illustration for this is given on Fig. \[fig:branch\_adiab\_et\_cr\_niv\]. ![Schematic energy level pattern as a function of interaction strength for a conducting Fermi system. Different levels correspond to different choices for the [Fermi surface]{}of the non-interacting system. The left figure represents what happens in the standard perturbation theory, where the level repulsion at avoided crossings can not be resolved, so that one obtains a non-adiabatic evolution of the system wave-function as interactions are increased, therefore generating an excited state. On the right we represent the effect of applying the standard perturbation theory in a finite size system. In this case an adiabatic generation of the interacting ground-state is possible.[]{data-label="fig:branch_adiab_et_cr_niv"}](branch_adiab_et_cr_niv){width="8cm"} At this stage, it is important to distinguish between two situations, which have both interesting physical realizations. For some simple models, such as a ladder of interacting spinless fermions, or a single chain of spin $1/2$ electrons, the total number of particles of a given species (transverse momentum in the ladder case, or the $z$ component of the spin for spin $1/2$ electrons) may be conserved. As a result of this symmetry, the level crossings just mentioned are an essential feature of the exact many-body spectrum. In more general situations, these level crossings appear at any [*finite*]{} order in a perturbative computation of the spectrum as a function of interaction strength, although they are expected to disappear in an exact treatment for a finite-size system. Let us first concentrate on the former case for a while, since it shows dramatically why and where difficulties arise. In such situations, the conventional assumption often made in many-body computations does not hold. It states that one can get the interacting many-body ground-state by adiabatically switching on the interactions, starting from the non-interacting ground-state. A trivial example where the adiabatic switching procedure most often generates an excited state is provided in the case of the free Hamiltonian: $$H_\lambda = \sum_k {\varepsilon}_{\lambda}(k){c^\dagger}(k)c(k),$$ where we arbitrarily split ${\varepsilon}_{\lambda}(k)$ in two parts: $${\varepsilon}_{\lambda}(k) = {\varepsilon}_{0}(k)+\lambda {\varepsilon}_{1}(k).$$ This induces a decomposition of $H_\lambda$ as a sum $H_\lambda=H_{0}+\lambda H_{1}$, where $H_{0}$ is the “unperturbed” Hamiltonian, and $\lambda H_{1}$ the perturbation. Since $H_{0}$ and $H_{1}$ commute, the eigenstates of $H_\lambda$ do not depend on the strength $\lambda$ of the perturbation. But energy levels as functions of $\lambda$ are free to cross, so the initial ground-state ([i.e.]{}for $\lambda = 0$) becomes in general an excited state for finite $\lambda$. This is reflected on the computation of the single particle Green’s function in the zero temperature formalism. Starting with the “bare” propagator $G^{(0)}(k,\omega)^{-1}=\omega-{\varepsilon}_{0}(k)+{{\mathrm i}}\eta{\mathrm{sgn}}\left({\varepsilon}_{0}(k)-\mu_{0}\right)$, the conventional algorithm yields a “dressed” propagator $\tilde{G}^{(\lambda)}(k,\omega)^{-1}=\omega-{\varepsilon}_{\lambda}(k)+{{\mathrm i}}\eta{\mathrm{sgn}}\left({\varepsilon}_{0}(k)-\mu_{0}\right)$ instead of the correct result: $G^{(\lambda)}(k,\omega)^{-1}=\omega-{\varepsilon}_{\lambda}(k)+{{\mathrm i}}\eta{\mathrm{sgn}}\left({\varepsilon}_{\lambda}(k)-\mu_{\lambda}\right)$, where $\mu_{0}$ and $\mu_{\lambda}$ denote the bare and the dressed chemical potentials respectively. Note that the problem would apparently disappear in a finite temperature approach using the Matsubara formalism. However, Kohn and Luttinger have shown that special care is needed in taking the zero temperature limit, since they have found a class of diagrams (they have called them anomalous diagrams) for which the zero temperature limit and the infinite volume limit do not commute. Taking the former limit first yields a vanishing contribution for those diagrams, and therefore the wrong result of the standard zero temperature formalism is obtained. The correct result for an infinite system is obtained by taking the other order of limits, where anomalous diagrams do provide finite contributions. For this reason, and also given the conceptual interest of this problem, we shall use only the zero temperature formalism throughout this paper. In this framework, a natural way to circumvent this problem with level crossings is to start the standard perturbation algorithm with any arbitrary eigenstate of the non-interacting Hamiltonian $H_{0}$. Intuitively, we believe in most cases it is sufficient to choose an initial state where the locus of occupied single particle states is singly connected ([i.e.]{}it has no isolated particle-hole excitations from the viewpoint of $H_{0}$), but with a deformed Fermi surface, as shown on Fig. \[fig:sf\_lib\_hab\]. The selection of the correct initial state is performed by minimizing the total energy of the dressed state it generates, after switching on the interactions. An example of this procedure is given below (Sec. \[sec:sub:2chaines\_ordre1\]) for a simple two-chain model. For practical purposes, it is important to note that this approach may also be implemented through a perturbative computation of the single particle Green’s function. Instead of using the free propagator $G^{(0)}(k,\omega)^{-1}=\omega-{\varepsilon}_{0}(k)+{{\mathrm i}}\eta{\mathrm{sgn}}\left({\varepsilon}_{0}(k)-\mu_{0}\right)$, we should first make a guess for the dressed Fermi surface. This allows us to define a function $\Phi (k)$ such that $\Phi (k)=1$ if $k$ does not belong to the trial Fermi sea, and $\Phi (k)=-1$ if $k$ belongs to it. ![Examples of possible initial states for the perturbation algorithm. These states are Slater determinants with occupied single-particle states depicted by the dashed areas in $k$-space. The dashed line denotes the non-interacting [Fermi surface]{}. In state (a), the [Fermi surface]{}is deformed, but no additional particle-hole excitations are present, unlike in state (b).[]{data-label="fig:sf_lib_hab"}](sf_lib_hab){width="8cm"} The locus of points in $k$-space where $\Phi (k)$ jumps from $-1$ to $+1$ is our trial Fermi surface, and points on this set will be generically denoted as $k_{F}$ in the present discussion. The corresponding bare propagator to be used in Feynman graph expansions is: $$G^{(0)}_{\Phi}(k,\omega)^{-1}=\omega-{\varepsilon}_{0}(k)+{{\mathrm i}}\eta \Phi (k).$$ As usual, the dressed propagator is obtained as $G_{\Phi}(k,\omega)^{-1}=\omega-{\varepsilon}_{0}(k)-\Sigma_{\Phi}(k,\omega)$, where the subscript $\Phi$ in $\Sigma_{\Phi}(k,\omega)$ is to stress the influence of the choice of a trial Fermi surface encoded in the function $\Phi$. If this trial Fermi surface is the correct one for the interacting Fermi system, we expect the [self-energy]{}satisfies the following well-known conditions: i\) There exists a well defined chemical potential $\mu$ so that for any $k_{F}$ belonging to the trial Fermi surface, we have: $$\label{eq:conditioon_re_fs} \mu-{\varepsilon}_{0}(k_{F})-\Re\Sigma_{\Phi}(k_{F},\mu)=0.$$ ii\) The inverse life-time of “quasiparticles” vanishes on the trial Fermi surface so that: $$\Im \Sigma_{\Phi}(k_{F},\mu)=0.$$ Of course, these conditions are not satisfied for most trial Fermi surfaces, as the reader will immediately notice on simple examples. We have checked on several examples that both procedures ([i.e.]{}minimizing the total energy, or satisfying conditions i) and ii) on the dressed single particle propagator) yield the same dressed Fermi surface. In Appendix \[app:equi\], we provide a formal proof of this equivalence, first in the finite volume case, and then in the case of an infinite volume. When the choice of $\Phi$ is not the correct one, it is impossible to satisfy both conditions i) and ii) simultaneously. In the case of standard perturbation theory $\Phi$ is taken to be $\Phi^{(0)}$ corresponding to the bare Fermi surface, obtained from $H_{0}$.[@Zlatic95; @Halboth97] The dressed Fermi surface is assumed to be determined from an equation which resembles condition i), namely: $$\mu-{\varepsilon}_{0}(k_{F})-\Re\Sigma_{\Phi^{(0)}}(k_{F},\mu)=0.$$ But doing this yields two severe flaws: as shown in Appendix \[app:diff\_re\], this does not generate the same dressed Fermi surface as the two procedures presented above and argued to be the correct ones do. Furthermore, in perturbation theory, $\Im\Sigma_{\Phi^{(0)}}(k_{F},\mu)$ changes sign on the non-interacting Fermi surface, and for $\omega$ equal to the non-interacting chemical potential $\mu_{0}$ as shown in Appendix \[app:diff\_im\]. This discussion holds clearly in the case where energy level crossings associated to various initial shapes of the Fermi surface are protected by some symmetries of the full Hamiltonian, as stated at the beginning of this section. Here, we would like to emphasize that a similar qualitative picture also holds in a more generic situation. On general grounds, we expect that energy levels of a finite system do not cross as the interaction strength is increased. This is the famous phenomenon of energy level repulsion which plays a key role in the field of “quantum chaos” (see for instance the book by Gutzwiller[@Gutzwiller]). So, standard perturbation theory starting from the unperturbed ground-state is expected to generate the correct interacting ground-state for a [*finite*]{} system. However, to get the full single energy level resolution in the spectrum with all the avoided level crossings clearly requires going to very high orders in perturbation theory. Instead, in most many-body computations, we first get formal expressions for various quantities such as Green’s functions for a chosen [*finite*]{} order in powers of the interaction, and we most often take the thermodynamical limit [*before*]{} summing the perturbation series. We believe this procedure is most likely to generate in the end an excited state of the interacting system, although the seed of the perturbation series is the non-interacting ground-state. This belief is confirmed by the simple computations in Appendix \[app:diff\_re\], which do not require any special symmetry of the full Hamiltonian. Two chains of spinless fermions: Energy minimization {#sec:sub:2chaines_ordre1} ---------------------------------------------------- ### Model and notations {#sec:sub:sub:model_notations} Let us first focus on the simplest possible model exhibiting the features described previously: a system of two chains of interacting spinless fermions. We will assume this system to be anisotropic, described by a tight-binding Hamiltonian, with a hopping ${t_\|}$ along the chain much larger than the transverse hopping ${t_\perp}$. Hence, we have two bands, named by the transverse momentum they correspond to, [i.e.]{}0 (bonding) and $\pi$ (anti-bonding). We suppose the filling is such that both bands are partially filled. We will furthermore focus on the low-energy properties, so that we can linearize the spectrum around the four Fermi points, giving rise to four types of fermions: $({{\mathrm R}},0)$, $({{\mathrm R}},\pi)$, $({{\mathrm L}},0)$ and $({{\mathrm L}},\pi)$. As usual, we extend the spectrum for arbitrary momenta. The low-energy free Hamiltonian is thus given by: $$\begin{aligned} \label{eq:ham_libre_lin} &&H_0=\sum_k \sum_{I=0,\pi}\\ &&\hspace{0.5cm}\Bigg\lbrace \left[\mu^{(0)}+v^{(0)}_{{{\mathrm F}},I} (k-k^{(0)}_{{{\mathrm F}},I})\right] {c^\dagger}_{{{\mathrm R}},I}(k) {c^{}}_{{{\mathrm R}},I}(k) + \nonumber\\ &&\hspace{1cm}\left[\mu^{(0)}-v^{(0)}_{{{\mathrm F}},I}(k+k^{(0)}_{{{\mathrm F}},I})\right] {c^\dagger}_{{{\mathrm L}},I}(k) {c^{}}_{{{\mathrm L}},I}(k) \Bigg\rbrace.\nonumber\end{aligned}$$ In the above expression, all the superscripts $^{(0)}$ denote free quantities. $\mu^{(0)}$ is the chemical potential, $v^{(0)}_{{{\mathrm F}},I}$ and $k^{(0)}_{{{\mathrm F}},I}$ the Fermi velocity and momentum on chain $I$. ${c^\dagger}_{{{\mathrm R}},I}(k)$ is the creation operator of a right fermion on chain $I$, with parallel momentum $k$. The sum over $k$ is to be understood as an integral for a system in the thermodynamic limit. In all that follows, we will simplify the problem and suppose that the Fermi velocities for both branches are equal, and they will simply be denoted as $v_{{\mathrm F}}^{(0)}$. We shall also make simplifying assumptions about the interactions. Thus, the only low-energy interaction processes we will be interested in, are of the forward scattering type ($g_2$), classified as $A$, $B$, $C$, $D$, and $F$. They are represented on Fig. \[fig:interactions\_2chaines\]. ![Selected low-energy interactions for the two-chain model.[]{data-label="fig:interactions_2chaines"}](interactions_2chaines){width="8cm"} We shall neglect the Umklapps, assuming the filling is not commensurate. $g_4$ interactions, involving four right or four left fermions, are also discarded, because we shall restrict ourselves to first and second order effects, to which these interactions give no contribution. In order to save space, we only give the $D$ type interaction Hamiltonian: $$\begin{aligned} &&H_\mathrm{int}^{(D)}=\frac{D}{L}\sum_{k,k',q}\\ &&\hspace{0.5cm}\Big\lbrace{c^\dagger}_{{{\mathrm R}},\pi}(k+q) {c^\dagger}_{{{\mathrm L}},0}(k'-q) {c^{}}_{{{\mathrm L}},\pi}(k') {c^{}}_{{{\mathrm R}},0}(k) +\mathrm{h.c.} \Big\rbrace,\nonumber\end{aligned}$$ where h.c. means the hermitic conjugate. ### First order {#sec:sub:sub:min_energy} We will here compute the energy to order one in the usual quantum mechanical perturbation theory, of eigenstates obtained from two types of free eigenstates. The first ones, denoted as ${| 0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi} \rangle}_0$, are free states for which the bonding (respectively anti-bonding) band is filled up to $k_{{{\mathrm F}},0}$ (respectively $k_{{{\mathrm F}},\pi}$). The ground-state of the free system is thus obviously ${| 0;k_{{{\mathrm F}},0}^{(0)},k_{{{\mathrm F}},\pi}^{(0)} \rangle}_0$. Of course, as the number of particles is fixed, the condition $k_{{{\mathrm F}},0}+k_{{{\mathrm F}},\pi}=k_{{{\mathrm F}},0}^{(0)}+k_{{{\mathrm F}},\pi}^{(0)}$ must be satisfied. As we wish to understand what happens if one adds a particle to the system, we will also consider states that are simply obtained from the first ones by adding a particle of momentum $q$ on branch 0 or $\pi$ (with $q\geqslant k_{{{\mathrm F}},0}$ or $q\geqslant k_{{{\mathrm F}},\pi}$). We will refer to these states as ${| 1,q,0(\pi);k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi} \rangle}_0$. We shall neither consider states with one hole, nor states with an arbitrary number of particles or holes. First of all we can compute the energies of these states, in the non-interacting case. Of course, because our linearized dispersion relations have been extended to include infinitely many single-particle states, there is strictly speaking an infinite particle density in these Dirac seas, which yields divergent expressions for the total energy. We will regularize these divergences by putting an ultra-violet cut-off $\Lambda_0$ on the momenta, around the four [*free*]{} Fermi momenta, as shown on Fig. \[fig:cut-off\] for one band. ![Here we show how the ultra-violet cut-off is chosen around the free [Fermi surface]{}, for one band.[]{data-label="fig:cut-off"}](cut-off){width="7cm"} For the sake of simplicity, we work in the thermodynamic limit, and after a bit of algebra we find: $$\begin{aligned} &&E^{(0)}(0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})=\frac{L}{\pi}(2\mu^{(0)}\Lambda_0-v_{{\mathrm F}}^{(0)}\Lambda_0^2)\nonumber\\ &&\hspace{2cm}+\frac{v_{{\mathrm F}}^{(0)} L}{\pi}\left(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}\right)^2, \mbox{ and }\\ &&E^{(0)}(1,q,0(\pi);k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})=E^{(0)}(0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})\nonumber\\ &&\hspace{2cm}+\mu^{(0)}+v_{{\mathrm F}}^{(0)}(q-k_{{{\mathrm F}},0(\pi)}^{(0)}).\end{aligned}$$ It is obvious that the minimum of the energy is obtained for the free [Fermi surface]{}. The value of $\mu^{(0)}$ does not play a role here since we have fixed the total particle number. To order one in the couplings, it is well known that the energy of a free state is simply shifted by the mean value of the interaction for this state. As a consequence, the $D$ and $F$ couplings do not give any contribution. They will only start playing a role to second order. It is a very simple matter to check that: $$\begin{aligned} &&\Delta E^{(1)}(0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})=\frac{L}{(2\pi)^2}\Bigg[ A \left(\Lambda_0+ (k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}) \right)^2\nonumber\\ \label{eq:en_1_sf} &&\hspace{2cm}+ B \left(\Lambda_0-( k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}) \right)^2\\ &&\hspace{0.3cm}+2C \left(\Lambda_0+ (k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}) \right) \left(\Lambda_0-( k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}) \right)\Bigg],\nonumber\\ &&\Delta E^{(1)}(1,q,0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})=\Delta E^{(1)}(0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})\nonumber\\ \label{eq:en_1_1part0} &&\hspace{1cm}+\frac{1}{2\pi}\Bigg[ A\left(\Lambda_0+ (k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}) \right)\\ &&\hspace{3cm}+ C \left(\Lambda_0-( k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}) \right)\Bigg],\nonumber\\ &&\Delta E^{(1)}(1,q,\pi;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})=\Delta E^{(1)}(0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})\nonumber\\ \label{eq:en_1_1partpi} &&\hspace{1cm}+\frac{1}{2\pi}\Bigg[ B\left(\Lambda_0- (k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}) \right)\\ &&\hspace{3cm}+ C \left(\Lambda_0+( k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)}) \right)\Bigg].\nonumber\end{aligned}$$ We have used the conservation of the number of particles so that the above expressions are expressed only in terms of the Fermi momenta on branch 0. Thus we minimize the energy $E^{(1)}=E^{(0)}+\Delta E^{(1)}$ simply by requiring for its derivative with respect to $k_{{{\mathrm F}},0}$ to vanish. This yields: $$\label{eq:dk0_vrai} k_{{{\mathrm F}},0}^{(1)}-k_{{{\mathrm F}},0}^{(0)}=(B-A)\frac{\Lambda_0}{4\pi v_{{\mathrm F}}^{(0)}}\left(1+\frac{A+B-2C}{4\pi v_{{\mathrm F}}^{(0)}}\right)^{-1}.$$ Let us show how the chemical potential can be computed, using the energies of the states with one added particle. First of all, we notice that the expressions $\Delta E^{(1)}(q)$ are independent of $q$. It implies the energy for adding a particle to the system on branch 0 ($\pi$) will be minimal if $q$ is as small as possible, [i.e.]{}$q=k_{{{\mathrm F}},0}$ ($q=k_{{{\mathrm F}},\pi}$). This confirms that $k_{{{\mathrm F}},0}$ and $k_{{{\mathrm F}},\pi}$ are the actual Fermi momenta. Now if we require this minimal energy to be the same on the two branches, equal to the renormalized chemical potential, we obtain the two following conditions: $$\begin{aligned} &&\mu^{(1)}=\mu^{(0)}+v_{{\mathrm F}}^{(0)}(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})\nonumber\\ &&\hspace{1cm}+\Delta E^{(1)}(1,q=k_{{{\mathrm F}},0},0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})\\ &&\hspace{2cm}-\Delta E^{(1)}(0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi}),\nonumber\\ &&\mu^{(1)}=\mu^{(0)}+v_{{\mathrm F}}^{(0)}(k_{{{\mathrm F}},\pi}-k_{{{\mathrm F}},\pi}^{(0)})\nonumber\\ &&\hspace{1cm}+\Delta E^{(1)}(1,q=k_{{{\mathrm F}},\pi},\pi;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi})\\ &&\hspace{2cm}-\Delta E^{(1)}(0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi}).\nonumber\end{aligned}$$ One can check these equations give the deformation (\[eq:dk0\_vrai\]) of the [Fermi surface]{}. This is physically desirable. Indeed, imposing that the minimum energies to add one particle on one branch or the other are identical, should be equivalent to the requirement that taking two particles at the [Fermi surface]{}on one branch and putting them at the [Fermi surface]{}on the other branch costs nothing (in the thermodynamical limit). Finally we find the chemical potential: $$\begin{aligned} \label{eq:dmu_vrai} &&\mu^{(1)}=\mu^{(0)}+(A+B+2C)\frac{\Lambda_0}{4\pi}\\ &&\hspace{1cm}-(B-A)^2\frac{\Lambda_0}{4\pi v_{{\mathrm F}}^{(0)}}\left(1+\frac{A+B-2C}{4\pi v_{{\mathrm F}}^{(0)}}\right)^{-1}.\nonumber\end{aligned}$$ To conclude this section about first order computations, we show two figures of what would happen for a total energy of the following simplified form: $E^{(1)}=(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})^2+(A-B)(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})$. Fig. \[fig:croisements\_1\] illustrates the level crossings: we represent the energy as a function of $(B-A)$ (assumed positive), for various values of $(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})$. ![Energies as functions of $(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})$ for different values of $B-A$ (0, 0.1, 0.2, 0.3, 0.35). The dashed curve gives the energy of the ground-state, and thus goes through the minima of all the different curves.[]{data-label="fig:croisements_2"}](croisements_2){width="8cm"} Fig. \[fig:croisements\_2\] proposes an alternative vision of the same thing (see the caption). ![Energies as functions of $B-A$, for states with different values of $(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})$ (0, 0.05, 0.1, 0.15 et 0.2). The crossings show us that the [Fermi surface]{}will be deformed. The dashed curve is the envelope of all these curves. It is thus the energy of the interacting ground-state, as a function of the interaction.[]{data-label="fig:croisements_1"}](croisements_1){width="8cm"} Eq. (\[eq:dk0\_vrai\]) shows us that the deformation of the [Fermi surface]{}at first order is due to the difference between the couplings on the branches: if $A=B$, no deformation takes place. We can understand the sign of the deformation very simply. Suppose the fermions repel each other ([i.e.]{}the couplings are positive), but that the repulsion is bigger on chain $\pi$ for example: $B-A>0$. It is then natural, in order to lower the energy of the system, that some fermions of chain $\pi$ go to chain $0$, so that $(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})$ should be positive. This is indeed what we find. We will now see that things are different at second order: the $D$ couplings tend to flatten the [Fermi surface]{}, whatever their sign, and without having to invoke a difference between two couplings. ### Second order and further {#sec:sub:2chaines_ordre2} We shall now discuss in detail perturbation theory to second order (this notion of order being simply the number of vertices in the corresponding Feynman graphs), and we will also see that some problems arise to third order and beyond, showing that another perturbation scheme is needed. We have already seen the effects of $A$, $B$ and $C$ interactions on the [Fermi surface]{}’s shape to first order. We let the reader check that these three interactions play no essential role in the deformation of the [Fermi surface]{}at second order. Indeed, when we compute the energy of the states ${| 0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi} \rangle}_0$, if we only keep contributions that diverge when $\Lambda_0\to\infty$, we get a quantity that is proportional to $\Lambda_0^2$, and [*independent*]{} of the dressed Fermi momenta. Only finite terms do depend on the dressed Fermi momenta. We shall thus neglect these contributions, and focus on $D$ and $F$ interactions. Let us begin with the effect of $D$ interaction, which is the only one that does not exist at zero energy if the [Fermi surface]{}is not strictly flat. This is due to the constraint of momentum conservation, and is easily visualized from Fig. \[fig:interactions\_2chaines\]. Second order perturbation theory tells us that the eigenenergy of an eigenstate ${| n \rangle}$, obtained from the free eigenstate ${| n \rangle}_0$, is obtained by shifting the free eigenenergy of a quantity: $$\label{eq:formule_th_pert_2} \sum_{k\neq n} \frac{V_{nk}V_{kn}}{E^{(0)}_n-E^{(0)}_k}, \mbox{ where } V_{nk}=\,_0{\langle n |} V {| k \rangle}_0,$$ and where $V$ is the interaction potential. This formula involves energy denominators. If these become smaller, the energy will decrease. When $D$ interactions are considered, we understand from these considerations that they will tend to flatten the [Fermi surface]{}, because this will allow for smaller energy denominators. That this is true can be checked by explicitly computing the energy shift, which is found to be: $$\Delta E^{(2)}_D=\frac{L D^2}{v_{{\mathrm F}}^{(0)} (2\pi)^3} (k_{{{\mathrm F}},0}-k_{{{\mathrm F}},\pi})^2\ln\left(\frac{\Lambda_0}{|k_{{{\mathrm F}},0}-k_{{{\mathrm F}},\pi}|}\right).$$ We stress that this result has been found computing (\[eq:formule\_th\_pert\_2\]), keeping only terms that are divergent when $\Lambda_0\to\infty$ and that depend on the Fermi momenta. If only $D$ terms are considered, it is now easy to show that the free and renormalized Fermi momenta are linked by the following formula: $$\label{eq:def_sf_D2} \Delta k_{{\mathrm F}}^{(0)}=\Delta k_{{\mathrm F}}\left[ 1+2\left(\frac{D}{2\pi v_{{\mathrm F}}^{(0)}}\right)^2 \ln\left( \frac{\Lambda_0}{|\Delta k_{{\mathrm F}}|} \right)\right],$$ where we have set $\Delta k_{{\mathrm F}}=k_{{{\mathrm F}},0}-k_{{{\mathrm F}},\pi}$ (and the same for free quantities). This clearly shows the tendency towards the flattening of the [Fermi surface]{}, induced by $D$ terms. ![Example of non-skeleton diagram giving rise to infrared divergences.[]{data-label="fig:diag_squelette_3_spinless"}](diag_squelette_3_spinless){width="7cm"} What about $F$ interactions? Perturbation theory at second order is divergent in the low-energy limit. Indeed ${| 0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi} \rangle}_0$ states that are not the free ground-states, are coupled to a continuum of excited states composed of two particles and two holes, which have kinetic energies arbitrarily close to the one of the seed state ${| 0;k_{{{\mathrm F}},0},k_{{{\mathrm F}},\pi} \rangle}_0$. This yields energy denominators that are very small in absolute value, and even zero. In the [self-energy]{}formalism (constructed from an excited state, see Appendix \[app:equi\] for details), this problem is regularized by the imaginary parts $\pm{{\mathrm i}}\eta$ in the [self-energy]{}approach. For the minimization of energy, one can similarly define the divergent integrals with a principal part, and one finds the same results as in this [self-energy]{}version. But this infrared divergence is only the first one, and is not the most problematic. Things become worse and worse for higher orders. This has already been discussed by Feldman, Salmhofer and Trubowitz[@Feldman96] so that we shall be brief. In the language of Feynman diagrams, the divergences come from repeated [self-energy]{}insertions, or to say it differently, with non-skeleton diagrams. An example of the lowest order diagrams of this type (apart from the Kohn-Luttinger diagram we have already discussed at second order, and which is zero) is given in Fig. \[fig:diag\_squelette\_3\_spinless\]. The problem with such a diagram is the following. Because of the inserted first order [self-energy]{}in the internal right propagator, we now have two right internal propagators. This gives a bad behavior of the integral over $q$ around $q=0$ for $\omega=0$ and $k=0$, once all other variables have been integrated out. It is clear that things get even worse if two or more such first or higher order [self-energies]{}are inserted. We thus have to find a way of getting rid of these infrared problems that plague our perturbation theory. This is achieved by the use of counterterms, that we will expose now. The use of counterterms in the two-chain model {#sec:sub:use_ct} ---------------------------------------------- ### Notations and first order calculation In order to simplify the notations, we will denote the Fermi velocity by $v_{{\mathrm F}}$ instead of $v_{{\mathrm F}}^{(0)}$. We will again suppose this velocity to be independent of the chain index and its renormalization will be neglected throughout this paper to simplify the discussion. The use of counterterms in interacting fermionic systems, for which the [Fermi surface]{}gets deformed, is quite old, and can for example be found in the beautiful discussion by Nozières,[@Nozieres_anglais] where the reader will find more details. The main idea, which has been illustrated very recently,[@Neumayr02; @Ledowski02], is to take the interacting Fermi sea as the starting point of perturbation theory. As it is a priori unknown, we must ensure in the end of the calculation, that the “guessed” [Fermi surface]{}is indeed the dressed one. In order to have a good starting point, the most natural idea is to split the free Hamiltonian into two bits: one that is a modified free Hamiltonian with the correct interacting [Fermi surface]{}, and another that will be the difference between the true free Hamiltonian, and the modified one. We will thus write: $$\begin{aligned} &&H_0=\sum_{I=0,\pi} \sum_k \Bigg\lbrace \left[\mu+v_{{\mathrm F}}(k-k_{{{\mathrm F}},I})\right] {c^\dagger}_{{{\mathrm R}},I}(k) {c^{}}_{{{\mathrm R}},I}(k)\nonumber\\ &&\hspace{2cm}+\left[\mu-v_{{\mathrm F}}(k+k_{{{\mathrm F}},I})\right] {c^\dagger}_{{{\mathrm L}},I}(k) {c^{}}_{{{\mathrm L}},I}(k) \Bigg\rbrace\nonumber\\ &&\hspace{3cm} + H_{0,\mathrm{ct}}^{(\mu)}+H_{0,\mathrm{ct}}^{(k)},\end{aligned}$$ with $$H_{0,\mathrm{ct}}^{(\mu)}=\delta\mu \sum_{I,k} {c^\dagger}_{{{\mathrm R}},I}(k) {c^{}}_{{{\mathrm R}},I}(k) + \delta\mu \sum_{I,k} {c^\dagger}_{{{\mathrm L}},I}(k) {c^{}}_{{{\mathrm L}},I}(k),$$ and $$\begin{aligned} H_{0,\mathrm{ct}}^{(k)}&=&-v_{{\mathrm F}}\delta k_I \sum_{I,k} {c^\dagger}_{{{\mathrm R}},I}(k) {c^{}}_{{{\mathrm R}},I}(k)\nonumber\\ &&- v_{{\mathrm F}}\delta k_I \sum_{I,k} {c^\dagger}_{{{\mathrm L}},I}(k) {c^{}}_{{{\mathrm L}},I}(k).\end{aligned}$$ ![Graphical representation of the chemical potential counterterm, at first order.[]{data-label="fig:dmu1_spinless"}](dmu1_spinless){width="5cm"} The counterterms $\delta\mu$ and $\delta k_I$ are found by the requirement that $H_0$ remains the true free Hamiltonian: $$\begin{aligned} &&\mu^{(0)}=\mu+\delta\mu, \mbox{ with }\\ &&\delta\mu=\delta\mu^{(1)}+\delta\mu^{(2)}+\ldots=G^\alpha \delta \mu_1^\alpha+\ldots\\ &&k_{{{\mathrm F}},I}^{(0)}=k_{{{\mathrm F}},I}+\delta k_I, \mbox{ with }\\ &&\delta k_I=\delta k_I^{(1)}+\delta k_I^{(2)}+\ldots=G^\alpha \delta k_{I,1}^\alpha+\ldots.\end{aligned}$$ Note that we have used a symbolic notation $G^\alpha$ for the couplings $A$ to $F$, and the sum over $\alpha$ is implicit. We stress that there is not only one counterterm for the chemical potential (or for the Fermi momentum of each chain), but an infinity, which are all the $\delta\mu^{(n)}$’s, for $n=1,2,\ldots$. The number $n$ gives the power in the couplings of the considered counterterm. Counterterms have to be computed order by order, one after the other, in a perturbation theory. When using the counterterms, the Luttinger theorem simply says that: $\sum_I \delta k_I=0$, or for each order $j$: $\sum_I \delta k_I^{(j)}=0$. Now the free (R,0) propagator with which Feynman diagrams are computed is: $$G^*_{{{\mathrm R}},0}(k,\omega)=\frac{1}{\omega-\left[\mu+v_{{\mathrm F}}(k-k_{{{\mathrm F}},I})\right]+{{\mathrm i}}\eta \,{\mathrm{sgn}}(k-k_{{{\mathrm F}},I})},$$ and similarly for other types of fermions. Both real and imaginary parts of these propagators refer to the interacting [Fermi surface]{}. We shall now see how to implement the use of counterterms in the perturbation theory of the two-chain model. For this, it is useful to associate a graphical representation to the counterterms. This is illustrated at first order in Fig. \[fig:dmu1\_spinless\] for the chemical potential, and in Fig. \[fig:dk1\_spinless\] for the Fermi momenta. The chemical potential counterterm is represented by a square, whereas the Fermi momenta counterterms are denoted by hexagons. In both cases, the number written inside the symbol is the order $n$ mentioned previously. Notice that for Fermi momenta, we do not need to explicitly write down the chain index $I$, because it would be redundant with the chain index of the propagators. The reader should also remark that counterterms for right or left fermions are exactly identical. Now that the general notations have been given, let us see what the counterterm approach gives to first order, for the two chains. In all that follows, we will not use an ultra-violet cut-off around the free [Fermi surface]{}, but [*around the interacting [Fermi surface]{}*]{}. This will slightly alter the results, but it makes the computation simpler, without involving a qualitatively different physics. The tadpole diagram of Fig. \[fig:tadpole\_spinless\], computed with the new free propagator $G_*^{(0)}$ and the new cut-off, gives the following contribution to the [self-energy]{}: $$\begin{aligned} &&\Sigma_{{{\mathrm R}},0}^{(1)}(k,\omega)=(A+C)\frac{\Lambda_0}{2\pi}, \mbox{ and }\\ &&\Sigma_{{{\mathrm R}},\pi}^{(1)}(k,\omega)=(B+C)\frac{\Lambda_0}{2\pi}.\end{aligned}$$ ![Graphical representation of the Fermi momenta counterterms, at first order.[]{data-label="fig:dk1_spinless"}](dk1_spinless){width="5cm"} ![First order contribution to the [self-energy]{}: the tadpole graph.[]{data-label="fig:tadpole_spinless"}](tadpole_spinless){width="6cm"} But we also have the counterterm contributions (diagrams of Figs. \[fig:dmu1\_spinless\] and \[fig:dk1\_spinless\]): $$\begin{aligned} &&\Sigma_{\mathrm{ct};{{\mathrm R}},0}^{(\mu;1)}(k,\omega)=\Sigma_{\mathrm{ct};{{\mathrm R}},\pi}^{(\mu;1)}(k,\omega)=\delta\mu^{(1)},\\ &&\Sigma_{\mathrm{ct};{{\mathrm R}},0}^{(k;1)}(k,\omega)=-v_{{\mathrm F}}\delta k_0^{(1)},\\ &&\Sigma_{\mathrm{ct};{{\mathrm R}},\pi}^{(k;1)}(k,\omega)=-v_{{\mathrm F}}\delta k_\pi^{(1)}.\end{aligned}$$ ![Sunrise diagram contributing to the second order [self-energy]{}.[]{data-label="fig:sunrise_spinless"}](sunrise_spinless){width="6cm"} ![Kohn-Luttinger diagram contributing to the second order [self-energy]{}.[]{data-label="fig:KL_spinless"}](KL_spinless){width="6cm"} ![“Mixed” contribution to the second order [self-energy]{}. This graph is a tadpole, with an insertion of the first order chemical potential counterterm.[]{data-label="fig:tadpole_dmu1_spinless"}](tadpole_dmu1_spinless){width="6cm"} The dressed propagators $G$ are such that they satisfy the Dyson equation: $G^{-1}={G_*^{(0)}}^{-1}-\Sigma-\Sigma_{\mathrm{ct}}$. The chemical potential and Fermi momenta are found by requiring that they vanish for $\omega=\mu$ and for $k$ on the interacting [Fermi surface]{}, and that the Luttinger theorem is satisfied: $$\begin{aligned} &&G_{{{\mathrm R}},0}^{-1}(k=k_{{{\mathrm F}},0},\omega=\mu)=0,\\ &&G_{{{\mathrm R}},\pi}^{-1}(k=k_{{{\mathrm F}},\pi},\omega=\mu)=0,\\ &&\sum_I \delta k_I^{(j)}=0,\end{aligned}$$ with $j=1$ here, because we’re working at first order for the moment. It is very easy to check that one finds: $$\begin{aligned} &&\delta\mu^{(1)}=-(A+B+2C)\frac{\Lambda_0}{4\pi},\\ &&\delta k_0^{(1)}=(A-B)\frac{\Lambda_0}{4\pi v_{{\mathrm F}}}.\end{aligned}$$ This is fully compatible with equations (\[eq:dk0\_vrai\]) and (\[eq:dmu\_vrai\]), except for second order terms that we do not find here, because we have changed the way we choose the cut-off. ### Second order calculation with counterterms and next-order considerations As for the first order calculation, we have “usual” contributions to the [self-energy]{}, namely the sunrise and Kohn-Luttinger diagrams of Figs. \[fig:sunrise\_spinless\] and \[fig:KL\_spinless\]. We also have “pure” counterterms contributions, as in Figs. \[fig:dmu1\_spinless\] and \[fig:dk1\_spinless\], with the index 1 replaced by an index 2. But now we also have two “mixed” contributions, involving counterterms of the previous order, shown in Figs. \[fig:tadpole\_dmu1\_spinless\] and \[fig:tadpole\_dk1\_spinless\]. ![“Mixed” contribution to the second order [self-energy]{}. This graph is a tadpole, with an insertion of the first order Fermi momentum counterterm.[]{data-label="fig:tadpole_dk1_spinless"}](tadpole_dk1_spinless){width="6cm"} In fact, these two graphs vanish, for the same reason the Kohn-Luttinger graph vanishes. This is consistent, because there is no divergence from the Kohn-Luttinger graph to cancel. Before studying the sunrise graph, let us see how useful the counterterms are for third order graphs, on the example of Fig. \[fig:diag\_squelette\_3\_spinless\]. It is now obvious that it will be completely canceled by the same graph, with the inserted tadpole replaced by the two first order counterterms. Notice that the fourth order graph, consisting of still the same graph, with an inserted sunrise instead of a tadpole, would not be canceled by the graph with inserted second-order counterterms. The reason is that the sunrise is frequency and momentum dependent, but the counterterms are not. However, the counterterms allow for the infrared divergence cancellation obtained at zero external momentum and frequency. The sunrise is easily computed, and one gets the following contributions (the interaction index $G$ refers to the interaction associated to the two black dots in the sunrise): $$\begin{aligned} &&\Sigma_{{{\mathrm R}},0;G}^{(2)}(k=k_{{{\mathrm F}},0}+\kappa,\omega=\mu+\nu)\\ &&\hspace{0.5cm}=\frac{1}{4}\left(\frac{G}{2\pi v_{{\mathrm F}}}\right)^2 (\nu-v_{{\mathrm F}}\kappa)\ln\left(\frac{| \nu^2-(v_{{\mathrm F}}\kappa)^2 |}{(2v_{{\mathrm F}}\Lambda_0)^2} \right),\nonumber\\ &&\hspace{3cm}\mbox{ for } G=A,C,F;\nonumber\\ \nonumber\\ &&\Sigma_{{{\mathrm R}},\pi;G}^{(2)}(k=k_{{{\mathrm F}},\pi}+\kappa,\omega=\mu+\nu)\\ &&\hspace{0.5cm}=\frac{1}{4}\left(\frac{G}{2\pi v_{{\mathrm F}}}\right)^2 (\nu-v_{{\mathrm F}}\kappa)\ln\left(\frac{| \nu^2-(v_{{\mathrm F}}\kappa)^2 |}{(2v_{{\mathrm F}}\Lambda_0)^2} \right),\nonumber\\ &&\hspace{3cm}\mbox{ for } G=B,C,F;\nonumber\\ \nonumber\\ &&\Sigma_{{{\mathrm R}},0;D}^{(2)}(k=k_{{{\mathrm F}},0}+\kappa,\omega=\mu+\nu)\nonumber\\ &&\hspace{0.5cm}=\frac{1}{4}\left(\frac{D}{2\pi v_{{\mathrm F}}}\right)^2 [\nu-v_{{\mathrm F}}(\kappa+2\Delta k_{{\mathrm F}})]\\ &&\hspace{1.5cm}\times\ln\left(\frac{| \nu^2-[v_{{\mathrm F}}(\kappa+2\Delta k_{{\mathrm F}})]^2 |}{(2v_{{\mathrm F}}\Lambda_0)^2} \right),\nonumber\\ \label{eq:contrib_se_DR0} \nonumber\\ &&\Sigma_{{{\mathrm R}},\pi;D}^{(2)}(k=k_{{{\mathrm F}},\pi}+\kappa,\omega=\mu+\nu)\nonumber\\ \label{eq:contrib_se_fin} &&\hspace{0.5cm}=\frac{1}{4}\left(\frac{D}{2\pi v_{{\mathrm F}}}\right)^2 [\nu-v_{{\mathrm F}}(\kappa-2\Delta k_{{\mathrm F}})]\\ &&\hspace{1.5cm}\times\ln\left(\frac{| \nu^2-[v_{{\mathrm F}}(\kappa-2\Delta k_{{\mathrm F}})]^2 |}{(2v_{{\mathrm F}}\Lambda_0)^2} \right)\nonumber.\end{aligned}$$ The second-order conditions ensuring that the trial [Fermi surface]{}is indeed the interacting one read: $$\begin{aligned} &&-\frac{1}{2}\left(\frac{D}{2\pi v_{{\mathrm F}}}\right)^2 \left( -2v_{{\mathrm F}}\Delta k_{{\mathrm F}}\right)\ln\left(\frac{|\Delta k_{{\mathrm F}}|}{\Lambda_0}\right)\nonumber\\ &&\hspace{3cm}-\delta\mu^{(2)}+v_{{\mathrm F}}\delta k_0^{(2)}=0,\\ &&-\frac{1}{2}\left(\frac{D}{2\pi v_{{\mathrm F}}}\right)^2 \left( 2v_{{\mathrm F}}\Delta k_{{\mathrm F}}\right)\ln\left(\frac{|\Delta k_{{\mathrm F}}|}{\Lambda_0}\right)\nonumber\\ &&\hspace{3cm}-\delta\mu^{(2)}+v_{{\mathrm F}}\delta k_\pi^{(2)}=0,\\ &&\delta k_0^{(2)}+\delta k_\pi^{(2)}=0.\end{aligned}$$ These equations lead to $\delta\mu^{(2)}=0$, and to $\delta k_0^{(2)}=-\left(\frac{D}{2\pi v_{{\mathrm F}}}\right)^2 \Delta k_{{\mathrm F}}\ln\left(|\Delta k_{{\mathrm F}}|/\Lambda_0\right)$, which is nothing but (\[eq:def\_sf\_D2\]). The dressed (R,0) propagator (and others as well) can finally be deduced from all this: $$\begin{aligned} &&{G_{{{\mathrm R}},0}^{(2)}}^{-1}(k=k_{{{\mathrm F}},0}+\kappa,\omega=\mu+\nu)=\nu-v_{{\mathrm F}}\kappa\nonumber\\ &&\hspace{1cm}-\frac{1}{4}\left[\left(\frac{A}{2\pi v_{{\mathrm F}}}\right)^2+\left(\frac{C}{2\pi v_{{\mathrm F}}}\right)^2+\left(\frac{F}{2\pi v_{{\mathrm F}}}\right)^2\right]\nonumber\\ \label{eq:dressed_prop_2c} &&\times(\nu-v_{{\mathrm F}}\kappa)\ln\left(\frac{| \nu^2-(v_{{\mathrm F}}\kappa)^2 |}{(2v_{{\mathrm F}}\Lambda_0)^2} \right)\\ &&-\frac{1}{4}\left(\frac{D}{2\pi v_{{\mathrm F}}}\right)^2 (\nu-v_{{\mathrm F}}\kappa)\ln\left(\frac{| \nu^2-[v_{{\mathrm F}}(\kappa+2\Delta k_{{\mathrm F}})]^2 |}{(2v_{{\mathrm F}}\Lambda_0)^2} \right)\nonumber\\ &&+\frac{1}{4}\left(\frac{D}{2\pi v_{{\mathrm F}}}\right)^2 (2v_{{\mathrm F}}\Delta k_{{\mathrm F}})\ln\left(\frac{| \nu^2-[v_{{\mathrm F}}(\kappa+2\Delta k_{{\mathrm F}})]^2 |}{(2v_{{\mathrm F}}\Delta k_{{\mathrm F}})^2} \right).\nonumber\end{aligned}$$ We could now define renormalized propagators, introducing a wave function renormalization, and show how to implement a RG calculation of the dressed [Fermi surface]{}. In order not to be too redundant, we will do this for the more general case of $N$ chains of spin 1/2 electrons, which is anyway physically motivated by the case of quasi 1D systems. Cut-off scaling RG calculation for a system of $N$ chains of spin 1/2 electrons: formalism {#sec:RG_formalism} ========================================================================================== Setting of the model {#sec:setting_model} -------------------- The free Hamiltonian is much like the one of Eq. (\[eq:ham\_libre\_lin\]), except that there are now $N$ chains instead of 2, and that the fermions carry a spin index $\sigma$: $$\begin{aligned} \label{eq:ham_libre_lin_N} &&H_0=\sum_k \sum_{I=1}^N \sum_{\sigma=\uparrow,\downarrow}\nonumber\\ &&\Bigg\lbrace \left[\mu^{(0)}+v^{(0)}_{{{\mathrm F}},I} (k-k^{(0)}_{{{\mathrm F}},I})\right] {c^\dagger}_{{{\mathrm R}},I,\sigma}(k) {c^{}}_{{{\mathrm R}},I,\sigma}(k)\\ &&+\left[\mu^{(0)}-v^{(0)}_{{{\mathrm F}},I}(k+k^{(0)}_{{{\mathrm F}},I})\right] {c^\dagger}_{{{\mathrm L}},I,\sigma}(k) {c^{}}_{{{\mathrm L}},I,\sigma}(k) \Bigg\rbrace.\nonumber\end{aligned}$$ We will in fact assume, as we did previously, that the Fermi velocity is independent of the chain index $I$, and that it remains unrenormalized. We will thus simply use the notation $v_{{\mathrm F}}$. As in the two-chain model, we select low-energy interaction processes. Those are of two types. The first one denoted by $G$, generalizes the interactions $A$ to $F$ of the two-chain model. They are forward or backward scattering interactions. We shall only be interested in interactions that are invariant under spin rotations. Thus, we will use the charge and spin couplings ${G^{{\mathrm c}}}$ and ${G^{{\mathrm s}}}$. We refer the reader to our previous paper[@Dusuel02] for more details about this parametrization. There is however one major difference between the situation described in this article, and the one we are interested in here. Because of periodic boundary conditions in the transverse direction, all indices $I$, $J$ and $\delta$ are defined modulo the number of chains $N$. This was not the case in our previous article, where the chains were obtained after considering patches on a nearly square [Fermi surface]{}, thus the $N$ chains had boundaries, and as a consequence the chains were not all equivalent. Furthermore, if the filling is not too far from one half, we have to consider Umklapp scatterings. These will be denoted by $U$. It is easy to convince oneself that due to the Pauli principle, there is no need to consider exchange couplings for the Umklapps. The interaction Hamiltonian is thus: $$\label{eq:H_int_basse_en} H_\mathrm{int}=H_\mathrm{int}^{(G)}+H_\mathrm{int}^{(U)}, \mbox{ with:}$$ $$\begin{aligned} &&H_\mathrm{int}^{(G)}=\frac{2\pi v_{{\mathrm F}}}{NL}\sum_{I,J,\delta}~ \sum_{k,k',q}~ \sum_{\tau,\tau'}~ \sum_{\rho,\rho'}\nonumber\\ \label{eq:H_int_basse_en_G} &&\Bigg\lbrace \Big[ {G^{{\mathrm c}}}_\delta(I,J) \mathbb{I}_{\tau,\tau'} \mathbb{I}_{\rho,\rho'} + {G^{{\mathrm s}}}_\delta(I,J) \boldsymbol{\sigma}_{\tau,\tau'} \cdot \boldsymbol{\sigma}_{\rho,\rho'} \Big]\\ &&\times{c^\dagger}_{{{\mathrm R}},I+\delta,\tau}(k+q) {c^\dagger}_{{{\mathrm L}},J-\delta,\rho}(k'-q) {c^{}}_{{{\mathrm L}},J,\rho'}(k') {c^{}}_{{{\mathrm R}},I,\tau'}(k)~ \Bigg\rbrace,\nonumber\end{aligned}$$ ![Graphical representation of the spin interaction ${G^{{\mathrm s}}}_\delta(I,J)$.[]{data-label="fig:gs"}](gs){width="6cm"} and $$\begin{aligned} &&H_\mathrm{int}^{(U)}=\frac{\pi v_{{\mathrm F}}}{NL}\sum_{I,J,\delta}~ \sum_{k,k',q}~ \sum_{\tau,\tau'}~ \sum_{\rho,\rho'} \nonumber\\ &&\Bigg\lbrace U_\delta(I,J) \mathbb{I}_{\tau,\tau'} \mathbb{I}_{\rho,\rho'}\\ \label{eq:H_int_basse_en_U} &&\times{c^\dagger}_{{{\mathrm R}},I+\delta,\tau}(k+q) {c^\dagger}_{{{\mathrm R}},J-\delta,\rho}(k'-q) {c^{}}_{{{\mathrm L}},J,\rho'}(k') {c^{}}_{{{\mathrm L}},I,\tau'}(k)\nonumber\\ &&\hspace{5cm} + \mbox{ h. c.}~ \Bigg\rbrace. \nonumber\end{aligned}$$ The factors $1/N$ are required to yield a good thermodynamical limit. The $2\pi v_{{\mathrm F}}$ terms have been factorized, so that the couplings are dimensionless, and this will suppress many $2\pi v_{{\mathrm F}}$ denominators in the following. In the case of $G$ couplings, the left-right symmetry requires ${G^{{{\mathrm c}}({{\mathrm s}})}}_\delta(I,J)={G^{{{\mathrm c}}({{\mathrm s}})}}_{-\delta}(J,I)$, and the hermiticity of $H_\mathrm{int}$ yields ${G^{{{\mathrm c}}({{\mathrm s}})}}_\delta(I,J)={G^{{{\mathrm c}}({{\mathrm s}})}}_{-\delta}(I+\delta,J-\delta)$. The first of these relations, [i.e.]{}$U_\delta(I,J)=U_{-\delta}(J,I)$, naturally holds for the Umklapps because of the Pauli principle, so that the interaction that destroys two left fermions on chains $I$ and $J$, and creates two right fermions on chains $I+\delta$ and $J-\delta$ is present twice. The difference of a 1/2 factor between the Umklapps and the $G$ interactions, is here to compensate this. We let the reader check that in the case of the Hubbard model with an interaction Hamiltonian ${\mathcal{U}}\sum_i n_{i,\uparrow} n_{i,\downarrow}$, one has (up to $2\pi v_{{\mathrm F}}$ factors) ${G^{{\mathrm c}}}={\mathcal{U}}/2$, ${G^{{\mathrm s}}}=-{\mathcal{U}}/2$ and $U={\mathcal{U}}$. Because of this last equality, we will simply give the value of $U$ when referring to Hubbard couplings. Of course the Hubbard model, in terms of right and left fermions, also contains $g_4$ interactions, but these have been set to zero, for the reasons already given in Sec. \[sec:sub:sub:model\_notations\]. ![Graphical representation of the Umklapp interaction $U_\delta(I,J)$.[]{data-label="fig:u"}](u){width="6cm"} In order to make our notations for the interactions a bit more concrete, we show two Feynman graphs in Figs. \[fig:gs\] and \[fig:u\], associated respectively with ${G^{{\mathrm s}}}$ and $U$ terms. The representation for ${G^{{\mathrm c}}}$ is the same as the one for ${G^{{\mathrm s}}}$, except it involves $\mathbb{I}$ matrices instead of $\sigma$ matrices. Notice we do not use the single dot notation as we did previously, because it is not suited for the Umklapps, but we have adopted the wiggly line instead. In these graphs we also show which external legs are numbered 1, 2, 3 and 4. Cut-off scaling calculation of the [Fermi surface]{} ---------------------------------------------------- ### General considerations {#sec:sub:sub:gen_th_an} One of the main conclusions of Sec. \[sec:csfs\] is the necessity to use a renormalized perturbation theory in situations where the [Fermi surface]{}changes as a function of interaction strength. In the standard many-body formalism, this is achieved by the introduction of counterterms which pin the dressed [Fermi surface]{}. The outcome is a precise connection such as Eq. (\[eq:def\_sf\_D2\]) between the bare and dressed [Fermi surfaces]{}which may in principle be computed to any order in perturbation theory. As noticed already long ago by Gell-Mann and Low,[@Gell-Mann54] it is possible to sum infinite classes of contributions using a renormalization group procedure. This idea has played a crucial role in building a consistent physical picture of quasi 1D conductors for instance.[@Solyom79_dans_articles] The most general and flexible way to implement a renormalization approach is based on Wilson’s idea of gradual mode elimination. Several groups have recently implemented Wilson’s approach to the RG, expressed via the Polchinski equation,[@Zanchi96; @Halboth00] or its one-particle irreducible version.[@Jungnickel96; @Honerkamp01] Although these equations are exact, they are quite complicated, since effective interactions involving an arbitrary number of particles are generated along the RG flow. Any numerical computation requires therefore drastic truncations in the effective action. For this reason, we have prefered to use a simplified version of RG which is known as “cut-off” scaling. This procedure has been initiated in the pioneering work by Anderson et al. [@Anderson70] for the Kondo problem, and put in a more mathematical form by Abrikosov and Migdal [@Abrikosov70] and Fowler and Zawadowski.[@Fowler71] A very extensive review on this method has been written by Sólyom.[@Solyom79_dans_articles] This scheme amounts to constructing a one parameter family of “bare” Hamiltonians. These are defined on the single particle states whose momentum lies in a strip of width $\Lambda$ away from the [Fermi surface]{}. It is therefore natural to parametrize these Hamiltonians as a function of $\Lambda$. Note that by contrast to Wilson’s effective action, which includes all the possible types of interactions (relevant, marginal [*and*]{} irrelevant ones), the cut-off scaling procedure only considers relevant and marginal couplings. So unlike what is achieved in Wilson’s RG, it is no longer possible to preserve invariance of the [*full*]{} set of low-energy correlation functions as $\Lambda$ is gradually decreased. The cut-off scaling approach only allows to preserve a restricted set of low-energy observables, for instance the first derivatives of the two-point function with respect to external momentum and frequency, and the value of the four-point function for external legs taken on-shell at the Fermi level. Actual computations within this scheme encounter a new difficulty when the [Fermi surface]{}is sensitive to the strength of interactions. As explained in detail in Sec. \[sec:sub:use\_ct\] the bare propagators used in Feynman graphs are required to be singular on the [*dressed*]{} [Fermi surface]{}. But clearly, this is not known until the whole computation has been performed. This calls for an iterative procedure. For a given microscopic model (defined with an initial cut-off $\Lambda_{0}$), the one-particle part of the corresponding Hamiltonian defines a [Fermi surface]{} which will be called the $\Lambda_{0}$-[Fermi surface]{}. This is a natural first choice for a trial dressed [Fermi surface]{}in the iterative computation. One can then construct the flow equations for the running bare [Fermi surface]{} (the $\Lambda$-[Fermi surface]{}) and the effective couplings using cut-off scaling. In the limit where $\Lambda$ goes to zero (or at least to its minimal value before a phase transition occurs), the $\Lambda$-[Fermi surface]{}goes towards a new dressed [Fermi surface]{}, which is to be used as the new trial dressed [Fermi surface]{}in the next step of the iteration. On physical grounds, such a computation is expected to converge although we have not embarked yet in checking this statement. Instead we have tried to bypass the intrinsic complication of an iterative procedure by appealing to the physical insights gained in Sec. \[sec:sub:2chaines\_ordre1\] devoted to the energetic approach. The main ideas are the following: firstly, in the high-energy regime, and in the logarithmic approximation, the dressed Fermi momenta do not appear in the flow couplings’ equations; secondly, in the low-energy part of the flow, the [Fermi surface]{}should not move too much, because the couplings that deform it are irrelevant in this regime. The first point will be checked on the RG equations. The second one has already been checked in the two-chain model, where only the $D$ coupling, that does not exist at very low energies because of the non-flatness of the [Fermi surface]{}, deforms the [Fermi surface]{}. We will furthermore check it remains true in the case of $N$ chains. Assuming what happens in the intermediate energy regime (defined by the curvature of the [Fermi surface]{}) is not essential, the computation of the dressed [Fermi surface]{}is now possible in a [*single*]{} step. Indeed, we do not need a priori knowledge of the dressed [Fermi surface]{}anymore, since it disappears from the RG couplings’ flow equations in the logarithmic approximation of the high-energy regime. The flow of the $\Lambda$-[Fermi surface]{}, will then be stopped when the cut-off $\Lambda$ becomes comparable to the maximal momentum scale defined by the running $\Lambda$-[Fermi surface]{}. Note that this is the least controlled step of this approximated scheme, because the [Fermi surface]{}does not define one single momentum scale, but rather a continuum of scales. This will for example prevent us from using this scheme when the Umklapp couplings are taken into account, in a system too far from half-filling. The couplings’ flow equations in the cut-off scaling are given in Appendix \[app:flow\_eq\], since their derivation is standard. We shall now focus on the RG computation of the [Fermi surface]{}. The basic equation is Eq. (\[eq:fs\_counterterm\]). The [self-energy]{}that appears in this equation can be found from Eq. (\[eq:se\_Nchaines\]), where, since we work in the cut-off scaling scheme, the cut-off $\Lambda_0$ should now be replaced by the running cut-off $\Lambda$, and where the couplings are to be understood as running bare couplings. Given that $\delta k_I=k^{(0)}_I(\Lambda)-k_I$, Eq. (\[eq:fs\_counterterm\]) allows us to express $k^{(0)}_I(\Lambda)$ as a function of $\Lambda$, the running bare couplings and the dressed Fermi momenta $k_I$. But it is the latter who are fixed independently of the value of $\Lambda$, so that it is more convenient to invert this relation, working only at a second order accuracy, to get $k_I$ as a function of $\Lambda$, the running bare couplings and the running Fermi momenta $k^{(0)}_I(\Lambda)$. Asking for the invariance of $k_I$ as $\Lambda$ is changed yields the [Fermi surface]{}flow equation that we give in Appendix \[app:flow\_eq\], Eq. (\[eq:flot\_fs\]). Only the couplings for which the curvature of the [Fermi surface]{}is felt, [i.e.]{}for which $\Delta k_\alpha(I,J)\neq 0$ or $\Delta k_\alpha^{U}(I,J)\neq 0$ contribute to the flow of $k_I^{(0)}(\Lambda)$. This is what we had already noticed in the two-chain system, where the $D$ coupling was the only one to give a deformation of the [Fermi surface]{}. This confirms that only couplings that will be irrelevant in the low-energy regime contribute to the deformation of the [Fermi surface]{}. Finally, notice that we have not taken the first order contribution into account. This is not justified in the general case, but for the initial condition we are interested in, [i.e.]{}with all charge, spin and Umklapp couplings equal to $G_{{\mathrm B}}^{{\mathrm c}}$, $G_{{\mathrm B}}^{{\mathrm s}}$ and $U_{{\mathrm B}}$, the first order contribution vanishes. We emphasize this is true only in the high-energy regime, where the couplings will have a purely one-dimensional (1D) flow (because of the logarithmic approximation), and thus will remain equal (with one value for each of the three types of couplings). Once the low-energy regime is reached the couplings become different, so that one should take the first-order deformation into account. The [self-energy]{}at one loop is given by the contribution of the tadpole diagram. It is easy to see that out of the three couplings ${G^{{\mathrm c}}}$, ${G^{{\mathrm s}}}$ and $U$, only the ${G^{{\mathrm c}}}$ couplings contribute. Compared to the spinless case, there will be a factor of 2, because of the two possible spin states of the propagator in the loop. We let the reader check that: $$\begin{aligned} &&\Sigma_{{{\mathrm R}},I}^{(1)}(k=k_{{{\mathrm F}},I}+\kappa,\omega=\mu+\nu)=\nonumber\\ &&\hspace{2cm}\left[\frac{1}{N}\sum_J {G^{{\mathrm c}}}_0(I,J)\right] (2 v_{{\mathrm F}}\Lambda),\end{aligned}$$ and that the corresponding first-order Fermi momenta counterterms read: $$\delta k_I^{(1)}=\frac{2v_{{\mathrm F}}\Lambda}{N}\left( \sum_J {G^{{\mathrm c}}}_0(I,J)-\frac{1}{N}\sum_{I,J} {G^{{\mathrm c}}}_0(I,J)\right).$$ However, because we are interested in systems for which ${t_\perp}$ is small, [i.e.]{}for systems that are nearly 1D, we know that all the chains are nearly equivalent, so that the RHS of the previous equation will be nearly independent of $I$, and thus, very small. That this is true can be checked on Fig. \[fig:evolution\_coupC\_charge\], that will be described later, and on which the couplings ${G^{{\mathrm c}}}_0(I,J)$ are represented after running the flow, into the low-energy regime. It is clear on this figure that the term $\sum_J {G^{{\mathrm c}}}_0(I,J)$ is nearly independent of $I$. ### Analytical study of the simplest example Let us illustrate all this on the simplest possible case, for which $G_{{\mathrm B}}^{{\mathrm s}}=0$ and $U_{{\mathrm B}}=0$. This physically corresponds to a system away from half filling, so that it is justified to neglect the Umklapps. Furthermore we have set the spin couplings to zero, which corresponds to the Luttinger liquid fixed point. This simplifies the flow, because the charge couplings then remain constant along it. Furthermore, this will allow us to compare our results to those obtained in the literature, starting from decoupled Luttinger liquids, coupled by a hopping term. It is easy to check that the general flow equation of the [Fermi surface]{}(\[eq:flot\_fs\]) can be simplified in: $$\begin{aligned} \partial_t k_{{{\mathrm F}},I}^{(0)}&=&\frac{{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}{N^2}\sum_{J,\alpha} \left[(k_{{{\mathrm F}},I+\alpha}^{(0)}+k_{{{\mathrm F}},J}^{(0)})-(k_{{{\mathrm F}},I}^{(0)}+k_{{{\mathrm F}},J-\alpha}^{(0)})\right]\nonumber\\ &=&-{{G^{{\mathrm c}}}_{{\mathrm B}}}^2 k_{{{\mathrm F}},I}^{(0)}+\frac{{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}{N}\sum_\alpha k_{{{\mathrm F}},I+\alpha}^{(0)}.\end{aligned}$$ If we denote by $\overline{k}=(\sum_I k_{{{\mathrm F}},I})/N=(\sum_I k_{{{\mathrm F}},I}^{(0)})/N$ the mean value of the Fermi momenta, and if we write $k_{{{\mathrm F}},I}^{(0)}(t)=\overline{k}+\delta k_{{{\mathrm F}},I}^{(0)}(t)$, the differential equation is easily solved and the solution is: $$\label{eq:lien_mult_kf_nu_hab} \delta k_{{{\mathrm F}},I}^{(0)}(\Lambda)=\delta k_{{{\mathrm F}},I}^{(0)}(\Lambda_0)\left( \frac{\Lambda}{\Lambda_0}\right)^{{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}.$$ The question that now arises, is how to determine at what scale $\Lambda^*$ the flow should be stopped. This scale cannot be determined precisely in the cut-off scaling scheme, which is only a very simple version of the Wilsonian approach. Only the latter approach could precisely describe the transition between the two regimes, which would not even occur at a single scale (because of the large number of different scales $K$ appearing in the RHS of the RG flow equations). We will thus adopt the simple and pragmatic following point of view: the flow of the [Fermi surface]{}will be stopped, when the biggest of these scales is reached, [i.e.]{}when the scale given by the difference between the biggest and the smallest Fermi momenta (denoted by $\Delta k_{{\mathrm F}}^{\max}(\Lambda)$) is reached: $$\label{eq:def_lambda_star} \Lambda^*=\Delta k_{{\mathrm F}}^{\max}(\Lambda^*).$$ Notice that if the [Fermi surface]{}flattens more quickly than the RG time decreases, this scale will never be reached. According to Eq. (\[eq:lien\_mult\_kf\_nu\_hab\]), the differences between the Fermi momenta and the mean value are all multiplied by the same factor. The biggest (respectively smallest) momentum will remain the biggest (respectively smallest) along the flow. We thus have $\Delta k_{{\mathrm F}}^{\max}(\Lambda)=\Delta k_{{\mathrm F}}^{\max}(\Lambda_0) (\Lambda/\Lambda_0)^{{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}$. We will stop the flow at the scale $\Lambda^*$ such that $\Lambda^*\simeq\Delta k_{{\mathrm F}}^{\max}(\Lambda_0) (\Lambda^*/\Lambda_0)^{{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}$. Finally we get the following link between high-energy and low-energy momenta: $$\begin{aligned} \delta k_{{{\mathrm F}},I}&\simeq&\delta k_{{{\mathrm F}},I}^{(0)}(\Lambda_0) \left( \frac{\Delta k_{{\mathrm F}}^{\max}(\Lambda_0)}{\Lambda_0}\right)^{\frac{{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}{1-{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}} \\ \label{eq:lien_DkfR_Dkfnu} \Rightarrow \Delta k_{{\mathrm F}}^{\max}&\simeq&\Delta k_{{\mathrm F}}^{\max}(\Lambda_0) \left( \frac{\Delta k_{{\mathrm F}}^{\max}(\Lambda_0)}{\Lambda_0}\right)^{\frac{{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}{1-{{G^{{\mathrm c}}}_{{\mathrm B}}}^2}}.\end{aligned}$$ As $\Delta k_{{\mathrm F}}^{\max}(\Lambda_0)=2{t_\perp}/{t_\|}$, we obtain the result already found in the literature:[@Prigodin79; @Bourbonnais85] $$\label{eq:def_sf_litterature} {t_\perp}^\mathrm{eff}\sim{t_\perp}\left( \frac{{t_\perp}}{{t_\|}}\right)^\frac{\alpha}{1-\alpha},$$ where $\alpha$ is the single-particle Green’s function’s exponent: $\alpha=(K_\rho+1/K_\rho)/4-1/2$, with $K_\rho=\sqrt{(1-2G^{{\mathrm c}}_{{\mathrm B}})/(1+2G^{{\mathrm c}}_{{\mathrm B}})}$. Perturbatively, $\alpha=\left.{G^{{\mathrm c}}_{{\mathrm B}}}\right.^2$, so that our result is indeed the same as Eq. (\[eq:def\_sf\_litterature\]), to lowest order. Let us mention that the power-law behavior of Eq. (\[eq:def\_sf\_litterature\]) has been confirmed numerically, for a two-chain system, using exact diagonalization techniques.[@Capponi98] Notice that according to Eq. (\[eq:lien\_DkfR\_Dkfnu\]), if ${{G^{{\mathrm c}}}_{{\mathrm B}}}^2\geqslant 1$, the effective transverse hopping vanishes, and the dressed [Fermi surface]{}is flat. Although this result is confirmed by the non-perturbative (in the coupling) result Eq. (\[eq:def\_sf\_litterature\]) after replacing ${G^{{\mathrm c}}_{{\mathrm B}}}^2$ by $\alpha$, we shall not use the perturbative RG in such situations that lay outside the validity range of this approach. ### Comparison to other previous results of the literature {#sec:sub:sub:comparison} Before turning to numerical calculations, we shall compare our equations describing the deformation of the [Fermi surface]{}to some more results of the recent literature. Let us begin with the article by Kishine and Yonemitsu,[@Kishine98] which treats exactly the same problem as ours. We shall compare our equations (\[eq:flot\_fs\]) to their flow equation for the effective transverse hopping (Eq. (4)). Note that they obtained this equation from the previous works by Bourbonnais and co-workers[@Bourbonnais84; @Bourbonnais91; @Bourbonnais93] and Kimura[@Kimura75] (see also the review article by Firsov, Prigodin and Seidel[@Firsov85]). We however choose the paper by Kishine and Yonemitsu because their formalism is the closest to ours. The comparison is easily achieved in two steps: first notice that our Fermi momenta do not flow when no interactions are turned on (which is desirable, since the [Fermi surface]{}should not get deformed in this case), while their ${t_\perp}$ flows in this case, because of the first term of their equation coming from the rescaling they have performed, so that we should simply forget this term if we want to compare our results. Second, we have to take the particular set of couplings they have chosen, namely local couplings. This is obtained when setting all charge couplings to the same value, and doing the same for spin couplings and Umklapps. Repeating exactly what we have done in the previous section, we find: $$\partial_t \delta k_{{{\mathrm F}},I}^{(0)}=-\delta k_{{{\mathrm F}},I}^{(0)} \left({{G^{{\mathrm c}}}_{{\mathrm B}}}^2+3{{G^{{\mathrm s}}}_{{\mathrm B}}}^2+\frac{U^2}{2}\right).$$ This in particular means that all Fermi momenta $\delta k_{{{\mathrm F}},I}^{(0)}$ will be scaled by the same factor. Next, we have to link our charge and spin couplings to the g-ology notation. It is easily checked that one simply has: ${G^{{\mathrm c}}}=g_2-g_1/2$ and ${G^{{\mathrm s}}}=-g_1/2$ (here all $g_\|$ and $g_\perp$ couplings of the g-ology are equal because we have restricted ourselves to spin-rotation invariant couplings). We also have the trivial identification $U=g_3$. The difference in the numerical factor 4 simply comes from a different normalization of the dimensionless couplings (we divided the couplings by $2\pi v_{{\mathrm F}}$ and they divided them by $\pi v_{{\mathrm F}}$). Finally, we have $\Delta k_{{\mathrm F}}^{\max}=2({t_\perp}/{t_\|})$ which shows the equivalence between the two approaches. We want to stress that this equivalence relies on our simple approximation according to which the [Fermi surface]{}is deformed only in the high energy regime, since this deformation is driven by irrelevant couplings. It would be possible to go beyond this approximation by implementing the iterative procedure outlined in Sec. \[sec:sub:sub:gen\_th\_an\]. To estimate the residual deformation of the [Fermi surface]{}induced by these irrelevant couplings in the low-energy regime remains an interesting open question, that could be addressed within the general framework discussed in this paper. Furthermore, such a calculation would enable us to take into account the deformation of the [Fermi surface]{}induced by the Hartree terms which are effective only when the forward scattering amplitudes significantly vary along the Fermi line. Such a dependence is only generated when the running cut-off becomes comparable to or smaller than the natural scale associated to the transverse dispersion. For the sake of completeness, we shall give a simple and quick derivation of the RG equation, which emphasizes the role of the 1D chains (see note 31 of Ref. [@Bourbonnais84]), and explains why the exponent obtained in Eq. (\[eq:def\_sf\_litterature\]) is the 1D propagator’s exponent. The idea is to assume that the full propagator at scale $\Lambda$ can be obtained by taking the corresponding purely 1D propagator at scale $\Lambda$, and correcting it with the dispersion relation induced by the bare ${t_\perp}$: $G^{-1}(\Lambda)=G^{-1}_{\mathrm{1D}}(\Lambda)+2{t_\perp}\cos(k_\perp)$. (In other words, this amounts to assume that when computing the effective action at scale $\Lambda$, one puts the inter-chain hopping aside, so that the flow is purely 1D, and the (unrenormalized) inter-chain hopping is reintroduced in the effective action at the end of the computation). But we can write $G^{-1}_{\mathrm{1D}}=Z^{-1}_{\mathrm{1D}}(\Lambda) [\omega-\tilde{{\varepsilon}}_\Lambda(k_\|)]$. Note that in the previous two formulas, we have denoted by $k_\perp$ and $k_\|$ the transverse and longitudinal momenta. $Z_{\mathrm{1D}}$ is the 1D wave-function renormalization, and $\tilde{{\varepsilon}}_\Lambda(k_\|)$ is the renormalized 1D dispersion relation. We thus get $G^{-1}(\Lambda)=Z^{-1}_{\mathrm{1D}}(\Lambda) [\omega-\tilde{{\varepsilon}}_\Lambda(k_\|)+2Z_{\mathrm{1D}}(\Lambda){t_\perp}\cos(k_\perp)]$, showing that the effective inter-chain hopping at scale $\Lambda$ reads: ${t_\perp}(\Lambda)=Z_{\mathrm{1D}}(\Lambda){t_\perp}$. The effective ${t_\perp}$ at two different scales are thus proportionally related by the 1D $Z$ function, whose flow equation can easily be deduced from Eq. (\[eq:flot\_phi\]) specialized to the 1D case. This yields the correct flow equation for ${t_\perp}$. Let us also mention that this way of taking into account the inter-chain tunneling has been recently adopted by Essler and Tsvelik,[@Essler02] except that they use the exact 1D Green’s function instead of the result of a perturbative RG computation. Let us now compare our results with those of Fabrizio,[@Fabrizio93] whose work is devoted to the two-chain model without longitudinal Umklapps, but with Fermi velocity renormalization. Fabrizio used a Wilsonian RG (at two loops) for the calculation of the deformation of the [Fermi surface]{}. As one can expect, this formalism allows to cross the energy scales coming from the non-flatness of the [Fermi surface]{}(see how the flows are defined piece-wise in his appendix A, and for which the various RHS never diverge). Our equations coincide with those of Fabrizio in the high-energy regime (when his function $C_2$ is expanded to lowest order in $h$, the dimensionless $\Delta k_{{\mathrm F}}$), and in the low-energy regime where the flow of the [Fermi surface]{}vanishes. The intermediate regime is of course different. Note for the comparison that Fabrizio’s coupling $g_b$ is our coupling $D$. Finally, we would like to note that our results at one loop are consistent with the article of Louis, Alvarez and Gros,[@Louis01] (see their Eqs. (9) and (13)) if we specialize these to the case of uniform Fermi velocities. Coupled cut-off scaling and field-theoretical RG calculations for a system of $N$ chains of spin 1/2 electrons: numerical results {#sec:RG_numerical} ================================================================================================================================= Motivation of the use of two RG schemes {#sec:sub:motiv_use_two_RG} --------------------------------------- Before we present our results for specific models, we wish to emphasize that one of our motivations besides the [Fermi surface]{}deformation was to describe precisely the connection between the essentially 1D high-energy regime, and the 2D low-energy physics where the system is sensitive to the warping of the [Fermi surface]{}. This can be viewed as a complement to the RG analysis of Lin et al.[@Lin97] who have focused exclusively on the low-energy side where the cut-off is much smaller than the scale associated to the transverse dispersion. In this work, they could relate the high and low energy regimes without actually solving RG flow equations for the former since they assumed very weak bare couplings (so that these couplings were barely renormalized in the high energy part of the flow). Although working with the full Wilsonian effective action allows one to get through such intermediate energy scales, it is not clear to us that this can be achieved in a reliable way with the cut-off scaling. Indeed, our view of this procedure is that it provides a simple approximation of the full Wilsonian RG, which is certainly well controlled when the running cut-off is much larger than the intrinsic low-energy scales of the system’s dynamics. Because of this we have decided to study the low-energy part of the flow in the field-theoretical framework. This latter scheme heavily relies on the existence of an infinite cut-off limit (continuum limit), or in other words the corresponding theory of 1D fermions with linear dispersion and point-like interactions is renormalizable. This statement is [*independent*]{} of the existence of intrinsic low-energy scales such as a mass term, or variations in Fermi wave vectors with the chain index. In this context RG equations are obtained by relating physical properties measured at different running energy scales. To avoid confusion with cut-off scaling this running scale has been denoted by $\nu$ in Appendix \[app:field\_th\] which presents some details on this approach. Since we have used a logarithmic approximation, the high-energy flows of the couplings in both cut-off scaling and field-theoretical RG are identical. It is an interesting question whether the two schemes give the same physical low-energy results or not. We plan to study this in more detail in a forthcoming work. A first numerical study: incommensurate nesting {#sec:sub:first_num_std} ----------------------------------------------- We will now show what information can be deduced from numerical computations. For this we choose to focus on a simple example, where the Umklapps are set to zero, but we still assume a perfect [Fermi surface]{}nesting. This situation is realized in several interesting systems as for instance in two dimensional molybdenum and tungsten bronzes. For a review, see for example the paper by Foury and Pouget.[@Foury93] ![Flow of the normalized charge (top) and spin (bottom) couplings, as a function of the “good” time $s$, for $N=8$. Initially the true couplings (not normalized) are ${G^{{\mathrm c}}}=0.3=-{G^{{\mathrm s}}}$, and the bare hopping is ${t_\perp}/{t_\|}=0.1$. We have indicated the value $s^*$ corresponding to the value $\Lambda^*$ (see Eq. (\[eq:def\_lambda\_star\])).[]{data-label="fig:flot_gcs"}](flot_gcs){width="8cm"} We have chosen an initial condition for which all charge couplings are equal, and all spin couplings too, with ${G^{{\mathrm c}}}=0.3=-{G^{{\mathrm s}}}$. The bare hopping is ${t_\perp}/{t_\|}=0.1$. The couplings are quite large so that the deformation of the [Fermi surface]{}will be visible. The [Fermi surface]{}could be deduced analytically, but we have computed it numerically as all other quantities. All the results are contained in Figs. \[fig:flot\_gcs\] to \[fig:Z32\_t\]. The first three (respectively last three) of these figures have been computed with $N=8$ (respectively $N=32$). The reasons for these choices are that we could not represent all the couplings (the first of the six figures) for a too high value of $N$, because the number of couplings grows like $N^3$. But this was no problem for the [Fermi surface]{}and the quasiparticle weights, except for a longer computation time. ![Flow of the norm $\mathcal{N}$ as $s$ grows, corresponding to Fig. \[fig:flot\_gcs\]. The inserted flow is a zoom on short times.[]{data-label="fig:norme"}](norme){width="8cm"} In these figures, that we shall comment one after the other, we have made use of some notions such as the norm of the couplings, the normalized couplings, and the adapted RG time $s$. All these notions, and some others (such as fixed directions, etc.) have been dealt with extensively in our previous paper,[@Dusuel02] so we shall simply give the few basic definitions. The norm is the Euclidean norm of the coupling vector, and the normalized couplings are the usual couplings divided by the norm (we give an explicit formula for the charge couplings only): $$\begin{aligned} &&\mathcal{N}=\sqrt{\sum_{I,J,\delta} {{G^{{\mathrm c}}}_\delta(I,J)}^2 + {{G^{{\mathrm s}}}_\delta(I,J)}^2 + {U_\delta(I,J)}^2},\quad\\ &&\widetilde{{G^{{\mathrm c}}}}_\delta(I,J)=\frac{{G^{{\mathrm c}}}_\delta(I,J)}{\mathcal{N}}.\end{aligned}$$ In all the figures in which flows will be represented (such as Fig. \[fig:flot\_gcs\]), we adopt the following convention: in the caption, we give the initial condition for [*usual*]{} couplings, while in the figures themselves we draw the [*normalized*]{} couplings, and suppress the $\tilde{\:}$ in the $y$-axis legends. The reader should not get confused by this abuse of notation. The time $s$ that we have used in the numerical simulation is defined by ${{\mathrm d}}s=\mathcal{N}(t){{\mathrm d}}t$, and is the time adapted for zooming on the flow singularities. ![Link between the “good” RG time $s$ and the “true” RG time $t$, corresponding to Fig. \[fig:flot\_gcs\]. We have indicated the values of $s^*$ and of $t^*=\ln(\Lambda_0/\Lambda^*)$ corresponding to $\Lambda^*$[]{data-label="fig:temps"}](temps){width="8cm"} The first of the six figures, Fig. \[fig:flot\_gcs\], represents the “field-theoretical” RG flow of the normalized charge and spin couplings, as functions of the RG time $s$. This flow is divided into three regions. In the first one ($0\leqslant s\leqslant s^*\simeq 15$), corresponding to the high-energy regime, all charge couplings and all spin couplings remain equal. Indeed, in this regime, the curvature of the [Fermi surface]{}is not felt at all, in the logarithmic approximation we use. All chains are thus identical (remember we use periodic boundary conditions in the transverse direction), the system is purely one-dimensional, so that the symmetry between the chains cannot be broken. ![Bare and dressed [Fermi surface]{}for $N=32$.[]{data-label="fig:defSF32"}](defSF32){width="8cm"} In this high-energy regime, the cut-off scaling flow is exactly the same as the “field-theoretical” one, so that we could use the latter for the computation of the deformation of the [Fermi surface]{}. After $s\simeq 15$, the flow of the [Fermi surface]{}is stopped, because the scale $\Lambda^*$ (as previously defined by $\Lambda^*=\Delta k_{{\mathrm F}}^{\max}(\Lambda^*)$) is reached. The dressed [Fermi surface]{}is the one obtained at that scale, and is then used in the “field-theoretical” flow of the couplings. This dressed [Fermi surface]{}, and the bare one, are represented on Fig. \[fig:defSF32\]. As discussed in Sec. \[sec:sub:sub:comparison\], within the approximation we use, the dressed [Fermi surface]{}is still given by ${t_\perp}\cos(k_\perp)$ but with the dressed value of the interchain hopping. Higher harmonics for $\Delta k_{{\mathrm F}}$ as a function of $k_\perp$, corresponding to longer range transverse hoppings, are expected to be generated only in the low-energy part of the [Fermi surface]{}flow. Indeed it is only in this regime that effective couplings acquire a dramatic dependence with respect to transverse momenta. But this goes beyond the scope of the simple ([i.e.]{}non iterative) procedure used for the numerical computations presented here. ![Flow of the quasiparticle weights $Z_I=1/{\varphi}_I$, for $N=32$. We have represented these for $I=16$ to $I=32$, as the ones for smaller values of $I$ can be found using the top-bottom ($I\leftrightarrow N-I$ because $N$ is even) symmetry. Note that we have indicated the value of $t^*=\ln(\Lambda_0/\Lambda^*)$ on the time axis.[]{data-label="fig:Z32"}](Z32){width="8cm"} ![Here we show the transverse momentum ([i.e.]{}chain number $I$) dependence of the quasiparticle weights $Z_I$, at four RG times $t=1$, $2$, $3$ and $4$ (see Fig. \[fig:Z32\] for a time reference).[]{data-label="fig:Z32_t"}](Z32_t){width="8cm"} We emphasize that [*by contrast*]{} to the flow of the couplings, for the quasiparticle weights $Z_I$ functions associated with the renormalized propagator, [*the whole flow must be computed with the fixed dressed [Fermi surface]{}*]{}, because the flow equations do depend on the [Fermi surface]{}in the high-energy regime (see Eq. (\[eq:flot\_phi\])). As a consequence it is necessary to first compute the dressed [Fermi surface]{}, and then use it to compute the flow of the $Z_I$’s. These flows are represented on Fig. \[fig:Z32\]. We furthermore show the variation of the $Z_I$’s along the Fermi line at four different RG times on Fig. \[fig:Z32\_t\]. We note that the dispersion in the $Z_I$’s is small, in qualitative agreement with the dynamical mean field theory results obtained by Biermann et al,[@Biermann01] and this provides a consistency check of Bourbonnais’ computation (see Sec. \[sec:sub:sub:comparison\]). Fig. \[fig:Z32\_t\] shows that the evolution of the quasiparticle weights with typical energy scale exhibits some similarity with the results of Kishine and Yonemitsu:[@Kishine99] in the early stages of the flow, the quasiparticle weight is larger for $k_y=\pm\pi$ ($I=0$ or $N$), than for $k_y=\pi/2$ ($I=N/2$), and this ordering is reversed at later stages. However, we stress that the two models are different since Kishine and Yonemitsu have considered a 2D model with flat [Fermi surface]{}segments, and it is not obvious that the end points of these segments should exhibit the same properties as the extremal points $k_y=\pm\pi$ in our quasi 1D model. The variation of the quasiparticle weight along the [Fermi surface]{}has also been investigated by D. Zanchi[@Zanchi01] for the 2D Hubbard model, where he found a much stronger reduction of the $Z$ factor in the vicinity of the Van-Hove singularities than for typical [Fermi surface]{}points. We believe this effect requires to take into account the variation of the Fermi velocity along the [Fermi surface]{}which we have not done here. The influence of these variations on the flow of couplings for a $N$-leg Hubbard ladder has been recently studied[@Ledermann00; @Ledermann01] (for ${t_\perp}<t$), with the conclusion that they play a dramatic role only below a cross-over scale which is extremely small as $N$ becomes large. The second regime ($15\lesssim s\lesssim 60$) is a transient between the 1D high-energy flow, and the low-energy regime, where the shape of the [Fermi surface]{}is felt, and where the differentiation between the couplings takes place. In more physical terms, it corresponds to a Fermi liquid regime, located between a Luttinger liquid state at higher energies, and an ordered phase at lower energies. One might have expected that in this Fermi liquid regime, the $Z_I$’s would remain constant, so that if this Fermi liquid regime was the final one, the quasiparticle residue would be finite. Instead of this, we see on Fig. \[fig:Z32\], that the time derivative of the $Z_I$’s decreases (in absolute value) but does not vanish. ![Flow of $Z_{N/4}$ as a function of the “true” RG time $t$, for different values of $N$, and for ${t_\perp}/{t_\|}=0.1$. These flows were obtained assuming that all coupling remain constant to their bare values (${G^{{\mathrm c}}}=-{G^{{\mathrm s}}}=0.3$). Note that we chose $I=N/4$ because $Z_{N/4}$ is about the mean value of the $Z_I$’s. The inserted graph is a base 10 log-log representation of the value of the time derivative of $-\ln(Z_{N/4})$ at time $t=10$ as a function of $N$. The solid curve is the numerical fit, which shows a $1/N$ behavior.[]{data-label="fig:verif_FL_Z"}](verif_FL_Z){width="8cm"} This is in fact a finite $N$ effect, as can be seen on the flows of the $Z_I$’s one obtains (see Fig. \[fig:verif\_FL\_Z\]), assuming that the couplings are constant, equal to their bare value, all along the flow. The inserted graph shows that the growth rate of the logarithm of the $Z_I$’s behaves like $1/N$ in the Fermi liquid regime. This is indeed consistent with the flow equations (\[eq:flot\_phi\]) in the low-energy regime, where only the terms with a vanishing $\Delta k_\alpha(I,J)$ contribute, and whose number is of order $N$. This factor $N$ combined with the $1/N^2$ denominator explains the numerical result. The relevance of these considerations with constant couplings is demonstrated by Figs. \[fig:norme\] and \[fig:temps\]. Indeed they show that the norm is almost constant for the interval $t\in[0,5]$ in which we are interested (see Fig. \[fig:Z32\]). (Note that Figs. \[fig:norme\] and \[fig:temps\] were obtained for $N=8$ not for $N=32$, but we have checked that on this interval $t\in[0,5]$, the flows of the two norms are identical, apart from a multiplicative factor of $8=(32/8)^{3/2}$, whose origin is the number $N^3$ of couplings for a given $N$, and which is irrelevant for our discussion). The final regime is one of a fixed direction, for which some normalized couplings are zero, and the others are gathered around specific values. It is in this final phase of the flow that the norm of the couplings explodes, as can be seen on Fig. \[fig:norme\]. In order to be complete, we have also represented the link between the two RG times $s$ and $t$ on Fig. \[fig:temps\]. Notice that because of the definition of $s$ (see the comment after Eq. (10) of our previous paper), and because the norm explodes in the end of the flow, the time $t$ “saturates”, as a function of $s$, to a value which is roughly the critical temperature at which the final phase sets in. Let us now study more precisely the final fixed direction, in the spirit of our previous paper. First of all, let us have a more precise look at the values of the couplings, on the fixed direction that is reached. These values are shown on Fig. \[fig:analyse\_df\_finale\], for the $N=16$ case. We did not choose $N=8$ as on Fig. \[fig:flot\_gcs\], because we wanted to have more values (which was manageable here since we represent the whole set of values only once). When briefly looking at these values, one can deduce that the couplings seem to be grouped into a few sets of similar values (with lots of couplings being equal to zero). Furthermore, forgetting about the zero value, it seems the three values (for charge or spin couplings) are not independent, but one is the sum of the other two. Finally, the values of the charge couplings seem to differ by a factor of three (and a minus sign) from the spin couplings. In fact, if we also look at Fig. \[fig:flot\_gcs\], we see that this will probably not be an exact statement for all values of $N$, but only in the limit of infinite $N$. ![Values of the charge and spin couplings, for the fixed direction that is finally reached on Fig. \[fig:flot\_gcs\], but for $N=16$ here. The inserted graph is a zoom on the values taken by the positive charge couplings, and shows that these can be grouped into two sets.[]{data-label="fig:analyse_df_finale"}](analyse_df_finale){width="8cm"} It is then interesting to study what types of couplings take non-zero values. For the system we study, the notation $G_\delta(I,J)$ which is best adapted to superconductivity, can favorably be changed for ${F^{{{\mathrm c}}({{\mathrm s}})}}_\delta(I,J)={G^{{{\mathrm c}}({{\mathrm s}})}}_{J-I-\delta}(I,J)$. $\delta$ is then the transferred transverse momentum, between the R particle that is destroyed, and the L particle that is created. In this notation, only $F$ couplings with $\delta=J-I$ or with $\delta=N/2$ are numerically found to have non-zero values. Notice that here, as $N$ is even, $N/2$ is an integer (we will discuss the odd $N$ case a bit further). The couplings for which $\delta=J-I$ and $J-I\neq N/2$ will be denoted as ${\mathcal{C}^{{{\mathrm c}}({{\mathrm s}})}}$ couplings, and correspond to the charge (respectively spin) couplings that are negative (respectively positive) on the fixed direction. They are the usual forward scattering couplings. The couplings that satisfy $\delta=N/2$ and $J-I\neq N/2$ will be denoted as ${\mathcal{D}^{{{\mathrm c}}({{\mathrm s}})}}$, whereas the ones for which $\delta=N/2$ and $J-I=N/2$ will be denoted as ${\mathcal{A}^{{{\mathrm c}}({{\mathrm s}})}}$ couplings. Both have a transferred transverse momentum $\delta$ which is half the number of chains ([i.e.]{}$\pi$ if we use the usual momentum units). The vector linking a point of the [Fermi surface]{}on the R side, to the one on the L side, and $N/2$ chains further is a [*nesting*]{} vector, which explains why these couplings are present in the final low-energy fixed direction. ![Flow of the charge and spin couplings, for the same values of the parameters as in Fig. \[fig:flot\_gcs\], but with $N=7$.[]{data-label="fig:flot_Nimpair"}](flot_Nimpair){width="8cm"} We let the reader write down the RG equations satisfied by the $F$ couplings, specialize these for the three types of couplings above, and deduce the equations satisfied for the final fixed direction, in the spirit of our previous paper.[@Dusuel02] The resulting equations are: $$\label{eq:fd} \left\lbrace \begin{array}{l} N{\mathcal{A}^{{\mathrm c}}}=(N-1)({{\mathcal{D}^{{\mathrm c}}}}^2+3{{\mathcal{D}^{{\mathrm s}}}}^2)\\ N{\mathcal{A}^{{\mathrm s}}}=4{{\mathcal{A}^{{\mathrm s}}}}^2 +2(N-1)({{\mathcal{D}^{{\mathrm s}}}}^2+{\mathcal{D}^{{\mathrm s}}}{\mathcal{D}^{{\mathrm c}}})\\ N{\mathcal{C}^{{\mathrm c}}}=-({{\mathcal{D}^{{\mathrm c}}}}^2+3{{\mathcal{D}^{{\mathrm s}}}}^2)\\ N{\mathcal{C}^{{\mathrm s}}}=4{{\mathcal{C}^{{\mathrm s}}}}^2 +2({{\mathcal{D}^{{\mathrm s}}}}^2-{\mathcal{D}^{{\mathrm s}}}{\mathcal{D}^{{\mathrm c}}})\\ N{\mathcal{D}^{{\mathrm c}}}=(N-2)({{\mathcal{D}^{{\mathrm c}}}}^2+3{{\mathcal{D}^{{\mathrm s}}}}^2)+2({\mathcal{A}^{{\mathrm c}}}-{\mathcal{C}^{{\mathrm c}}}){\mathcal{D}^{{\mathrm c}}}+6({\mathcal{A}^{{\mathrm s}}}-{\mathcal{C}^{{\mathrm s}}}){\mathcal{D}^{{\mathrm s}}}\\ N{\mathcal{D}^{{\mathrm s}}}=2(N-2)({{\mathcal{D}^{{\mathrm s}}}}^2+{\mathcal{D}^{{\mathrm s}}}{\mathcal{D}^{{\mathrm c}}})+2({\mathcal{A}^{{\mathrm s}}}-{\mathcal{C}^{{\mathrm s}}}){\mathcal{D}^{{\mathrm c}}}+2({\mathcal{A}^{{\mathrm c}}}-{\mathcal{C}^{{\mathrm c}}}){\mathcal{D}^{{\mathrm s}}}+4({\mathcal{A}^{{\mathrm s}}}+{\mathcal{C}^{{\mathrm s}}}){\mathcal{D}^{{\mathrm s}}}\end{array} \right..$$ This set of coupled equations is nothing but Eq. (46) of our previous paper, with usual letters replaced by calligraphic letters. The condition $A^{\rm c,s}=C^{\rm c,s}+D^{\rm c,s}$ was satisfied, and ensured the SU$(N)$ symmetry of the interaction Hamiltonian. Here we will thus also be able to fulfill the relation ${\mathcal{A}^{{{\mathrm c}}({{\mathrm s}})}}={\mathcal{C}^{{{\mathrm c}}({{\mathrm s}})}}+{\mathcal{D}^{{{\mathrm c}}({{\mathrm s}})}}$, which was previously guessed when looking at the fixed direction obtained numerically. The values of the couplings can be found in Table II of our previous paper. It is clear that in the present situation, it is the so called $(+,-)$ fixed direction that is selected since in the infinite $N$ limit, it is the one for which the charge couplings equal minus three times the spin couplings. The effective low-energy interaction Hamiltonian has the following schematic form (we drop the charge and spin structure): $$\begin{aligned} H_\mathrm{int}^\mathrm{eff} &\sim &\mathcal{C}\sum_{q} \Big[ \sum_{I,k} {c^\dagger}_{{{\mathrm R}},I} (k+q){c^{}}_{{{\mathrm R}},I} (k) \Big]\nonumber\\ \label{eq:Hint_schematic} &&\hspace{1cm}\times\Big[ \sum_{J,k'} {c^\dagger}_{{{\mathrm L}},J} (k'-q){c^{}}_{{{\mathrm L}},J} (k') \Big]\\ &&- \mathcal{D} \sum_{q,k,k'} \Big[ \sum_{J} {c^\dagger}_{{{\mathrm R}},J-N/2} (k+q){c^{}}_{{{\mathrm L}},J} (k') \Big]\nonumber\\ &&\hspace{1cm}\times\Big[ \sum_{I} {c^\dagger}_{{{\mathrm L}},I+N/2} (k'-q){c^{}}_{{{\mathrm R}},I} (k) \Big]\;.\nonumber\end{aligned}$$ Let us describe the physics associated with such an effective Hamiltonian. We will assume that $N$ is large, so that we can neglect all finite $N$ corrections. Thus, for example, only the $\mathcal{D}$ terms (Peierls couplings) survive, as the forward couplings $\mathcal{C}$ are a correction of order $1/N$. Furthermore, in the infinite $N$ limit, the couplings take the values ${\mathcal{D}^{{\mathrm c}}}=3/4$ and ${\mathcal{D}^{{\mathrm s}}}=-1/4$, so that ${\mathcal{D}^{{\mathrm c}}}=-3{\mathcal{D}^{{\mathrm s}}}$. This relation implies that the interaction exists only in the triplet channel, and the effective Hamiltonian can be written as ($g>0$): $$\begin{aligned} \label{eq:Hint_eff_esp_reel} &&H_\mathrm{int}^\mathrm{eff} = -\frac{g}{N}\int_0^L {{\mathrm d}}x \nonumber\\ &&\hspace{1cm}: \left[ \sum_J {\psi^\dagger}_{{{\mathrm R}},J-N/2,\rho}(x) \boldsymbol{\sigma}_{\rho,\rho'} {\psi^{}}_{{{\mathrm L}},J,\rho'}(x) \right]\\ &&\hspace{1.7cm}\times \left[ \sum_I {\psi^\dagger}_{{{\mathrm L}},I+N/2,\tau}(x) \boldsymbol{\sigma}_{\tau,\tau'}{\psi^{}}_{{{\mathrm R}},I,\tau'}(x) \right]:.\nonumber\end{aligned}$$ The only difference (except for changes of notation) between this effective Hamiltonian and the one we arrived at in Eq. (70) of our previous paper, is in the shift of the creation operators’ chain number by an amount of $N/2$. We thus expect the physics to be essentially the same as we had discussed in our previous paper, apart from a different SDW’s wave vector that will now be $(2\overline{k},\pi)$ (remember $\overline{k}$ is the average Fermi momentum). We refer the reader to our previous paper for details. We thus have shown that after a high-energy 1D regime where the [Fermi surface]{}’s shape is not felt in the flow of the couplings, and after the crossing of the typical energy-scale given by the curvature of the [Fermi surface]{}, the system goes to a strong coupling phase of the SDW type, with the above effective low-energy Hamiltonian. The $(2\overline{k},\pi)$ nesting vector naturally arises from the RG flow, and there is no need to artificially introduce it. Let us make a final remark, about the odd $N$ case. The flow of the charge and spin couplings for $N=7$ is shown on Fig. \[fig:flot\_Nimpair\]. This figure is obviously different from Fig. \[fig:flot\_gcs\]. The reason for this is that there is no [*exact*]{} nesting vector anymore when $N$ is odd. We have analysed which couplings are non-zero in the low-energy phase of Fig. \[fig:flot\_Nimpair\], and these turn out to be BCS type couplings, indicating a superconducting low-energy phase. This is not in contradiction with what has been said before in the even $N$ case. When $N$ grows, the nesting is better and better in the odd $N$ case, so that the RG flow will first be towards the same fixed direction as in the even $N$ case. Then, there will be a shift from this fixed direction to another one, corresponding to superconductivity. But, this will take place at very low energies, and in regimes where the norm of the couplings has exploded. The conclusion is that the low-energy phase, in the thermodynamical limit, is always the one we have observed in the even $N$ case. This discussion has been quite brief, but we refer the reader to our previous paper where we had analysed in detail how the observed shift from one fixed direction to another one, in a finite $N$ situation, slows down as $N$ increases and finally disappears in the infinite $N$ limit. We have checked all this numerically, but unfortunately it requires quite a large value of $N$ (more than 30) to be visible, so that we could not depict it in this paper. Taking account of the Umklapps and limitations of the method {#sec:sub:umklapps_limitations} ------------------------------------------------------------ In the above two simple examples we have studied, we encountered no real limitation of the computation scheme we have proposed. Of course, we had to use a non rigorous (but plausible) argument to define the scale at which we had to stop the flow of the couplings. We will see that there are some cases where it is not possible to use such a simple point of view. The well known main limitation of a RG approach is its perturbative nature. We cannot fully trust the RG flows when they go to strong couplings, even if the fixed direction that is reached in this regime gives an insight of what physics takes place. For the two computations of the [Fermi surface]{}we have given previously, it is clear that this was no limitation, because the deformation of the [Fermi surface]{}occurred in a weak coupling regime (remember Figs. \[fig:flot\_gcs\] and \[fig:norme\], where the norm diminishes between $0\leqslant s \lesssim 20$, which is the time interval where the deformation of the [Fermi surface]{}takes place). The half-filled system is interesting, because it exhibits a variety of behaviors, depending on the strength of the bare couplings (compared to the value of the bare transverse hopping). We will discuss the strong, intermediate and weak coupling situations, which do not give the same low-energy physics. After this, we will consider the case of a nearly half-filled system. ### Half-filled system in strong coupling By strong coupling, we mean the initial couplings are large enough for the behavior of the system to remain purely one-dimensional. To make this statement more precise, let us study the flows for one of these strong coupling initial conditions. We will assume, as we always did, that ${t_\perp}/{t_\|}=0.1$. In this case, the Hubbard-condition ${G^{{\mathrm c}}}=0.2=-{G^{{\mathrm s}}}$ and $U=0.4$ is a strong coupling condition, for which the RG leads to a fully flat [Fermi surface]{}. In fact, as the couplings are large and grow quickly (because of the Umklapps), the [Fermi surface]{}flattens quickly, so that the decreasing cut-off never catches the scale of the [Fermi surface]{}, and there is no non-zero value of $\Lambda^*$. The flow of the couplings is thus purely 1D all along the flow and is well known, so that we do not show it. However, to be concrete, we show the evolution of the Fermi momenta on Fig. \[fig:defSF\_um\_coup\_fort\]. ![Flow of the Fermi momenta, for $N=16$ chains, ${t_\perp}/{t_\|}=0.1$ and Hubbard initial condition ${G^{{\mathrm c}}}=0.2=-{G^{{\mathrm s}}}$ and $U=0.4$.[]{data-label="fig:defSF_um_coup_fort"}](defSF_um_coup_fort){width="8cm"} About the norm, let us say that its value at the beginning of the flow is about 30, and at $s=140$, it is about 1400, so that it is around 50 times bigger. This means the couplings have grown a lot. For example for the Umklapps, $U\simeq 19$ which is very big, so that the RG is not valid anymore. However, if we believe the RG is qualitatively valid, the flow of the Fermi momenta seems to show the existence of a confined phase (the effective ${t_\perp}$ is zero). As the system goes to strong coupling, and remains purely 1D, it would thus be natural to directly start from a system of decoupled (no hopping) Luttinger liquids or 1D Mott insulators. We shall simply direct the reader to some papers on this line of approach.[@Boies95; @Arrigoni00; @Essler02] ### Half-filled system in weak coupling {#sec:sub:sub:hfwc} ![Flow of the three types of couplings, for $N=8$ chains, ${t_\perp}/{t_\|}=0.1$, and the initial condition ${G^{{\mathrm c}}}=0.03=-{G^{{\mathrm s}}}$ and $U=0.06$.[]{data-label="fig:flots_gcsu"}](flots_gcsu1 "fig:"){width="8cm"} ![Flow of the three types of couplings, for $N=8$ chains, ${t_\perp}/{t_\|}=0.1$, and the initial condition ${G^{{\mathrm c}}}=0.03=-{G^{{\mathrm s}}}$ and $U=0.06$.[]{data-label="fig:flots_gcsu"}](flots_gcsu2 "fig:"){width="8cm"} A weak coupling initial condition is one for which the [Fermi surface]{}does not get completely flat during the RG flow ([i.e.]{}a non-zero value of $\Lambda^*$ exists), and for which all couplings that are irrelevant ([i.e.]{}that do not exist at zero energy) do go to zero (after dividing by the norm) during the flow. As a consequence, the cross-over scale $\Lambda^*$ between the Luttinger and the Fermi liquid behavior is much larger than the typical scale for the onset of long range order. The system is therefore in a deconfined regime, in the sense that it allows for coherent transverse motion of electron-like excitations. An example of this is obtained while using initial Hubbard couplings ${G^{{\mathrm c}}}=0.03=-{G^{{\mathrm s}}}$ and $U=0.06$, and as usual ${t_\perp}/{t_\|}=0.1$. It is not worth representing the deformation of the [Fermi surface]{}in this case, for it is very small (for $N=8$, the effective ${t_\perp}/{t_\|}$ is about 0.0995, so that the correction is of the order of half a percent). Let us however represent the flow of the couplings, on Fig. \[fig:flots\_gcsu\], in the $N=8$ case, and the flow of the quasiparticle weight (and of the norm of the couplings) on Fig. \[fig:comp\_Z\_um\_N\]. As in the incommensurate case, the time derivative of the quasiparticle weights becomes smaller (in absolute value) in an intermediate regime, and this effect is more and more visible as $N$ gets bigger. This decrease can be understood from the flow of the norm, which shows a tendency towards a plateau behavior at intermediate scales, so that the arguments previously given in Sec. \[sec:sub:first\_num\_std\] still apply. For the comparison with Fig. \[fig:flots\_gcsu\], let us simply say that for $0\leqslant t\leqslant 6$ the link between $s$ and $t$ is approximately linear, and for $t=6$, one has $s=14$. Before studying the fixed direction, let us make a remark about the scale $\Lambda^*$. Because of the presence of the Umklapps, not only the scales defined by all the $\Delta k_{{{\mathrm F}},\alpha}(I,J)$ play a role, but also the scales $\Delta k^U_{{{\mathrm F}},\alpha}(I,J)$. However, as the filling is one-half, the average Fermi momentum is $\pi/2$, and one can check that in this case, the biggest $\Delta k^U_{{{\mathrm F}},\alpha}(I,J)$ is, as the biggest $\Delta k_{{{\mathrm F}},\alpha}(I,J)$, equal to twice the difference between the biggest and the smallest Fermi momenta. $\Lambda^*$ will then be defined exactly as we did when the Umklapps were zero. The couplings on the fixed direction have very well defined values, with many being equal to zero. For the charge and spin couplings, the only non-zero couplings are the same as previously discussed in Sec. \[sec:sub:first\_num\_std\], namely of the type $\mathcal{A}$, $\mathcal{C}$ and $\mathcal{D}$. The condition ${\mathcal{A}^{{{\mathrm c}}({{\mathrm s}})}}={\mathcal{C}^{{{\mathrm c}}({{\mathrm s}})}}+{\mathcal{D}^{{{\mathrm c}}({{\mathrm s}})}}$ still seems valid, and one can furthermore check that ${\mathcal{C}^{{\mathrm c}}}=0$, so that ${\mathcal{A}^{{\mathrm c}}}={\mathcal{D}^{{\mathrm c}}}$. For the study of the Umklapps, it is also interesting to introduce $V$ couplings, which are the equivalent of the $F$ couplings: $V_\delta(I,J)=U_{J-I-\delta}(I,J)$. An analysis of the non-zero Umklapps reveals that there are only three types of such couplings: ${\mathcal{U}}=U_{N/2}(I,J\neq I)$, ${\mathcal{V}}=V_{N/2}(I,J\neq I)$ and ${\mathcal{W}}=U_{N/2}(I,I)=V_{N/2}(I,I)$. Furthermore, these Umklapps also seem not to be independent, but linked by the ${\mathcal{W}}={\mathcal{U}}+{\mathcal{V}}$ relation. We let the reader check that the fixed direction is found by solving the following set of coupled equations (which is just a generalization of Eq. (\[eq:fd\])): $$\label{eq:fd_um} \left\lbrace \begin{array}{l} N{\mathcal{A}^{{\mathrm c}}}=(N-1)({{\mathcal{D}^{{\mathrm c}}}}^2+3{{\mathcal{D}^{{\mathrm s}}}}^2+{\mathcal{U}}^2+{\mathcal{V}}^2-{\mathcal{U}}{\mathcal{V}})\\ N{\mathcal{A}^{{\mathrm s}}}=4{{\mathcal{A}^{{\mathrm s}}}}^2 +2(N-1)\left[{{\mathcal{D}^{{\mathrm s}}}}^2+{\mathcal{D}^{{\mathrm s}}}{\mathcal{D}^{{\mathrm c}}}+\frac{1}{2}({\mathcal{U}}^2-{\mathcal{U}}{\mathcal{V}})\right]\\ N{\mathcal{C}^{{\mathrm c}}}=-({{\mathcal{D}^{{\mathrm c}}}}^2+3{{\mathcal{D}^{{\mathrm s}}}}^2+{\mathcal{U}}{\mathcal{V}}-{\mathcal{U}}^2-{\mathcal{V}}^2)\\ N{\mathcal{C}^{{\mathrm s}}}=4{{\mathcal{C}^{{\mathrm s}}}}^2 +2\left[{{\mathcal{D}^{{\mathrm s}}}}^2-{\mathcal{D}^{{\mathrm s}}}{\mathcal{D}^{{\mathrm c}}}+\frac{1}{2}({\mathcal{V}}^2-{\mathcal{U}}{\mathcal{V}})\right]\\ N{\mathcal{D}^{{\mathrm c}}}=(N-2)({{\mathcal{D}^{{\mathrm c}}}}^2+3{{\mathcal{D}^{{\mathrm s}}}}^2+{\mathcal{U}}^2+{\mathcal{V}}^2-{\mathcal{U}}{\mathcal{V}})+2({\mathcal{A}^{{\mathrm c}}}-{\mathcal{C}^{{\mathrm c}}}){\mathcal{D}^{{\mathrm c}}}+6({\mathcal{A}^{{\mathrm s}}}-{\mathcal{C}^{{\mathrm s}}}){\mathcal{D}^{{\mathrm s}}}+{\mathcal{W}}({\mathcal{U}}+{\mathcal{V}})\\ N{\mathcal{D}^{{\mathrm s}}}=2(N-2)\left[{{\mathcal{D}^{{\mathrm s}}}}^2+{\mathcal{D}^{{\mathrm s}}}{\mathcal{D}^{{\mathrm c}}}+\frac{1}{2}({\mathcal{U}}^2-{\mathcal{U}}{\mathcal{V}})\right]+2({\mathcal{A}^{{\mathrm s}}}-{\mathcal{C}^{{\mathrm s}}}){\mathcal{D}^{{\mathrm c}}}+2({\mathcal{A}^{{\mathrm c}}}-{\mathcal{C}^{{\mathrm c}}}){\mathcal{D}^{{\mathrm s}}}+4({\mathcal{A}^{{\mathrm s}}}+{\mathcal{C}^{{\mathrm s}}}){\mathcal{D}^{{\mathrm s}}}+{\mathcal{W}}({\mathcal{U}}-{\mathcal{V}})\\ N{\mathcal{U}}=2(N-2)\left[({\mathcal{D}^{{\mathrm c}}}+3{\mathcal{D}^{{\mathrm s}}}){\mathcal{U}}-2{\mathcal{D}^{{\mathrm s}}}{\mathcal{V}}\right]+2({\mathcal{D}^{{\mathrm c}}}+{\mathcal{D}^{{\mathrm s}}}){\mathcal{W}}+2({\mathcal{A}^{{\mathrm c}}}+3{\mathcal{A}^{{\mathrm s}}}){\mathcal{U}}-4{\mathcal{A}^{{\mathrm s}}}{\mathcal{V}}+2({\mathcal{C}^{{\mathrm c}}}-{\mathcal{C}^{{\mathrm s}}}){\mathcal{U}}\\ N{\mathcal{V}}=2(N-2)({\mathcal{D}^{{\mathrm c}}}-{\mathcal{D}^{{\mathrm s}}}){\mathcal{V}}+2({\mathcal{D}^{{\mathrm c}}}-{\mathcal{D}^{{\mathrm s}}}){\mathcal{W}}+2({\mathcal{A}^{{\mathrm c}}}-{\mathcal{A}^{{\mathrm s}}}){\mathcal{V}}-4{\mathcal{C}^{{\mathrm s}}}{\mathcal{U}}+2({\mathcal{C}^{{\mathrm c}}}+3{\mathcal{C}^{{\mathrm s}}}){\mathcal{V}}\\ N{\mathcal{W}}=2(N-1)\left[ {\mathcal{D}^{{\mathrm c}}}({\mathcal{U}}+{\mathcal{V}})+3{\mathcal{D}^{{\mathrm s}}}({\mathcal{U}}-{\mathcal{V}})\right]+4{\mathcal{A}^{{\mathrm c}}}{\mathcal{W}}\end{array} \right..$$ It is easy to solve this set of equations (with the relations between the couplings), order by order in $N$. One finds a few fixed directions, but the one of interest is the following (that we give to order 3, for the independent couplings): $$\begin{aligned} &&{\mathcal{C}^{{\mathrm s}}}=0+\frac{1}{4N}-\frac{1}{4N^2}+\frac{47}{96N^3}+{\mathcal{O}}\left(\frac{1}{N^4}\right),\nonumber\\ &&{\mathcal{D}^{{\mathrm c}}}=\frac{3}{8}-\frac{3}{16N}-\frac{1}{16N^2}+\frac{33}{64N^3}+{\mathcal{O}}\left(\frac{1}{N^4}\right),\nonumber\\ \label{eq:coup_dir_fix_av_um} &&{\mathcal{D}^{{\mathrm s}}}=-\frac{1}{8}+\frac{1}{16N}+\frac{1}{16N^2}-\frac{11}{64N^3}+{\mathcal{O}}\left(\frac{1}{N^4}\right),\hspace{0.9cm}\\ &&{\mathcal{U}}=\frac{1}{4}-\frac{1}{8N}-\frac{7}{24N^2}+\frac{89}{96N^3}+{\mathcal{O}}\left(\frac{1}{N^4}\right),\nonumber\\ &&{\mathcal{V}}=\frac{1}{2}-\frac{1}{4N}+\frac{1}{6N^2}+\frac{13}{24N^3}+{\mathcal{O}}\left(\frac{1}{N^4}\right).\nonumber\end{aligned}$$ ![The top figure represents the evolution of $Z_{N/4}$ as a function of $t$ for $N$=8 (squares), 16 (circles) and 32 (triangles). The bottom figure shows how the norm varies with time $t$. In fact, in order to allow the comparison, we have divided the norm for $N=16$ and $32$ by a constant $C$ so as to make all the norms equal at time $t=0$. The squares thus represent the norm $\mathcal{N}$ for $N=8$, the circles represent $\mathcal{N}/C$ ($N=16$) with $C=(16/8)^{3/2}=2\sqrt{2}$ and the triangles represent $\mathcal{N}/C$ ($N=32$) with $C=(32/8)^{3/2}=8$.[]{data-label="fig:comp_Z_um_N"}](comp_Z_um_N){width="8cm"} We let the reader check that even the order 0 reproduces quite accurately the values of the couplings of Fig. \[fig:flots\_gcsu\] (of course, up to an overall normalization factor). Here again, the effective Hamiltonian contains forward interactions that are $1/N$ corrections, and interactions for which the transferred momentum is the nesting vector. The $\mathcal{D}$ couplings satisfy the relation ${\mathcal{D}^{{\mathrm c}}}+3{\mathcal{D}^{{\mathrm s}}}=0$ (in the infinite $N$ limit), so that the part of the effective low-energy Hamiltonian containing the $\mathcal{D}$ couplings is the same (apart from a numerical factor) as the one we previously obtained (see Sec. \[sec:sub:first\_num\_std\]). It is non-zero only in the triplet channel (we again consider the particle-hole parametrization of the couplings). Let us see what form the effective Umklapp Hamiltonian takes in this singlet and triplet parametrization. From Eq. (\[eq:coup\_dir\_fix\_av\_um\]), we see that we have a relation between ${\mathcal{U}}$ and ${\mathcal{V}}$ which reads ${\mathcal{V}}=2{\mathcal{U}}$ (this is valid up to $\mathcal{O}(1/N^2)$ terms). The corresponding interaction involving chains $I$ and $J$ on one side of the [Fermi surface]{}, and $I+\delta$ and $J-\delta$ on the other side, may typically be written as (using Pauli’s principle): $$\begin{aligned} &&\frac{{\mathcal{U}}}{2NL}\sum_{I,J}~ \sum_{\tau,\tau'}~ \sum_{\rho,\rho'} \left(\mathbb{I}_{\tau,\tau'} \mathbb{I}_{\rho,\rho'}-2\mathbb{I}_{\tau,\rho'} \mathbb{I}_{\rho,\tau'}\right)\nonumber\\ &&\times\left({c^\dagger}_{{{\mathrm R}},I+N/2,\tau} {c^\dagger}_{{{\mathrm R}},J-N/2,\rho} {c^{}}_{{{\mathrm L}},J,\rho'} {c^{}}_{{{\mathrm L}},I,\tau'}+\mbox{ h. c.}\right).\hspace{0.5cm}\end{aligned}$$ As $\mathbb{I}_{\tau,\tau'} \mathbb{I}_{\rho,\rho'}-2\mathbb{I}_{\tau,\rho'} \mathbb{I}_{\rho,\tau'}=-\boldsymbol{\sigma}_{\tau,\tau'}\cdot \boldsymbol{\sigma}_{\rho,\rho'}$, it is easy to rewrite the Umklapp interaction in the triplet channel only. If we define the generalized current $$\boldsymbol{J}_{{{\mathrm R}}{{\mathrm L}}}(x)=\sum_I {\psi^\dagger}_{{{\mathrm R}},I+N/2,\tau}(x) \boldsymbol{\sigma}_{\tau,\tau'}{\psi^{}}_{{{\mathrm L}},I,\tau'}(x)=\boldsymbol{J}^\dagger_{{{\mathrm L}}{{\mathrm R}}}(x),$$ the total effective Hamiltonian takes the simple following form ($g>0$): $$\label{eq:Hint_eff_esp_reel_U} H_\mathrm{int}^\mathrm{eff} = -\frac{g}{N}\int_0^L {{\mathrm d}}x : \left[\boldsymbol{J}_{{{\mathrm R}}{{\mathrm L}}}(x)+\boldsymbol{J}_{{{\mathrm L}}{{\mathrm R}}}(x)\right]^2:.$$ The low-energy physics can again be described by the fluctuations of the massless modes associated to the order parameter (which is $\boldsymbol{\xi}(x)=\langle\boldsymbol{J}_{{{\mathrm R}}{{\mathrm L}}}(x)\rangle$). The difference with the non half-filled case studied in Sec. \[sec:sub:first\_num\_std\], is that the spin-density wave will be pinned to the lattice by the Umklapps. That this is indeed what happens can be seen by computing the effective action of the gapless modes, and one finds (we drop less relevant terms): $$\label{eq:eff_action} S_\mathrm{eff}(\boldsymbol{n})=\frac{N}{4\pi}\int {\rm d}x{\rm d}t \,\partial_\mu \boldsymbol{n} \partial^\mu \boldsymbol{n}.$$ with $\boldsymbol{\xi}(x)=\rho \boldsymbol{n}(x)$, $\rho$ being a positive number found by solving mean-field equations, and $\boldsymbol{n}(x)$ is a real unit vector, giving the direction of the staggered magnetization. This time, there is no gapless mode associated to the “phason” field (see the discussion around Eqs. (70) and (71) of our previous paper). This is physical, for a shift $\theta(x)\to\theta(x)+\Theta$ in the “phason” field corresponds roughly to a uniform translation of the spin-density wave condensate. As the physics of Eq. (\[eq:eff\_action\]) has already been discussed in our previous paper, we do not consider it further. ![Evolution of the smallest and largest charge (squares), spin (circles) and Umklapp (triangles) irrelevant couplings, for $N=16$, ${t_\perp}/{t_\|}=0.1$ and initial condition $U=0.02$.[]{data-label="fig:irr"}](irr){width="8cm"} Let us now study more precisely the fate of the irrelevant couplings. We defined the initial coupling as a weak coupling if the normalized irrelevant couplings go to zero during the flow. In fact it is interesting to check that the final fixed direction that is reached is the same whether we run the complete flow, or we run the flow in which the irrelevant couplings are initially set to zero. We have done this in a system of $N=16$ chains, with ${t_\perp}/{t_\|}=0.1$ and an initial Hubbard coupling $U=0.02$. The evolution of the irrelevant couplings is shown on Fig. \[fig:irr\]. In fact we have not represented all the couplings, because there are too many of them. We have decided to show only the smallest and largest charge, spin and Umklapp couplings. In order to make sure that the final fixed direction is the same as the one we would have obtained when initially setting irrelevant couplings to zero, we show the different values of the couplings on this fixed direction, in both cases, on Fig. \[fig:comp\_moi\_lbf\]. ![Values of all the couplings on the final fixed direction, for $N=16$, ${t_\perp}/{t_\|}=0.1$ and weak initial Hubbard coupling $U=0.02$. On the left (respectively right), we represented the values obtained when computing the whole flow (respectively initially setting the irrelevant couplings to zero).[]{data-label="fig:comp_moi_lbf"}](comp_moi_lbf){width="8cm"} ![Representation of the numerical values of $U_{{\mathrm c}}\ln(N)$ as functions of $N$, where $U_{{\mathrm c}}$ is the value above which the irrelevant couplings do not flow to zero.[]{data-label="fig:verif_critere_wc"}](verif_critere_wc){width="8cm"} A natural question that arises from this discussion is how small should the couplings be for being weak couplings according to the definition given above ? This question has already been answered by Lin, Balents and Fisher[@Lin97]. They have done so on a theoretical ground (and for a situation that is not the half-filled system, but this should not change anything), and found that the weak coupling condition reads $U \ln(N)\ll 1$ (for large $N$). Thanks to our ability to take the irrelevant couplings into account, we have tried to check this numerically. To do so, we have determined the critical coupling $U_{{\mathrm c}}$ for which the irrelevant couplings do not flow to zero anymore, for $N=$8, 12, 16, 24 and 32, and represented the values $U_{{\mathrm c}}\ln(N)$ as functions of $N$. The result is shown on Fig. \[fig:verif\_critere\_wc\]. Because of the small values of $N$ we have used, we do not observe an horizontal line as could have been inferred from the $U \ln(N)\ll 1$ criterion. But this criterion is in fact a sufficient condition (maybe not a necessary one) to observe a weak coupling behavior, since it implies that the effective Hamiltonian hardly changes during the high-energy part of the RG flow, for scales above $\Lambda^*$. ### Half-filled system in intermediate coupling ![Evolution of the (normalized) charge forward scattering couplings ${G^{{\mathrm c}}}_0(I,J)$, when the strength of the initial Hubbard coupling $U$ is increased ($U=0.12$ for (a), $U=0.17$ for (b), $U=0.19$ for (c) and $U=0.202$ for (d)). The number of chains is $N=16$, and ${t_\perp}/{t_\|}=0.1$. The flow has been stopped at the RG time for which the biggest of the whole set of couplings is equal to 1. The charge forward scattering pictured here are the ones obtained at this time. In order to keep the figures clear, we have not put any indication on the $z$ axis. Let us simply say that the values of the couplings range from $3\cdot10^{-3}$ to $1.3\cdot10^{-2}$ in (a), and from $6.75\cdot10^{-3}$ to $7.05\cdot10^{-3}$ in (d), so that (d) is in reality much flatter than (a).[]{data-label="fig:evolution_coupC_charge"}](evolution_coupC_charge){width="8cm"} When the couplings are neither strong nor weak, that is intermediate, we suspect the system will behave more and more like a 1D system, as the initial Hubbard coupling $U$ grows. Before we check that this is the case, let us clarify this notion of intermediate coupling. We have just seen at the end of the previous section (\[sec:sub:sub:hfwc\]), that the intermediate coupling should typically be characterized by $U\simeq U_{{\mathrm c}}$ (remember Fig. \[fig:verif\_critere\_wc\]). In the case $N=16$ and ${t_\perp}/{t_\|}=0.1$ on which we shall focus, this means $U\simeq 0.15$. We should also have $U<0.206$, value above which the system is in the confined phase. If we expect the system’s behavior to change and become nearly one-dimensional for these typical values of $U$, this should mean that the effective hopping is of the same order of magnitude as the critical temperature. This will be discussed in Sec. \[sec:sub:sub:phase\_diagram\], when we study the phase diagram of the system. Let us simply say here that for the minimum (respectively maximum) value of the coupling $U$ we will consider, namely $U=0.12$ (respectively $U=0.202$), the effective hopping ${t_\perp}^\mathrm{eff}=$ is about 8 times (respectively 0.9 times) the critical temperature. These values confirm the previous expectation. ![Same as Fig. \[fig:evolution\_coupC\_charge\], for the spin forward scattering couplings ${G^{{\mathrm s}}}_0(I,J)$. Here, the values of the couplings range from $-1.8\cdot10^{-3}$ to $-2\cdot10^{-4}$ in (a), and from $7.95\cdot10^{-4}$ to $7.7\cdot10^{-4}$ in (d).[]{data-label="fig:evolution_coupC_spin"}](evolution_coupC_spin){width="8cm"} ![Same as Fig. \[fig:evolution\_coupC\_charge\], for the Umklapp couplings $U_\delta(I,I)$. Here, the values of the couplings range from $8\cdot10^{-3}$ to $2.4\cdot10^{-2}$ in (a), and from $1.3954\cdot10^{-2}$ to $1.3961\cdot10^{-2}$ in (d).[]{data-label="fig:evolution_coupC_umklapp"}](evolution_coupC_umklapp){width="8cm"} We have numerically studied the evolution of some special couplings, as $U$ becomes bigger. It is not possible to consider the couplings on the final fixed direction. Indeed, the norm is huge even before it is reached. We had neglected this problem in the strong and weak coupling regimes. In the first case, we anyway knew that the RG is not valid anymore and should be replaced by a non-perturbative analysis. In the second case, we can make the norm as small as we want when reducing the initial coupling, because in this case, the RG flow is scale invariant (the RHS of the RG equations quickly become $\nu$ independent, as $\nu$ rapidly goes to very small values). In this last case we refer the reader to our previous paper[@Dusuel02] for more details about the implications of this scale invariance. We thus have chosen to stop the RG flows at the time when the biggest of all couplings (the true couplings, not the normalized ones) reaches the value 1. In this regime, the RG should be valid (of course, the two-loops contributions are not negligible when the couplings approach 1). This gives us the results shown on Figs. \[fig:evolution\_coupC\_charge\] to \[fig:evolution\_coupC\_umklapp\], which were obtained for $N=16$ and initial couplings $U=$0.12, 0.17, 0.19 and 0.202. For the charge and spin couplings, we have represented forward scattering couplings ${G^{{{\mathrm c}}({{\mathrm s}})}}_0(I,J)$. In the weak coupling regime, we would have obtained ${G^{{\mathrm c}}}_0(I,J)\sim \delta_{J,I+N/2}$ (remember the fixed direction we found in Sec. \[sec:sub:sub:hfwc\]). Here we also obtain peaks around $J=I+N/2$ values, but these peaks progressively disappear when $U$ grows, as is expected because the system looks more and more one-dimensional. We have chosen to represent $U_\delta(I,I)$ couplings, in the case of the Umklapps. The reason for this choice is that the biggest of all Umklapps is (numerically) found in this subset of couplings. Again, the weak coupling would give a peak $U_\delta(I,I)\sim \delta_{\delta,N/2}$, which is smeared in the case of intermediate couplings, and disappears in the strong coupling limit. ### Phase diagram {#sec:sub:sub:phase_diagram} As a conclusion of this investigation, we shall summarize the numerical results we obtained on a single figure, which is the phase diagram of the system. It is depicted in Fig. \[fig:phase\_diag\]. The solid curves represent ${t_\perp}^\mathrm{eff}/{t_\|}$ as a function of ${t_\perp}/{t_\|}$, for $U$=0.05, 0.06 (indicated by the arrows), 0.08, 0.1, 0.15 and 0.2 (notice that both $x$ and $y$ scales are expressed in base 10 logarithms). For a given $U$, ${t_\perp}^\mathrm{eff}$ is zero in the confined phase, for ${t_\perp}$ smaller than a critical value ${t_\perp}^{{\mathrm c}}$ (this explains the vertical lines), and it takes non-zero values as soon as ${t_\perp}>{t_\perp}^{{\mathrm c}}$. When ${t_\perp}$ is much larger than ${t_\perp}^{{\mathrm c}}$, ${t_\perp}^\mathrm{eff}\simeq{t_\perp}$, so that all solid curves asymptotically go to the first bisector. The dashed curves give the value of the scale at which the couplings diverge, which is the critical temperature $T_{{\mathrm c}}$. They are horizontal when ${t_\perp}<{t_\perp}^{{\mathrm c}}$, since in our approach the RG flows are purely one-dimensional in this regime. The upper (respectively lower) dotted curve is a straight line of slope 1 (numerically found to be 1.004), going through the points of coordinates $({t_\perp}^{{\mathrm c}}(U),T_{{\mathrm c}}(U,{t_\perp}^{{\mathrm c}}(U))$ (respectively $({t_\perp}^{{\mathrm c}}(U),{t_\perp}^\mathrm{eff}({t_\perp}^{{\mathrm c}}(U)^+)$), as the one represented by a diamond (respectively circle) in the inserted figure. This inserted figure is a zoom of the interesting region where both scales meet, for $U=0.05$. We have indicated the different phases (Luttinger Liquid, Fermi Liquid, Mott Insulator and Spin-Density Wave). The dash-dotted curve is the first bisector, that we did not represent in the global figure to keep it readable. ![Phase diagram of the system, computed for $N=8$ chains. See text for a detailed description of this diagram.[]{data-label="fig:phase_diag"}](phase_diag){width="8cm"} Let us now study the physical implications of the value 1 taken by the slopes of the two dotted lines. The upper one tells us that the critical value of ${t_\perp}$ is proportional to the value of the charge gap $\Delta$ of the Mott insulating phase, and we numerically find ${t_\perp}^{{\mathrm c}}\simeq 1.14 \Delta$. The lower curve gives the following relation between the value of the effective hopping for ${t_\perp}={{t_\perp}^{{\mathrm c}}}^+$, that we will denote by ${{t_\perp}^\mathrm{eff}}^*$, and the critical value of the bare hopping: ${{t_\perp}^\mathrm{eff}}^*\simeq0.68{t_\perp}^{{\mathrm c}}$. Let us remark that this also implies: ${{t_\perp}^\mathrm{eff}}^*\simeq 0.78\Delta$. These relations show that the confinement-deconfinement transition takes place when the bare and the effective hopping become of the order of the charge gap. These results have natural interpretations. Imagine the system is in the Mott insulating phase, with zero effective hopping. It is clear that we can only compare the bare hopping to the charge gap. If on the contrary the system is in the deconfined regime, the low-energy physics is dominated by two scales: the critical temperature and the effective hopping. When decreasing the bare hopping, we expect a phase transition to occur when these two low-energy scales become of the same order of magnitude. These results are in quantitative agreement with those obtained by Tsuchiizu et al[@Tsuchiizu99_2] for a two-chain model, see for instance the insert in their Fig. 1. For values of ${t_\perp}$ which are not too small, they indeed find a proportionality between $\Delta$ and ${t_\perp}$ with a slope compatible with our results. For smaller values of ${t_\perp}$, they obtain a sizeable deviation away from a linear behavior. But this difference with our conclusions comes from their choice of a fixed value for the intra-chain forward scattering, while they let the Umklapp scattering go to zero. In this regime, they observed a significant renormalization of the hopping amplitude, so that the transition is finally given by the balance between the charge gap and the renormalized hopping. ### Nearly half-filled system Up to now, we have not encountered any real difficulty when choosing the $\Lambda^*$ scale. Of course, the choice was purely pragmatic, as we simply chose the biggest of the scales $K$ appearing in the RHS of the RG flows. There was no problem in the half-filled case, because the biggest scale was the same for all the nine sorts of $K$ (remember the definitions in Eqs. (\[eq:car\_length\_1\]) to (\[eq:car\_length\_9\])). Let us consider the non half-filled case, for which the mean Fermi momentum is $\overline{k}=\pi/2+\delta k$. The filling does not change the biggest $K^{\mathrm{pp}}$ and $K^{\mathrm{ph}}$, which is still the difference between the biggest and the smallest Fermi momenta, $\Delta k_{{\mathrm F}}^{\max}=k_{{\mathrm F}}^{\max}-k_{{\mathrm F}}^{\min}$. We let the reader check that the biggest $K^{UU}$ is now $2|\delta k|+\Delta k_{{\mathrm F}}^{\max}$ and the biggest $K^{GUi}$ is $|\delta k|+\Delta k_{{\mathrm F}}^{\max}$. Those last two scales are obviously always bigger than the first one. What are the consequences of the existence of these three scales? At the formal level there is no real consequence. Indeed, even in the half-filled case, there were a lot of different scales, given by all the possible $K$’s, so that introducing more scales does not make much change. But, practically, we have used the pragmatic point of view that we should stop the cut-off scaling flow when the biggest scale is reached. This relied on the hypothesis that the flow of the [Fermi surface]{}nearly stops at that scale. It is clear that if the filling is not too far from one-half, the three scales are not qualitatively different, so that we can afford stopping the flow when the biggest scale is reached. When the filling is quite far from one-half, things become more involved. Of course, we could simply forget about the Umklapps and perform the analysis of Sec. \[sec:sub:first\_num\_std\] devoted to the non half-filled system. But we expect that in an intermediate to strong coupling regime, the Umklapps could play a non-negligible role in the high-energy part of the flow, where there are not yet irrelevant. We would thus like to be able to take them into account. Intuitively, we expect that the [Fermi surface]{}deformation is caused by both $U$ and $G$ couplings for a cut-off $\Lambda>\Delta k_{{\mathrm F}}^{\max}+(2)|\delta k|$ and by the $G$ couplings only for $\Delta k_{{\mathrm F}}^{\max}<\Lambda<\Delta k_{{\mathrm F}}^{\max}+(2)|\delta k|$. It is however not possible to implement this idea in a simple manner. Indeed, once the scale $\Delta k_{{\mathrm F}}^{\max}+(2)|\delta k|$ is reached, we could drop the Umklapp contribution to the flow of the [Fermi surface]{}, but the flow of the Umklapp couplings will not stop and will depend on the shape of the fixed dressed [Fermi surface]{}. This flow of the Umklapps will affect the flow of the $G$ couplings and as these latter still deform the [Fermi surface]{}, we see that the flow of the Umklapps indirectly affects the flow of the [Fermi surface]{}. As the shape of the dressed [Fermi surface]{}comes into play [*before*]{} the flow of the running [Fermi surface]{}stops, there is here no simple way to circumvent the inversion problem we have discussed in the introduction. As a consequence, in what follows, we will restrict ourselves to situations where our pragmatic scheme works, [i.e.]{}to the nearly half-filled system. We have considered a filling slightly less than 1/2, setting the chemical potential to -0.01. For this value, the difference between the initial values of $\Delta k_{{\mathrm F}}^{\max}$ and $\Delta k_{{\mathrm F}}^{\max}+(2)|\delta k|$ is about 10%, which is reasonable. As in the half-filled case, we have observed different regimes, when changing the strength of the initial coupling namely weak, intermediate and strong coupling regimes. ![Flow of the three types of couplings, for $N=8$ chains, ${t_\perp}/{t_\|}=0.1$, $\mu=-0.01$, and the initial condition ${G^{{\mathrm c}}}=0.05=-{G^{{\mathrm s}}}$ and $U=0.1$. We also have represented the flow of the norm, for $0\leqslant s\leqslant 60$.[]{data-label="fig:flots_gcsu_pdr"}](flots_gcsu1_pdr "fig:"){width="8cm"} ![Flow of the three types of couplings, for $N=8$ chains, ${t_\perp}/{t_\|}=0.1$, $\mu=-0.01$, and the initial condition ${G^{{\mathrm c}}}=0.05=-{G^{{\mathrm s}}}$ and $U=0.1$. We also have represented the flow of the norm, for $0\leqslant s\leqslant 60$.[]{data-label="fig:flots_gcsu_pdr"}](flots_gcsu2_pdr "fig:"){width="8cm"} In the strong coupling regime, as before, the system behaves as a purely 1D system, and the [Fermi surface]{}gets completely flat. The intermediate coupling regime is the same as the one observed in the half-filled case, where the Umklapps are not suppressed, because the scale at which the phase transition takes place is bigger than the scale that measures the distance from half-filling, namely $(2)\delta k$. In the weak coupling regime, the Umklapps are irrelevant, and all vanish (for the normalized couplings) at low energies. The final fixed direction is simply the one we previously found in Sec. \[sec:sub:first\_num\_std\], where we had set the Umklapps to zero at the beginning of the flow. The flows of the couplings are represented on Fig. \[fig:flots\_gcsu\_pdr\], and were obtained for $N=8$ and initial coupling $U=0.1$. We also have represented the evolution of the norm of the couplings, because its behavior changes drastically at the precise time the Umklapps become irrelevant ($s\simeq24$). The plateau observed in the norm’s evolution just after this time reveals the existence of an intermediate Fermi liquid phase. These flows show how our RG, taking account of the different scales of the system, is able to get rid of the irrelevant couplings, such as the Umklapps, and leads to the correct final fixed direction. This is not of a purely academic interest. Indeed, if one takes the same value of the initial couplings (${G^{{\mathrm c}}}=0.05=-{G^{{\mathrm s}}}$), but set $U$ to zero from the beginning (with a flow very much like the one in Fig. \[fig:flot\_gcs\]), the scale at which the phase transition occurs is found to be $4\cdot 10^{12}$ times smaller as the one found when the Umklapps are incorporated and vanish along the flow. This can in part be explained because neglecting the Umklapps from the beginning of the flow is a very crude approximation. We could also take the Umklapps into account in the 1D part of the flow, and then take them to zero. However, a look at Fig. \[fig:flots\_gcsu\_pdr\] shows that the Umklapps do not vanish very fast, and their influence has thus no reason to be small. Indeed, we performed this comparison, and found that this prediction is correct. In fact both methods neglecting the Umklapps at one time or another, give approximately the same critical temperature, because the Hubbard coupling $U$ is small so that the couplings do not change much in the 1D region (we found ${G^{{\mathrm c}}}=0.074$ and ${G^{{\mathrm s}}}=-0.038$ in the end of the 1D flow). Conclusion ========== We have attempted to develop a simple physical picture to understand the forces which drive the deformations of the [Fermi surface]{}of an interacting electron system. Using considerations from second-order time-independent perturbation theory, we showed that the shape of the dressed [Fermi surface]{}controls the quantum zero-point motion correction to the ground-state energy. We demonstrated that a given coupling tends to deform the [Fermi surface]{}so as to have all its four momenta precisely on the [Fermi surface]{}, because this allows for smaller energy denominators (in the second order contribution) and thus decreases the total energy. As a consequence, we find that the [Fermi surface]{}is deformed by irrelevant couplings, which are characterized by the impossibility of choosing all four momenta on the [Fermi surface]{}. Because of this, the [Fermi surface]{}deformation induced by these couplings can only occur in the high-energy regime where the kinematical constraints associated to the [Fermi surface]{}do not play much role. Once the energy is low enough and the warping of the [Fermi surface]{}is felt, these irrelevant couplings do not flow any more and no longer contribute to the [Fermi surface]{}deformation. For the quasi 1D materials at half-filling, the Umklapp couplings belong to this category. As they undergo a strong renormalization in the high-energy regime, they have a much more drastic effect than the charge or spin couplings in the doped system. Our numerical simulation provide a description of the cross-over from the confined regime to the Fermi liquid, which is in overall good agreement with previous works.[@Prigodin79; @Bourbonnais85; @Kishine98; @Tsuchiizu99_2; @Biermann01] We have presented a detailed analysis of the evolution of the quasiparticle weights as a function of the typical energy scale. This confirms the existence of a Fermi liquid regime at intermediate energies, for the deconfined systems. It would be very interesting to compute the longitudinal and transverse optical conductivities, which could be done by adapting some existing methods[@Honerkamp01_2] to quasi 1D systems. Another problem is to investigate the nature of spin correlations in the confined regime. This can not be achieved within the present formalism since the couplings diverge at a scale associated to the charge gap which is much larger than the Néel temperature in the limit of small transverse hopping. This problem has been addressed by Kishine and Yonemitsu[@Kishine98] who used RG equations to two-loop order for the couplings. But it is not clear that the two-loop corrections provide a reliable description since the couplings do not remain small. This limitation of our method is certainly connected to the fact that we are using a physical picture in which the fermion fields remain the elementary objects. This is valid at sufficiently high energies and thus generically adapted to the study of the [Fermi surface]{}deformations. However in the confined regime, elementary excitations are likely to be very different from the Fermi liquid-like quasiparticles, but rather some soliton-like objects. In this case, it seems a deeper understanding of the corresponding phases should be obtained by expanding around the exact solution for a system of uncoupled chains.[@Boies95; @Arrigoni00; @Essler02] Nevertheless this raises the important issue of the validity of an adiabatic principle for generating the ground-state at finite transverse hopping from the ground-state of uncoupled chains. At half-filling this adiabatical principle certainly holds in the confined phase where the charge gap is finite. But its validity beyond the critical value of the transverse hopping is questionable. This may explain the qualitative discrepancy between the perturbation approach of Essler and Tsvelik,[@Essler02] which leads to a disconnected [Fermi surface]{}with electron and hole pockets, and the dynamical mean-field theory of Biermann et al, who have obtained a conventional [Fermi surface]{}(see Fig. 5 of the latter work[@Biermann01]). Away from half-filling, this notion of adiabatic continuity is even less obvious to prove since the energy gap vanishes for uncoupled Luttinger liquids. However we believe the use of a skeleton expansion by Arrigoni[@Arrigoni00] likely provides a way to circumvent this potential difficulty. ACKNOWLEDGMENTS {#acknowledgments .unnumbered} =============== We would like to thank Fernão Vistulo de Abreu for providing the initial impetus to this work. We have also benefitted from many valuable remarks by Dražen Zanchi and Benedikt Binz. Finally we are grateful to Bertrand Delamotte and Dominique Mouhanna for sharing their experience with various aspects of the renormalization group. Equivalence between minimizing the total energy and stability criteria for the dressed single particle propagator {#app:equi} ================================================================================================================= Two-chain system at first order {#sec:sub:sub:pert_ex} ------------------------------- We wish to check that the [self-energy]{}computed by using a free excited state, with a trial Fermi sea which should be determined only at the end of the calculation, gives the correct results of Sec. \[sec:sub:sub:min\_energy\]. The sole difference with the calculation of Appendix \[sec:sub:sub:std\_pert\] below is that we shall consider here the following free propagators: $$\begin{aligned} \widetilde{G}^{(0)}_{{{\mathrm R}},I}(k,\omega)=\hspace{5.8cm}\nonumber\\ \label{eq:free_propR_tilde} \frac{1}{\omega-\left[\mu^{(0)}+v_{{\mathrm F}}^{(0)}(k-k_{{{\mathrm F}},I}^{(0)})\right]+{{\mathrm i}}\eta{\mathrm{sgn}}(k-k_{{{\mathrm F}},I})},\hspace{0.4cm}\\ \widetilde{G}^{(0)}_{{{\mathrm L}},I}(k,\omega)=\hspace{5.8cm}\nonumber\\ \label{eq:free_propL_tilde} \frac{1}{\omega-\left[\mu^{(0)}-v_{{\mathrm F}}^{(0)}(k+k_{{{\mathrm F}},I}^{(0)})\right]-{{\mathrm i}}\eta{\mathrm{sgn}}(k+k_{{{\mathrm F}},I})}.\hspace{0.4cm}\end{aligned}$$ These formulas differ with (\[eq:free\_propR\]) and (\[eq:free\_propL\]) below only in the imaginary parts. We let the reader check (some more details about one-loop [self-energy]{}calculations can be found in Appendix \[sec:sub:sub:std\_pert\] devoted to the standard perturbation theory) that the [self-energies]{}read: $$\begin{aligned} &&\Sigma_{{{\mathrm R}},0}^{(1)}(k,\omega)=A\frac{\Lambda_0+(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})}{2\pi}\nonumber\\ &&\hspace{2.5cm}+C\frac{\Lambda_0-(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})}{2\pi},\\ &&\Sigma_{{{\mathrm R}},\pi}^{(1)}(k,\omega)=B\frac{\Lambda_0-(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})}{2\pi}\nonumber\\ &&\hspace{2.5cm}+C\frac{\Lambda_0+(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})}{2\pi}.\end{aligned}$$ It is sufficient to compare these results with (\[eq:en\_1\_1part0\]) and (\[eq:en\_1\_1partpi\]) to understand that we will [*exactly*]{} find (\[eq:dk0\_vrai\]) and (\[eq:dmu\_vrai\]). The energy minimization method or this self-consistent computation of the [self-energy]{}carried formally to first order in interaction do generate the same higher order terms. These appear since corresponding contributions are sensitive to the shape of the trial [Fermi surface]{}. As shown in Appendix \[app:diff\_re\], this feedback effect is missing in the standard perturbative approach. Formal proof for a finite size system ------------------------------------- Let us choose a trial Fermi surface, with the corresponding occupation numbers $n(k) \in \{0,1\}$ and $\Phi(k)=1-2n(k)$. This generates a free particle state (Slater determinant) which may be used as a starting point for perturbative expansions of the total energy $E(\{n\})$ and the single particle propagator $G_{\Phi}(k,\omega)$. Suppose we add one particle to the system, so that the total momentum is increased by $k_{0}$. This is achieved by using $n'(k)=n(k)+\delta_{k,k_{0}}$ instead of $n(k)$, assuming that $n(k_{0})=0$. This induces a change in the total energy of the system $\Delta E = E(\{n'\})-E(\{n\})$. For a finite size system, it is easy to connect this energy shift to the single particle propagator $G_{\Phi}(k_{0},\omega)$. Indeed, we have the well-known spectral decomposition: $$\begin{aligned} &&G_{\Phi}(k_{0},\omega)=\sum_{\alpha} \frac{\left|{\langle N+1,k_{0},\alpha |}{c^\dagger}_{k_{0}}{| N \rangle}\right|^{2}} {\omega - E(N+1,k_{0},\alpha)+E(N)+{{\mathrm i}}\eta}\nonumber\\ &&\hspace{0.5cm}+ \sum_{\beta} \frac{\left|{\langle N-1,-k_{0},\beta |}{c^{}}_{k_{0}}{| N \rangle}\right|^{2}} {\omega + E(N-1,-k_{0},\beta)-E(N)-{{\mathrm i}}\eta}.\end{aligned}$$ Here, ${| N \rangle}$ is the eigenstate with $N$ particles obtained perturbatively from the free-particle state with distribution $n(k)$, and the kets ${| M,k,\alpha \rangle}$ denote eigenstates with $M$ particles and a total momentum $k$ with respect to the total momentum of state ${| N \rangle}$. The total energy of these states is of course $E(M,k,\alpha)$. So the energy shift $\Delta E$ is one of the poles of $G_{\Phi}(k_{0},\omega)$ seen as a function of $\omega$. Denoting the typical interaction strength by $V$, it is sensible to assume that for a finite-size system, energy differences like $E(N+1,k_{0},\alpha)-E(N)$ (and $\Delta E$ in particular) can be expanded as power series in $V$. In the non-interacting case, $\Delta E={\varepsilon}_{0}(k_{0})$, and $G^{(0)}_{\Phi}(k_{0},\omega)^{-1} =\omega- {\varepsilon}_{0}(k_{0})+{{\mathrm i}}\eta$. Therefore, $\Delta E$ is the pole of $G_{\Phi}(k_{0},\omega)$ (as a function of $\omega$) which goes smoothly towards ${\varepsilon}_{0}(k_{0})$ as $V$ goes to zero. Writing $G_{\Phi}(k_{0},\omega)^{-1}=\omega- {\varepsilon}_{0}(k_{0}) -\Sigma_{\Phi}(k_{0},\omega)$, and given the fact that $\Sigma_{\Phi}(k_{0},\omega)$ has a well-defined power series expansion in $V$ which vanishes as $V$ goes to zero, we may conclude that $\Delta E$ may be obtained as a formal power series in $V$ from the solution of the following equation for $\omega$: $$\omega- {\varepsilon}_{0}(k_{0})-\Re \Sigma_{\Phi}(k_{0},\omega)=0.$$ Indeed, let us denote by $\omega (k_{0})$ the solution of this equation which goes to ${\varepsilon}_{0}(k_{0})$ as $V$ goes to zero. Then, we have: $$\Delta E = \omega (k_{0}).$$ Now, if the trial state ${| N \rangle}$ obtained from $n(k)$ minimizes the total energy $E(\{n\})$, it means that removing a particle at $k_{1}$ on the Fermi surface associated to the distribution $n(k)$ and adding another particle at $k_{2}$ also on the Fermi surface does not change the total energy (up to corrections which are negligible for very large systems). This yields $\omega(k_{1})=\omega(k_{2})$, implying that quasiparticle energies are constant (equal the the dressed chemical potential $\mu$) on the Fermi surface associated to $n(k)$. This is exactly condition i) (see Eq. (\[eq:conditioon\_re\_fs\])) for the dressed propagator discussed in Sec. \[sec:sub:gen\_cons\]. Assuming condition i) holds, condition ii) on the imaginary part follows from standard phase-space arguments and analyticity considerations developed already long ago by Luttinger[@Luttinger61] or Langer[@Langer61]. The main idea is to use an expression for the self-energy in terms of skeleton graphs. Condition i) suggests that the full one-particle spectral function (which determines completely the internal lines of these graphs) is qualitatively similar to the one of a Fermi liquid with the Fermi surface obtained from $n(k)$. Extension to an infinite system ------------------------------- Applying the previous argument to infinite systems requires some care. In fact, we have to prove that the coefficients of perturbative series in powers of $V$ for $\Delta E$ or $\omega(k_{0})$ have a well-defined infinite volume limit. Our experience with other systems including models for an unstable state coupled to a continuum suggests that such a limit does not exist in general. However, perturbation theory in powers of the interaction strength for fermion systems with local two-body interactions is likely to be a favorable case for which this limit may be safely taken. For instance, the perturbative expansion of the ground-state energy involves connected Feynman graphs with no external lines, which contributions are easily shown to be proportional to the volume. Similarly, standard techniques based on the Luttinger-Ward energy functional (see for instance the text by Nozières,[@Nozieres_anglais] pages 222 to 229) show that: $$\begin{aligned} \Delta E &=& E(\{n'\})-E(\{n\})\nonumber\\ &=& {\varepsilon}_{0}(k_{0}) + \int_{-\infty}^{\infty} \frac{{{\mathrm d}}\omega}{2\pi {{\mathrm i}}} \log \Bigg[\frac{\omega -{\varepsilon}_{0}(k_{0})-{{\mathrm i}}\eta} {\omega -{\varepsilon}_{0}(k_{0})+{{\mathrm i}}\eta}\\ &&\hspace{2cm}\times \frac{\omega -{\varepsilon}_{0}(k_{0})-\Sigma_{\Phi}(k_{0},\omega)+{{\mathrm i}}\eta} {\omega -{\varepsilon}_{0}(k_{0})-\Sigma_{\Phi}(k_{0},\omega)-{{\mathrm i}}\eta}\Bigg].\nonumber\end{aligned}$$ up to terms which vanish in the thermodynamical limit. If $k_{0}$ is close enough to the dressed Fermi surface so that the inverse life time of the corresponding “quasiparticle” is small compared to $\eta$, it is easy to show that $\Delta E = \omega(k_{0})$. This shows that the series expansion of $\omega(k_{0})$ has a well-defined infinite volume limit. This fact is a priori non trivial since any perturbative algorithm for $\omega(k_{0})$ involves partial derivatives at any order for $\Sigma_{\Phi}(k_{0},\omega)$ with respect to $\omega$, taken at $\omega = {\varepsilon}_{0}(k_{0})$. Although $\Sigma_{\Phi}(k_{0},\omega)$ has a good thermodynamical limit, some difficulties arise while considering derivatives with respect to $\omega$. Indeed, their expressions for finite size systems involve sums of rational functions of $\omega$ with multiple poles, and these are not easily converted into converging integrals in the infinite volume limit. But the above connection between $\Delta E$ (which has a thermodynamical limit) and $\omega(k_{0})$ shows that all the wild terms which are expected to appear in a perturbative expression of $\omega(k_{0})$, eventually cancel. Difficulties with the real part of $\Sigma$ in the traditional perturbation scheme {#app:diff_re} ================================================================================== Two-chain system at first order {#sec:sub:sub:std_pert} ------------------------------- Let us begin by the calculation of the [self-energy]{}in the usual case where one starts from the free ground-state. The free propagators are simply given by: $$\begin{aligned} G^{(0)}_{{{\mathrm R}},I}(k,\omega)=\hspace{5.8cm}\nonumber\\ \label{eq:free_propR} \frac{1}{\omega-\left[\mu^{(0)}+v_{{\mathrm F}}^{(0)}(k-k_{{{\mathrm F}},I}^{(0)})\right]+{{\mathrm i}}\eta{\mathrm{sgn}}(k-k_{{{\mathrm F}},I}^{(0)})},\hspace{0.4cm}\\ G^{(0)}_{{{\mathrm L}},I}(k,\omega)=\hspace{5.8cm}\nonumber\\ \label{eq:free_propL} \frac{1}{\omega-\left[\mu^{(0)}-v_{{\mathrm F}}^{(0)}(k+k_{{{\mathrm F}},I}^{(0)})\right]-{{\mathrm i}}\eta{\mathrm{sgn}}(k+k_{{{\mathrm F}},I}^{(0)})}.\hspace{0.4cm}\end{aligned}$$ We will restrict ourselves to the study of the right propagators, because the left ones can be analyzed in an analogous way. The first order correction to the right propagators is given by the tadpole graph represented on Fig. \[fig:tadpole\_spinless\], where the solid (respectively dashed) lines represent right (respectively left) propagators, and where the black dot denotes one of the couplings. The [self-energy]{}for the (R,0) fermions is given by two terms: either the interaction is $A$, in which case the left propagator in the loop is on branch 0, or the interaction is $C$, and the left propagator is on branch $\pi$. It is a simple matter to evaluate the tadpole, and to show that in the thermodynamic limit the [self-energy]{}given by the $A$ interaction reads: $$\Sigma_{{{\mathrm R}},0;A}^{(1)}=A \int\frac{{{\mathrm d}}q}{2\pi} n_{{{\mathrm L}},0}^{(0)}(q),$$ where $n_{{{\mathrm L}},0}^{(0)}(q)$ is the particle distribution on branch (L,0), [i.e.]{}it is 1 if $q\geqslant -k_{{{\mathrm F}},0}$ and 0 otherwise. Of course, exactly as in the energy minimization scheme described in Sec. \[sec:sub:2chaines\_ordre1\], we get infinite results because our linearized dispersion relations have been extended to include infinitely many states. We will thus here too regularize these divergences by putting an ultra-violet cut-off $\Lambda_0$ on the momenta, around the four [*free*]{} Fermi momenta (remember Fig. \[fig:cut-off\] for one band). It is then easy to show that $\Sigma_{{{\mathrm R}},0;A}^{(1)}=A\Lambda_0/(2\pi)$. We let the reader check that the final results for the [self-energies]{}of right fermions are: $$\begin{aligned} \Sigma_{{{\mathrm R}},0}^{(1)}(k,\omega)&=&(A+C)\frac{\Lambda_0}{2\pi},\\ \Sigma_{{{\mathrm R}},\pi}^{(1)}(k,\omega)&=&(B+C)\frac{\Lambda_0}{2\pi}.\end{aligned}$$ The renormalized chemical potential $\mu$ and Fermi momenta $k_{{{\mathrm F}},0(\pi)}$ can now be deduced from the condition that the inverse propagators vanishes for $\omega=\mu$ and $k=k_{{{\mathrm F}},0}$ or $k=k_{{{\mathrm F}},\pi}$, and from the conservation of the number of particles. This last condition is nothing but the Luttinger theorem. We thus have to solve for the following system of three equations for three unknown quantities: $$\begin{aligned} &&\mu-\left[\mu^{(0)}+v_{{\mathrm F}}^{(0)}(k_{{{\mathrm F}},0}-k_{{{\mathrm F}},0}^{(0)})\right]-(A+C)\frac{\Lambda_0}{2\pi}=0,\hspace{0.8cm}\\ &&\mu-\left[\mu^{(0)}+v_{{\mathrm F}}^{(0)}(k_{{{\mathrm F}},\pi}-k_{{{\mathrm F}},\pi}^{(0)})\right]-(B+C)\frac{\Lambda_0}{2\pi}=0,\hspace{0.8cm}\\ &&k_{{{\mathrm F}},0}+k_{{{\mathrm F}},\pi}=k_{{{\mathrm F}},0}^{(0)}+k_{{{\mathrm F}},\pi}^{(0)}.\end{aligned}$$ The chemical potential is found by summing the first two equations and making use of the third one. Then one gets the difference between interacting and free Fermi momenta at one loop: $$\begin{aligned} \label{eq:dmu_faux} &&\mu^{(1)}-\mu^{(0)}=(A+B+2C)\frac{\Lambda_0}{4\pi},\\ \label{eq:dk0_faux} &&k_{{{\mathrm F}},0}^{(1)}-k_{{{\mathrm F}},0}^{(0)}=(B-A)\frac{\Lambda_0}{4\pi v_{{\mathrm F}}^{(0)}}.\end{aligned}$$ These are the results given by the standard perturbation theory. This last result is to be compared with (\[eq:dk0\_vrai\]). To first order in the couplings, both results are equal. But (\[eq:dk0\_vrai\]) contains next order contributions that are not present in (\[eq:dk0\_faux\]). This happens although both computations assume the same physics, namely the validity of the Hartree approximation. As has been shown in Appendix  \[sec:sub:sub:pert\_ex\], consistency between both viewpoints is recovered only if the electron [self-energy]{}is computed with free propagators corresponding to the [*dressed*]{} [Fermi surface]{}. A similar conclusion also holds for the chemical potential shift, as a comparison between Eqs. (\[eq:dmu\_vrai\]) and  (\[eq:dmu\_faux\]) readily shows. Formal calculation to second order ---------------------------------- Let us consider a system of interacting spinless Fermions in $d=3$ dimensions. In fact, the actual value of $d$ does not have much influence in the following discussion, the main point is that $d \geqslant 2$ so the Fermi surface is in general a smooth manifold of codimension one in [$\boldsymbol{k}$]{}-space. We take the following Hamiltonian: $$\begin{aligned} &&H = \int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k}}}}{(2\pi)^{3}}{\varepsilon}({\ensuremath{\boldsymbol{k}}}){c^\dagger}({\ensuremath{\boldsymbol{k}}})c({\ensuremath{\boldsymbol{k}}})\\ &&\hspace{0.8cm}+\frac{V}{2}\int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k}}}}{(2\pi)^{3}}\int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k'}}}}{(2\pi)^{3}} \int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{q}}}}{(2\pi)^{3}}f({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k'}}},{\ensuremath{\boldsymbol{q}}})\nonumber\\ &&\hspace{3cm}\times{c^\dagger}({\ensuremath{\boldsymbol{k}}}+{\ensuremath{\boldsymbol{q}}}){c^\dagger}({\ensuremath{\boldsymbol{k'}}}-{\ensuremath{\boldsymbol{q}}})c({\ensuremath{\boldsymbol{k'}}})c({\ensuremath{\boldsymbol{k}}}).\nonumber\end{aligned}$$ We assume the Fermi surface for $V=0$ is connected and that each half-line starting from the origin in [$\boldsymbol{k}$]{}-space intersects it only once. For any unit vector [$\boldsymbol{u}$]{}, we thus define a positive number $k_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})$ such that $k_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}}){\ensuremath{\boldsymbol{u}}}$ belongs to the Fermi surface. The Fermi sea is then the set of [$\boldsymbol{k}$]{} points such that ${\ensuremath{\boldsymbol{k}}}= k{\ensuremath{\boldsymbol{u}}}$ with [$\boldsymbol{u}$]{} unit vector and $0 \leqslant k \leqslant k_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})$. The total particle number is assumed to be fixed, independently of the coupling strength $V$. We now consider eigenstates obtained by adiabatic switching of the interaction $V$ on free particle states with a deformed Fermi surface ${\ensuremath{\boldsymbol{u}}}\mapsto k_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})$. Denoting $k_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})- k_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})=\delta k_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})$, the constraint on the total particle number reads: $$\begin{aligned} &&\int {{\mathrm d}}^{2}{\ensuremath{\boldsymbol{u}}}\Big[k^{2}_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})\delta k_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})\\ &&\hspace{2cm}+k_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})\delta k^{2}_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})+\frac{1}{3}\delta k^{3}_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})\Big]=0,\nonumber\end{aligned}$$ where ${{\mathrm d}}^{2}{\ensuremath{\boldsymbol{u}}}$ is the usual area element on the unit sphere, for instance ${{\mathrm d}}^{2}{\ensuremath{\boldsymbol{u}}}=\sin\theta\, {{\mathrm d}}\theta {{\mathrm d}}\phi$ in spherical coordinates. We now wish to choose $\delta k_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})$ in order to minimize the total energy of the corresponding eigenstate, while keeping a constant particle number. To second order in $V$, this total energy is given by: $$\begin{aligned} &&E(\{k_{{{\mathrm F}}}\}) = \int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k}}}}{(2\pi)^{3}}n({\ensuremath{\boldsymbol{k}}}){\varepsilon}({\ensuremath{\boldsymbol{k}}})\\ &&\hspace{0.5cm}+ \frac{V}{2}\int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k}}}}{(2\pi)^{3}}\int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k'}}}}{(2\pi)^{3}} n({\ensuremath{\boldsymbol{k}}})n({\ensuremath{\boldsymbol{k'}}})g({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k'}}},0)\nonumber\\ && \hspace{0.5cm}+ \frac{V^{2}}{4}\int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k}}}}{(2\pi)^{3}} \int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k'}}}}{(2\pi)^{3}}\int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{q}}}}{(2\pi)^{3}} g({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k'}}},{\ensuremath{\boldsymbol{q}}})^{2}\nonumber\\ &&\hspace{1.5cm}\times\frac{n({\ensuremath{\boldsymbol{k}}})n({\ensuremath{\boldsymbol{k'}}})\big(1-n({\ensuremath{\boldsymbol{k}}}+{\ensuremath{\boldsymbol{q}}})\big)\big(1-n({\ensuremath{\boldsymbol{k'}}}-{\ensuremath{\boldsymbol{q}}})\big)} {{\varepsilon}({\ensuremath{\boldsymbol{k}}})+{\varepsilon}({\ensuremath{\boldsymbol{k'}}})-{\varepsilon}({\ensuremath{\boldsymbol{k}}}+{\ensuremath{\boldsymbol{q}}})-{\varepsilon}({\ensuremath{\boldsymbol{k'}}}-{\ensuremath{\boldsymbol{q}}})} \nonumber\\ &&\hspace{0.5cm}+\mathcal{O}(V^{3}).\nonumber\end{aligned}$$ Here, $n(k{\ensuremath{\boldsymbol{u}}})=1$ if $k$ is smaller than $k_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})$ and $n(k{\ensuremath{\boldsymbol{u}}})=0$ otherwise. We have also defined $g({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k'}}},{\ensuremath{\boldsymbol{q}}})\equiv f({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k'}}},{\ensuremath{\boldsymbol{q}}})-f({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k'}}},{\ensuremath{\boldsymbol{k'}}}-{\ensuremath{\boldsymbol{k}}}-{\ensuremath{\boldsymbol{q}}})$. After some algebra, we find that $\delta k_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})=V\delta k_{{{\mathrm F}},1}({\ensuremath{\boldsymbol{u}}})+V^{2}\delta k_{{{\mathrm F}},2}({\ensuremath{\boldsymbol{u}}})+ \mathcal{O}(V^{3})$, and $\mu = \mu_{0}+ V\mu_{1}+V^{2}\mu_{2}+\mathcal{O}(V^{3})$, where $\delta k_{{{\mathrm F}},1}({\ensuremath{\boldsymbol{u}}})$, $\delta k_{{{\mathrm F}},2}({\ensuremath{\boldsymbol{u}}})$, $\mu_{1}$ and $\mu_{2}$ are given by the following coupled linear equations: $$v_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})\delta k_{{{\mathrm F}},1}({\ensuremath{\boldsymbol{u}}})+\Sigma^{(1)}\big({\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}})\big)-\mu_{1} = 0,$$ $$\int {{\mathrm d}}^{2}{\ensuremath{\boldsymbol{u}}}k^{2}_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})\delta k_{{{\mathrm F}},1}({\ensuremath{\boldsymbol{u}}}) = 0,$$ $$\begin{aligned} &&v_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})\delta k_{{{\mathrm F}},2}({\ensuremath{\boldsymbol{u}}})+\Re\Sigma^{(2)}\Big({\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}}),{\varepsilon}\big({\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}})\big)\Big)\nonumber\\ &&\hspace{0.3cm}+{\ensuremath{\boldsymbol{u}}}\cdot\nabla_{{\ensuremath{\boldsymbol{k}}}}\Sigma^{(1)}\big({\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}})\big)+\frac{1}{2}v_{{{\mathrm F}}}'({\ensuremath{\boldsymbol{u}}}) \delta k^{2}_{{{\mathrm F}},1}({\ensuremath{\boldsymbol{u}}})\\ &&\hspace{0.6cm}+\Delta\Sigma^{(1)}\big({\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}})\big)-\mu_{2} = 0, \mbox{ and}\nonumber\end{aligned}$$ $$\int {{\mathrm d}}^{2}{\ensuremath{\boldsymbol{u}}}\Big[k^{2}_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})\delta k_{{{\mathrm F}},2}({\ensuremath{\boldsymbol{u}}})+ k_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})\delta k^{2}_{{{\mathrm F}},1}({\ensuremath{\boldsymbol{u}}})\Big] = 0.$$ In these expressions, $v_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})$ and $v_{{{\mathrm F}}}'({\ensuremath{\boldsymbol{u}}})$ denote the first and second derivatives of the function $x \mapsto {\varepsilon}\big(\big(k_{{{\mathrm F}},0}({\ensuremath{\boldsymbol{u}}})+x\big){\ensuremath{\boldsymbol{u}}}\big)$, taken at $x=0$. $\Sigma^{(1)}({\ensuremath{\boldsymbol{k}}})$ and $\Sigma^{(2)}({\ensuremath{\boldsymbol{k}}},\omega)$ are the self-energies computed to first and second order in $V$, using the standard algorithm: $$\Sigma^{(1)}({\ensuremath{\boldsymbol{k}}}) = \int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k'}}}}{(2\pi)^{3}} n^{(0)}({\ensuremath{\boldsymbol{k'}}}) g({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k'}}},0),\mbox{ and}$$ $$\begin{aligned} &&\Re\Sigma^{(2)}({\ensuremath{\boldsymbol{k}}},\omega) = \frac{1}{2}\int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k}}}}{(2\pi)^{3}} \int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{k'}}}}{(2\pi)^{3}} \int \frac{{{\mathrm d}}^{3}{\ensuremath{\boldsymbol{q}}}}{(2\pi)^{3}} \;g({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k'}}},{\ensuremath{\boldsymbol{q}}})^{2}\\ &&\hspace{4cm}\times\frac{n^{(0)}({\ensuremath{\boldsymbol{k'}}})\big(1-n^{(0)}({\ensuremath{\boldsymbol{k}}}+{\ensuremath{\boldsymbol{q}}})\big)\big(1-n^{(0)}({\ensuremath{\boldsymbol{k'}}}-{\ensuremath{\boldsymbol{q}}})\big)+ \big(1-n^{(0)}({\ensuremath{\boldsymbol{k'}}})\big)n^{(0)}({\ensuremath{\boldsymbol{k}}}+{\ensuremath{\boldsymbol{q}}})n^{(0)}({\ensuremath{\boldsymbol{k'}}}-{\ensuremath{\boldsymbol{q}}})} {\omega+{\varepsilon}({\ensuremath{\boldsymbol{k'}}})-{\varepsilon}({\ensuremath{\boldsymbol{k}}}+{\ensuremath{\boldsymbol{q}}})-{\varepsilon}({\ensuremath{\boldsymbol{k'}}}-{\ensuremath{\boldsymbol{q}}})}.\nonumber \end{aligned}$$ The most interesting quantity in these formulae is $\Delta\Sigma^{(1)}({\ensuremath{\boldsymbol{k}}})$. It is the change in the first-order (with respect to $V$) self-energy due to the fact that the Fermi surface has changed by an amount $\delta k_{{{\mathrm F}},1}$. More precisely, we have: $$\begin{aligned} &&\Delta\Sigma^{(1)}({\ensuremath{\boldsymbol{k}}})= \nonumber\\ &&\hspace{0.5cm}\int \frac{{{\mathrm d}}^{2}{\ensuremath{\boldsymbol{u'}}}}{(2\pi)^{3}} g\big({\ensuremath{\boldsymbol{k}}},{\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u'}}}),0\big) k_{{{\mathrm F}},0}^{2}({\ensuremath{\boldsymbol{u'}}})\delta k_{{{\mathrm F}},1}({\ensuremath{\boldsymbol{u'}}}).\hspace{1cm}\end{aligned}$$ It turns out this term is [*not*]{} recovered in the naive perturbation algorithm. This latter procedure is based on solving for $k_{{{\mathrm F}}}({\ensuremath{\boldsymbol{u}}})$ in the equations: $${\varepsilon}\big({\ensuremath{\boldsymbol{k}_{{{\mathrm F}}}}}({\ensuremath{\boldsymbol{u}}})\big)+\Re\Sigma\big({\ensuremath{\boldsymbol{k}_{{{\mathrm F}}}}}({\ensuremath{\boldsymbol{u}}}),\mu\big) = \mu,$$ where $\Sigma({\ensuremath{\boldsymbol{k}}},\omega)$ is computed with the free propagators associated to the non-interacting Fermi surface. As before, $\mu$ is chosen to keep a constant total particle number. This second approach yields the same set of equations as before, except that the term $\Delta\Sigma^{(1)}({\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}}))$ is missing in the first equation for $\delta k_{{{\mathrm F}},2}({\ensuremath{\boldsymbol{u}}})$. This shows that the naive algorithm is not able to keep track of the first-order Fermi surface deformation while evaluating the Hartree-Fock corrections to second order. Intuitively, these effects are expected to be associated to the four second order graphs for $\Sigma$ shown on Fig. \[fig:KL\_diag\]. ![The four anomalous Kohn-Luttinger graphs contributing to the [self-energy]{}at two loops.[]{data-label="fig:KL_diag"}](KL_diag){width="7cm"} However, these graphs are anomalous according to Kohn and Luttinger, and their contribution vanishes in the $T=0$ perturbation theory scheme. We believe this illustrates the crucial problem with naive perturbation theory. As we go to higher orders in $V$, lower order graphs for $\Sigma$ are modified by the changes already induced on the occupation numbers of single particle states. One would expect to capture these changes thanks to self-energy insertions in the internal lines of the lower-order graphs for $\Sigma$. But the example of anomalous graphs shows this does not work so well in general. As discussed in section \[sec:sub:use\_ct\] the natural cure for this problem is to fix the dressed Fermi surface, thanks to counterterms which gradually modify the single particle dispersion of the original Hamiltonian, as $V$ is increased. In this approach, we choose the dressed Fermi surface ${\ensuremath{\boldsymbol{u}}}\mapsto {\ensuremath{\boldsymbol{k}_{{{\mathrm F}}}}}({\ensuremath{\boldsymbol{u}}})$ and the dressed dispersion relation ${\varepsilon}({\ensuremath{\boldsymbol{k}}})$. We therefore compute the self-energy $\widetilde\Sigma({\ensuremath{\boldsymbol{k}}},\omega)$ with respect to this dressed Fermi surface. Denoting by $\mu={\varepsilon}({\ensuremath{\boldsymbol{k}_{{{\mathrm F}}}}}({\ensuremath{\boldsymbol{u}}}))$ the dressed chemical potential, the counterterms $\Sigma_{CT}({\ensuremath{\boldsymbol{u}}})$ are defined by: $$\widetilde\Sigma({\ensuremath{\boldsymbol{k}_{{{\mathrm F}}}}}({\ensuremath{\boldsymbol{u}}}),\mu)+\Sigma_{CT}({\ensuremath{\boldsymbol{u}}}) = 0.$$ In this third approach, the “bare” Fermi surface ${\ensuremath{\boldsymbol{u}}}\mapsto {\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}})$ becomes a function of $V$ and $\{k_{{{\mathrm F}}}\}$. It is obtained from: $${\varepsilon}({\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}}))+\Sigma_{CT}({\ensuremath{\boldsymbol{u}}}) = \mu_{0},$$ where, as always, $\mu_{0}$ is chosen in order to conserve the total particle number. We therefore have to solve: $${\varepsilon}({\ensuremath{\boldsymbol{k}_{{{\mathrm F}}}}}({\ensuremath{\boldsymbol{u}}}))-{\varepsilon}({\ensuremath{\boldsymbol{k}_{{{\mathrm F}},0}}}({\ensuremath{\boldsymbol{u}}}))+\widetilde\Sigma({\ensuremath{\boldsymbol{k}_{{{\mathrm F}}}}}({\ensuremath{\boldsymbol{u}}}),\mu) = \mu-\mu_{0}.$$ Since $\widetilde\Sigma$ is computed with free propagators whose singularities lie on the dressed Fermi surface, it is easy to check that this yields the same expressions for $\delta k_{{{\mathrm F}},1}$ and $\delta k_{{{\mathrm F}},2}$ as the energy minimizing procedure, in complete agreement with the general conclusions of Appendix \[app:equi\]. Difficulties with the imaginary part of $\Sigma$ in the traditional perturbation scheme {#app:diff_im} ======================================================================================= Let us consider the traditional perturbation scheme around the unperturbed Fermi surface. With the same notation as before, this corresponds to the choice of free propagator: $G^{(0)}(k,\omega)^{-1}=\omega- {\varepsilon}_{0}(k)+ {{\mathrm i}}\eta {\mathrm{sgn}}({\varepsilon}_{0}(k)-\mu_{0})$. In the discussion, we shall use the spectral densities $\rho_{p,h}(k,\omega)$ for excited states involving $p$ particles and $h$ holes, with a total momentum $k$ and a total energy $\omega$. We have: $$\begin{aligned} &&\rho_{p,h}(k,\omega)=\prod_{i=1}^{p}\int \frac{{{\mathrm d}}k_{i}}{(2\pi)^d}\theta ({\varepsilon}_{0}(k_{i})-\mu_{0})\prod_{j=1}^{h}\int \frac{{{\mathrm d}}k_{j}'}{(2\pi)^d}\theta (\mu_{0}-{\varepsilon}_{0}(k_{j}'))\\ &&\hspace{6cm}(2\pi)^{d}\delta(k-\sum_{i=1}^{p}k_{i} +\sum_{j=1}^{h}k_{j}') \times(2\pi)\delta(\omega-\sum_{i=1}^{p}{\varepsilon}_{0}(k_{i}) +\sum_{j=1}^{h}{\varepsilon}_{0}(k_{j}')).\nonumber \end{aligned}$$ Let us consider the simple second order diagram for $\Sigma(k,\omega)$ shown on Fig. \[fig:sunrise\_spinless\] (the sunrise diagram). It is simple to check that the imaginary part of this diagram is proportional to $\rho_{2,1}(k,\omega)-\rho_{1,2}(-k,-\omega)$. As is well known since Landau, this quantity vanishes at the bare chemical potential $\mu_{0}$, and behaves as $(\omega-\mu_{0})^{2}$ in magnitude for $\omega$ close to $\mu_{0}$. The effect of Fermi surface deformation on $\Sigma(k,\omega)$ arises via the replacement of bare propagators by sequences of these propagators separated by lower order self-energy insertions. The main point we wish to emphasize here is that there is no simple way to predict the influence of these insertions on the frequency dependence of $\Im \Sigma(k,\omega)$. For some graphs, and some patterns of insertions, the resulting $\Im \Sigma(k,\omega)$ will continue to vanish at $\omega=\mu_{0}$, whereas some other combinations will produce a finite contribution to $\Im \Sigma(k,\omega)$ at $\omega=\mu_{0}$. Therefore, the traditional scheme does not allow for a good control of the analytical structure of the self-energy. Let us show this on a typical example. The simplest interesting situation is obtained for the sunrise graph, for [self-energy]{}insertions which are assumed not to depend on $k$ nor on $\omega$, since it is then easy to perform the frequency integrals. In the more general case of an arbitrary frequency dependence for the insertions, the natural procedure would be to Taylor-expand them in the vicinity of $\mu_{0}$. The strongest effect is obtained for the constant term in these expansions, and this leads to our toy example. For a total number $n$ of constant insertions, we get a contribution to $\Im \Sigma(k,\omega)$ proportional to $\rho_{2,1}^{(n)}(k,\omega)-(-1)^{n}\rho_{1,2}^{(n)}(-k,-\omega)$, where $\rho_{p,h}^{(n)}(k,\omega)$ stands for the $n$-th partial derivative of $\rho_{p,h}(k,\omega)$ with respect to $\omega$. Using the fact that $\rho_{2,1}(k,\omega)$ and $\rho_{1,2}(-k,-\omega)$ behave as $(\omega-\mu_{0})^{2}$ for $\omega$ close to $\mu_{0}$, we notice that a single self-energy insertion in the sunrise graph preserves the property that $\Im \Sigma(k,\omega)$ vanishes for $\omega=\mu_{0}$. More generally, this implies that $\Im \Sigma(k,\omega=\mu_{0})=0$ up to third order in perturbation theory. However, a Fermi surface deformation already occurs usually for the simplest Hartree and Fock graphs, which are first order in the coupling strength. In the standard perturbation scheme, the dressed chemical potential and the dressed Fermi surface are determined by solving the infinite set of equations: $$\mu- {\varepsilon}_{0}(k_{F})-\Re \Sigma(k_{F},\mu)=0,$$ with the constraint that the total volume of the Fermi surface does not change as interactions are switched on. We therefore see that, in this scheme, there is no reason for which $\Im \Sigma(k_{{\mathrm F}},\omega=\mu)$ should vanish. This is rather unsatisfactory on physical grounds, since it would imply a finite life-time for particle-like excitations lying just on the dressed Fermi surface. Field-theoretical RG {#app:field_th} ==================== This Appendix is devoted to a detailed derivation of the field-theoretical RG equations which we have gathered in Appendix \[app:flow\_eq\]. The two main considerations we wish to stress here are: i) the choice of external momenta in the renormalization prescriptions, which has to be adapted to the [Fermi surface]{}shape, and ii) the use of the logarithmic approximation. Motivation and general idea {#sec:sub:sub:motivation_gal_idea} --------------------------- In the usual “field-theoretical” RG, the high-energy Hamiltonian is given and fixed, and one parametrizes the theory by low-energy values of proper Green’s functions (for instance the interaction vertices) at a typical scale $\nu$. Requiring that all these theories at different energy scales should correspond to one and the same high-energy theory yields RG flows, when $\nu$ is varied. This approach is physically natural, because it is based on the calculation of low-energy observables. Furthermore, as we will see, it allows for a study of crossovers between high and low-energy regimes. One of its limitations is that it requires renormalizable interactions ([i.e.]{}the existence of a continuous limit). But it is not really a severe drawback, since non-renormalizable interactions are expected to be irrelevant (by power counting) in the low-energy limit, which is the most interesting to us. Note that the renormalizability constraint disappears in RG schemes based on Wilson’s idea of gradual mode elimination. Several groups have recently implemented Wilson’s approach to the RG, expressed via the Polchinski equation,[@Zanchi96; @Halboth00] or its one-particle irreducible version.[@Jungnickel96; @Honerkamp01] Although these equations are exact, they are quite complicated, since effective interactions involving an arbitrary number of particles are generated along the RG flow. Any numerical computation therefore requires drastic truncations in the effective action. By contrast, the field-theory approach involves only a much smaller set of effective or running couplings, which is a good feature for practical implementations. Renormalization of the interactions {#sec:sub:sub:ren_int} ----------------------------------- First of all, we have to define the renormalized couplings. The two corresponding Green’s functions (in real space for the direction parallel to the chains) are: $$\begin{aligned} &&G^{I\!\!I}(X_4,X_3,X_2,X_1)=\nonumber\\ &&\hspace{1cm}-{\langle 0 |} {\mathrm{T}}\Big[ {\psi^{}}_{{{\mathrm R}},I+\delta,\tau}(X_4) {\psi^{}}_{{{\mathrm L}},J-\delta,\rho}(X_3)\\ &&\hspace{3cm}\times{\psi^\dagger}_{{{\mathrm L}},J,\rho'}(X_2) {\psi^\dagger}_{{{\mathrm R}},I,\tau'}(X_1)\Big] {| 0 \rangle},\nonumber\\ &&U^{I\!\!I}(X_4,X_3,X_2,X_1)=\nonumber\\ &&\hspace{1cm}-{\langle 0 |} {\mathrm{T}}\Big[ {\psi^{}}_{{{\mathrm R}},I+\delta,\tau}(X_4) {\psi^{}}_{{{\mathrm R}},J-\delta,\rho}(X_3)\\ &&\hspace{3cm}\times{\psi^\dagger}_{{{\mathrm L}},J,\rho'}(X_2) {\psi^\dagger}_{{{\mathrm L}},I,\tau'}(X_1)\Big] {| 0 \rangle}.\nonumber\end{aligned}$$ In the above equations, ${| 0 \rangle}$ is the interacting ground-state, T is the time ordering operator, and $X$ is a shorthand notation for $(t,x)$. The renormalized couplings are the values of the amputated one-particle irreducible parts of these Green’s functions, divided by ${{\mathrm i}}$. In fact, the charge and spin couplings are the coefficients obtained from the Fourier transform of $G^{I\!\!I}$, factor of $\mathbb{I}_{\tau,\tau'} \mathbb{I}_{\rho,\rho'}$ and $\boldsymbol{\sigma}_{\tau,\tau'} \cdot \boldsymbol{\sigma}_{\rho,\rho'}$ respectively. The Umklapp coupling $U_\delta(I,J)$ is defined in the same way from $U^{I\!\!I}$, as the coefficient of $\mathbb{I}_{\tau,\tau'} \mathbb{I}_{\rho,\rho'}$ (the one in front of $\mathbb{I}_{\rho,\tau'} \mathbb{I}_{\tau,\rho'}$ being $-U_{J-I-\delta}(I,J)$). The set of external frequencies is chosen to be the same for all types of couplings. We have decided to take: $$\label{eq:energies_pattes_externes} \omega_1={\frac{\nu}{2}},\quad \omega_2={\frac{\nu}{2}},\quad \omega_3={\frac{3\nu}{2}},$$ and by energy conservation we have of course $\omega_4=\omega_1+\omega_2-\omega_3=-{\frac{\nu}{2}}$. $\nu$ is the typical energy scale of the interaction process, and is the quantity to be varied to get RG flows. It is a bit more difficult to choose the external momenta, because of the warping of the [Fermi surface]{}. Our choice has been dictated by a few natural requirements. First the symmetries of the [Fermi surface]{}should be respected, as the right-left symmetry, the up-down symmetry ([i.e.]{}$k_y\leftrightarrow -k_y$, in terms of the original transverse momenta). Interactions processes for which it is possible to choose all external momenta on the [Fermi surface]{}, should be computed for this special choice, because it would otherwise mean the introduction of a spurious energy scale. Let us first consider ${G^{{{\mathrm c}}({{\mathrm s}})}}_\delta(I,J)$. It is possible to choose $k_1=k_{{{\mathrm F}},I}$, $k_2=-k_{{{\mathrm F}},J}$, $k_3=-k_{{{\mathrm F}},J-\delta}$, and $k_4=k_{{{\mathrm F}},I+\delta}$, only if momentum conservation $k_{{{\mathrm F}},I}-k_{{{\mathrm F}},J}=k_{{{\mathrm F}},I+\delta}-k_{{{\mathrm F}},J-\delta}$ is respected. In general, this will not be possible, for we will have $\Delta k_\delta(I,J)=(k_{{{\mathrm F}},I+\delta}+k_{{{\mathrm F}},J})-(k_{{{\mathrm F}},I}+k_{{{\mathrm F}},J-\delta})\neq 0$. Notice that up to a minus sign and a factor of 2, $\Delta k_\delta(I,J)$ is simply the generalization of $\Delta k_{{\mathrm F}}$ in the two-chain model. It is then natural to split this quantity equally among the four momenta. One can check that the following choice fulfills all the conditions we have mentioned: $$\begin{aligned} \label{eq:choix_4impulsions} &&k_1=k_{{{\mathrm F}},I}+\frac{\Delta k_\delta(I,J)}{4},\nonumber\\ &&k_2=-\left(k_{{{\mathrm F}},J}-\frac{\Delta k_\delta(I,J)}{4}\right),\\ &&k_3=-\left(k_{{{\mathrm F}},J-\delta}+\frac{\Delta k_\delta(I,J)}{4}\right),\mbox{ and }\nonumber\\ &&k_4=k_{{{\mathrm F}},I+\delta}-\frac{\Delta k_\delta(I,J)}{4}.\nonumber\end{aligned}$$ For the Umklapps, the choice of external momenta is dictated by the same requirements. The equivalent of $\Delta k_\delta(I,J)$ is now $\Delta k^U_\delta(I,J)=2\pi-(k_{{{\mathrm F}},I}+k_{{{\mathrm F}},J}+k_{{{\mathrm F}},I+\delta}+k_{{{\mathrm F}},J-\delta})$, and the natural choice of momenta reads: $$\begin{aligned} \label{eq:choix_4impulsions_U} &&k_1=-\left(k_{{{\mathrm F}},I}+\frac{\Delta k^U_\delta(I,J)}{4}\right),\nonumber\\ &&k_2=-\left(k_{{{\mathrm F}},J}+\frac{\Delta k^U_\delta(I,J)}{4}\right),\\ &&k_3=k_{{{\mathrm F}},J-\delta}+\frac{\Delta k^U_\delta(I,J)}{4},\mbox{ and }\nonumber\\ &&k_4=k_{{{\mathrm F}},I+\delta}+\frac{\Delta k^U_\delta(I,J)}{4}.\nonumber\end{aligned}$$ The next task is to draw all possible Feynman diagrams, and compute them. One can then establish the field theoretical RG flows, requiring the couplings measured at two different scales should correspond to the same high-energy theory. The couplings’ flow equations are given in Appendix \[app:flow\_eq\_coup\], and are obtained in the one-loop approximation. As argued in Sec. \[sec:csfs\], the most interesting effects connected to [Fermi surface]{}deformation appear at the two-loop level for the single-electron propagator. For the sake of simplicity, we shall use a hybrid scheme, involving a one-loop approximation for the couplings and a two-loop approximation for the electronic [self-energy]{}. In the case where all the couplings remain weak, it is reasonable to keep only the dominant term in the corresponding flow equation. If by contrast couplings have a tendency to grow and become large at low energies, experience from the Kondo problem suggests adding the subleading terms to the couplings’ flow does not provide a better physical picture. For the Kondo problem, the two-loop approximation predicts an intermediate coupling fixed point,[@Fowler71; @Abrikosov70] whereas the low-energy physics corresponds to an infinite coupling fixed point.[@Anderson70; @Wilson75] We shall not give any technical detail on the derivation of these couplings’ flow equations which is standard.[@Solyom79_dans_articles] The main new feature is the use of special sets of external momenta, described in Eqs. (\[eq:choix\_4impulsions\]) and (\[eq:choix\_4impulsions\_U\]) above. However, it is worth focusing on the $f$ function that appears in these equations (the $f$ function is defined by $f(t=\ln(\Lambda_0/\nu),\delta)=1$ if $\nu\geqslant |\delta|$ and 0 otherwise, see Appendix \[app:flow\_eq\_coup\]). In fact, the “true” RG equations do not involve this function, but rather: $$\widetilde{f}(t=\ln(\Lambda_0/\nu),\delta)=\frac{1}{2}\left( \frac{\nu}{\nu-\delta+{{\mathrm i}}\eta} + \frac{\nu}{\nu+\delta-{{\mathrm i}}\eta} \right).$$ It is obvious that these $\widetilde{f}$ diverge for $\nu=|\delta |$, so that some couplings will diverge or vanish singularly at the scales given in (\[eq:car\_length\_1\]) to (\[eq:car\_length\_9\]). Though this physically signals the crossing of the characteristic scales, it is practically unpleasant for the numerical simulations. Furthermore, if we did not work at zero temperature, the energy scale given by the temperature $T$ would suppress these divergences. Notice that the presence of ${{\mathrm i}}\eta$ factors also implies that the couplings will not remain real. For all these reasons, it is thus natural to try to find a way to get rid of these singular behaviors. This can be achieved by replacing $\widetilde{f}$ by a function that extends its asymptotic behaviors (for $\nu\gg|\delta|$ and $\nu\ll|\delta|$) up to $\nu=|\delta|$. This is exactly what the function $f$ does. The quality of the approximation can be checked on a very simple flow equation, for which one knows the exact solution: $$\label{eq:test_1coup} \partial_\nu g(\nu)=\frac{1}{2}\left( \frac{1}{\nu-\delta+{{\mathrm i}}\eta} + \frac{1}{\nu+\delta-{{\mathrm i}}\eta} \right) g^2(\nu).$$ This is simply a RPA-like flow, for only one coupling $g$. We will not study this in detail, but we show the good quality of the approximate solution for a positive initial coupling, on Fig. \[fig:sol\_div\_pos\_regularisee\] ![The dashed curve is the real part of the coupling $g$ satisfying (\[eq:test\_1coup\]), with initial condition $g(0)=0.1$. Here $\delta/\Lambda_0=10^{-2}$, so that the coupling goes to zero singularly for $t\simeq 4.6$. The solid line represents the solution of the approximate equation where $\widetilde{f}$ is replaced by $f$. The inserted plot is a zoom on the end of the flows.[]{data-label="fig:sol_div_pos_regularisee"}](sol_div_pos_regularisee){width="8cm"} The approximation we use is in fact nothing but a logarithmic approximation. Indeed, $\widetilde{f}$ appears in the flow equations, after we have differentiated $\ln[(\nu-\delta+{{\mathrm i}}\eta)(\nu+\delta-{{\mathrm i}}\eta)/(2v_{{\mathrm F}}\Lambda_0)^2]$ factors (with respect to the scale $\nu$), coming from the Feynman graphs’ logarithmic divergences. Changing $\widetilde{f}$ in $f$ just amounts to replacing this logarithm by the approximation $2\ln[\mathrm{Max}(\nu,|\delta|)/(2v_{{\mathrm F}}\Lambda_0)]$. Renormalization of the propagator --------------------------------- For reasons explained at the end of Sec. \[sec:sub:sub:gen\_th\_an\], the one-loop [self-energy]{}correction does not have much influence on the [Fermi surface]{}deformation for the quasi 1D Hubbard systems considered in this paper. Therefore, we will only focus on the two-loop sunrise diagram (the Kohn-Luttinger diagram being equal to zero). We only show the Feynman diagram here, with all the information about the internal lines, for the “$G^2$” contribution, on Fig. \[fig:se\_gg\_sunrise\]. ![Sunrise diagram involving two $G$ interactions.[]{data-label="fig:se_gg_sunrise"}](se_gg_sunrise){width="8cm"} There is of course a “$U^2$” contribution, and the spin algebra has to be taken into account. In order to simplify the expressions, we have computed the [self-energy]{}at $k=k_{{{\mathrm F}},I}$, and not for a general momentum, since we have decided not to take Fermi velocity renormalization into account. We give the expression of the [self-energy]{}in Appendix \[app:flow\_eq\_prop\]. From this, the first thing to do is to compute the counterterms, so that the inverse propagators vanish on the dressed [Fermi surface]{}. Focusing on second order terms only, this amounts to require: $$\begin{aligned} &&\forall I\in\{1,\ldots,N\},\nonumber\\ &&-\Sigma_{{{\mathrm R}},I}^{(2)}(k=k_{{{\mathrm F}},I},\omega=\mu)-\delta\mu^{(2)}+v_{{\mathrm F}}\delta k_I^{(2)}=0,\hspace{0.8cm}\end{aligned}$$ together with the conservation of the total particle number. $\delta\mu^{(2)}$ is found by summing all these equations: $\delta\mu^{(2)}=-\left[\sum_I \Sigma_{{{\mathrm R}},I}^{(2)}(k=k_{{{\mathrm F}},I},\omega=\mu)\right]/N$. We let the reader check that it is in fact sufficient to take the Umklapp contribution in this last equation, because the $G$ contribution sums to zero. This is true because of the following properties: $\Delta k_\alpha(I,J)=-\Delta k_{-\alpha}(J,I)$, ${G^{{{\mathrm c}}({{\mathrm s}})}}_\alpha(I,J)={G^{{{\mathrm c}}({{\mathrm s}})}}_{-\alpha}(J,I)$ and ${G^{{{\mathrm c}}({{\mathrm s}})}}_{-\alpha}(I+\alpha,J-\alpha)={G^{{{\mathrm c}}({{\mathrm s}})}}_\alpha(J-\alpha,I+\alpha)$, so that in the sum over $I$, $J$ and $\alpha$, the terms $(I,J,\alpha)$ and $(J,I,-\alpha)$ will cancel each other. Once the chemical potential is known, the Fermi momenta counterterms can be found: $$\begin{aligned} \label{eq:fs_counterterm} &&\delta k_I^{(2)}=\frac{1}{v_{{\mathrm F}}} \Big[ \Sigma_{{{\mathrm R}},I}^{(2)}(k=k_{{{\mathrm F}},I},\omega=\mu)\\ &&\hspace{2.5cm}-\frac{1}{N}\sum_I \Sigma_{{{\mathrm R}},I}^{(2)}(k=k_{{{\mathrm F}},I},\omega=\mu) \Big].\nonumber\end{aligned}$$ In order to save space, we will not give the full expressions of these counterterms. From all this, we can deduce a dressed propagator, as we did in Eq. (\[eq:dressed\_prop\_2c\]) for the two-chain model. As in Eq. (\[eq:dressed\_prop\_2c\]), the result is still divergent when $\Lambda_0$ is sent to infinity. This simply means that the counterterms are not sufficient. Something more is needed, and as is well known, this is wave function renormalization. The renormalized (R,I) propagator is defined as usual by: $${G_{{{\mathrm R}},I}^{(\mathcal{R})}}^{-1}(k,\omega)=Z_{{{\mathrm R}},I} G^{-1}_{{{\mathrm R}},I}(k,\omega),$$ and the wave function renormalization factor $Z_{{{\mathrm R}},I}$ is found by imposing the following renormalization prescription: $$\begin{aligned} &&{G_{{{\mathrm R}},I}^{(\mathcal{R})}}^{-1}(k=k_{{{\mathrm F}},I},\omega=\mu+\nu)=\nonumber\\ \label{eq:prescr_renorm_prop} &&\hspace{1cm}Z_{{{\mathrm R}},I} G^{-1}_{{{\mathrm R}},I}(k=k_{{{\mathrm F}},I},\omega=\mu+\nu)=\nu.\hspace{0.8cm}\end{aligned}$$ We have not written which variables $Z_{{{\mathrm R}},I}$ depends on, in order to make the equations lighter, but it should be clear that it is a function of the couplings at scale $\nu$, of the dressed Fermi momenta, of $\nu$ and of $\Lambda_0$. The calculation of the renormalized propagator is achieved thanks to the standard observation that Eq. (\[eq:prescr\_renorm\_prop\]) implies: $$\begin{aligned} \label{eq:lien_prop_2echelles} &&{G_{{{\mathrm R}},I}^{(\mathcal{R})}}^{-1}(k,\omega;g;\nu,\Lambda_0)=\\ &&\hspace{1cm}{\varphi}_{{{\mathrm R}},I}(g;\nu,\nu',\Lambda_0)\, {G_{{{\mathrm R}},I}^{(\mathcal{R})}}^{-1}(k,\omega;{g}';\nu',\Lambda_0), \mbox{ with }\nonumber\\ &&{\varphi}_{{{\mathrm R}},I}(g;\nu,\nu',\Lambda_0)=\frac{Z_{{{\mathrm R}},I}(g;\nu,\Lambda_0)}{Z_{{{\mathrm R}},I}({g}';\nu',\Lambda_0)}\nonumber\\ \label{eq:definition_phi} &&\hspace{2.4cm}=\frac{Z_{{{\mathrm R}},I}(g;\nu,\Lambda_0)}{Z_{{{\mathrm R}},I}(\overline{g}(g;\nu,\nu',\Lambda_0);\nu',\Lambda_0)}.\end{aligned}$$ In these equations, $g$ is a shorthand notation for all the couplings, at scale $\nu$, and $g'$ is the same at scale $\nu'$. In the last equation, the function $\overline{g}$ is relating the value of the couplings at two different scales by $g'=\overline{g}(g;\nu,\nu',\Lambda_0)$. Finally, the RG flow equations for the ${\varphi}$ functions are found by differentiating the multiplicative relation ${\varphi}_{{{\mathrm R}},I}(g;\nu,\nu'',\Lambda_0)={\varphi}_{{{\mathrm R}},I}(g;\nu,\nu',\Lambda_0) {\varphi}_{{{\mathrm R}},I}({g}';\nu',\nu'',\Lambda_0)$, and one obtains: $$\begin{aligned} \label{eq:eq_flot_phi_gale} &&\frac{\partial}{\partial \nu'}{\varphi}_{{{\mathrm R}},I}(g;\nu,\nu',\Lambda_0)={\varphi}_{{{\mathrm R}},I}(g;\nu,\nu',\Lambda_0)\\ &&\hspace{1cm}\times\left.\left[ \frac{\partial}{\partial \nu''} {\varphi}_{{{\mathrm R}},I}(\overline{g}(g^\beta;\nu,\nu',\Lambda_0);\nu',\nu'',\Lambda_0)\right]\right|_{\nu''=\nu'}.\nonumber\end{aligned}$$ Notice that the importance of the ${\varphi}$ functions lay in the close link between these and the renormalized propagator, that one can deduce from Eq.(\[eq:lien\_prop\_2echelles\]) and the renormalization prescription of Eq. (\[eq:prescr\_renorm\_prop\]): $$\label{eq:expression_prop_renorm} {G_{{{\mathrm R}},{{\mathrm L}}}^{(\mathcal{R})}}^{-1}(k=k_{{\mathrm F}},\omega;g^\alpha;\nu,\Lambda_0)=\omega\, {\varphi}_{{{\mathrm R}},{{\mathrm L}}}(g^\alpha;\nu,\omega,\Lambda_0).$$ For the $N$ chains we are interested in, the flow equations of ${\varphi}_{{{\mathrm R}},I}$ are given in Appendix \[app:flow\_eq\_prop\], Eq. (\[eq:flot\_phi\]). The flow equations of ${\varphi}_{{{\mathrm L}},I}$ can be checked to be exactly the same (this is due to the Left-Right symmetry of the system we study). RG flow equations {#app:flow_eq} ================= Flows of the couplings {#app:flow_eq_coup} ---------------------- The flow equations for the couplings that are given below are the field-theoretical RG equations, obtained after the general analysis of Appendix \[app:field\_th\] has been performed. In these, the RG time $t$ is given by $t=\ln(\Lambda_0/\nu)$ ($\Lambda_0$ being the ultra-violet cut-off and $\nu$ the typical energy scale of the interaction), and the running couplings are low-energy interactions $G(\nu)$. The high-energy flow equations of the couplings in the cut-off scaling scheme are easily deduced from the ones in the field theoretical version. They are in fact the same, with the sole difference that the $f$ functions appearing in the flows are to be replaced by 1 (which is the value these $f$ functions take at high energies). Notice however that the quantities entering the flow equations now have different physical meanings. The cut-off scaling time is $t=\ln(\Lambda_0/\Lambda)$ ($\Lambda$ being the running cut-off), and the couplings are running bare couplings $G_{{\mathrm B}}(\Lambda)$. The field theoretical RG flow equations for the charge, spin and Umklapp couplings are given below: $$\begin{aligned} &&\partial_t {G^{{\mathrm c}}}_\delta(I,J)=\frac{1}{N}\sum_\alpha\nonumber\\ &&\hspace{0.5cm}\Biggl\lbrace f\left(t,2v_{{\mathrm F}}K^{\mathrm{ph}}_{\alpha;\delta}(I,J)\right)\Big[{G^{{\mathrm c}}}_\alpha(I,J+\alpha-\delta) {G^{{\mathrm c}}}_{\delta-\alpha}(I+\alpha,J)+3 {G^{{\mathrm s}}}_\alpha(I,J+\alpha-\delta) {G^{{\mathrm s}}}_{\delta-\alpha}(I+\alpha,J)\Big]\nonumber\\ \label{eq:eqRGNchaines_c} &&\hspace{0.5cm}-f\left(t,2v_{{\mathrm F}}K^{\mathrm{pp}}_{\alpha;\delta}(I,J)\right)\Big[ {G^{{\mathrm c}}}_\alpha(I,J) {G^{{\mathrm c}}}_{\delta-\alpha}(I+\alpha,J-\alpha)-3 {G^{{\mathrm s}}}_\alpha(I,J) {G^{{\mathrm s}}}_{\delta-\alpha}(I+\alpha,J-\alpha)\Big]\\ &&\hspace{0.5cm}+f\left(t,2v_{{\mathrm F}}K^{UU}_{\alpha;\delta}(I,J)\right)\Big[ U_\alpha(I,J+\alpha-\delta) U_{\delta-\alpha}(I+\alpha,J)+ U_{J-I-\delta}(I,J+\alpha-\delta) U_{J-I-\delta}(I+\alpha,J) \nonumber\\ &&\hspace{4cm}-\frac{1}{2} U_{\delta-\alpha}(I+\alpha,J) U_{J-I-\delta}(I,J+\alpha-\delta)-\frac{1}{2} U_{J-I-\delta}(I+\alpha,J) U_\alpha(I,J+\alpha-\delta) \Big] \Biggr\rbrace.\nonumber \end{aligned}$$ $$\begin{aligned} &&\partial_t {G^{{\mathrm s}}}_\delta(I,J)=\frac{1}{N}\sum_\alpha\nonumber\\ &&\hspace{0.5cm}\Biggl\lbrace f\left(t,2v_{{\mathrm F}}K^{\mathrm{ph}}_{\alpha;\delta}(I,J)\right)\Big[2 {G^{{\mathrm s}}}_\alpha(I,J+\alpha-\delta) {G^{{\mathrm s}}}_{\delta-\alpha}(I+\alpha,J)\nonumber+ {G^{{\mathrm s}}}_\alpha(I,J+\alpha-\delta) {G^{{\mathrm c}}}_{\delta-\alpha}(I+\alpha,J) \nonumber\\ &&\hspace{4cm}+{G^{{\mathrm c}}}_\alpha(I,J+\alpha-\delta) {G^{{\mathrm s}}}_{\delta-\alpha}(I+\alpha,J)\Big]\nonumber\\ \label{eq:eqRGNchaines_s} &&\hspace{0.5cm}+f\left(t,2v_{{\mathrm F}}K^{\mathrm{pp}}_{\alpha;\delta}(I,J)\right)\Big[2 {G^{{\mathrm s}}}_\alpha(I,J) {G^{{\mathrm s}}}_{\delta-\alpha}(I+\alpha,J-\alpha) - {G^{{\mathrm s}}}_\alpha(I,J) {G^{{\mathrm c}}}_{\delta-\alpha}(I+\alpha,J-\alpha)\\ &&\hspace{4cm}-{G^{{\mathrm c}}}_\alpha(I,J) {G^{{\mathrm s}}}_{\delta-\alpha}(I+\alpha,J-\alpha)\Big]\nonumber\\ &&\hspace{0.5cm}+f\left(t,2v_{{\mathrm F}}K^{UU}_{\alpha;\delta}(I,J)\right)\Big[ U_{J-I-\delta}(I,J+\alpha-\delta) U_{J-I-\delta}(I+\alpha,J)-\frac{1}{2} U_{\delta-\alpha}(I+\alpha,J) U_{J-I-\delta}(I,J+\alpha-\delta)\nonumber\\ &&\hspace{4cm} -\frac{1}{2} U_{J-I-\delta}(I+\alpha,J) U_\alpha(I,J+\alpha-\delta) \Big] \Biggr\rbrace.\nonumber \end{aligned}$$ $$\begin{aligned} &&\partial_t U_\delta(I,J) = \frac{1}{N}\sum_\alpha\nonumber\\ &&\hspace{0.5cm}\Biggl\lbrace f\left(t,2v_{{\mathrm F}}K^{GU1}_{\alpha;\delta}(I,J)\right)\big[ {G^{{\mathrm c}}}_{-\alpha}(J+\alpha-\delta,I)-{G^{{\mathrm s}}}_{-\alpha}(J+\alpha-\delta,I) \big] U_{\delta-\alpha}(I+\alpha,J) \nonumber\\ &&\hspace{0.5cm}+f\left(t,2v_{{\mathrm F}}K^{GU2}_{\alpha;\delta}(I,J)\right)\big[ {G^{{\mathrm c}}}_{\delta-\alpha}(I+\alpha,J)-{G^{{\mathrm s}}}_{\delta-\alpha}(I+\alpha,J) \big] U_\alpha(I,J+\alpha-\delta) \nonumber\\ \label{eq:eqRGNchaines_u} &&\hspace{0.5cm}-f\left(t,2v_{{\mathrm F}}K^{GU3}_{\alpha;\delta}(I,J)\right)\big[2{G^{{\mathrm s}}}_{I-\alpha}(\alpha+\delta,I) U_{J-\alpha-\delta}(\alpha,J)\big]\\ &&\hspace{0.5cm}-f\left(t,2v_{{\mathrm F}}K^{GU4}_{\alpha;\delta}(I,J)\right) \big[2{G^{{\mathrm s}}}_{J-\alpha-\delta}(\alpha,J) U_{I-\alpha}(\alpha+\delta,I)\big] \nonumber\\ &&\hspace{0.5cm}+f\left(t,2v_{{\mathrm F}}K^{GU5}_{\alpha;\delta}(I,J)\right)\big[ {G^{{\mathrm c}}}_{-\alpha}(I+\delta+\alpha,I)+3{G^{{\mathrm s}}}_{-\alpha}(I+\delta+\alpha,I)\big] U_\delta(I+\alpha,J) \nonumber\\ &&\hspace{0.5cm}+f\left(t,2v_{{\mathrm F}}K^{GU6}_{\alpha;\delta}(I,J)\right)\big[ {G^{{\mathrm c}}}_{J-\alpha}(\alpha-\delta,J)+3{G^{{\mathrm s}}}_{J-\alpha}(\alpha-\delta,J)\big] U_\delta(I,\alpha) \Biggr\rbrace.\nonumber \end{aligned}$$ In the above three flow equations, we have used the following notations: $$\begin{aligned} \label{eq:car_length_1} K^{\mathrm{pp}}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[ \Delta k_\delta(I,J)-2\Delta k_\alpha(I,J)\right]\\ K^{\mathrm{ph}}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[ \Delta k_\delta(I,J)-2\Delta k_\alpha(I,J+\alpha-\delta)\right]\\ K^{UU}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[\Delta k_\delta(I,J)-2\Delta k^U_\alpha(I,J+\alpha-\delta)\right]\quad\quad\end{aligned}$$ $$\begin{aligned} K^{GU1}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[ \Delta k^U_\delta(I,J)-2\Delta k_\alpha(I,J+\alpha-\delta)\right]\\ K^{GU2}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[ \Delta k^U_\delta(I,J)-2\Delta k_{\alpha-\delta}(J,I+\alpha)\right]\\ K^{GU3}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[ \Delta k^U_\delta(I,J)-2\Delta k_{\alpha-I}(I,\alpha+\delta)\right]\quad\quad\end{aligned}$$ $$\begin{aligned} K^{GU4}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[ \Delta k^U_\delta(I,J)-2\Delta k_{\alpha+\delta-J}(J,\alpha)\right]\\ K^{GU5}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[ \Delta k^U_\delta(I,J)-2\Delta k_\alpha(I,I+\alpha+\delta)\right]\quad\quad\\ \label{eq:car_length_9} K^{GU6}_{\alpha;\delta}(I,J)&\!\!=\!\!&\frac{1}{4}\left[ \Delta k^U_\delta(I,J)-2\Delta k_{\alpha-J}(J,\alpha-\delta)\right]\end{aligned}$$ We refer the reader to Sec. \[sec:sub:sub:ren\_int\] for the definitions of $\Delta k_\delta(I,J)$ and $\Delta k^U_\delta(I,J)$. The $f$ function is defined as follows: $f(t=\ln(\Lambda_0/\nu),\delta)=1$ if $\nu\geqslant |\delta|$ and 0 otherwise. Renormalization of the propagator {#app:flow_eq_prop} --------------------------------- The two-loop [self-energy]{}has the following expression: $$\begin{aligned} &&\Sigma_{{{\mathrm R}},I}^{(2)}(k=k_{{{\mathrm F}},I},\omega=\mu+\nu)=\frac{1}{4N^2}\sum_{J,\alpha}\nonumber\\ \label{eq:se_Nchaines} &&\hspace{1.5cm}\Biggl\lbrace 2\left[ {G^{{\mathrm c}}}_\alpha(I,J) {G^{{\mathrm c}}}_{-\alpha}(I+\alpha,J-\alpha) + 3 {G^{{\mathrm s}}}_\alpha(I,J) {G^{{\mathrm s}}}_{-\alpha}(I+\alpha,J-\alpha) \right]\\ &&\hspace{6cm}\times\Big(\nu+v_{{\mathrm F}}\Delta k_\alpha(I,J) \Big) \ln\left[ \frac{\big| \nu^2-\big(v_{{\mathrm F}}\Delta k_\alpha(I,J)\big)^2\big|}{(2v_{{\mathrm F}}\Lambda_0)^2}\right]\nonumber\\ &&\hspace{1.5cm}+U_\alpha(I,J) \big[ 2U_\alpha(I,J)-U_{J-I-\alpha}(I,J) \big]\Big( \nu+v_{{\mathrm F}}\Delta k^U_\alpha(I,J) \Big) \ln\left[ \frac{\big| \nu^2-\big(v_{{\mathrm F}}\Delta k^U_\alpha(I,J)\big)^2\big|}{(2v_{{\mathrm F}}\Lambda_0)^2}\right]\Biggr\rbrace.\nonumber \end{aligned}$$ The ${\varphi}$ functions relating renormalized propagators at two different scales satisfy the general Eq.(\[eq:eq\_flot\_phi\_gale\]), which in the case of $N$ chains reads: $$\begin{aligned} &&\partial_t \ln({\varphi}_I)=\frac{1}{2N^2}\sum_{J,\alpha}\nonumber\\ &&\hspace{1.5cm}\Biggl\lbrace 2\left[ {G^{{\mathrm c}}}_\alpha(I,J) {G^{{\mathrm c}}}_{-\alpha}(I+\alpha,J-\alpha) + 3 {G^{{\mathrm s}}}_\alpha(I,J) {G^{{\mathrm s}}}_{-\alpha}(I+\alpha,J-\alpha) \right]\nonumber\\ &&\hspace{3cm}\times\left[ \left(1+\frac{v_{{\mathrm F}}\Delta k_\alpha(I,J)}{\nu}\right)f\big(\nu,v_{{\mathrm F}}\Delta k_\alpha(I,J)\big)-\frac{v_{{\mathrm F}}\Delta k_\alpha(I,J)}{\nu}l\big(\nu,v_{{\mathrm F}}\Delta k_\alpha(I,J)\big)\right]\nonumber\\ \label{eq:flot_phi} &&\hspace{1.5cm}+U_\alpha(I,J) \big[ 2U_\alpha(I,J)-U_{J-I-\alpha}(I,J) \big]\\ &&\hspace{3cm}\times\left[ \left(1+\frac{v_{{\mathrm F}}\Delta k^U_\alpha(I,J)}{\nu}\right)f\big(\nu,v_{{\mathrm F}}\Delta k^U_\alpha(I,J)\big)-\frac{v_{{\mathrm F}}\Delta k^U_\alpha(I,J)}{\nu}l\big(\nu,v_{{\mathrm F}}\Delta k^U_\alpha(I,J)\big)\right]\Biggr\rbrace,\nonumber \end{aligned}$$ where the function $l$ is defined by $l(\nu,\delta)=\ln(|\nu/\delta|)$ if $\nu\geqslant |\delta|$ and $=0$ otherwise. This function is, as $f$, a logarithmic approximation of a more complex function, diverging at scale $|\delta|$. From the definition of ${\varphi}_I$ (see Eq. \[eq:definition\_phi\]), it is clear that if the initial high-energy quasiparticle weight is equal to one, then one simply has : $Z_I(t)=1/{\varphi}_I(t)$. The flow equation for the running [Fermi surface]{}in the cut-off scaling scheme is also obtained thanks to the [self-energy]{}, and we find: $$\begin{aligned} \partial_t k_{{{\mathrm F}},I}^{(0)}&=&\frac{1}{2N^2}\sum_{J,\alpha} \Biggl\lbrace 2\Delta k_\alpha^{(0)}(I,J) \Big[ {G^{{\mathrm c}}}_{{{\mathrm B}},\alpha}(I,J) {G^{{\mathrm c}}}_{{{\mathrm B}},-\alpha}(I+\alpha,J-\alpha) + 3 {G^{{\mathrm s}}}_{{{\mathrm B}},\alpha}(I,J) {G^{{\mathrm s}}}_{{{\mathrm B}},-\alpha}(I+\alpha,J-\alpha) \Big]\nonumber\\ \label{eq:flot_fs} &&\hspace{1.5cm}+ {\Delta k^U_\alpha}^{(0)}(I,J) \, U_\alpha(I,J) \big[ 2U_\alpha(I,J)-U_{J-I-\alpha}(I,J) \big]\\ &&\hspace{5cm}-\frac{1}{N}\sum_I {\Delta k^U_\alpha}^{(0)}(I,J) \, U_\alpha(I,J) \big[ 2U_\alpha(I,J)-U_{J-I-\alpha}(I,J) \big]\Biggr\rbrace.\nonumber \end{aligned}$$ [56]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{} , ****, (). , ****, (). , ****, (). , , , , ****, (). , ****, (). , ****, (). , ****, (). , , , , , , ****, (). , ****, (). , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ** (, ). , , , ****, (). , , , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). , ****, (). . , ** (, ). 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--- abstract: 'We study for the first time a three-dimensional octahedron constellation for a space-based gravitational wave detector, which we call the Octahedral Gravitational Observatory (OGO). With six spacecraft the constellation is able to remove laser frequency noise and acceleration disturbances from the gravitational wave signal without needing LISA-like drag-free control, thereby simplifying the payloads and placing less stringent demands on the thrusters. We generalize LISA’s time-delay interferometry to displacement-noise free interferometry (DFI) by deriving a set of generators for those combinations of the data streams that cancel laser and acceleration noise. However, the three-dimensional configuration makes orbit selection complicated. So far, only a halo orbit near the Lagrangian point L1 has been found to be stable enough, and this allows only short arms up to 1400km. We derive the sensitivity curve of OGO with this arm length, resulting in a peak sensitivity of about 2$\times10^{-23}\,\mathrm{Hz}^{-1/2}$ near 100Hz. We compare this version of OGO to the present generation of ground-based detectors and to some future detectors. We also investigate the scientific potentials of such a detector, which include observing gravitational waves from compact binary coalescences, the stochastic background and pulsars as well as the possibility to test alternative theories of gravity. We find a mediocre performance level for this short-arm-length detector, between those of initial and advanced ground-based detectors. Thus, actually building a space-based detector of this specific configuration does not seem very efficient. However, when alternative orbits that allow for longer detector arms can be found, a detector with much improved science output could be constructed using the octahedron configuration and DFI solutions demonstrated in this paper. Also, since the sensitivity of a DFI detector is limited mainly by shot noise, we discuss how the overall sensitivity could be improved by using advanced technologies that reduce this particular noise source.' author: - Yan Wang - David Keitel - Stanislav Babak - Antoine Petiteau - Markus Otto - Simon Barke - Fumiko Kawazoe - Alexander Khalaidovski - Vitali Müller - Daniel Schütze - Holger Wittel - Karsten Danzmann - 'Bernard F. Schutz' title: 'Octahedron configuration for a displacement noise-canceling gravitational wave detector in space' --- Introduction ============ The search for gravitational waves (GWs) has been carried out for more than a decade by ground-based detectors. Currently, the LIGO and Virgo detectors are being upgraded using advanced technologies [@aLIGO; @adVIRGO]. The ground-based detectors are sensitive in quite a broad band from about 10Hz to a few kHz. In this band possible GW sources include stellar-mass compact coalescing binaries [@Abadie2010b], asymmetric core collapse of evolved heavy stars [@FryerNew2011], neutron stars with a nonzero ellipticity [@Owen2009] and, probably, a stochastic GW background from the early Universe or from a network of cosmic strings [@Allen99; @Maggiore00]. In addition, the launch of a space-based GW observatory is expected in the next decade, such as the classic LISA mission concept [@LISA] (or its recent modification known as evolved LISA (eLISA) / NGO [@eLISA]), and DECIGO [@Ando2010]. LISA has become a mission concept for any heliocentric drag-free configuration that uses laser interferometry for detecting GWs. The most likely first GW observatory in space will be the eLISA mission, which has an arm length of $10^9$m and two arms, with one “mother” and two “daughter” spacecraft exchanging laser light in a V-shaped configuration to sense the variation of the metric due to passing GWs. The eLISA mission aims at mHz frequencies, targeting other sources than ground-based detectors, most importantly supermassive black hole binaries. In a more ambitious concept, DECIGO is supposed to consist of a set of four smaller triangles (12 spacecraft in total) in a common orbit, leading to a very good sensitivity in the intermediate frequency region between LISA and advanced LIGO (aLIGO). Here we want to present a concept for another space-based project with quite a different configuration from what has been considered before. The concept was inspired by a three-dimensional interferometer configuration in the form of an octahedron, first suggested in Ref. [@chen2006] for a ground-based detector, based on two Mach-Zehnder interferometers. The main advantage of this setup is the cancellation of timing, laser frequency and displacement noise by combining multiple measurement channels. We have transformed this detector into a space-borne observatory by placing one LISA-like spacecraft (but with four telescopes and a single test mass) in each of the six corners of the octahedron, as shown in Fig. \[F:orbit\]. Therefore, we call this project the *Octahedral Gravitational Observatory* (OGO). Before going into the mathematical details of displacement-noise free interferometry (DFI), we first consider possible orbits for a three-dimensional octahedron constellation in Sec. \[S:Orbit\]. As we will find later on, the best sensitivities of an OGO-like detector are expected at very long arm lengths. However, the most realistic orbits we found that can sustain the three-dimensional configuration with stable distances between adjacent spacecraft for a sufficiently long time are so-called “halo” and “quasihalo” orbits around the Lagrange point L1 in the Sun-Earth system. These orbits are rather close to Earth, making a mission potentially cheaper in terms of fuel and communication, and corrections to maintain the formation seem to be reasonably low. On the other hand, a constellation radius of only 1000km can be supported, corresponding to a spacecraft-to-spacecraft arm length of approximately 1400km. We will discuss this as the standard configuration proposal for OGO in the following, but ultimately we still aim at using much longer arm lengths. As a candidate, we will also discuss OGO orbits with $2\times10^9\,$m arm lengths in Sec. \[S:Orbit\]. However, such orbits might have significantly varying separations and would require further study of the DFI technique in such circumstances. {width="\textwidth"}\ The octahedron configuration gives us 24 laser links, each corresponding to a science measurement channel of the distance (photon flight-time) variation between the test masses on adjacent spacecraft. The main idea is to use a sophisticated algorithm called *displacement-noise free interferometry* (DFI, [@kawamura2004; @chen2006; @chenkawa2006]), which proceeds beyond conventional Time-Delay Interferometry techniques (TDI, [@TintoDhurandhar; @otto2012]), and in the right circumstances can improve upon them. It can cancel both timing noise and acceleration noise when there are more measurements than noise sources. In three dimensions, the minimum number of spacecraft for DFI is 6, which we therefore use for OGO: this gives $6-1$ relative timing (clock) noise sources and $3\times 6 = 18$ components of the acceleration noise, so that $24 > 5+18$ and the DFI requirement is fulfilled. On the one hand, this required number of links increases the complexity of the detector. On the other hand, it provides some redundancy in the number of shot-noise-only configurations, which could be very useful if one or several links between spacecraft are interrupted. After applying DFI, we assume that the dominant remaining noise will be shot noise. For the case of an equal-arm-length three-dimensional constellation, we analytically find a set of generators for the measurement channel combinations that cancel simultaneously all timing and acceleration noise. We assume that all deviations from the equal-arm configuration are small and can be absorbed into a low-frequency part of the acceleration noise. We describe the procedure of building DFI combinations in Sec. \[S:TDI\]. This will also allow us to quantify the redundancy inherent in the six-spacecraft configuration. The technical details of the derivation can be found in Appendix \[S:Appendix\]. In Sec. \[S:Sens\], we compute the response functions of the octahedron DFI configuration and derive the sensitivity curve of the detector. We assume the conservative 1400km arm length, a laser power of 10W and a telescope diameter of 1m, while identical strain sensitivity is achievable for smaller telescopes and higher power. Unfortunately, those combinations that cancel acceleration and timing noise also suppress the GW signal at low frequencies. This effect shows up as a rather steep slope $\sim f^{2}$ in the response function. We present sensitivity curves for single DFI combinations and find that there are in principle 12 such noise-uncorrelated combinations (corresponding to the number of independent links) with similar sensitivity, leading to an improved network sensitivity of the full OGO detector. We find that the best sensitivity is achieved around 78Hz, in a range similar to that of ground-based detectors. The network sensitivity of OGO is better than that of initial LIGO at this frequency, but becomes better than that of aLIGO only below 10Hz. The details of these calculations are presented in Sec. \[sec:transfer\_function\]. At this point, in Sec. \[sec:performance\], we briefly revisit the alternative orbits with a longer arm length, which would result in a sensitivity closer to the frequency band of interest for LISA and DECIGO. For this variant of OGO, we assume LISA-like noise contributions (but without spacecraft jitter) and compare the sensitivity of an octahedron detector using DFI with one using TDI, thus directly comparing the effects of these measurement techniques. Actually, we find that the $2\times10^9\,$m arm length is close to the point of equal sensitivity of DFI and TDI detectors in the limit of vanishing jitter. This implies that DFI would be preferred for even longer arm lengths, but might already become competitive at moderate arm lengths if part of the jitter couples into the displacement noise in such a way that it can also be canceled. A major advantage of the OGO concept lies in its rather moderate requirement on acceleration noise, as detailed in Sec. \[sec:feasibility\]. For other detectors, this limits the overall performance, but in this concept it gets canceled out by the DFI combinations. Assuming some improvements in subdominant noise sources, our final sensitivity thus depends only on the shot-noise level in each link. Hence, we can improve the detector performance over all frequencies by reducing solely the shot noise. This could be achieved, for example, by increasing the power of each laser, by introducing cavities (similar to DECIGO), or with nonclassical (squeezed) states of light. We briefly discuss these possibilities in Sec. \[sec:shot\_noise\_reduction\]. In Sec. \[S:Sources\], we discuss the scientific potentials OGO would have even using the conservative short-arm-length orbits. First, as a main target, the detection rates for inspiraling binaries are higher than for initial LIGO, but fall short of aLIGO expectations. However, joint detections with OGO and aLIGO could yield some events with greatly improved angular resolution. Second, due to the large number of measurement channels, OGO is good for probing the stochastic background. Furthermore, the three-dimensional configuration allows us to test alternative theories of gravity by searching for additional GW polarization modes. In addition, we briefly consider other source types such as pulsars, intermediate mass ($10^2 < M/M_{\odot} < 10^4$) black hole (IMBH) binaries and supernovae. Finally, in Sec. \[S:Summary\], we summarize the description and abilities of the Octahedral Gravitational Observatory and mention additional hypothetical improvements. In this article, we use geometric units, $c = G = 1$, unless stated otherwise. Orbits {#S:Orbit} ====== The realization of an octahedral constellation of spacecraft depends on the existence of suitable orbits. Driving factors, apart from separation stability, are assumed to be (i) fuel costs in terms of velocity $\Delta v$ necessary to deploy and maintain the constellation of six spacecraft, and (ii) a short constellation-to-Earth distance, required for a communication link with sufficient bandwidth to send data back to Earth. As described in the introduction, OGO features a three-dimensional satellite constellation. Therefore, using heliocentric orbits with a semimajor axis $a = 1$AU similar to LISA would cause a significant drift of radially separated spacecraft and is in our opinion not feasible. However, in the last decades orbits in the nonlinear regime of Sun/Earth-Moon libration points L1 and L2 have been exploited, which can be reached relatively cheaply in terms of fuel [@Gomez1993]. A circular constellation can be deployed on a torus around a halo L1 orbit. The radius is limited by the amount of thrust needed for keeping the orbit stable. A realistic $\Delta v$ for orbit maintenance allows a nominal constellation radius of $r=1000$km [@Howell1999]. We assume the spacecraft B, C, E and F in Fig. \[F:orbit\] to be placed on such a torus, whereby the out-of-plane spacecraft A and D will head and trail on the inner halo. The octahedron formation then has a base length $L=\sqrt{2}\,r\approx1400$km. The halo and quasihalo orbits have an orbital period of roughly 180 days and the whole constellation rotates around the A-D line. We already note at this point that a longer baseline would significantly improve the detector strain sensitivity. Therefore, we also propose an alternative configuration with an approximate average side length of $2\times10^9\,$m, where spacecraft A and D are placed on a small halo or Lissajous orbit around L1 and L2, respectively. The remaining spacecraft are arranged evenly on a (very) large halo orbit around either L1 or L2. However, simulations using natural reference trajectories showed that this formation is slightly asymmetric and that the variations in the arm lengths (and therefore in the angles between the links) are quite large. Nevertheless, we will revisit this alternative in Sec. \[sec:performance\] and do a rough estimation of its sensitivity. To warrant a full scientific study of such a long-arm-length detector would first require a more detailed study of these orbits. Hence, we assume the 1400 km constellation to be a more realistic baseline, especially since the similarity of the spacecraft orbits is advantageous for the formation deployment, because large (and expensive) propulsion modules for each satellite are not required as proposed in the LISA/NGO mission [@NGOYellowBook; @LisaYellow]. The $2\times10^9\,$m formation will be stressed only to show the improvement of the detector sensitivity with longer arms. Formation flight in the vicinity of Lagrange points L1 and L2 is still an ongoing research topic [@Folta2004]. Detailed (numerical) simulations have to be performed to validate these orbit options and to figure out appropriate orbit and formation control strategies. In particular the suppression of constellation deformations using non-natural orbits with correction maneuvers and required $\Delta v$ and fuel consumption needs to be investigated. Remaining deformations and resizing of the constellation will likely require a beam or telescope steering mechanism on the spacecraft. In addition, the formation will have a varying Sun-incidence angle, leading to further issues for power supply, thermal shielding and blinding of interferometer arms. These points need to be targeted at a later stage of the OGO concept development as well as the effect of unequal arms on the DFI scheme. Measurements and noise-canceling combinations {#S:TDI} ============================================= In this section we will show how to combine the available measurement channels of the OGO detector to cancel laser and acceleration noise. Each spacecraft of OGO is located at a corner of the octahedron, as shown in Fig. \[F:orbit\], and it exchanges laser light with four adjacent spacecraft. We consider interference between the beam emitted by spacecraft $I$ and received by spacecraft $J$ with the local beam in $J$, where $I,J = \{\mathrm{A,B,C,D,E,F}\}$ refer to the labels in Fig. \[F:orbit\]. For the sake of simplicity, we assume a rigid and nonrotating constellation. In other words, all arm lengths in terms of light travel time are equal, constant in time and independent of the direction in which the light is exchanged between two spacecraft. This is analogous to the first generation TDI assumptions [@TintoDhurandhar]. If the expected deviations from the equal arm configuration are small, then they can be absorbed into the low-frequency part of the acceleration noise. This imposes some restrictions on the orbits and on the orbit correction maneuvers. We also want to note that the overall breathing of the constellation (scaling of the arm length) is not important if the breathing time scale is significantly larger than the time required for the DFI formation, which is usually true. All calculations below are valid if we take the arm length at the instance of DFI formation, which is the value that affects the sensitivity of the detector. The measurement of the fractional frequency change for each link is then given by $$s^{\mathrm{tot}}_{IJ} = h_{IJ} + b_{IJ} + \mathcal{D} p_{I} - p_{J} + \mathcal{D} \left( \vec{a}_{I}\cdot\hat{n}_{IJ} \right) - \left( \vec{a}_{J}\cdot\hat{n}_{IJ} \right) \, , \label{E:Mes1Gen}$$ where we have neglected the factors to convert displacement noise to optical frequency shifts. Here, we have the following: (i) $h_{IJ}$ is the influence of gravitational waves on the link $I \rightarrow J$, (ii) $b_{IJ}$ is the shot noise (and other similar noise sources at the photo detector and phase meter of spacecraft $J$) along the link $I \rightarrow J$. (iii) $p_{I}$ is the laser noise of spacecraft $I$. (iv) $\vec{a}_{I}$ is the acceleration noise of spacecraft $I$. (v) $\hat{n}_{IJ} = ( \vec{x}_{J} - \vec{x}_{I}) / L$ is the unit vector along the arm $I \rightarrow J$ (with length $L$). Hence, the scalar product $\vec{a}_I \cdot \hat{n}_{IJ}$ is the acceleration noise of spacecraft $I$ projected onto the arm characterized by the unit vector $\hat{n}_{IJ}$. This is similar to TDI considerations, but in addition to canceling the laser noise $p_I$, we also want to eliminate the influence of the acceleration noise, that is all terms containing $a_I$. Following Ref. [@TintoDhurandhar], we have introduced a delay operator $\mathcal{D}$, which acts as $$\mathcal{D} y(t) = y( t - L) \, . \label{E:Delay}$$ Note that we use a coordinate frame associated with the center of the octahedron, as depicted in Fig. \[F:orbit\]. The basic idea is to find combinations of the individual measurements (Eq. \[E:Mes1Gen\]) which are free of acceleration noise $\vec{a}_{I}$ and laser noise $p_{I}$. In other words, we want to find solutions to the following equation: $$\sum_{\textrm{all} \ IJ \ \textrm{links}} q_{IJ} \ s_{IJ} = 0\,. \label{E:GenCondNoiseNull}$$ In Eq. (\[E:GenCondNoiseNull\]), $q_{IJ}$ denotes an unknown function of delays $\mathcal{D}$ and $ s_{IJ}$ contains only the noise we want to cancel: $$\begin{aligned} s_{IJ} &\equiv& s^{\mathrm{tot}}_{IJ}(b_{IJ} = h_{IJ} = 0) \nonumber \\ &=& \mathcal{D} p_{I} - p_{J} + \mathcal{D} \left( \vec{a}_{I}\cdot\hat{n}_{IJ} \right) - \left( \vec{a}_{J}\cdot\hat{n}_{IJ} \right). \label{E:MesN} \end{aligned}$$ If a given $q_{IJ}$ is a solution, then $f(\mathcal{D})q_{IJ}$ is also a solution, where $f(\mathcal{D})$ is a polynomial function (of arbitrary order) of delays. The general method for finding generators of the solutions for this equation is described in Ref. [@TintoDhurandhar] and we will follow it closely. Before we proceed to a general solution for Eq. (\[E:GenCondNoiseNull\]), we can check that the solution corresponding to Mach-Zehnder interferometers suggested in Ref. [@chen2006] also satisfies Eq. (\[E:GenCondNoiseNull\]): $$\begin{aligned} Y_1 &= [ \,( s_{CD} + \mathcal{D} s_{AC} ) - ( s_{CA} + \mathcal{D} s_{DC} ) + ( s_{FD} + \mathcal{D} s_{AF} ) \nonumber \\ &- ( s_{FA} + \mathcal{D} s_{DF} ) \, ] - [ \, ( s_{BD} + \mathcal{D} s_{AB} ) - ( s_{BA} + \mathcal{D} s_{DB} ) \nonumber \\ & + ( s_{ED} + \mathcal{D} s_{AE} ) - ( s_{EA} + \mathcal{D} s_{DE} )\, ] \,. \end{aligned}$$ Using the symmetries of an octahedron, we can write down two other solutions: $$\begin{aligned} Y_2 &= [\, (s_{CE} + \mathcal{D} s_{BC}) - (s_{CB} + \mathcal{D} s_{EC}) + (s_{FE} + \mathcal{D} s_{BF}) \nonumber \\ &- (s_{FB} + \mathcal{D} s_{EF})\,] - [\,(s_{AE} + \mathcal{D} s_{BA}) - (s_{AB} + \mathcal{D} s_{EA})\nonumber\\ & + (s_{DE} + \mathcal{D} s_{BD}) - (s_{DB} + \mathcal{D} s_{ED}) \,]\, , \\ & \nonumber \\ Y_3 &= [\,(s_{DF} + \mathcal{D} s_{CD}) - (s_{DC} + \mathcal{D} s_{FD}) + (s_{AF} + \mathcal{D} s_{CA}) \nonumber\\ &- (s_{AC} + \mathcal{D} s_{FA})\,] - [\,((s_{EF} + \mathcal{D} s_{CE}) - (s_{EC} + \mathcal{D} s_{FE})\nonumber\\ &+ (s_{BF} + \mathcal{D} s_{CB}) - (s_{BC} + \mathcal{D} s_{FB}) \,]\,. \end{aligned}$$ We can represent these solutions as 24-tuples of coefficients for the delay functions $q_{IJ}$: $$\begin{aligned} q_1 &= & \{ 1, 1, -1, -1, -1, -1, 1, 1, -\mathcal{D}, \mathcal{D}, 0, 0, -\mathcal{D}, \mathcal{D}, 0, 0, \nonumber\\ & & \mathcal{D},-\mathcal{D}, 0, 0, \mathcal{D}, -\mathcal{D}, 0, 0 \} \, , \\ q_2 &= & \{ -\mathcal{D} , \mathcal{D} , 0 , 0 , -\mathcal{D} , \mathcal{D} , 0 , 0 , 1 , 1 , -1 , -1 , -1 , -1 , 1 , 1, \nonumber\\ & & 0 , 0 , \mathcal{D} , -\mathcal{D} , 0 , 0 , \mathcal{D} , -\mathcal{D} \} \, , \\ q_3 &= & \{ 0 , 0 , \mathcal{D} , -\mathcal{D} , 0 , 0 , \mathcal{D} , -\mathcal{D} , 0 , 0 , -\mathcal{D} , \mathcal{D} , 0 , 0 , -\mathcal{D} , \mathcal{D}, \nonumber\\ & & -1 , -1 , 1 , 1 , 1 , 1 , -1 , -1 \} \, . \end{aligned}$$ The order used in the 24-tuples is $\{ BA$, $EA$, $CA$, $FA$, $BD$, $ED$, $CD$, $FD$, $AB$, $DB$, $CB$, $FB$, $AE$, $DE$, $CE$, $FE$, $AC$, $DC$, $BC$, $EC$, $AF$, $DF$, $BF$, $EF \}$, so that, for example, the first entry in $q_1$ represents the $s_{BA}$ coefficient in the $Y_1$ equation. These particular solutions illustrate that not all links are used in producing a DFI stream. Multiple zeros in the equations for $q_1, q_2, q_3$ above indicate those links which do not contribute to the final result, and each time we use only 16 links. We will come back to the issue of “lost links” when we discuss the network sensitivity. In the following, we will find generators of all solutions. The first step is to use Gaussian elimination (without division by delay operators) in Eq. (\[E:GenCondNoiseNull\]), and as a result, we end up with a single (master) equation which we need to solve: $$\begin{aligned} 0 &=& (\mathcal{D}-1)^2 q_{BC} + (\mathcal{D}-1)\mathcal{D} q_{CE} + (1-\mathcal{D})(\mathcal{D}-1)\mathcal{D} q_{DB} \nonumber \\ &+& (\mathcal{D}-1)((1-\mathcal{D})\mathcal{D}-1) q_{DC} \nonumber\\ &+& (\mathcal{D}-1) q_{DF} + (\mathcal{D}-1) q_{EF} \, . \label{E:TDIAcc_FinalEq} \end{aligned}$$ In the next step, we want to find the so-called “reduced generators” of Eq. (\[E:TDIAcc\_FinalEq\]), which correspond to the reduced set $( q_{BC}, q_{CE}, q_{DB}, q_{DC}, q_{DF}, q_{EF} )$. For this we need to compute the Gröbner basis [@Buchberger1970], a set generating the polynomial ideals $q_{IJ}$. Roughly speaking, the Gröbner basis is comparable to the greatest common divisor of $q_{IJ}$. Following the procedure from Ref. [@TintoDhurandhar], we obtain seven generators: $$\begin{aligned} \label{E:S1} S_1 &=& \{ 0, \mathcal{D}^2 + \mathcal{D}, 0, - \mathcal{D} - \mathcal{D}^2, 1 - \mathcal{D},\mathcal{D}^2 + 1, -1 + \mathcal{D}, -1 - \mathcal{D}^2, \mathcal{D} - \mathcal{D}^2, 0, -\mathcal{D},\mathcal{D}^2, -\mathcal{D}^2 - 1, -\mathcal{D} - 1, 1, \nonumber \\ & & 1 + \mathcal{D} + \mathcal{D}^2, -\mathcal{D} + \mathcal{D}^2,0, \mathcal{D}, -\mathcal{D}^2, \mathcal{D}^2 + 1, 1 + \mathcal{D}, -1, -\mathcal{D} - \mathcal{D}^2 - 1 \},\\ & &\nonumber \\ S_2 &=& \{ \mathcal{D} + 1, \mathcal{D} + 1, -\mathcal{D} -1,-\mathcal{D}-1, -1+\mathcal{D},\mathcal{D}-1, 1-\mathcal{D}, 1 - \mathcal{D}, -2\mathcal{D},0,\mathcal{D},\mathcal{D},-2\mathcal{D},0, \mathcal{D},\mathcal{D}, 2\mathcal{D}, 0, -\mathcal{D}, \nonumber \\ & & -\mathcal{D},2\mathcal{D},0,-\mathcal{D},-\mathcal{D}\} , \\ & &\nonumber \\ S_3 &=& \{ 0, \mathcal{D}, -\mathcal{D}, 0, - 1, \mathcal{D} - 1, 1- \mathcal{D},1, 1 - \mathcal{D}, 1, -1 + \mathcal{D}, -1, -\mathcal{D},0, \mathcal{D}, 0, \mathcal{D}, 0, 0, -\mathcal{D}, \mathcal{D} -1,-1, 1, -\mathcal{D} + 1 \},\\ & &\nonumber \\ S_4 &=& \{ \mathcal{D}, -\mathcal{D} + \mathcal{D}^2, \mathcal{D}, -\mathcal{D}-\mathcal{D}^2, 2, -2\mathcal{D}+\mathcal{D}^2+2, -2+2\mathcal{D}, -2-\mathcal{D}^2,2\mathcal{D} - 2 -\mathcal{D}^2,-2,2 - 2\mathcal{D},2+\mathcal{D}^2,\mathcal{D} - \mathcal{D}^2, \nonumber \\ & & -\mathcal{D},-\mathcal{D},\mathcal{D} + \mathcal{D}^2,-2\mathcal{D} + \mathcal{D}^2,0,0,2\mathcal{D} - \mathcal{D}^2,-\mathcal{D} + \mathcal{D}^2 + 2, 2 + \mathcal{D},-2-\mathcal{D},\mathcal{D} - \mathcal{D}^2 - 2\} , \\ & &\nonumber \\ S_5 &=& \{ 0, \mathcal{D}^2 + \mathcal{D}, -\mathcal{D}^2, - \mathcal{D}, 1 - \mathcal{D}, \mathcal{D}^2 + 1, \mathcal{D} - \mathcal{D}^2 - 1, -1, \mathcal{D} - \mathcal{D}^2, 0, -\mathcal{D} + \mathcal{D}^2, 0, -1 - \mathcal{D}^2, -\mathcal{D} - 1, 1 + \mathcal{D}^2, \nonumber \\ & & 1 + \mathcal{D}, \mathcal{D}^2, \mathcal{D}, 0, -\mathcal{D}^2 - \mathcal{D}, -\mathcal{D} + \mathcal{D}^2 +1, 1, \mathcal{D} - 1, -1 - \mathcal{D}^2\},\\ & & \nonumber \\ S_6 &=& \{ \mathcal{D} + 2 + \mathcal{D}^2,\mathcal{D}+\mathcal{D}^3+2,-\mathcal{D} + \mathcal{D}^2 - 2, -\mathcal{D} - 2 - 2\mathcal{D}^2 - \mathcal{D}^3,-2 + 2\mathcal{D},2\mathcal{D} - \mathcal{D}^2 + \mathcal{D}^3 - 2,\nonumber \\ & & -2\mathcal{D} + 2\mathcal{D}^2 + 2,2 - 2\mathcal{D} - \mathcal{D}^2 - \mathcal{D}^3,\mathcal{D}^2 - 4\mathcal{D} - \mathcal{D}^3, 0, 2\mathcal{D} -2\mathcal{D}^2,2\mathcal{D} + \mathcal{D}^2 + \mathcal{D}^3,-3\mathcal{D} - \mathcal{D}^3,\mathcal{D} - \mathcal{D}^2, \nonumber \\ & &\mathcal{D} - \mathcal{D}^2, 2\mathcal{D}^2 + \mathcal{D} + \mathcal{D}^3, -\mathcal{D}^2 + 2\mathcal{D} + \mathcal{D}^3, -2\mathcal{D},0,\mathcal{D}^2 - \mathcal{D}^3,5\mathcal{D} + \mathcal{D}^3,\mathcal{D} + \mathcal{D}^2,-3\mathcal{D} - \mathcal{D}^2,-3\mathcal{D} - \mathcal{D}^3 \} , \\ & &\nonumber \\ S_7 &=& \{ 1, 1 + \mathcal{D}, -1, -1 - \mathcal{D}, 0, \mathcal{D}, 0, -\mathcal{D}, -\mathcal{D}, 0, 0, \mathcal{D}, -1 - \mathcal{D}, -1, 1, 1 + \mathcal{D}, \mathcal{D}, 0, 0, -\mathcal{D}, 1 + \mathcal{D}, 1, -1, -1 - \mathcal{D} \}. \label{E:S7} \end{aligned}$$ As before, these operators have to be applied to $s_{IJ}$, using the same ordering as given above. All other solutions can be constructed from these generators. A detailed derivation of expressions (\[E:S1\])–(\[E:S7\]) is given in Appendix \[S:Appendix\]. Before we proceed, let us make several remarks. The generators found here are not unique, just like in the case of TDI [@TintoDhurandhar]. The set of generators does not necessarily form a minimal set, and we can only guarantee that the found set of generators gives us a module of syzygies and can be used to generate other solutions. The combinations $S_1$ to $S_7$ applied on 24 raw measurements $s_{IJ}^{\mathrm{tot}}$ eliminate both laser and displacement noise while mostly preserving the gravitational wave signal. Note that again in those expressions we do not use all links – for example, if the link $BA$ is lost due to some reasons, we still can use $S_1, S_3, S_5$ to produce DFI streams. Response functions and sensitivity {#S:Sens} ================================== In the previous section we have found generators that produce data streams free of acceleration and laser noise. Now we need to apply these combinations to the shot noise and to the GW signal to compute the corresponding response functions. Shot noise level and noise transfer function -------------------------------------------- We will assume that the shot noise is independent (uncorrelated) in each link and equal in power spectral density, based on identical laser sources and telescopes on each spacecraft. We denote the power spectral density of the shot noise by $\widetilde{S}_{\rm sn}$. A lengthy but straightforward computation shows that the spectral noise $\tilde{S}_{\mathrm{n},i}$ corresponding to the seven combinations $S_i$, $i=1,\ldots,7$ from Eqs. (\[E:S1\]–\[E:S7\]) is given by $$\begin{aligned} \widetilde{S}_{\rm n, 1} &=& \0 16\, \widetilde{S}_{\rm sn} \,\epsilon^2\,( 9 + 2\cos2\epsilon + 3\cos4\epsilon)\,,\\ \widetilde{S}_{\rm n, 2} &=& 160\, \widetilde{S}_{\rm sn} \,\epsilon^2\,,\\ \widetilde{S}_{\rm n, 3} &=& \0 48\, \widetilde{S}_{\rm sn} \,\epsilon^2 \,( 2 - \cos2\epsilon)\,,\\ \widetilde{S}_{\rm n, 4} &=& \0 16\, \widetilde{S}_{\rm sn} \,\epsilon^2 \,(24 -13\cos2\epsilon + 6\cos4\epsilon)\,,\\ \widetilde{S}_{\rm n, 5} &=& \0 16\, \widetilde{S}_{\rm sn} \,\epsilon^2(\, 9 - 2\cos 2\epsilon + 3\cos 4\epsilon)\,,\\ \widetilde{S}_{\rm n, 6} &=& \0 16\, \widetilde{S}_{\rm sn} \,\epsilon^2 \,(45 -6\cos2\epsilon+17\cos4\epsilon)\,,\\ \widetilde{S}_{\rm n, 7} &=& \048\, \widetilde{S}_{\rm sn} \,\epsilon^2 \,(2 + \cos2\epsilon)\,, \end{aligned}$$ where $\epsilon \equiv \omega L/2$, with the GW frequency $\omega$. In the low frequency limit ($\epsilon \ll 1$), the noise $\widetilde{S}_{{\rm n }, i}$ for each combination $S_i$ is proportional to $\epsilon^2$. Let us now compute the shot noise in a single link. We consider for OGO a configuration with LISA-like receiver-transponder links and the following parameters: spacecraft separation $L=1414\,$km, laser wavelength $\lambda=532$nm, laser power $P=10\,$W and telescope diameter $D=1\,$m. For this arm length and telescope size, almost all of the laser power from the remote spacecraft is received by the local spacecraft. Hence, the shot-noise calculation for OGO is different from the LISA case, where an overwhelming fraction of the laser beam misses the telescope [@LisaYellow]. For a Michelson interferometer, the sensitivity to shot noise is usually expressed as [@Maggiore_book] $$\sqrt{\widetilde{S}_h(f)}=\frac{1}{2L}\sqrt{\frac{\hbar c \lambda}{\pi P}}\,\,\, [1/\sqrt{\rm Hz}] \, ,$$ where we have temporarily restored the speed of light $c$ and the reduced Planck constant $\hbar$. Notice that the effect of the GW transfer function is not included here yet. For a single link $I \rightarrow J$ of OGO as opposed to a full two-arm Michelson with dual links, $\sqrt{\widetilde{S}_{h,IJ}}$ is a factor of 4 larger. However, our design allows the following two improvements: (i) Since there is a local laser in $J$ with power similar to the received laser power from $I$, the power at the beam splitter is actually $2P$, giving an improvement of $1/\sqrt{2}$. This is also different from LISA, where due to the longer arm length the received power is much smaller than the local laser power. (ii) If we assume that the arm length is stable enough to operate at the dark fringe, then we gain another factor of $1/\sqrt{2}$. So, we arrive at the following shot-noise-only sensitivity for a single link: $$\label{E:shotnoise1} \sqrt{\widetilde{S}_{h ,IJ}(f)}=\frac{1}{L}\sqrt{\frac{\hbar c \lambda}{\pi P}}\,\,\, [1/\sqrt{\rm Hz}] \, .$$ GW signal transfer function and sensitivity {#sec:transfer_function} ------------------------------------------- Next, we will compute the detector response to a gravitational wave signal. We assume a GW source located in the direction $\hat{n} = -\hat{k} = \left( \sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta \right)$ as seen from the detector frame. We choose unit vectors $$\hat{u} = \left[ \begin {array}{c} \cos \theta \cos \phi \\ \cos \theta \sin \phi \\ -\sin \theta \end {array} \right],\;\;\;\;\; \hat{v} = \left[ \begin {array}{c} \sin \phi \\ -\cos \phi \\ 0 \end {array} \right]$$ orthogonal to $\hat{k}$ pointing tangentially along the $\theta$ and $\phi$ coordinate lines to form a polarization basis. This basis can be described by polarization tensors $\mathbf{e}_+$ and $\mathbf{e}_\times$, given by $$\mathbf{e}_+ \equiv \hat{u}\otimes \hat{u} - \hat{v}\otimes \hat{v}\,, \ \ \ \mathbf{e}_\times \equiv \hat{u}\otimes \hat{v} + \hat{v}\otimes \hat{u}\,. \label{E:polbas}$$ The single arm fractional frequency response to a GW is [@EW1975] $$\begin{aligned} h_{IJ} = \frac{H_{IJ} (t - \hat{k}\cdot\vec{x}_{I} - L) - H_{IJ} (t - \hat{k}\cdot\vec{x}_{J} )}{2 \left( 1 - \hat{k}\cdot\hat{n}_{IJ} \right)} , \label{E:E:GWlink} \end{aligned}$$ where $\vec{x}_I$ is the position vector of the $I$-th spacecraft, $L$ the (constant) distance between two spacecraft and $$H_{IJ } (t) \equiv h_{+} (t) \ \xi_{+} (\hat{u}, \hat{v}, \hat{n}_{IJ}) + h_{\times} (t) \ \xi_{\times} (\hat{u}, \hat{v}, \hat{n}_{IJ}) \, .$$ Here $h_{+,\times}(t)$ are two GW polarizations in the basis (\[E:polbas\]) and $$\begin{aligned} \xi_{+}(\hat{u}, \hat{v}, \hat{n}_{IJ}) &\equiv \hat{n}_{IJ}^\textsf{T}\mathbf{e}_+ \hat{n}_{IJ} = {\left(\hat{u}\cdot \hat{n}_{IJ} \right)}^{2} - {\left(\hat{v}\cdot \hat{n}_{IJ} \right)}^{2} \, , \nonumber\\ \xi_{\times}(\hat{u}, \hat{v}, \hat{n}_{IJ}) &\equiv \hat{n}_{IJ}^\textsf{T}\mathbf{e}_\times \hat{n}_{IJ} = 2 \left(\hat{u}\cdot \hat{n}_{IJ} \right)\left(\hat{v}\cdot \hat{n}_{IJ} \right) \, . \label{E:E:GWStrain}\end{aligned}$$ In order to find the arm response for arbitrary incident GWs, we can compute the single arm response to a monochromatic GW with Eq. (\[E:E:GWlink\]) and then deduce the following general response in the frequency domain, $$\begin{aligned} h_{IJ}(f) & =& \epsilon \, {\rm sinc}\left[\epsilon(1-\hat{k}\cdot\hat{n}_{IJ})\right] \mathrm{e}^{-\mathrm{i}\epsilon[\hat{k}\cdot(\vec{x}_{I} + \vec{x}_{J})/L +1]} \nonumber \\ & & \ \ \ \times \left[\xi_{+}(\hat{n}_{IJ}) h_{+}(f) + \xi_{\times}(\hat{n}_{IJ}) h_{\times}(f)\right] \,, \end{aligned}$$ where we used the normalized sinc function, conventionally used in signal processing: ${\rm sinc}(x) := {\sin(\pi x)}/(\pi x)$. Hence, the transfer function for a GW signal is $$\begin{aligned} \mathcal{T}_{IJ+,\times}^{\rm GW}(f) &=& \epsilon \, {\rm sinc}\left[\epsilon(1-\hat{k}\cdot\hat{n}_{IJ})\right] \nonumber\\ & & \ \ \ \times\, \mathrm{e}^{-\mathrm{i}\epsilon[\hat{k}\cdot(\vec{x}_{I} + \vec{x}_{J})/L +1]} \xi_{+, \times}(\hat{n}_{IJ}) \, . \label{Eq:TF} \end{aligned}$$ For the sake of simplicity, we will from now on assume that the GW has “+” polarization only. This simplification will not affect our qualitative end result. Substituting the transfer function for a single arm response into the above 7 generators \[Eqs. (\[E:S1\])-(\[E:S7\])\], we can get the transfer function $\mathcal{T}_{i}^{\rm GW}$ for each combination. The final expressions are very lengthy and not needed here explicitly. Having obtained the transfer function, we can compute the sensitivity for each combination $i=1,\ldots,7$ as $$\label{E:sensitivity1} \sqrt{\widetilde{S}_{h,i}} = \sqrt{\frac{\widetilde{S}_{{\rm n}, i}}{\langle(\mathcal{T}^{\rm GW}_i)^2\rangle}} \ ,$$ where the triangular brackets imply averaging over polarization and source sky location. We expect up to 12 independent round trip measurements, corresponding to the number of back-and-forth links between spacecraft. It is out of the scope of this work to explicitly find all noise-uncorrelated combinations (similar to the optimal channels $A, E, T$ in the case of LISA [@TintoDhurandhar]). However, if we assume approximately equal sensitivity for each combination (which is almost the case for the combinations $S_1,\ldots,S_7$), we expect an improvement in the sensitivity of the whole network by a factor $1/\sqrt{12}$. ![Sensitivities for two single DFI combinations ($S_1$, *blue crosses* and $S_5$, *green plus signs*) of OGO (with $L\approx1400$km) and for the full OGO network sensitivity (scaled from $S_5$, *red solid line*). For comparison, the dashed lines show sensitivities for initial LIGO (H1 during science run S6, from Ref. [@Abadie2010a], *cyan dashed line*) and aLIGO (design sensitivity for high-power, zero detuning configuration, from Ref. [@aLIGO_sensitivity], *magenta dash-dotted line*).[]{data-label="F:Sens"}](ogo_sensitivity "fig:"){width="\columnwidth"}\ Therefore, we simply approximate the network sensitivity of the full detector as $\sqrt{\widetilde{S}_{h, {\rm net}}} = \sqrt{\widetilde{S}_{h,5}/12}$. Note that the potential loss of some links would imply that not all generators can be formed. We can lose up to 6 links and still be able to form a DFI stream (but probably only one). The number of lost links (and which links are lost exactly) will affect the network sensitivity. In our estimations below we deal with the idealized situation and assume that no links are lost. We plot the sensitivity curves for individual combinations and the network sensitivity in Fig. \[F:Sens\]. For comparison we also show the design sensitivity curves of initial LIGO (S6 science run [@Abadie2010a]) and advanced LIGO (high laser power configuration with zero detuning of the signal recycling mirror [@aLIGO_sensitivity]). Indeed one can see that the sensitivities of the individual OGO configurations are similar to each other and close to initial LIGO. The network sensitivity of OGO lies between LIGO and aLIGO sensitivities. OGO as expected outperforms aLIGO below 10Hz, where the seismic noise on the ground becomes strongly dominant. General performance of the DFI scheme {#sec:performance} ------------------------------------- ![Network sensitivities, scaled from $S_5$, of standard OGO (with DFI, arm length 1414 km, *red solid line*) compared to an OGO-like detector with spacecraft separation of $2\cdot10^9$ m, with either full DFI scheme (*blue crosses*) or standard TDI only (*green plus signs*). Also shown for comparison are (classic) LISA ($5\cdot10^9$ m, network sensitivity, *magenta dashed line*, from Ref. [@Larson2000]) and DECIGO (using the fitting formula Eq. (20) from Ref. [@Yagi2013], *cyan dash-dotted line*).[]{data-label="F:ogo_TDI_comparison_2Mkm"}](ogo_TDI_comparison_2Mkm_nojitter "fig:"){width="\columnwidth"}\ Having derived the full sensitivity curve of the OGO mission design with $L\approx1400$km as an exemplary implementation of the three-dimensional DFI scheme in space, let us take a step back and analyze the general performance of a DFI-enabled detector. These features are also what led us to consider the octahedron configuration in the first place. Specifically, let us look in more detail at the low frequency asymptotic behavior of the transfer functions and sensitivity curves. We consider a LISA-like configuration with two laser noise free combinations: an unequal arm Michelson () and a Sagnac combination (). Let us assume for a moment that the only noise source is shot noise, which at low frequencies ($\epsilon \ll 1$) scales as $\sqrt{\widetilde{S}_{\mathrm{n},X}} \sim \epsilon^2$ and $\sqrt{\widetilde{S}_{\mathrm{n},\alpha}} \sim \epsilon^1$ for those two combinations, respectively. The GW transfer function, for both TDI combinations, scales as $\mathcal{T}_\alpha,\mathcal{T}_X \sim \epsilon^2$; therefore, the sensitivity curves scale as $\sqrt{\widetilde{S}_{h,\alpha}} \sim \sqrt{\widetilde{S}_{\mathrm{n},\alpha}}/\mathcal{T}_\alpha \sim \epsilon^{-1}$ for and $\sqrt{\widetilde{S}_{h,X}} \sim \sqrt{\widetilde{S}_{\mathrm{n},X}}/\mathcal{T}_X \sim \epsilon^{0}$ for . We see that a LISA-like -combination has a *flat* shot-noise spectrum at low frequencies, corresponding to a flat total detector sensitivity if all other dominant noise sources can be canceled – which looks extremely attractive. Thus, a naive analysis suggests that the acceleration and laser noise free combinations for an octahedron detector could yield a flat sensitivity curve at low frequencies. Checking this preliminary result with a more careful analysis was the main motivation for the research presented in this article. In fact, as we have seen in Sec. \[sec:transfer\_function\], the full derivation delivers transfer functions that, in leading order of $\epsilon$, go as $\mathcal{T}_{1,2,\ldots,7} \sim \epsilon^3$. This implies that the sensitivity for laser and acceleration noise free combinations behaves as $\sqrt{\widetilde{S}_{h,1,2,\ldots,7}}/\mathcal{T}_{1,2,\ldots,,7} \sim \epsilon^{-2}$, which is similar to the behavior of acceleration noise. In other words, the combinations eliminating the acceleration noise also cancel a significant part of the GW signal at low frequencies. In fact, we find that a standard LISA-like TDI-enabled detector of the same arm length and optical configuration as OGO could achieve a similar low-frequency sensitivity (at few to tens of Hz) with an acceleration noise requirement of only $\sim10^{-12}$ m/s${}^2\,\sqrt{\mathrm{Hz}}$. This assumes negligible spacecraft jitter and that no other noise sources (phase-meter noise, sideband noise, thermal noise) limit the sensitivity, which at this frequency band would behave differently than in the LISA band. In fact, the GOCE mission [@Drinkwater2003] has already demonstrated such acceleration noise levels at mHz frequencies [@Sechi2011], and therefore this seems a rather modest requirement at OGO frequencies. We therefore see that such a short-arm-length OGO would actually only be a more complicated alternative to other feasible mission designs. In addition, it is hard to see from just the comparison with ground-based detectors in Fig. \[F:Sens\] how exactly the DFI method itself influences the final noise curve of OGO, and how much of its shape is instead determined by the geometrical and technical parameters of the mission concept (arm length, laser power, telescope size). Also, the secondary technological noise sources of a space mission in the comparatively high-frequency band of this exemplary OGO implementation are somewhat different from more well-studied missions like LISA and DECIGO. Therefore, to disentangle these effects, we will now tentatively study a different version of OGO based on the alternative orbit with an average arm length of $2\cdot10^9$m, as mentioned in Sec. \[S:Orbit\]. It requires further study to determine whether a stable octahedron constellation and the DFI scheme are possible on such an orbit, but assuming they are, we can compute its sensitivity as before. In Fig. \[F:ogo\_TDI\_comparison\_2Mkm\], we then compare this longer-baseline DFI detector with another detector with the same geometry and optical components, but without the DFI technique, using instead conventional TDI measurements. Here, we are in a similar frequency range as LISA and therefore assume similar values for the acceleration noise of $3 \cdot 10^{-15}$ m/s${}^2\,\sqrt{\mathrm{Hz}}$ [@LisaYellow] and secondary noise sources (phase meter, thermal noise, etc.; see Sec. \[sec:feasibility\]). However, there is another noise source, spacecraft jitter, which is considered subdominant for LISA, but might become relevant for both the TDI and DFI versions of the $2\cdot10^9$m OGO-like detector. Jitter corresponds to the rotational degrees of freedom between spacecraft, and its coupling into measurement noise is not fully understood. We have therefore computed both sensitivities without any jitter. It seems possible that at least the part of jitter that couples linearly into displacement noise could also be canceled by DFI, or that an extension of DFI (e.g. more links) could take better care of this, and therefore that the full OGO with DFI would look more favorable compared to the TDI version when nonvanishing jitter is taken into account. Generally, as one goes for longer arm lengths, the DFI scheme will perform better in comparison to the TDI scheme. At the high-frequency end of the sensitivity curves, both schemes are limited by shot noise and the respective GW transfer functions. Since the shot-noise level does not depend on the arm length, it remains the same for all relevant frequencies. Therefore, as the arm length increases, the high-frequency part of the sensitivity curves moves to the low-frequency regime in parallel (i.e. the corner frequency of the transfer function is proportional to $1/L$). This is the same for both schemes. On the other hand, in the low-frequency regime of the sensitivity curves the two schemes perform very differently. For TDI, the low-frequency behavior is limited by acceleration noise, while for DFI this part is again limited by shot noise and the GW transfer function. When the arm length increases, the low-frequency part of the sensitivity curve in the TDI scheme moves to lower frequencies in proportion to $1/\sqrt{L}$; while for DFI, it moves in proportion to $1/L$. Graphically, when the arm length increases, the high-frequency parts of the sensitivity curves in both schemes move toward the lower-frequency regime in parallel, while the low-frequency part of the sensitivity curve for DFI moves faster than for TDI. Under the assumptions given above, we find that an arm length of $2\cdot10^9$m is close to the transition point where the sensitivities of TDI and DFI are almost equal, as shown in Fig. \[F:ogo\_TDI\_comparison\_2Mkm\]. At even longer arm lengths, employing DFI would become clearly advantageous. Of course, these considerations show that a longer-baseline detector with good sensitivity in the standard space-based detector frequency band of interest would make a scientifically much more interesting case than the default short-arm OGO which we presented first. However, as no study on the required orbits has been done so far, we consider such a detector variant to be highly hypothetical and not worthy of a detailed study of technological feasibility and scientific potential yet. Instead, for the remainder of this paper, we concentrate again on the conservative 1400km version of OGO. Although the sensitivity curve in Fig. \[F:Sens\] already demonstrates its limited potential, we will attempt to neutrally assess its advantages, limitations and scientific reach. Technological feasibility {#sec:feasibility} ------------------------- Employing DFI requires a large number of spacecraft but on the other hand allows us to relax many of the very strict technological requirements of other space-based GW detector proposals such as (e)LISA and DECIGO. Specifically, the clock noise is canceled by design, so there is no need for a complicated clock tone transfer chain [@barke2010]. Furthermore, OGO does not require a drag-free technology, and the configuration has to be stabilized only as much as required for the equal arm length assumption to hold. This strongly reduces the requirements on the spacecraft thrusters. Also, for the end mirrors, which have to be mounted on the same monolithic structure for all four laser links per spacecraft, it is not required that they are freefalling. Instead, they can be fixed to the spacecraft. Still, to reach the shot-noise-only limited sensitivity shown in Fig. \[F:Sens\], the secondary noise contributions from all components of the measurement system must be significantly below the shot-noise level. Considering a shot-noise level of about $2\cdot 10^{-17}\,\mathrm{m}/\sqrt{\mathrm{Hz}}$ – which is in agreement with the value derived earlier for the 1400km version of OGO – this might be challenging. When actively controlling the spacecraft position and hence stabilizing the distance and relative velocity between the spacecraft, we will be able to lower the heterodyne frequency of the laser beat notes drastically. Where LISA will have a beat note frequency in the tens of MHz, with OGO’s short arm length we could be speaking of kHz or less and might even consider a homodyne detection scheme as in LIGO. This might in the end enable us to build a phase meter capable of detecting relative distance fluctuations with a sensitivity of $10^{-17}\,\mathrm{m}/\sqrt{\mathrm{Hz}}$ or below as required by OGO. As mentioned before, temperature noise might be a relevant noise source for OGO: The relative distance fluctuations on the optical benches due to temperature fluctuations and the test mass thermal noise must be significantly reduced in comparison to LISA. But even though the LISA constellation is set in an environment which is naturally more temperature stable, stabilization should be easier for the higher-frequency OGO measurement band. A requirement of $10^{-17}\,\mathrm{m}/\sqrt{\mathrm{Hz}}$ could be reached by actively stabilizing the temperature down to values of 1nK/$\sqrt{\mathrm{Hz}}$ at the corner frequency. Assuming future technological progress, optimization of the optical bench layout could also contribute to mitigating this constraint, as could the invention of thermally more stable materials for the optical bench. Most likely, this challenge can be solved only with a combination of the mentioned approaches. The same is true for the optical path length stability of the telescopes. We estimate the required pointing stability to be roughly similar to the LISA mission requirements. Shot-noise reduction {#sec:shot_noise_reduction} -------------------- Assuming the requirements from the previous section can be met, the timing and acceleration noise free combinations of the OGO detector are dominated by shot noise, and any means of reducing the shot noise will lead to a sensitivity improvement over all frequencies. In this subsection, we discuss possible ways to achieve such a reduction. The most obvious solution is to increase laser power, with an achievable sensitivity improvement that scales with $\sqrt{P}$. However, the available laser power is limited by the power supplies available on a spacecraft. Stronger lasers are also heavier and take more place, making the launch of the mission more difficult. Therefore, there is a limit to simply increasing laser power, and we want to shortly discuss more advanced methods of shot-noise reduction. One such hypothetical possibility is to build cavities along the links between spacecraft, similar to the DECIGO design [@Ando2010]. The shot noise would be decreased due to an increase of the effective power stored in the cavity. Effectively, this also results in an increase of the arm length. Note, however, that the sensitivity of OGO with cavities cannot simply be computed by inserting effective power and arm length into our previously derived equations. Instead, a rederivation of the full transfer function along the lines of Ref. [@Rakhmanov2005] is necessary. Alternatively, squeezed light [@Schnabel10] is a way to directly reduce the quantum measurement noise, which has already been demonstrated in ground-based detectors [@SqueezedGEO; @Khalaidovski12]. However, squeezing in a space-based detector is challenging in many aspects due to the very sensitive procedure and would require further development. Scientific perspectives {#S:Sources} ======================= In this section, we will discuss the science case for our octahedral GW detector (with an arm length of 1400km) by considering the most important potential astrophysical sources in its band of sensitivity. Using the full network sensitivity, as derived above, the best performance of OGO is at 78Hz, between the best achieved performance of initial LIGO during its S6 science run and the anticipated sensitivity for advanced LIGO. OGO outperforms the advanced ground-based detectors below 10Hz, where the seismic noise strongly dominates. In this analysis, we will therefore consider sources emitting GWs with frequencies between 1Hz and 1kHz, concentrating on the low end of this range. Basically, those are the same sources as for ground-based detectors, which include compact binaries coalescences (CBCs), asymmetric single neutron stars (continuous waves, CWs), binaries containing intermediate-mass black holes (IMBHs), burst sources (unmodeled short-duration transient signals), and a cosmological stochastic background. We will go briefly through each class of sources and consider perspectives of their detection. As was to be expected from the sensitivity curve in Fig. \[F:Sens\], in most categories OGO performs better than initial ground-based detectors, but does not even reach the potential of the advanced generation currently under commissioning. Therefore, this section should be understood not as an endorsement of actually building and flying an OGO-like mission, but just as an assessment of its (limited, but existing) potentials. This demonstrates that an octahedral GW detector employing DFI in space is in principle capable of scientifically interesting observations, even though improving its performance to actually surpass existing detectors or more mature mission proposals still remains a subject of further study. In addition, we put a special focus on areas where OGO’s design offers some specific advantages. These include the triangulation of CBCs through joint detection with ground-based detectors as well as searching for a stochastic GW background and for additional GW modes. Note that the hypothetical $2\cdot10^9$m variant of OGO (see Secs. \[S:Orbit\] and \[sec:performance\]) would have a very different target population of astrophysical sources due to its sensitivity shift to lower frequencies. Such a detector would still be sensitive to CBCs, IMBHs, and stochastic backgrounds, probably much more so. But instead of high-frequency sources like CW pulsars and supernova bursts, it would start targeting supermassive black holes, investigating the merging history of galaxies over cosmological scales. However, as this detector concept relies on an orbit hypothesis not studied in any detail, we do not consider it mature enough to warrant a study of potential detection rates in any detail, and we therefore only refer to established reviews of the astrophysical potential in the frequency band of LISA and DECIGO, e.g. Ref. [@Sathya2009]. Coalescing compact binaries --------------------------- Heavy stars in binary systems will end up as compact objects (such as NSs or BHs) inspiralling around each other, losing orbital energy and angular momentum through gravitational radiation. Depending on the proximity of the source and the detector’s sensitivity, we could detect GWs from such a system a few seconds up to a day before the merger and the formation of a single spinning object. These CBCs are expected to be the strongest sources of GWs in the frequency band of current GW detectors. To estimate the event rates for various binary systems, we will follow the calculations outlined in Ref. [@Abadie2010b]. To compare with predictions for initial and advanced LIGO (presented in Ref. [@Abadie2010b]), we also use only the inspiral part of the coalescence to estimate the *horizon distance* (the maximum distance to which we can observe a given system with a given signal-to-noise ratio (SNR)). We use here the same detection threshold on signal-to-noise ratio, a SNR of $\rho=8$, as in Ref. [@Abadie2010b] and consider the same fiducial binary systems: NS-NS (with $1.4 \msun$ each), BH-NS (BH mass $10 \msun$, NS with $1.4 \msun$), and BH-BH ($10 \msun$ each). For a binary of given masses, the sky-averaged horizon distance is given by $$D_{h} = \frac{4 \sqrt{5} \, G^{\frac{5}{6}} \, \mu^{\frac{1}{2}} \, M^{\frac{1}{3}}}{\sqrt{96} \, \pi^{\frac{2}{3}} \, c^{\frac{3}{2}} \, \rho} \sqrt{\int_{f_{\rm min}}^{f_{\rm ISCO}} \frac{f^{-\frac{7}{3}}}{\widetilde{S}_{\rm h}(f)}\,\mathrm{d}f} \; . \label{eq:Dhorizon}$$ Here, $M=M_1+M_2$ is the total mass and $\mu={M_1M_2}/{M}$ is the reduced mass of the system. We have used a lower cutoff of $f_{\rm min} = 1$Hz, and at the upper end the frequency of the innermost stable circular orbit is $f_{\rm ISCO} = c^3/(6^{3/2}\pi\ G\ M)$ Hz, which conventionally is taken as the end of the inspiral. Now, for any given type of binary (as characterized by the component masses), we obtain the observed event rate (per year) using $\dot{N}=R \cdot N_{\mathrm{G}}$, where we have adopted the approximation for the number of galaxies inside the visible volume from Eq. (5) of Ref. [@Abadie2010b]: $$\label{NG} N_{\mathrm{G}}=\frac{4}{3}\pi \left(\frac{D_{\mathrm{h}}}{\mathrm{Mpc}}\right)^3 (2.26)^{-3} \cdot 0.0116 \, ,$$ and the intrinsic coalescence rates $R$ per Milky-Way-type galaxy are given in Table 2 of Ref. [@Abadie2010b]. A single DFI combination $S_i$ has annual rates similar to initial LIGO, and the results for the network sensitivity of full OGO are summarized in Table \[T:cbc\_rates\]. For each binary, we give three numbers following the uncertainties in the intrinsic event rate (“pessimistic”, “realistic”, “optimistic”) as introduced in Ref. [@Abadie2010b]. NS-NS NS-BH BH-BH ------- ------------------- -------------------- -------------------- OGO (0.002, 0.2, 2.2) (0.001, 0.06, 2.0) (0.003, 0.1, 9) LIGO (2e-4, 0.02, 0.2) (7e-5, 0.004, 0.1) (2e-4, 0.007, 0.5) aLIGO (0.4, 40, 400) (0.2, 10, 300) (0.4, 20, 1000) : Estimated yearly detection rates for CBC events, given in triplets of the form (lower limit, realistic value, upper limit) as defined in Ref. [@Abadie2010b].[]{data-label="T:cbc_rates"} From this, we see that OGO achieves detection rates an order of magnitude better than initial LIGO. But we still expect to have only one event in about three years of observation assuming “realistic” intrinsic coalescence rates. The sensitivity of aLIGO is much better than for OGO above 10 Hz, and the absence of seismic noise does not help OGO much because the absolute sensitivities below 10 Hz are quite poor and only a very small fraction of SNR is contributed from the lower frequencies. This is the reason why OGO cannot compete directly with aLIGO in terms of total CBC detection rates, which are about two orders of magnitude lower. However, OGO does present an interesting scientific opportunity when run in parallel with aLIGO. If OGO indeed detects a few events over its mission lifetime, as the realistic predictions allow, it can give a very large improvement to the sky localization of these sources. Parameter estimation by aLIGO alone typically cannot localize signals enough for efficient electromagnetic follow-up identification. However, in a joint detection by OGO and aLIGO, triangulation over the long baseline between space-based OGO and ground-based aLIGO would yield a fantastic angular resolution. As signals found by OGO are very likely to be picked up by aLIGO as well, such joint detections indeed seem promising. Additionally, the three-dimensional configuration and independent channels of OGO potentially allow a more accurate parameter estimation than a network of two or three simple L-shaped interferometers could achieve. Stochastic background --------------------- There are mainly two kinds of stochastic GW backgrounds [@Allen99; @Maggiore00]: The first is the astrophysical background (sometimes also called astrophysical foreground), arising from unresolved astrophysical sources such as compact binaries [@Farmer03] and core-collapse supernovae [@Ferrari99]. It provides important statistical information about distribution of the sources and their parameters. The second is the cosmological background which was generated by various mechanisms in the early Universe [@Brustein95; @Turner97; @Ananda07]. It carries unique information about the very beginning of the Universe ($\sim 10^{-28}$s). Thus, the detection of the GW stochastic background is of great interest. Currently, there are two ways to detect the stochastic GW background. One of them [@Hogan01] takes advantage of the null stream (e.g. the Sagnac combination of LISA). By definition, the null stream is insensitive to gravitational radiation, while it suffers from the same noise sources as the normal data stream. A comparison of the energy contained in the null stream and the normal data stream allows us to determine whether the GW stochastic background is present or not. The other way of detection is by cross-correlation [@Allen99; @Seto06] of measurements taken by different detectors. In our language, this uses the GW background signal measured by one channel as the template for the other channel. In this sense, the cross-correlation can be viewed as matched filtering. Both ways require redundancy, i.e. more than one channel observing the same GW signal with independent noise. Luckily, the octahedron detector has plenty of redundancy, which potentially allows precise background detection. There are in total 12 dual-way laser links between spacecraft, forming 8 LISA-like triangular constellations. Any pair of two such LISA-like triangles that does not share common links can be used as an independent correlation. There are 16 such pairs within the octahedron detector. Within each pair, we can correlate the orthogonal TDI variables A, E and T (as they are denoted in LISA [@TintoDhurandhar]). Altogether, there are $16\times3^2=144$ cross-correlations. And we have yet more information encoded by the detector, which we can access by considering that any two connected links form a Michelson interferometer, thus providing a Michelson-TDI variable. Any two of these variables that do not share common links can be correlated. There are in total 36 such variables, forming 450 cross-correlations, from which we can construct the optimal total sensitivity. Furthermore, each of these is sensitive to a different direction on the sky. So the octahedron detector has the potential to detect anisotropy of the stochastic background. However, describing an approach for the detection of anisotropy is beyond the scope of this feasibility study. Instead, we will present here only an order of magnitude estimation of the total cross-correlation SNR. Usually, it can be expressed as $$\mathrm{SNR} = \frac{3H_0^2}{10\pi^2}\sqrt{T_{\mathrm{obs}}}\left[ 2 \sum_{k,l}\int_0^\infty {\rm d}f\frac{\gamma_{kl}^2(f)\Omega_{\rm gw}^2(f)} {f^6\widetilde{S}_{{\rm h}, k}(f)\widetilde{S}_{{\rm h}, l}(f)} \right]^{\frac{1}{2}}\,,$$ where $T_{\mathrm{obs}}$ is the observation time, $\Omega_{\rm gw}$ is the fractional energy-density of the Universe in a GW background, $H_0$ the *Hubble constant*, and $\widetilde{S}_{{\rm h}, k}(f)$ is the effective sensitivity of the $k$-th channel. $\gamma_{kl}(f)$ denotes the *overlap reduction function* between the $k$-th and $l$-th channels, introduced by Flanagan [@Flanagan93]. $$\gamma_{kl}(f) = \frac{5}{8\pi}\sum_{p=+,\times}\int {\rm d}\hat{\Omega}\,\mathrm{e}^{2\pi {\rm i} f \hat{\Omega}\cdot \Delta \mathbf{x}/c}F_k^p(\hat{\Omega})F_l^p(\hat{\Omega})\, ,$$ where $F_k^p(\hat{\Omega})$ is the antenna pattern function. As mentioned in the previous section, there might be 12 independent DFI solutions. These DFI solutions can form $12\times 11/2 = 66$ cross-correlations. According to Ref. [@Allen99], we know $\gamma_{kl}^2(f)$ varies between $0$ and $1$. As a rough estimate, we approximate $\sum_{k,l}\gamma_{kl}^2(f)\sim 10$; hence, we get the following result for OGO: $${\rm SNR} = 2.57\left(\frac{H_0}{72\,\frac{{\rm km} \, {\rm s^{-1}}}{{\rm Mpc}}}\right)^2\left(\frac{\Omega_{\rm gw}}{10^{-9}}\right)\left(\frac{T_{\mathrm{obs}}}{10\,{\rm yr}}\right)^{\frac{1}{2}}\,.$$ Initial LIGO has set an upper limit of $6.9\cdot 10^{-6}$ on $\Omega_{\rm gw}$ [@nature09], and aLIGO will be able to detect the stochastic background at the $1\cdot 10^{-9}$ level [@nature09]. Hence, our naive estimate of OGO’s sensitivity to the GW stochastic background is similar to that of aLIGO. Actually, an optimal combination of all the previously-mentioned possible cross-correlations would potentially result in an even better detection ability for OGO. Testing alternative theories of gravity --------------------------------------- ![Relative sensitivity of the full OGO network (scaled from S5 combination) to alternative polarizations: $+$ mode (*blue solid line*), $x$ mode (*red crosses*), vector-$x$ mode (*green dash-dotted line*), vector-$y$ mode (*black stars*), longitudinal mode (*magenta dashed line*), and breathing mode (*cyan plus signs*). []{data-label="F:altern_polar"}](ogo_polar_modes_all){width="\columnwidth"} In this section we will consider OGO’s ability to test predictions of General Relativity against alternative theories. In particular, we will estimate the sensitivity of the proposed detector to all six polarization modes that could be present in (alternative) metric theories of gravitation [@Hohmann2012]. We refer to Ref. [@Eardley1973] for a discussion on polarization states, which are (i) two transverse-traceless (tensorial) polarizations usually denoted as $+$ and $\times$, (ii) two scalar modes called breathing (or common) and longitudinal and (iii) two vectorial modes. We also refer to Refs. [@Will_LR; @Gair_LR] for reviews on alternative theories of gravity. We have followed the procedure for computing the sensitivity of OGO, as outlined above, for the four modes not present in General Relativity, and we compare those sensitivities to the results for the $+, \times$ modes as presented in Fig. \[F:Sens\]. The generalization of the transfer function used in this paper \[Eq. \[Eq:TF\]\] for other polarization modes is given in Ref. [@Chamberlin_2012]. We have found that all seven generators show similar sensitivity for each mode. OGO is not sensitive to the common (breathing) mode, which is not surprising as it can be attributed to a common displacement noise, which we have removed by our procedure. The sensitivity to the second (longitudinal) scalar mode scales as $\epsilon^{-4}$ at low frequencies and is much worse than the sensitivity to the $+, \times$ polarizations below 200Hz. However, OGO is more sensitive to the longitudinal mode (by about an order of magnitude) above 500Hz. The sensitivity of OGO to vectorial modes is overall similar to the $+, \times$ modes: it is by a few factors less sensitive to vectorial modes below 200Hz and by similar factors more sensitive above 300Hz. These sensitivities are shown in Fig. \[F:altern\_polar\]. Pulsars – Continuous Waves -------------------------- CWs are expected from spinning neutron stars with nonaxisymmetric deformations. Spinning NSs are already observed as radio and gamma-ray pulsars. Since CW emission is powered by the spindown of the pulsar, the strongest emitters are the pulsars with high spindowns, which usually are young pulsars at rather high frequencies. Note that the standard emission model [@Jaranowski1998] predicts a gravitational wave frequency $f_{\mathrm{gw}}=2f$, while alternative models like free precession [@Jones2001] and $r$-modes [@Andersson1998] also allow emission at $f_{\mathrm{gw}}=f$ and $f_{\mathrm{gw}}=\tfrac{4}{3}f$, where $f$ is the NS spin frequency. OGO has better sensitivity than initial LIGO below 133Hz, has its best sensitivity around 78Hz, and is better than aLIGO below 9Hz. This actually fits well with the current radio census of the galactic pulsar population, as given by the ATNF catalog [@ATNF]. As shown in Fig. \[fig:pulsars\_atnf\], the bulk of the population is below $\sim$ 10Hz, and also contains many low-frequency pulsars with decent spindown values, even including a few down to $\sim$ 0.1Hz. We estimate the abilities of OGO to detect CW emission from known pulsars following the procedure outlined in Ref. [@Abadie2011] for analysis of the Vela pulsar. The GW strain for a source at distance $D$ is given as $$h_0 = \frac{4 \pi^2 G I_{zz} \epsilon f^2}{c^4 D} \, ,$$ where $\epsilon$ is the ellipticity of the neutron star and we assume a canonical momentum of inertia $I_{zz} = 10^{38}$kgm$^2$. After an observation time $T_{\rm obs}$, we could detect a strain amplitude $$h_0 = \Theta\sqrt{\frac{S_\mathrm{h}}{T_{\mathrm{obs}}}} \, .$$ The statistical factor is $\Theta\approx11.4$ for a fully coherent targeted search with the canonical values of 1% and 10% for false alarm and false dismissal probabilities, respectively [@Abbott2004]. We find that, for the Vela pulsar (at a distance of 290pc and a frequency of $f_{\mathrm{Vela,gw}}=2 \cdot 11.19$Hz), with $T_{\mathrm{obs}}=30$ days of observation, we could probe ellipticities as low as $\epsilon \sim 5\cdot10^{-4}$ with the network OGO configuration. Several known low-frequency pulsars outside the aLIGO band would also be promising objectives for OGO targeted searches. ![Population of currently known pulsars in the frequency-spindown plain ($f$-$\dot{f}$). OGO could beat initial LIGO left of the red solid line and Advanced LIGO left of the green dashed line. Data for this plot were taken from Ref. [@ATNF] on March 2, 2012.[]{data-label="fig:pulsars_atnf"}](pulsars_atnf_f_fdot){width="\columnwidth"} All-sky searches for unknown pulsars with OGO would focus on the low-frequency range not accessible to aLIGO with a search setup comparable to current Einstein@Home LIGO searches [@Aasi2013]. As seen above, the sensitivity estimate factors into a search setup related part $\Theta / \sqrt{T_{\mathrm{obs}}}$ and the sensitivity $\sqrt{S_\mathrm{h}}$. Therefore, scaling a search with parameters identical to the Einstein@Home S5 runs to OGO’s best sensitivity at 76Hz would reach a sensitivity of $h_0 \approx 3 \cdot 10^{-25}$. This would, for example, correspond to a neutron star ellipticity of $\epsilon \sim 4.9 \cdot 10^{-5}$ at a distance of 1kpc. Since the computational cost of such searches scales with $f^2$, low-frequency searches are actually much more efficient and would allow very deep searches of the OGO data, further increasing the competitiveness. Note, however, that for low-frequency pulsars the ellipticities required to achieve detectable GW signals can be very high, possibly mostly in the unphysical regime. On the other hand, for “transient CW”-type signals [@Prix2011], low-frequency pulsars might be the strongest emitters, even with realistic ellipticities. Other sources ------------- Many (indirect) observational evidences exist for stellar mass BHs, which are the end stages of heavy star evolution, as well as for supermassive BHs, the result of accretion and galactic mergers throughout the cosmic evolution, in galactic nuclei. On the other hand, there is no convincing evidence so far for a BH of an intermediate mass in the range of $10^2-10^4\msun$. These IMBHs might, however, still exist in dense stellar clusters [@Miller2004; @Pasquato2010]. Moreover, stellar clusters could be formed as large, gravitationally bound groups, and collision of two clusters would produce inspiralling binaries of IMBHs [@IMBH_pau2006; @IMBH_pau2009]. The ISCO frequency of the second orbital harmonic for a $300 \msun$-$300 \msun$ system is about 7Hz, which is outside the sensitivity range of aLIGO. Still, those sources could show up through the higher harmonics (the systems are expected to have non-negligible eccentricity) and through the merger and ring-down gravitational radiation [@Fregeau2006; @Mandel2008; @Yagi2012]. The ground-based LIGO and VIRGO detectors have already carried out a first search for IMBH signals in the $100\msun$ to $450\msun$ mass range [@Abadie2012_imbh]. With its better low-frequency sensitivity, OGO can be expected to detect a GW signal from the inspiral of a $300 \msun$-$300 \msun$ system in a quasicircular orbit up to a distance of approximately 245Mpc, again using Eq. (\[eq:Dhorizon\]). This gives the potential for discovery of such systems and for estimating their physical parameters. As for other advanced detectors, unmodeled searches (as opposed to the matched-filter CBC and CW searches; see Ref. [@Abadie2012_burst] for a LIGO example) of OGO data have the potential for detecting many other types of gravitational wave sources, including, but not limited to, supernovae and cosmic string cusps. However, as in the case for IMBHs, the quantitative predictions are hard to produce due to uncertainties in the models. Summary and Outlook {#S:Summary} =================== In this paper, we have presented for the first time a three-dimensional gravitational wave detector in space, called the Octahedral Gravitational wave Observatory (OGO). The detector concept employs displacement-noise free interferometry (DFI), which is able to cancel some of the dominant noise sources of conventional GW detectors. Adopting the octahedron shape introduced in Ref. [@chen2006], we put spacecraft in each corner of the octahedron. We considered a LISA-like receiver-transponder configuration and found multiple combinations of measurement channels, which allow us to cancel both laser frequency and acceleration noise. This new three-dimensional result generalizes the Mach-Zehnder interferometer considered in Ref. [@chen2006]. We have identified a possible halolike orbit around the Lagrange point L1 in the Sun-Earth system that would allow the octahedron constellation to be stable enough. However, this orbit limits the detector to an arm length of $\approx1400$ km. Much better sensitivity and a richer astrophysical potential are expected for longer arm lengths. Therefore, we also looked for alternative orbits and found a possible alternative allowing for $\approx 2\cdot10^{9}$ m arms, but is is not clear yet if this would be stable enough. Future studies are required to relax the equal-arm-length assumption of our DFI solutions, or to determine a stable, long-arm-length constellation. Next, we have computed the sensitivity of OGO-like detectors – and have shown that the noise-cancelling combinations also cancel a large fraction of the GW signal at low frequencies. The sensitivity curve therefore has a characteristic slope of $f^{-2}$ at the low-frequency end. However, the beauty of this detector is that it is limited by a single noise source at all frequencies: shot noise. Thus, any reduction of shot noise alone would improve the overall sensitivity. This could, in principle, be achieved with DECIGO-like cavities, squeezing or other advanced technologies. Also, OGO does not require drag-free technology and has moderate requirements on other components so that it could be realized with technology already developed for LISA Pathfinder and eLISA. When comparing a DFI-enabled OGO with a detector of similar design, but with standard TDI, we find that at $\approx1400$ km, the same sensitivity could be reached by a TDI detector with very modest acceleration noise requirements. However, at longer arm lengths DFI becomes more advantageous, reaching the same sensitivity as TDI under LISA requirements but without drag-free technology and clock transfer, at $\approx 2\cdot10^{9}$ m. Such a DFI detector would have its best frequency range between LISA and DECIGO, with peak sensitivity better than LISA and approaching DECIGO without the latter mission concept’s tight acceleration noise requirements and with no need for cavities. Finally, we have assessed the scientific potentials of OGO, concentrating on the less promising, but more mature short-arm-length version. We estimated the event rates for coalescing binaries, finding that OGO is better than initial LIGO, but does not reach the level of advanced LIGO. Any binary detected with both OGO and aLIGO could be localized in the sky with very high accuracy. Also, the three-dimensional satellite constellation and number of independent links makes OGO an interesting mission for detection of the stochastic GW background or hypothetical additional GW polarizations. Further astrophysically interesting sources such as low-frequency pulsars and IMBH binaries also lie within the sensitive band of OGO, but again the sensitivity does not reach that of aLIGO. However, we point out that the improvement in the low-frequency sensitivity with increasing arm length happens faster for DFI as compared to the standard TDI. Therefore, searching for stable three-dimensional (octahedron) long-baseline orbits could lead to an astrophysically much more interesting mission. Regarding possible improvements of the presented setup, there are several possibilities to extend and improve the first-order DFI scheme presented here. One more spacecraft could be added in the middle, increasing the number of usable links. Breaking the symmetry of the octahedron could modify the steep response function at low frequencies. This should be an interesting topic for future investigations. In principle, the low-frequency behavior of OGO-like detectors could also be improved by more advanced DFI techniques such as introducing artificial time delays [@Somiya2007a; @Somiya2007b]. This would result in a three-part power law less steep than the shape derived in Sec. \[sec:transfer\_function\]. On the other hand, this would also introduce a new source of time delay noise. Therefore, such a modification requires careful investigation. Acknowledgments =============== We would like to thank Gerhard Heinzel for very fruitful discussions, Albrecht Rüdiger for carefully reading through the paper and helpful comments, Sergey Tarabrin for discussions on the optical layout, Masaki Ando for kindly sharing DECIGO simulation tools and Guido Müller for helpful comments on the final manuscript. Moreover, Berit Behnke, Benjamin Knispel, Badri Krishnan, Reinhard Prix, Pablo Rosado, Francesco Salemi, Miroslav Shaltev and others helped us with their knowledge regarding the astrophysical sources. We would also like to thank the anonymous referee for very insightful and detailed comments on the original manuscript. The work of the participating students was supported by the International Max-Planck Research School for Gravitational Waves (IMPRS-GW) grant. The work of S. B. and Y. W. was partially supported by DFG Grant No. SFB/TR 7 Gravitational Wave Astronomy and DLR (Deutsches Zentrum für Luft- und Raumfahrt). Furthermore, we want to thank the Deutsche Forschungsgemeinschaft (DFG) for funding the Cluster of Excellence QUEST – Centre for Quantum Engineering and Spacetime Research. We thank the LIGO Scientific Collaboration (LSC) for supplying the LIGO and aLIGO sensitivity curves. Finally, we would like to emphasize that the idea of a three-dimensional GW detector in space is the result of a student project from an IMPRS-GW lecture week. This document has been assigned LIGO document number and LIGO-P1300074 and AEI-preprint number AEI-2013-261. Details on calculating the displacement and laser noise free combinations {#S:Appendix} ========================================================================= Here we will give details on building the displacement (acceleration) and laser noise free configurations. The derivations closely follow the method outlined in [@TintoDhurandhar]. We want to find the generators solving Eq. (\[E:TDIAcc\_FinalEq\]), so called reduced generators because they correspond to the reduced set $( q_{BC}, q_{CE}, q_{DB}, q_{DC}, q_{DF}, q_{EF} )$. We start with building the ideal $Z$: $$\begin{aligned} Z = \left\{ \begin{array}{rcl} f_1 &=& (\mathcal{D}-1)^2 \\ f_2 &=& (\mathcal{D}-1)\mathcal{D} \\ f_3 &=& (1-\mathcal{D})(\mathcal{D}-1) \\ f_4 &=& (\mathcal{D}-1)((1-\mathcal{D})\mathcal{D}-1) \\ f_5 &=& \mathcal{D}-1 \\ f_6 &=& \mathcal{D}-1 \\ \end{array} \right.\,. \label{E:TDIAcc_ideal} \end{aligned}$$ The corresponding Gröbner basis to this ideal is: $$\mathcal{G}=\{ g_1 = \mathcal{D} - 1 \}. \label{E:E:TDIAcc_Groebner}$$ The connection between $f_i$ and $g_j$ is defined by two transformation matrices $$\begin{aligned} d & =& \left( \begin{array}{c} \mathcal{D}-1 \\ \mathcal{D} \\ 1-\mathcal{D} \\ (1-\mathcal{D})\mathcal{D}-1 \\ 1 \\ 1 \\ \end{array} \right) \end{aligned}$$ and $c$ with (at least) two possible solutions $$c^{(1)} = \left( 0 \ 0 \ 0 \ 0 \ 1 \ 0 \right)\; \textrm{or} \; c^{(2)} = \left( 0 \ 0 \ 0 \ 0 \ 0 \ 1 \right).$$ The resulting basis is not unique and not necessarily independent. The first 6 reduced generators are given by the row vectors of the matrix $A^{(1)} = a^{(1)}_i = I - d\cdot c^{(1)}$ : $$\begin{aligned} a^{(1)}_1 & = \left\{ 1 , 0 , 0 , 0 , 0 , 1-\mathcal{D} \right\}, \\ a^{(1)}_2 & = \left\{ 0 , 1 , 0 , 0 , 0 , -\mathcal{D} \right\}, \\ a^{(1)}_3 & = \left\{ 0 , 0 , 1 , 0 , 0 , (\mathcal{D}-1)\mathcal{D} \right\}, \\ a^{(1)}_4 & = \left\{ 0 , 0 , 0 , 1 , 0 , 1+(\mathcal{D}-1)\mathcal{D} \right\}, \\ a^{(1)}_5 & = \left\{ 0 , 0 , 0 , 0 , 1 , -1 \right\}, \\ a^{(1)}_6 & = \left\{ 0 , 0 , 0 , 0 , 0 , 0 \right\}. \end{aligned}$$ These reduced generators correspond directly to values for $( q_{BC}, q_{CE}, q_{DB}, q_{DC}, q_{DF}, q_{EF} )$. As the Gröbner basis contains only one element, we cannot form other generator from $S$-polynomial. We can form 6 other generators using $c^{(2)}$ instead of $c^{(1)}$. After applying those generators we have the following acceleration-free combinations: $$\begin{aligned} a^{(1)}_1 s^{n} & = 2 (p_{B} - p_{C} + p_{E} - p_{F} + \mathcal{D}(-p_{A} + p_{B} - p_{D} + p_{E} \nonumber \\ & + (p_{B} - p_{C} + p_{E} - p_{F}) q_{BA})), \\ a^{(1)}_2 s^{n} & = -2 \mathcal{D} (p_{A} + p_{D} + p_{C} (-1 + q_{BA}) + p_{F} (-1 + q_{BA}) \nonumber \\ & - (p_{B} + p_{E}) q_{BA}), \\ a^{(1)}_3 s^{n} & = 2 \mathcal{D} ((1 + \mathcal{D}) p_{A} + p_{D} - p_{E} - \mathcal{D} (p_{C} - p_{D} + p_{F}) \nonumber \\ & + p_{B} (-1 + q_{BA}) - (p_{C} - p_{E} + p_{F}) q_{BA}), \\ a^{(1)}_4 s^{n} & = 2 (p_{B} - p_{C} + p_{E} + \mathcal{D}^2 (p_{A} - p_{C} + p_{D} - p_{F}) \nonumber \\ & - p_{F} + \mathcal{D} (p_{B} - p_{C} + p_{E} - p_{F}) q_{BA}), \\ a^{(1)}_5 s^{n} & = 2 \mathcal{D} (p_{A} + p_{D} + p_{B} (-1 + q_{BA}) + p_{E} (-1 + q_{BA}) \nonumber \\ & - (p_{C} + p_{F}) q_{BA}), \\ a^{(1)}_6 s^{n} & = 2 \mathcal{D} (p_{B} - p_{C} + p_{E} - p_{F}) q_{BA}, \end{aligned}$$ where $s^{n}_{IJ}$ are given by Eq. (\[E:MesN\]). Note that we have a free (polynomial) function of delay $q_{BA}$ which we can choose arbitrary. We will omit subscripts $BA$ and use $q\equiv q_{BA}$. The arbitrariness of this function implies that terms which contain $q$ and terms free of $q$ are two independent sets of generators. We will keep $q$ until we obtain laser noise free combinations, and then split each generator in two. After some analysis only two out of six acceleration free generators are independent, so we can rewrite them as $$\begin{aligned} s_1 &= y_{12} + \mathcal{D}(y_{13} + qy_{12}),\\ s_3 &= -y_{13} + \mathcal{D}(y_{12} -y_{13}) + qy_{12},\\ s_4 &= y_{12} + \mathcal{D}q y_{12} + \mathcal{D}^2(y_{12} - y_{13}),\\ s_2+s_5 &= y_{12} - 2y_{13},\label{E:s2ps5}\\ s_2-s_5 &= (2q-1)y_{12},\label{E:s2ms5}\\ s_6 &= q y_{12}, \end{aligned}$$ where $$\begin{aligned} s_1 &=& \frac{a^{(1)}_1 s^{n}}{2}, s_2 = -\frac{\mathcal{D}^{-1}( a^{(1)}_2 s^{n})}{2}, s_3 = \frac{\mathcal{D}^{-1}( a^{(1)}_3 s^{n})}{2} \nonumber \\ s_4 &=& \frac{a^{(1)}_4 s^{n}}{2}, s_5 = \frac{\mathcal{D}^{-1}( a^{(1)}_5 s^{n})}{2} , s_6 = \frac{\mathcal{D}^{-1}( a^{(1)}_6 s^{n})}{2}\end{aligned}$$ and $$y_{12} = p_B+p_E-p_C-p_F,\, y_{13} = p_B+p_E-p_A-p_D\,.$$ We have introduced the inverse delay operator, $ \mathcal{D}^{-1}$, for mathematical convenience, which obeys $\mathcal{D}\mathcal{D}^{-1} = \mathds{1}$. One can easily get rid of it by applying the delay operator on both sides. The final result will not contain the operator $\mathcal{D}^{-1}$. Next we use Eqs. (\[E:s2ps5\]) and (\[E:s2ms5\]) to express $y_{12}, y_{13}$ and eliminate them from the other equations. The resulting combinations that eliminate both acceleration and laser noise are $$\begin{aligned} &(1-2q)s_1 + (-1-2\mathcal{D}q)s_2 + (1+\mathcal{D})s_5\\ &(1-2q)s_3 + \mathcal{D}(q-1)s_2 + (-1+2q+q\mathcal{D})s_5\\ &(1-2q)s_4 - (1+ \mathcal{D}q)(s_2-s_5) - \mathcal{D}^2((1-q)s_2 - qs_5)\\ &(1-2q)s_6 - q(s_2 - s_5). \end{aligned}$$ Out of these solutions we obtain seven independent generators which we have rewritten in the final form similar to the $Y$-equations from Sec. \[S:TDI\]. They are explicitly given by Eqs. (\[E:S1\])–(\[E:S7\]). [99]{} G. M. Harry \[LIGO Scientific Collaboration\], Class. Quant. Grav.  [**27**]{}, 084006 (2010). The Virgo Collaboration, VIR–027A–09 (2009). <https://tds.ego-gw.it/itf/tds/file.php?callFile=VIR-0027A-09.pdf> J. Abadie [*et al.*]{} \[LIGO Scientific and Virgo Collaborations\], Class. Quant. Grav.  [**27**]{}, 173001 (2010) [\[arXiv:1003.2480\]](http://www.arxiv.org/abs/1003.2480). C.L. Fryer and K.C.B. New, Living Rev. Relativity 14 (2011), 1. <http://www.livingreviews.org/lrr-2011-1> B. Owen, LIGO-T0900053 (2009), [\[arXiv:0903.2603\]](http://www.arxiv.org/abs/0903.2603) B. Allen and J. D. Romano, Phys. Rev. D [**59**]{}, 102001 (1999) [\[arXiv:gr-qc/9710117\]](http://www.arxiv.org/abs/gr-qc/9710117). M. Maggiore, Phys. Rept.  [**331**]{}, 283 (2000)\ [\[arXiv:gr-qc/9909001\]](http://www.arxiv.org/abs/gr-qc/9909001). K. Danzmann and the LISA Study Team, Adv. Space Res. [**32**]{}, 12331242 (2003). P. Amaro-Seoane [*et al.*]{}, GW Notes, Vol. 6, p. 4-110 (2013) [\[arXiv:1201.3621\]](http://www.arxiv.org/abs/1201.3621). M. Ando [*et al.*]{}, Class. Quant. Grav.  [**27**]{}, 084010 (2010). Y. Chen [*et al.*]{}, Phys. Rev. Lett.  [**97**]{}, 151103 (2006) [\[arXiv:gr-qc/0603054\]](http://www.arxiv.org/abs/gr-qc/0603054). S. Kawamura and Y. Chen, Phys. Rev. Lett.  [**93**]{}, 211103 (2004) [\[arXiv:gr-qc/0405093\]](http://www.arxiv.org/abs/gr-qc/0405093). Y. Chen and S. Kawamura, Phys. Rev. Lett.  [**96**]{}, 231102 (2006) [\[arXiv:gr-qc/0504108\]](http://www.arxiv.org/abs/gr-qc/0504108). M. Tinto and S. V. Dhurandhar, Living Rev. Relativity 8 (2005), 4 [\[arXiv:gr-qc/0409034\]](http://www.arxiv.org/abs/gr-qc/0409034).\ <http://www.livingreviews.org/lrr-2005-4> M. Otto, G. Heinzel and K. Danzmann, Class. Quant. Grav.  [**29**]{}, 205003 (2012). G. Gomez, A. Jorba, J. Masdemont and C. Simo, CeMDA **56**, 4, pp. 541-562 (1993). K.C. Howell and B.T. Barden, 50th International Astronautical Federation Congress, IAF/IAA Paper 99-A707 (1999). NGO science working team, ESA/SRE(2011)19 (2011) European Space Agency, ESA/SRE(2011)3 (2011). <http://sci.esa.int/science-e/www/object/index.cfm?fobjectid=48364> D. Folta, Ann. N. Y. Acad. Sci. **1017** (2004). B. Buchberger, Aequationes mathematicae **4**, 374-383 (1970). M. Maggiore, “Gravitational waves”, *Oxford University Press, New York* (2007). F. B. Estabrook and H. D. Wahlquist, Gen. Rel. Grav. [**6**]{}, 439 (1975). J. Abadie [*et al.*]{} \[LIGO and Virgo Collaborations\], LIGO-T0900499-v19 (2010) [\[arXiv:1003.2481\]](http://www.arxiv.org/abs/1003.2481). D. Shoemaker *et al.*, LIGO-T0900288-v3 (2010).\ <https://dcc.ligo.org/LIGO-T0900288-v3/public> M. R. Drinkwater [*et al.*]{}, Space Science Reviews, v. 108, Issue 1, p. 419-432 (2003) G. Sechi [*et al.*]{}, Preprints of the 18th IFAC World Congress, Milano, pp.733-740 (2011) S. Barke, M. Tröbs, B. Sheard, G. Heinzel and K. Danzmann, Appl. Phys. B **98**, 1, 33-39 (2010). M. Rakhmanov, LIGO-T060237-00-D (2005).\ <https://dcc.ligo.org/T060237-x0/public> R. Schnabel, N. Mavalvala, D.E. McClelland and P.K. Lam, Nature Comm. **1**, 121 (2010) A. Khalaidovski, H. Vahlbruch, N. Lastzka, C. Graf, H. Luck, K. Danzmann, H. Grote and R. Schnabel, J. Phys. Conf. Ser.  [**363**]{}, 012013 (2012) [\[arXiv:1112.0198\]](http://www.arxiv.org/abs/1112.0198). The LIGO Scientific Collaboration, Nature Phys.  [**7**]{}, 962 (2011) [\[arXiv:1109.2295\]](http://www.arxiv.org/abs/1109.2295). B. S. Sathyaprakash and B. F. Schutz, Living Rev. Relativity, 12 (2009), 2 [\[arXiv:0903.0338\]](http://www.arxiv.org/abs/0903.0338).\ <http://www.livingreviews.org/lrr-2009-2> A. J. Farmer and E. S. Phinney, Mon. Not. Roy. Astron. Soc.  [**346**]{}, 1197 (2003) [\[arXiv:astro-ph/0304393\]](http://www.arxiv.org/abs/astro-ph/0304393). V. Ferrari, S. Matarrese and R. Schneider, Mon. Not. Roy. Astron. Soc.  [**303**]{}, 247 (1999)\ [\[arXiv:astro-ph/9804259\]](http://www.arxiv.org/abs/astro-ph/9804259). R. Brustein, M. Gasperini, M. Giovannini and G. Veneziano, Phys. Lett. B [**361**]{}, 45 (1995)\ [\[arXiv:hep-th/9507017\]](http://www.arxiv.org/abs/hep-th/9507017). M. S. Turner, Phys. Rev. D [**55**]{}, 435 (1997)\ [\[arXiv:astro-ph/9607066\]](http://www.arxiv.org/abs/astro-ph/9607066). K. N. Ananda, C. Clarkson and D. Wands, Phys. Rev. D [**75**]{}, 123518 (2007) [\[arXiv:gr-qc/0612013\]](http://www.arxiv.org/abs/gr-qc/0612013). C. J. Hogan and P. L. Bender, Phys. Rev. D [**64**]{}, 062002 (2001) [\[arXiv:astro-ph/0104266\]](http://www.arxiv.org/abs/astro-ph/0104266). N. Seto, Phys. Rev. D [**73**]{}, 063001 (2006)\ [\[arXiv:gr-qc/0510067\]](http://www.arxiv.org/abs/gr-qc/0510067). E. E. Flanagan, Phys. Rev. D [**48**]{}, 2389 (1993) [\[arXiv:astro-ph/9305029\]](http://www.arxiv.org/abs/astro-ph/9305029). B. P. Abbott [*et al.*]{} \[LIGO Scientific and VIRGO Collaborations\], Nature [**460**]{}, 990 (2009) [\[arXiv:0910.5772\]](http://www.arxiv.org/abs/0910.5772). M. Hohmann, Phys. Rev. D [**85**]{}, 084024 (2012) [\[arXiv:1105.2555\]](http://www.arxiv.org/abs/1105.2555). D. M. Eardley, D. L. Lee and A. P. Lightman, Phys. Rev. D [**8**]{}, 3308 (1973). C. M. Will, Living Rev. Relativity 9 (2006), 3\ [\[arXiv:gr-qc/0510072\]](http://www.arxiv.org/abs/gr-qc/0510072).\ <http://www.livingreviews.org/lrr-2009-2> J. R. Gair, M. Vallisneri, S. L. Larson and J. G. Baker, Living Rev. Relativity 16 (2013), 7 [\[arXiv:1212.5575\]](http://www.arxiv.org/abs/1212.5575). <http://www.livingreviews.org/lrr-2013-7> S. J. Chamberlin and X. Siemens, Phys. Rev. D [**85**]{}, 082001 (2012) [\[arXiv:1111.5661\]](http://www.arxiv.org/abs/1111.5661). P. Jaranowski, A. Krolak and B. F. Schutz, Phys. Rev. D [**58**]{}, 063001 (1998) [\[arXiv:gr-qc/9804014\]](http://www.arxiv.org/abs/gr-qc/9804014). D. I. Jones and N. Andersson, Mon. Not. Roy. Astron. Soc.  [**331**]{}, 203 (2002) [\[arXiv:gr-qc/0106094\]](http://www.arxiv.org/abs/gr-qc/0106094). N. Andersson, K. D. Kokkotas and N. Stergioulas, Astrophys. J.  [**516**]{}, 307 (1999) [\[arXiv:astro-ph/9806089\]](http://www.arxiv.org/abs/astro-ph/9806089). R.N. Manchester, G.B. Hobbs, A. Teoh and M. Hobbs, Astron. J.  [**129**]{}, 1993 (2005) [\[arXiv:astro-ph/0412641\]](http://www.arxiv.org/abs/astro-ph/0412641). <http://www.atnf.csiro.au/research/pulsar/psrcat> J. Abadie [*et al.*]{} \[LIGO Scientific and Virgo Collaborations\], Astrophys. J.  [**737**]{}, 93 (2011) [\[arXiv:1104.2712\]](http://www.arxiv.org/abs/1104.2712). B. Abbott [*et al.*]{} \[LIGO Scientific Collaboration\], Phys. Rev. D [**69**]{}, 082004 (2004) [\[arXiv:gr-qc/0308050\]](http://www.arxiv.org/abs/gr-qc/0308050). J. Aasi [*et al.*]{} \[The LIGO Scientific and the Virgo Collaboration\], Phys. Rev. D [**87**]{}, 042001 (2013) [\[arXiv:1207.7176\]](http://www.arxiv.org/abs/1207.7176). R. Prix, S. Giampanis and C. Messenger, Phys. Rev. D [**84**]{}, 023007 (2011) [\[arXiv:1104.1704\]](http://www.arxiv.org/abs/1104.1704). M. C. Miller and E. J. M. Colbert, Int. J. Mod. Phys. D [**13**]{}, 1 (2004) [\[arXiv:astro-ph/0308402\]](http://www.arxiv.org/abs/astro-ph/0308402). M. Pasquato, [\[arXiv:1008.4477\]](http://www.arxiv.org/abs/1008.4477). P. Amaro-Seoane and M. Freitag, Astrophys. J.  [**653**]{}, L53 (2006) [\[arXiv:astro-ph/0610478\]](http://www.arxiv.org/abs/astro-ph/0610478). P. Amaro-Seoane, M. C. Miller and M. Freitag, Astrophys. J.  [**692**]{}, L50 (2009) [\[arXiv:0901.0604\]](http://www.arxiv.org/abs/0901.0604). J. M. Fregeau, S. L. Larson, M. C. Miller, R. W. O’Shaughnessy and F. A. Rasio, Astrophys. J.  [**646**]{}, L135 (2006) [\[arXiv:astro-ph/0605732\]](http://www.arxiv.org/abs/astro-ph/0605732). I. Mandel, D. A. Brown, J. R. Gair and M. C. Miller, Astrophys. J.  [**681**]{}, 1431 (2008) [\[arXiv:0705.0285\]](http://www.arxiv.org/abs/0705.0285). K. Yagi, Class. Quant. Grav.  [**29**]{}, 075005 (2012) [\[arXiv:1202.3512\]](http://www.arxiv.org/abs/1202.3512). J. Abadie [*et al.*]{} \[LIGO Scientific and Virgo Collaborations\], Phys. Rev. D [**85**]{}, 102004 (2012) [\[arXiv:1201.5999\]](http://www.arxiv.org/abs/1201.5999). J. Abadie [*et al.*]{} \[LIGO Scientific and Virgo Collaborations\], Phys. Rev. D [**85**]{}, 122007 (2012) [\[arXiv:1202.2788\]](http://www.arxiv.org/abs/1202.2788). K. Somiya, K. Goda, Y. Chen and E. E. Mikhailov, J. Phys. Conf. Ser.  [**66**]{}, 012053 (2007)\ [\[arXiv:gr-qc/0610117\]](http://www.arxiv.org/abs/gr-qc/0610117). K. Somiya, Y. Chen, K. Goda and E. E. Mikhailov, Phys. Rev. D [**76**]{}, 022002 (2007). S. L. Larson, W. A. Hiscock and R. W. Hellings, Phys. Rev. D [**62**]{}, 062001 (2000) [\[arXiv:gr-qc/9909080\]](http://www.arxiv.org/abs/gr-qc/9909080). K. Yagi, Int. J. Mod. Phys. D [**22**]{}, 1341013 (2013) [\[arXiv:1302.2388\]](http://www.arxiv.org/abs/1302.2388).

--- abstract: 'The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the [*same*]{} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal “thin sandwich” viewpoint coupled with the transformation properties of the lapse function.' address: 'Department of Physics, North Carolina State University, Raleigh, NC 27695-8202' author: - 'James W. York, Jr.[@address]' date: 'October 15, 1998' title: 'Conformal “thin sandwich” data for the initial-value problem of general relativity[^1]' --- In this paper I propose a new interpretation of the four Einstein vacuum initial-value constraints. (The presence of matter would add nothing new to the analysis.) Partly in the spirit of a “thin sandwich” viewpoint, I base this approach on prescribing the [*conformal*]{} metric [@York72] on each of two nearby spacelike hypersurfaces (“time slices” $t=t^\prime \mbox{ and } t=t^\prime + \delta t$) that make a “thin sandwich” (TS). Essential use is made of a new understanding of the role of the lapse function in general relativity [@AAJWY98; @YorkFest]. The new formulation could prove useful both conceptually, and in practice, as a way to construct initial data in which one has a hold on the input data different from that in the currently accepted approach. The new approach allows us to [*derive*]{} from its dynamical and metrical foundations the important scaling law $\bar{A}^{i j} = \psi^{-10} A^{i j}$ for the traceless part of the extrinsic curvature. This rule is simply postulated in the one-hypersurface approach. The new formulation differs from the well-known TS conjecture of Baierlein, Sharp, and Wheeler (BSW), in which the [*full*]{} spatial Riemannian metric $\bar{g}_{i j}$ is given on each of two infinitesimally separated hypersurfaces [@BSW; @Wheeler; @MTW]. (The orthogonal separation $\bar{N} \delta t$ between the slices is assumed never to change signs in the BSW proposal and also here.) The four unknowns needed to solve the constraints were taken by BSW to be the “lapse function” $\bar{N}(x)$ and the spatial “shift vector” $\bar{\beta}^i(x)$ (see below). By using a known vacuum spacetime solution of Einstein’s equations from which to obtain BSW data, one sees that their proposal must sometimes work. However, an analysis of the BSW proposal by Bartnik and Fodor [@Bartnik] describes the general situation clearly, and one can only conclude that the BSW proposal is unsatisfactory. For example, an infinite number of non-trivial counterexamples to the BSW conjecture, based on compact three-geometries of negative scalar curvature with one (not $\infty^1$) constraint (fixed volume), have been described in [@York83]. The initial-value problem (IVP), that is, satisfying the four constraints, is fundamentally a [*one*]{}-hypersurface embedding problem. The four constraints are the Gauss-Codazzi embedding equations for a time slice in a Ricci-flat spacetime. They limit the allowed values of the metric $\bar{g}_{i j}$ and extrinsic curvature $\bar{K}_{i j}$ of an “initial” time slice in a yet-to-be constructed vacuum spacetime. This basic form will be referred to as the $(\Sigma, \bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$ form, where $\Sigma$ is the slice, say $t=t^\prime$. In this case, the constraints have already been posed as a semi-linear elliptic system for spatial scalar and spatial vector potentials, generalizations of the Newtonian potential [@CBYHeld; @OMY; @York79]. A significant virtue of the formulation in this paper is that the constraints again become a semi-linear elliptic system with the [*same*]{} essential mathematical structure as has the $(\Sigma, \bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$ form. This surprising result, as we shall see, arises from the behavior of the lapse function [@AAJWY98; @YorkFest]. The constraint equations on $\Sigma$ are, in vacuum, $$\begin{aligned} \bar{\nabla}_j(\bar{K}^{i j}-\bar{K}\bar{g}^{i j})&=&0 \; , \label{Eq:MomCon}\\ R(\bar{g})-\bar{K}_{i j}\bar{K}^{i j}+\bar{K}^2&=&0\;, \label{Eq:HamCon}\end{aligned}$$ where $R(\bar{g})$ is the scalar curvature of $\bar{g}_{i j}$, $\bar{\nabla}_j$ is the Levi-Civita connection of $\bar{g}_{i j}$; and $\bar{K}$ is the trace of $\bar{K}_{i j}$, also called the “mean curvature” of the slice. (A review of this geometry is given in [@York79].) The overbar is used to denote quantities that satisfy the constraints. The time derivative of the spatial metric $\bar{g}_{i j}$ is related to $\bar{K}_{i j}$, $\bar{N}$, and the shift vector $\bar{\beta}^{i}$ by $$\partial_t \bar{g}_{i j} \equiv \dot{\bar{g}}_{i j} = -2\bar{N} \bar{K}_{i j} + (\bar{\nabla}_i \bar{\beta}_{j}+\bar{\nabla}_j \bar{\beta}_{i}) \; , \label{Eq:gdot}$$ where $\bar{\beta}_{j}=\bar{g}_{j i} \bar{\beta}^{i}$. The fixed spatial coordinates $\vec{x}$ of a point on the “second” hypersurface, as evaluated on the “first” hypersurface, are displaced by $\bar{\beta}^i (\vec{x}) \delta t$ with respect to those on the first hypersurface, with an orthogonal link from the first to the second surface as a fiducial reference: $\bar{\beta}_{i}= \mbox{\boldmath $\frac{\partial}{\partial t} $} * \mbox{\boldmath $\frac{\partial}{\partial x^i} $}$, where $*$ is the physical spacetime inner product of the indicated natural basis four-vectors. The essentially arbitrary direction of [$\frac{\partial}{\partial t}$]{} is why $\bar{N}(x)$ and $\bar{\beta}^{i}(x)$ appear in the TS formulation. In contrast, the tensor $\bar{K}_{i j}$ is always determined by the behavior of the unit normal on one slice and therefore does not possess the kinematical freedom, [*i.e.*]{} the gauge variance, of [$\frac{\partial}{\partial t}$]{}. Therefore, $\bar{N}$ and $\bar{\beta}^i$ do not appear in the one-hypersurface IVP for $(\Sigma, \bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$. Turning now to the conformal metrics in the IVP, we recall that two metrics $g_{i j}$ and $\bar{g}_{i j}$ are conformally equivalent if and only if there is a scalar $\psi > 0$ such that $\bar{g}_{i j} = \psi^4 g_{i j}$. The conformally invariant representative of the entire conformal equivalence class, in three dimensions, is the weight $(-2/3)$ unit-determinant “conformal metric” $\hat{g}_{i j}=\bar{g}^{-1/3} \bar{g}_{i j}=g^{-1/3} g_{i j}$ with $\bar{g}=\det(\bar{g}_{i j})$ and $g=(\det g_{i j})$. Note particularly that for any small perturbation, $\bar{g}^{i j} \delta \hat{g}_{i j}=0$. We will use the important relation $$\bar{g}^{i j} \partial_t \hat{g}_{i j} = g^{i j} \partial_t \hat{g}_{i j} = \hat{g}^{i j} \partial_t \hat{g}_{i j} = 0\;. \label{Eq:ggdot}$$ In the following, rather than use the mathematical apparatus associated with conformally weighted objects such as $\hat{g}_{i j}$, we find it simpler to use ordinary scalars and tensors to the same effect. Thus, let the role of $\hat{g}_{i j}$ on the first surface be played by a given metric $g_{i j}$ such that the physical metric that satisfies the constraints is $\bar{g}_{i j} = \psi^4 g_{i j}$ for some scalar $\psi > 0$. (This corresponds to “dressing” the initial unimodular conformal metric $\hat{g}_{i j}$ with the correct determinant factor $\bar{g}^{1/3} = \psi^4 g^{1/3}$. This process does not alter the conformal equivalence class of the metric.) The role of the conformal metric on the second surface is played by the metric $g^\prime_{i j} = g_{i j} + u_{i j} \delta t$, where, in keeping with (\[Eq:ggdot\]), the velocity tensor $u_{i j}= \dot{g}_{i j}$ is chosen such that $$g^{i j} u_{i j} = g^{i j} \dot{g}_{i j} = 0 \; .$$ Then, to first order in $\delta t$, $g^\prime_{i j}$ and $g_{i j}$ have equal determinants, as desired; but $g_{i j}$ and $g^\prime_{i j}$ are not in the same conformal equivalence class in general. We now examine the relation between the covariant derivative operators $\nabla_i$ of $g_{i j}$ and $\bar{\nabla}_i$ of $\bar{g}_{i j}$. The relation is determined by $$\bar{\Gamma}^i\mathstrut_{j k}(\bar{g}) = \Gamma^i\mathstrut_{j k}(g) + 2 \psi^{-1} \left( 2 \delta^i_{( j} \partial_{k )} \psi - g^{i l} g_{j k} \partial_l \psi \right) \; ,$$ from which follows the scalar curvature relation first used in an initial-value problem by Lichnerowicz [@Lich], $$R(\bar{g}) = \psi^{-4} R(g) - 8 \psi^{-5} \Delta_g \psi \; ,$$ where $\Delta_g \psi \equiv g^{k l} \nabla_k \nabla_l \psi$ is the “rough” scalar Laplacian associated with $g_{i j}$. Next, we solve (\[Eq:gdot\]) for its traceless part $$\dot{\bar{g}}_{i j} - \frac{1}{3} \bar{g}_{i j} \bar{g}^{k l} \dot{\bar{g}}_{k l} \equiv \bar{u}_{i j} = -2 \bar{N} \bar{A}_{i j} + (\bar{L} \bar{\beta})_{i j} \label{Eq:traceless}$$ with $\bar{A}_{i j} \equiv \bar{K}_{i j} - \frac{1}{3} \bar{K} \bar{g}_{i j}$ and $$(\bar{L} \bar{\beta})_{i j} \equiv \bar{\nabla}_i \bar{\beta}_j + \bar{\nabla}_j \bar{\beta}_i - (2/3) \bar{g}_{i j} \bar{\nabla}^k \bar{\beta}_k \; . \label{Eq:LB}$$ Expression (\[Eq:LB\]) vanishes, for non-vanishing $\bar{\beta}^i$, if and only if $\bar{g}_{i j}$ admits a conformal Killing vector $\bar{\beta}^i = k^i$. Clearly, $k^i$ would also be a conformal Killing vector of $g_{i j}$, or of any metric conformally equivalent to $\bar{g}_{i j}$, with no scaling of $k^i$. This teaches us that in general $\bar{\beta}^i = \beta^i$, while $\bar{\beta}_i = \bar{g}_{i j} \bar{\beta}^j = \psi^4 g_{i j} \beta^j=\psi^4 \beta_i$. That $\bar{\beta}^i=\beta^i$ also follows because $\beta^i$, generator of a spatial diffeomorphism, is not a dynamical variable. The latter “rule” was inferred as a matter of principle. It is clear in (\[Eq:traceless\]) that the left hand side $\bar{u}_{i j}$ satisfies $\bar{u}_{i j} = \psi^4 u_{i j}$ because the terms in $\dot{\psi}$ cancel out. Furthermore, a straightforward calculation shows that $$(\bar{L} \bar{\beta})_{i j} = \psi^4 (L \beta)_{i j} \; ; \qquad (\bar{L} \beta)^{i j} = \psi^{-4} (L \beta)^{i j} \; .$$ Next, we note, perhaps surprisingly, that the lapse function $\bar{N}$ has essential non-trivial conformal behavior. Furthermore, this is [*the*]{} new element in the IVP analysis. In [@Teitel; @Ashtekar; @AAJWY98; @YorkFest] the “slicing function” $\alpha(t,x) > 0$ replaces the lapse function $\bar{N}$, $$\bar{N} = \bar{g}^{1/2} \alpha \; , \label{Eq:slicingfunction}$$ with important improvements then appearing in Teitelboim’s path integral [@Teitel], in Ashtekar’s new variables program [@Ashtekar], in the canonical action principle [@AAJWY98; @YorkFest], and in making clear the role of the contracted Bianchi identities [@AAJWY98; @YorkFest]. The lapse is now a dynamical variable because of the $\bar{g}^{1/2}$ factor [@Ashtekar; @AAJWY98; @YorkFest]. Furthermore, in the construction of mathematically hyperbolic systems for the Einstein [*evolution*]{} equations with explicitly physical characteristics, and only such, (for example [@CBY97; @ACBY97; @CBYAnew]), it turns out to be $\alpha(t,x)$, not the usual lapse function $\bar{N}$, that can be freely specified. This use of $\bar{N} = \bar{g}^{1/2} \alpha$ is Choquet-Bruhat’s “algebraic gauge” [@CBRug; @CBY95] with, in general, a “gauge source” [@Friedrich]. Actually, $\bar{N} = \bar{g}^{1/2} \alpha$ should be seen as a change of variables in which one specifies freely $\alpha(t,x)>0$ rather than $N$. For these reasons, we conclude that $\alpha$ is not a dynamical variable, $\bar{\alpha} = \alpha$. For the lapse, we have from (\[Eq:slicingfunction\]), with $N$ given and positive, $$\bar{N} = \psi^6 N \; .$$ Finally, we recall from the standard initial value problem for $(\Sigma,\bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$, that the separation of the extrinsic curvature into (its irreducible) trace and traceless parts is fundamental, as it is here, and that $\bar{K}=K$ [@York72]: the trace is not transformed even though it is dynamical. It “anchors” the construction, setting a reference scale by fixing an observable dimensionful dynamical variable. (In closed worlds $K$ is like a “time” variable, in that it may “locate” the thin sandwich. In cosmology, $K$ is essentially the inverse mean “Hubble time.”) There is no underlying geometrical derivation of $\bar{K}=K$, unlike the case of $\bar{A}_{i j}$ below. The conformal invariance of $K$ is primitive. See the result in (\[Eq:DotLogPsi\]) below. Now we solve (\[Eq:traceless\]) for $\bar{A}^{i j}$ and find $$\begin{aligned} \bar{A}^{i j} &=& \psi^{-6} (2N)^{-1} \left[ \psi^{-4} (L \bar{\beta})^{i j} - \psi^{-4} u^{i j} \right] \nonumber\\ &=& \psi^{-10} \left\{ (2N)^{-1} \left[ (L \bar{\beta})^{i j} - u^{i j} \right] \right\} = \psi^{-10} A^{i j} \; , \label{Eq:Abar}\end{aligned}$$ the same conformal scaling that was postulated by Lichnerowicz [@Lich] and others [@CBYHeld; @OMY; @York79] for the traceless part of $\bar{K}^{i j}$ in the one-hypersurface problem. One now has a derivation of this fundamental transformation from its metrical foundations. The momentum constraint (\[Eq:MomCon\]) becomes $$\begin{aligned} \nabla_j \left[ (2N)^{-1} (L \bar{\beta})^{i j}\right] &=& \nabla_j \left[ (2N)^{-1} u^{i j} \right] \nonumber\\ & & + (2/3) \psi^6 \nabla^i K \; , \label{Eq:NewMomCon}\end{aligned}$$ for unknown $\bar{\beta}^i$ and known $N$, $g_{i j}$, $u_{i j}$, and $K$. The operator on the left, being in elliptic “divergence form” with $N>0$, does not differ in any important property from its counterpart in the $(\Sigma, \bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$ analysis [@CBYHeld; @OMY; @York79]. The Hamiltonian constraint (\[Eq:HamCon\]) becomes [@York73] $$8 \Delta_g \psi - R(g) \psi + A_{i j} A^{i j} \psi^{-7} - (2/3) K \psi^5 = 0 \; , \label{Eq:NewHamCon}$$ for unknown $\psi$, where $A^{i j}$ is given in (\[Eq:Abar\]). This equation has precisely the same form in the one-hypersurface and two-hypersurfaces constraint problems. Note that (\[Eq:NewMomCon\]) and (\[Eq:NewHamCon\]) are not coupled if $K = \mbox{constant}$, [*i.e.*]{}, one solves (\[Eq:NewMomCon\]), then (\[Eq:NewHamCon\]). [*No tensor splittings*]{} [@York73; @York74] are needed in the new formulation of the constraints. Thus, the free data are $\left\{g_{i j}, u_{i j}, N, K\right\}$ and the solution is $\left\{ \psi, \bar{\beta}^i \right\}$. Mathematical analysis of the corresponding elliptic system (\[Eq:NewMomCon\], \[Eq:NewHamCon\]) has been carried out elsewhere, for example, [@CBYHeld; @OMY; @Isenberg; @CBBrill; @CBIJWY], and will not be repeated here. The corresponding situation in the $(\Sigma,\bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$ analysis is that the free data are $\left\{g_{i j}, A_{i j}, K\right\}$ and the solution is $\left\{ \varphi, W^i \right\}$, where $W^i$ is obtained from a tensor splitting of $A_{i j}$ [@York73; @York74]. Note that $\varphi \neq \psi$ and $W^i \neq \bar{\beta}^i$. Only part of $A_{i j}$, found in the splitting, is free. The conformal covariance of the new method, [*i.e.*]{}, starting with different representatives of a given conformal equivalence class is [*unique*]{} and clear. On the other hand, that of the $(\Sigma,\bar{\mbox{\bf g}}, \bar{\mbox{\bf K}})$ analysis can follow two inequivalent routes because there are two slightly different conformal analyses possible for construction of $\left( \Sigma, \bar{\mbox{\bf g}}, \bar{\mbox{\bf K}} \right)$. This non-uniqueness arises because conformal scaling and tensor splittings are not commutative in a straightforward way. The method of tensor splitting in [@York79] gives the Hamiltonian constraint in the form of (\[Eq:NewHamCon\]). These data are not in perfect analogy to those conjectured by BSW, because $K$ and $N$ can be thought of as belonging to the thin sandwich as a whole. The role of $K$ has been described. The role of $N = g^{1/2} \alpha$ is to give the thickness of the sandwich, $\bar{N} \delta t$, in proper time measured orthogonally from $t=t^\prime$ to $t=t^{\prime\prime}$: $$\bar{N} \delta t = (\bar{g}^{1/2} \alpha) \delta t = (\psi^6 g^{1/2}) \alpha \delta t = \psi^6 (N \delta t) \; .$$ The final relationships between the two physical Riemannian metrics $\bar{g}_{i j}$ and $\bar{g}^\prime_{i j} = \bar{g}_{i j} + \dot{\bar{g}}_{i j} \delta t$ and the given data $g_{i j}$ and $g^\prime_{i j} = g_{i j} + u_{i j} \delta t$ are quite interesting. Of course, $\bar{g}_{i j} = \psi^4 g_{i j}$ is clear. But we have to calculate the relationship between $\bar{g}_{i j}$ and $\bar{g}^\prime_{i j}$ as $\bar{g}^\prime_{i j} = \bar{g}_{i j} + \dot{\bar{g}}_{i j} \delta t$, where, as in (\[Eq:gdot\]), $$\begin{aligned} \dot{\bar{g}}_{i j} &=& \partial_t \left( \psi^4 g_{i j} \right) \nonumber\\ &=& -2 \bar{N} \left( \bar{A}_{i j} + \frac{1}{3} \bar{g}_{i j} K \right) + \left( \bar{\nabla}_i \bar{\beta}_j + \bar{\nabla}_j \bar{\beta}_i \right) \; . \label{Eq:gbardot}\end{aligned}$$ Working out (\[Eq:gbardot\]) gives a key result, namely, $$\begin{aligned} \dot{\bar{g}}_{i j} &=& \psi^4 \left[ u_{i j} + g_{i j} \partial_t \left(4 \log \psi\right) \right] \nonumber\\ &=& \bar{u}_{i j} + \bar{g}_{i j} \partial_t \left(4 \log \psi\right) \; ,\end{aligned}$$ where $$\begin{aligned} \partial_t \left( 4 \log \psi \right) &=& \frac{2}{3} \left( \nabla_k \beta^k + 6 \beta^k \partial_k \log \psi - N K \psi^6 \right) \nonumber\\ &=& \partial_t \left( \bar{g}/g \right)^{1/3} = \frac{2}{3} \left( \bar{\nabla}_k \bar{\beta}^k - \bar{N} \bar{K} \right) \; . \label{Eq:DotLogPsi}\end{aligned}$$ Therefore, $$\begin{aligned} \dot{\bar{g}}_{i j} &=& \psi^4 \left[ u_{i j} + \frac{2}{3} g_{i j} \left( \nabla_k \beta^k + 6 \beta^k \partial_k \log \psi - N K \psi^6 \right) \right] \nonumber\\ &=& \bar{u}_{i j} + \frac{1}{3} \bar{g}_{i j} \left( 2 \bar{\nabla}_k \bar{\beta}^k - 2 \bar{N} K \right) \; . \label{Eq:Bargdot}\end{aligned}$$ We see that $\dot{\psi}$ and $\dot{\bar{g}}_{i j}$ are fully determined by the constraints and, in the last equality of (\[Eq:Bargdot\]), that the conformal invariance of $\beta^k$ ($=\bar{\beta}^k$) and $K$ ($= \bar{K}$) are fully consistent, having led to the precisely geometrically correct form of $\dot{\bar{g}}_{i j}$ by virtue also of $\bar{N} = \psi^6 N$. This interpretation of the semi-linear elliptic constraint system has interesting differences from earlier ones because the data and solutions are related more simply to the spacetime metric, though not in the manner that would be implied by ordinary conformal transformations of the spacetime metric. In this “conformal” TS form one can see explicitly the role of every part of the metric. The new formulation shows that the one-hypersurface and two-hypersurfaces initial-value problems are both viable once the full implications in general relativity of the “dynamical conformal structures” are understood. The two viewpoints can be thought of as roughly analogous to a Hamiltonian and to a Lagrangian view of the constraints; the former because using $\bar{K}_{i j}$ directly [@CBYHeld; @OMY; @York79] is equivalent to using the initial canonical momentum $\bar{\pi}^{i j} = \bar{g}^{1/2} \left( \bar{K} \bar{g}^{i j} - \bar{K}^{i j} \right)$, and the latter because $\dot{\bar{g}}_{i j}$ is the initial velocity. This striking correspondence hangs on the subtle role of the lapse function through the Choquet-Bruhat relation $\bar{N} = \bar{g}^{1/2} \alpha$ and on the corresponding conformal invariance of $K$ postulated by the author [@York73] in going beyond Lichnerowicz’s choice $K=0$. The “conformal thin sandwich” aspect of the results reflects Wheeler’s approach. I am grateful to A. Anderson, J. David Brown, N. O’Murchadha, and especially Y. Choquet-Bruhat for encouragement and to Sarah and Mark Rupright for technical assistance. I owe special thanks to Dean Jerry Whitten of the College of Physical and Mathematical Sciences of North Carolina State University for making my Leave of Absence possible, and to J. A. Isenberg for his advice on the presentation. Research support has been received by the author from National Science Foundation Grants No. PHY 94-13207 to the University of North Carolina, Chapel Hill, and No. PHY 93-18152/ASC 93-18152 (ARPA supplemented). [^1]: **Dedicated to John Archibald Wheeler and to the memory of André Lichnerowicz.**

--- abstract: | Scholars have wondered for a long time whether quantum mechanics (QM) subtends a quantum concept of truth which originates quantum logic (QL) and is radically different from the classical (Tarskian) concept of truth. We show in this paper that QL can be interpreted as a pragmatic language $\mathcal{L}_{QD}^{P}$ of pragmatically decidable assertive formulas, which formalize statements about physical systems that are empirically *justified* or *unjustified* in the framework of QM. According to this interpretation, QL formalizes properties of the metalinguistic concept of empirical justification within QM rather than properties of a quantum concept of truth. This conclusion agrees with a general integrationist perspective, according to which nonstandard logics can be interpreted as theories of metalinguistic concepts different from truth, avoiding competition with classical notions and preserving the globality of logic. By the way, some elucidations of the standard concept of quantum truth are also obtained. **Key words:** pragmatics, quantum logic, quantum mechanics, justifiability, decidability, global pluralism. author: - | Claudio Garola\ Dipartimento di Fisica, Università di Lecce e Sezione INFN\ 73100 Lecce, Italy\ E-mail: Garola@le.infn.it title: A Pragmatic Interpretation of Quantum Logic --- Introduction ============ The formal structure called *quantum logic* (QL) springs out in a natural way from the formalism of quantum mechanics (QM). Scholars have debated for a long time on it, wondering whether it subtends a concept of quantum truth which is typical of QM, and a huge literature exists on this topic. We limit ourselves here to quote the classical book by Jammer,$^{(1)}$ which provides a general review of QL from its birth to the early seventies, and the recent books by Rèdei$^{(2)}$ and Dalla Chiara *et al.*,$^{(3)}$ which contain updated bibliographies. Whenever the existence of a quantum concept of truth is accepted, one sees at once that it has to be radically different from the classical (Tarskian) concept, since the set of propositions of QL has an algebraic structure which is different from the structure of classical propositional logic. Thus, a new problem arises, *i.e*. the problem of the “correct” logic to be adopted when reasoning in QM. We want to show in the present paper that the above problem can be avoided by adopting an *integrated perspective*, which preserves both the globality of logic (in the sense of *global pluralism*, which admits the existence of a plurality of mutually compatible logical systems, but not of systems which are mutually incompatible$^{(4)}$) and the classical notion of *truth as correspondence*, which we consider as explicated rigorously by Tarski’s semantic theory.$^{(5,6)}$ This perspective reconciliates non-Tarskian theories of truth with Tarski’s theory by reinterpreting them as theories of metalinguistic concepts that are different from truth, and can be fruitfully applied to QL. Indeed, we prove in this paper that QL can be interpreted as a theory of the concept of *empirical justification* within QM. In order to grasp intuitively our results, let us anticipate briefly some remarks that will be discussed more extensively in Sec. 2. First of all, it must be noted that QM usually avoids making statements about properties of individual samples of a physical system (*physical objects*). Rather, it is concerned with probabilities of results of measurements on physical objects (*standard interpretation*, as espounded in any manual of QM; see, *e.g*., Refs. 7, 8 and 9), or with statistical predictions about ensembles of identically prepared physical objects (*statistical interpretation*; see, *e.g*., Refs. 1, 10 and 11). Yet, QM also distinguishes between properties that are real (or *actual*) and properties that are not real (or *potential*) in a given state $S$ of the physical system that is considered (briefly, the property $E$ is actual in $S$ whenever a test of $E$ on any physical object $x$ in $S$ would show that $E$ is possessed by $x$ without changing $S^{(12)}$). This amounts to introduce implicitly a concept of truth that also applies to statements about individuals. Indeed, asserting that a property $E$ is actual in the state $S$ is equivalent to asserting that the statement $E(x)$ that attributes $E$ to a physical object $x$ is *true* for every $x$ in the state $S$. Moreover, according to QM, $E(x)$ is true, for a given $x$ in the state $S$, if and only if (briefly, *iff*) $E$ is actual in the state $S$.$^{(12)}$ Falsity is then defined by considering a complementary property $E^{\bot }$ of $E$, so that $E(x)$ is false for a given $x$ in the state $S$ iff $E^{\bot }$ is actual in $S$. It follows in particular that $E(x)$ is true (false) for a given $x$* *in the state* *$S$ iff it is true (false) for every $x$ in $S$, or, equivalently, iff it is *certainly true* (*certainly false*) in $S$. This result explains the notion of true as *certain* introduced in some well known approaches to QM$^{(13,14)}$. More important, it shows that the notion of truth has very peculiar features in QM. Indeed, the truth and falsity of a statement $E(x)$ about an individual are equivalent to the truth of two universally quantified statements. Both these statements may be false. In this case $E(x)$ has no truth value, hence it is meaningless. The existence of meaningless statements implies, in particular, that no Tarskian set-theoretical semantics can be introduced in QM. The quantum notion of truth and meaning pointed out above is typical of the standard interpretation of QM, and it is inspired by a verificationist position which identifies truth and verifiability, meaning and verifiability conditions. These identifications are rather doubtful from an epistemological viewpoint, yet it is commonly maintained in the literature that the standard quantum conception of truth has no alternatives, since it seems firmly rooted in the formalism of QM itself. The mathematical apparatus of QM would imply indeed the impossibility of defining an assignment function associating a truth value with every individual statement of the form $E(x)$ by referring only to the property $E$ and the state $S$ of $x$. The outcomes obtained in a concrete experiment whenever $E$ or $E^{\bot }$ are not actual in $S$ would depend on the set of observations that are carried out simultaneously, not only on $S$ (*contextuality*).$^{(15-18)}$ Notwithstanding the arguments supporting it, the standard viewpoint can be criticized, and an alternative *SR interpretation* of QM can be constructed based on an epistemological position (*semantic realism*, or, briefly *SR*) which allows one to define a truth value for every statement of the form $E(x)$ according to a Tarskian set-theoretical model.$^{(19-26)}$ Of course, all statements that are certainly true (equivalently, true) or certainly false (equivalently, false) according to the standard interpretation with its quantum concept of truth, are also certainly true or certainly false, respectively, according to the SR interpretation with its Tarskian concept of truth. The remaining statements are meaningless according to the former interpretation, while they have truth values according to the latter: these values, however, may change when different objects in the same state are considered, and cannot be predicted in QM (which is, in this sense, an incomplete theory). Because of its intuitive, philosophical and technical advantages, we adopt the SR interpretation in the present paper. It is then important to observe that our definitions and reasonings take into account only statements that are certainly true (certainly false) in the sense explained above, hence they actually do not depend on the choice of the interpretation of QM (standard or SR). Thus, our reinterpretation of QL should be acceptable also for logicians and physicists who do not agree with our epistemological position. Of course, if the SR interpretation is not accepted one loses all philosophical advantages of the integrated perspective mentioned at the beginning of this section. Let us come now to empirical justification. Whenever a statement $E(x)$ is certainly true (certainly false), its truth (falsity) can be predicted within QM if the property $E$ and the state $S$ of $x$ are known, and can be checked (by means of nontrivial physical procedures, see Sec. 2.6). Hence, we can say that the assertion of $E(x)$ ($E^{\bot }(x)$) is empirically justified, since we can both deduce the truth of $E(x)$ ($E^{\bot }(x)$) inside QM and provide an empirical proof of it. More formally, one can introduce an assertion sign $\vdash $ and say that $E(x)$ is certainly true (certainly false) iff $\vdash E(x)$ ($\vdash E^{\bot }(x)$) is empirically justified. In this way a semantic notion (certainty of truth) is translated into a pragmatic notion (empirical justification). Now, we remind that a pragmatic extension of a classical language and some general properties of the concept of justification have been studied by Dalla Pozza and by the author$^{(27)}$ and note that all results obtained in this research apply to the notion of empirical justification introduced above. Moreover, further results can be obtained which are typical of the case under consideration, since the notion of justification is now specified (empirical justification in QM). Thus, a pragmatic language $\mathcal{L}_{Q}^{P}$ can be constructed (Sec. 3) in which assertions of the form $\vdash E(x)$ are taken as elementary *assertive formulas* (*afs*) and pragmatic connectives are introduced, for which a *set-theoretical pragmatics* is defined basing on the concept of empirical justification in QM. This pragmatics defines a justification value for every elementary or complex af of $\mathcal{L}_{Q}^{P}$, yet not all complex afs of $\mathcal{L}_{Q}^{P}$ are *pragmatically decidable*, that is, such that an empirical procedure of justification exists (it obviously exists for all elementary afs of $\mathcal{L}_{Q}^{P}$ because of our arguments above). However, one can single out a subset of pragmatically decidable afs of $\mathcal{L}_{Q}^{P}$ and consider a sublanguage $\mathcal{L}_{QD}^{P}$ of $\mathcal{L}_{Q}^{P}$ which contains only afs in this subset. It is then easy to see that our set-theoretical pragmatics, when restricted to $\mathcal{L}_{QD}^{P} $, endows it with the structure of QL. The above result is highly interesting in our opinion. Indeed, it provides the desired reinterpretation of QL as a theory of the metalinguistic concept of empirical justification in QM, allowing us to place it within an integrationist perspective that avoids any conflict with classical logic (we stress again that this conclusion can be accepted also by scholars who want to maintain the standard interpretation of QM). We conclude this Introduction by observing that our results suggest that the standard partition of properties in two subsets (actual properties and potential properties) should be substituted by a partition in three subsets, as follows. *Actual properties*. A property $E$ is actual in the state $S$ iff the assertion $\vdash E(x)$, with $x$ in $S$, is justified. *Nonactual properties*. A property $E$ is nonactual in the state $S$ iff the assertion $\vdash E^{\bot }(x)$, with $x$ in $S$, is justified. *Potential properties*. A property $E$ is potential in the state $S$ iff both assertions $\vdash E(x)$ and $\vdash E^{\bot }(x)$, with $x$ in $S$, are unjustified. Physical preliminaries ====================== We introduce in this section a number of symbols, definitions and physical concepts that will be extensively used in Sec. 3 in order to supply an intuitive support and an intended interpretation for the pragmatic language that will be introduced there. Basic notions and mathematical representations ---------------------------------------------- The following notions will be taken as primitive. *Physical system* $\Omega $*.* *Pure state* $S$* of* $\Omega $, and *set $\mathcal{S}$ of all pure states of* $\Omega $ (the word *pure* will be usually implied in the following). *Testable property* $E$* of* $\Omega $, and *set $\mathcal{E}$ of all testable properties of* $\Omega $ (the word *testable* will be usually implied in the following).[^1] States and properties will be interpreted operationally as follows. A state $S\in \mathcal{S}$ is a class of physically equivalent[^2] preparing devices (briefly, *preparations*) which may prepare individual samples of $\Omega $ (*physical objects*). A physical object $x$ *is in the state* $S$ iff it is prepared by a preparation $\pi \in S$. A property $E\in \mathcal{E}$ is a class of physically equivalent ideal dichotomic (outcomes 1, 0) registering devices (briefly, *registrations*) which may test physical objects.[^3] The above notions do not distinguish between classical and quantum mechanics. The mathematical representation of physical systems, states and properties are different, however, in the two theories. Let us resume these representations in the case of QM. Every physical system $\Omega $ is associated with a Hilbert space $\mathcal{H}$ over the field of complex numbers (we use the Dirac notation $\mid \cdot \rangle $ in order to denote vectors of $\mathcal{H}$ in the following). Let us denote by $(\mathcal{L(H)},\subset )$ the partially ordered set (briefly, *poset*) of all closed subspaces of $\mathcal{H}$ (here $\subset $ denotes set-theoretical inclusion), and let $\mathcal{A}\subset \mathcal{L(H)}$ be the set of all one-dimensional subspaces of $\mathcal{H}$. Then (in absence of superselection rules) a mapping $\varphi :S\in \mathcal{S\longrightarrow \varphi }(S)\in \mathcal{A}$ exists which maps bijectively the set $\mathcal{S}$ of all states of $\Omega $ onto $\mathcal{A}$,[^4] and a mapping $\chi :E\in \mathcal{E\longrightarrow \chi }(E)\in \mathcal{L(H)}$ exists which maps bijectively the set $\mathcal{E}$ of all properties of $\Omega $ onto $\mathcal{L(H)}$.[^5] Physical Quantum Logic ---------------------- The poset $(\mathcal{L(H)},\subset )$ is characterized by a set of mathematical properties. In particular, it is a complete, orthocomplemented, weakly modular, atomic lattice which satisfies the covering law$^{(13,27-30)} $. We denote by $^{\bot }$, $\Cap $ and $\Cup $ orthocomplementation, meet and join, respectively, in $(\mathcal{L(H)},\subset )$, and remind that $\Cap $ coincides with the set-theoretical intersection $\cap $ of subspaces of $\mathcal{H}$, while $^{\bot }$ does not generally coincide with the set-theoretical complementation $^{\prime }$, nor $\Cup $ coincides with the set-theoretical union $\cup $. Furthermore, we denote the minimal element $\{\mid 0\rangle \}$ and the maximal element $\mathcal{H}$ of $(\mathcal{L(H)},\subset )$ by $O$ and $I$, respectively. Finally, we note that $\mathcal{A}$ obviously coincides with the set of all atoms of $(\mathcal{L(H)},\subset )$. Let us denote by $\prec $ the order induced on $\mathcal{E}$, via the bijective representation $\chi $, by the order $\subset $ defined on $\mathcal{L(H)}$. Then, the poset $(\mathcal{E},\prec )$ is order-isomorphic to $(\mathcal{L(H)},\subset )$, hence it is characterized by the same mathematical properties characterizing $(\mathcal{L(H)},\subset )$. In particular, the unary operation induced on it, via $\chi $, by the orthocomplementation defined on $(\mathcal{L(H)},\subset )$, is an orthocomplementation, and $(\mathcal{E},\prec )$ is an orthomodular (i.e., orthocomplemented and weakly modular) lattice, usually called *the lattice of properties* of $\Omega $. By abuse of language, we denote the lattice operations on $(\mathcal{E},\prec )$ by the same symbols used above in order to denote the corresponding lattice operations on $(\mathcal{L(H)},\subset )$. Orthomodular lattices are said to characterize semantically *orthomodular QLs* in the literature.$^{(3)}$ The lattice of properties has a less general structure in QM, since it inherits a number of further properties from $(\mathcal{L(H)},\subset )$. Therefore, $(\mathcal{E},\prec ) $ will be called *physical QL* in this paper. A further lattice, isomorphic to $(\mathcal{E},\prec )$, will be used in the following. In order to introduce it, let us consider the mapping $\rho :E\in \mathcal{E}\longrightarrow \mathcal{S}_{E}=\{S\in \mathcal{S}\mid \varphi (S)\subset \chi (E)\}\in \mathcal{L(S)}$, where $\mathcal{L(S)}=\{\mathcal{S}_{E}\mid E\in \mathcal{E}\}$ is the range of $\rho $, which generally is a proper subset of the power set $\mathcal{P(S)}$ of $\mathcal{S}$. The poset $(\mathcal{L(S)},\subset )$ is order-isomorphic to $(\mathcal{L(H)},\subset )$, hence to $(\mathcal{E},\prec )$, since $\varphi $ and $\chi $ are bijective, so that $\rho $ is bijective and order-preserving. Therefore $(\mathcal{L(S)},\subset )$ is characterized by the same mathematical properties characterizing $(\mathcal{E},\prec )$. In particular, the unary operation induced on it, via $\rho $, by the orthocomplementation defined on $(\mathcal{E},\prec )$, is an orthocomplementation, and $(\mathcal{L(S)},\subset )$ is an orthomodular lattice. We denote orthocomplementation, meet and join on $(\mathcal{L(S)},\subset )$ by the same symbols $^{\bot }$, $\Cap $, and $\Cup $, respectively, that we have used in order to denote the corresponding operations on $(\mathcal{L(H)},\subset )$ and $(\mathcal{E},\prec )$, and call $(\mathcal{L(S)},\subset )$ *the lattice of closed subsets of* $\mathcal{S}$ (the word *closed* refers here to the fact that, for every $\mathcal{S}_{E}\in $ $\mathcal{L(S)}$, $(\mathcal{S}_{E}^{\bot })^{\bot }=\mathcal{S}_{E}$). We also note that the operation $\Cap $ coincides with the set-theoretical intersection $\cap $ on $\mathcal{L(S)}$ because of the analogous result holding in $(\mathcal{L(H)},\subset )$.[^6] To close up, let us observe that the unary operation $^{\bot }$ defined on $\mathcal{L(S)}$ can be extended to $\mathcal{P(S)}$ by setting, for every $\mathcal{T\in P(S)}$, $\mathcal{T}^{\bot }=($*min*$\{\mathcal{S}_{E}\in \mathcal{L(S)\mid T\subset S}_{E}\})^{\bot }$ (the symbol *min* obviously refers to the order $\subset $ defined on $\mathcal{L(S)}$). This extension will be needed indeed in Sec. 3.2. Actual and potential properties ------------------------------- We say that a property $E$ is* actual* (*nonactual*) in the state $S$ iff one can perform a test of $E$ on any physical object $x$ in the state $S$ by means of a registration $r\in E$, obtaining outcome 1 (0) without modifying $S$.[^7] Basing on the above definition, for every state $S\in \mathcal{S}$ three subsets of $\mathcal{E}$ can be introduced. $\mathcal{E}_{S}$ : the set of all properties that are actual in $S$. $\mathcal{E}_{S}^{\bot }$ : the set of all properties that are nonactual in $S$. $\mathcal{E}_{S}^{I}$ : the set $\mathcal{E}\setminus \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$ (called the set of all properties that are *indeterminate*, or* potential*, in $S$). By using the mathematical apparatus of QM, the sets $\mathcal{E}_{S}$ and $\mathcal{E}_{S}^{\bot }$ can be characterized as follows. $\mathcal{E}_{S}=\{E\in \mathcal{E}\mid \varphi (S)\subset \chi (E)\}=\{E\in \mathcal{E}\mid S\in \mathcal{S}_{E}\}$. $\mathcal{E}_{S}^{\bot }=\{E\in \mathcal{E}\mid \varphi (S)\subset \chi (E)^{\bot }\}=\{E\in \mathcal{E}\mid S\in \mathcal{S}_{E}^{\bot }\}$. It can also be proved that $\mathcal{E}_{S}$ ($\mathcal{E}_{S}^{\bot }$) coincides with the set of all properties that have probability 1 (0), according to QM, for every $x$ in the state $S$, and that the set $\mathcal{E}_{S}^{I}$ (which is non-void in QM, while it would be void in classical physics) coincides with the set of all properties that have probability different from 0 and 1 for every $x$ in the state $S$. Further characterizations of the above sets can be obtained as follows.$^{(12)}$ Since the mapping $\rho $ is bijective, while every singleton $\{S\}$, with $S\in \mathcal{S}$, obviously is an atom of $\mathcal{L(S)}$, one can associate a property $E_{S}=\rho ^{-1}(\{S\})$ (equivalently, $E_{S}=\chi ^{-1}(\varphi (S))$) with every $S\in \mathcal{S}$. This property is an atom of $(\mathcal{E},\prec )$, and is usually called the *support* of $S$. The mapping $\rho ^{-1}$ thus induces a one-to-one correspondence between (pure) states and atoms of $(\mathcal{E},\prec )$. Then, one can prove the following equalities. $\mathcal{E}_{S}=\{E\in \mathcal{E}\mid E_{S}\prec E\}$. $\mathcal{E}_{S}^{\bot }=\{E\in \mathcal{E}\mid E\prec E_{S}^{\bot }\}$. $\mathcal{E}_{S}^{I}=\{E\in \mathcal{E}\mid E_{S}\nprec E$ and $E\nprec E_{S}^{\bot }\}$. Finally, the following equality also follows from the above definitions. $\mathcal{S}_{E}=\{S\in \mathcal{S}\mid E_{S}\prec E\}.$ Truth in standard QM -------------------- No mention has been done of truth values (*true/false*) in the foregoing sections. However, we will be concerned with logical structures in Sec. 3, hence it is natural to wonder what QM says about the truth of a sentence as “the physical object $x$ has the property $E$” (briefly, $E(x)$ in the following). We have already noted in the Introduction that QM usually avoids making explicit statements regarding individual samples of physical systems. Yet, a sentence as “the property $E$ is actual in the state $S$” (Sec. 2.3) intuitively means that all physical objects in the state $S$ have the property $E$. Hence, it can be translated, in terms of truth, into the sentence “for every physical object $x$ in the state $S$, $E(x)$ is true”. This translation shows that QM is concerned also with truth values of individual statements. Moreover, by considering the literature on the subject, one can argue that QM more or less implicitly adopts the following verificationist criterion of truth.$^{(12)}$ EV (empirical verificationism). *The sentence* $E(x)$* has truth value* true* (*false*) for a physical object* $x$* in the state* $S$* iff* $E$* is actual (nonactual) in* $S$*, while it is meaningless otherwise.* Criterion EV is obviously at odds with standard definitions in classical logic (CL), and is suggested by the fact that $E$ can be attributed (not attributed) to a physical object $x$ in the state S on the basis of an experimental procedure only when it is actual (nonactual) for $x$ (see Sec. 2.6). Hence, we say that $E(x)$ is *Q-true* (*Q-false*) whenever its truth value is true (false) according to criterion EV, in order to stress the difference between the truth values introduced in QM and those introduced in CL. Because of the foregoing translation, criterion EV implies the following proposition. TF. *The sentence* $E(x)$* is* Q-true* (*Q-false*) for a physical object x in the state* $S$* iff it is* Q-true* (*Q-false*) for every physical object x in the state* $S$*.* Loosely speaking, proposition TF can be rephrased by saying that $E(x)$ is true (false) in the sense established by criterion EV iff it is *certainly* true (*certainly* false) in the same sense, which explains the intuitive terminology that we have adopted in the Introduction. Furthermore, criterion EV implies that $E(x)$ has a truth value in standard QM iff $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$ (of course, $E(x)$ is Q-true iff $E\in \mathcal{E}_{S}$, Q-false iff $E\in \mathcal{E}_{S}^{\bot }$). It is then important to observe that the characterizations of $\mathcal{E}_{S}$ and $\mathcal{E}_{S}^{\bot }$ provided in Sec. 2.3 show that, for every $S\in \mathcal{S}$, one can deduce from theoretical laws of QM whether a property $E$ belongs to $\mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$. In particular, $E$ belongs to $\mathcal{E}_{S}$ ($\mathcal{E}_{S}^{\bot }$) iff it has probability 1 (0) for every $x$ in the state $S$. Hence, one can predict, for every $E\in \mathcal{E}$ and $x$ in the state $S$, whether $E(x)$ is Q-true, Q-false or meaningless. This result shows that standard QM is a semantically complete theory$^{(12)}$ and, together with proposition TF, explains the definition of *true* as *certain*, or *predictable*, which occurs in some approaches to QM.$^{(13,14)}$ Nonobjectivity versus objectivity in QM --------------------------------------- The position expounded in Sec. 2.4 about the truth value of sentences of the form $E(x)$, with $E\in \mathcal{E}$, is sometimes summarized by saying, briefly, that physical properties are *nonobjective* in standard QM (to be precise, only the properties in $\mathcal{E}_{S}^{I}$ should be classified as nonobjective for a given $S\in \mathcal{S}$). Nonobjectivity of properties is supported by a number of arguments. Some of them are based on empirical results (e.g., the two-slits experiment), some follow from seemingly reasonable epistemological choices (e.g., the adoption of a verificationist position, together with the indeterminacy principle) and some take the form of theorems deduced from the mathematical apparatus of QM. These last arguments are usually considered conclusive in the literature. We remind here the Bell-Kochen-Specker and Bell’s theorems$^{(15-18)}$ which seem to prove that it is impossible to assign classical truth values to all sentences of the form $E(x)$, with $E\in \mathcal{E}$, without contradicting the predictions of QM. However, all arguments which show that nonobjectivity of properties is an unavoidable feature of QM can be criticized (this of course does not make the claim of nonobjectivity wrong, but only proves that there are alternatives to it). In particular, one can observe that a *no-go theorem* as Bell-Kochen-Specker’s is certainly correct from a mathematical viewpoint, but rests on implicit assumptions that are problematic from a physical and epistemological viewpoint.$^{(22-25)}$ Basing on this criticism, an alternative interpretation (*semantic realism*, or *SR*, interpretation) has been propounded by the author, together with other authors.$^{(19-23,25,26)}$ As we have already observed in the Introduction, the SR interpretation adopts a Tarskian theory of truth as correspondence, and all properties are objective according to it (equivalently, the sentence $E(x)$ has a truth value defined in a classical way for every physical object $x$ and property $E$). According to this interpretation $E(x)$ is *certainly true* (*certainly false*) in the state $S$, that is, it is true (false) in a classical sense for every $x$ is in the state $S$, iff $E\in \mathcal{E}_{S}$ ($E\in \mathcal{E}_{S}^{\bot }$), hence iff it is Q-true (Q-false) according to the standard interpretation. The SR interpretation of QM has some definite advantages. Firstly, it makes QM compatible with a realistic perspective without requiring any change of its mathematical apparatus and preserving all statistical predictions following from the standard interpretation, hence it provides a solution of the quantum measurement problem.$^{(26)}$ Secondly, it rests on a classical conception of truth and meaning. Thirdly, it leads one to consider QM as an incomplete theory,$^{(12)}$ and provides some suggestions about the way in which a more general theory embodying QM could be constructed. Also within the SR interpretation one can deduce from theoretical laws of QM whether $E\in \mathcal{E}_{S}$ ($E\in \mathcal{E}_{S}^{\bot }$), for a given $S\in \mathcal{S}$. Moreover, for every $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$, the sentence $E(x)$ obviously is certainly true, hence true (certainly false, hence false) iff $E\in \mathcal{E}_{S}$ ($E\in \mathcal{E}_{S}^{\bot }$). On the contrary, no prediction of the truth value of $E(x)$ can be done if $E\notin \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$. Thus, the difference between the standard and the SR interpretation reduces to the fact that, whenever $E\in \mathcal{E}_{S}^{I}$, $E(x)$ is meaningless within the former, while it has a truth value that cannot be predicted by QM within the latter. Empirical proof in QM --------------------- The results at the end of Secs. 2.4 and 2.5 show that, whenever $x$ is in the state $S$, the truth value of the sentence $E(x)$ can be predicted (or *theoretically proved*) iff $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$, both in the standard and in the SR interpretation. One is thus led to wonder whether and when the truth value of $E(x)$ can be determined empirically. At first glance, it seems sufficient to test $x$ by means of a registering device belonging to $E$ (Sec. 2.1). This is untrue according to the standard as well as the SR interpretation. Indeed, both interpretations maintain that a single test modifies, whenever $E\notin \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$, the state $S$ of the physical object $x$, so that its result refers to the final state after the test, which is different from $S$ (moreover, within the standard interpretation, $E(x)$ has no truth value whenever $E\notin \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$). Thus, a test of $E(x)$ is physically meaningful iff $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$, since only in this case it does not modify the state $S$. It follows that an *empirical proof* of the truth value of $E(x)$ can be given iff a theoretical proof of this value exists, and it consists in checking whether $E\in \mathcal{E}_{S}$ or $E\in \mathcal{E}_{S}^{\bot }$. Then, the characterizations of $\mathcal{E}_{S}$ and $\mathcal{E}_{S}^{\bot }$ in Sec. 2.3 suggest the empirical procedures to be adopted. Indeed, they show that $E\in \mathcal{E}_{S}$ ($E\in \mathcal{E}_{S}^{\bot }$) iff $E_{S}\prec E$ ($E\prec E_{S}^{\bot }$), or, equivalently, iff $S\in \mathcal{S}_{E}$ ($S\in \mathcal{S}_{E}^{\bot }$). Hence, one can get an empirical proof that $E(x)$ is Q-true (Q-false) within the standard interpretation, or equivalently, that $E(x)$ is certainly true, hence true (certainly false, hence false) within the SR interpretation, by checking whether the state $S$ of $x$ belongs to the set $\mathcal{S}_{E}$ ($\mathcal{S}_{E}^{\bot }$). The empirical procedure required by this check is rather complex, since it does not reduce to a test of $E$ on the physical object $x$, but consists in testing a huge number of physical objects in the state $S$ by means of registrations belonging to $E$, in order to show that all of them yield outcome 1 (0) (it has been proven elsewhere$^{(26)}$ that this procedure actually tests a quantified statement, or a second order physical property). We conclude by noticing that truth and empirical provability of truth coincide within the standard interpretation of QM, which expresses the verificationist position that characterizes this interpretation. On the contrary, within the SR interpretation of QM the concepts of truth and empirical provability of truth are different, in accordance with the well known distinction between truth and epistemic accessibility of truth in classical logic. QL as a pragmatic language ========================== We aim to show in this section that physical QL can be recovered as a pragmatic language in the sense established in Ref. 27. It is noteworthy that, by weakening slightly the assumptions introduced in Ref. 27, one could perform this task without choosing between the standard and the SR interpretation of QM (see footnotes 8 and 9). We adopt however the SR interpretation in this section, since we maintain that the verificationist attitude of the standard interpretation is epistemologically and philosophically doubtful, but we point out by means of footnotes the simple changes to be introduced in order to attain the same results within the standard interpretation. The general pragmatic language $\mathcal{L}^{P}$ ------------------------------------------------ Let us summarize briefly the construction of the general pragmatic language $\mathcal{L}^{P}$ introduced in Ref. 27. The alphabet $\mathcal{A}^{P}$ of $\mathcal{L}^{P}$ contains as *descriptive signs* the propositional letters $p$, $q$, $r$,...; as*logical-semantic signs* the connectives $\urcorner $, $\wedge $, $\vee $, $\rightarrow $ and $\leftrightarrow $; as * logical-pragmatic signs* the assertion sign $\vdash $ and the connectives $N$, $K$, $A$, $C$ and* *$E$; as* auxiliary signs* the round brackets $(.)$.* *The set $\psi _{R}$ of all *radical formulas* (*rfs*) of $\mathcal{L}^{P}$ is made up by all formulas constructed by means of descriptive and logical-semantic signs, following the standard recursive rules of classical propositional logic (a rf consisting of a propositional letter only is then called *atomic*). The set $\psi _{A}$ of all *assertive formulas* (*afs*) of $\mathcal{L}^{P}$ is made up by all rfs preceded by the assertive sign $\vdash $ (*elementary* afs), plus all formulas constructed by using elementary afs and following standard recursive rules in which $N$, $K$, $A$, $C$ and $E$ take the place of $\urcorner $, $\wedge $, $\vee $, $\rightarrow $ and $\leftrightarrow $, respectively. A *semantic interpretation* of $\mathcal{L}^{P}$ is then defined as a pair $(\{1,0\},\sigma )$, where $\sigma $ is an* assignment function* which maps $\psi _{R}$ onto the set $\{1,0\}$ of* truth values* (1 standing for *true* and 0 for *false*), following the standard truth rules of classical propositional calculus. Whenever a semantic interpretation $\sigma $ is given, a *pragmatic interpretation* of $\mathcal{L}^{P}$ is defined as a pair $(\{J,U\},\pi _{\sigma })$, where $\pi _{\sigma }$ is a *pragmatic evaluation function* which maps $\psi _{A}$ onto the set $\{J,U\}$ of *justification values* following * justification rules* which refer to $\sigma $ and are based on the informal properties of the metalinguistic concept of proof in natural languages. In particular, the following justification rules hold. JR$_{1}$. *Let* $\alpha \in \psi _{R}$*; then,* $\pi _{\sigma }(\vdash \alpha )=J$* iff a proof exists that* $\alpha $* is true, i.e., that* $\sigma (\alpha )=1$* (hence,* $\pi _{\sigma }(\vdash \alpha )=U$* iff no proof exists that* $\alpha $*is true).* JR$_{2}$.* Let* $\delta \in \psi _{A}$*; then,* $\pi _{\sigma }(N\delta )=J$* iff a proof exists that* $\delta $* is unjustified, i.e., that* $\pi _{\sigma }(\delta )=U$*.* JR$_{3}$.* Let* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}$*; then,* *(i)* $\pi _{\sigma }(\delta _{1}K\delta _{2})=J$* iff* $\pi _{\sigma }(\delta _{1})=J$* and* $\pi _{\sigma }(\delta _{2})=J$*,* *(ii)* $\pi _{\sigma }(\delta _{1}A\delta _{2})=J$* iff* $\pi _{\sigma }(\delta _{1})=J$* or* $\pi _{\sigma }(\delta _{2})=J$*,* *(iii)* $\pi _{\sigma }(\delta _{1}C\delta _{2})=J$* iff a proof exists that* $\pi _{\sigma }(\delta _{2})=J$* whenever* $\pi _{\sigma }(\delta _{1})=J$*,* *(iv)* $\pi _{\sigma }(\delta _{1}E\delta _{2})=J$* iff* $\pi _{\sigma }(\delta _{1}C\delta _{2})=J$* and* $\pi _{\sigma }(\delta _{2}C\delta _{1})=J$*.* Furthermore, the following *correctness criterion* holds in $\mathcal{L}^{P}$. CC\. *Let* $\alpha \in \psi _{R}$*; then,* $\pi _{\sigma }(\vdash \alpha )=J$* implies* $\sigma (\alpha )=1.$ Finally, the set of all pragmatic evaluation functions that can be associated with a given semantic interpretation $\sigma $ is denoted by $\Pi _{\sigma }$. The quantum pragmatic language $\mathcal{L}_{Q}^{P}$ ---------------------------------------------------- The quantum pragmatic language $\mathcal{L}_{Q}^{P}$ that we want to introduce here is obtained by specializing syntax, semantics and pragmatics of $\mathcal{L}^{P}$. Let us begin with the syntax. We introduce the following assumptions on $\mathcal{L}_{Q}^{P}$. A$_{1}$. *The propositional letters* $p$*,* $q$*, ... are substituted by the symbols* $E(x)$*,* $F(x)$*, ..., with* $E$*,* $F$*, ...* $\in \mathcal{E}$*.* A$_{2}$. *The set* $\psi _{R}^{Q}$* of all rfs of* $\mathcal{L}_{Q}^{P}$* reduces to the set of all atomic rfs of* $\mathcal{L}_{Q}^{P}$* (in different words, if* $\alpha $* is a rf of* $\mathcal{L}_{Q}^{P}$*, then* $\alpha =E(x)$*, with* $E\in \mathcal{E}$*).* A$_{3}$. *Only the logical-pragmatic signs* $\vdash $*,* $N$*,* $K$* and* $A$* appear in the afs of* $\mathcal{L}_{Q}^{P}$*.* The substitution in A$_{1}$ aims to suggest the *intended interpretation* that we adopt in the following. To be precise, the rfs $E(x)$, $F(x)$, ... are interpreted as sentences stating that the physical object $x$ has the properties $E$, $F$, ..., respectively (Sec. 2.4). The restriction in A$_{2}$ aims to select rfs that are interpreted as *testable* sentences, i.e., sentences stating testable physical properties (Sec. 2.1), so that physical procedures exist for testing their truth values (which may not occur in the case of a rf of the form, say, $E(x)\vee F(x)$; note that a similar restriction has been introduced in Ref. 27 when recovering intuitionistic propositional logic within $\mathcal{L}^{P} $). The restriction in A$_{3}$ is introduced for the sake of simplicity, since only the pragmatic connectives $N$, $K$ and $A$ are relevant for our goals in this paper. Because of A$_{1}$, A$_{2}$ and A$_{3}$, the set $\psi _{A}^{Q}$ of afs of $\mathcal{L}_{Q}^{P}$ is made up by all formulas constructed by means of the following recursive rules. \(i) *Let* $E(x)$* be a rf. Then* $\vdash E(x)$* is an af.* \(ii) *Let* $\delta $* be an af. Then,* $N\delta $* is an af.* \(iii) *Let* $\delta _{1}$* and* $\delta _{2}$* be afs. Then,* $\delta _{1}K\delta _{2}$* and* $\delta _{1}A\delta _{2}$* are afs.* Let us come now to the semantics of $\mathcal{L}_{Q}^{P}$. We introduce the following assumption on $\mathcal{L}_{Q}^{P}$. A$_{4}$. *Every assignment function* $\sigma $* defined on* $\psi _{R}^{Q}$* is induced by an interpretation* $\xi $* of the variable x that appears in the rfs into a universe* $\mathcal{U}$* of physical objects, hence* $\sigma =\sigma (\xi )$* and the values of* $\sigma $ *on* $\psi _{R}^{Q}$ *are consistent with (not necessarily determined by) the laws of QM within the intended interpretation established above.* Let us comment briefly on assumption A$_{4}$. Firstly, note that the interpretation $\xi $ was understood in Sec. 2.1, when we introduced the informal expression “the physical object $x$ is in the state $S$”. Secondly, observe that the requirement that $\sigma =\sigma (\xi )$ be consistent with the laws of QM (briefly, *QM-consistent*) obviously follows from the fact that these laws, via intended interpretation, establish relations among the truth values of elementary rfs of $\mathcal{L}_{Q}^{P}$ whenever a specific physical object is considered. We denote by $\Sigma $ in the following the set of all QM-consistent assigment functions. Thirdly, note that, since $\sigma =$ $\sigma (\xi )$, there may be many interpretations of the variable x that lead to the same assigment function. Finally, observe that the universe $\mathcal{U}$ can be partitioned into (disjoint) subsets of physical objects, each of which consists of physical objects in the same state (different subsets corresponding to different states). Thus, specifying the state $S$ of $x$ means requiring that the interpretation $\xi $ of $x$ that is considered maps $x$ on a physical object in the subset corresponding to the state $S$, hence it singles out a subclass $\Sigma _{S}\subset \Sigma $ of assigment functions. All functions in $\Sigma _{S}$ assign truth value 1 (0) to a sentence $E(x)\in \psi _{R}^{Q}$ whenever $E\in \mathcal{E}_{S}$ ($\mathcal{E}_{S}^{\bot }$), while the truth values assigned by different functions in $\Sigma _{S}$ to $E(x)$ may differ if $E$ $\notin $ $\mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$.[^8] Let us come now to the pragmatics of $\mathcal{L}_{Q}^{P}$. We introduce the following assumption on $\mathcal{L}_{Q}^{P}$. A$_{5}$. *Let a mapping* $\xi $* be given which interpretes the variable* $x$* in the rfs of* $\mathcal{L}_{Q}^{P}$* on a physical object in the state* $S$*. A proof that the rf* $E(x)$* is true (false) consists in performing one of the empirical procedures mentioned in Sec. 2.6 and showing that* $E\in \mathcal{E}_{S}$* (*$E\in \mathcal{E}_{S}^{\bot }$*).* Assumption A$_{5}$ is obviously suggested by the intended interpretation discussed above. Taking into account A$_{1}$ and JR$_{1}$ in Sec. 3.1, it implies the following statement. P.* Let* $E(x)$* be a rf of* $\mathcal{L}_{Q}^{P}$*, let* $\xi $* be an interpretation of the variable* $x$* on a physical object in the state* $S$*, and let* $S_{E}$* be defined as in Sec. 2.2. Then,* $\pi _{\sigma (\xi )}(\vdash E(x))=J$* iff* $S\in $* *$S_{E}$*,* $\pi _{\sigma (\xi )}(\vdash E(x))=U$* iff* $S\notin $* *$S_{E}$*.* The above result specifies $\pi _{\sigma (\xi )}$ on the set of all elementary afs of $\mathcal{L}_{Q}^{P}$ and shows that it depends only on the state $S$. Hence, we write $\pi _{S}$ in place of $\pi _{\sigma (\xi )}$ in the following (for the sake of brevity, we also agree to use the intuitive statement “the physical object $x$ is in the state $S$” introduced in Sec. 2.1 in place of the more rigorous statement “the variable $x$ is interpreted on a physical object in the state $S$”). Statement P provides the starting point for introducing a *set-theoretical pragmatics* for $\mathcal{L}_{Q}^{P}$, as follows. Firstly, we introduce a mapping $f:\delta \in \psi _{A}^{Q}\longrightarrow \mathcal{S}_{\delta }\in \mathcal{P(S)}$ which associates a *pragmatic extension* $\mathcal{S}_{\delta }$ with every assertive formula $\delta \in \psi _{A}^{Q}$, defined by the following recursive rules. \(i) *For every* $E(x)\in \psi _{R}^{Q}$*,* $f(\vdash E(x))=S_{\vdash E(x)}=S_{E}$*.* \(ii) *For every* $\delta $* *$\in \psi _{A}^{Q}$*,* $f(N\delta )=S_{N\delta }=S_{\delta }^{\bot }$*.* (iii)* For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $f(\delta _{1}K$ $\delta _{2})=\mathcal{S}_{\delta _{1}K\delta _{2}}=\mathcal{S}_{\delta _{1}}\cap \mathcal{S}_{\delta _{2}}$. \(iv) *For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q} $*,* $f(\delta _{1}A$* *$\delta _{2})=S_{\delta _{1}A\delta _{2}}=S_{\delta _{1}}\cup S_{\delta _{2}}$*.* Secondly, we rewrite statement P above substituting $\mathcal{S}_{\vdash E(x)}$ to $\mathcal{S}_{E}$ in it. P$^{\prime }$.* Let* $\vdash E(x)$* be an elementary af of* $\mathcal{L}_{Q}^{P}$* and let* $x$ *be in the state* $S$*. Then,* $\pi _{S}(\vdash E(x))=J$* iff* $S\in $* *$S_{\vdash E(x)}$*,* $\pi _{S}(\vdash E(x))=U$* iff* $S\notin $* *$S_{\vdash E(x)} $*.* Thirdly, we note that statement P$^{\prime }$ defines the pragmatic evaluation function $\pi _{S}$ on all elementary afs of $\mathcal{L}_{Q}^{P}$. Finally, for every $S\in \mathcal{S}$, we extend $\pi _{S}$ from the set of all elementary afs of $\mathcal{L}_{Q}^{P}$ to the set $\psi _{A}^{Q}$ of all afs of $\mathcal{L}_{Q}^{P}$ bearing in mind JR$_{2}$ and JR$_{3}$ in Sec. 3.1, hence introducing the following recursive rules. \(i) *For every* $\delta $* *$\in \psi _{A}^{Q}$*,* $\pi _{S}(N\delta )=J$* iff* $S\in S_{N\delta }=S_{\delta }^{\bot }$*.* (ii)* For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\pi _{S}(\delta _{1}K$* *$\delta _{2})=J$* iff* $S\in S_{\delta _{1}K\delta _{2}}=S_{\delta _{1}}\cap S_{\delta _{2}}$*.* \(iii) *For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\pi _{S}(\delta _{1}A$* *$\delta _{2})=J$* iff* $S\in S_{\delta _{1}A\delta _{2}}=S_{\delta _{1}}\cup S_{\delta _{2}}$*.* The above procedure defines, for every $S\in \mathcal{S}$, a pragmatic evaluation function $\pi _{S}:\delta \in \psi _{A}^{Q}\longrightarrow \pi _{S}(\delta )\in \{J,U\}$ which provides a set-theoretical pragmatics for $\mathcal{L}_{Q}^{P}$, as stated. On the notion of justification in $\mathcal{L}_{Q}^{P}$ ------------------------------------------------------- The notion of justification introduced in Sec. 3.2 is basic in our approach and must be clearly understood. So we devote this section to comments on it. Whenever an elementary af $\vdash E(x)$ of $\mathcal{L}_{Q}^{P}$ is considered, the notion of justification obviously coincides with the notion of existence of an empirical proof of the truth of $E(x)$ because of assumption A$_{5}$ and proposition P in Sec. 3.2, which fits in with JR$_{1}$ in Sec. 3.1. Whenever molecular afs of $\mathcal{L}^{P}$ are considered, one can grasp intuitively the meaning of the notion of justification for them by considering simple instances. Indeed, let $E(x)$ be a rf and let $x$ be in the state $S$. We get $\pi _{S}(N\vdash E(x))=J$ iff $S\in \mathcal{S}_{E}^{\bot }$, which means, shortly, that it is justified to assert that $E(x)$ cannot be asserted iff MQ entails that the truth value of $E(x)$ is *false* for every $x$ in the state $S$. This result, of course, fits in with JR$_{2}$ in Sec. 3.1. Furthermore, let $E(x)$ and $F(x)$ be rfs, and let $x$ be in the state $S$. We get $\pi _{S}(\vdash E(x)K\vdash F(x))=J$ iff $S\in \mathcal{S}_{E}\cap \mathcal{S}_{F}$, $\pi _{S}(\vdash E(x)A\vdash F(x))=J$ iff $S\in \mathcal{S}_{E}\cup \mathcal{S}_{F}$. The first equality shows that asserting $E(x)$ and $F(x)$ conjointly is justified iff both assertions are justified. The second equality shows that asserting $E(x)$ or asserting $F(x)$ is justified iff one of these assertions is justified. Both these results, of course, fit in with JR$_{3}$ in Sec. 3.1. We add that $\pi _{S}(\vdash E(x))=J$ implies $\pi _{S}(N\vdash E(x))=U$ and $\pi _{S}(N\vdash E(x))=J$ implies $\pi _{S}(\vdash E(x))=U$ since $\mathcal{S}_{E}\cap \mathcal{S}_{E}^{\bot }=\emptyset $. Nevertheless, $\pi _{S}(\vdash E(x))=U$ and $\pi _{S}(N\vdash E(x))=U$ iff $S\notin \mathcal{S}_{E}\cup \mathcal{S}_{E}^{\bot }$, which shows that a *tertium non datur* principle does not hold for the pragmatic connective $N$ in $\mathcal{L}_{Q}^{P}$ (it has already been proved in Ref. 27 that this principle does not hold in the general language $\mathcal{L}^{P}$). It is also interesting to note that the justification values of different elementary afs, say $\vdash E(x)$ and $\vdash F(x)$, must be different for some state $S$, since $\mathcal{S}_{E}\neq \mathcal{S}_{F}$ if $E\neq F$ (Sec. 2.2), hence $\mathcal{S}_{\vdash E(x)}\neq \mathcal{S}_{\vdash F(x)}$. Finally, we remind that the general theory of $\mathcal{L}^{P}$ associates an assignment function $\sigma $ with a set $\Pi _{\sigma }$ of pragmatic evaluation functions (Sec. 3.1), hence this also occurs within $\mathcal{L}_{Q}^{P}$. One may then wonder whether $\Pi _{\sigma }$ is necessarily nonvoid and, if this is the case, whether it may contain more than one pragmatic evaluation function. In order to answer these questions, let us consider an interpretation $\xi $ of the variable $x$ that maps $x$ on a physical object in the state $S$. Then, $\xi $ determines a unique assignment function $\sigma (\xi )$ and a unique pragmatic evaluation function associated with it, that we have denoted by $\pi _{S}$, for it depends only on the state $S$. Since every assigment function in $\Sigma $ is induced by an interpretation $\xi $ because of A$_{4}$ in Sec. 3.2, this proves that $\Pi _{\sigma }$ is necessarily nonvoid for every $\sigma \in \Sigma $. Moreover, note that an interpretation $\xi ^{\prime }$ of $x$ may exist within the SR interpretation of QM that maps $x$ on a physical object in the state $S^{\prime }$, with $S^{\prime }\neq S$, yet such that $\sigma (\xi ^{\prime })=\sigma (\xi )$. The pragmatic evaluation functions $\pi _{S} $ and $\pi _{S^{\prime }}$ are then different, but they are both associated with the assignment function $\sigma =\sigma (\xi )=\sigma (\xi ^{\prime })$, so that they both belong to $\Pi _{\sigma }$. Hence, $\Pi _{\sigma }$ may contain many pragmatic evaluation functions.[^9] Pragmatic validity and order in $\mathcal{L}_{Q}^{P}$ ----------------------------------------------------- Coming back to the general language $\mathcal{L}^{P}$, we remind that a notion of pragmatic validity (invalidity) is introduced in it by means of the following definition. *Let* $\delta \in \psi _{A}$*. Then,* $\delta $* is* pragmatically valid*, or* p-valid* (*pragmatically invalid*, or* p-invalid*) iff for every* $\sigma \in \Sigma $* and* $\pi _{\sigma }\in \Pi _{\sigma }$*,* $\pi _{\sigma }(\delta )=J$* (*$\pi _{\sigma }(\delta )=U$*).* By using the notions of justification in $\mathcal{L}_{Q}^{P}$, one can translate the notion of p-validity (p-invalidity) within $\mathcal{L}_{Q}^{P} $ as follows. *Let* $\delta \in \psi _{A}^{Q}$*. Then,* $\delta $*is p-valid (p-invalid) iff, for every* $S\in S$*,* $\pi _{S}(\delta )=J$* (*$\pi _{S}(\delta )=U$*).* The notion of p-validity (p-invalidity) can then be characterized as follows. *Let* $\delta \in \psi _{A}^{Q}$*. Then,* $\delta $*is p-valid (p-invalid) iff* $S_{\delta }=S$* (*$S_{\delta }=\emptyset $*).* The set of all p-valid afs plays in $\mathcal{L}_{Q}^{P}$ a role similar to the role of tautologies in classical logic, and some afs in it can be selected as axioms if one tries to construct a p-correct and p-complete calculus for $\mathcal{L}_{Q}^{P}$. We will not deal, however, with this topic in the present paper. Furthermore, let us observe that a binary relation can be introduced in the general language $\mathcal{L}^{P}$ by means of the following definition. *For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}$*,* $\delta _{1}\prec $* *$\delta _{2}$* iff a proof exists that* $\delta _{2}$* is justified whenever* $\delta _{1}$*is justified (equivalently,* $\delta _{1}\prec \delta _{2}$* iff* $\delta _{1}C\delta _{2}$* is justified*). The set-theoretical pragmatics introduced in Sec. 3.2 allows one to translate the above definition in $\mathcal{L}_{Q}^{P}$ as follows. *For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\prec $* *$\delta _{2}$* iff for every* $S\in S$*,* $\pi _{S}(\delta _{1})=J$* implies* $\pi _{S}(\delta _{2})=J$*.* The binary relation $\prec $ can then be characterized as follows. *For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\prec $* *$\delta _{2}$* iff* $S_{\delta _{1}}\subset S_{\delta _{2}}$*.* The relation $\prec $ is obviously a pre-order relation on $\psi _{A}^{Q}$, hence it induces canonically an equivalence relation $\approx $ on $\psi _{A}^{Q}$, defined as follows. *For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\approx $* *$\delta _{2}$* iff* $\delta _{1}\prec $* *$\delta _{2}$* and* $\delta _{2}\prec $* *$\delta _{1}$*.* The equivalence relation $\approx $ can then be characterized as follows. *For every* $\delta _{1}$*,* $\delta _{2}\in \psi _{A}^{Q}$*,* $\delta _{1}\approx $* *$\delta _{2}$* iff* $S_{\delta _{1}}=S_{\delta _{2}}$*.* Decidability versus justifiability in $\mathcal{L}_{Q}^{P}$ ----------------------------------------------------------- We have commented rather extensively in Sec. 3.3 on the notion of justification formalized in $\mathcal{L}_{Q}^{P}$, for every $S\in \mathcal{S}$, by the pragmatic evaluation function $\pi _{S}$. It must still be noted, however, that the definition of $\pi _{S}$ on all afs in $\psi _{A}^{Q}$ does not grant that an empirical procedure of proof exists which allows one to establish, for every $S\in \mathcal{S}$, the justification value of every af of $\mathcal{L}_{Q}^{P}$. In order to understand how this may occur, note that the notion of empirical proof is defined by A$_{5}$ for atomic rfs of $\mathcal{L}_{Q}^{P}$ and makes explicit reference, for every $E(x)\in \psi _{R}^{Q}$, to the closed subset $\mathcal{S}_{E}\in \mathcal{L(S)}$ associated with $E$ by the function $\rho $ introduced in Sec. 2.2. Basing on this notion, the justification value $\pi _{S}(\vdash E(x))$ of an elementary af $\vdash E(x)\in \psi _{A}^{Q}$ can be determined by means of the same empirical procedure, making reference to the closed subset $\mathcal{S}_{\vdash E(x)}=\mathcal{S}_{E}$ associated to $\vdash E(x)$ by the function $f$ (Sec. 3.2). Yet, whenever $\pi _{S}$ is recursively defined on the whole $\psi _{A}^{Q}$, new subsets of states are introduced (as $\mathcal{S}_{\delta _{1}}\cup \mathcal{S}_{\delta _{2}}$) which do not necessarily belong to $\mathcal{L(S)}$. If an af $\delta $ is associated by $f$ with a subset that does not belong to $\mathcal{L(S)}$, no empirical procedure exists in QM which allows one to determine the justification value $\pi _{S}(\delta )$. We are thus led to introduce the subset $\psi _{AD}^{Q}$ $\subset \psi _{A}^{Q}$ of all *pragmatically decidable*, or *p-decidable*, afs of $\mathcal{L}_{Q}^{P}$. An af $\delta $ of $\mathcal{L}_{Q}^{P}$ is p-decidable iff an empirical procedure of proof exists which allows one to establish whether $\delta $ is justified or unjustified, whatever the state $S$ of $x$ may be. Because of the remark above, the subset of all p-decidable afs of $\mathcal{L}_{Q}^{P}$ can be characterized as follows. $\psi _{AD}^{Q}=\{\delta \in \psi _{A}^{Q}\mid $ $\mathcal{S}_{\delta }\in \mathcal{L(S)}\}$. Let us discuss some criteria for establishing whether a given af $\delta \in $ $\psi _{A}^{Q}$ belongs to $\psi _{AD}^{Q}$. C$_{1}$. *All elementary afs of* $\psi _{A}^{Q}$* belong to* $\psi _{AD}^{Q}$*.* C$_{2}$. *If* $\delta \in $* *$\psi _{AD}^{Q}$*, then* $N\delta \in $* *$\psi _{AD}^{Q}$* * Indeed, $S_{\delta }\in \mathcal{L(S)}$ implies $S_{\delta }^{\bot }\in \mathcal{L(S)}$. C$_{3}$. *If* $\delta _{1}$*,* $\delta _{2}\in $* *$\psi _{AD}^{Q}$*, then* $\delta _{1}K$* *$\delta _{2}\in $* *$\psi _{AD}^{Q}$ Indeed, $S_{\delta _{1}}\in \mathcal{L(S)}$ and $S_{\delta _{2}}\in \mathcal{L(S)}$ imply $S_{\delta _{1}}\cap S_{\delta _{2}}\in \mathcal{L(S)} $, since $S_{\delta _{1}}\cap S_{\delta _{2}}=S_{\delta _{1}}\Cap S_{\delta _{2}}$ because of known properties of the lattice $(\mathcal{L(S)},\subset ) $ (Sec. 2.2). C$_{4}$. *If* $\delta _{1}$*,* $\delta _{2}\in $* *$\psi _{AD}^{Q}$*, then* $\delta _{1}A$* *$\delta _{2}$* may belong or not to* $\psi _{AD}^{Q}$*. To be precise, it belongs to* $\psi _{AD}^{Q}$* iff* $S_{\delta _{1}}\subset S_{\delta _{2}}$* or* $S_{\delta _{2}}\subset S_{\delta _{1}}$ Indeed, $S_{\delta _{1}}\cup S_{\delta _{2}}\in \mathcal{L(S)}$ or, equivalently, $S_{\delta _{1}}\cup S_{\delta _{2}}=S_{\delta _{1}}\Cup S_{\delta _{2}}$, iff one of the conditions in C$_{4}$ is satisfied. It is apparent from criteria C$_{2}$ and C$_{3}$ that $\psi _{AD}^{Q}$ is closed with respect to the pragmatic connectives $N$ and $K$, in the sense that $\delta \in \psi _{AD}^{Q}$ implies $N\delta \in \psi _{AD}^{Q}$, and $\delta _{1}$, $\delta _{2}\in \psi _{AD}^{Q}$ implies $\delta _{1}K\delta _{2}\in \psi _{AD}^{Q}$. On the contrary, $\psi _{AD}^{Q}$ is not closed with respect to $A$, since it may occur that $\delta _{1}A$ $\delta _{2}\notin \psi _{AD}^{Q}$ even if $\delta _{1}$, $\delta _{2}\in \psi _{AD}^{Q}$. In order to obtain a closed subset of afs of $\mathcal{L}_{Q}^{P} $, one can consider the set $\phi _{AD}^{Q}=\{\delta \in \psi _{A}^{Q}\mid $ the pragmatic connective $A$ does not occur in $\delta \}$. The set $\phi _{AD}^{Q}$ obviously contains all elementary afs of $\mathcal{L}_{Q}^{P}$, plus all afs of $\psi _{A}^{Q}$ in which only the pragmatic connectives $N$ and $K$ occur. We can thus consider a sublanguage of $\mathcal{L}_{Q}^{P}$ whose set of afs reduces to $\phi _{AD}^{Q}$. This new language is relevant since all its afs are p-decidable, hence we call it* the* *p-decidable sublanguage* of $\mathcal{L}_{Q}^{P}$ and denote it by $\mathcal{L}_{QD}^{P}$. The p-decidable sublanguage $\mathcal{L}_{QD}^{P}$ -------------------------------------------------- As we have anticipated in the Introduction, we aim to show in this paper that the sublanguage $\mathcal{L}_{QD}^{P}$ has the structure of a physical QL, hence it provides a new pragmatic interpretation of this relevant physical structure. However, this interpretation will be more satisfactory from an intuitive viewpoint if we endow $\mathcal{L}_{QD}^{P}$ with some further derived pragmatic connectives which can be made to correspond with connectives of physical QL. To this end, we introduce the following definitions. D$_{1}$. *We call* quantum pragmatic disjunction *the connective* $A_{Q}$* defined as follows.* *For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi _{AD}^{Q}$*,* $\delta _{1}A_{Q}\delta _{2}=N((N\delta _{1})K(N\delta _{2}))$*.* D$_{2}$. *We call* quantum pragmatic implication* the connective* $I_{Q}$* defined as follows.* *For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi _{AD}^{Q}$*,* $\delta _{1}I_{Q}\delta _{2}=(N\delta _{1})A_{Q}(\delta _{1}K\delta _{2})$*.* Let us discuss the justification rules which hold for afs in which the new connectives $A_{Q}$ and $I_{Q}$ occur. By using the function $f$ introduced in Sec. 3.2 we get (since the set-theoretical operation $\cap $ coincides with the lattice operation $\Cap $ in $(\mathcal{L(S)},\subset )$, see Sec. 2.2), $\mathcal{S}_{\delta _{1}A_{Q}\delta _{2}}=\mathcal{S}_{(N\delta _{1})K(N\delta _{2})}^{\bot }=(\mathcal{S}_{N\delta _{1}}\cap \mathcal{S}_{N\delta _{2}})^{\bot }=(\mathcal{S}_{\delta _{1}}^{\bot }\Cap \mathcal{S}_{\delta _{2}}^{\bot })^{\bot }=(\mathcal{S}_{\delta _{1}}\Cup \mathcal{S}_{\delta _{2}})$. Hence, for every $S\in \mathcal{S}$, $\pi _{S}(\delta _{1}A_{Q}\delta _{2})=J$ iff $S\in \mathcal{S}_{\delta _{1}}\Cup \mathcal{S}_{\delta _{2}}$. Let us come to the quantum pragmatic implication $I_{Q}$. By using the definition of $A_{Q}$, one gets $\delta _{1}I_{Q}\delta _{2}=N((NN\delta _{1})K(N(\delta _{1}K\delta _{2}))$. By using the function $f$ and the above result about $A_{Q}$, one then gets $\mathcal{S}_{\delta _{1}I_{Q}\delta _{2}}=\mathcal{S}_{N\delta _{1}}\Cup \mathcal{S}_{\delta _{1}K\delta _{2}}=\mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$. It follows that, for every $S\in \mathcal{S}$, $\pi _{S}(\delta _{1}I_{Q}\delta _{2})=J$ iff $S\in \mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$. Let us observe now that $\mathcal{L}_{QD}^{P}$ obviously inherits the notions of p-validity and order defined in $\mathcal{L}_{Q}^{P}$ (Sec. 3.4). Hence, we can illustrate the role of the connective $I_{Q}$ within $\mathcal{L}_{QD}^{P}$ by means of the following *pragmatic deduction lemma*. PDL.* Let* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi _{AD}^{Q}$*. Then,* $\delta _{1}\prec $* *$\delta _{2}$* iff for every* $S\in S$*,* $\pi _{S}(\delta _{1}I_{Q}\delta _{2})=J$* (equivalently, iff* $\delta _{1}I_{Q}\delta _{2}$*is p-valid).* Proof. The following sequence of equivalences holds. For every $S\in \mathcal{S}$, $\pi _{S}(\delta _{1}I_{Q}\delta _{2})=J$ iff for every $S\in \mathcal{S}$, $S\in \mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})$ iff $\mathcal{S}_{\delta _{1}}^{\bot }\Cup (\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}})=\mathcal{S}$ iff $\mathcal{S}_{\delta _{1}}\Cap \mathcal{S}_{\delta _{2}}=$ $\mathcal{S}_{\delta _{1}}$ iff $\mathcal{S}_{\delta _{1}}\subset \mathcal{S}_{\delta _{2}}$ iff $\delta _{1}\prec $ $\delta _{2}$.$\blacksquare \smallskip $ PDL shows that the quantum pragmatic implication $I_{Q}$ plays within $\mathcal{L}_{QD}^{P}$ a role similar to the role of material implication in classical logic. Interpreting QL onto $\mathcal{L}_{QD}^{P}$ ------------------------------------------- In order to show that the physical QL $(\mathcal{E},\prec )$ introduced in Sec. 2.2 can be interpreted into $\mathcal{L}_{QD}^{P}$, a further preliminary step is needed. To be precise, let us make reference to the preorder introduced on $\psi _{A}^{Q}$ in Sec. 3.4 and consider the pre-ordered set $(\phi _{AD}^{Q},\prec )$ of all afs of $\mathcal{L}_{QD}^{P} $. Furthermore, let us denote by $\approx $ (by abuse of language) the restriction of the equivalence relation introduced on $\psi _{A}^{Q}$ in Sec. 3.4 to $\phi _{AD}^{Q}$, and let us denote by $\prec $ (again by abuse of language) the partial order induced on $\phi _{AD}^{Q}/\approx $ by the preorder defined on $\phi _{AD}^{Q}$. Then, let us show that $(\phi _{AD}^{Q}/\approx ,\prec )$ is order isomorphic to $(\mathcal{L(S)},\subset ) $. Let us consider the mapping $f_{\approx }:[\delta ]_{\approx }\in \psi _{AD}^{Q}/\approx \;\longrightarrow $ $\mathcal{S}_{\delta }\in \mathcal{L(S)}$. This mapping is obviously well defined because of the characterization of $\approx $ in Sec. 3.4. Furthermore, the following statements hold. \(i) *For every* $\delta \in \phi _{AD}^{Q}$*, one and only one elementary af* $\vdash E(x)$* exists such that* $\vdash E(x)\in \lbrack \delta ]_{\approx }$. \(ii) *The mapping* $f_{\approx }$* is bijective.* \(iii) *For every* $\delta _{1}$*,* $\delta _{2}\in $* *$\phi _{AD}^{Q}$*,* $[\delta _{1}]_{\approx }\prec \lbrack \delta _{2}]_{\approx }$* iff* $S_{\delta _{1}}\subset S_{\delta _{2}}$*.* Let us prove (i). Consider $[\delta ]_{\approx }$. Since $\mathcal{S}_{\delta }\in \mathcal{L(S)}$ and $\rho $ is bijective (Sec. 2.2), a property $E\in \mathcal{E}$ exists such that $E=\rho ^{-1}(\mathcal{S}_{\delta })$, hence $\mathcal{S}_{\delta }=\mathcal{S}_{E}$. It follows that $[\delta ]_{\approx }$ contains the af $\vdash E(x)$, for $\mathcal{S}_{\vdash E(x)}=\mathcal{S}_{E}$ (Sec. 3.2). Moreover, $[\delta ]_{\approx }$ does not contain any further elementary af. Indeed, let $\vdash F(x)$ be an elementary af of $\phi _{AD}^{Q}$ with $E\neq F$: then, $\mathcal{S}_{E}\neq \mathcal{S}_{F}$, hence $\mathcal{S}_{\vdash E(x)}\neq \mathcal{S}_{\vdash F(x)}$, which implies $\vdash F(x)\notin \lbrack \delta ]_{\approx }$. Thus, statement (i) is proved. The proofs of statements (ii) and (iii) are then immediate. Indeed, statement (ii) follows from (i) and from the definition of $f_{\approx }$, while statement (iii) follows from (ii) and from the definition of $\prec $ on $\phi _{AD}^{Q}/\approx $. Because of (ii) and (iii), the poset $(\phi _{AD}^{Q}/\approx ,\prec )$ is order-isomorphic to $(\mathcal{L(S)},\subset )$, as stated. Let us come now to physical QL. We have seen in Sec. 2.2 that $(\mathcal{L(S)},\subset )$ is order-isomorphic to $(\mathcal{E},\prec )$. We can then conclude that $(\mathcal{E},\prec )$ is order-isomorphic to $(\phi _{AD}^{Q}/\approx ,\prec )$, which provides the desired interpretation of a physical QL into $\mathcal{L}_{QD}^{P}$. Let us comment briefly on the pragmatic interpretation of physical QL provided above. Firstly, we note that our interpretation maps $\mathcal{E}$ on the quotient set $\phi _{AD}^{Q}/\approx $, not onto $\phi _{AD}^{Q}$. Yet, the set of the (well formed) formulas of the lattice $(\mathcal{E},^{\bot },\Cap ,\Cup ) $ can be mapped bijectively onto $\phi _{AD}^{Q}$ by means of the mapping induced by the following formal correspondence. \(i) $E\in \mathcal{E}$ $\longleftrightarrow \vdash E(x)\in \phi _{AD}^{Q}$. \(ii) $^{\bot }\longleftrightarrow N$ \(iii) $\Cap \longleftrightarrow K$ \(iv) $\Cup \longleftrightarrow A_{Q}$. Thus, the formal language of QL, for which the lattice $(\mathcal{L(S)},\subset )$ can be considered as an *algebraic semantics*,$^{(3)}$ can be substituted by the language $\mathcal{L}_{QD}^{P}$, for which $(\mathcal{L(S)},\subset )$ can be considered as an *algebraic pragmatics* (by the way, we also note that the above correspondence makes $I_{Q}$ correspond to a *Sasaki hook*, the role of which is well known in QL). This reinterpretation is relevant from a philosophical viewpoint, since it avoids all problems following from the standard concept of quantum truth (Sec. 2.4) considering physical QL as formalizing properties of a quantum concept of justification rather than a quantum concept of truth. This makes physical QL consistent also with the classical concept of truth adopted with the SR interpretation of QM (Sec. 2.5). Furthermore, as we have already observed in the Introduction, it places physical QL within a general *integrated perspective*, according to which non-Tarskian theories of truth can be integrated with Tarski’s theory by reinterpreting them as theories of metalinguistic concepts that are different from truth (in the case of physical QL, the concept of *empirical justification* in QM). Secondly, we observe that our interpretation has some consequences that are intuitively satisfactory. For instance, for every state $S\in \mathcal{S}$, it attributes a justification value to every af in $\phi _{AD}^{Q}$, while it is well known that there are formulas in physical QL which have no truth value according to the standard interpretation of QL (Sec. 2.4). Some remarks on a possible calculus for $\mathcal{L}_{QD}^{P}$ -------------------------------------------------------------- One may obviously wonder whether a calculus can be given for the language $\mathcal{L}_{QD}^{P}$ which is *pragmatically correct* (*p-correct*) and *pragmatically complete* (*p-complete*). This is not a difficult task if we limit ourselves to the general lattice structure of $(\phi _{AD}^{Q}/\approx ,\prec )$. Indeed, a set of axioms and/or inference rules which endow $\phi _{AD}^{Q}/\approx $ of the structure of orthomodular lattice can be easily obtained by using the formal correspondence introduced in Sec. 3.7, since this correspondence allows one to translate the axioms and/or inference rules that are usually stated in order to provide a calculus for orthomodular QL into $\phi _{AD}^{Q}$ (of course, all the afs produced by this translation are p-valid afs of $\mathcal{L}_{QD}^{P}$). Here is a sample set of axioms of this kind (where, of course, $\delta $, $\delta _{1}$, $\delta _{2}$ and $\delta _{3}$ are afs of $\phi _{AD}^{Q}$) obtained by translating a set of rules provided by Dalla Chiara and Giuntini.$^{(32)}$ A$_{1}$. $\delta I_{Q}\delta $. A$_{2}$. $($ $\delta _{1}K$ $\delta _{2})I_{Q}\delta _{1}$. A$_{3}$. $($ $\delta _{1}K$ $\delta _{2})I_{Q}\delta _{2}$. A$_{4}$. $\delta I_{Q}(NN\delta )$. A$_{5}$. $(NN\delta )I_{Q}\delta $. A$_{6}$. $((\delta _{1}I_{Q}\delta _{2})K(\delta _{1}I_{Q}\delta _{3}))I_{Q}(\delta _{1}I_{Q}(\delta _{2}K\delta _{3}))$. A$_{7}$. $((\delta _{1}I_{Q}\delta _{2})K(\delta _{2}I_{Q}\delta _{3}))I_{Q}(\delta _{1}I_{Q}\delta _{3})$. A$_{8}$. $(\delta _{1}I_{Q}\delta _{2})I_{Q}((N\delta _{2})I_{Q}(N\delta _{1}))$. A$_{9}$. $(\delta _{1}I_{Q}\delta _{2})I_{Q}(\delta _{2}I_{Q}(\delta _{1}A_{Q}((N\delta _{1})K\delta _{2})))$. However, in order to obtain physical QL one needs a number of further axioms, since the structure of $(\mathcal{L(H)},\subset )$ must be recovered (Sec. 2.2). Providing a complete calculus for such a structure is a much more complicate task, which must take into account a number of mathematical results in lattice theory (in particular, Soler’s theorem$^{(33)}$). Therefore we will not discuss this problem in the present paper. **ACKNOWLEDGEMENT** The author is greatly indebted to Carlo Dalla Pozza, Jaroslaw Pykacz and Sandro Sozzo for reading the manuscript and providing many useful suggestions. [10]{} M. Jammer, *The Philosophy of Quantum Mechanics* (Wiley, New York, 1974). M. Rédei, *Quantum Logic in Algebraic Approach* (Kluwer, Dordrecht, 1998). M. Dalla Chiara, R. Giuntini and R. Greechie, *Reasoning in Quantum Theory* (Kluwer, Dordrecht, 2004). S. Haak, *Philosophy of Logics*, (Cambridge University Press, Cambridge, 1978). A. Tarski, “The concept of truth in formalized languages,” in Tarski, A., *Logic, Semantics, Metamathematics*, (Blackwell, Oxford, 1956). A. Tarski, “The semantic conception of truth and the foundations of semantics,” in L. Linski (ed.), *Semantics and the Philosophy of Language* (University of Illinois Press, Urbana, 1944). A. Messiah, *Quantum Mechanics* (North Holland, Amsterdam, 1961). Cohen-Tannoudij, B. Dieu and F. Laloe, *Mecanique Quantique* (Hermann, Paris, 1973). W. Greiner, *Quantum Mechanics, an Introduction* (Verlag, Berlin, 1989). B. D’Espagnat, *Conceptual Foundations of Quantum Mechanics* (Benjamin, Reading, MA, 1976). G. Ludwig, *Foundations of Quantum Mechanics I* (Springer, New York, 1983). C. Garola and S. Sozzo, “A semantic approach to the completeness problem in quantum mechanics,” *Found. Phys.* **34**, 1249-1266 (2004). C. Piron, *Foundations of Quantum Physics* (Benjamin, Reading, MA, 1976). D. Aerts,* *“Description of many separated entities without the paradoxes encountered in quantum mechanics,”*Found. Phys.* **12**, 1131-1170* *(1982). J. S. Bell, “On the problem of hidden variables in quantum mechanics,” *Rev. Mod. Phys.* **38**, 447-452 (1966). S. Kochen and E. P. Specker, “The problem of hidden variables in quantum mechanics,” *J. Math. Mech.* **17**, 59-87 (1967). J. S. Bell, “On the Einstein-Podolski-Rosen paradox,” *Physics* **1**, 195-200 (1964). N. D. Mermin, “Hidden variables and the two theorems of John Bell,” *Rev. Mod. Phys.* **65**, 803-815 (1993). C. Garola, “Classical foundations of quantum logic,” *Int. J. Theor. Phys.* **30**, 1-52 (1991). C. Garola, “Truth versus testability in quantum logic,” *Erkenntnis* **37**, 197-222 (1995). C. Garola and L. Solombrino, “The theoretical apparatus of semantic realism: A new language for classical and quantum physics,” *Found. Phys.* **26**, 1121-1164 (1996a). C. Garola and L. Solombrino, “Semantic realism versus EPR-like paradoxes: The Furry, Bohm-Aharonov and Bell paradoxes” *Found. Phys.* **26**, 1329-1356 (1996b). C. Garola, “Against ‘paradoxes’: A new quantum philosophy for quantum physics,” in *Quantum Physics and the Nature of Reality*, D. Aerts and J. Pykacz, eds. (Kluwer, Dordrecht,1999). C. Garola, “Objectivity versus nonobjectivity in quantum mechanics,” *Found. Phys.* **30**, 1539-1565 (2000). C. Garola, “A simple model for an objective interpretation of quantum mechanics,” *Found. Phys.* **32**, 1597-1615 (2002). C. Garola and J. Pykacz, “Locality and measurements within the SR model for an objective interpretation of quantum mechanics,” *Found. Phys.* **34**, 449-475 (2004). C. Dalla Pozza and C. Garola, “A pragmatic interpretation of intuitionistic propositional logic,” *Erkenntnis* **43**, 81-109 (1995). E. Beltrametti and G. Cassinelli, *The Logic of Quantum Mechanics* (Addison-Wesley, Reading, MA, 1981). G. W. Mackey, *The Mathematical Foundations of Quantum Mechanics* (Benjamin, New York, 1963). J. M. Jauch, *Foundations of Quantum Mechanics* (Addison-Wesley, Reading, MA, 1968). G. Birkhoff and J. von Neumann, “The logic of quantum mechanics,” *Ann. Math.*** 37**, 823-843 (1936). M. Dalla Chiara and R. Giuntini, “La Logica Quantistica,” in G. Boniolo (ed.), *Filosofia della Fisica* (B. Mondadori, Milan, 1997) M. P. Soler, “Characterization of Hilbert spaces by orthomodular spaces,” *Comm. Algebra* **23**, 219-243 (1995). [^1]: It must be noted that the physical properties considered here are first order properties from a logical viewpoint.$^{(26)}$ Higher order properties obviously occur in physics and will be encountered later on (Sec. 2.6), but they need not be considered here. [^2]: The notion of physical equivalence for preparing or registering devices is not trivial.$^{(11,21)}$ We do not discuss it here for the sake of brevity. [^3]: Note that a registration may act as a new preparation of the physical object $x$, so that the state of $x$ may change after a test on it. [^4]: It follows easily that every state S can also be represented by any vector $\mid \psi \rangle \in \varphi (S)\in \mathcal{A}$, which is the standard representation adopted in elementary QM. Moreover, a state S is usually represented by an (orthogonal) projection operator on $\varphi (S)$ in more advanced QM. However, the representation $\varphi $ introduced here is more suitable for our purposes in the present paper. [^5]: Equivalently, a property is often represented in QM as a pair $(A,\Delta )$, where is $A$ a self-adjoint operator on $\mathcal{H}$ representing a physical observable, and $\Delta $ a Borel set on the real line.$^{(28)}$ We do not use this representation, however, in the present paper. [^6]: Whenever the dimension of $\mathcal{H}$ is finite, the lattice $(\mathcal{L(H)},\subset )$ and/or the lattice $(\mathcal{L(S)},\subset )$ can be identified with Birkhoff and von Neumann’s lattice of *experimental propositions*, which was introduced in the 1936 paper that started the research on QL.$^{(31)}$ This identification is impossible, however, if $\mathcal{H}$ is not finite-dimensional, since Birkhoff and von Neumann’s lattice is modular, not simply weakly modular. The requirement of modularity has deep roots in the von Neumann concept of probability in QM according to some authors.$^{(2)}$ [^7]: One can provide an intuitive support to this definition by noticing that the result obtained in a test of $E$ on a physical object $x$ in the state $S$ can be attributed to $x$ only whenever $S$ is not modified by the test. Moreover, only in this case the test is *repeatable*, i.e., it can be performed again obtaining the same result. It is well known that classical physics assumes that tests which do not modify the state $S$ are always possible, at least as ideal limits of concrete procedures, while this assumption does not hold in QM. [^8]: Assumption A$_{4}$ can be stated unchanged whenever the standard interpretation of QM is adopted instead of the SR interpretation. In this case, however, for every $\xi $, $\sigma (\xi )$ is defined only on a subset of rfs, not on the whole $\psi _{R}^{Q}$ (which requires a weakening of the assumptions on $\sigma $ if one wants to recover this case within the general perspective in Sec. 3.1). Furthermore, $\Sigma _{S}$ reduces to a singleton. Indeed, for every interpretation $\xi $, a state $S=S(\xi )$ exists such that $\xi (x)\in S$. Then, $\sigma (\xi )$ is defined on a rf $E(x)$ iff $E\in \mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$ (Sec. 2.4), and does not change if $\xi $ is substituted by an interpretation $\xi ^{\prime } $ such that $\xi ^{\prime }(x)\in S$. [^9]: Assumption A$_{5}$ in Sec. 3.2 can be stated unchanged if the standard interpretation of QM is adopted instead of the SR interpretation. In this case, however, it is impossible that a mapping $\xi ^{\prime }$ exists such that $\xi ^{\prime }(x)\in S^{\prime }$, with $S\neq S^{\prime }$ and $\sigma (\xi )=\sigma (\xi ^{\prime })$, since $\sigma (\xi )$ and $\sigma (\xi ^{\prime })$ are defined on different domains ($\mathcal{E}_{S}\cup \mathcal{E}_{S}^{\bot }$ and $\mathcal{E}_{S^{\prime }}\cup \mathcal{E}_{S^{\prime }}^{\bot }$, respectively). Hence, an assigment function $\sigma $ is associated with a unique state $S$, and $\Pi _{\sigma }$ reduces to the singleton $\{\pi _{S}\}$.

--- abstract: 'The paper presents the background for Toeplitz and Hankel operators acting between distinct Hardy type spaces over the unit circle ${\mathbb{T}}$. We characterize possible symbols of such operators and prove general versions of Brown-Halmos and Nehari theorems. The lower bound for measure of noncomactness of Toeplitz operator is also found. Our approach allows Hardy spaces associated with arbitrary rearrangement invariant spaces, but a main part of results is new even for the classical case of $H^p$ spaces.' address: 'Institute of Mathematics, Poznań University of Technology, ul. Piotrowo 3a, 60-965 Poznań, Poland' author: - Karol Leśnik title: Toeplitz and Hankel operators between distinct Hardy spaces --- Introduction ============ Classical Toeplitz $T_a$ and Hankel $H_a$ operators on Hardy space $H^2$ (on the unit circle ${\mathbb{T}}$) are defined by $$\label{TH} T_a\colon f\mapsto P(af)\ {\rm\ and\ } H_a\colon f\mapsto P(aJf),$$ where $P$ is the Riesz projection, $J$ is the flip operator and the function $a\in L^{\infty}$ is called the symbol of $T_a$ and $H_a$, respectively. Theory of Toeplitz and Hankel operators acting on $H^p$ spaces, as well as on a number of another function spaces is very well developed and still widely investigated. Moreover, such operators are interesting not only from the point of view of operator theory, but they are intimately connected with harmonic analysis, prediction theory and approximation theory (see for example [@Pel03]). However, in the literature Toeplitz and Hankel operators are mainly considered to act from one to the same space. Suppose now we leave the above definition (\[TH\]) unchanged, but take a symbol $a\in L^r$ for some $1<r<\infty$. In such a case $T_a$ and $H_a$ need not be bounded on any $H^p$ space, but they act boundedly from $H^p$ to $H^q$ if $1<q<p<\infty$ and $\frac{1}{p}+\frac{1}{r}=\frac{1}{q}$. It appears that almost nothing is known about such operators. Among a huge number of papers considering Toeplitz and Hankel operators we were able to find only few, where they act between distinct spaces. This number includes papers of Tolokonnikov [@Tol87] and of Tolokonnikov and Volberg [@TV87]. In the first the symbols of Toeplitz and Hankel operators acting between distinct $H^p$ spaces were determined, while the second is devoted to approximation problem connected with the representation of Hankel operators considered between abstract Hardy type spaces. Except these two papers one can find investigations of Toeplitz and Hankel operators acting from some Hardy type space into $H^1$ in the Janson, Peetre and Semmes paper [@JPS84] and a generalization of these investigations for Hardy spaces over more complicated domains in [@BG10]. The goal of this paper is to present an unified background for Toeplitz and Hankel operators acting between distinct Hardy spaces, i.e. $T_a,H_a\colon H[X]\to H[Y]$, where $X,Y$ are rearrangement invariant spaces. The main results are general versions of Brown–Halmos and Nehari theorems. In such a general situation symbols $a$ belong to the space of pointwise multipliers $M(X,Y)$. In consequence, a deeper theory of function spaces, pointwise multipliers, pointwise products and factorization comes into play. The paper is organized as follows. In the second section we collect required definitions and notation, that will be used through the paper. The third section contains a number of technical results describing basic properties of Hardy type spaces built upon rearrangement invariant function spaces on the unit circle ${\mathbb{T}}$. The fourth section is devoted to Toeplitz operators. In the classical case of $H^2$ the following theorem characterizes bounded Toeplitz operators. It not only identifies possible symbols of bounded Toeplitz operators on $H^2$, but mainly says that each operator satisfying (\[Toep0\]), i.e. having Toeplitz matrix with respect to the standard basis of $H^2$, has the representation of the form $T_a$, where $a\in L^{\infty}$ is uniquely determined. We give an analogue of the Brown–Halmos theorem for the case of operators acting from $H[X]$ to $H[Y]$, under some mild assumptions on spaces $X,Y$. The result seems to be new even in the case of $T_a\colon H^p\to H^q$. Let us mention also, that the version of Brown–Halmos theorem for the case $X=Y$ has been already proved in [@K04], but even in this particular case our assumptions are less restrictive. Moreover, we discuss also the case of nonseparable spaces $X$ and $Y$. In the main, fifth section, Hankel operators are taken into account. While the previous section is rather analogous to the classical case, except some technicalities, situation for Hankel operators makes much more interesting. Let us recall the statement of the classical Nehari theorem. Thus, the theorem characterizes operators with Hankel matrices and their symbols. However, we point out that, in contrast to Brown–Halmos theorem, a symbol $a$ is not unique (i.e. the operator remains the same if $a$ is modified by adding arbitrary function $b$ satisfying $Pb=0$, since only Fourier coefficients of $a$ for $n>0$ appears in (\[Hankel condition0\])). In this section we prove a general Nehari theorem for Hankel operators acting from $H[X]$ to $H[Y]$, under some assumptions on spaces $X,Y$. Let us mention, that modern proofs of Nehari theorem base on the (strong) factorization of $H^1$ function $f$ into product $f=gh$, where $g,h\in H^2$ (see for example [@BS06 Theorem 2.11] or [@Pel03 Theorem 1.1]). A direct translation of this idea together with the Lozanovskii factorization theorem for Hardy spaces (i.e. $H[X]\odot H[X']=H^1$, where $X'$ is the Köthe dual of $X$) was used in [@K04] to prove the Nehari theorem for $H_a:H[X]\to H[X]$ (see also [@Ha98], where the same subject was undertaken). Of course, the generalized Lozanovskii-like factorization would do the job also in our setting, however the assumption that $X$ factorizes $Y$ (i.e. $X\odot M(X,Y)=Y$) is rather restrictive (see [@KLM14] for extensive studies of this problem) and we expect weaker assumptions for the general Nehari theorem. On the other hand, as it was noticed by Coifman, Rochberg and Weiss [@CRW76] (see also [@JPS84] and [@TV87]) the strong factorization may be replaced by the weak one (i.e. $f=\sum_kg_kh_k$ instead of $f=gh$). However, theory of such factorization is not very well developed and it is not at all applicable in a general setting (the space of symbols of Hankel operators were described in terms of weak factorization in [@TV87], but it appeared that the authors were able to give concrete representation only in cases when strong factorization holds). Therefore, instead of weak factorization, we base our proof of general Nehari theorem on the concept of Banach envelopes, which works pretty well in this setting and, indeed, gives a weak factorization, as a byproduct (see discussion after Lemma \[Ban-env-Cor\]). This section is finished by an extensive discussion on assumptions of the main theorem and we give some examples for concrete types of spaces, like Orlicz and Lorentz spaces. We finish the paper estimating the measure of noncompactness of Toeplitz operator $T_a$ in terms of Fourier coefficients of its symbol $a$. Notions and notations ===================== Let ${\mathbb{T}}$ be the unit circle equipped with the normalized Lebesgue measure $dm(t)=|dt|/(2\pi)$. Let $L^0:=L^0({\mathbb{T}},m)$ be the space of all measurable complex-valued almost everywhere finite functions on ${\mathbb{T}}$. As usual, we do not distinguish functions, which are equal almost everywhere (for the latter we use the standard abbreviation a.e.). The characteristic function of a measurable set $E\subset{\mathbb{T}}$ is denoted by $\chi_E$. A complex quasi-Banach space $X\subset L^0({\mathbb{T}},m)$ is called a quasi-Banach function space (q-B.f.s for short) if\ (a) $f\in X$, $g\in L^0$ and $|g| \le |f|$ a.e. $\Rightarrow \ g\in X$ and $\|g\|_X \le \|f\|_X$ (the ideal property),\ (b) $L^{\infty}\subset X$,\ (c) $X$ has the semi-Fatou property, i.e. $(f_n)\subset X$, $f\in X$ and $0\le f_n\uparrow f$ a.e. implies $\|f\|_X = \sup_{n\in{\mathbb{N}}}\|f_n\|_X$. If $X$ as above is a Banach space, we will call it the Banach function space (B.f.s. for short). Spaces as above are also called Köthe spaces (see for example [@LT79 page 28], [@Za67 Chapter 15]), while the name B.f.s. comes from [@BS88 Chapter 1], [@MN91 p. 114] or [@Mal89 p. 161] (cf. [@KPS82 pages 40–43]). Notice however, that our definition differs a little from respective definitions in the mentioned books. In fact, assumptions in our definition are weaker than in [@BS88], but stronger than in [@LT79; @Za67; @KPS82; @Mal89; @MN91]. The point (a) is crucial in all of them. Point (b) is satisfied by each rearrangement invariant spaces and we will focus only on such spaces, thus we assumed it already for B.f. spaces to simplify presentation. Finally, semi-Fatou property from the point (c) will be crucial in few places, but we cannot replace it by the stronger Fatou property (which is assumed for B.f. spaces in [@BS88]), because we will work a lot with subspaces of order continuous elements, which, in general, need not satisfy the Fatou property. Finally, notice that classical spaces, such as Lebesgue, Orlicz and Lorentz spaces, fulfill conditions of our definition. It is known, that for q-B.f. spaces $X,Y$ inclusions $X\subset Y$ are always continuous, i.e. there is $C>0$ such that $\|f\|_Y\leq C\|f\|_X$ for each $f\in X$. It follows, for example, from continuity of embedding $X,Y\subset L^0$ (see for example [@Ro85 Proposition 2.7.2]) and the closed graph theorem (see [@KPR84 pages 9–11] for discussion on classical theorems in quasi-Banach case). We will write $X=Y$ if $X$ and $Y$ coincide as sets and there are positive constants $c_1,c_2$ such that $c_1\|f\|_X\le \|f\|_Y\le c_2\|f\|_X$ for all $f\in X$ (the latter inequalities will be also denoted as $\|f\|_Y\approx \|f\|_X$), and $X\equiv Y$ if $c_1=c_2=1$. A q-B.f.s. $X$ has the Fatou property ($X\in (FP)$ for short) when given a sequence $(f_n)_{n\in{\mathbb{N}}}\subset X$ and $f\in L^0$ satisfying $0\le f_n\uparrow f$ a.e. as $n\to\infty$ and $\sup_{n\in{\mathbb{N}}}\|f_n\|_X<\infty$ there holds $f\in X$ and $\|f\|_X = \sup_{n\in{\mathbb{N}}}\|f_n\|_X$. Recall that $f\in X$ is said to be an order continuous element, if for each $(f_n)_{n\in{\mathbb{N}}}\subset X$ satisfying $0\le f_n\le |f|$ for all $n\in{\mathbb{N}}$ and $f_n\to 0$ a.e. as $n\to\infty$, there holds $\|f_n\|_X\to 0$ as $n\to\infty$. The subspace of order continuous elements of $X$ is denoted by $X_o$. Evidently, $X_o$ enjoys the semi-Fatou property. We say that $X$ is order continuous, when $X=X_o$, which is equivalent with separability of $X$. For a q-B.f.s. $X$, its associate space (Köthe dual) $X'$ is defined as the space of functions $g\in L^0$ satisfying $$\|g\|_{X'}:=\sup\left\{ \int_{\mathbb{T}}|f(t)g(t)|dm(t)\colon \|f\|_X \le 1 \right\}<\infty.$$ Notice that $X'$ is nontrivial and $X'\in (FP)$ for each B.f.s. $X$. However, $X'$ may be trivial, i.e. $X'=\{0\}$, when $X$ is just a q-B.f.s.. For example, $(L^p)'=\{0\}$ when $0<p<1$. It is known that a B.f.s. $X$ has the Fatou property if and only if $X''\equiv X$ (see [@LT79 p. 30]). Moreover, when $X$ is a B.f.s., the property (c) of definition implies that $$\label{eq:pol-norm-0} \|f\|_X=\sup\{|\langle f,g\rangle| : g\in X',\ \|g\|_{X'}\le 1\},$$ i.e. $\|f\|_X=\|f\|_{X''}$ for each $f\in X$ (see [@LT79 Proposition 1.b.18]). Finally, a B.f.s. $X$ satisfies $$\label{drugi dual} (X_o)'\equiv X',$$ provided $L^{\infty}\subset (X')_o$. It follows directly from definitions of norms in $X',(X_o)'$ and the Lebesgue dominated convergence theorem, since the assumption $L^{\infty}\subset X_o$ implies that simple functions are in $X_o$ and each function from $X$ is a pointwise (a.e.) limit of an increasing sequence of simple functions. The distribution function $\mu_f$ of $f\in L^0$ is given by $$\mu_f(\lambda)=m\{t\in{\mathbb{T}}\colon |f(t)|>\lambda\},\quad\lambda\ge 0.$$ Two functions $f,g\in L^0$ are said to be equimeasurable if $\mu_f(\lambda)=\mu_g(\lambda)$ for all $\lambda\ge 0$. The non-increasing rearrangement $f^*$ of $f\in L^0$ is defined by $$f^*(x)=\inf\{\lambda \colon \mu_f(\lambda)\le x\},\quad x\ge 0.$$ A q-B.f.s. $X$ is called rearrangement-invariant (r.i. for short) if for every pair of equimeasurable functions $f,g \in L^0$, $f\in X$ implies that $g\in X$ and $\|f\|_X=\|g\|_X$. Lebesgue, Orlicz and Lorentz spaces are examples of r.i. q-B.f. spaces. In general, each r.i. B.f.s. $X$ satisfies inclusion $X\subset L^1$. Moreover, if $X$ is r.i. B.f.s on ${\mathbb{T}}$ and $X\not =L^{\infty}$, then $L^{\infty}\subset X_o$. We refer to [@KPS82] and [@BS88] for more informations on non-increasing rearrangements and r.i. spaces. Let $X$ be a r.i. q-B.f. space. For each $s\in{\mathbb{R}}_+$ the dilation operator $D_s$ is defined as $$(D_s f)(e^{i\theta})= \left\{ \begin{array}{ll} f(e^{i\theta s}), & \theta s\in[0,2\pi),\\ 0, &\theta s\not\in[0,2\pi), \end{array} \right. \quad \theta \in[0,2\pi).$$ It is known (see, for example, [@KPS82]) that $D_s$ is bounded on $X$ for each $s>0$ and limits $$\alpha_X=\lim_{s\to 0}\frac{\log \|D_{1/s}\|_{X\to X}}{\log s}, \quad \beta_X=\lim_{s\to \infty}\frac{\log \|D_{1/s}\|_{X\to X}}{\log s}$$ exist. The numbers $\alpha_X$ and $\beta_X$ are called lower and upper Boyd indices of $X$, respectively. For an arbitrary r.i. B.f.s. $X$, its Boyd indices belong to $[0,1]$ and $\alpha_X\le\beta_X$. Moreover, $$\label{indeksysuma} \alpha_X+\beta_{X'}=1,$$ when $X$ is a B.f.s., thanks to the semi-Fatou property of $X$ (see [@KPS82 Theorem 4.11, p. 106]). We say that Boyd indices are nontrivial if $\alpha_X,\beta_X\in(0,1)$. More informations on Boyd indices of r.i. B.f. spaces may be found in [@BS88; @KPS82; @LT79], while the quasi-Banach case was considered in [@MS96; @Di15]. For two B.f. spaces $X$ and $Y$, let $M(X,Y)$ denote the space of pointwise multipliers from $X$ to $Y$ defined by $$M(X,Y)=\{f\in L^0 \colon fg\in Y\text{ for all } g\in X\},$$ equipped with the natural operator norm $$\|f\|_{M(X,Y)}=\sup_{\|g\|_X\le 1}\|fg\|_Y.$$ It is known ([@KLM13 Theorem 2.2]) that $M(X,Y)$ is r.i. space when $X,Y$ are so. Note that it may happen that $M(X,Y)$ contains only the zero function. For instance, if $1\le p<q<\infty$, then $M(L^p,L^q)=\{0\}$. In general, for two r.i. B.f.s. $X,Y$, $M(X,Y)\not =\{0\}$ if and only if $X\subset Y$. On the other hand, if $1\le q\le p\le\infty$ and $1/r=1/q-1/p$, then $M(L^p,L^q)\equiv L^r$. Also $M(X,X)\equiv L^\infty$ for arbitrary B.f.s. $X$. We will need one more easy fact about space $M(X,Y)$, for which we cannot give any reference, thus let us state it and prove. \[order multipliers\] Let $X,Y$ be B.f. spaces such that $X\subset Y$, $L^{\infty}\subset X_o$ and $Y\in (FP)$. Then $M(X_o,Y)\equiv M(X,Y)$. It is known (see [@KLM13 (vii) on p. 879]) that $M(Z,Y)\equiv M(Y',Z')$, when $Y\in (FP)$. Applying this together with (\[drugi dual\]) we get $$M(X_o,Y)\equiv M(Y',(X_o)')\equiv M(Y',X')\equiv M(X'',Y'')\equiv M(X'',Y).$$ On the other hand, the following inclusions always hold $X_o\subset X\subset X''$, thus $$\|f\|_{M(X_o,Y)}\leq \|f\|_{M(X,Y)}\leq \|f\|_{M(X'',Y)},$$ which means that $M(X_o,Y)\equiv M(X,Y)\equiv M(X'',Y)$. The space $M(X,Y)$ may be regarded as division of $Y$ by $X$. In virtue of this point of view we define an opposite construction, that is the pointwise product space. Given two q-B.f. spaces $X$ and $Y$, the pointwise product $X\odot Y$ is defined by $$\label{product} X\odot Y=\{gh\colon g\in X,\ h\in Y\}$$ equipped with the functional $\|\cdot\|_{X\odot Y}$ $$\|f\|_{X\odot Y}=\inf\{\|g\|_X\|h\|_Y\colon f=gh,\ g\in X,\ h\in Y\}.$$ It follows from the ideal property of B.f. spaces that $X\odot Y$ is a linear space (products of sequence spaces without the ideal property have been investigated in [@Bu87; @BG87]). Given two B.f. spaces $X,Y$ we say that $X$ factorizes $Y$ when $X\odot M(X,Y)=Y$ (factorization of function spaces is widely discussed in [@KLM14]). Using this notion, the classical Lozanovskii factorization theorem reads as follows $$\label{Loz fact} X\odot X'\equiv L^1,$$ where $X$ is a B.f.s. ([@Lo69 Theorem 6], cf. [@Re88 Proposition 6]). The Calderón–Lozanovskii construction $X^{1-\theta}Y^{\theta}$ is defined for $0<\theta <1$ and two q-B.f. spaces $X,Y$ by $$X^{1-\theta}Y^{\theta}=\{f\in L^0:|f|=g^{1-\theta}h^{\theta},\ g\in X,\ h\in Y\},$$ with the (quasi) norm given by $$\label{CLnorm} \|f\|_{X^{1-\theta}Y^{\theta}}=\inf\{\max\{\|g\|_X,\|h\|_Y\}:|f|=g^{1-\theta}h^{\theta},\ g\in X,\ h\in Y\},$$ (for more informations see [@Lo69; @Re88], [@KPS82 Chapter IV], [@Mal89 Chapter 15]). The second Lozanovskii’s theorem that we shall need is the duality theorem, which states that $$\label{lozan dual} [X^{1-\theta}Y^{\theta}]'\equiv (X')^{1-\theta}(Y')^{\theta},$$ for B.f. spaces $X,Y$ ([@Lo69 Theorem 1], cf. [@Re88 Theorem 1]). For a q-B.f.s. $X$ and $p>1$ one defines its $p$-convexification ($p$-concavication when $0<p<1$) $X^{(p)}$ as $$X^{(p)}=\{f\in L^0\colon |f|^p\in X\}$$ with the quasi-norm given by $\|f\|_{X^{(p)}}=\||f|^p\|_{X}^{1/p}$ (see [@LT79 pages 40–59]). The product space $X\odot Y$ is intimately related with Calderón–Lozanovskii construction. In fact, $X\odot Y$ may be represented in the following way $$\label{repres} X\odot Y\equiv (X^{1/2}Y^{1/2})^{(1/2)},$$ (see [@KLM14 Theorem 1 (iv)]). In particular, it explains that $X\odot Y$ is a quasi-Banach space. We will use (\[repres\]) few times in the sequel, since it allows us to apply the known theory of Calderón–Lozanovskii construction to product spaces. For $n\in{\mathbb{Z}}$ and $t\in{\mathbb{T}}$, let $\chi_n(t):=t^n$. The Fourier coefficients of a function $f\in L^1$ are given by $$\widehat{f}(n)=\langle f,\chi_n\rangle, \quad n\in{\mathbb{Z}},$$ where $$\langle f,g\rangle= \int_{\mathbb{T}}f(t)\overline{g(t)}\,dm(t).$$ Let further $\mathcal{P}=\{\sum_{i=-n}^n\alpha_i\chi_i: \alpha_i\in{\mathbb{C}},\ n\geq 0\}$ and $\mathcal{P}_A=\{\sum_{i=0}^n\alpha_i\chi_i : \alpha_i\in{\mathbb{C}},\ n\geq 0\}$ denote the sets of all trigonometric polynomials and all analytic trigonometric polynomials, respectively. The Riesz projection $P$ is defined for $f\in L^1$, as $$P\colon f\mapsto \frac{1}{2}(f+i\tilde f+\hat f(0)),$$ where $\tilde f$ is the conjugate function of $f$ (see [@Kat76 Chapter III] or [@Gar06 Chapter III] for precise definitions). For each $f\in L^p$, $p>1$ equivalent definition of $P$, i.e. $$P\colon \sum_{n=-\infty}^{\infty}\widehat f(n)t^n\mapsto \sum_{n=0}^{\infty}\widehat f(n)t^n,$$ is meaningful. It is known that $P$ is bounded on r.i. q-B.f.s. $X$ if and only if $X$ has nontrivial Boyd indices (in the case of Banach spaces it follows directly from the Boyd theorem [@LT79; @KPS82], while the quasi-Banach case was considered in [@Di15] and, before, in [@MS96] with an additional assumption of the Fatou property). In the paper $P$ will always stand for the Riesz projection. Let $X$ be a r.i. q-B.f.s. such that $X\subset L^1$ (inclusion $X\subset L^1$ holds for each r.i. B.f.s.). The Hardy space $H[X]$ is defined by $$H[X]=\big\{f\in X\colon \widehat{f}(n)=0\quad\mbox{for all}\quad n<0\big\},$$ with the quasi-norm inherited from $X$ (see for example [@Xu92], [@MM09] or [@MRP15], where this kind of Hardy spaces is considered). Let us mention that Hardy spaces may be equivalently regarded as spaces of analytic functions on the unit disc ${\mathbb{D}}$, since the convolution with the Poisson kernel gives the analytic extension of each function from $H[X]$ on ${\mathbb{T}}$ to the whole ${\mathbb{D}}$. If $1\le p\le\infty$, then $H^p:=H[L^p]$ is the classical Hardy space (see for example [@Hof62; @Du70; @Kat76]). We shall use also the following variants of Hardy spaces $$\overline{H[X]}=\{\bar f\colon f\in H[X]\}$$ and $$H_-[X]=\{f\chi_{-1}\colon f\in \overline{H[X]}\}.$$ Finally, we can introduce main actors of the paper – Toeplitz and Hankel operators. We will extend the definition (\[TH\]) to allow possibly large class of symbols, but at this moment we say nothing about boundedness. Thus, for a given $a\in L^1$ the Toeplitz $T_a$ operator may be formally defined on $\mathcal{P}_A$ (or on $H^{\infty}$) by $$T_a\colon f\to P(af).$$ The flip operator $J: L^1\to L^1$ is defined as $$Jf(t)=t^{-1}f(t^{-1}).$$ Of course, it is isometry on each r.i. B.f.s. $X$. Consequently, the Hankel operator $H_a$ may be defined on $\mathcal{P}_A$ (or on $H^{\infty}$) by $$H_a\colon f\to P(aJf).$$ Notice that the definition of Toeplitz operator is rather the same through literature, while definitions of Hankel operator vary. The definition proposed above corresponds to the one from [@BS06] and $H_a$ acts into the space of analytic functions, while, for example in [@Pel03], $H_a$ maps analytic functions into anti-analytic ones. Anyhow, the merit is preserved in any case and Toeplitz operators have Toeplitz matrices (i.e. $\langle T_a\chi_j,\chi_k\rangle=\hat a(k-j)$ for all $k,j\geq 0$), while Hankel operators are representable by Hankel matrices (i.e. $\langle H_a\chi_j,\chi_k\rangle=\hat a(k+j+1)$ for all $k,j\geq 0$). We will also consider Toeplitz and Hankel operators on nonseparable spaces. In such a case the above definition of Toeplitz and Hankel operators has to be done more precise, since behavior on polynomials will not determine them. Namely, if $X$ is r.i. B.f.s., then assumption $a\in X'$ ensures that $af, aJf\in L^1$ for each $f\in H[X]$ thus definitions of $T_a$ and $H_a$ make sense. Preliminaries ============= Before we will be ready to state the main results, we need to collect a sequence of technical results concerning the structure of $H[X]$ spaces. Recall that the Fejér kernel $(K_n)$ is defined as $$K_n(t)=\sum_{k=-n}^n\left(1-\frac{|k|}{n+1}\right)\chi_k(t), \quad t\in{\mathbb{T}}.$$ \[le:dens\] Let $X$ be a r.i. B.f. space. If $X$ is separable, then 1. for every $f\in X$ $$\lim_{n\to\infty}\|f-f*K_n\|_X=0;$$ 2. $\mathcal{P}$ is dense in $X$; 3. $\mathcal{P}_A$ is dense in $H[X]$. \(a) It follows from [@BS88 Chap. 3, Lemma 6.3] that continuous functions are dense in $X$, since separability of $X$ means that $X=X_o$. Consequently, $X$ is the homogeneous Banach space (in the sense of [@Kat76 Chap. I, Definition 2.10]) and the claim follows by [@Kat76 Chap. I, Theorem 2.11]. Part (b) is an immediate consequence of part (a) and the fact that $f* K_n\in\mathcal{P}$ if $f\in X\subset L^1$. Part (c) follows from part (a) and the observation that $f*K_n\in\mathcal{P}_A$ if $f\in H[X]$. \[le:pol-norm\] Let $X$ be a r.i. B.f.s.. Then $$\label{eq:pol-norm-1} \|f\|_X=\sup\{|\langle f,p\rangle|\colon p\in\mathcal{P},\ \|p\|_{X'}\le 1\}.$$ We know by (\[eq:pol-norm-0\]) that $$\label{eq:pol-norm-2} \|f\|_X=\sup\{|\langle f,g\rangle|\colon g\in X',\ \|g\|_{X'}\le 1\}.$$ First of all notice that in the above supremum we may restrict to simple functions from $X'$, i.e. $$\label{eq:pol-norm-3} \|f\|_X=\sup\{|\langle f,g\rangle|\colon g {\rm \ is\ simple\ function\ and\ } \|g\|_{X'}\le 1\}.$$ In fact, for each $g\in X'$ there is a sequence of simple functions $(g_n)$ such that $|g_n|\leq |g|$ and $g_n\to g$ a.e.. Then the Lebesgue dominated convergence theorem implies that $\langle f,g_n\rangle\to \langle f,g\rangle$. In particular, if $\|g\|_{X'}\le 1$ then also $\|g_n\|_{X'}\le 1$. Since $X'$ is r.i. and enjoys the Fatou property, it is an exact interpolation space between $L^1$ and $L^{\infty}$ (see [@KPS82 Theorem 4.9, p. 105]). In consequence, for each $g\in X'$ $$\label{eq:pol-norm-4} \|g*K_n\|_{X'}\le\|K_n\|_{L^1}\|g\|_{X'}\le\|g\|_{X'}, \quad n\in{\mathbb{N}},$$ where $(K_n)$ is the Fejér kernel. Moreover, $g*K_n\to g$ a.e. (in fact in each Lebesgue point of $g$, since Fejér kernel is approximative unity). However, if we choose $g$ to be simple function then also $|g*K_n|\leq \|g\|_{\infty}\chi_{{\mathbb{T}}}$. Therefore, using once again the Lebesgue dominated convergence theorem we conclude that $\langle f,g*K_n\rangle\to \langle f,g\rangle$ for each $f\in X$ and each simple function $g\in X'$. Together with (\[eq:pol-norm-4\]) it proves our claim. The idea of the proof of lemma below is analogously as for $H^p$ spaces in [@Du70]. We believe it is known, but cannot find any reference. Moreover, it was proved in [@K04] with additional assumption, that $X$ is reflexive. To avoid the impresion that this assumption is necessary, we present a short proof. \[duality\] Let $X$ be separable r.i. B.f.s. with nontrivial Boyd indices. Then $H[X]^*$ is isomorphic with $H[X']$, i.e. $H[X]^*\simeq H[X']$. Moreover, each functional $G\in H[X]^*$ is of the form $$G(f)=\langle f,g \rangle=\int_{{\mathbb{T}}} f(t)\bar g(t)dm(t),$$ for some unique $g\in H[X']$. Moreover, for such $G$ there holds $$\|G\|_{H[X]^*}\leq \|g\|_{H[X']}\leq \|P\|_{X'\to X'}\|G\|_{H[X]^*}.$$ Once we know that $H[X]^*\simeq X^*/H[X]^{\perp}$ (since $H[X]$ is closed subspace of $X$) and $X^*\simeq X'$ (by separability of $X$ and [@LT79 p. 29]), it is enough to prove that $H[X]^{\perp}\simeq H_-[X']$. In fact, since $X'$ has nontrivial Boyd indices when $X$ has, it follows that $P$ is bounded on $X'$, $P(X')=H[X']$ and $H_-[X']$ is complement of $H[X']$ in $X'$. Since $X^*\simeq X'$ we may regard elements of $H[X]^{\perp}$ as functions in $X'$. Let $f\in H[X]^{\perp}$. Then $$\langle \chi_n,f\rangle =0\ {\rm for\ each\ }n\geq 0,$$ since $\chi_n\in H[X]$. But it means that $f\in H_-[X']$. For the opposite inclusion let $g\in H_-[X']$. Then for each polynomial $p=\sum_{k=1}^np_k\chi_k\in H[X]$ there holds $$\langle p,g\rangle =\sum_{k=1}^np_k\langle \chi_k,g\rangle=0$$ and, in view of density of analytic polynomials in $H[X]$, we conclude that $g\in H[X]^{\perp}$. The remaining inequalities for norms may be explained exactly as in [@Du70 Section 7.2]. Toeplitz operators ================== \[le:multiplication\] Let $X,Y$ be r.i. B.f.s. and suppose $X$ is separable. If a linear operator $A:X\to Y$ is bounded and there exists a sequence $(a_n)_{n\in{\mathbb{N}}}$ of complex numbers such that $$\label{eq:multiplication-1} \langle A\chi_j,\chi_k\rangle=a_{k-j} \quad\text{for all}\quad j,k\in{\mathbb{Z}},$$ then there exists a function $a\in M(X,Y)$ such that $A=M_a$ and $\widehat{a}(n)=a_n$ for all $n\in{\mathbb{Z}}$. Put $a:=A\chi_0\in Y$. Since $Y\subset L^1$, we infer from that $$\widehat{a}(n) = \langle a,\chi_n\rangle = \langle A\chi_0,\chi_n\rangle = a_n, \quad n\in{\mathbb{Z}}.$$ If $f=\sum_{k=-m}^m \widehat{f}(k)\chi_k\in\mathcal{P}$, then $af\in Y\subset L^1$ and the $j$-th Fourier coefficient of $af$ is $$\label{eq:multiplication-2} (af)\widehat{\hspace{2mm}}(j) = \sum_{k\in{\mathbb{Z}}}\widehat{a}(j-k)\widehat{f}(k) = \sum_{k=-m}^m a_{j-k}\widehat{f}(k).$$ On the other hand, from we get for $j\in{\mathbb{Z}}$, $$\label{eq:multiplication-3} (Af)\widehat{\hspace{2mm}}(j) = \langle Af,\chi_j\rangle = \sum_{k=-m}^m \widehat{f}(k)\langle A\chi_k,\chi_j\rangle = \sum_{k=-m}^m a_{j-k}\widehat{f}(k).$$ By and , $(af)\widehat{\hspace{2mm}}(j)=(Af)\widehat{\hspace{2mm}}(j)$ for all $j\in{\mathbb{Z}}$. Therefore, $Af=af$ for all $f\in\mathcal{P}$ in view of the uniqueness of the Fourier series. Since the space $X$ is separable, the set $\mathcal{P}$ is dense in $X$ by Lemma \[le:dens\]. In consequence $Af=af$ for all $f\in X$. This means that $A=M_a$ and $a\in M(X,Y)$ by the definition of $M(X,Y)$. \[Tw-BH\] Let $X,Y$ be two separable r.i. B.f. spaces, such that $X\subset Y$, $Y$ has nontrivial Boyd indices and the Fatou property. A continuous linear operator $A:H[X]\to H[Y]$ satisfies $$\label{Toep1} \langle A\chi_j,\chi_k\rangle=a_{k-j}$$ for some sequence $(a_k)_{k\in \mathbb{Z}}$ and all $j,k\geq 0$ if and only if there exists $a\in M(X,Y)$ such that $A=T_a$ and $\widehat{a}(n)=a_n$ for all $n\in{\mathbb{Z}}$. Moreover, $$\label{Toep1norm} \|a\|_{M(X,Y)}\leq \|T_a\|_{H[X]\to H[Y]}\leq \|P\|_{Y\to Y}\|a\|_{M(X,Y)}.$$ Of course, we need to prove only necessity. For $n\geq 0$ put $$b_n=\chi_{-n}A\chi_n.$$ Then $b_n\in Y$ and $\|b_n\|_Y\leq \|A\|_{H[X]\to H[Y]}$. Notice that under our assumptions on $Y$, $(Y')_o\not = \{0\}$ and $[(Y')_o]^*=[(Y')_o]'\equiv Y''\equiv Y$ by (\[drugi dual\]), which means that $Y$ is a dual of separable space. In consequence, relative weakly\* topology of $B(Y)$ is metrizable (see for example [@Meg98 Corollary 2.6.20] p. 231). Thus the Banach-Alaoglu theorem implies that there is $a\in Y$, $\|a\|_Y\leq \|A\|_{H[X]\to H[Y]}$ and a sequence $(n_k)$ such that $b_{n_k}\to a$ weakly\*. In particular, for each $j\in \mathbb{Z}$ $$\langle b_{n_k},\chi_j\rangle \to \langle a,\chi_j\rangle.$$ On the other hand, $$\langle b_{n_k},\chi_j\rangle=\langle A\chi_{n_k},\chi_{n_k+j}\rangle=a_j,$$ when $n_k+j\geq 0$. This means that for each $j\in \mathbb{Z}$ $$\langle a,\chi_j\rangle=a_j.$$ Consider $B\colon X\to Y$ given by $B\colon f\mapsto af$. Then we have $$\langle Bf,g \rangle=\langle \chi_{-n}A(\chi_nf),g\rangle$$ for polynomials $f,g\in \mathcal{P}$ and $n>\max\{\deg f,\deg g\}$. Also for these $n$’s there holds $$\|A(\chi_nf)\|_Y \leq \|A\|_{H[X]\to H[Y]}\|\chi_n f\|_{H[X]} = \|A\|_{H[X]\to H[Y]}\|f\|_{X}.$$ Thus $$|\langle \chi_{-n}A(\chi_nf),g\rangle|\leq \|A\|_{H[X]\to H[Y]}\|f\|_{X}\|g\|_{Y'}$$ and $$|\langle Bf,g \rangle|\leq \limsup_{n\to \infty}|\langle \chi_{-n}A(\chi_nf),g\rangle| \leq \|A\|_{H[X]\to H[Y]}\|f\|_{X}\|g\|_{Y'}.$$ Taking supremum over $\|f\|_{X}\leq 1,\|g\|_{Y'}\leq 1$, $f,g\in\mathcal{P}$, by density of $\mathcal{P}$ in $X$ and by Lemma \[le:pol-norm\] we conclude $$\|B\|_{X\to Y}\leq \|A\|_{H[X]\to H[Y]}.$$ Furthermore, for $k,j\in \mathbb{Z}$ $$\langle B\chi_j,\chi_k \rangle=\langle a,\chi_{k-j} \rangle=a_{k-j}.$$ Consequently, Lemma \[le:multiplication\] implies that $a\in M(X,Y)$. On the other hand $$\langle T_a\chi_j,\chi_k \rangle=a_{k-j}=\langle A\chi_j,\chi_k\rangle$$ for $j,k\geq 0$. Moreover, $A\chi_j, T_a\chi_j\in H[Y]\subset H^1$, thus $$A\chi_j=T_a\chi_j$$ for each $j\geq 0$, by uniqueness of Fourier series. Finally, since $\mathcal{P}_A$ is dense in $H[X]$, we conclude that $T_a=A$ and $$\|a\|_{M(X,Y)} = \|B\|_{X\to Y} \leq \|A\|_{H[X]\to H[Y]}=\|T_a\|_{H[X]\to H[Y]},$$ as claimed. Indeed, we can slightly relax assumptions from the previous theorem, allowing $X$ to be nonseparable. However, then the condition (\[Toep1\]) no more determines an operator, so Theorem \[Tw-BH\] rather reads as follows. \[BHnonsep\] Let $X,Y$ be r.i. B.f. spaces, such that $X\subset Y$, $Y$ has nontrivial Boyd indices and the Fatou property. Then the Toeplitz operator $T_a:f\mapsto P(af)$ is bounded from $H[X]$ to $H[Y]$ if and only if $a\in M(X,Y)$ and then $$\|a\|_{M(X,Y)}\leq \|T_a\|_{H[X]\to H[Y]}\leq \|P\|_{Y\to Y}\|a\|_{M(X,Y)}.$$ Suppose first $X\not= L^{\infty}$. Then $T_a:H[X_o]\to H[Y]$ and has Toeplitz matrix representation, i.e. satisfies (\[Toep1\]). Then applying Theorem \[Tw-BH\] (we can, since $X_o$ is separable) we conclude that $a\in M(X_o,Y)\equiv M(X,Y)$ (see Lemma \[order multipliers\]). Moreover, respective inequalities are preserved, since $\|T_a\|_{H[X_o]\to H[Y]}\leq \|T_a\|_{H[X]\to H[Y]}$. In the case of $X=L^{\infty}$ we cannot use the previous argument, since $X_o=\{0\}$. However, we may take ${\mathcal{C}}:={\mathcal{C}}(\mathbb{T})$ instead, which gives disc algebra ${\mathcal{A}}$ in place of $H[X_o]$. Then the proof of Theorem \[Tw-BH\] follows the same lines, once we know that $M({\mathcal{C}},Y)\equiv M(L^{\infty},Y)\equiv Y$, but it is evident since $\chi_0\in {\mathcal{C}}$. When $X=Y$ we get another corollary of Theorem \[Tw-BH\], which improves assumptions of [@K04 Theorem 4.5], since we do not require that $X$ is reflexive. Let $X$ be a separable r.i. B.f.s. with nontrivial Boyd indices and the Fatou property. If a linear operator $A$ is bounded on $H[X]$ and there exists a sequence $(a_n)_{n\in{\mathbb{Z}}}$ of complex numbers satisfying , then there exists a function $a\in L^\infty$ such that $A=T_a$ and $\widehat{a}(n)=a_n$ for all $n\in{\mathbb{Z}}$. Moreover, $$\|a\|_{L^\infty} \leq \|T_a\|_{H[X]\to H[X]} \leq \|P\|_{X\to X}\|a\|_{L^\infty}.$$ Hankel operators ================ In order to prove generalized Nehari theorem we need to state some results on pointwise products of Hardy type spaces. The theorem below may be regarded as a kind of regularization for the Lozanovskii’s type factorization (see forthcoming paper [@LMM17] for more general treating of this subject). The case of $H[X]\odot H[X']=H^1$ was already considered in [@K04 Theorem 5.2]. The proof below goes similar lines, but we provide it for the sake of convenience. The pointwise product $H[X]\odot H[Y]$ of two Hardy spaces is defined analogously as in (\[product\]), this is $$H[X]\odot H[Y]=\{ h\in L^0\colon h=fg,f\in H[X],g\in H[Y]\},$$ with $$\| h\| _{H[X]\odot H[Y]}=\inf \{ \| f\|_{H[X]}\| g\| _{H[Y]}\colon h=fg\}.$$ For the moment we do not even know that such a product is a linear space, but it follows at once from the lemma below. \[Tw-prod\] Let $X,Y$ be r.i. B.f. spaces with $X\odot Y\subset L^1$. Then $$H[X]\odot H[Y]\equiv H[X\odot Y].$$ Suppose that $f\in H[X]$, $g\in H[Y]$, regarded as functions on ${\mathbb{T}}$. Then, we allow $F,G$ to be extensions of $f$ and $g$ to the unit disc $\mathbb{D}$ by convoluting $f$ and $g$ with the Poisson kernel. Evidently, $F,G$ are analytic, their radial limits exist and are equal a.e. to $f,g$, respectively. In consequence, radial limit of $FG$ is equal a.e. to $fg$ and belongs to $H[X\odot Y]$. Thus $H[X]\odot H[Y]\subset H[X\odot Y]$. Let $0\not =h\in H[X\odot Y]$. In particular $h\in X\odot Y$, thus for each $\epsilon>0$ there are $f\in X, g\in Y$ such that $h=fg$ and $\|f\|_X\|g\|_Y-\epsilon\leq \|h\|_{X\odot Y}\leq \|f\|_X\|g\|_Y$. But $ h\in H[X\odot Y]\subset H^1$ and therefore assumptions of Proposition 5.1 from [@K04] are satisfied. Thus $$\tilde F(z)=\exp{\int_{{\mathbb{T}}}\frac{t+z}{t - z}\log{|f(t)|}dt}$$ and $$\tilde G(z)=\exp{\int_{{\mathbb{T}}}\frac{t+z}{t - z}\log{|g(t)|}dt}$$ are well defined outer functions of $F$ and $G$, respectively (see [@Du70 Section 2.4]). It means that radial limits $\tilde f, \tilde g$ of $\tilde F, \tilde G$ satisfy $|\tilde f|=|f|$ and $|\tilde g|=|g|$. Letting $\Phi =\frac{H}{\tilde F\tilde G}$ we see that $\Phi$ is analytic ($H$ is extension of $h$ to ${\mathbb{D}}$), since $\tilde F\tilde G$ have no zeros in $\mathbb{D}$ (in particular, $\Phi$ is an inner function). In consequence, taking $\phi$ as radial limit of $\Phi$ we see that $$x=\phi \tilde f {\rm\ and\ } y= \tilde g,$$ which gives the required factorization of $h$, i.e. $h=xy$, $\|x\|_{H[X]}=\|f\|_X$ and $\|y\|_{H[Y]}=\|g\|_Y$. The subsequent lemmas will be used in the proof of Nehari theorem. The second one is rather technical, while the first one is of independent interest and may be regarded as complement of considerations from [@KLM14]. \[cancel\] Let $X,Y$ be two r.i. B.f. spaces with the Fatou property. If $X\odot M(X,Y)=Y$, then $$M(X,Y)'=X\odot Y'.$$ Of course, the assumption $X\odot M(X,Y)=Y$ implies that $M(X,Y)\not =\{0\}$ and thus $X\subset Y$. The Lozanovskii factorization theorem (\[Loz fact\]) applied twice gives $$\label{skrac1} M(X,Y)\odot M(X,Y)'\equiv L^1\equiv Y\odot Y'= X\odot M(X,Y) \odot Y'.$$ Thus applying Theorem 1 from [@KLM14] we may write $$[M(X,Y)\odot M(X,Y)']^{(4)}= [M(X,Y)^{(2)}]^{1/2}[(M(X,Y)')^{(2)}]^{1/2}.$$ At the same time we have $$[X\odot M(X,Y) \odot Y']^{(4)}=[M(X,Y)^{(2)}]^{1/2}[X^{1/2}(Y')^{1/2}]^{1/2}.$$ Thus equality (\[skrac1\]) gives $$[M(X,Y)^{(2)}]^{1/2}[(M(X,Y)')^{(2)}]^{1/2}=[M(X,Y)^{(2)}]^{1/2}[X^{1/2}(Y')^{1/2}]^{1/2}$$ and applying uniqueness of the Calderón–Lozanovskii construction ([@CN03] or [@BM05]), since all spaces $M(X,Y)^{(2)},(M(X,Y)')^{(2)}$ and $X^{1/2}(Y')^{1/2}$ are B.f. spaces with the Fatou property, we conclude that $$X^{1/2}(Y')^{1/2}=(M(X,Y)')^{(2)},$$ or, equivalently, $$M(X,Y)'= X\odot Y',$$ which proves our claim. \[Le-dens\] Let $X,Y$ be two r.i. B.f. spaces such that $X$ is separable and $X\subset Y$. Then the set $$S=\{pq\colon p,q\in \mathcal{P}_A {\rm\ and\ } \|p\|_{H[X]}\leq 1, \|q\|_{H[Y']}\leq 1\}$$ is dense in the unit ball of $H[X]\odot H[Y']$. If $Y=L^{1}$, then $X\odot Y'\equiv X\odot L^{\infty}\equiv X$ and $H[X]\odot H^{\infty}\equiv H[X]$. Since $H[X]$ is separable, the claim follows. We can therefore assume that $Y\not = L^1$, which means that $Y'\not =L^{\infty}$ and thus $L^{\infty}\subset (Y')_o$. First of all we need to explain that $$X\odot Y'\equiv X\odot (Y')_o.$$ In order to do it, we use representation $$X\odot Y'\equiv (X^{1/2} (Y')^{1/2})^{(1/2)}$$ (see [@KLM14 Theorem 1(iv)]). Thus, it is enough to prove that $X^{1/2} (Y')^{1/2}\equiv X^{1/2} (Y')_o^{1/2}$. However, both spaces $X^{1/2}(Y')^{1/2}$ and $X^{1/2}[(Y')_o]^{1/2}$ are order continuous, since $X$ is order continuous (see [@Re88 Proposition 4] or [@KL10 Theorem 13]). Therefore, both have the semi-Fatou property, as order continuous spaces. It follows that their norms are realized by duality as in (\[eq:pol-norm-2\]) in Lemma \[le:pol-norm\]. On the other hand, Lozanovskii duality theorem (\[lozan dual\]), together with the equality (\[drugi dual\]), tells that their Köthe duals are both equal $X'^{1/2}Y''^{1/2}$. Thus both spaces have to be equal, because simple functions belong to both of them and are dense there. In consequence, also equality $X\odot Y'\equiv X\odot (Y')_o$ is proved. Continuing, we see that $X\odot Y'\subset L^1$, because $X\subset Y$. Hence, by Theorem \[Tw-prod\], we get $$H[X]\odot H[Y']\equiv H[X]\odot H[(Y')_o].$$ Therefore, it is enough to prove density of $S$ in the unit ball of $H[X]\odot H[(Y')_o]$. However, both spaces $X$ and $(Y')_o$ are separable, therefore the set $\mathcal{P}_A\cap B(H[X])$ is dense in $B(H[X])$ and the set $\mathcal{P}_A\cap B(H[(Y')_o])$ is dense in $B(H[(Y')_o])$ in view of Lemma \[le:dens\]. Let $\epsilon>0$ and $f\in B(H[X]\odot H[(Y')_o])$. Then $(1-\epsilon)f=gh$ for some $g\in H[X]$ and $h\in H[(Y')_o]$ satisfying $\|g\|_{H[X]}<1$ and $\|h\|_{H[Y]}<1$. Furthermore, there are $p,q\in \mathcal{P}_A$ such that $\|p\|_{H[X]}<1$, $\|q\|_{H[Y]}<1$ and $\|g-p\|_{H[X]}<\epsilon$, $\|h-q\|_{H[Y]}<\epsilon$. It means that $$\|f- pq\|_{H[X]\odot H[Y']}\leq 2(\epsilon+\|gh- pq\|_{H[X]\odot H[Y']})$$ $$\leq 2\epsilon +4\|g\|_{H[X]}\|h-q\|_{H[Y']}+4\|g-p\|_{H[X]}\|q\|_{H[Y']}\leq 8\epsilon,$$ where the constant $2$ appears when we apply triangle inequality to the quasi norm $\|\cdot \|_{H[X]\odot H[Y']}$ (see [@KLM14 Corollary 1]). The following lemma is a key for the general Nehari theorem. Let us however postpone its proof to the next part of this section, because we will be able to comment it and its assumptions better, once we know how it works in the proof of Theorem \[extTw-Neh\]. \[Ban-env-Cor\] Let $X,Y$ be two r.i. B.f. spaces, such that $X$ is separable, $X\subset Y$, $Y$ has nontrivial Boyd indices and one of the following conditions holds:\ i) $X\odot M(X,Y)=Y$ and $X,Y$ have the Fatou property,\ ii) $\beta_X<\alpha_Y$.\ Then for each bounded linear functional $\phi$ on $\overline{H[X\odot Y']}$ there is $g\in M(X,Y)$ (not unique) such that $$\phi(f)=\int_{{\mathbb{T}}}g(t)f(t)dm(t)$$ for all $f\in \overline{H[X\odot Y']}$. Furthermore, $$\label{normrepr} \|\phi\|_{(\overline{H[X\odot Y']})^*}\approx \operatorname{dist}_{M(X,Y)}(\chi_1g,\overline{H[M(X,Y)]}).$$ \[extTw-Neh\] Let $X,Y$ be two r.i. B.f. spaces, such that $X$ is separable, $X\subset Y$, $Y$ has nontrivial Boyd indices and one of the following conditions holds:\ i) $X\odot M(X,Y)=Y$ and $X,Y$ have the Fatou property,\ ii) $\beta_X<\alpha_Y$.\ A continuous linear operator $A:H[X]\to H[Y]$ satisfies $$\label{Hankel condition} \langle A\chi_j,\chi_k\rangle=a_{k+j+1} {\rm \ for\ all\ } j,k\geq 0$$ and some sequence $(a_k)_{k>0}$ if and only if there exists $a\in M(X,Y)$ such that $\hat a(n)=a_n$ for each $n>0$ and $A=H_a$, i.e. $A\colon f\mapsto PaJf$. Moreover, $$c\operatorname{dist}_{M(X,Y)}(a,\overline{ H[M(X,Y)]}) \leq \|H_a\|_{H[X]\to H[Y]}$$ $$\leq \|P\|_{Y\to Y}\operatorname{dist}_{M(X,Y)}(a,\overline{ H[M(X,Y)]}),$$ where the constant $c>0$ depends only on spaces $X,Y$. If $a\in M(X,Y)$ and $j,k\geq 0$, then $$\langle H_a\chi_j,\chi_k\rangle = \langle PM_a\chi_{-1-j},\chi_k\rangle = \hat a(k+j+1),$$ as required, while $\|H_a\|_{H[X]\to H[Y]}\leq \|P\|_{Y\to Y}\|a\|_{M(X,Y)}$. Notice however, that only $\hat a(k)$’s for positive $k$ play in the definition of $H_a$, so that we may write $$\begin{aligned} \|H_a\|_{H[X]\to H[Y]} &\leq \|P\|_{Y\to Y}\inf\{\|b\|_{M(X,Y)}\colon \hat b(k)=\hat a(k) {\rm\ for \ each\ } k>0\} \\ &= \|P\|_{Y\to Y}\operatorname{dist}_{M(X,Y)}(a,\overline{ H[M(X,Y)]}),\end{aligned}$$ as required. Let $A$ satisfy condition (\[Hankel condition\]). We need to find $a\in M(X,Y)$ such that $\hat a(n)=a_n$ for each $n>0$. Define $$\gamma(A)=\sup\{\langle A\chi_0, f\rangle\colon \|f\|_{H[X]\odot H[Y']}\leq 1\}.$$ Since $S$, as defined in Lemma \[Le-dens\], is dense in $B(H[X]\odot H[Y'])$, we have $$\label{fiA} \gamma(A)=\sup\{ \langle A\chi_0, pq\rangle\colon \|p\|_{H[X]}\leq 1, \|q\|_{H[Y']}\leq 1, p,q\in \mathcal{P}_A \}.$$ On the other hand, $$\|A\|_{H[X]\to H[Y]}= \sup\{\varphi(Ag)\colon \|g\|_{H[X]}\leq 1, \|\varphi\|_{H[Y]^*}\leq 1\}$$ $$\geq \sup\{\langle Ag, h\rangle\colon \|g\|_{H[X]}\leq 1, \|h\|_{H([Y'])}\leq 1\},$$ since each $h\in H[Y']$ defines functional $\varphi_h(f)=\langle f,h\rangle$ on $H[Y]$ and evidently $\|\varphi_h\|_{H[Y]^*}\leq \|h\|_{H[Y']}$. In consequence, $$\label{normA} \|A\|_{H[X]\to H[Y]} \geq \sup\{\langle Ap, q\rangle\colon \|p\|_{H[X]}\leq 1, \|q\|_{H([Y'])}\leq 1, p,q\in \mathcal{P}_A\}.$$ For a polynomial $p=\sum{p_k}\chi_k$ define $p^c=\sum \overline{p_k}\chi_k$. Then $$p^c(t)= \overline{p(1/t)}$$ and, since $t\to 1/t$ is the measure preserving transformation on ${\mathbb{T}}$, we see that $p\to p^c$ is isometry on the set of analytic polynomials in $H[X]$ and $p^{cc}=p$. Therefore, for $f=pq$, $p,q\in \mathcal{P}_A$, simple calculations shows that $$\langle A\chi_0, pq\rangle = \langle Ap^c, q\rangle.$$ Applying the above to formulas (\[fiA\]) and (\[normA\]) we get $$\label{han1} \gamma(A)\leq \|A\|_{H[X]\to H[Y]}.$$ Define now the functional $L_A$ on $\overline{H[X]\odot H[Y']}$ by $$L_A(f)=\int_{{\mathbb{T}}} A\chi_0(t) f(t)dm(t).$$ Inequality (\[han1\]) gives $\|L_A\|=\gamma(A)\leq \|A\|_{H[X]\to H[Y]}$. Therefore, by Lemma \[Ban-env-Cor\], there is $d\in M(X,Y)$ such that $$L_A(\bar f)=\int_{{\mathbb{T}}} A\chi_0(t)\bar f(t)dm(t)=\int_{{\mathbb{T}}} d(t)\bar f(t)dm(t),$$ for each $f\in H[X]\odot H[Y']$. Finally, for each $n\geq 0$ $$a_{n+1}=\langle A\chi_0, \chi_n\rangle=\hat d(n)$$ and, taking $a=\chi_1d\in M(X,Y)$, we see that $\hat a(n)=a_n$ for each $n>0$. Moreover, from (\[normrepr\]) it follows $$\gamma(A)=\|L_A\|\geq c\operatorname{dist}_{M(X,Y)}(a,\overline{ H[M(X,Y)]}),$$ where $c>0$ depends only on spaces $X,Y$. To finish the proof it is enough to prove the remaining Lemma \[Ban-env-Cor\]. In order to do it, we will need some informations on Banach envelopes of quasi-Banach spaces. Given a quasi-Banach space $X$ with separating dual one defines a functional on $X$ by $$\|f\|_{X^{\wedge}}=\inf\{\sum_{k=1}^{n}\|f_k\|_X\colon f=\sum_{k=1}^{n}f_k, f_k\in X,n\in {\mathbb{N}}\}.$$ Under assumption that $X^*$ separates points of $X$, it is a norm. In this case, the Banach envelope $X^{\wedge}$ of $X$ is defined as the completion of $X$ with respect to the norm $\|\cdot\|_{X^{\wedge}}$. More informations on Banach envelopes may be found in [@KPR84; @KK16; @KC17; @KM07; @Sh76]. Let us collect some properties of $X^{\wedge}$ for the special kind of spaces $X$ that will appear in our proofs. - $X$ and $X^{\wedge}$ have the same dual spaces (see [@KPR84 page 27]). - If $X\subset L^1$ is separable q-B.f.s. then $X^{\wedge}$ is the closure of $X$ in $X''$. In fact, assumption $X\subset L^1$ means that $L^{\infty}\subset X'$ and thus $X^*$ separates points of $X$. Then $X^{\wedge}$ is the closure in $X^{**}$ of the image of $X$ under $J$, where $(Jx)(x^*)=x^*x$ is the natural canonical embedding (see [@KPR84 page 27]). However, by Propositions 3.2 and 3.4 in [@KK16] and by separability of $X$ it follows that $X^*=(X^{\wedge})^*\simeq X'$. On the other hand, it is known that the dual $Z^*$ of a B.f.s. may be represented as $Z'\oplus S$, where $S$ is the space of singular functionals (see [@Za67 Theorem 2, p. 467]). Thus $X^{**}\simeq X''\oplus S$. Finally, for separable q-B.f.s. $X$, its image under $J$ is in $X''$, which explains the claim (cf. [@KM07 p. 232]). - If $X\subset L^1$ is a separable r.i. q-B.f.s., then $$(X'')_o\equiv \overline{X}^{X''},$$ where $\overline{X}^{X''}$ denotes the closure of $X$ in $X''$. In fact, since $L^{\infty}\subset X$ and $X$ is order continuous, it follows that $X''\not = L^{\infty}$. Therefore $L^{\infty}\subset(X'')_o$ and $(X'')_o$ is the closure of simple functions in $X''$. On the other hand, simple functions are contained also in $X$. Thus, evidently $(X'')_o\subset \overline{X}^{X''}$. To see that the second inclusion also holds it is enough to notice that $\|f\|_{X''}=\|f\|_{X^{\wedge}}\leq \|f\|_{X}$ for each $f\in X$, which implies that simple functions are dense in $X$ equipped with the norm of $X''$. Of course, $X^{\wedge}$ is also r.i. B.f.s. (cf. [@KM07 Lemma 2.1]). The next lemma gives representation of the Banach envelope of Hardy space $H[Z]$, which will be used in the proof of Lemma \[Ban-env-Cor\]. It seems to be of independent interest that such a simple representation is possible. Notice that the crucial assumption here is $Z\subset L^1$, which gives that $Z$ has nontrivial dual, in contrast to the situation of $H^p=H[L^p]$, $p<1$, where $(L^p)^*=\{0\}$, but $(H^p)^*\not=\{0\}$ (see [@Du70 page 115]). \[Ban-env\] Let $Z$ be a separable r.i. q-B.f.s. such that $Z\subset L^1$ and $Z$ has nontrivial Boyd indices. Then $$\label{envrep} H[Z]^{\wedge}=H[Z^{\wedge}].$$ First of all notice that for r.i. q-B.f.s. the assumption $Z\subset L^1$ implies $L^{\infty}\subset Z'$. Therefore, $Z'=Z^*$ separates points of $Z$, because $L^{\infty}$ separates points of $L^1$ and it follows by properties 2) and 3) of Banach envelopes that $Z^{\wedge}=(Z'')_o$, where we know that $(Z'')_o\not =\{0\}$, since $Z$ has nontrivial Boyd indices. In particular, $\|f\|_{Z^{\wedge}}= \|f\|_{Z''}$ for $f\in Z^{\wedge}$ and the space $H[Z^{\wedge}]$ is well defined. Secondly, $P$ is bounded on $Z^{\wedge}$, since for each $f\in Z$ there holds $$\|Pf\|_{Z^{\wedge}}\leq \inf\{\sum_{k=1}^{n} \|Pf_k\|_Z\colon f=\sum_{k=1}^{n} f_k, f_k\in Z, n\in {\mathbb{N}}\}\leq \|P\|_{Z\to Z}\|f\|_{Z^{\wedge}}$$ and $Z$ is dense in $Z^{\wedge}$. Then it follows that $H[Z]$ is dense in $H[Z^{\wedge}]$, because $Z$ is dense in $Z^{\wedge}$ and $P$ is bounded on both $Z$ and $Z^{\wedge}$. Moreover, $H[Z]$ is dense in $H[Z]^{\wedge}$ just by definition. Now let $f\in H[Z]$. Then its norm in $H[Z^{\wedge}]$ is given by $$\|f\|_{H[Z^{\wedge}]}=\|f\|_{Z^{\wedge}}=\inf\{\sum_{k=1}^{n} \|f_k\|_Z\colon f=\sum_{k=1}^{n} f_k, f_k\in Z, n\in {\mathbb{N}}\}.$$ On the other hand, the norm of the same $f$ regarded as an element of $H[Z]^{\wedge}$ is $$\|f\|_{H[Z]^{\wedge}}=\inf\{\sum_{k=1}^{n} \|f_k\|_Z\colon f=\sum_{k=1}^{n} f_k, f_k\in H[Z], n\in {\mathbb{N}}\}.$$ Evidently, $\|f\|_{H[Z^{\wedge}]}\leq \|f\|_{H[Z]^{\wedge}}$, while the opposite inequality follows from boundedness of $P$ on $Z$. This is $$\|f\|_{H[Z^{\wedge}]}=\|f\|_{Z^{\wedge}}=\inf\{\sum_{k=1}^{n} \|f_k\|_Z\colon f=\sum_{k=1}^{n} f_k, f_k\in Z, n\in {\mathbb{N}}\}$$ $$\geq 1/\|P\|_{Z\to Z} \inf\{\sum_{k=1}^{n} \|Pf_k\|_Z\colon f=\sum_{k=1}^{n} f_k, f_k\in Z, n\in {\mathbb{N}}\}$$ $$\geq 1/\|P\|_{Z\to Z} \inf\{\sum_{k=1}^{n} \|g_k\|_Z\colon f=\sum_{k=1}^{n} g_k, g_k\in H[Z], n\in {\mathbb{N}}\}$$ $$= 1/\|P\|_{Z\to Z}\|f\|_{H[Z]^{\wedge}}.$$ Consequently, $H[Z^{\wedge}]=H[Z]^{\wedge}$ as completions of the same space under equivalent norms. Assume that the condition i) is satisfied. Lemma \[cancel\] gives $$M(X,Y)'= X\odot Y'.$$ Consequently, $M(X,Y)\equiv M(X,Y)''= (X\odot Y')'$ and, since $X\odot Y'$ is separable (see the proof of Lemma \[Le-dens\]), we get $$(X\odot Y')^*=M(X,Y).$$ Furthermore, $\overline{H[X\odot Y']}$ is the closed subspace of the Banach space $X\odot Y'$, thus the claim follows by the Hahn-Banach theorem. Suppose now that condition ii) holds. This time $X\odot Y'$ need not be a Banach space, therefore we will use Lemma \[Ban-env\]. In order to do it we need to see that $X\odot Y'$ has nontrivial Boyd indices. First of all, it is easy to see that $$\label{indeksyZ} \alpha_{Z^{(1/2)}}=2\alpha_{Z} {\rm \ and\ } \beta_{Z^{(1/2)}}=2\beta_{Z},$$ for an arbitrary r.i. q-B.f.s. $Z$. Furthermore, $\beta_{X^{1/2}(Y')^{1/2}}\leq 1/2\beta_{X}+1/2\beta_{Y'}$ and together with the representation $X\odot Y'\equiv (X^{1/2} (Y')^{1/2})^{(1/2)}$ we get $$\beta_{X\odot Y'}\leq \beta_X+\beta_{Y'}=1+\beta_X-\alpha_{Y}<1.$$ Using once again equalities (\[indeksyZ\]) and representation of $X\odot Y'$ we see that $\alpha_{X\odot Y'}>0$ if and only if $\alpha_{X^{1/2}Y'^{1/2}}>0$, which, in turn, is equivalent to $\beta_{[X^{1/2}Y'^{1/2}]'}<1$ (notice that $X^{1/2}Y'^{1/2}$ is order continuous, thus satisfies the semi-Fatou property and we are free to use (\[indeksysuma\])). Using Lozanvskii duality theorem and assumption that $Y$, and so $Y''$, has nontrivial Boyd indices we conclude $$\beta_{[X^{1/2}Y'^{1/2}]'}=\beta_{X'^{1/2}Y''^{1/2}}\leq \frac{1}{2} \beta_{X'}+\frac{1}{2} \beta_{Y''}<1,$$ which proves that $X\odot Y'$ has nontrivial Boyd indices. Applying now Lemma \[Ban-env\] to $Z=X\odot Y'$ we get $\overline{H[X\odot Y']}^{\wedge}=\overline{H[X\odot Y']^{\wedge}}=\overline{H[(X\odot Y')^{\wedge}]}$. It follows, by properties of Banach envelope that $$(\overline{H[X\odot Y']})^*=(\overline{H[X\odot Y']}^{\wedge})^*=(\overline{H[(X\odot Y')^{\wedge}]})^*.$$ On the other hand $$((X\odot Y')^{\wedge})^*=(X\odot Y')^*=(X\odot Y')'=M(X,Y).$$ Consequently, each functional on $\overline{H[X\odot Y']}$ extends to a functional on $(X\odot Y')^{\wedge}$, which proves the claim. It remains to explain (\[normrepr\]). We will do it only for the case ii), since it works similarly, but easier for i). Let $\phi\in (\overline{H[X\odot Y']})^*$. Then $$\|\phi\|_{(\overline{H[X\odot Y']})^*}\approx \|\phi\|_{(\overline{H[(X\odot Y')^{\wedge}]})^*} =\inf\|\tilde\phi\|_{((X\odot Y')^{\wedge})^*},$$ where the infimum runs over all extensions $\tilde\phi$ of $\phi$ to $(X\odot Y')^{\wedge}$. Of course, each such extension corresponds to some $\tilde g\in M(X,Y)$, thus we get $$\|\phi\|_{(\overline{H[X\odot Y']})^*}\approx \operatorname{dist}_{M(X,Y)}(g,H_-[M(X,Y)])=\operatorname{dist}_{M(X,Y)}(\chi_1g,\overline{H[M(X,Y)]}).$$ Notice that we have lost equality of norms because of Lemma \[Ban-env\]. The second part of the above proof could be done without Lemma \[Ban-env\]. In fact, we have explained that under assumption ii), $P$ is bounded on $X\odot Y'$. Consequently, if $\phi$ is continuous functional on $\overline{H[X\odot Y']}$, then $$\phi\circ P:X\odot Y'\to {\mathbb{C}}$$ defines its extension to the whole $X\odot Y'$. Since $(X\odot Y')^*=M(X,Y)$, we see that there is $f\in M(X,Y)$ that represents $\phi\circ P$, as well as its restriction $\phi$ to $\overline{H[X\odot Y']}$. We are therefore obliged to explain our choice of argument. First of all this argument does not imply directly the lower estimate of the norm of Hankel operator. On the other hand the author believes that Lemma \[Ban-env\] holds without assumption on Boyd indices of $Z$, or, at least, boundedness of $P$ on $Z$ is not necessary. Thus, if we can prove (\[envrep\]) for some space $Z=X\odot Y'$ without boundedness of $P$, then the general Nehari theorem holds for $X,Y$ with the same proof as in Theorem \[extTw-Neh\]. Is it true that $H[Z]^{\wedge}=H[Z^{\wedge}]$, for each separable r.i. q-B.f.s. $Z$ with $Z\subset L^1$? It was already known since the paper of Janson, Peetre and Semmes [@JPS84] (see also the classical paper [@CRW76] where usefulness of weak factorization in harmonic analysis was exhibited for the first time) that strong factorization from the original proof of Nahari theorem may be replaced by weak factorization, i.e. instead of factorization $f=gh$, we have only $f=\sum_{n=1}^{\infty} g_nh_n$ (this idea was also used in [@TV87] and [@BG10]). It is worth to notice that in our argument with Banach envelope of Hardy spaces the weak factorization is hidden as well. Namely, we have that each $f\in \overline{H[X\odot Y']}^{\wedge}=\overline{H[X]\odot H[ Y']}^{\wedge}$, by Theorem \[Tw-prod\] and properties of Banach envelopes, admits weak factorization of the form $$f=\sum_{n=1}^{\infty} g_nh_n,$$ where $g_n\in \overline{H[X]}$, $h_n\in \overline{H[Y']}$ and $$\|f\|_{\overline{H[X\odot Y']}^{\wedge}}\approx \sum_{n=1}^{\infty} \|g_n\|_{\overline{H[X]}}\|h_n\|_{\overline{H[Y']}}.$$ Similarly, as in the previous section, we may remove assumption on separability of $X$ from Theorem \[extTw-Neh\], but then it has a slightly different form. \[nonsepNeh\] Let $X,Y$ be two r.i. B.f. spaces, such that $X\subset Y$, $Y\in (FP)$, $Y$ has nontrivial Boyd indices and one of the following conditions holds:\ i) $X\odot Y'=X_o\odot Y'$ is a Banach space,\ ii) $\beta_X<\alpha_Y$ and $X\not=L^{\infty}$,\ iii) $X=L^{\infty}$ and $Y'$ is separable.\ If the operator $H_b=PM_bJ\colon H[X]\to H[Y]$ is bounded, then there exists $a\in M(X,Y)$ such that $\hat a(n)=\hat b(n)$ for $n>0$ and $H_b=H_a$. Moreover, $$c\operatorname{dist}_{M(X,Y)}(a,\overline{ HM(X,Y)}) \leq \|H_a\|_{H[X]\to H[Y]} \leq \|P\|_{Y\to Y}\operatorname{dist}_{M(X,Y)}(a,\overline{ H[M(X,Y)]}),$$ where the constant $c$ depends only on spaces $X,Y$. It is, of course, enough to consider only the case of nonseparable $X$. Suppose firstly, that $X\not= L^{\infty}$ and $X\odot Y'=X_o\odot Y'$ is a Banach space or $\beta_X<\alpha_Y$. Then $X_o\not =\{0\}$ and the thesis of Lemma \[Ban-env-Cor\] follows for $X_o$ in place of $X$. Consequently, we just apply Theorem \[extTw-Neh\] with $X_o$ and $Y$ in place of $X,Y$. In the case of $X= L^{\infty}$ we take disc algebra ${\mathcal{A}}$ in place of $H[X_o]$. It is evident that $ L^{\infty}\odot Y'={\mathcal{C}}\odot Y'=Y'$ as well as $ H^{\infty}\odot H[Y']={\mathcal{A}}\odot H[Y']=H[Y']$. Once we know that $Y'$ is separable, Lemma \[Le-dens\] holds and we may follow the proof of Theorem \[extTw-Neh\] with ${\mathcal{A}}$ and ${\mathcal{C}}$ in place of $H[X]$ and $X$, respectively. Recall that Lorentz space $L^{p,q}$, where $0< q\leq \infty$ and $0<p< \infty$, is defined by the (quasi-) norm $$\|f\|_{L^{p,q}}=(\int_{0}^1 [f^*(s)s^{1/p}]^q\frac{ds}{s})^{1/q},$$ with the standard modification when $q=\infty$. A function $\varphi\colon [0,\infty)\to [0,\infty]$ is called the Orlicz function when it is convex, nondecreasing and $\varphi(0)=0$. Then the Orlicz space $L^{\varphi}$ is defined by the norm $$\|f\|_{\varphi}=\inf\{\lambda>0\colon \int_{{\mathbb{T}}}\varphi(|f(t)|/\lambda)dm(t)\leq 1\}.$$ We will use the standard notion $H^{p,q}:=H[L^{p,q}]$ and $H^{\varphi}=H[L^{\varphi}]$ for Hardy–Lorentz and Hardy–Orlicz spaces, respectively. Let us also mention, that description of the space of multipliers between two Orlicz spaces was already described in full generality in [@LT17]. This reads as follows $$M(L^{\varphi_1},L^{\varphi})=L^{\varphi\ominus\varphi_1},$$ where the generalized Legendre transform $\varphi\ominus\varphi_1$ is defined as $$\varphi\ominus\varphi_1(u)=\sup_{v>0}\{\varphi(uv)-\varphi(v)\}\ {\rm for\ } u\geq0.$$ It was also proved therein that $L^{\varphi_1}$ factorizes $L^{\varphi}$ if and only if there are constants $c,C,u_0>0$ such that for each $u>u_0$ there holds $$\label{Orfac} c \varphi_1^{-1}(u)(\varphi\ominus\varphi_1)^{-1}(u)\leq \varphi^{-1}(u)\leq C \varphi_1^{-1}(u)(\varphi\ominus\varphi_1)^{-1}(u),$$ where $\varphi^{-1}$ stands for the right continuous inverse of $\varphi$. Recall finally that Boyd indices of the Orlicz space $L^{\varphi}$ coincides with Matuszewska–Orlicz indices of the function $\varphi$, i.e. $$\alpha_{L^{\varphi}}=a_{\varphi}\ {\rm and\ } \beta_{L^{\varphi}}=b_{\varphi},$$ where $a_{\varphi}$ is the lower and $b_{\varphi}$ - the upper Matuszewska–Orlicz index of $\varphi$ (we refer to [@LT79; @Mal89] for respective definitions). \[Orlicz\] Let $\varphi,\varphi_1$ be two Orlicz functions such that $\varphi$ has nontrivial Matuszewska–Orlicz indices.\ a) The Toeplitz operator $T_a=PM_a$ is bounded from $H^{\varphi_1}$ to $H^{\varphi}$ if and only if $a\in L^{\varphi\ominus \varphi_1}$ and $$\|a\|_{L^{\varphi\ominus \varphi_1}}\leq \|T_a\|_{H^{\varphi_1}\to H^{\varphi}}\leq \|P\|_{Y\to Y}\|a\|_{L^{\varphi\ominus \varphi_1}}.$$ b) If additionally one of the conditions holds:\ i) inequalities (\[Orfac\]) are satisfied and $L^{\varphi_1}\not =L^{\infty}$,\ ii) $b_{\varphi_1}<a_{\varphi}$ and $L^{\varphi_1}\not =L^{\infty}$,\ iii) $L^{\varphi_1} =L^{\infty}$ and $L^{\varphi}$ is separable,\ then the Hankel operator $H_b=PM_bJ$ is bounded from $H^{\varphi_1}$ to $H^{\varphi}$ if and only if there exists $a\in L^{\varphi\ominus \varphi_1}$ such that $\hat a(n)=\hat b(n)$ for each $n>0$. In this case $H_a=H_b$ and $$c\operatorname{dist}_{ L^{\varphi\ominus \varphi_1}}(a,\overline{ H^{\varphi\ominus \varphi_1}}) \leq \|H_a\|_{H^{ \varphi_1}\to H^{ \varphi}} \leq \|P\|_{L^{\varphi}\to L^{\varphi}}\operatorname{dist}_{ L^{\varphi\ominus \varphi_1}}(a,\overline{ H^{\varphi\ominus \varphi_1}}).$$ The proof is a routine verification of assumptions of Theorem \[nonsepNeh\] together with [@LT17 Theorem 2]. \[Lorentz\] Let $1\leq p_1, q_1, q\leq \infty$ and $1<p<\infty$. Assume also that either $p< p_1$, or $p=p_1$ and $q>q_1$. Put $p_2=\frac{pp_1}{p_1-p}$ when $p<p_1$, or $p_2=\infty$ when $p=p_1$ and $q_2=\frac{qq_1}{q_1-q}$ when $q< q_1$, or $q_2=\infty$ when $q\geq q_1$.\ a) The Toeplitz operator $T_a=PM_a$ is bounded from $H^{p_1,q_1}$ to $H^{p,q}$ if and only if $a\in L^{p_2,q_2}$ and $$\|a\|_{L^{p_2,q_2}}\leq \|T_a\|_{H^{p_1,q_1}\to H^{p,q}}\leq \|P\|_{L^{p,q}\to L^{p,q}}\|a\|_{L^{p_2,q_2}}.$$ b) If $p<p_1$ or $p=p_1$, $q=q_1$, then then the Hankel operator $H_b=PM_bJ$ is bounded from $H^{p_1,q_1}$ to $H^{p,q}$ if and only if there exists $a\in L^{p_2,q_2}$ such that $\hat a(n)=\hat b(n)$ for each $n>0$. In this case $H_a=H_b$ and $$c\operatorname{dist}_{L^{p_2,q_2}}(a,\overline{ H^{p_2,q_2}}) \leq \|H_a\|_{H^{p_1,q_1}\to H^{p,q}} \leq \|P\|_{L^{p,q}\to L^{p,q}}\operatorname{dist}_{L^{p_2,q_2}}(a,\overline{ H^{p_2,q_2}}).$$ The condition $p< p_1$ or $p=p_1$ and $q>q_1$ guarantees that $L^{p_1,q_1}\subset L^{p,q}$ and $1<p<\infty$ means that $P$ is bounded on $L^{p,q}$. It is also known that in such a case $M(L^{p_1,q_1},L^{p,q})=L^{p_2,q_2}$ (see [@KLM18 Theorem 4]). Therefore, application of Theorem \[BHnonsep\] proves the case of Toeplitz operators. It follows that the assumption $p< p_1$ implies point ii) of Theorem \[nonsepNeh\], while the case of $p=p_1$, $q=q_1$ is evident. Thus point b) is also proved. The following example illustrates the independence of assumptions i) and ii) in Theorem \[extTw-Neh\]. Consider $X=L^{p_1,1}$, $Y=L^{p,\infty}$ with $1<p<p_1<\infty$. Then $X$ is separable, $P$ is bounded on $Y$ and $X\subset Y$, as required in Theorem \[extTw-Neh\]. Furthermore, from [@KLM14] Theorem 10 we know that $X$ does not factorize $Y$, i.e. assumption i) is not satisfied. On the other hand, $Y'=L^{p',1}$ and applying [@KLM14 Theorem 7], we have $$X\odot Y'=L^{p_1,1}\odot L^{p',1}=L^{p_2,1/2},$$ where $1/p_2=1/p_1+1/p'=1+1/p_1-1/p<1$. This means that $L^{p_2,1/2}$ satisfies condition ii) in Theorem \[extTw-Neh\]. On the other hand, taking a separable r.i. B.f.s. $Y$ with nontrivial Boyd indices and $X\equiv Y$, condition i) of Theorem \[extTw-Neh\] is trivially satisfied, since we have by Lozanovskii factorization theorem $X\odot X'\equiv X_o\odot X'\equiv L^1$. Evidently, condition ii) cannot be satisfied in this case. Measure of noncompactness of Toeplitz operators =============================================== It is well known (see [@BH63/64; @BS06]) that Toeplitz operator $T_a$ acting on $H^2$ is never compact, unless the trivial case of $a=0$. It follows from the fact that $(\chi_n)_{n=0}^{\infty}$ is weakly null in $H^2$ while $T_a\chi_n\not \to 0$ in $H^2$. This proof could be generalized to the case of $T_a:H[X]\to H[Y]$ when $P$ is bounded on $Y$ and $(\chi_n)_{n=0}^{\infty}$ is weakly null sequence in $H[X]$ (it happens when characteristic functions are order continuous in $X$ - see [@MRP15]). However, $(\chi_n)_{n=0}^{\infty}$ is not weakly null for example in $H^{\infty}$ (neither in the disc algebra $\mathcal{A}$) and therefore we need a new argument that applies in the general situation of our considerations. It apears that our argument gives estimation of the measure of noncompactness of Toeplitz operator. Let us recall that for a given set $A$ in a Banach space, its measure of noncompactness $\alpha(A)$ is defined as $$\alpha(A)=\inf\{\delta>0\colon A\subset \sum_{k=1}^{N}B_k, diam(B_k)\leq \delta \ {\rm and\ } N<\infty\}.$$ Then the measure of noncompactness $\alpha(T)$ of a given operator $T:X\to Y$ is just defined as the measure of noncompactness of the set $T(B(X))$ in $Y$, where $B(X)$ is the unit ball of $X$. \[comp-Toepl\] Let $X,Y$ be r.i. B.f. spaces such that $X\subset Y$ and $Y$ has nontrivial Boyd indices. Suppose $a\in M(X,Y)$. Then the measure of noncompactness of the Toeplitz operator $T_a$ satisfies $$\alpha(T_a)\geq m\max_{n\in {\mathbb{Z}}}|\hat a(n)|,$$ where $m$ is a constant of inclusion $Y\subset L^1$. In particular, operator $T_a:H[X]\to H[Y]$ is compact if and only if $a=0$. Assume that $a\not =0$. We will show that for each $\epsilon>0$ there is a sequence of indices $(k_n)_{n=0}^{\infty}$ such that $\|T_a\chi_{k_n}- T_a\chi_{k_l}\|_{H[Y]}\geq (1-\epsilon)m\|(\hat a(n))\|_{\infty}$ for each $0\leq n,l$ with $n\not = l$. Of course, it will imply our claim. First of all notice that $a\in M(X,Y)$ implies that $a\in Y\subset L^1$. Since $a\chi_k\in Y$, for each $k$, and $P$ is bounded on $Y$ we have $T_a\chi_k=P(a\chi_k)\in Y$. In consequence $T_a\chi_k\in L^1$ for each $k$. Let $c=\sup_{n\in {\mathbb{Z}}}|\hat a(n)|$. Then, by the Riemann–Lebesgue theorem, there is $s\in {\mathbb{Z}}$ such that $c=|\hat a(s)|$. We put $k_0=-\min\{0,s\}$. Notice that $\widehat{T_a\chi_k}(n)=\hat a(n-k)$ for $n\geq 0$ and $\widehat{T_a\chi_k}(n)=0$ for $n< 0$. In particular, for each $k\geq k_0$ we have $\widehat{T_a\chi_k}(s+k)=\hat a(s)$ and for these $k$ there holds $$\|T_a\chi_k\|_1\geq c=|\hat a(s)|.$$ We are in a position to find the announced sequence. We have already determined $k_0$. Without lost of generality we may assume that $s<0$, i.e. $k_0=-s$ (if $k_0=0$ the proof is analogous). Fix $\epsilon>0$. Thanks to the Riemann–Lebesgue theorem there is $k_1>k_0$ such that for each $k\geq k_1$ there holds $|\hat a(-k)|\leq c\epsilon$. Then $$\|T_a\chi_{k_0}-T_a\chi_{k_1}\|_1\geq |\widehat{T_a\chi_{k_0}}(0)-\widehat{T_a\chi_{k_1}}(0)|=|\hat a(-k_0)-\hat a(-k_1)|\geq (1-\epsilon)c.$$ Then we choose $k_n=k_0+n(k_1-k_0)$. Thus for $0\leq d<n$ there holds $$\|T_a\chi_{k_d}-T_a\chi_{k_n}\|_1\geq |\widehat{T_a\chi_{k_d}}(d(k_1-k_0))-\widehat{T_a\chi_{k_n}}(d(k_1-k_0))|$$ $$=|\hat a(-k_0)-\hat a(-k_0-(n-d)(k_1-k_0))|\geq (1-\epsilon)c,$$ where the last inequality follows from the fact that $k_0+(n-d)(k_1-k_0)\geq k_1$, which in turn implies $|\hat a(-k_0-(n-d)(k_1-k_0))|\leq \epsilon$. Finally, since $\|f\|_Y\geq m\|f\|_1$ for each $f\in Y$ we conclude that $\|T_a\chi_{k_n}- T_a\chi_{k_l}\|_{H[Y]}\geq (1-\epsilon)mc$ for each $0\leq n,l$ with $n\not = l$. [**Acknowledgments**]{} The author wish to thank Professor Alexei Karlovich for inspiring discussions on the subject and Professors Lech Maligranda and Anna Kamińska for valuable remarks and advices. He is also grateful to the anonymous referee for many valuable remarks that significantly improved presentation of the paper. [KLM14]{} C. Bennett and R. Sharpley, *Interpolation of Operators*, Academic Press, Boston, 1988. E. I. Berezhno[ĭ]{} and L. Maligranda, [*Representation of Banach ideal spaces and factorization of operators*]{}, Canad. J. Math. [**57**]{} (2005), no. 5, 897–940. A. Bonami and S. Grellier, [*Hankel operators and weak factorization for Hardy-Orlicz spaces,*]{} Colloq. Math. [**118**]{} (2010), no. 1, 107–132. A. Böttcher and B. Silbermann, *Analysis of Toeplitz Operators*, 2nd ed., Springer, Berlin, 2006. A. Brown and P.R. Halmos, *Algebraic properties of Toeplitz operators*, J. Reine Angew. Math. **213** (1963/64), 89–102. M. Buntinas, [*Products of sequence spaces*]{}, Analysis [**7**]{} (1987), 293–304. M. Buntinas and G. Goes, [*Products of sequence spaces and multipliers*]{}, Radovi Mat. [**3**]{} (1987), 287–300. R. R. Coifman, R. Rochberg, and G. Weiss, *Factorization theorems for Hardy spaces in several variables*, Ann. Math., **103** (1976), 611–635. M. Cwikel and P.G. Nilsson, [*Interpolation of weighted Banach lattices*]{}, in: M. Cwikel, P.G. Nilsson, G. Schechtman (Eds.), Interpolation of Weighted Banach Lattices. A Characterization of Relatively Decomposable Banach Lattices, in: Mem. Amer. Math. Soc., [**165**]{}, 2003, pp. 1–105. S. Dirksen, [*Noncommutative Boyd interpolation theorems*]{}, Trans. Am. Math. Soc. [**367**]{} (2015), no. 6, 4079–4110. P. Duren, *Theory of $H^p$ Spaces*, Academic Press, Boston, 1970. J. B. Garnett, [*Bounded analytic functions*]{}, Graduate Texts in Mathematics 236, Springer, New York, 2006. A. Hartmann, [*The generalized Carleson condition in certain spaces of analytic functions,*]{} Proceedings of the 13th International Ionference on Banach Algebras, University of Tübingen, Blaubeuren, Germany, July 20–August 3, 1997 Berlin: Walter de Gruyter. 245–260 (1998). K. Hoffman, *Banach Spaces of Analytic Functions*, Dover Publications, Inc., New York, 1962. S. Janson, J. Peetre, and S. Semmes, *On the action of Hankel and Toeplitz operators on some function spaces*, Duke Math. J. **51** (1984), 937–958. N. J. Kalton, N. T. Peck and J. W. Roberts, [*An $F$-space Sampler*]{}, London Math. Soc. Lecture Note Series, vol. 89, Cambridge Univ. Press, Cambridge 1984. A. Kamińska and M. M. Czerwińska, [*Banach envelopes in symmetric spaces of measurable operators*]{}, Positivity [**21**]{} (2017), no. 1, 473–492. A. Kamińska and D. Kubiak, [*Banach envelopes of some quasi-Banach function spaces*]{}, Comm. Math. [**56**]{} (2016), no. 1, 87–101. A. Kaminska and M. Mastyło, Abstract duality Sawyers formula and its applications, Monatsh. Math [**151**]{} (2007), 223–245. A. Yu. Karlovich, *Norms of Toeplitz and Hankel operators on Hardy type subspaces of rearrangement-invariant spaces*, Integr. Equ. Oper. Theory **49** (2004), 43–64. Y. Katznelson, *An Introduction to Harmonic Analysis*, Dower Publications, Inc., New York, 1976. P. Kolwicz and K. Leśnik, *Topological and geometrical structure of Calderón-Lozanovski[ĭ]{} construction*, Math. Inequal. Appl. **13** (2010), 175–196. P. Kolwicz, K. Leśnik, and L. Maligranda, *Pointwise multipliers of Calderón-Lozanovskiĭ spaces*, Math. Nachr. [**286**]{} (2013), 876–907. P. Kolwicz, K. Leśnik, and L. Maligranda, *Pointwise products of some Banach function spaces and factorization*, J. Funct. Anal. **266** (2014), 616–659. P. Kolwicz, K. Leśnik, and L. Maligranda, *Symmetrization, factorization and arithmetic of quasi-Banach function spaces*, preprint available at arXiv: https://arxiv.org/pdf/1801.05799.pdf. S. G. Krein, Yu. I. Petunin, and E. M. Semenov, *Interpolation of Linear Operators*, Amer. Math. Soc., Providence, 1982. K. Leśnik, L. Maligranda and P. Mleczko, *Regularization for Lozanovskii’s type factorization with applications*, preprint. K. Leśnik and J. Tomaszewski, [*Pointwise mutipliers of Orlicz function spaces and factorization*]{}, Positivity, DOI: 10.1007/s11117-017-0485-x. J. Lindenstrauss and L. Tzafriri, *Classical Banach Spaces II*, Springer, Berlin, 1979. G. Ya. Lozanovskiĭ, *On some Banach lattices,* Sibirsk. Mat. Zh. 10 (1969), 584–599; English transl. in: Siberian. Math. J. 10 (1969), 419-431. L. Maligranda, [*Orlicz Spaces and Interpolation*]{}, Seminars in Mathematics 5, University of Campinas, Campinas SP, Brazil 1989. L. Maligranda and L. E. Persson, *Generalized duality of some Banach function spaces*, Indag. Math. **51** (1989), 323–338. R. A. Martinez-Avendano and P. Rosenthal, *An Introduction to Operators on Hardy-Hilbert Spaces*, Springer, Berlin, 2007. M. Mastyło and P. Mleczko, *Absolutely summing multipliers on abstract Hardy spaces*, Acta Math. Sin. **25** (2009), 883–902. M. Mastyło and L. Rodriguez-Piazza, [*Carleson measures and embeddings of abstract Hardy spaces into function lattices*]{}, J. Funct. Anal. [**268**]{} (2015), no. 4, 902–928. R. E. Megginson, [*An Introduction to Banach Space Theory*]{}, Springer-Verlag, New York, 1998. P. Meyer-Nieberg, [*Banach Lattices*]{}, Springer-Verlag, Berlin, 1991. S. Montgomery-Smith, [*The Hardy operator and Boyd indices.*]{} In: Interaction between Functional Analysis, Harmonic Analysis, and Probability (Columbia, MO, 1994), volume 175 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1996, pages 359–364. Z. Nehari, *On bounded bilinear forms*, Ann. of Math. (2) **65** (1957), 153–162. V. V. Peller, *Hankel Operators and Their Applications*, Springer, Berlin, 2003. S. Reisner, [*On two theorems of Lozanovski[ĭ]{} concerning intermediate Banach lattices*]{}, Lecture Notes in Math. 1317, Springer, Berlin 1988, 67–83. S. Rolewicz, [*Metric Linear Spaces*]{}, 2nd Edition, Mathematics and its Applications, vol. 20, D. Reidel Publishing Co., Dordrecht; PWN - Polish Scientific Publisher, Warsaw 1985. J. H. Shapiro, [*Mackey topologies, reproducing kernels, and diagonal maps on the Hardy and Bergman spaces*]{}, Duke Math. J. [**41**]{} (1976), no. 1, 187–202. V. A. Tolokonnikov, *Hankel and Toeplitz operators in Hardy spaces*, J. Soviet Math. **37** (1987), 1359–1364. V. A. Tolokonnikov and A. Vol’berg, [*Hankel operators and problems of best approximation of unbounded functions*]{}, Investigations on linear operators and function theory. Part XIV, Zap. Nauchn. Sem. LOMI, 141, “Nauka”, Leningrad. Otdel., Leningrad, 1985, 5–17; J. Soviet Math., 37:5 (1987), 1269–1275. Q. Xu, *Notes on interpolation of Hardy spaces*, Ann. Inst. Fourier **42** (1992), 875–889. A.C. Zaanen, [*Integration*]{}, North–Holland, Amsterdam, 1967.

--- abstract: 'We present constraints on the dark energy equation-of-state parameter, $w=P/(\rho c^2)$, using [60]{} Type Ia supernovae ([SNe Ia]{}) from the ESSENCE supernova survey. We derive a set of constraints on the nature of the dark energy assuming a flat Universe. By including constraints on ([${\Omega}_{\rm M}$]{}, $w$) from baryon acoustic oscillations, we obtain a value for a static equation-of-state parameter $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{} and [${\Omega}_{\rm M}$]{}=[$0.274^{+0.033}_{-0.020}~{\rm (stat}~1\sigma{)}$]{} with a best-fit [$\chi^2/{\rm DoF}$]{} of [$0.96$]{}. These results are consistent with those reported by the SuperNova Legacy Survey in a similar program measuring supernova distances and redshifts. We evaluate sources of systematic error that afflict supernova observations and present Monte Carlo simulations that explore these effects. Currently, the largest systematic currently with the potential to affect our measurements is the treatment of extinction due to dust in the supernova host galaxies. Combining our set of ESSENCE [SNe Ia]{} with the SuperNova Legacy Survey [SNe Ia]{}, we obtain a joint constraint of $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, [${\Omega}_{\rm M}$]{}=[$0.267^{+0.028}_{-0.018}~{\rm (stat}~1\sigma{)}$]{}with a best-fit [$\chi^2/{\rm DoF}$]{} of [$0.91$]{}. The current [SNe Ia]{} data are fully consistent with a cosmological constant.' author: - '[W. M. Wood-Vasey]{}, [G. Miknaitis]{}, [C. W. Stubbs]{}, [S. Jha]{}, [A. G. Riess]{}, [P. M. Garnavich]{}, [R. P. Kirshner]{}, [C. Aguilera]{}, [A. C. Becker]{}, [J. W. Blackman]{}, [S. Blondin]{}, [P. Challis]{}, [A. Clocchiatti]{}, [A. Conley]{}, [R. Covarrubias]{}, [T. M. Davis]{}, [A. V. Filippenko]{}, [R. J. Foley]{}, [A. Garg]{}, [M. Hicken]{}, [K. Krisciunas]{}, [B. Leibundgut]{}, [W. Li]{}, [T. Matheson]{}, [A. Miceli]{}, [G. Narayan]{}, [G. Pignata]{}, [J. L. Prieto]{}, [A. Rest]{}, [M. E. Salvo]{}, [B. P. Schmidt]{}, [R. C. Smith]{}, [J. Sollerman]{}, [J. Spyromilio]{}, [J. L. Tonry]{}, [N. B. Suntzeff]{}, and [A. Zenteno]{}' bibliography: - 'apj-jour.bib' - 'cos\_paper.bib' title: 'Observational Constraints on the Nature of Dark Energy: First Cosmological Results from the ESSENCE Supernova Survey' --- Introduction: Supernovae and Cosmology {#sec:introduction} ====================================== We report the analysis of [60]{} Type Ia supernovae ([SNe Ia]{}) discovered in the course of the ESSENCE program (Equation of State: Supernovae trace Cosmic Expansion—an NOAO Survey Program) from 2002 to 2005. The aim of ESSENCE is to measure the history of cosmic expansion over the past 5 billion years with sufficient precision to distinguish whether the dark energy is different from a cosmological constant at the $\sigma_w=\pm0.1$ level. Here we present our first results and show that we are well on our way towards that goal. Our present data are fully consistent with a $w=-1$, flat Universe, and our uncertainty in $w$, the parameter that describes the cosmic equation of state, analyzed in the way we outline here, will shrink below $0.1$ for models of constant $w$ as the ESSENCE program is completed. Other approaches to using the luminosity distances have been suggested to constrain possible cosmological models. We here provide the ESSENCE observations in a convenient form suitable for a testing a variety of models.[^1] As reported in a companion paper [@miknaitis07], ESSENCE is based on a supernova search carried out with the 4-m Blanco Telescope at the Cerro Tololo Inter-American Observatory (CTIO) with the prime-focus MOSAIC II 64 Megapixel CCD camera. Our search produces densely sampled $R$-band and $I$-band light curves for supernovae in our fields. As described in that paper, we optimized the search to provide the best constraints on $w$, given fixed observing time and the properties of the MOSAICII camera and CTIO 4-m telescope. Spectra from a variety of large telescopes, including Keck, VLT, Gemini, and Magellan, allow us to determine supernova types and redshifts. We have paid particular attention to the central problems of calibration and systematic errors that, when the survey is complete in 2008, will be more important to the final precision of our cosmological inferences than statistical sampling errors for about 200 objects. This first cosmological report from the ESSENCE survey derives some properties of dark energy from the sample presently in hand, which is still small enough that the statistics of the sample size make a noticeable contribution to the uncertainty in dark-energy properties. But our goal is to set out the systematic uncertainties in a clear way so that these are exposed to view and so that we can concentrate our efforts where they will have the most significant effect. To infer luminosity distances to the ESSENCE supernovae over the redshift interval $0.15$–$0.70$, we employ the relations developed for [SNe Ia]{} at low redshift [@jha06c] among their light-curve shapes, colors, and intrinsic luminosities. The expansion history from $z\approx0.7$ to the present provides leverage to constrain the equation-of-state parameter for the dark energy as described below. In §\[sec:introduction\] we sketch the context of the ESSENCE program. In §\[sec:distances\] we show from a set of simulated light curves that this particular implementation of light-curve analysis is consistent, with the same cosmology emerging from the analysis as was used to construct the samples, and that the statistical uncertainty we ascribe to the inference of the dark-energy properties is also correctly measured. This modeling of our analysis chain gives us confidence that the analysis of the actual data set is reliable and its uncertainty is correctly estimated. Section \[sec:systematics\] delineates the systematic errors we confront, estimates their present size, and indicates some areas where improvement can be achieved. Section \[sec:cosmology\] describes the sample and provides the estimates of dark energy properties using the ESSENCE sample. The conclusions of this work are given in §\[sec:conclusions\]. Context ------- Supernovae have been central to cosmological measurements from the very beginning of observational cosmology. @shapley19 employed supernovae against the “island universe” hypothesis arguing that objects such as SN 1885A in Andromeda would have $M=-16$ which was “out of the question.” Edwin Hubble [@hubble29b] noted “a mysterious class of exceptional novae which attain luminosities that are respectable fractions of the total luminosities of the systems in which they appear.” These extra-bright novae were dubbed “supernovae” by @baade34 and divided into two classes, based on their spectra, by @minkowski41. Type I supernovae (SNe I) have no hydrogen lines while Type II supernovae (SNe II) show H$\alpha$ and other hydrogen lines. The high luminosity and observed homogeneity of the first handful of SN I light curves prompted @wilson39 to suggest that they be employed for fundamental cosmological measurements, starting with time dilation of their characteristic rise and fall to distinguish true cosmic expansion from “tired light.” After the [SN Ib]{} subclass was separated from the [SNe Ia]{} [see @filippenko97 for a review] this line of investigation has grown more fruitful as techniques of photometry have improved and as the redshift range over which supernovae have been well observed and confirmed to have standard light-curve shapes and luminosities has increased [@rust74; @leibundgut96; @riess97; @goldhaber01; @riess04b; @foley05; @hook05; @conley06; @blondin06]. Within the uncertainties, the results agree with the predictions of cosmic expansion and provide a fundamental test that the underlying assumption of an expanding universe is correct. Evidence for the homogeneity of [SNe Ia]{} comes from their small scatter in the Hubble diagram. @kowal68 compiled data for the first well-populated Hubble diagram of SNe I. The 1$\sigma$ scatter about the Hubble line was $0.6$ mag, but Kowal presciently speculated that supernova distances to individual objects might eventually be known to 5-10% and suggested that “\[i\]t may even be possible to determine the second-order term in the redshift-magnitude relation when light curves become available for very distant supernovae.” Precise distances to [SNe Ia]{} enable tests for the linearity of the Hubble law and provide evidence for local deviations from the local Hubble flow, attributed to density inhomogeneities in the local universe [@riess95; @riess97; @zehavi98; @bonacic00; @radburn-smith04; @jha06c]. While [SN Ia]{} cosmology is not dependent on the value of $H_0$, it is sensitive to deviations from a homogeneous Hubble flow and these regional velocity fields may limit our ability to estimate properties of dark energy, as emphasized by @hui06 and by @cooray06. Whether the best strategy is to map the velocity inhomogeneities thoroughly or to skip over them by using a more distant low-redshift sample remains to be demonstrated. We have used a lower limit of redshift $z>0.015$ in constructing our sample of [SNe Ia]{}. The utility of [SNe Ia]{} as distance indicators results from the demonstration that the intrinsic brightness of each [SN Ia]{} is closely connected to the shape of its light curve. As the sample of well-observed [SNe Ia]{} grew, some distinctly bright and faint objects were found. For example, SN 1991T [@filippenko92a; @phillips92] and SN 1991bg [@filippenko92b; @leibundgut93] were of different luminosity, and their light curves were not the same, either. The possible correlation of the shapes of supernova light curves with their luminosities had been explored by @pskovskii77b. More homogeneous photometry from CCD detectors, more extreme examples from larger samples, and more reliable distance estimators enabled @phillips93 to establish the empirical relation between light-curve shapes and supernova luminosities. The Calán-Tololo sample [@hamuy96] and the CfA sample [@riess99; @jha06a], have been used to improve the methods for using supernova light curves to measure supernova distances. Many variations on Phillips’ idea have been developed, including [${\Delta{m_{15}}}$]{} [@phillips99], MLCS [@riess96; @jha06c], DM15 [@prieto06], stretch [@goldhaber01], CMAGIC [@wang03], and SALT [@guy05]. These methods are capable of achieving the 10% precision for supernova distances that [@kowal68] foresaw 40 years ago. In the ESSENCE analysis, we have used a version of the @jha06c method called MLCS2k2. We have compared it with the results of the SALT [@guy05] light-curve fitter used by the SuperNova Legacy Survey [SNLS; @astier06]. This comparison provides a test: if the two approaches do not agree when applied to the same data they cannot both be correct. As shown in §\[sec:distances\], SALT and this version of MLCS2K2, with our preferred extinction prior, are in excellent accord when applied to the same data. While gratifying, this agreement does not prove they are both correct. Moreover, as described in §\[sec:cosmology\], the cosmological results depend somewhat on the assumptions about SN host-galaxy extinction that are employed. This has been an ongoing problem in supernova cosmology. The work of @lira95 demonstrated the empirical fact that although SN Ia have a range of colors at maximum light, they appear to reach the same intrinsic color about 30–90 days past maximum light, independent of light curve shape. @riess96 used de-reddened [SN Ia]{} data to show that near maximum light intrinsic color differences existed with fainter [SNe Ia]{}appearing redder than brighter objects and then used this information to construct an absorption-free Hubble diagram. Given a good set of observations in several bands, the reddening for individual supernovae can then be determined and the general relations between supernova luminosity and the light curve shapes in many bands can be established [@hamuy96; @riess99; @phillips99]. The initial detections of cosmic acceleration employed either these individual absorption corrections [@riess98] or a full-sample statistical absorption correction [@perlmutter99]. Finding the best approach to this problem, whether by shifting observations to the infrared, limiting the sample to low-extinction cases, or making other restrictive cuts on the data, is an important area for future work. Some ways to explore this issue are sketched in §\[sec:cosmology\]. @kowal68 recognized that second-order terms in cosmic expansion might be measured with supernovae once the precision and redshift range grew sufficiently large. More direct approaches with the [*Hubble Space Telescope (HST)*]{} were imagined by @colgate79 and with special clarity by @tammann79. Tammann anticipated that [*HST*]{} photometry of [SNe Ia]{} at $z\approx0.5$ would lead to a direct determination of cosmic deceleration and that the time dilation of [SN Ia]{} light curves would be a fundamental test of the expansion hypothesis. While [*HST*]{} languished on the ground after the Challenger disaster, this line of research was attempted from the ground at the European Southern Observatory (ESO) by a Danish group in 1986–1988. Their cyclic CCD imaging of the search fields used image registration, convolution and subtraction, and real-time data analysis [@hansen87]. Alas, the rate of [SNe Ia]{} in their fields was lower than they had anticipated, and only one [SN Ia]{}, SN 1988U was discovered and monitored in two years of effort [@hansen87; @norgaard-nielsen89]. More effective searches by the Lawrence Berkeley Lab (LBL) group exploiting larger CCD detectors and sophisticated detection software showed that this approach could be made practical and used to find significant numbers of high-redshift [SNe Ia]{} [@perlmutter95]. By 1995, two groups, the LBL-based Supernova Cosmology Project (SCP) and the High-Z Supernova Search Team [HZT; @schmidt98]) were working in this field. The first [SN Ia]{} cosmology results using 7 high-redshift [SNe Ia]{}[@perlmutter97] found a Universe consistent with [${\Omega}_{\rm M}$]{}$=1$ but subsequent work by the SCP [@perlmutter98] and by the HZT [@garnavich98] revised this initial finding to favor a lower value of [${\Omega}_{\rm M}$]{}. At the January 1998 meeting of the American Astronomical Society both teams reported that the [SN Ia]{} results favored a universe that would expand without limit, but at that time neither team claimed the Universe was accelerating. The subsequent publication of stronger results based on larger samples by the HZT [@riess98] and by the SCP [@perlmutter99] provided a big surprise. The supernova data showed that [SNe Ia]{} at $z\approx0.5$ were about $0.2$ mag dimmer than expected in an open universe and pointed firmly at an accelerating universe [for first-hand accounts, see @overbye99; @riess00b; @filippenko01b; @kirshner02; @perlmutter03]; reviews are given by @leibundgut01, @filippenko05a and others. The supernova route to cosmological understanding continues to improve. One source of uncertainty has been the small sample of very well observed low-redshift supernovae [@hamuy96; @riess99]. The most recent contribution is the summary of CfA data obtained in 1997–2001 [@jha06a], but significantly enhanced samples from the CfA [@hicken06] together with new data from the Katzman Automatic Imaging Telescope [KAIT; @li00; @filippenko01a; @filippenko05b], from the Carnegie SN Program [@hamuy06], from the Supernova Factory [@wood-vasey04; @copin06], and from the Sloan Digital Sky Survey II Supernova Survey [SDSS II; @frieman04; @dilday05] are in prospect. As the low-$z$ sample approaches 200 objects, the size of the sample will cease to be a source of statistical uncertainty for the determination of cosmological parameters. As described in §\[sec:systematics\], systematic errors of calibration and K-correction will ultimately impose the limits to understanding dark energy’s properties, and we are actively working to improve these areas [@stubbs06]. Some of the potential sources of systematic error in the high-$z$ sample have been examined. The fundamental assumption is that distant [SNe Ia]{} can be analyzed using the methods developed for the low-$z$ sample. Since nearby samples show that the [SNe Ia]{} in elliptical galaxies have a different distribution in luminosity than the [SNe Ia]{} in spirals [@hamuy00; @gallagher05; @neill06; @sullivan06b], morphological classification of the distant sample may provide some useful clues to help improve the cosmological inferences [@williams03]. For example, @sullivan03 showed that restricting the SCP sample to [SNe Ia]{} in elliptical galaxies gave identical cosmological results to the complete sample, which is principally in spiral galaxies. The possibility of grey dust raised by @aguirre99a [@aguirre99b] was examined by @riess00a and and by @nobili05 through infrared observations of high-$z$ supernovae and was put to rest by the very high-redshift observations of @riess04b. Improved methods for handling the vexing problems of absorption by dust have been developed by @knop03 and by @jha06c. These questions are described in more detail in §\[sec:extinction\]. The question of whether distant supernovae have spectra that are the same as nearby supernovae has been investigated by @coil00, @lidman05, @matheson05, @hook05, @howell05, and @blondin06. The more telling question of whether these spectra evolve in the same way as those of nearby objects was approached by @foley05. In all cases, the evidence points toward nearby supernovae behaving in the same way as distant ones, bolstering confidence in the initial results. This observed consistency does not mean that the samples are identical, only that the variations between the nearby and distant samples are successfully accounted for by the methods currently in hand. We do not know whether this will continue to be the case as future investigations press for more stringent limits on cosmological parameters [@albrecht06]. The highest redshift [SN Ia]{} data [@riess04b] show the qualitative signature expected from a mixed dark-energy/dark-matter cosmology. That is, they show cosmic deceleration due to dark matter preceded the current era of cosmic acceleration due to dark energy. The sign of the observed effect on supernova apparent magnitudes reverses—[SNe Ia]{} at $z\sim 0.5$ appear $0.2$ mag dimmer than expected in a coasting cosmology but the very distant supernovae whose light comes from $z>1$ appear brighter than they would in that cosmology. By itself, this turnover is a very encouraging sign that supernova cosmology does not founder on grey dust or even on a simple evolution of supernova properties with cosmic epoch. As part of this analysis, @riess04b constructed the “gold” sample of high-$z$ and low-$z$ supernovae whose observations met reasonable criteria for inclusion in an analysis of all of the published light curves and spectra using a uniform method of deriving distances from the light curves. The analysis of the gold sample provided an estimate of the time derivative of the equation-of-state parameter, $w$, for dark energy. These observations are very important conceptually because the simplest fact about the cosmological constant as a candidate for dark energy is that it should be constant (i.e., $w' = dw/dz = 0$). The observations are consistent with a constant dark energy over the redshift range out to $z\approx 1.6$. Other forms of dark energy could satisfy the observed constraints, but this observational test is one that the cosmological constant could have failed. In the analysis of the ESSENCE data presented in §\[sec:cosmology\], we use the supernova data to constrain the properties of $w$, as first carried out by @white98 and by @garnavich98. This parameterization of dark energy by $w$ is not the only possible approach. A more detailed approach is to compare the observational data to a specific model and, for example, try to reconstruct the dark energy scalar-field potential [see, for example, @li06]. A more agnostic view is that we are simply measuring the expansion history of the universe, and a kinematic description of that history in terms of expansion rate, acceleration, and jerk [@riess04b; @rapetti06] covers the facts without assuming the nature of dark energy. The ESSENCE project was conceived to tighten the constraints on dark energy at $z\approx0.5$ to reveal any discrepancy between the observations and the leading candidate for dark energy, the cosmological constant. A simple way to express this is that we aim for a 10% uncertainty in the value of $w$. This program is similar to the approach of the SuperNova Legacy Survey (SNLS) being carried out at the Canada-France-Hawaii Telescope, and we compare our methods and results to theirs [@guy05; @astier06] at several points in the analysis below. The SNLS has taken the admirable step of publishing their light curves online and making the code of their light-curve fitting program, SALT, available for public inspection and use[^2]. Making the light curves public, as was done for the results of the HZT and its successors @riess98 [@tonry03; @barris04; @krisciunas05; @clocchiatti06], by @knop03, by @riess04b for the very high redshift [*HST*]{} supernova program, and for the low-$z$ data of @hamuy96, @riess99, and @jha06a, provides the opportunity for others to perform their own analysis of the results. In addition to exploring a variety of approaches to analyzing our own [SN Ia]{} observations, we show the first joint constraints from ESSENCE and SNLS, and some joint constraints derived from combining these with the @riess04b gold sample in §\[sec:cosmology\]. Luminosity Distance Determination {#sec:distances} ================================= The physical quantities of interest in our cosmological measurements are the redshifts and distances to a set of space-time points in the Universe. The redshifts come from spectra and the luminosity distances, $D_L$, come from the observed flux of the supernova combined with our understanding of [SN Ia]{} light curves from nearby objects. Extracting a luminosity distance to a supernova from observations of its light curve necessitates a number of assumptions. We use the observations of nearby supernovae to establish the relations between color, light-curve shape in multiple bands, and peak luminosity. These nearby observations attain high signal-to-noise ratios, and the nearby objects can be observed in more passbands (including infrared) than faint distant objects. We assume that the resulting method of converting light curves to luminosity distances applies at all redshifts. The observed spectral uniformity of supernovae over a range of redshift [@lidman05; @hook05; @blondin06] supports this approach. We assume that $R_V$, the ratio of selective to absolute extinction, is independent of redshift. Below in §\[sec:extinction\], we test the potential systematic effect of departures from this assumption. We adopt an astrophysically sensible prior distribution of host-galaxy extinction properties, with a redshift dependence that is derived from the simulations we present below. Our approach is to conduct comprehensive simulations of the ESSENCE data and analysis. As described by @miknaitis07, we use this same approach to explore our photometric performance. For the aspects of our analysis that are “downstream” of the light-curve generation, we generate sets of synthetic light curves and subject them to our analysis pipeline. In this way we can test the performance of our distance-fitting tools, and by exaggerating various systematic errors (zeropoint offsets, etc.) we can assess the impact of these effects on our determination of $w$. We must recognize and emphasize that in the era of precision [SN Ia]{}cosmology (constraining dark energy properties, rather than just detecting its existence), careful attention to systematic errors is of paramount importance: shifts of a few hundredths of a magnitude can lead to constraints on $w$ that change by $0.1$. Different, yet defensible, choices in the analysis chain may show such effects. Extracting Luminosity Distances from Light Curves: Distance Fitters ------------------------------------------------------------------- We use the MLCS2k2 method of @jha06c as the primary tool to derive relative luminosity distances to our [SNe Ia]{}. For comparison, we also provide the results obtained using the Spectral Adaptive Lightcurve Template (SALT) fitter of @guy05 on the ESSENCE light curves. SALT was used in the recent cosmological results paper from the SNLS [@astier06, hereafter A06]. We provide a consistent and comprehensive set of distances obtained to nearby, ESSENCE, and SNLS supernovae for each luminosity-distance fitting technique. The ESSENCE light curves used in this analysis were presented by @miknaitis07 and we provide them online, together with our set of previously published light curves for nearby [SNe Ia]{}, for the convenience of those interested.[^3] Additional [SN Ia]{} light-curve fitting methods will be further explored in future ESSENCE analyses. Understanding the behavior of our distance determination method is critical to our goal of quantifying the uncertainties of our analysis chain. MLCS2k2 and SALT, as well as the light-curve “stretch” approach used by @perlmutter97 [@perlmutter99], @goldhaber01 and @knop03, exploit the fact that the rate of decline, the color, and the intrinsic luminosity of [SNe Ia]{}are correlated. At present we treat [SNe Ia]{} as a single-parameter family, and the distance fitting techniques use multi-color light curves to deduce a luminosity distance and host-galaxy reddening for each supernova. Previous papers have shown that the different techniques produce relative luminosity distances that scatter by $\sim 0.10$ mag for an individual [SN Ia]{} [e.g., @tonry03], but this scatter is uncorrelated with redshift. As a consequence, the cosmology results are insensitive to the distance fitting technique. However, as described by @miknaitis07, the measurement of the equation-of-state parameter hinges on subtle distortions in the Hubble diagram, so we have undertaken a comprehensive set of simulations to understand potential biases introduced by MLCS2k2. The MLCS2k2 approach [@riess96; @riess98; @jha06c] to determining luminosity distances uses well-observed nearby [SNe Ia]{} to establish a set of light-curve templates in multiple passbands. The parameters $\Delta$ (roughly equivalent to the variation in peak visual luminosity, this parameter characterizes intrinsic color, rate of decline, and peak brightness), $A_V$ (the $V$-band extinction of the supernova light in its host galaxy), and $\mu$ (the distance modulus) are then determined by fitting each multi-band set of distant supernova light curves to redshifted versions of these templates. Jha et al. (2006c) present results from MLCS2k2 based on nearby SN Ia. Here we have modified MLCS2k2 for application to both high and low-redshift [SNe Ia]{}. We begin with a rest-frame model of the [SN Ia]{} in its host galaxy, and then propagate the model light curves through the host-galaxy extinction, K-correction, Milky Way extinction to the detector, incorporating the measured passband response (including the atmosphere for ground-based observations). We then fit this model directly to the natural-system observations. This forward-modeling approach has particular advantages in application to the more sparsely sampled (in color and time) data typical of high-redshift SN searches. The SALT method of @guy05, which was used for the SNLS first-results analysis of A06, constructs a fiducial [SN Ia]{} template using combined spectral and photometric information, then transforms this template into the rest frame of the [SN Ia]{}, and finally calculates a flux, stretch, and generalized color. The color parameter in SALT is notable in that it includes both the intrinsic variation in [SN Ia]{} color and the extinction from dust in the host galaxy within a single parameter (in contrast, MLCS2k2 attempts to separate these components of the observed colors for each supernova). While the reddening vector (attenuation vs. color excess) is similar to the [SN Ia]{} color vs. absolute magnitude relation, the two sources of correlated color and luminosity variation are not identical. The stretch and color parameters of SALT were used by A06 to estimate luminosity distances by fitting for the stretch-luminosity and color-luminosity relationships in the nearby sample and applying those to the full [SNe Ia]{} sample. Given that the SALT color parameter conflates the two physically distinct phenomena of host-galaxy extinction and [SN Ia]{} color variation, it is remarkable and perhaps a source of deep insight that this treatment works as well as it does. Because of both survey selection effects and possible demographic shifts in the host environments of [SNe Ia]{}we would not expect that the proportion of reddening from dust and from intrinsic variation would remain constant with redshift as this approach assumes. However, the SALT/A06 method does seem to work quite well in practice. Sensitivity to Assumptions about the Host-Galaxy Extinction Distribution: Extinction Priors {#sec:extinction_prior} ------------------------------------------------------------------------------------------- The best way to treat host-galaxy extinction is a serious question for this work and for the field of supernova cosmology. The Bayesian approach we use is detailed in §\[sec:priors\]. Here we describe simulations that are designed to evaluate the effects of those methods. There have been four basic approaches to combining reddening measurements with astrophysical knowledge to determine the host galaxy extinction along the line-of sight: (1) assume that linear $A_V$ is the natural space for extinction and assume a flat prior [@perlmutter99; @knop03]; (2) use models of the dust distribution in galaxies [@hatano98; @commins04; @riello05] to model line-of-sight extinction values [@riess98; @tonry03; @riess04b]; (3) assume that the distribution of host-galaxy $A_V$ follows an exponential form [@jha06c], based on observed distributions of $A_V$ in nearby [SNe Ia]{}; and (4) self-calibrate within a set of low-$z$ [SNe Ia]{} to obtain a consistent color+$A_V$ relationship and assume that relation for the full set [@astier06]. Approach (1) assumes the least prior knowledge about the distribution of $A_V$ and produces a Gaussian probability distribution for the fitted luminosity distance. However, this approach weakens the ability to separate intrinsic [SN Ia]{}color from $A_V$ and results in a fit parameter $A_V$ that is a mixture of the two. An $A_V$ that is truly related to the dust extinction should never be negative. The probability prior with $-\infty < A_V < +\infty$ is not the natural range over which to assume a flat distribution. The physically reasonable prior on $A_V$ should be strictly positive. One approach is to base the prior for absorption on the distribution of dust in galaxies. Theoretical modeling of dust distributions in galaxies, such as that of @hatano98, @commins04, and @riello05, provides a physically motivated dust distribution. This method represents approach (2) above and is the method we adopt here. In contrast, @jha06c empirically derived an exponential $A_V$ distribution from MLCS2k2 fits to nearby [SNe Ia]{} by assuming a particular color distribution of [SNe Ia]{}. This distribution was derived using the empirical fact that [SNe Ia]{}  reach a common color about 40 days past maximum light [@lira95]. They found an exponential distribution of $A_V$, $$p(A_V) \propto \exp \left(\frac{-A_V}{\tau}\right), \label{eq:default}$$ where $\tau=0.46$ mag. Unfortunately, the highest-extinction objects drive the tail of this exponential and significantly affect the fit, resulting in a prior sensitive to sample selection, which differs significantly in high-redshift searches compared to the nearby objects studied by @jha06c. A06 analyzed the results of the SALT [SN Ia]{} light-curve fitter with approach (4) and have systematic sensitivities that are similar to those of approach (1). We use MLCS2k2 as our main analysis tool. We designate approach (1) the “flatnegav” prior and approach (3) the “default” prior and discuss both of these further in §\[sec:priors\]. Approach (2) is based on a galactic line-of-sight or “glos” prior on $A_V$: $$\hat{p}(A_V) \propto \frac{A}{\tau} \exp\left({\frac{-A_V}{\tau}}\right) + \frac{2B}{\sqrt{2\pi}\sigma} \exp\left(\frac{-A_V^2}{2\sigma^2}\right), \label{eq:glos}$$ where $A=1$, $B=0.5$, $\tau=0.4$, $\sigma=0.1$, and $\hat{p}(A_V) \equiv 0$ for $A_V < 0$. This exponential plus one-sided narrow Gaussian “glos” prior is based on the host-galaxy dust models of @hatano98, @commins04, and @riello05. As described below, we have modeled our selection effects with redshift to adapt the “glos” prior into the “glosz” prior that is the basis for our analysis. We feel this approach leverages our best understanding of the effects of extinction and selection. Figs. \[fig:mlcs\_fits\_prior\_glosz\] and \[fig:salt\_fits\] show the distribution of the fit parameters and overlay the prior distribution assumed for each of these approaches. Fig. \[fig:mlcs\_vs\_salt\] compares the fit distances and extinction/color parameters of the MLCS2k2 “glosz” and SALT fit results for the ESSENCE, SNLS, and nearby samples. The distribution of recovered $\Delta$ and $A_V$ match their imposed priors for MLCS2k2 “glosz” while the stretch and color fit parameters from SALT show a consistent distribution for the three different sets of [SNe Ia]{}. ESSENCE Selection Effects and the Motivation for a Redshift-Dependent Extinction Prior {#sec:selection} -------------------------------------------------------------------------------------- We examined the effect of the survey selection function on the expected demographics of the ESSENCE [SNe Ia]{} and explored the interplay between extinction, Malmquist bias, and our observed light curves. To determine the impact of the selection bias, we developed a Monte Carlo simulation of the ESSENCE search. We created a range of supernova light curves that match the properties of the nearby sample, added noise based on statistics from actual ESSENCE photometry, and then fit the resulting light curves in the same way the real events are analyzed. In this way we estimated the impact of subtle biases, although this simulation cannot test for errors in our light-curve model or population drift with redshift. Based on its low-redshift training set, MLCS2K2 is able to output a finely sampled light curve given a redshift ($z$), distance modulus ($\mu$), light-curve shape parameter ($\Delta$), host extinction ($A_V$), host extinction law ($R_V$), date of rest-frame $B$-band maximum light ($t_0$), Milky Way reddening ($E(B-V)_\mathrm{MW}$), and the bandpasses of the observations. At a given redshift we calculated a distance modulus, $\mu_\mathrm{true}$, from the luminosity distance for the standard cosmology ($\Omega_m=0.3$, $\Omega_\Lambda =0.7$) and that distance modulus plus an assumed $M_B=-19.5$ for [SNe Ia]{} set the brightness for our simulated supernovae. Varying the assumed cosmology does not significantly impact the simulation results since we are comparing the input distance modulus with the recovered distance modulus, $\mu_\mathrm{obs}$, which is independent of the cosmology. At each of a series of fixed redshifts, we created $\sim1000$ simulated light curves with parameters chosen from random distributions. The light-curve width, $\Delta$, was selected from the @jha06c distribution measured from the low-$z$ sample. The $\Delta$ distribution is approximately a Gaussian peaking at $\Delta =-0.15$ with an extended tail out to $\Delta =1.5$. The host extinction for each simulated event, $A_V$, was selected from either the @jha06c distribution (“default”) estimated from the local sample or from a “galaxy line-of-sight” estimation (“glos”). The “default” distribution was an exponential decay with index $0.46$ mag and set to zero for $A_V < 0.0$ mag. The “glos” distribution is also set to zero for $A_V < 0.0$ mag and combines a narrow Gaussian with a exponential tail for $A_V > 0.0$ mag (see Eq. \[eq:glos\]). The extinction law is assumed to be $R_V = 3.1$. The Milky Way reddening \[$E(B-V)_\mathrm{MW}$\] distribution was constructed from the @schlegel98 (hereafter SFD) reddening maps that cover the ESSENCE fields. The $E(B-V)_\mathrm{MW}$ was measured for 10,000 random locations in each ESSENCE field and the reddening was selected from the sum of the histograms (see Figure 2). The dates of observation for a simulated [SN Ia]{} were based on the actual dates of ESSENCE 4-m observations. An ESSENCE field was chosen at random from the list of monitored fields and a date of maximum, $t_0$, selected to fall randomly between the Modified Julian Date (MJD) of the first and last observation of an observing season. The simulated light curve was then interpolated for only those dates that ESSENCE took images. With each ESSENCE field observation, we estimated the magnitude in $R$ and $I$ that provided a 10$\sigma$ photometric detection based on the seeing and sky brightness. The signal-to-noise ratio (SNR) for each simulated light-curve point was then scaled from the 10$\sigma$ detection magnitude, assuming the noise was dominated by the sky background. For each date of ESSENCE observation, we have a simulated noiseless magnitude and an estimate of the SNR of the observation. To each simulated observation we added an appropriate random value in flux space selected from a normal distribution with a width corresponding to the predicted SNR. MLCS2k2 was then used to fit the simulated light curves and provide estimates of $\mu$, $\Delta$, $A_V$, and $t_0$, assuming a fixed $R_V = 3.1$. MLCS2k2 required an initial guess of the date of maximum, an estimate achieved by selecting from a normal distribution about the true date with a $1\sigma$ width of 2 days. The SFD Milky Way reddening was also required in MLCS2k2 and was provided from the true reddening after adding an uncertainty of 10%. Finally, in the real ESSENCE data we discarded supernovae when the MLCS2k2 reduced $\chi^2$ indicated a very poor fit. For the simulated light curves, we dropped events from the sample if the reduced $\chi^2$ exceeded 2. ### Deriving an Extinction Prior from the Simulation Results Simulated ESSENCE samples were created at a range of redshifts out to $z=0.70$ and the light curves that passed the detection criteria from the actual ESSENCE search were fit with MLCS2k2. The fitting was done with the “default” prior and the “glos” prior (with corresponding $A_V$ distributions). The difference between the “true” (input) distance modulus and recovered (fit) distance modulus, $\Delta\mu$, was calculated for each event and the mean, median, and dispersion for the ensemble were calculated at each redshift. The median $\Delta\mu$ of the simulations was within $0.03$ mag for $z < 0.45$, but at higher redshift the simulated supernovae were estimated to be brighter than the input supernovae by more than $0.2$ mag. This bias results from the loss of faint events (large $A_V$ and large $\Delta$) from the sample as the distance increases. In a sense, this is a classic Malmquist bias, but here it is caused by an uninformed prior. These results are shown in Fig. \[fig:simulatepriors\]. The decreasing ability to observe large $A_V$ events as the redshift increases (see Fig. \[fig:cut\]) makes it clear that a using single $A_V$ prior for all redshifts is not correct. Because events with large $A_V$ and large $\Delta$ are lost at large redshift due to the magnitude limits of the search, we should adjust the prior as a function of $z$ to account for these predictable losses. Applying redshift-dependent window functions to the basic “glos” prior provides a much better prior as a function of redshift. We fit the recovered $A_V$ distributions derived from the simulations, which start with a uniform $A_V$, to a window function based on the error function (integral of a Gaussian), and two parameters describe where that function drops to half its peak value ($A_{1/2}$) and the width of the transition ($\sigma_A$). The window function $W$ has the form $$W(A_V,A_{1/2},\sigma_A) = 1- {1\over{\sqrt{\pi}}}\int_{-\infty}^{(A_V-A_{1/2})\sigma_A} e^{-x^2}\;dx \label{eq:gloszwindow}$$ where $A_{1/2}$ and $\sigma_A$ are functions of $z$ and estimated from the simulations. A similar process was applied to the $\Delta$ distribution and a table providing the parameters is given in Table \[tab:gloszwindow\]. We embody this prescription in the “glosz” prior we use for our main MLCS2k2 light-curve fitting. The “glosz” prior is the “glos” prior modified by the window functions in $A_V$ and $\Delta$. The simulations using the “glosz” prior provide a median $\Delta\mu$ within 0.03 mag for $z < 0.7$, which we judge to be satisfactory performance. Comparison of MLCS2k2 and SALT Luminosity Distance Fitters ---------------------------------------------------------- The release of the source code to the SALT fitter [@guy05] makes a modern [SN Ia]{} light-curve fitter fully accessible and available to the community. This public release of SALT allows us to compare the results of our MLCS2k2 distance fitter with the SALT fitter used in the SNLS first results paper [@astier06]. We present the results of SALT fits to our nearby and ESSENCE samples in Table \[tab:salt\_fits\]. To compute the distance moduli we quote in that table, we assume the $\alpha=1.52$, $\beta=1.57$ values from A06. To calibrate the additional dispersion to add to the distance moduli of MLCS2k2 and SALT, we fit a [$\Lambda{\rm CDM}$]{} model to the nearby sample alone and derived the additional [$\sigma_{\rm add}$]{} to added in quadrature to recover [$\chi^2/{\rm DoF}$]{}$=1$ for the nearby sample. This [$\sigma_{\rm add}$]{} is related to the intrinsic dispersion of the absolute luminosity of [SNe Ia]{}, but is not precisely the same both because the light-curve fitters include varying degrees of model uncertainty and because the light curves of the [SNe Ia]{} are subject to photometric uncertainty. We find [$\sigma_{\rm add}$]{}$=0.10$ for MLCS2k2 with the “glosz” prior and [$\sigma_{\rm add}$]{}$=0.13$ for SALT. These values should be added to the $\sigma_\mu$ uncertainties given Tables \[tab:mlcs\_fits\_prior\_glosz\] and \[tab:salt\_fits\] Fig. \[fig:mlcs\_vs\_salt\] visually demonstrates that the relative luminosity distances using the SALT light-curve fitter agree, within uncertainties, with the MLCS2k2 distances when the latter are fit using the “glosz” $A_V$ prior. Testing the Recovery of Cosmological Models Using Simulations of the ESSENCE Dataset {#sec:simulation} ------------------------------------------------------------------------------------ In order to assess the reliability with which we recover cosmological parameters, we have simulated 100 sets of 100 light curves representing both the nearby and the ESSENCE light curves. Table \[tab:mlcs\_cuts\] presents the quality cuts for MLCS2k2 we derived from these simulated light-curve sets. Our light-curve goodness-of-fit cuts, when applied to these simulated light curves (see Table \[tab:mlcs\_cuts\]) and combined with the same external constraints of baryon acoustic oscillations [BAO; @eisenstein05] and flatness, allow us to recover our input cosmology of $(\Omega_M=0.3, \Omega_\Lambda=0.7, w=-1)$ to within $\pm0.11$ in $w$. This $\pm0.11$ uncertainty on an individual measurement of $w$ is matched by the $\sigma=0.11$ distribution of recovered $w$ values from the 100 sets of simulated light curves. This confirms our statistical error estimate on $w$; the estimated uncertainty matches the distribution, and within the self-consistent realm of synthetic and analyzed light curves based on MLCS2k2 our estimates of luminosity distance are not biased. Potential Sources of Systematic Error {#sec:systematics} ===================================== Here we identify and assess sources of systematic error that could afflict our measurements. These can be divided into two groups. Certain sources of systematic error may introduce perturbations either to individual photometric data points or to the distances or redshifts estimated to the [SNe Ia]{}. Others affect the data in a more or less random fashion and produce excess [*scatter*]{} in the Hubble diagram. Errors that are uncorrelated with either distance or redshift will not bias the cosmological result. These sources of photometric error are detailed by @miknaitis07; we summarize those results here in Table \[tab:photscatter\]. We add these effects in quadrature to the statistical uncertainties given by the luminosity distance fitting codes for each [SN Ia]{} distance measurement: $\sigma_\mu^{\rm phot scatter}=0.026$ mag. In §\[sec:distances\] we discussed our testing of the MLCS2k2 fitter on simulated data sets that replicate the data quality of the ESSENCE and nearby [SNe Ia]{}. We explore the issue of host-galaxy extinction further in §\[sec:extinction\] & \[sec:priors\]. The interaction of Malmquist bias and selection effects with the extinction and color distribution of [SNe Ia]{}is discussed in §\[sec:malmquist\]. Any non-cosmological difference in measurements of nearby and distant [SNe Ia]{} has the potential to perturb our measurement of $w$. Table \[tab:systematics\] lists potential systematic effects of this sort. We present both our estimate of the sensitivity $dw/dx$ of the equation-of-state parameter to each potential systematic effect and our best estimate of the potential size of the perturbation, ${\Delta}x$. The upper bound on the bias introduced in $w$ is then ${\Delta}w = dw/dx\times{\Delta}x$. @miknaitis07 discusses the systematic uncertainties on $\mu$, which we convert here to systematic uncertainties on $w$, due to photometric errors from astrometric uncertainty on faint objects (${\Delta}w = 0.005$), potential biases from the difference imaging (${\Delta}w = 0.001$), and linearity of the MOSAIC II CCD (${\Delta}w = 0.005$). None of these contributed noticeably to the systematic uncertainty in our measurement of $w$. The rest of this section describes how we appraised our additional potential sources of systematic uncertainty. The conclusion of this section is that our current overall estimate for the 1$\sigma$ equivalent systematic uncertainty in a static equation-of-state parameter is ${\Delta}w =$ [$0.13$]{} for our “glosz” analysis. Photometric Zeropoints {#sec:zeropoint} ---------------------- Supernova cosmology fundamentally depends on the ability to accurately measure fluxes of objects over a range in redshift. Errors in photometric calibration translate to errors in cosmology in two basic ways. Nearby objects at redshifts $<0.1$ play a crucial role in establishing a comparison reference for cosmological measurements. ESSENCE is inefficient at finding and observing low-redshift objects with the same telescope and detector system, so we use photometry of low-redshift [SNe Ia]{} in the literature from our own work and that of others [for the full list see @jha06c]. Using these external [SNe Ia]{} requires understanding the photometric calibration of our high-redshift sample relative to this low-redshift sample. Every supernova cosmology result to date has made use of more or less the same low-redshift photometry, so any inaccuracies in the nearby sample are a source of common systematic error for all [SN Ia]{} cosmology experiments. Calibration of photometry at the $\sim1\%$ level required to make precise inferences about the nature of dark energy is notoriously difficult [@stubbs06]. Photometric miscalibration can result in a second, more insidious systematic error if there is an error in the relative flux scaling between the broad-band passbands. This offset would distort the observed colors for the entire sample. Since these colors are used to infer the extinction, even small color errors result in significant biases in the measured distances. After all, the inferred host galaxy extinction, $A_{V}$, is related to the measured color excess, $E(B-V)$, by $A_{V}\approx3.1 E(B-V)$ (for Milky Way-like dust). A color error in rest-frame $B-V$ (observer-frame $R$, $I$ for ESSENCE) of $0.01$ mag can result in $0.03$ mag error in extinction, an inaccuracy that would lead directly to a $3\%$ error in the distance modulus, or a $1.5\%$ error in the distance. We currently estimate our color zeropoint uncertainty at $0.02$ mag and our absolute zeropoint (relative to the nearby [SNe Ia]{}) uncertainty to be $0.02$ mag. These respectively translate to $0.04$ and $0.02$ shifts in $w$ (see Table \[tab:systematics\]). @miknaitis07 describe the calibration program we undertook to measure the transmission of the CTIO 4-m MOSAIC II system with the $R$ and $I$ filters of the ESSENCE survey. The calibration of the ESSENCE survey fields will be further improved by an intensive calibration program we are undertaking on the CTIO 4-m in 2006. Together with the improved calibration of the SDSS Southern Stripe by the SDSSII project, which overlaps 25% of our ESSENCE fields, we aim to achieve 1% photometric calibration of our CTIO 4-m MOSAIC II BVRI natural system. We here use MLCS2k2 v004 with the @bohlin04 values for the magnitudes of Vega: i.e., `alpha_lyr_stis_002.fits` with $R_{\rm Vega}=0.033$ mag. This value for $R_{\rm Vega}$ comes from @bessell98 but has been shifted down by $0.004$ mag as @bohlin04 suggest (from their $V_{\rm Vega}=0.026$ mag compared to @bessell98 $V_{\rm Vega}=0.030$ mag). K-Corrections and Bandpass Uncertainty {#sec:kcorrection} -------------------------------------- Uncertainty in the transmission function, typically called the bandpass, of the optical path of the telescope+detector is an important and potentially systematic effect. In this context, bandpass refers to the wavelength-dependent throughput of the entire optical path, including atmospheric transmission, mirror reflectivity, filter function, and CCD response. Since an error in the assumed bandpasses translates into a redshift-dependent error in the supernova flux, it is important to account for possible errors in the bandpass estimates. The *relative* error due to bandpass miscalibration is small for objects with similar spectra, such as [SNe Ia]{}. Bandpass shape errors are largely accounted for by the filter zeropoint calibration, with residual errors corresponding to the difference between the spectral energy distribution of the objects of interest and those of the calibration sources. In the case of [SN Ia]{} observations, any residual zeropoint error is absorbed when we marginalize over the “nuisance parameter,” [$\cal{M}$]{}$=M_B - 5\log_{10} (H_0) + 25$ [@kim04]. This relative comparison results in a very small systematic error in the cosmological parameters from a global calibration error across bandpasses. Moreover, variations in atmospheric transmission are expected to contribute only random uncertainty. However, the bandpass uncertainty becomes important when we compare [SNe Ia]{} at different redshifts for which the bandpass samples different spectral regions. In order to compare [SNe Ia]{} at multiple redshifts, we need to perform a K-correction [@leibundgut90; @hamuy93a; @kim96; @nugent02]. That is, we assume a spectral distribution for the supernova and convert the observed magnitude to what it would have been had the supernova been at another redshift. This process involves performing synthetic photometry of the assumed spectral distribution over the assumed bandpass. We address the issue of systematics arising from errors in the assumed spectral distribution in the supernova evolution section, §\[sec:snevolution\]. Here we address systematics arising from errors in our determination of the CTIO 4-m MOSAIC II $R$ and $I$ bandpass functions. Systematic effects on supernova cosmology that result from bandpass uncertainties are discussed more thoroughly by @davis06. \[sec:bandpass\] Calculating the effect of bandpass uncertainty is fairly difficult because of the arbitrary nature of the shape changes that might affect the bandpass. However, we can make several general calculations. As a first step, we take standard bandpasses and add white noise to represent a miscalibrated filter. White noise contributes power on all scales, so this approach adds small-scale discrepancies as well as large-scale warps or shifts in the filter. By averaging over many such miscalibrated filters, we can estimate the effect of filter miscalibration. Fig. 16 of @davis06 shows photometric error as a function of noise amplitude. A noise amplitude of $0.02$ produces a typical deviation of $2$% from the nominal filter shape at any wavelength. Calibrating the bandpass to better than $3\%$ allows us to keep the K-correction error introduced from a mismeasurement of our effective bandpass to less than 0.005 mag (0.5% in flux) and a systematic uncertainty of ${\Delta}w = 0.005$. Extinction {#sec:extinction} ---------- The most significant cause of variation in luminosity of [SNe Ia]{} is the extinction experienced by the light from the [SN Ia]{}due to scattering and absorption from dust in the host galaxy. Dust introduces a wavelength-depended diminution of a supernova’s light. In the case of Milky Way dust, we correct for its effects by using tabulated values as a function of Galactic longitude and latitude measured by other means [SFD @schlegel98], being sure in our MLCS2k2 fits to properly account for its uncertainty and correlation across all observations. For dust in the supernova’s host galaxy, we infer the extinction from the reddening of each supernova’s light curve. However, the slope of differential reddening, characterized in the @cardelli89 extinction model by the parameter $R_V$, may vary. The nominal value of $R_V$ for the Milky Way is $3.1$, but different lines of sight within our galaxy have values of $R_V$ that vary from $2.1$ to $5.1$. Studies of $R_V$ in other galaxies have been more limited because we lack sources of known color and luminosity with which to probe the dust. Because we use the supernova rest-frame $B-V$ color to determine the reddening of each [SN Ia]{}, and the distance modulus to a supernova is corrected by a value approximately three times the inferred reddening, extinction correction magnifies any source of systematic error in a supernova’s observed effective color. Systematic color errors can result from photometry errors, redshift-dependent K-correction errors, and evolution in the colors of supernovae. Using the IR-emission maps of the Galaxy from the all-sky COBE/DIRBE and IRAS/ISSA maps, SFD have estimated the dust column density around the sky, which can then be translated to a color excess. This analysis has largely superseded the work of @burstein78, who used radio HI measurements and a relationship between gas and extinction to estimate the color excess across the sky. @burstein03 has reanalyzed the IR and HI measurements and finds that Milky Way extinctions are more precisely derived using the IR method. However, @burstein03 still finds a discrepant value for extinction at the poles, with SFD providing extinctions that are $E(B-V)=0.02$ mag higher than what the HI measurements indicate. @burstein03 suggests as a possible explanation for the discrepancy that SFD may predict too large an extinction in areas with high gas-to-dust ratios. @finkbeiner99 precisely estimated their sensitivities to these systematics and concluded they had controlled them to $0.01$ mag. The ESSENCE program targets fields at high Galactic latitude to minimize Galactic extinction. Although nearby and distant [SNe Ia]{}are both affected by the assumed Milky Way extinction, the nearby objects are observed in $B-V$, whereas the $z\approx0.5$ objects are observed in $R-I$. An $E(B-V)=0.02$ difference in extinction at the pole leads to approximately a $0.02$ mag difference in the relative distances between $z=0$ and $z=0.5$ objects, assuming a Galactic reddening law, host-galaxy corrections based on rest-frame $B-V$ color, and distances based on $V$. For this analysis, we use the SFD extinction map values with an uncertainty of 16% for each individual [SN Ia]{}  but assume an additional $0.01$ mag of systematic uncertainty in our distance moduli to account for the known source of uncertainty of extinction at the pole. In most supernova work we assume the Galactic reddening law [@cardelli89] applies to external galaxies ($R_V=3.1$), but studies of individual [SNe Ia]{} have found a range of values extending to much smaller values of $R_V$ [@riess96; @tripp98; @phillips99; @krisciunas00; @wang03; @altavilla04; @reindl05; @elias-rosa06]. These measurements are dominated by objects with large extinction values, where a significant measurement can be made can be made of the extinction law (lessening the effects of intrinsic color scatter and systematic color variations with luminosity), and it is possible that $R_V$ is correlated with total extinction [@jha06c]. In principle, with photometry in three or more passbands, it is possible to fit for $R_V$, but in practice, at $z>0.2$, there are only a few [SNe Ia]{} in the literature with the requisite high-precision photometry extending from the rest-frame UV to the near-IR. The systematic error on our measurement of $D_L$ caused by assuming a particular value of $R_V$ depends on the average extinction as a function redshift, assuming $R_V$ is constant with $z$, except for a small correction caused by the rest-frame effective bandpass of our filters drifting away from the low-$z$ values, depending on the precise redshift of each object. To quantify this effect, we fit our complete distance set with three different values of $R_V$: $2.1$, $3.1$, and $4.1$. Color and Extinction Distributions and Priors {#sec:priors} --------------------------------------------- To evaluate the systematic effects produced by various prior assumptions about extinction, we have fit the entire data set with a variety of plausible priors: the “exponential” prior of @jha06c, a flat prior from $-\infty$ to $+\infty$ (the “flatnegav” prior), and an exponential prior with an added Gaussian around zero that is based on models of the dust distribution in galaxies (“glos” and the redshift-dependent “glosz”). These results are presented in §\[sec:cosmology\] and form the basis for Table \[tab:w\_rv\_sys\]. To separate the effects of color and extinction, @jha06c noted that the distribution of color excess in their nearby sample was consistent with a Gaussian distribution of $\sigma=0.2$ convolved with a one-sided exponential, $\exp{(-A_V/\tau)}$, where $\tau=0.46$ mag. As discussed in §\[sec:extinction\_prior\], the “glosz” prior we adopt here is derived from models of line-of-sight dust distributions in galaxies. It has more parameters than the simple exponential model of @jha06c, but we believe these additional parameters are well motivated. The power of MLCS2k2 to distinguish between color and extinction lies in the ability to treat the two phenomena independently. A06 uses SALT and makes the assumption that the color$+$extinction distribution is the same in the nearby and in the high-redshift samples; the separation of the $A_V$ component in the MLCS2k2 model allows us to model our expected distribution of $A_V$ based on both models of dust in galaxies and selection effects of the ESSENCE survey. This separation allows us to take the nominal “glos” model and create the “glosz” prior that combines the distribution of dust in galaxies with the redshift-dependent selection effects. The difference in the mean estimated parameter for a constant $w$ is given in Table \[tab:w\_rv\_sys\] for the different MLCS2k2 $A_V$ priors discussed above. For the main MLCS2k2 “glosz” analysis we present here, we find a slope of ${\Delta}w / {\Delta}R_V=0.02$ in the dependence of $w$ on the assumed $R_V$. The effect on $w$ of varying $R_V$ is substantially greater for the less restrictive $A_V$ priors because the covariance between $A_V$ and $\mu$ is substantially greater for these priors. A reasonable variation of $0.5$ in the value for $R_V$ contributes a systematic uncertainty of ${\Delta}w=0.01$ Differences in the inferred value of $w$ for various assumed absorption priors shows that this is a significant systematic effect. The maximum difference between two priors, “exponential” and “glosz,” for the nominal $R_V=3.1$ case is ${\Delta}w=0.165$. While we have conducted careful simulations to determine the most appropriate prior for our sample (see §\[sec:selection\]) and it is clear that the “exponential” is not appropriate for this analysis, we nonetheless take half of the difference between the two as representative of our systematic uncertainty, $\Delta_w^{\rm prior}=0.08$, due to the choice of prior. The residual $0.02$ mag shift of the simulations with the “glosz” prior shown in Fig. \[fig:cut\] for $z\approx0.65$ results in a very small shift in ${\Delta}w$ of only $0.001$. Since we use an $A_V$, that obviously interacts strongly with our understanding of the intrinsic color distribution of [SNe Ia]{}. We estimate this contribution to our systematic error budget at ${\Delta}w=0.06$ We have not undertaken a similar analysis with the SALT fitter, but the underlying assumption that the color, extinction, luminosity relationship for [SNe Ia]{} is constant with redshift is subject to uncertainties analogous to those considered here in the context of the MLCS2k2 $A_V$ prior. The issue of color and extinction distributions clearly needs to be addressed for substantial further progress to be made in the field of supernova cosmology. Malmquist Bias and Other Selection Effects {#sec:malmquist} ------------------------------------------ As with all magnitude-limited surveys, at the faint limits of the survey we are more likely to observe objects drawn from the bright end of the [SN Ia]{} luminosity distribution. This Malmquist bias is particularly dangerous for inferences about cosmology based on supernova observations. However, it is not necessarily troubling that we may observe more luminous, broad events at high redshift, as long as the known empirical luminosity-width relation is valid at those redshifts. Rather, the concern for cosmological measurements is that at high redshift, we may preferentially find [SNe Ia]{} which are bright *for their light curve shape*. A second and more subtle concern is that at higher redshifts we are also less likely to detect [SNe Ia]{} whose light suffers significant absorption due to dust in their host galaxies. We have modeled both of these effects (see §\[sec:selection\] & \[sec:extinction\]) and have controlled for their impact. Our current limits on systematics due to uncontrolled selection effects is $\Delta_w^{\rm selection}=0.02$. A thorough study of the efficiency of the ESSENCE survey will be presented by @pignata07. We aim for this future work to allow us to reduce this contribution to our systematic error to no more than 1%. Type Ia Supernova Evolution {#sec:snevolution} --------------------------- A persistent concern for any standard-candle cosmology is the possibility that the distant candles may differ slightly from their low-redshift counterparts. In a recent paper [@blondin06] we compare the spectra of the high-redshift [SNe Ia]{} in this sample with low-redshift [SNe Ia]{} and demonstrate that there is no evidence for any systematic difference in their properties. This conclusion is based on line-profile morphology and measurements of the phase-evolution of the velocity location of maximum absorption and peak emission. These results confirm a number of other studies of distant [SNe Ia]{}[e.g., @coil00; @sullivan03; @lidman04] that all confirm that, to the accuracy of current observations, the high and low redshift supernova populations are indistinguishable. Recently @hook05 used spectral dating, spectral time sequences, and measurements of expansion velocities to compare distant and nearby [SNe Ia]{}; they also find no evidence for evolution in [SN Ia]{} properties up to $z\approx0.8$. Although we are confident that the subtypes of distant [SNe Ia]{} are well represented by the subtypes seen nearby, we cannot rule out a subtle shift in the population demographics that may yet bias the estimates of cosmological parameters. This potential bias is of particular concern for future experiments that plan to measure the equation-of-state parameter, $w$, with an accuracy of a few percent. There is now evidence that [SN Ia]{} properties are correlated with host-galaxy morphology. @hamuy96 and @riess99 show that the brightest [SNe Ia]{} occur only in galaxies with on-going star formation. However, they observe no residual correlation after light-curve shape correction. Because the galactic demographics over the redshift range of interest change less than current variations in stellar population of [SN Ia]{}host galaxies, we remain confident that our one-parameter correction for supernova luminosity adequately corrects any shift in the average luminosity of [SNe Ia]{} to the same precision as in the nearby Universe, $\sigma_\mu < 0.02$ mag. We thus estimate a systematic uncertainty from possible [SN Ia]{}evolution on our measurement of $w$ of ${\Delta}w=0.02$. One way to verify this confidence is to search for additional parameters that allow tighter luminosity groupings of the low-redshift population. In a first, reassuring step, Hubble diagrams for subsets of [SNe Ia]{} based on host-galaxy type separately confirm the accelerated expansion of the Universe [@sullivan03]. Hubble Bubble and Local Large-Scale Structure {#sec:bubble} --------------------------------------------- The local large-scale structure and associated correlated flows of the Universe should not yet present a significant contribution to the systematic error budget of the current survey [@hui06; @cooray06]. However, at the lowest multipoles we are sensitive to local correlated flows, and, at the most extreme, our cosmological results would be sensitive to a local velocity monopole or “Hubble bubble.” @jha06c see such an effect in their analysis of nearby [SNe Ia]{}. We use only the subset of [SNe Ia]{} from @jha06c with $z>0.015$ and find that this effect could contribute as much as $0.065$ to our systematic error budget in $w$. We will rely on future sets of nearby [SNe Ia]{} ($0.01<z<0.05$) that are now being acquired at the CfA, by the Carnegie Supernova program, by the Lick Observatory Supernova Search, and by the SNfactory to reduce this uncertainty below 2% to help achieve the desired systematic uncertainty required for the final ESSENCE analysis. Gravitational Lensing {#sec:lensing} --------------------- Gravitational lensing can increase or decrease the observed flux from a distant object. The expected distribution is asymmetric about the average flux multiplier of unity. @holz05 calculate the effect for [SN Ia]{}surveys and determine that any systematic effect from neglecting the asymmetry of the probability distribution function for magnification (as we do here) decreases quickly with the number of [SNe Ia]{} per effective bin. Roughly speaking, at a $z\approx0.5$, in a redshift bin width of ${\Delta}z \sim 0.1$, ten [SNe Ia]{} per bin are sufficient to reduce any systematic effect in luminosity distance to less than $0.3\%$, which makes no noticeable contribution to our systematic error budget. For the redshifts of interest in the ESSENCE survey, lensing has a more significant effect in the scatter it adds to the observed brightness of [SNe Ia]{}. @holz05 calculate a 3% increase in the dispersion in distance modulus at $z\approx0.5$. We include the effect of lensing in our analysis by adding a statistical dispersion of $\sigma^{\rm lensing}_{\mu}=0.03$ to our luminosity distance uncertainty for the ESSENCE and SNLS [SNe Ia]{}. Grey Dust {#sec:greydust} --------- When the first cosmological results with [SNe Ia]{} were announced, that distant [SNe Ia]{} were dimmer than they would be in a decelerating Universe, @aguirre99a [@aguirre99b] suggested various models for intergalactic grey dust that could explain this dimming without producing observable reddening. To explain [SNe Ia]{}becoming consistently dimmer with distance, this dust would need to be distributed throughout intergalactic space beginning at least at $z=2$ [@goobar02]. The most naive model of such dust distribution and creation would predict that [SNe Ia]{} should continue to get dimmer relative to a flat, [${\Omega}_{\rm M}$]{}$=1$, cosmology all the way up to at least a redshift of $2$. The high-redshift [SN Ia]{} work of @riess04b demonstrated that this continued dimming is not what is observed: the apparent magnitudes of [SNe Ia]{} become first a little dimmer and then a little brighter with redshift than they would in an empty Universe. This is exactly what we expect from an early phase of deceleration followed by a recent phase of acceleration in a mixed, dark-matter/dark-energy cosmology. A more complicated model of dust was contrived by @goobar02. It involves the creation of intergalactic dust at just the right rate to match the decrease in opacity due to expansion of the Universe. This carefully constructed model mimics the signal of an accelerating universe and is difficult to distinguish from a universe that is presently dominated by dark energy. This model does not have a strong underpinning in the behavior of known dust and represents a form of fine-tuning. In the larger context of converging cosmological evidence, this particular scheme for matching the data seems less plausible than a universe with dark energy. Recent observational constraints from non-[SN Ia]{} sources have independently placed significant constraints on the amount of intergalactic dust [@petric06; @ostman06]. In particular, the observations of @petric06 limit intergalactic dust to contributing no more than one percent to potential dimming of light out to a redshift of $0.5$, based on upper limits to X-ray scattering by dust along the line of sight to a quasar at $z=4.3$. Cosmological Results from the ESSENCE Four-Year Data {#sec:cosmology} ==================================================== The ESSENCE [SNe Ia]{} allow us test the hypothesis of a [$\Lambda{\rm CDM}$]{} concordance model and constrain flat, constant-$w$ models of the Universe. We use our MLCS2k2 light-curve fitting technique to measure luminosity distances to nearby and ESSENCE [SNe Ia]{} (Table \[tab:mlcs\_fits\_prior\_glosz\]). When then fit cosmological models to constrain the properties of the dark energy. We compare the results we obtain using MLCS2k2 with those obtained using the SALT light-curve fitter [@guy05]. The SALT fitter was used to fit the nearby light curves, our ESSENCE light curves, and the SNLS light curves.[^4] To verify that we were making appropriate use of the fitter, we fit the nearby and SNLS light curves with SALT, taking the same $\alpha=1.52$ and $\beta=1.57$ width and color parameters used in A06. We recovered the $w$ result of A06 to within $0.01$ in best-fit constant $w$ in a model with a flat Universe using the cosmology fitter that we employ here[^5]. We have compiled our light curves of nearby [SNe Ia]{} from the literature independently of the SNLS analysis and used slightly different quality cuts, so it is quite encouraging that we can replicate these results. Table \[tab:salt\_fits\] gives the SALT fit parameters for the nearby, ESSENCE, and SNLS [SNe Ia]{} discussed here. ESSENCE [SN Ia]{} Sample {#sec:essence} ------------------------ For the ESSENCE project we find that using photometric selection criteria based on the color and rise time of the candidate object, similar to those used by the SNLS [@howell05; @sullivan06a], and in good weather and seeing conditions, 80% of the candidates we take spectra of are [SNe Ia]{}. We use a deterministic analysis [@blondin07], as described in @miknaitis07, to determine final types and redshifts for our SNe and to cull objects that are not [SNe Ia]{} from our sample. All of the ESSENCE supernovae used in this analysis were spectroscopically confirmed as [SNe Ia]{}. From 2002–2005 the ESSENCE project discovered and spectroscopically confirmed 113 [SNe Ia]{}. As discussed by @miknaitis07, which gives full details of these [SNe Ia]{} including their RA and Dec, only 4 of the 15 [SNe Ia]{} from 2002 have been fully analyzed so that leaves us with 102 [SNe Ia]{}. Although we kept 91T-like [SNe Ia]{} such as d083, d085, and d093, we rejected the peculiar [SN Ia]{} d100 [@matheson05]. Three [SNe Ia]{} were rejected from the nearby+ESSENCE only fits because they were at redshifts greater than $0.67$ (see below). After we applied the cuts in Tables \[tab:mlcs\_cuts\] and Tables \[tab:salt\_cuts\], we were left with 57 and 60 [SNe Ia]{} for MLCS2k2 and SALT respectively. With the MLCS2k2 fitter, the largest cut was the 32 [SNe Ia]{} rejected because they had fewer than 8 data points with an SNR $> 5$, no such points after +9 days, or no such points before +4 days. Two of the 102 [SNe Ia]{} were located near edges of the detector field-of-view that we later determined were photometrically less reliable. Due to high [$\chi^2/{\rm DoF}$]{} or related poor light-curve goodness-of-fit values, we eliminated an additional 6 [SNe Ia]{}. This left us with a total of 57 [SNe Ia]{} for our main MLCS2k2 nearby+ESSENCE analysis. The SALT fitter successfully fit three more [SNe Ia]{} than MLCS2k2, but, in general, our SALT quality cuts accepted the same [SNe Ia]{} as our MLCS2k2 quality cuts. The requirements we imposed here on the light curves were stringent cuts to ensure reliable fit parameters. We are currently engaged in an active program to improve the sensitivity of [SN Ia]{} light-curve fitters and we anticipate recovering 50% of the [SNe Ia]{} rejected here in the final ESSENCE analysis. Nearby [SN Ia]{} Sample {#sec:nearby} ----------------------- The [SN Ia]{} cosmological measurement is fundamentally a comparison of the luminosity distance vs. redshift relation at low redshift and high redshift. The ESSENCE [SNe Ia]{} alone provide a homogeneous set of luminosity distance vs. redshift measurements covering the redshift range $0.15<z<0.7$. We selected our nearby [SNe Ia]{} from the set compiled by @jha06c. We applied the light-curve criteria from Tables \[tab:mlcs\_cuts\] and \[tab:salt\_cuts\] and also rejected known peculiar [SNe Ia]{} such as SN 2000cx [@li01] and SN 2002cx [@li03; @jha06b]. Our list of nearby [SNe Ia]{} has 41 [SNe Ia]{} in common with the set used by A06. To minimize complications from loosely constrained local peculiar and coordinated flows, we only considered [SNe Ia]{} with CMB-frame redshifts of $z>0.015$. Our final sample consists of [45]{} nearby [SNe Ia]{} as listed in the fit parameter tables (Tables \[tab:mlcs\_fits\_prior\_glosz\] and \[tab:salt\_fits\]). We used the re-derived Landolt/Vega calibration of A06 to determine the passbands for this set of nearby [SNe Ia]{}. The light curves we used for these [SNe Ia]{} are also included with the ESSENCE light curves available on our website.[^6] External Constraints -------------------- To provide complementary cosmological constraints on our [SN Ia]{} observations, we include external information from baryon acoustic oscillations [BAO; @eisenstein05]. The BAO constraints on ([${\Omega}_{\rm M}$]{}, $w$) from @eisenstein05 are the most complementary measurement in the ([${\Omega}_{\rm M}$]{}, $w$) plane to our [SN Ia]{} measurements, relying only on the observed redshift and angular size of the first doppler peak in the CMB and not on $H_0$. In addition, because the BAO constraints on [${\Omega}_{\rm M}$]{} are similar in precision (and value) to those derived from large scale structure [@percival01; @percival02], WMAP directly [@spergel06], and from the study of X-ray clusters [for a review see @voit05], we choose to combine our results only with the BAO results. We compare the specific model of a flat Universe with either $w=-1$ or constant $w$ of any value to our data. [SNe Ia]{} have very little leverage on the global flatness of the Universe because they effectively measure the difference between [${\Omega}_{\rm M}$]{} and [${\Omega}_{\Lambda}$]{}, and flatness depends on the sum. @eisenstein05 have constrained curvature to be within [${\Omega}_{\rm K}$]{}$=\pm0.01$ of flat. The results presented here (from the [SNe Ia]{}) on $w$ are not significantly affected by variation of [${\Omega}_{\rm K}$]{} by this amount, because the effects of curvature are not noticeable until looking back to much higher redshift. However, non-flat models will significant alter the BAO results on ([${\Omega}_{\rm M}$]{}, $w$) and therefore our joint constraints. For our analysis of the ESSENCE and nearby [SNe Ia]{}, we have chosen to additionally limit our redshift range to $z<0.670$ to avoid using the rest-frame $U$ band. Since this remove just three ESSENCE [SNe Ia]{} from our sample, the tradeoff is worthwhile to minimize this source of uncertainty (see §\[sec:distances\]). When we add in the SNLS or Riess gold samples, we relax this constraint to incorporate those higher-redshift [SNe Ia]{}. In Figs. \[fig:joint\_mlcs\_prior\_glosz\_hubble\_diagram\] and \[fig:joint\_salt\_hubble\_diagram\] we show Hubble diagrams of the nearby, ESSENCE, and SNLS samples for the two different fitters we consider in this paper. We overplot an empty Universe ([${\Omega}_{\rm M}$]{},[${\Omega}_{\Lambda}$]{},$w$) = $(0,0,-1)$, a matter-only open Universe $(0.3,0,-1)$, and a [$\Lambda{\rm CDM}$]{} concordance cosmology $(0.27,0.73,-1)$. The residuals in luminosity distance are then shown with respect to the [$\Lambda{\rm CDM}$]{} model. MLCS2k2 appears to be more suited for the ESSENCE data sample than SALT, although the latter benefits from its flux-based fitting by being able to extract useful luminosity distances from a few more [SNe Ia]{}. One [SN Ia]{}, “d083,” is a particular outlier in both fitters at $\sim0.5$ mag brighter than expected in the best-fit or [$\Lambda{\rm CDM}$]{} cosmologies. @matheson05 found the spectrum of this object to be like that of SN 1991T, which is the archetype of over-luminous [SNe Ia]{}. This [SN Ia]{} is likely an interesting object worthy of further study and is potentially similar to a similarly super-luminous object, SN 2003fg, found in the SNLS survey [@howell06]. However, given that our sample comprises [60]{} objects, we certainly allow for the reasonable statistical possibility of a 3$\sigma$ outlier such as “d083” and thus retain it in our sample. In Fig. \[fig:mlcs\_prior\_glosz\_salt\_om\_w\] we show the 1$\sigma$, 2$\sigma$, and 3$\sigma$ probability contours for our measurement of $w$ vs. [${\Omega}_{\rm M}$]{} for ESSENCE+nearby alone, the BAO constraints from @eisenstein05, and the combination of the [SN Ia]{} and BAO constraints. Table \[tab:results\] shows the cosmological parameters $w$ and [${\Omega}_{\rm M}$]{} for each of these sets for flat models of the Universe with a constant $w$ as well as the [$\chi^2/{\rm DoF}$]{} for a concordance cosmology and the 1-D marginalized values. A [$\Lambda{\rm CDM}$]{} model of the Universe fits the MLCS2k2-analyzed ESSENCE+nearby sample with a [$\chi^2/{\rm DoF}$]{} of [$0.96$]{}and a residual standard deviation of [$0.20$]{} mag. Thus, while the estimated $w$ parameter in the constant-$w$ models is $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, a flat, $w=-1$ model of the Universe is consistent with our data. Our results from these [60]{} [SNe Ia]{} from the ESSENCE survey are consistent with the results of A06. It is reassuring that two independent teams using different telescopes and studying different regions of the sky find that [SNe Ia]{} at high redshift exhibit the same luminosity distance vs. redshift relationship. These samples strengthen and extend the evidence from [SNe Ia]{} for dark energy and, together with complementary constraints on [${\Omega}_{\rm M}$]{}, point toward simple [$\Lambda{\rm CDM}$]{} models for our Universe. Joint ESSENCE+SNLS Cosmological Constraints ------------------------------------------- A new opportunity presents itself with the release of the SNLS light curves from A06 and the light curves presented in this paper. For the first time it is possible to do a proper, self-consistent joint fit of two large, independent sets of distant [SNe Ia]{}. When fitting the SNLS [SNe Ia]{} with MLCS2k2 and the “glosz” prior we shift the assumed $A_V$ and $\Delta$ prior selection window functions by ${\Delta}z=+0.20$ to represent the greater depth of the SNLS survey. The proper way to derive this prior for SNLS would be to model the SNLS survey efficiency and and fit simulated [SNe Ia]{} with MLCS2k2 as we presented in §\[sec:selection\] for the ESSENCE survey. Similar concerns apply for possible selection effects in the heterogeneously nearby sample. Nevertheless, we believe our use of the “glosz” prior is appropriate for the low-redshift sample (where it is just the “glos” prior) and the simple extension in redshift to be a reasonable approach for the SNLS sample. The additional systematic errors introduced by this joint comparison center on the photometric calibration of the distant sample relative to the nearby [SNe Ia]{}. We estimate that uncertainty to be the same as the calibration uncertainty to the nominal Vega system used by each project: $\Delta{\rm zpt}=0.02$ mag. We have not modeled different offsets between the two data sets, but merely express the uncertainty as an additional uncertainty in our inferred cosmological parameters. This relative zeropoint uncertainty adds an additional ${\Delta}w=0.02$ to our overall systematic uncertainty on $w$. With our combined analysis, we start with the traditional [${\Omega}_{\rm M}$]{}-[${\Omega}_{\Lambda}$]{}contour plot that was the first clear evidence for dark energy. Table \[tab:joint\_results\] shows the cosmological parameters $w_0$ and [${\Omega}_{\rm M}$]{} for each of these sets for flat models of the Universe with a constant $w$ as well as the [$\chi^2/{\rm DoF}$]{} for a concordance cosmology. A [$\Lambda{\rm CDM}$]{} model of the Universe fits the SNLS+ESSENCE+nearby sample analyzed using MLCS2k2 “glosz” with a [$\chi^2/{\rm DoF}$]{} of [$0.90$]{} from [162]{} [SNe Ia]{}and a residual standard deviation of [$0.23$]{} mag. A joint analysis of the luminosity distances from the SALT fitter results in a [$\chi^2/{\rm DoF}$]{} of [$2.76$]{} from [178]{} [SNe Ia]{}and a residual standard deviation of [$0.28$]{} mag.. Fig. \[fig:joint\_mlcs\_prior\_glosz\_salt\_om\_w\] show the joint MLCS2k2 and SALT results for this joint sample. The estimated $w$ parameter in the constant-$w$ models is $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}, and a flat, $w=-1$ model of the Universe remains consistent with the current generation of [SN Ia]{} data. Joint ESSENCE+SNLS+Riess Gold Sample Cosmological Constraints ------------------------------------------------------------- In order to explore models with varying $w$, we now include the gold sample from @riess04b to extend our reach out to $z\approx1.5$. The high-quality intermediate-redshift samples of the ESSENCE and SNLS surveys provide an excellent complement to the high-redshift [SNe Ia]{}in this set. The heterogeneous nature of the collection of [SNe Ia]{} in the gold sample makes it beyond the scope of this paper to produce definite estimates of the systematic errors that result from including this additional set, but it is tempting to add these [SNe Ia]{} and examine the new constraints on cosmological parameters. We used the 39 nearby [SNe Ia]{} in common between the nearby [SN Ia]{} sample we discuss here and the gold sample to normalize the luminosity distances between the two sets. To avoid double-counting of [SNe Ia]{} in this joint analysis, we then drop the nearby [SNe Ia]{} from the gold sample and use only the nearby [SNe Ia]{} fit in this paper. We first compute the [${\Omega}_{\rm M}$]{}-[${\Omega}_{\Lambda}$]{} contours to update the case for dark energy from [SNe Ia]{}. Fig. \[fig:OMOL\_OMw\_joint\_riess04\] represents the most stringent demonstration to date of the existence of dark energy. The [SNe Ia]{} data alone rule out an empty Universe at $4.5$ $\sigma$, an ([${\Omega}_{\rm M}$]{}, [${\Omega}_{\Lambda}$]{}) = $(0.3, 0)$ Universe at $10$ $\sigma$, and an ([${\Omega}_{\rm M}$]{}, [${\Omega}_{\Lambda}$]{}) = $(1, 0)$ $\sigma$ Universe at $>20$ $\sigma$. The joint constraints on constant-$w$ models from this full set are $w=-1.09^{+0.09}_{-0.10}$. The highest-redshift data do not noticeably improve constraints for these models over the set of intermediate-redshift [SNe Ia]{} from ESSENCE+SNLS. It is for models with variable $w$ that the high-redshift data summarized by @riess04b provide the most utility. We here provide the global constraints on models characterized by $w=w_0+w_a(1-a)$ [@linder03; @albrecht06]. Using the BAO constraints on variable $w$ models would require integration from $z=0.35$ to $z\sim1089$ and the corresponding assumption that $w=w_0+w_a(1-a)$ is the proper parameterization over this stretch. If one is willing to make this assumption, then BAO+CMB already places significant constraints on the allowed $(w_0,w_a)$ parameter space. However, given that our multi-variable parameterizations of $w$ are arbitrary models with no clear theoretical motivation, we instead choose to regard $w=w_0+w_a(1-a)$ as a local expansion valid out to a redshift of $\sim2$ but not necessarily to $z\sim1089$. We then explicitly assume [${\Omega}_{\rm M}$]{}$=0.27\pm0.03$. Fig. \[fig:w0wa\_joint\_riess04\] shows the $(w_0, w_a)$ contours for this combined analysis. These constraints represent the advances of our understanding of dark energy. It is clear that work remains to constrain models of variable $w$. Conclusions {#sec:conclusions} =========== The ESSENCE survey has successfully discovered, confirmed, and followed 119 [SNe Ia]{} in our first four years of operation. We presented results from an analysis of [60]{} of those [SNe Ia]{} here, chosen so as to maximize insight while minimizing susceptibility to systematic errors. We have expended considerable effort to make quantitative estimates of various sources of systematic uncertainty that may afflict the ESSENCE results; of these, host-galaxy extinction and a potential local velocity monopole are currently the predominant concerns. We are working to devise ways to better estimate extinction, using both spectroscopic and photometric observations. Ideally, we would use all available information to arrive at an extinction prior customized for each supernova (e.g., different priors for elliptical and spiral host galaxies), rather than applying a single prior to the collection of all light curves. The ESSENCE photometric calibration uncertainties will be reduced by an intensive calibration campaign this fall on the CTIO 4-m telescope in conjunction with the improved calibration of the SDSS southern stripe from the SDSS II project [@frieman04; @dilday05]. We hope to reduce our overall systematic uncertainty to the 5% level through this improved photometric calibration and an improved external nearby [SN Ia]{} sample from KAIT, the Nearby Supernova Factory, CfA, SDSS II, and the Carnegie SN Program to reduce our systematic sensitivity to a potential velocity monopole in the local [SN Ia]{} sample. Combining our [SN Ia]{} observations with the BAO results of @eisenstein05 we find that a fit to a constant-$w$, flat-Universe model to our data finds an estimated parameter value of $w=$[$-1.05^{+0.13}_{-0.12}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{} with a [$\chi^2/{\rm DoF}$]{}$=$[$0.96$]{} using our full set analyzed with the MLCS2k2 fitter of @jha06c. A $w=-1$, flat-Universe model is consistent with our data. A combined analysis of ESSENCE+SNLS+nearby results in a estimated mean parameter of $w=$[$-1.07^{+0.09}_{-0.09}~{\rm (stat}~1\sigma{)} \pm 0.13~{\rm (sys)}$]{}. We have no reliable estimate of the systematic effects involving the SALT fitter but take our general systematic uncertainty of [$0.13$]{} as representative of the issues currently confronting supernova cosmology. The statistical increase from the [SNe Ia]{} from the entire 6-year ESSENCE data set plus improved photometric calibration of our detector and photometric measurements will reduce our statistical uncertainty to 7% and, together with an improvement in our systematic uncertainties to the level 5%, allow us to surpass our goal of a 10% measurement of a constant $w$ in a flat Universe. Establishing the nature of dark energy is among the most pressing issues in the physical sciences today. The emerging impression that the equation-of-state parameter is close to $w=-1$ makes it difficult to determine whether the underlying physics arises in the particle physics sector or from the classical cosmological constant of general relativity. A value of $w=-1$ is perhaps the least informative possible outcome. In our view, this state of affairs motivates a vigorous effort to push the observational constraints to improve our sensitivity to the value and derivative of $w$ and strongly encourages searching for other indications of new physics, as we well may need another piece to solve the puzzle handed us by Nature. Acknowledgments =============== Based in part on observations obtained at the Cerro Tololo Inter-American Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc. (AURA) under cooperative agreement with the National Science Foundation (NSF); the European Southern Observatory, Chile (ESO Programmes 170.A-0519 and 176.A-0319); the Gemini Observatory, which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the NSF (United States), the Particle Physics and Astronomy Research Council (United Kingdom), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), CNPq (Brazil), and CONICET (Argentina) (Programs GN-2002B-Q-14, GS-2003B-Q-11, GN-2003B-Q-14, GS-2004B-Q-4, GN-2004B-Q-6, GS-2005B-Q-31, GN-2005B-Q-35); the Magellan Telescopes at Las Campanas Observatory; the MMT Observatory, a joint facility of the Smithsonian Institution and the University of Arizona; and the F. L. Whipple Observatory, which is operated by the Smithsonian Astrophysical Observatory. Some of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California, and the National Aeronautics and Space Administration; the Observatory was made possible by the generous financial support of the W. M. Keck Foundation. The ESSENCE survey team is very grateful to the scientific and technical staff at the observatories we have been privileged to use. [*Facilities:*]{} , , , , , , , , . The survey is supported by the US National Science Foundation through grants AST-0443378, AST-057475, and AST-0607485. The Dark Cosmology Centre is funded by the Danish National Research Foundation. SJ thanks the Stanford Linear Accelerator Center for support via a Panofsky Fellowship. AR thanks the NOAO Goldberg fellowship program for its support. PMG is supported in part by NASA Long-Term Astrophysics Grant NAG5-9364 and NASA/HST Grant GO-09860. RPK enjoy support from AST06-06772 and PHY99-07949 to the Kavli Institute for Theoretical Physics. AC acknoledges the support of CONICYT, Chile, under grants FONDECYT 1051061 and FONDAP Center for Astrophysics 15010003. Our project was made possible by the survey program administered by NOAO, and builds upon the data reduction pipeline developed by the SuperMacho collaboration. IRAF is distributed by the National Optical Astronomy Observatory, which is operated by AURA under cooperative agreement with the NSF. The data analysis in this paper has made extensive use of the Hydra computer cluster run by the Computation Facility at the Harvard-Smithsonian Center for Astrophysics. We also acknowledge the support of Harvard University. This paper is dedicated to the memory of our friend and colleague Bob Schommer. [^1]: <http://www.ctio.noao.edu/essence/> [^2]: <http://snls.in2p3.fr/conf/release/> [^3]: <http://www.ctio.noao.edu/essence/> [^4]: <http://snls.in2p3.fr/conf/release/> [^5]: <http://qold.astro.utoronto.ca/conley/simple_cosfitter/> [^6]: <http://www.ctio.noao.edu/essence/>

--- author: - 'Takuya Sugimoto$^1$, Daiki Ootsuki$^2$, Takashi Mizokawa$^{1,2}$' title: ' Impact of Local Lattice Disorder on Spin and Orbital Orders in Ca$_{2-x}$Sr$_x$RuO$_4$ ' --- Introduction ============ The layered perovskite Ca$_{2-x}$Sr$_x$RuO$_4$ (CSRO) system has been attracting considerable interest due to the interesting evolution from the spin-triplet superconducting state in Sr$_2$RuO$_4$ to the Mott insulating state in Ca$_2$RuO$_4$ [@Maeno94; @Nakatsuji00a; @Nakatsuji00b; @Imada98]. The structural phase diagram of CSRO exhibits an interesting interplay between titling, rotation, and Jahn-Teller distortion of RuO$_6$ octahedron [@Friedt01]. The magnetic and electronic properties of CSRO strongly correlate with the structural distortions. The Mott transition at $x = 0.0$ (Ca$_2$RuO$_4$) is accompanied by the Ru 4$d$ orbital change due to the Jahn-Teller distortion [@Mizokawa01; @Mizokawa04]. For $x \leq 0.2$, CSRO is an antiferromagnetic insulator at low temperature due to the Jahn-Teller driven compression of RuO$_6$ octahedron along the $c$-axis. For $0.2 \leq x \leq 0.5$, the tilting of RuO$_6$ octahedron provides orthorhombic distortion, and the magnetic susceptibility shows a heavy Fermion behavior at low temperature. The orthorhombic distortion seems to suppress the ferromagnetic and/or small-$q$ antiferromagnetic fluctuation while it still enhances the mass renormalization towards $x= 0.2$. Although the mass enhancement around $x= 0.2$ is claimed to be explained by orbital selective Mott transition [@Anisimov02], existence of orbital selective Mott transition in multi-band Hubbard models depends on the details of parameters in the Hubbard Hamiltonians [@Liebsch03; @Koga04]. It is still controversial whether the orbital selective Mott transition of the multi-band Ru 4$d$ electrons is relevant for the electronic phase diagram of CSRO or not. As for the tiny Sr substitution in the Mott insulating state of Ca$_2$RuO$_4$, the transport properties are dramatically changed by the Sr doping [@Nakatsuji04]. Very recently, a systematic $\mu$SR study has revealed that static antiferromagnetic order exists at low temperature even in the Sr-rich region ($1.5 \leq x \leq 2.0$) of its phase diagram [@Carlo12]. This indicates that the local distortion of RuO$_6$ octahedron introduced by the Ca substitution plays important roles to stabilize the antiferromagnetic state. On the other hand, in the Ca-rich region ($0.0 \leq x \leq 0.5$), the Sr-substitution reduces the magnitude of the Jahn-Teller distortion, tilting, and rotation of the RuO$_6$ octahedron in Ca$_2$RuO$_4$ and, consequently, destroys the spin and orbital orders of Ca$_2$RuO$_4$. In order to gain deeper understandings of the phase diagram, it is important to study the Ru 4$d$ spin-orbital states using a realistic model in which the effects of spin-orbit interaction and lattice distortions are considered. In this paper, we investigate the Sr or Ca doping effects on the electronic structure of CSRO system at both ends of its phase diagram (which are Sr$_2$RuO$_4$ and Ca$_2$RuO$_4$) by means of unrestricted Hartree-Fock (HF) calculation, which includes the spin-orbit interaction and lattice distortion induced by the chemical substitution. Method of calculation ===================== We use the multiband $d$-$p$ model where full degeneracy of the Ru $4d$ orbitals and the O $2p$ orbitals are taken into account. The Hamiltonian is given by $$\begin{aligned} \hat{\mathrsfs{H}} =& \hat{\mathrsfs{H}}_p + \hat{\mathrsfs{H}}_d + \hat{\mathrsfs{H}}_{pd} \notag \\ \hat{\mathrsfs{H}}_p =& \sum_{kl\sigma} \epsilon^p_k p^{\dagger}_{kl\sigma} p_{kl\sigma} + \sum_{kll'\sigma}V^{pp}_{kll'}p^{\dagger}_{kl\sigma} p_{kl'\sigma} + \text{h.c.} \notag \\ \hat{\mathrsfs{H}}_d =& \epsilon^0_d \sum_{i \alpha m \sigma} d^{\dagger}_{i \alpha m \sigma}d_{i \alpha m \sigma} + \sum_{i \alpha mm'\sigma \sigma'}h_{mm'\sigma \sigma'}d^{\dagger}_{i \alpha m \sigma}d_{i \alpha m' \sigma'} \notag \\ &+ u\sum_{i \alpha m} d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow} \notag \\ & +u'\sum_{i \alpha m m'} d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow} \notag \\ &+ (u'-j)\sum_{i \alpha mm'\sigma} d^{\dagger}_{i \alpha m \sigma}d_{i \alpha m \sigma} d^{\dagger}_{i \alpha m' \sigma}d_{i \alpha m' \sigma} \notag \\ &+ j \sum_{i \alpha mm'} d^{\dagger}_{i \alpha m \uparrow} d_{i \alpha m' \uparrow} d^{\dagger}_{i \alpha m' \downarrow}d_{i \alpha m \downarrow} \notag \\ &+ j' \sum_{i \alpha mm'} d^{\dagger}_{i \alpha m \uparrow} d_{i \alpha m' \uparrow} d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m' \downarrow} \notag \\ \hat{\mathrsfs{H}}_{pd} =& \sum_{kml\sigma}V^{pd}_{kml}d^{\dagger}_{km\sigma} p_{kl\sigma} + \text{h.c.} \notag \end{aligned}$$ Here, $d^{\dagger}_{i \alpha m \sigma}$ are creation operators for the Ru $4d$ electrons at site $\alpha$ of the $i^{\text{th}}$ unit cell and $d^{\dagger}_{km\sigma}$ and $p^{\dagger}_{kl\sigma}$ are creation operators for Bloch electrons which are constructed from the $m^{\text{th}}$ component of the $4d$ orbitals and from the $l^{\text{th}}$ component of the O $2p$ orbitals, respectively, with wave vector $\bm{k}$. The matrix $h_{mm'\sigma \sigma'}$ denotes the spin-orbit interaction and the effects of crystal field splitting. The magnitude of the spin-orbit interaction for the Ru $4d$ orbital is fixed as 0.15 eV. The transfer integrals between the O $2p$ orbitals $V^{pp}_{kll'}$ are given by Slater-Koster parameters $(pp\sigma)$ and $(pp\pi)$ which are fixed at $0.60$ eV and $-0.15$ eV respectively. The transfer integrals between the Ru $4d$ and O $2p$ orbitals $V^{pd}_{kml}$ are represented by $(pd\pi)$ and $(pd\sigma)$. They are fixed as $(pd\sigma) = -2.8$ eV and $(pd\pi) = 1.26$ eV for the longer in-plane Ru-O bond of Ca$_2$RuO$_4$ whereas $(pd\sigma) = -3.4$ eV and $(pd\pi) = 1.53$ eV for the shorter in-plane Ru-O bond of Sr$_2$RuO$_4$. The summary of the material-dependent parameters are shown in Table \[parameters\]. The tilting of the RuO$_6$ octahedron is included for Ca$_2$RuO$_4$. The distortion parameter $\delta_{\text{JT}}$ is defined as $\delta_{\text{JT}} = d_{\text{apical}}/d_{\text{in-plane}}$ which is the ratio between the apical and in-plane Ru-O bond distances. The distortion parameter $\delta_{\text{JT}}$ is utilized to express the elongation/compression of RuO$_6$ octahedron as in Fig. \[fig:deltaJT\](a). In the Sr-rich (Ca-rich) region, $\delta_{\text{JT}} = 1.07$ (0.95) for the host lattice and $\delta_{\text{JT}} = 0.95$ (1.07) for the locally distorted site. When the RuO$_6$ octahedron is distorted, the transfer integrals are scaled by Harrison’s rule. The intra-atomic Coulomb interactions between Ru $4d$ electrons are given by Kanamori parameters. They are fixed as $u = u'+ j + j' = 3.0$ eV and $j = j' = 0.5$ eV. The charge-transfer energy $\Delta$ (fixed as $-0.4$ eV) is defined by $\epsilon_d -\epsilon_p +4U$, where $\epsilon_d$ and $\epsilon_p$ are the energies of the bare Ru $4d$ and O $2p$ orbitals and $U [=u-(20/9)j]$ is the multiplet-averaged $d$-$d$ Coulomb interaction fixed at 1.89 eV. We set the $8\times 8$ supercell with periodic boundary conditions and put the Ru $4d$ and O $2p$ electrons on its each site. The local distortion is introduced at one Ru site of the supercell as shown in Figure \[fig:deltaJT\](b). The total number of electron in the supercell is 1792 (256 Ru $4d$ and 1536 O $2p$ electrons). The HF mean-field treatment is applied to the two-body part in $\hat{\mathrsfs{H}}_d$ by replacing its average values, for instance, $$\begin{aligned} u \sum_{i \alpha m} & d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} d^{\dagger}_{i \alpha m \downarrow} d_{i \alpha m \downarrow} \notag \\ \rightarrow &\notag \\ & u\sum_{i \alpha m}\langle d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} \rangle d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow} \notag \\ &+ u\sum_{i \alpha m} d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} \langle d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow}\rangle \notag \\ &- u\sum_{i \alpha m} \langle d^{\dagger}_{i \alpha m \uparrow}d_{i \alpha m \uparrow} \rangle \langle d^{\dagger}_{i \alpha m \downarrow}d_{i \alpha m \downarrow}\rangle \notag\end{aligned}$$ In this Hartree-Fock calculation, we input the initial values of the order parameters such as $\langle d^{\dagger}d \rangle$ and diagonalize the mean-field Hamiltonian to get a set of eigen functions. Then the order parameters can be calculated using the obtained eigen functions. This self-consistency cycle is iterated until the successive difference of all the order parameters converge less than $10^{-4}$. Results and Discussion ====================== We calculate the following four cases to see the doping effects. For the Ca-rich region, we calculate the orbital population of Ca$_2$RuO$_4$ and Ca$_{2-x}$Sr$_{x}$RuO$_4$ with a single-site distortion by the tiny Sr doping (let us label these two cases as case (i) and (ii), respectively). For the Sr-rich region, we calculate Sr$_2$RuO$_4$ and Ca$_{x}$Sr$_{2-x}$RuO$_4$ with a single-site distortion by the Ca doping (let us label these two cases as case (iii) and (iv), respectively). Since we set the $8\times8$ supercell, the ratio of one-site transposition is equal to $1/64 \simeq 0.0156$. In terms of the doping amount in this CSRO system, $1.56\%$ is equivalent to $x=0.0312$ in Ca$_{2-x}$Sr$_{x}$RuO$_4$ and Ca$_{x}$Sr$_{2-x}$RuO$_4$, respectively. Ca-rich region -------------- The present HF calculation using the reasonable parameter set predicts that Ca$_2$RuO$_4$ is an antiferromagnetic insulator. The compression of the RuO$_6$ octahedra stabilizes the Ru 4$d$ $xy$ orbital compared to the Ru 4$d$ $yz/zx$ orbitals (Jahn-Teller type energy splitting). The combination of the Jahn-Teller type energy splitting and the Ru 4$d$ spin-orbit interaction can provide an interesting magnetic anisotropy. The antiferromagnetic insulating state with the in-plane ($x$ or $y$ axis) spin direction is lower in energy than that with the out-of-plane ($z$ axis) spin direction. The energy difference is 37 meV per Ru site for the present parameter set. The expectation value of $d^{\dagger}d$ for each orbital and spin component of the Ru $4d$ $t_{2g}$ states for Ca$_2$RuO$_4$ \[case (i)\] is shown in Table \[tab:orbpop2\] for the in-plane and out-of-plane antiferromagnetic states. In the out-of-plane case, the complex orbitals with type of $yz \pm izx$ are unoccupied \[namely, occupied by the two Ru 4$d$ $t_{2g}$ holes of the Ru$^{4+}$ ($d^4$) configuration\] while the $xy$ orbitals are almost fully occupied by the Ru 4$d$ $t_{2g}$ electrons. Consequently the orbital angular momentum along the $z$-axis is formed to align the spin moment along the $z$-axis. As for the in-plane case, the complex orbitals with type of $xy \pm izx$ or $xy \pm iyz$ are unoccupied to give orbital angular momentum in the $xy$-plane. Assuming that one of the RuO$_6$ octahedra is elongated due to the tiny Sr doping in the case (ii), the orbital population of the elongated site would be affected due to the reverse of the Jahn-Teller energy splitting between the $xy$ and $yz/zx$ orbitals. Interestingly, the orbital population of the elongated site is dramatically changed for the in-plane antiferromagnetic state whereas the impact of the elongation is limited for the out-of-plane antiferromagnetic state. As shown in Table \[tab:orbpop2\], the orbital population of the elongated site is almost the same as that of the compressed sites for the out-of-plane antiferromagnetic state. On the other hand, the $xy$ orbitals accommodate more holes at the elongated site for the in-plane antiferromagnetic state. The impact of local reverse of the Jahn-Teller energy splitting strongly depends on the global spin direction due to the strong Ru 4$d$ spin-orbit coupling. Since the in-plane antiferromagnetic state is realized in Ca$_2$RuO$_4$, the present calculation indicates that the local elongation of the RuO$_6$ octahedron produces the Ru 4$d$ $xy$ hole which can affect the global spin and orbital orders of Ca$_2$RuO$_4$. The density of states (DOS) calculated for the cases (i) and (ii) (which are Ca$_2$RuO$_4$ and Ca$_{2-x}$Sr$_{x}$RuO$_4$ at $x=0.0312$) with the in-plane antiferromagnetic states are shown in Figs. \[fig:CaRich\](a) and (b), respectively. One can see that the Mott gap of Ca$_2$RuO$_4$ becomes narrower by the local elongation of RuO$_6$ octahedron ($\delta_{\text{JT}}=1.07$) induced by the tiny Sr doping. The reduction of the Mott gap is related to the local orbital change induced by the reverse of the Jahn-Teller energy splitting. Sr-rich region -------------- The present HF calculation using the reasonable parameter set predicts that Sr$_2$RuO$_4$ is a paramagnetic or ferromagnetic metal. For the present parameter set, the ferromagnetic metallic state is slightly lower in energy than the paramagnetic metallic state, and the energy difference is about 0.4 meV per Ru site. In the HF approximation, stability of ferromagnetic or antiferromagnetic states tends to be overestimated. Therefore, the present calculation indicates that the parameter set is reasonable to analyze the paramagnetic metallic state of Sr$_2$RuO$_4$. The expectation value of $d^{\dagger}d$ for each orbital and spin component of the Ru $4d$ $t_{2g}$ states for Sr$_2$RuO$_4$ \[case (iii)\] is shown in Table \[tab:orbpop\] for the paramagnetic metallic state and the ferromagnetic metallic state. In the paramagnetic state, the spin-up and spin-down states have the same population for the $xy$, $yx$, and $zx$ orbitals, respectively. In the ferromagnetic state, the spin polarization for the $yz/zx$ orbitals is larger than that of the $xy$ orbital, indicating that the $yz/zx$ states stabilized by the elongation of the RuO$_6$ octahedra are playing important roles to provide the ferromagnetic interaction. As for the effect of tiny Ca doping to Sr$_2$RuO$_4$ \[case (iv)\], it is assumed that one of the RuO$_6$ octahedra is compressed due to the Ca doping. The local compression of the RuO$_6$ octahedron changes the stability of the paramagnetic state and the nature of the magnetic state. The paramagnetic metallic state becomes unstable and only the ferromagnetic and/or antiferromagnetic states are obtained as stable solutions. The $xy$ orbitals are more occupied at the compressed site to induce antiferromagnetic superexchange interaction. The global antiferromagnetic state is stabilized by the local orbital change. The spin direction is intermediate between the in-plane and out-of-plane directions probably due to the competition between the Ru 4$d$ spin-orbit interaction and the local Jahn-Teller splitting of the Ru 4$d$ levels. In the stable magnetic state, the in-plane components of the Ru 4$d$ spins are ferromagnetically aligned whereas the out-of-plane components are antiferromagnetically arranged. The DOS calculated for the cases (iii) and (iv) (which are Sr$_2$RuO$_4$ and Ca$_{x}$Sr$_{2-x}$RuO$_4$ at $x=0.0312$) are shown in Figs. \[fig:SrRich\](a) and (b), respectively. The local compression of RuO$_6$ octahedron by the tiny Ca doping induces the small band gap at the Fermi level, which can be assigned to the magnetic ordering. Conclusion ========== Investigating the electronic structure of CSRO by unrestricted HF calculation, we find the effects of local distortions on spin and orbital orders of the Ru $4d$ $t_{2g}$ states. As for the Ca-rich region, we find that the antiferromagnetism survives when the single-site distortion in Ca$_2$RuO$_4$ does carry on. The orbital state is locally disturbed for the in-plane antiferromagnetic state resulting in the reduction of the Mott gap. As for the Sr-rich region, on the other hand, the single-site distortion in Sr$_2$RuO$_4$ changes the entire system to the ferromagnetic/antiferromagnetic state from the paramagnetic state. The present study captures the trend of effects on Sr$_2$RuO$_4$ by the tiny amount of Ca doping, which could explain the recent $\mu$SR study [@Carlo12]. Acknowledgements {#acknowledgements .unnumbered} ================ The authors would like to thank Profs. Y. J. Uemura and N. L. Saini for fruitful discussions. [99]{} Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fujita, J. G. Bednorz, and F. Lichtenberg: Nature [**372**]{} (1994) 532. S. Nakatsuji and Y. Maeno: Phys. Rev. Lett. [**84**]{} (2000) 2666. S. Nakatsuji and Y. Maeno: Phys. Rev. B [**62**]{} (2000) 6458. M. Imada, A. Fujimori, and Y. Tokura: Rev. Mod. Phys. [**70**]{} (1998) 1039. O. Friedt, M. Braden, G. Andre, P. Adelmann, S. Nakatsuji, and Y. Maeno: Phys. Rev. B [**63**]{} (2001) 174432. T. Mizokawa, L. H. Tjeng, G. A. Sawatzky, G. Ghiringhelli, O. Tjernberg, N. B. Brookes, H. Fukazawa, S. Nakatsuji, and Y. Maeno: Phys. Rev. Lett. [**87**]{} (2001) 077202. T. Mizokawa, L. H. Tjeng, H.-J. Lin, C. T. Chen, S. Schuppler, S. Nakatsuji, H. Fukazawa, and Y. Maeno: Phys. Rev. B [**69**]{} (2004) 132410. V. I. Anisimov, I. A. Nekrasov, D. E. Kondakov, T. M. Rice, and M. Sigrist: Euro. Phys. J. B [**25**]{} (2002) 191. A. Liebsch: Phys. Rev. Lett. [**90**]{} (2003) 096401. A. Koga, N. Kawakami, T. M. Rice, and M. Sigrist: Phys. Rev. Lett. [**92**]{} (2004) 216402. S. Nakatsuji, V. Dobrosavljevic, D. Tanaskovic, M. Minakata, H. Fukazawa, and Y. Maeno: Phys. Rev. Lett. [**93**]{} (2004) 146401. J. P. Carlo, T. Goko, I. M. Gat-Malureanu, P. L. Russo, A. T. Savici, A. A. Aczel, G. J. MacDougall, J. A. Rodriguez, T. J. Williams, G. M. Luke, C. R. Wiebe, Y. Yoshida, S. Nakatsuji, Y. Maeno, T. Taniguchi, and Y. J. Uemura: Nature Materials [**11**]{} (2012) 323. T. Mizokawa and A. Fujimori: Phys. Rev. B [**54**]{} (1996) 5368. Tables:\ \ ($pd\sigma$) for in-plane Ru-O bond tilting angle $\delta_{\text{JT}}\;\;\;$ --------------- ------------------------------------- --------------- ---------------------------- Sr$_2$RuO$_4$ $-3.4$ eV 0 deg. 1.07 Ca$_2$RuO$_4$ $-2.8$ eV 12.5 deg. 0.95 : Parameter sets for Sr$_2$RuO$_4$ and Ca$_2$RuO$_4$.[]{data-label="parameters"} \ \ case (i) Ca$_2$RuO$_4$ ----------------------------------- -------------- ---------------- -------------- ---------------- -------------- ---------------- -- in-plane ($xy$) AFM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ 0.95 0.95 0.53 0.76 0.52 0.77 out-of-plane ($z$) AFM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ 0.98 0.98 0.98 0.28 0.99 0.26 case (ii) Ca$_{2-x}$Sr$_x$RuO$_4$ in-plane ($xy$) AFM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ compressed site (Ca site) 0.97 0.96 0.64 0.64 0.64 0.64 in-plane ($xy$) AFM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ elongated site (Sr site) 0.66 0.66 0.64 0.64 0.96 0.96 out-of-plane ($z$) AFM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ compressed site (Ca site) 0.98 0.98 0.98 0.28 0.99 0.26 out-of-plane ($z$) AFM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ elongated site (Sr site) 0.99 0.98 0.99 0.28 0.99 0.26 : Expectation values of $<d^{\dagger}_{m\sigma}d_{m\sigma}>$ with $m=xy, yz, zx$ and $\sigma =\uparrow, \downarrow$ in the Ru $4d$ $t_{2g}$ states for case (i) and case (ii).[]{data-label="tab:orbpop2"} \ \ case (iii) Sr$_2$RuO$_4$ ---------------------------------------------- -------------- ---------------- -------------- ---------------- -------------- ---------------- -- PM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ 0.71 0.71 0.88 0.88 0.88 0.88 FM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ 0.67 0.77 0.80 0.94 0.80 0.94 case (iv) Ca$_x$Sr$_{2-x}$RuO$_4$ (Ca-doped) AFM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ elongated site (Sr site) 0.60 0.85 0.85 0.88 0.86 0.87 AFM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ compressed site (Ca site) 0.79 0.94 0.61 0.89 0.61 0.89 FM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ elongated site (Sr site) 0.67 0.78 0.80 0.94 0.80 0.94 FM $xy\uparrow$ $xy\downarrow$ $yz\uparrow$ $yz\downarrow$ $zx\uparrow$ $zx\downarrow$ compressed site (Ca site) 0.88 0.91 0.68 0.81 0.68 0.81 : Expectation values of $<d^{\dagger}_{m\sigma}d_{m\sigma}>$ with $m=xy, yz, zx$ and $\sigma =\uparrow, \downarrow$ in the Ru $4d$ $t_{2g}$ states for case (iii) and case (iv).[]{data-label="tab:orbpop"} Figure captions:\ Figure 1:\ (Color online) Schematic diagram of (a) RuO$_6$ octahedron with the definition of $\delta_{\text{JT}}$ and (b) the supercell with a single-site distortion.\ \ Figure 2:\ (Color online) DOS (calculated with the Ru 4$d$ spin-orbit interaction) of (a) Ca$_2$RuO$_4$ and (b) Ca$_{2-x}$Sr$_{x}$RuO$_4$ at $x=0.0312$.\ \ Figure 3:\ (Color online) DOS (calculated with the Ru 4$d$ spin-orbit interaction) of (a) Sr$_2$RuO$_4$ and (b) Ca$_{x}$Sr$_{2-x}$RuO$_4$ at $x=0.0312$. {width="90.00000%"} \[fig:deltaJT\] {width="90.00000%"} \[fig:CaRich\] {width="90.00000%"} \[fig:SrRich\]

--- abstract: | Given a set of points, we define a minimum Steiner point tree to be a tree interconnecting these points and possibly some additional points such that the length of every edge is at most 1 and the number of additional points is minimized. We propose using Steiner minimal trees to approximate minimum Steiner point trees. It is shown that in arbitrary metric spaces this gives a performance difference of at most $2n-4$, where $n$ is the number of terminals. We show that this difference is best possible in the Euclidean plane, but not in Minkowski planes with parallelogram unit balls. We also introduce a new canonical form for minimum Steiner point trees in the Euclidean plane; this demonstrates that minimum Steiner point trees are shortest total length trees with a certain discrete-edge-length condition.\ *Keywords: minimum Steiner point trees, bounded edge-length, Minkowski planes* author: - 'M. Brazil' - 'C. J. Ras' - 'D. A. Thomas' title: 'Approximating Minimum Steiner Point Trees in Minkowski Planes[^1]' --- Introduction ============ Given a metric space $(S,d)$ with metric $d$, a set of points $N \subseteq S$, and an $R\in \mathbb{R}$, the *minimum Steiner point tree problem* (MSPT problem) asks for a set $U\subset S$ and a tree connecting $N\cup U$ such that no edge is longer than $R$ and $|U|$ is minimized. Clearly we may assume that $R=1$. An optimal solution is called a $d$-MSPT, or just an MSPT if the context is clear. MSPTs have applications in the deployment and augmentation of wireless sensor networks, VLSI design, and wavelength-division multiplexing networks - see for instance [@bib10; @bib8; @bib7; @bib9]. The MSPT problem was first described by Sarrafzadeh and Wong in [@bib3], where they showed that it is NP-complete in both the $\ell_1$ and $\ell_2$ metric. Consequently a fair amount of research has been directed towards finding good heuristics. In [@bib4] the minimum spanning tree (MST) heuristic was introduced (note that there they refer to the MSPT problem as the *Steiner tree problem with minimum number of Steiner points and bounded edge length*, or STP-MSPBEL). This heuristic simply subdivides all edges of an MST that are longer than one unit, resulting in an approximate MSPT solution within polynomial time. Mandoiu and Zelikovsky [@bib6] prove that, in any metric space, the performance ratio of the MST heuristic is always one less than the maximum possible degree of a minimum-degree MST spanning points from the space. This gives an approximation ratio of four in the Euclidean plane and three in the rectilinear plane. Chen et al. [@bib5] provide an improved approximation scheme, partly based on the MST heuristic, which has a performance ratio of three in the Euclidean plane. The MSPT problem may be seen as a variant of the classical Steiner tree problem, which asks for a shortest tree interconnecting $N\subseteq S$ where any number of additional points may be introduced. An optimal solution to this problem is called a Steiner minimal tree ($d$-SMT or just SMT). As $R$ tends to zero an SMT with subdivided edges becomes an optimal solution to the MSPT problem. This leads us to the question: would the SMT approximation for the MSPT problem be a practical and accurate heuristic? Certainly we do not have effective algorithms for calculating SMT’s in every metric, in fact the problem is NP-hard. However, in the Euclidean plane and other fixed orientation metrics, Warme, Winter, and Zachariasen [@bib1; @bib2] have developed practical, fast and optimal SMT algorithms, namely the GeoSteiner algorithms. These algorithms can comfortably solve most instances of up to a few thousand uniformly distributed terminals. However, as should be expected, it is possible to construct terminal-sets that take much longer to process; for instance, GeoSteiner cannot efficiently find an SMT when just one hundred terminals are located at the vertices of a regular square lattice in the Euclidean plane (although these instances can be solved in polynomial time by the algorithms of Brazil et al. [@Brazil2]). In this paper we define and analyze the *SMT heuristic* for MSPTs. We provide a small upper bound (in terms of $|N|$) for the performance difference of the SMT heuristic in any normed plane, and show that this bound is best possible in the Euclidean plane. We then show that, in the special case $|N|=3$, the upper bound is tight in a Minkowski plane with unit ball $B$ if and only if $B$ is not a parallelogram. For the Euclidean and rectilinear planes a brief comparison between the SMT heuristic and current best possible heuristics is given. Then we prove that the performance ratio of the SMT heuristic improves as $R$ decreases (or equivalently, as the terminals become further apart). This paper also explores the possibility of restating the Euclidean MSPT problem in terms of shortest total length, leading to a new MSPT canonical form. Finally, we state a number of strong conjectures on the relationship between the Steiner tree problem and the MSPT problem. Preliminaries ============= Let $(S,d)$ be a metric space with metric $d$, and consider a set $N \subseteq S$. The *Steiner tree problem* asks for a shortest tree interconnecting $N$, where extra nodes $W \subset S$ are introduced if they reduce the total length. Introducing degree-one or degree-two nodes will not reduce total length, henceforth for the Steiner tree problem we assume all added nodes are of degree at least three. The nodes in $N$ are called *terminal points* and the nodes in $W$ are called *Steiner points*. In general metric spaces there may be instances of the MSPT problem that have no solution; consider, for instance, the case when $N=S$ and $\min\{d(x,y):x,y\in S\}>1$. Henceforth we will assume the following: $S=\mathbb{R}^2$, $\vert N\vert$ is finite, and $d$ is a norm. In other words, we will only be considering the finite MSPT problem in *Minkowski planes*. In our discussions we distinguish between the concept of a *free node* and an *embedded node*. In other words any tree may be considered as a topological graph structure only, or as an embedded network. Embedded nodes are denoted by bold letters (as is common when representing vectors). An embedded set of terminals *admits* a tree with property $P$ if there exists a tree $T$ interconnecting the terminals such that $T$ has property $P$. Two standard techniques for shortening an embedded tree are *splitting* and Steiner point *displacements*. To *split* a node $v$ one disconnects two or more of the edges at $v$ and connects them instead to a new Steiner point, connected to $v$ by an extra edge. To *displace* a Steiner point one simply embeds it at any new point in the plane without changing the topology of the tree. If no shortening of a tree is possible when splitting or Steiner point displacements are allowed, then the tree is called a *Steiner tree*. Note that an SMT is always a Steiner tree. A *full Steiner tree* is a Steiner tree where every terminal is of degree one and every Steiner point is of degree three. A full Steiner tree has exactly $\vert N\vert -2$ Steiner points and $2\vert N \vert-3$ edges. A *cherry* of a full Steiner tree is the subtree induced by two terminals and their mutually adjacent Steiner point. Every full Steiner tree has at least two cherries. We refer the reader to [@bib13] and [@bib14] for more background on Steiner trees. Given two points $x,y \in S$, we denote the edge $e$ between them by $e=xy$, and we use the standard notation $\vert e \vert$ to denote $d(x,y)$. Any Steiner tree can be viewed as a candidate MSPT if we simply subdivide, or *bead*, edges that are longer than one unit. Formally, *beading* is the process whereby for every edge $e$, $\lceil \vert e \vert \rceil - 1$ equally spaced degree-two nodes lying on $e$ are included (along with the elements of $W$) in the set $U$ of extra MSPT nodes. In general, any tree can be viewed as an MSPT candidate if we partition its nodes into a set $N$ of terminals and a set $W$ of Steiner points of degree at least three, and then bead any edges that are too long. Consequently, when constructing an MSPT on a given set $N$, we are mainly concerned with finding the elements of $W$, i.e., the elements of $U$ that have degree at least three; clearly degree-one nodes will not occur in $U$ and degree-two nodes in $U$ only arise from beading. Henceforth, degree-two nodes in $U$ will not be considered as part of the topology of the MSPT. All nodes in $U$ will be referred to as *beads* and, specifically, the nodes in $W$ will be called *Steiner beads*. The procedure of constructing an SMT in order to approximate an MSPT will be referred to as the *SMT heuristic*. Let $T$ be any tree with node-set partitioned into terminals $N$ and Steiner beads $W$. Let $n=\vert N \vert$. Then $T^*$ is the tree that results by splitting nodes of $T$ until every terminal is of degree one and every Steiner bead is of degree three (i.e., $T^*$ is a full Steiner tree). New nodes are not displaced from their original positions, in other words some zero edge-lengths may be introduced and the total length of $T$ does not change. See Figure \[figureSat\] as an example; here $t$ is a degree-four terminal, $s$ is a Steiner point, and after splitting $t$ we have three zero-length edges (depicted by broken lines). ![Conversion to a full Steiner tree. \[figureSat\]](Ras01.eps) Let the edge-set of $T^*$ be $E(T^*)=\{e_1,...,e_m\}$, where $m=2n-3$. Then the *bead count* of $T$ is $\mathrm{beads}(T)=\vert U \vert= n-2+ \sum_{i=1}^m \left(\lceil \vert e_{i} \vert \rceil -1 \right)=1-n+\sum_{i=1}^m \lceil \vert e_{i} \vert \rceil$. In other words, by considering $T^*$ rather than $T$ we get a formula for $\mathrm{beads}(T)$ that does not depend on the number of Steiner beads of $T$; this formula works because every time a node is split (creating a new Steiner bead) we introduce a zero-length edge which in effect cancels the count of this Steiner bead. We can now reformulate the MSPT problem as follows. Let $N$ be a subset of $S$. Find a $W\subset S$ and a tree $T$ interconnecting $N \cup W$ such that every node in $W$ is of degree at least three and $\mathrm{beads}(T)$ is a minimum over all trees interconnecting $N$. The Upper Bound in any Normed Plane {#sec1} =================================== In this section we provide an upper bound for the performance difference of the SMT heuristic in any normed plane. Let $N$ be a set of $n$ terminals in a normed plane $(\mathbb{R}^2,d)$. We use $T_{\mathrm{opt}}$ to denote an MSPT on $N$ and $T_{S}$ to denote an SMT on $N$. We need the following lemma before we prove our main result: \[Lemma1\] If $i,k$ are real numbers then $\lceil i+k \rceil - \lceil i \rceil = \lceil k \rceil$ or $\lceil k \rceil -1$ (equivalently $\lfloor k \rfloor$ or $\lfloor k \rfloor +1$), with $\lceil i+k \rceil -\lceil i \rceil = k$ if $k$ is an integer. Suppose that $\lceil i \rceil = i +\varepsilon_i$ and $\lceil k \rceil = k + \varepsilon_k$ where $0 \leq \varepsilon_i,\varepsilon_k < 1$. Then $\lceil i+k \rceil = \lceil i \rceil + \lceil k \rceil - \lfloor \varepsilon_i + \varepsilon_k \rfloor$, from which the result follows. Suppose that $E(T_{\mathrm{opt}}^*)=\{e_1,...,e_m\}$ and $E(T_{S}^*)=\{a_{1},...,a_m\}$. Since $T_S$ is a shortest total length tree interconnecting $N$, we have: $$\begin{aligned} \sum\limits_{i=1}^m \vert a_i \vert \leq \sum\limits_{i=1}^m \vert e_i \vert \label{eq0}\end{aligned}$$ We can therefore partition the set $\{1,...,m\}$ as follows: let $\{1,...,m\}= I \cup D$ such that $\vert e_i \vert = \vert a_i \vert+p_i$ for $i \in I$ and $\vert e_i \vert = \vert a_i \vert -p_i$ for $i\in D$. Here each $p_i$ is a non-negative real number and the cardinality of $D$, but not $I$, may be zero. We further partition $I$ into $I_Z$ and $I_{Z}^{\prime}$ (where $I_Z$ may be empty) such that $i \in I_Z$ if and only if $\vert a_i \vert$ is an integer. We similarly partition $D$ into $D_Z$ and $D_{Z}^{\prime}$. From (\[eq0\]) it follows that $\sum \limits_{i\in I} p_i \geq \sum\limits_{i\in D} p_i$ - a result that is central to the proof of Proposition \[mainUpperProp\]. \[Lemma2\] If all edges of both $T_{S}$ and $T_{\mathrm{opt}}$ have integer length, then $T_{\mathrm{opt}}$ is also an SMT on $N$, and $\mathrm{beads}(T_S)=\mathrm{beads}(T_{\mathrm{opt}})$ . Let $T$ be any Steiner tree on $N$ such that all edges of $T$ have integer length, and let $L(T)$ be the total length of $T$. Then $\mathrm{beads}(T)= L(T) - n +1$. The lemma immediately follows since $L(T_{\mbox{\scriptsize opt}}) \geq L(T_S)$. \[mainUpperProp\] $\mathrm{beads}(T_S)-\mathrm{beads}(T_{\mathrm{opt}}) \leq \max\{2n-4-j, 0\}$, where $j$ is the number of integer-length edges in $E(T_{S}^*)$. $$\begin{aligned} \mathrm{beads}(T_S)-\mathrm{beads}(T_{\mbox{\scriptsize opt}}) &=& \left[1-n+\sum\limits_{i=1}^m \lceil \vert a_{i} \vert \rceil\right]-\left[1-n+\sum\limits_{i=1}^m \lceil \vert e_{i} \vert\rceil\right]\\ &=& \sum\limits_{i=1}^m \lceil \vert a_{i} \vert \rceil-\sum\limits_{i=1}^m \lceil \vert e_{i} \vert \rceil\\ &=& \sum\limits_{i \in D}\{ \lceil \vert a_{i} \vert\rceil-\lceil\vert a_{i} \vert-p_i\rceil \}- \sum\limits_{i \in I}\{ \lceil \vert a_{i} \vert+p_i\rceil-\lceil\vert a_{i} \vert\rceil \}.\\\end{aligned}$$ Using Lemma \[Lemma1\] we obtain: $$\begin{aligned} \mathrm{beads}(T_S)-\mathrm{beads}(T_{\mbox{\scriptsize opt}}) &\leq& \sum\limits_{i\in D_Z} \lfloor p_i \rfloor + \sum\limits_{i\in D_{Z}^{\prime}} \left( \lfloor p_i \rfloor+1\right) - \sum\limits_{i \in I_Z} \lceil p_i\rceil - \sum\limits_{i \in I_{Z}^{\prime}} \left(\lceil p_i\rceil -1\right) \nonumber \\ &=& \vert D_{Z}^{\prime} \vert + \vert I_{Z}^{\prime} \vert +\sum\limits_{i \in D} \lfloor p_i \rfloor - \sum\limits_{i\in I} \lceil p_i\rceil \nonumber \\ &\leq & m-j+\sum\limits_{i \in D} p_i - \sum\limits_{i \in I} \lceil p_i\rceil \label{eq1} \\ &\leq & m-j+\sum\limits_{i \in I} p_i - \sum\limits_{i \in I} \lceil p_i\rceil \nonumber \\ &\leq & m-j \label{eq2} \\ &=& 2n-3-j. \nonumber\end{aligned}$$ We now consider a number of cases showing that either (\[eq1\]) or (\[eq2\]) is a strict inequality or $\mathrm{beads}(T_S)=\mathrm{beads}(T_{\mbox{\scriptsize opt}})$. Together, these imply the statement of the lemma. <span style="font-variant:small-caps;">Case 1:</span> Suppose there exists an edge in $T_S$ that is longer than some edge in $T_{\mbox{\scriptsize opt}}$ and such that the difference between the lengths of the two edges is not an integer. Then there exists an assignment of labels $\{e_i\}$ to the edges of $T_{\mbox{\scriptsize opt}}$ and labels $\{a_i\}$ to the edges of $T_S$ such that $p_i \not\in \mathbb{Z}$ for some $i \in D$. Hence Inequality (\[eq1\]) is strict. <span style="font-variant:small-caps;">Case 2:</span> If there exists an edge in $T_S$ that is shorter than some edge in $T_{\mbox{\scriptsize opt}}$ and such that the difference between the lengths of the two edges is not an integer, then by the same argument as in Case 1, we can assume Inequality (\[eq2\]) is strict. <span style="font-variant:small-caps;">Case 3:</span> The only remaining possibility is that the difference in length between each edge in $T_S$ and each edge in $T_{\mbox{\scriptsize opt}}$ is an integer. This means there exists an $\varepsilon \in [ 0, 1)$ such that the length of every edge in both trees is an integer plus $\varepsilon$. If $\varepsilon=0$ then $\mathrm{beads}(T_S)-\mathrm{beads}(T_{\mbox{\scriptsize opt}})=0$ by Lemma \[Lemma2\]. If $\varepsilon \not= 0$ then we can move any Steiner point in $T_{\mbox{\scriptsize opt}}$ by a sufficiently small distance ($>0$) such that the length of at least one edge changes without changing the bead count of $T_{\mbox{\scriptsize opt}}$. Hence we can then apply Case 1 or 2. $\mathrm{beads}(T_S) - \mathrm{beads}(T_{\mathrm{opt}}) \leq 2n-c-3$ where $c$ is the number of full components of $T_S$. Note that every terminal $x$ of degree $\deg(x)$ is split $\deg(x)-1$ times to produce $T_{S}^*$, i.e., each terminal $x$ produces $\deg(x)-1$ zero-length edges after all splits. Clearly also $c=\sum\limits_{x \in N}\{\deg(x)-1\}+1$. \[EqCorollary\] If $T_S$ has at most one edge with non-integer length then $\mathrm{beads}(T_S)=\mathrm{beads}(T_{\mathrm{opt}})$. Du et al. [@bib5; @bib15] provide approximations for the MSPT problem that give performance ratios with upper bounds of three in the Euclidean plane and two in the rectilinear plane. Their algorithms are based on the MST heuristic and therefore run in polynomial time. If we rewrite our performance difference to get the bounded ratio $\frac{\mathrm{beads}(T_S)}{\mathrm{beads}(T_{\mathrm{opt}})} \leq 1+\frac{2n-4}{\mathrm{beads}(T_{\mathrm{opt}})}$ we see that the performance ratio of the SMT heuristic has a smaller upper bound than the heuristics of Du et al. when $\mathrm{beads}(T_{\mathrm{opt}}) > n-2$ in the Euclidean plane, and $\mathrm{beads}(T_{\mathrm{opt}}) > 2n-4$ in the rectilinear plane. Since $\mathrm{beads}(T_{\mathrm{opt}})$ increases as the minimum distance between any pair of terminals increases, we arrive at the intuitive fact that the performance of the SMT heuristic improves as the terminal configuration becomes more sparse. If $R$ was not fixed then we would arrive at the same result by decreasing $R$. During this limiting process the upper bound of the ratio $\frac{\mathrm{beads}(T_S)}{\mathrm{beads}(T_M)}$, where $T_M$ is an MST, tends towards the well-known Steiner ratio. This gives a limiting upper bound of $\frac{\mathrm{beads}(T_S)}{\mathrm{beads}(T_M)} \leq \frac{\sqrt{3}}{2}$ in the Euclidean plane, which serves as a comparison between the performances of the SMT heuristic and the standard MST heuristic. We mention once again that the SMT heuristic does not run in polynomial time. However, for $n$ up to a few thousand nodes (uniformly distributed in a square) the GeoSteiner algorithms will produce solutions in reasonable running time for the Euclidean and rectilinear plane [@bib1]. This makes the SMT heuristic a tool worthy of consideration for applications where optimization is required during an initialization process (such as deployment). In fact, one should consider this heuristic for any process where the cost benefit of a more accurate algorithm justifies a possible time delay. It should also be noted that SMTs can be approximated arbitrarily closely in polynomial time. The polynomial-time approximation scheme (PTAS) developed by Arora [@biba1; @biba2] works for any norm, and allows one to construct a solution to the SMT problem that is within a factor of $1 + \epsilon$ from optimality in polynomial time for any fixed $\epsilon > 0$. In theory this gives a good polynomial-time heuristic for the MSPT problem, obtained by replacing the SMT by its $1 + \epsilon$ approximation. There is, however, a difficulty with this approach in that the degree of the polynomial for small values of $\epsilon$ is too large to make the algorithm practical. The Euclidean Plane {#sec2} =================== The aim of this section is to show that the performance difference from Proposition \[mainUpperProp\] is best possible in the Euclidean plane. We begin with a few definitions and preliminary results. Due to minimality of total length, any two adjacent edges of a Euclidean Steiner tree meet at an angle of at least $120^\circ$. This implies that the degree of any terminal is no more than $3$, and the degree of any Steiner point is exactly $3$. Let $T$ be a full Euclidean Steiner tree on a set of embedded terminals. To *sprout* new terminals from a given terminal $\mathbf{t}$ of $T$ with incident edge $e$ one replaces $\mathbf{t}$ by a Steiner point $\mathbf{s}$ and embeds two new terminals $\mathbf{t}_1,\mathbf{t}_2$ adjacent to $\mathbf{s}$ such that the two new edges $\mathbf{st}_1$ and $\mathbf{st}_2$ each form $120^\circ$ angles with $e$ and with each other - see Figure \[figureSplit\]. We denote by $L(T)$ the total Euclidean edge length of $T$. If $N$ is a set of embedded terminals then $T_S$ will denote a Euclidean SMT on $N$ and $T_{\mathrm{opt}}$ will denote a Euclidean MSPT on $N$. As usual we let $n=\vert N \vert$. The next proposition shows that we can use sprouting to create full SMTs with any given topology. It is a fundamental result and is almost certainly known, but does not appear to have been explicitly written up in the literature before now. ![Sprouting new terminals. \[figureSplit\]](Ras02.eps) Given any full Steiner topology, there exists a set of embedded terminals $N$ such that the SMT for $N$ has the given topology and is unique. Furthermore, such trees can be explicitly constructed for any given topology. Let $G_n$ be a full Steiner topology on $n$ terminals. We will show how a suitable set of embedded terminals $N_n$ can be constructed by induction on $n$, where the inductive step involves sprouting new terminals. Note that the construction is trivial if $n=1,2$ or $3$. The inductive claim is as follows. **Claim**: For any full Steiner topology $G_i$ on $i$ terminals (with $i\geq 4$), there exists a set of embedded terminals $N_i$ and a real number $f_i >0$ such that 1. an SMT, $T_i$, for $N_i$ has topology $G_i$, and 2. if $T_i'$ is a Steiner tree for $N_i$ such that the topology of $T_i'$ is not $G_i$, then $L(T_i') - L(T_i) \geq f_i$. For the base case of the claim ($i=4$), choose $N_4$ to be the four points with coordinates $(\pm 1, \pm \sqrt{3}/2)$. It is easily checked that the SMT $T_4$ for $N_4$ has Steiner points $(\pm 1/2, 0)$ and length $5$ (see Figure \[figureBC\]). The shortest Steiner tree $T'_4$ with a different topology is full with Steiner points $(0, \pm (\sqrt{3}/2- 1/\sqrt(3)))$ and length $L(T'_4)= 3\sqrt{3}$. So we can choose $f_4= 3\sqrt{3}-5>0$. Up to relabelling of the terminals, there is only one full topology for $i=4$, so this completes the base case. ![Base case. \[figureBC\]](Ras03.eps) We now establish the inductive step for ($i=n$), where we assume that the claim holds for $i=n-1$. Given a full Steiner topology $G_n$ ($n>4$), this topology contains at least one cherry. Replacing such a cherry by a single terminal $t^*$ gives a full Steiner topology $G_{n-1}$ on $n-1$ terminals. By the inductive assumption there exists an embedded terminal set $N_{n-1}$ with unique SMT $T_{n-1}$ which has topology $G_{n-1}$ and a corresponding constant $f_{n-1} > 0$. Let $\mathbf{t}$ be the embedded terminal corresponding to $t^*$ and create a new Steiner tree as follows. We sprout new terminals $\mathbf{t}_n$ and $\mathbf{t}_{n-1}$ from $\mathbf{t}$, with $\mathbf{t}$ replaced by a Steiner point $\mathbf{s}$, such that $\vert\mathbf{st}_{n-1}\vert=\vert\mathbf{st}_n\vert= f_{n-1}/4$. Let this new tree be $T_n$ with embedded terminal set $N_n$. By construction, $T_n$ has the correct topology $G_n$. Let $T_n'$ be any Steiner tree (but not necessarily an SMT) on $N_n$ with topology *not* $G_n$. Suppose we collapse $\mathbf{t}_n$ and $\mathbf{t}_{n-1}$ onto the point $\mathbf{s}$ (fixing all other nodes in the network), and consider the topology $G$ of the resulting network. $G$ is a topology on $n-1$ terminals, but may be different from $G_{n-1}$, indeed $G$ is not necessarily a tree. If $G=G_{n-1}$, then $T_n'$ also has the same topology as $T_n$, which by convexity and the fact that $T_n'$ is a Steiner tree implies that $T_n'=T_n$ (see Theorem 1.3 of [@bib14]); this is a contradiction and hence $G\not=G_{n-1}$. It follows from this that, if we consider the network $T_n' \cup \{\mathbf{st}_n\}$ (which interconnects $N_{n-1}$), we have $L(T_n') + \vert \mathbf{st}_n \vert \geq L(T_{n-1})+ f_{n-1}$. This implies that $$\begin{aligned} L(T_n') & \geq & L(T_{n-1})+ f_{n-1}- \vert \mathbf{st}_n \vert \\ &=& L(T_{n-1})+ 3\vert \mathbf{st}_n \vert = L(T_{n})+ \vert \mathbf{st}_n \vert.\end{aligned}$$ Hence, we can choose $f_n= f_{n-1}/4 < L(T_n') - L(T_n)$. The claim (and lemma) now follow. Furthermore, the iterative algorithm for constructing a suitable set of embedded terminals for any required Steiner topology is constructive with $f_i= (3\sqrt{3}-5)/4^{i-4}$ for each $i \geq 4$. Let $G_n$ be a full Steiner topology on $n$ terminals. There exists an embedded set of terminals $N$ in the Euclidean plane such that $\mathrm{beads}(T_S)=\mathrm{beads}(T_{\mathrm{opt}})+2n-4$ and $T_S$ has topology $G_n$. We construct an SMT $T_S$ with topology $G_n$ by repeatedly sprouting terminals, starting from a full Steiner tree on three terminals called the *base*. By the previous proposition any full Steiner topology can be produced in this way. Note that we can create a base with edges of any length by simply intersecting the end-points of three line segments at one common point such that every pair of segments forms an angle of $120^\circ$ (and we have complete freedom to do this since we are constructing an SMT by choosing positions for the terminals). By making the edges of the base large enough, it is clear that we can construct $T_S$ such that every edge-length has the form $a_i \pm \varepsilon_i$, where $a_i$ is an integer of order at least two and $\varepsilon_i$ has any predefined value between zero and one. $T_S$ is then converted into an MSPT by a sequence of displacements (which we describe below) of the Steiner points, where displacements do not change the original topology $G_n$. In $T_S$, let $\mathbf{s}_0$ be a Steiner point adjacent to a terminal $\mathbf{t}$ and two other nodes $\mathbf{v}_1,\mathbf{v}_2$ where edge-lengths are preselected as follows: $\vert \mathbf{ts}_0 \vert=a_1-\varepsilon$, $\vert \mathbf{s}_0\mathbf{v}_1 \vert = \vert \mathbf{s}_0\mathbf{v}_2 \vert = b_1+\varepsilon_1$ for large integers $a_1,b_1$ and $0<\varepsilon,\varepsilon_1<1$. In the first step (Figure \[figureS1Perturb\]) we displace $\mathbf{s}_0$ along the line through $\mathbf{t}$ and $\mathbf{s}_0$ and in the direction of the vector $\overrightarrow{\mathbf{ts}_0}$. We displace until $\vert \mathbf{ts}_0 \vert=a_1-\varepsilon^\prime$ and $\vert \mathbf{s}_0\mathbf{v}_1 \vert = \vert \mathbf{s}_0\mathbf{v}_2 \vert = b_1-\varepsilon_1^\prime$ for some $0<\varepsilon^\prime,\varepsilon_1^\prime<1$. Clearly this is possible as long as we preselect $\varepsilon_1$ to be small enough compared to $\varepsilon$. ![First step of the displacement sequence. \[figureS1Perturb\]](Ras04.eps) We now displace all other Steiner points in a depth-first or breadth-first order rooted at $\mathbf{t}$. Suppose that in the process we have reached the Steiner point $\mathbf{s}$ with parent $\mathbf{s}^\prime$ and children $\mathbf{u}_1,\mathbf{u}_2$. We displace $\mathbf{s}$ along the line through $\mathbf{s}$ and the point $\mathbf{p}$ and in the direction $\overrightarrow{\mathbf{ps}}$, where $\mathbf{p}$ is the position $\mathbf{s}^\prime$ had *before* its displacement; see Figure \[figurePerturb\]. If $\vert \mathbf{ss}^\prime \vert =a-\varepsilon_1$ then we preselect $\vert \mathbf{su}_1 \vert = \vert \mathbf{su}_2 \vert =b+\varepsilon_2$ for $0<\varepsilon_2<1$. We select $\varepsilon_2$ small enough so that the displacement of $\mathbf{s}$ produces the lengths $\vert \mathbf{ss}^\prime \vert =a-\varepsilon_1^\prime$ and $\vert \mathbf{su}_1 \vert = \vert \mathbf{su}_2 \vert =b-\varepsilon_2^\prime$, for some $0<\varepsilon_1^\prime,\varepsilon_2^\prime<1$. We continue this process until we have displaced all Steiner points. Call the resultant tree $T$. Note that the edges of $T_S$ were preselected so that one edge has length $a_1-\varepsilon$ and every other edge $e_i$ has length $b_i+\varepsilon_i$. After all displacements the first edge has length $a_1-\varepsilon^\prime$ and every other $e_i$ has length $b_i-\varepsilon_i^\prime$. Clearly then $\mathrm{beads}(T_S)=\mathrm{beads}(T)+2n-4$ and $T$ is an MSPT. ![General step of the displacement sequence. \[figurePerturb\]](Ras05.eps) Minkowski Planes {#sec3} ================ A *Minkowski plane* is a two-dimensional real normed space $M=(\mathbb{R}^2,||\cdot||_M)$ with *unit ball* $B=\{\mathbf{x}:||\mathbf{x}||_M\leq1\}$. We denote the metric induced by $M$ by $d_M(\mathbf{x},\mathbf{y})=||\mathbf{x}-\mathbf{y}||_M$. Examples of Minkowski planes include the Euclidean plane and the rectilinear plane, where the unit balls are the circle and the $45^\circ$ rotated square respectively. The unit ball of a Minkowski plane is always convex, centrally symmetric and bounded in the Euclidean norm. Conversely, any such convex body is the unit ball of a Minkowski plane. The boundary of a ball $B$ is denoted by $\mathrm{bd}(B)$ and its interior by $\mathrm{int}(B)$. The question arises as to whether the upper bound from Proposition \[mainUpperProp\] is best possible in all Minkowski planes. In the three-terminal case we show that the upper bound can be improved for a given Minkowski plane if and only if the unit ball is a parallelogram. Well-known Minkowski planes with this property are the $L_1$ (rectilinear) and $L_\infty$ planes. Let $N=\{\mathbf{t}_i:i=1,2,3\}$ be a set of embedded terminals and let $T_S$ be a $d_M$-SMT on $N$. Many of the propositions below will refer to $T_S^*$ instead of $T_S$ in order to maintain generality. Recall that this convention may lead to zero-length edges, and consequently also to balls of zero radius. Throughout this section we let $L$ denote a Minkowski plane with a parallelogram unit ball. The corresponding metric is denoted by $d_L$. If $\mathbf{t}$ is a terminal point in the plane, we denote by $l_1(\mathbf{t})$ and $l_2(\mathbf{t})$ the Euclidean straight lines passing through $\mathbf{t}$ and parallel to the major diagonal and minor diagonal, respectively, of the parallelogram defining the unit ball of $L$. Minkowski balls $\{B_i:i=1,2,3\}$ *tessellate* at $\mathbf{x}$ if $B_i \cap B_j$ is a point or a Euclidean line segment whenever $i\neq j$, and $\bigcap B_i=\{\mathbf{x}\}$. The point $\mathbf{x}$ is called a *tessellation point* of $N$ if there exists a set of balls $\{B_i:i=1,2,3\}$, with $B_i$ centered at $\mathbf{t}_i$, that tessellate at $\mathbf{x}$. The next proposition is a generalization of a well-known result (*cf*. [@bib11]) on three-terminal rectilinear SMTs. \[tessellate\]Let $N=\{\mathbf{t}_i:i=1,2,3\}$ be a set of embedded terminals. Then there exists a $d_L$-SMT $T_S$ on $N$ such that the Steiner point of $T_S^*$ coincides with the intersection of the median of $\{l_1(\mathbf{t}_i)\}$ and the median of $\{l_2(\mathbf{t}_i)\}$. Let $\mathbf{x}$ be a point in the plane. We wish to minimize the function $f=|\mathbf{t}_1\mathbf{x}|+|\mathbf{t}_2\mathbf{x}|+ |\mathbf{t}_3\mathbf{x}|$ where all inequalities $|\mathbf{t}_i\mathbf{x}|+|\mathbf{t}_j\mathbf{x}| \geq |\mathbf{t}_i\mathbf{t}_j|$, with $1\leq i<j\leq 3$, hold by the triangle inequality. Therefore a minimum would occur if $|\mathbf{t}_i\mathbf{x}|+|\mathbf{t}_j\mathbf{x}| = |\mathbf{t}_i\mathbf{t}_j|$ for every $1\leq i<j\leq 3$; equivalently, a minimum would occur if the balls $\{B_i:i=1,2,3\}$, with $B_i$ centered at $\mathbf{t}_i$ and of radius $|\mathbf{t}_i\mathbf{x}|$, tessellate at $\mathbf{x}$. We show that this happens if $\mathbf{x}$ is the intersection of the median of $\{l_1(\mathbf{t}_i)\}$ and the median of $\{l_2(\mathbf{t}_i)\}$. Suppose, without loss of generality, that $l_1(\mathbf{t}_2)$ is the median of $\{l_1(\mathbf{t}_i)\}$ and $l_2(\mathbf{t}_3)$ is the median of $\{l_2(\mathbf{t}_i)\}$, and let $\mathbf{x}$ be the intersection of these two lines. Let $l_0$ be the line through $\mathbf{x}$ and parallel to the side of $B_3$ that intersects $B_2$ at $\mathbf{x}$ only - see Figure \[figureMedians\]. Then $\mathbf{t}_1$ must lie on the opposite side of $l_0$ to $\mathbf{t}_2$ and $\mathbf{t}_3$ (since $l_1(\mathbf{t}_2)$ and $l_2(\mathbf{t}_3)$ are medians). Therefore $B_1 \cap B_2 \subset l_0$ and $B_1 \cap B_3 \subset l_0$ and the result follows. ![Median diagonals $l_1(\mathbf{t}_2)$ and $l_2(\mathbf{t}_3)$ intersect at Steiner point $\mathbf{x}$. \[figureMedians\]](Ras06.eps) \[corTessellate\]Let $N=\{\mathbf{t}_i:i=1,2,3\}$ be a set of embedded terminals and let $T_S$ be a $d_L$-SMT on $N$. Then the Steiner point of $T_S^*$ is the unique tessellation point of $N$. The set of linear equations $|\mathbf{t}_i\mathbf{x}|+|\mathbf{t}_j\mathbf{x}| = |\mathbf{t}_i\mathbf{t}_j|$, for $1\leq i < j \leq 3$, has a unique solution. Given three terminals in the plane and a unit ball parallelogram $B$, the *enclosing diagonalized parallelogram* is the smallest parallelogram whose sides are parallel to the major and minor diagonals of $B$ and which includes all the terminals on its boundary. The next result also has an analogue in the rectilinear plane (*cf*. [@bib11]). The total length of a three-terminal $d_L$-SMT is equal to half the perimeter of the enclosing diagonalized parallelogram. The following proposition is the first main result of this section. It shows that the upper bound from Proposition \[mainUpperProp\] with $n=3$ is not strict for parallelogram-based Minkowski planes. \[propPara\]Let $T_S$ be a $d_L$-SMT and let $T_{\mathrm{opt}}$ be a $d_L$-MSPT on the embedded terminals $\{\mathbf{t}_i:i=1,2,3\}$. Then $\mathrm{beads}(T_S)\leq \mathrm{beads}(T_{\mathrm{opt}})+1$. Suppose that the Steiner point of $T_S^*$ is $\mathbf{s}$ and its edges are $a_i=\mathbf{t}_i\mathbf{s}$ for $i \in \{1,2,3\}$. Let $\mathbf{s^\prime}$ be the Steiner bead of $T_{\mathrm{opt}}^*$ and let $e_i=\mathbf{t}_i\mathbf{s^\prime}$ for $i \in \{1,2,3\}$. Since $\mathbf{s}$ is the tessellation point of $\{\mathbf{t}_i\}$, at most one inequality from $\vert e_i\vert < \vert a_i \vert$, $i\in \{1,2,3\}$ can be true (note that if none of these inequalities are true then $\mathrm{beads}(T_S)= \mathrm{beads}(T_{\mathrm{opt}})$, and we are done). Suppose w.l.o.g that $\vert e_1 \vert< \vert a_1 \vert$. Let $p_1=\vert a_1 \vert - \vert e_1\vert$ and let $p_i=\vert e_i \vert - \vert a_i \vert$ for $i=\{2,3\}$. **Claim**: $p_1 \leq \min{\{p_2,p_3 \}}$.\ Clearly $\vert a_1 \vert + \vert a_3\vert=\vert \mathbf{t}_1\mathbf{t}_3\vert$ since $\mathbf{t}_1,\mathbf{s},\mathbf{t}_3$ is a shortest path between $\mathbf{t}_1$ and $\mathbf{t}_3$. Also, by using the triangle inequality in $\triangle \mathbf{t}_1\mathbf{s}^\prime\mathbf{t}_3$ we obtain $\vert \mathbf{t}_1\mathbf{t}_3\vert\leq \vert e_1\vert+\vert e_3\vert$. Therefore $\vert a_1\vert+\vert a_3\vert \leq (\vert a_1\vert-p_1)+(\vert a_3\vert+p_3)$ so that $p_1\leq p_3$. Similarly, $p_1 \leq p_2$ and this proves the claim. We now have: $$\begin{aligned} \mathrm{beads}(T_S)- \mathrm{beads}(T_{\mathrm{opt}}) &=& \sum\limits_{i=1}^3\lceil\vert a_i \vert\rceil-\sum\limits_{i=1}^3\lceil\vert e_i \vert\rceil\\ &=& \lceil\vert a_1 \vert\rceil - \lceil\vert e_1\vert\rceil-\sum\limits_{i\in\{2,3\}}\{\lceil \vert e_i \vert\rceil - \lceil \vert a_i\vert\rceil \}\\ &=& \lceil\vert e_1 \vert+p_1\rceil - \lceil\vert e_1\vert\rceil-\sum\limits_{i\in\{2,3\}}\{\lceil \vert e_i \vert\rceil - \lceil \vert e_i\vert-p_i\rceil \}\\ &\leq & \lceil p_1 \rceil -\sum\limits_{i\in\{2,3\}}\{\lceil p_i \rceil -1\}\\ &\leq & 0-\lceil \max\{p_2,p_3\} \rceil +2 \hspace{0.5 in} \mathrm{(since \ } 0<p_1 \leq \min{\{p_2,p_3 \}}\mathrm{)}\\ &<& 2.\end{aligned}$$ To prove a converse of the previous proposition we first show that Corollary \[corTessellate\] is unique to Minkowski planes with parallelogram unit balls. We find that some three-terminal sets in Minkowski planes with hexagon unit balls do have tessellation points, but that this is not true in general. For any non-parallelogram based Minkowski plane we then construct a three-terminal example that achieves the upper bound from Proposition \[mainUpperProp\]. Two points on the boundary of a ball $B$ form a *diametric pair* if the Euclidean straight line passing through the points also passes through the center of $B$. A point $\mathbf{z}$ on a ball $B_i$ is *equivalent* to a point $\mathbf{z}^\prime$ on a ball $B_j$ if and only if, by translating $B_i$ so that its center coincides with the center of $B_j$, it is possible to rescale $B_i$ so that $\mathbf{z}$ coincides with $\mathbf{z}^\prime$. Let $\{\mathbf{t}_i:i=1,2,3\}$ be a set of embedded terminals with tessellation point $\mathbf{s}$ in an arbitrary non-parallelogram-based Minkowski plane $M$, and suppose that the balls $\{B_i\}$ tessellate at $\mathbf{s}$. Suppose that $B_i \cap B_j=\{\mathbf{s}\}$ for some $i,j \in \{1,2,3\}$. Then the point that forms a diametric pair with $\mathbf{s}$ on $B_i$ is equivalent to $\mathbf{s}$ when considered as a point on $B_j$. This follows from the convexity and central symmetry of the balls. As a consequence of the previous lemma there exist distinct $i,j,k \in \{1,2,3\}$ such that $B_i \cap B_j$ and $B_i \cap B_k$ are Euclidean line segments (as opposed to single points only corresponding to $\mathbf{s}$). Suppose w.l.o.g that $B_1 \cap B_2$ and $B_1 \cap B_3$ are line segments. $B_2 \cap B_3$ is also a line segment. With the aim of producing a contradiction we assume that $B_2 \cap B_3=\{\mathbf{s}\}$. Note that there exist exactly four maximal-length line segments in $\bigcup \{\mathrm{bd}\left(B_i\right)\}$ that have $\mathbf{s}$ as an endpoint. We list these segments in any clockwise order as $\{S_i:0 \leq i \leq 3\}$. Since $\mathbf{s}$ forms a diametric pair on $B_2$ and $B_3$, $S_i$ and $S_{(i+2)\mathrm{mod}4}$ have the same gradient for any $i \in \{0,...,3\}$. By central symmetry, the only possible balls that can produce such a configuration of line segments are parallelograms, which is a contradiction. A point $\mathbf{x}$ on $\mathrm{bd}(B)$ is called a *corner* if the intersection of any neighborhood of $\mathbf{x}$ with $\mathrm{bd}(B)$ is not a Euclidean straight line segment. The point $\mathbf{s}$ is a corner of every member of $\{B_i\}$. Clearly $\mathbf{s}$ is a corner of at least two members of $\{B_i\}$. Suppose, for a contradiction, that $\mathbf{s}$ is not a corner of some $B_i$. In this case there exist exactly three maximal-length line segments in $\bigcup \{\mathrm{bd}\left(B_i\right)\}$ that have $\mathbf{s}$ as an endpoint. Furthermore, exactly two of these segments have the same gradient. By central symmetry such a configuration of line segments can only be produced by parallelogram balls. By combining the previous lemmas, we know there exist exactly three maximal-length line segments in $\bigcup \{\mathrm{bd}\left(B_i\right)\}$ that have $\mathbf{s}$ as an endpoint, all with distinct gradients. From this fact and central symmetry we conclude the following lemma: \[hexagon\]The balls $\{B_i\}$ are hexagons. We also need the following two lemmas which follow directly from results by Martini, Swanepoel and Weiss [@bib12]. Suppose that $\mathbf{s}$ is a degree-three Steiner point of a $d_M$-SMT on embedded terminal set $\{\mathbf{t}_i:i=1,2,3\}$, where $\mathbf{s}$ does not coincide with a terminal. Then $\mathbf{s}$ is also a Steiner point of the terminal set $\{\mathbf{t}_i^\prime\}$ where $\mathbf{t}_i^\prime$ is any point lying on the Euclidean ray with origin $s$ and passing through $\mathbf{t}_i$. There exists a set $\{\mathbf{t}_i:i=1,2,3\}$ of embedded terminals such that some $d_M$-SMT on $\{\mathbf{t}_i\}$ has a degree-three Steiner point that does not coincide with a terminal. We can now prove the final proposition of this section. If the unit ball defining $M$ is not a hexagon then, by using the previous two lemmas, we can construct a *critical* $d_M$-SMT. A critical $d_M$-SMT on an embedded terminal set $\{\mathbf{t}_i:i=1,2,3\}$ has the following properties: 1. The Steiner point does not coincide with a terminal: i.e., the Steiner point is of degree three and there are no edges of zero length, 2. Each edge $e_i$ has length $a_i+\varepsilon_i$ where $a_i$ is an integer and $0<\varepsilon_i<1$ has any predefined value, 3. The balls centered at the terminals and meeting the Steiner point do not tessellate. Otherwise, if $M$’s unit ball is a hexagon we first use the previous two lemmas to find a terminal set satisfying properties (1) and (2). We then destroy the tessellation property (if necessary) by performing a rotational displacement around the Steiner point of one of the terminals. Let $T_S$ be a critical $d_M$-SMT on $\{\mathbf{t}_i\}$. The final result of this section is a converse to Proposition \[propPara\]. $\mathrm{beads}(T_S)= \mathrm{beads}(T_{\mathrm{opt}})+2$. By Properties (1) and (3) we may displace the Steiner point $\mathbf{s}$ into a region corresponding to the intersection of the interiors of two balls, say $\mathrm{int}(B_1) \cap \mathrm{int}(B_2)$. By Property (2) we can preselect each $\varepsilon_i$ so that after displacement we have $\vert \mathbf{t}_1\mathbf{s} \vert< a_1$, $\vert \mathbf{t}_2\mathbf{s} \vert< a_2$ and $\vert \mathbf{t}_3\mathbf{s} \vert \leq a_3+1$. We conclude this section with two conjectures. Let $N$ be a set of $n$ terminals in a Minkowski plane $M$ with unit ball $B$, and let $T_S$ and $T_{\mathrm{opt}}$ be a $d_M$-SMT and $d_M$-MSPT on $N$ respectively. Suppose first that $B$ is a parallelogram. Let $\mathbf{s}^\prime$ be the Steiner point of some cherry of $T_S$, and let $\mathbf{u}_1,\mathbf{u}_2$ be terminals adjacent to $\mathbf{s}^\prime$. The proof of Proposition \[propPara\] implies that a displacement of $\mathbf{s}^\prime$ can shorten at most one of $\mathbf{s}^\prime\mathbf{u}_1,\mathbf{s}^\prime\mathbf{u}_2$. Since every Steiner tree has a minimum of two cherries, displacements of Steiner points can shorten at most $2n-5$ edges of $T_S$ (note, of course, that it may be possible to shorten up to $2n-4$ edges of $T_S$ if we change its topology). The upper bound $\mathrm{beads}(T_S)-\mathrm{beads}(T_{\mathrm{opt}}) \leq 2n-4$ is tight if and only if $B$ is not a parallelogram. The upper bound $\mathrm{beads}(T_S)-\mathrm{beads}(T_{\mathrm{opt}}) \leq 2n-5$ is tight if and only if $B$ is a parallelogram. A Canonical Form for Euclidean MSPTs {#sec4} ==================================== Throughout this section we only consider Euclidean MSPTs. In general there are many possible ways to embed an MSPT in Euclidean space. Here we introduce a canonical form for MSPTs (over all possible embeddings) which allows us to reformulate the MSPT problem as that of finding a shortest total length tree in which almost all edges have integer length. Understanding this canonical form provides a valuable first step towards finding an efficient exact algorithm for the problem (like the canonical forms for Steiner trees in fixed-orientation metrics [@bib16], and those used in the previously mentioned GeoSteiner algorithms for rectilinear Steiner trees). In other words, we show that in order to find MSPTs, it suffices to explore a class of trees with strong structural restrictions. The canonical form we describe is also interesting from a more theoretical point of view as it gives an insight into the geometry of MSPTs. An MSPT where every terminal is of degree one and every Steiner bead is of degree three is called a *full MSPT*. This term refers to MSPTs that have this property “naturally", i.e., not through a splitting process. Recall that a full MSPT contains $n-2$ Steiner beads and $2n-3$ edges. The Steiner bead of a cherry will be referred to as a *cherry bead*. A *level-region* of a function $f$ is a set of points satisfying $f=k$ for some constant $k$. Let $N$ be a set of three embedded terminals admitting a full MSPT. Then there exists an MSPT on $N$ such that at least two of its edges are of integer length. Let $T_{\mathrm{opt}}$ be a full MSPT on the embedded terminal set $N=\{\mathbf{t}_i:i=1,2,3\}$ and let $\mathbf{s}$ be the Steiner bead. Then $\mathrm{beads}(T_{\mathrm{opt}})$ is the minimum value of the function $f=\lceil \vert \mathbf{t}_1\mathbf{s} \vert \rceil + \lceil \vert \mathbf{t}_2\mathbf{s} \vert \rceil + \lceil \vert \mathbf{t}_3\mathbf{s} \vert \rceil - 2$, where the position of $\mathbf{s}$ is variable. The level-region $k=\lceil \vert \mathbf{t}_1\mathbf{s} \vert \rceil + \lceil \vert \mathbf{t}_2\mathbf{s} \vert \rceil + \lceil \vert \mathbf{t}_3\mathbf{s} \vert \rceil - 2$, where $k$ is a positive integer, consists of regions that are bounded by at least one and at most six integer-radius circular arcs. Since $\mathbf{s}$ does not correspond to a terminal, the region $L(\mathbf{s})$ containing $\mathbf{s}$ must be bounded by at least two arcs. Displacing $\mathbf{s}$ to coincide with an intersection point of the arcs bounding $L(\mathbf{s})$ will lead to an MSPT of the desired form; see Figure \[figureLevs\]. ![Level-regions of $f$. \[figureLevs\]](Ras08.eps) A tree $T$ connecting $n$ embedded terminals and some Steiner points is called $\mathbb{Z}$-packed if $T$ has a full Steiner topology and at least $2n-4$ of its edges are of integer length. Let $N$ be a set of three embedded terminals in the Euclidean plane admitting a full MSPT. Then a shortest total length $\mathbb{Z}$-packed tree on $N$ is an MSPT. By the previous lemma, $N$ must admit a $\mathbb{Z}$-packed MSPT. Note that Proposition \[mainUpperProp\] still holds if we modify it slightly by constraining $T_{\mathrm{opt}}$ to be $\mathbb{Z}$-packed and by letting $T_S$ be a shortest $\mathbb{Z}$-packed tree. Corollary \[EqCorollary\] now gives us our result. We wish to generalize the previous result to any number of terminal points, but for this we need another condition. If two edges of an MSPT are incident to the same Steiner bead and are collinear then these edges are said to form a *Steiner bond*. An MSPT $T$ on a set $N$ of embedded terminals is called *bond-free* if every MSPT on $N$ with the same topology as $T$ is free of Steiner bonds. Figure \[figureBond\] provides an example of an MSPT that is not bond-free; the fact that the depicted tree is an MSPT on the three solid nodes follows from Proposition \[mainUpperProp\] once it is noted that the SMT on the same terminals has $4$ beads. ![An MSPT that is not bond-free. \[figureBond\]](Ras12.eps) Let $N$ be a set of four embedded terminals admitting a bond-free full MSPT $T_{\mathrm{opt}}$. Then it is possible to find an MSPT on $N$, with the same topology as $T_{\mathrm{opt}}$, such that all edges incident to terminals are of integer length.\[FourPRep\] Let $N=\{\mathbf{t}_i:i=1,..,4 \}$ be the terminal set for $T_{\mathrm{opt}}$ and let $\mathbf{s}_1,\mathbf{s}_2$ be the Steiner beads of $T_{\mathrm{opt}}$ with $\mathbf{s}_1$ adjacent to $\mathbf{t}_1$ and $\mathbf{t}_2$. Let $T_0$ be the subtree of $T_{\mathrm{opt}}$ induced by the nodes $\mathbf{t}_1,\mathbf{t}_2,\mathbf{s}_1,\mathbf{s}_2$. Clearly $T_0$ is an MSPT on the nodes $\mathbf{t}_1,\mathbf{t}_2,\mathbf{s}_2$. By fixing the position of $\mathbf{s}_2$ we convert $T_0$ into a $\mathbb{Z}$-packed tree by displacing $\mathbf{s}_1$. We now fix $\mathbf{s}_1$ at its new position and displace $\mathbf{s}_2$ until the subtree induced by $\mathbf{t}_3,\mathbf{t}_4,\mathbf{s}_1,\mathbf{s}_2$ is $\mathbb{Z}$-packed. The modified $T_{\mathrm{opt}}$ must now have at least three integer length edges. Suppose that the edges $\mathbf{t}_1\mathbf{s}_1,\mathbf{s}_1\mathbf{s}_2,\mathbf{t}_3\mathbf{s}_2$ have integer lengths (the other cases are handled similarly). We fix $\mathbf{s}_2$ and displace $\mathbf{s}_1$ along the circle centered at $\mathbf{t}_1$ and of radius $\vert \mathbf{t}_1\mathbf{s}_1 \vert$. Note that the smallest value of $\vert \mathbf{s}_1\mathbf{s}_2 \vert$ occurs when $\mathbf{s}_1$ is displaced until it reaches the line connecting $\mathbf{t}_1$ and $\mathbf{s}_2$. Therefore displacement until $\vert \mathbf{t}_2\mathbf{s}_1 \vert$ is an integer is possible due to the bond-free condition; see Figure \[figureFourP\]. We fix the position of $\mathbf{s}_1$ and repeat the process for $\mathbf{s}_2$. ![Bond creation. \[figureFourP\]](Ras09.eps) A *caterpillar* is a tree with the property that the removal of its degree-one nodes results in a path. Let $N$ be a set of $n \geq 4$ embedded terminals admitting a bond-free full MSPT $T_{\mathrm{opt}}$ and suppose that $T_{\mathrm{opt}}$ is a caterpillar. Let $\mathbf{s}_0$ be a cherry bead of $T_{\mathrm{opt}}$ connected to another Steiner bead $\mathbf{s}_1$. If $T_{\mathrm{opt}}$ is bond-free then there exists a $\mathbb{Z}$-packed MSPT on $N$ with the same topology as $T_{\mathrm{opt}}$ such that every edge, other than possibly $\mathbf{s}_0\mathbf{s}_1$, has integer length. This follows readily from repeated application of the previous lemma. Let $T$ be a non-caterpillar full MSPT rooted at two terminals connected to a cherry bead $\mathbf{r}$, and let $\mathbf{s}$ be any Steiner bead of $T$ with children $\mathbf{v}_1,\mathbf{v}_2$. Then $\mathbf{s}$ is called a *junction* of $T$ if, for each $i\in\{1,2\}$, the subtree induced by $\mathbf{s}$,$\mathbf{v}_i$ and all descendants of $\mathbf{v}_i$ (if they exist) is a caterpillar. A Steiner bead is a *maximal* junction if it is a junction but its parent is not a junction - see Figure \[figureJunc\]. ![Maximal junction $\mathbf{s}$. \[figureJunc\]](Ras10.eps) Let $N$ be a set of $n \geq 4$ embedded terminals admitting a bond-free full MSPT. Then it is possible to find an MSPT, say $T$, on $N$ such that at most one of its edges are of non-integer length. Furthermore, either all $T$’s edges will be of integer length, or we will be able to choose which internal edge of $T$ has non-integer length. Root $T_{\mathrm{opt}}$ at two terminals connected to a cherry bead $\mathbf{r}$. Let $\mathbf{s}$ be a maximal junction of $T_{\mathrm{opt}}$ with children $\mathbf{v}_1,\mathbf{v}_2$ and for each $i \in \{1,2\}$ let $T_i$ be the subtree of $T_{\mathrm{opt}}$ induced by $\mathbf{s}$,$\mathbf{v}_1$,$\mathbf{v}_2$, the parent of $\mathbf{s}$ and the descendants of $\mathbf{v}_i$. Note that $T_i$ is a caterpillar. By fixing the positions of the parent of $\mathbf{s}$ and the child of $\mathbf{s}$ not equal to $\mathbf{v}_i$, and applying the previous lemma, we force all edges of $T_i$ (except possibly $\mathbf{sv}_i$) to be of integer length. We do this for all maximal junctions of $T_{\mathrm{opt}}$. For the resultant tree, say $T$, we now consider $\mathbf{v}_1$ and $\mathbf{v}_2$ as terminals and ignore the subtrees induced by the descendants of $\mathbf{v}_1$ and $\mathbf{v}_2$ (similarly for other junctions). We select the maximum junctions with respect to $T$ and continue the process. Once there are no junctions left (i.e., only a caterpillar remains) we force all edges, except possibly the edge between $\mathbf{r}$ and its child, to be of integer length. Repeated application of Lemma \[FourPRep\] now allows us to choose which internal edge should possibly be of non-integer length, i.e., we now “move" the non-integer property to any other internal edge. Let $N$ be any set of terminals in the Euclidean plane admitting a bond-free full MSPT $T_{\mathrm{opt}}$. Then a shortest total length $\mathbb{Z}$-packed tree on $N$ is an MSPT. This follows, as before, from the previous result and from Corollary \[EqCorollary\] after a slight modification to Proposition \[mainUpperProp\]. Concluding Remarks and Conjectures ================================== We suspect that, at least in the Euclidean case, the performance difference $2n-4$ of the SMT heuristic can be improved by supplementing it with an algorithm that involves relatively small displacements of the Steiner points. The question is: by how much can we improve the performance? If for any set of embedded terminals it is possible to find an MSPT with the same topology (or a degeneracy thereof) as an SMT on the terminals, then it would be theoretically possible to improve the performance to optimality in this way. If an SMT $T_S$ is not full then certainly the topology of an MSPT on the same terminals is generally not a degeneracy of $T_S$ (i.e., an MSPT topology cannot be obtained simply by collapsing edges of $T_S$). Consider for instance the embedded terminals $\{\mathbf{t}_i:i=1,2,3\}$ in the Euclidean plane where $\angle \mathbf{t}_1\mathbf{t}_2\mathbf{t}_3=120^\circ$ and $\vert \mathbf{t}_1\mathbf{t}_2\vert=\vert \mathbf{t}_2\mathbf{t}_3\vert=5.1$ units. Clearly the SMT $T_S$ on $\{\mathbf{t}_i\}$ is not full and $\mathrm{beads}(T_S)=10$. However, $\mathrm{beads}(T_{\mathrm{opt}})=9$ as shown in Figure \[figureTop\]. ![$T_{\mathrm{opt}}$ with nine beads. \[figureTop\]](Ras11.eps) To conclude we now state some conjectures that we hope will inspire further research into the relationship between SMTs and MSPTs. Let $N$ be a set of terminals embedded in the Euclidean plane and let $T_S$ be an SMT on $N$. If $T_S$ is full then there exists an MSPT on $N$ that is a degeneracy of $T_S$. This conjecture implies that an algorithm based on displacing Steiner points of an SMT has the potential for generating optimal MSPTs for a very large class of terminal configurations. The next conjecture would allow such an algorithm to run in polynomial time. Let $G$ be any tree topology on $N$ where Steiner points are of degree at least three. Then finding a tree $T$ on $N$ with the same topology as $G$ and minimizing $\mathrm{beads}(T)$ can be done in polynomial time. **Acknowledgement.** The authors wish to thank Jamie Evans for partaking in many fruitful discussions during the development of this paper. We would also like to thank the referees for their insightful comments. [l]{} S. Arora, Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems, Journal of the ACM 45 (1998), 753–782. S. Arora, Approximation schemes for NP-hard geometric optimization problems: A survey, Math Program Ser B 97 (2003), 43–69. M. Brazil, J.H. Rubinstein, D.A. Thomas, J.F. Weng, and N.C. Wormald, Minimal Steiner trees for rectangular arrays of lattice points, J Combin Theory Series A 79 (1997), 181–208. M. Brazil, D. A. Thomas, J. F. Weng, and M. Zachariasen, Canonical forms and algorithms for Steiner trees in uniform orientation metrics, Algorithmica 44 (2006), 281–300. D. Chen, D.–Z. Du, X.–D. Hu, G.–H. Lin, L. Wang, and G. Xue, Approximations for Steiner trees with minimum number of Steiner points, Theoretical Computer Science 262 (2001), 83–99. X. Cheng, D.–Z. Du, L. Wang, and B. Xu, Relay sensor placement in wireless sensor networks, Wireless Networks 14 (2008), 347–355. D.–Z. Du and X. Hu, Steiner tree problems in computer communication networks, World Scientific Publishing Company, Singapore, 2008. R.L. Francis, A note on the optimum location of new machines in existing plant layouts, J Indust Eng 14 (1963), 57–59. E.N. Gilbert and H.O. Pollak, Steiner minimal trees, SIAM J Appl Math 16 (1968), 1–29. F.K. Hwang, D.S. Richards, and P. Winter, The Steiner tree problem, Annals of Discrete Mathematics 53, Elsevier Science Publishers B V, Amsterdam, 1992. C.–S. Li, F.F. Tong, C.J. Georgiou, and M. Chen, Gain equalization in metropolitan and wide area optical networks using optical amplifers, 13th Proc IEEE INFOCOM, Toronto, Canada, June 1994, pp. 130–-137. G.–H. Lin and G.L. Xue, Steiner tree problem with minimum number of Steiner points and bounded edge–length, Inform Process Lett 69 (1999), 53–-57. B. Lu, J. Gu, X. Hu, and E. Shragowitz, Wire segmenting for buffer insertion based on RSTP-MSP, Theoretical Computer Science 262 (2001), 257–-267. I.I. Mandoiu and A.Z. Zelikovsky, A note on the MST heuristic for bounded edge–length Steiner trees with minimum number of Steiner points, Inform Process Lett 75 (2000), 165–-167. H. Martini, K.J. Swanepoel, and G. Weiss, The Fermat-–Torricelli problem in normed planes and spaces, Journal of Optimization Theory and Applications 115 (2002), 283–314. B. Ramamurthy, J. Iness, and B. Mukherjee, Minimizing the number of optical amplifiers needed to support a multi–wavelength optical LAN/MAN, Proc IEEE INFOCOM, 16th Ann Joint Conf IEEE Computer and Communications Societies, Kobe, Japan, April 1997, pp. 261–268. M. Sarrafzadeh and C.K. Wong, Bottleneck Steiner trees in the plane, IEEE Trans Comput 41 (1992), 370–374. D.M. Warme, P. Winter, and M. Zachariasen, “Exact algorithms for plane Steiner tree problems: A computational study", Advances in Steiner Trees, D.–Z. Du, J. M. Smith, and J. H. Rubinstein (Editors), Kluwer Academic Publishers, Boston, 2000, pp. 81-–116. P. Winter and M. Zachariasen, Euclidean Steiner minimum trees: An improved exact algorithm, Networks 30 (1997), 149–166. [^1]: This research was supported by an ARC Discovery Grant.

--- address: | Laboratoire de Physique Subatomique et de Cosmologie, Université Grenoble-Alpes, CNRS/IN2P3,\ 53 avenue des Martyrs,\ 38026 Grenoble Cedex, France author: - SELIM TOUATI title: ELECTRIC DIPOLE MOMENTS AND NEUTRINO MASS MODELS --- EDMs generated by the CKM phase =============================== In the standard model (SM), the only source of weak CP-violation is the complex phase of the CKM matrix. In order to measure the strength of CP-violation, one can construct a flavor invariant (basis-independent) which is sensitive to this phase, called the Jarlskog invariant [@Jarl]. A non-vanishing Jarlskog invariant is a necessary condition for having CP-violation. In the SM, all CP-violating effects are proportional to this invariant. However, this invariant is adequate for estimating CP-violation from closed fermion loops. For example, let us consider the CKM-induced lepton EDMs. Because the leptons cannot feel directly the complex phase of the CKM matrix, we need to go through a closed quark loop. The dominant diagram is: ![CKM-induced lepton EDM[]{data-label="fig:CKMleptonEDM"}](figures/FigCKMLeptonEDM) This EDM is tuned by the Jarlskog invariant $\det[Y_{u}^{\dagger}Y_{u},Y_{d}^{\dagger}Y_{d}]$ which is proportional to the imaginary part of a quartet $Im(V_{us}V_{cb}V_{ub}^{\ast}V_{cs}^{\ast})$. As for the quarks, they can feel directly the complex phase of the CKM matrix and then there are non-invariants structures which arise from rainbow-like processes. Indeed, the dominant diagrams for the CKM-induced quark EDMs have a rainbow topology. For instance, for the d-quark EDM: ![CKM-induced d-quark EDM[]{data-label="fig:CKMquarkEDM"}](figures/FigCKMQuarkEDM) This EDM is tuned by the imaginary part of the 1-1 entry of a non-invariant commutator $Im(\textbf{X}^{dd}_{q})$, where: $$\textbf{X}_{q}=\mathbf{[Y}_{u}^{\dagger}\mathbf{Y}_{u},\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}\mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{d}\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}], \label{eq:Xq}$$ which is also proportional to $Im(V_{us}V_{cb}V_{ub}^{\ast}V_{cs}^{\ast})$ as for the lepton EDMs (because we are in the SM), but not with the same proportionality factor. It turns out to be much larger by 10 orders of magnitude: $$Im\mathbf{[Y}_{u}^{\dagger}\mathbf{Y}_{u},\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}\mathbf{Y}_{d}^{\dagger}\mathbf{Y}_{d}\mathbf{Y}_{u}^{\dagger}\mathbf{Y}_{u}]^{dd}\gg\det[Y_{u}^{\dagger}Y_{u},Y_{d}^{\dagger}Y_{d}].$$ In the SM, the rainbow-like flavor structures are typically much larger than the invariant determinants and they are correlated (strictly proportional). Now, let us turn on neutrino masses (beyond the SM) and check whether this behavior is confirmed or not. As we do not know yet the nature of the neutrino (Dirac or Majorana particle), we will consider both scenarios for generating neutrino masses. EDMs in the presence of neutrino masses ======================================= Dirac neutrino masses --------------------- The simplest way of including neutrino masses to the SM is to extend its particles content by adding three right-handed (RH) fully neutral neutrinos (one for each generation). They belong to the trivial representation of the SM gauge group: $N=\nu_{R}^{\dagger}\sim(1,1)_{0}$. We add to the SM Yukawa Lagrangian an extra Yukawa interaction for the neutrinos: $$\mathcal{L}_{Yukawa}=\mathcal{L}_{Yukawa}^{SM}-N^{I}Y_{\nu}^{IJ}L^{J}H^{\dagger C}+h.c.$$ We have a new neutrinos-related flavor structure $Y_{\nu}$ ($3\times3$ matrix in flavor space). In the presence of neutrino masses, we get an additional source of weak CP-violation coming from the complex phase of the PMNS matrix. In complete analogy with the quark sector, we can construct new CP-violating flavor structures which tune the PMNS-induced quark and lepton EDMs. In this case, quark EDMs have a bubble topology whereas lepton EDMs have a rainbow topology. For instance, the dominant diagrams for the PMNS-induced quark and lepton EDMs are shown in figure \[fig:DiracEDMs\]. ![PMNS-induced quark (on the left) and lepton (on the right) EDMs[]{data-label="fig:DiracEDMs"}](figures/FigDiracEDMs) They are tuned respectively by $J_{\mathcal{CP}}^{Dirac}$ and $Im(\textbf{X}_{e}^{Dirac})^{11}$, where $$\begin{aligned} J_{\mathcal{CP}}^{Dirac}= & \frac{1}{2i}\det\left[Y_{\nu}^{\dagger}Y_{\nu},Y_{e}^{\dagger}Y_{e}\right]\\ \textbf{X}_{e}^{Dirac}= & \left[Y_{\nu}^{\dagger}Y_{\nu},Y_{\nu}^{\dagger}Y_{\nu}Y_{e}^{\dagger}Y_{e}Y_{\nu}^{\dagger}Y_{\nu}\right]. \label{eq:XeDirac}\end{aligned}$$ In this scenario, $Im(\textbf{X}_{e}^{Dirac})^{11}$ is 11 orders of magnitude larger than $J_{\mathcal{CP}}^{Dirac}$ and they are correlated (strictly proportional). Majorana neutrino masses ------------------------ Another way for generating neutrino masses is possible if we consider Majorana masses. In this mechanism, there is no additional RH neutrinos, we get directly a gauge-invariant but lepton-number violating mass term for the left-handed (LH) neutrinos. Indeed, we add to the SM Yukawa Lagrangian the effective dimension-five Weinberg operator: $$\mathcal{L}_{Yukawa}=\mathcal{L}_{Yukawa}^{SM}-\frac{1}{2v}(L^{I}H)(\Upsilon_{\nu})^{IJ}(L^{J}H)+h.c,$$ which after spontaneous symmetry breaking collapses to a Majorana mass term for the LH neutrinos: $$\frac{1}{2v}(L^{I}H)(\Upsilon_{\nu})^{IJ}(L^{J}H)\overset{SSB}{\longrightarrow}\frac{v}{2}(\Upsilon_{\nu})^{IJ}\nu_{L}^{I}\nu_{L}^{J}.$$ $\Upsilon_{\nu}$ (3$\times$3 matrix in flavor space) is a new flavor structure purely of the Majorana type. In this model, we must redefine the PMNS matrix in order to add two new CP-violating phases, called Majorana phases, $$U_{PMNS}\rightarrow U_{PMNS}\cdot diag(1,e^{i\alpha_{M}},e^{i\beta_{M}}).$$ Let us consider the PMNS-induced quark and lepton EDMs in this scenario. The dominant diagrams are: ![PMNS-induced quark (on the left) and lepton (on the right) EDMs[]{data-label="fig:MajoEDMs"}](figures/FigMajoEDMs) The CP-violating flavor structures which tune these EDMs are $J_{\mathcal{CP}}^{\mathrm{Majo}}$ [@Branco] and $Im(\mathbf{X}_{e}^{\mathrm{Majo}})^{11}$, where: $$\begin{aligned} J_{\mathcal{CP}}^{\mathrm{Majo}}= & \frac{1}{2i}Tr[\mathbf{\Upsilon}_{\nu}^{\dagger}\mathbf{\Upsilon}_{\nu}\cdot\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e}\cdot\mathbf{\Upsilon}_{\nu}^{\dagger}(\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e})^{T}\mathbf{\Upsilon}_{\nu}-\mathbf{\Upsilon}_{\nu}^{\dagger}(\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e})^{T}\mathbf{\Upsilon}_{\nu}\cdot\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e}\cdot\mathbf{\Upsilon}_{\nu}^{\dagger}\mathbf{\Upsilon}_{\nu}]\\ \mathbf{X}_{e}^{\mathrm{Majo}}= & [\mathbf{\Upsilon}_{\nu}^{\dagger}\mathbf{\Upsilon}_{\nu},\mathbf{\Upsilon}_{\nu}^{\dagger}(\mathbf{Y}_{e}^{\dagger}\mathbf{Y}_{e})^{T}\mathbf{\Upsilon}_{\nu}].\end{aligned}$$ We find that $Im(\textbf{X}_{e}^{Majo})^{11}$ is 4 orders of magnitude larger than $J_{\mathcal{CP}}^{Majo}$ but in this scenario they are not correlated. In figure \[fig:CorrelationMajo\], we can see the values that can take the PMNS-induced quark and lepton EDMs (tuned respectively by $J_{\mathcal{CP}}^{Majo}$ and $Im(\textbf{X}_{e}^{Majo})^{11}$). The lightest neutrino mass $m_{\nu 1}$ is set to $1eV$ and the CP-violating phases (PMNS phase $\delta_{13}$ and the Majorana phases $\alpha_{M}$ and $\beta_{M}$) are allowed to take on any values. ![Area spanned by $J_{\mathcal{CP}}^{Majo}$ and $Im(\textbf{X}_{e}^{Majo})^{11}$[]{data-label="fig:CorrelationMajo"}](figures/CorrelationMajo) The lines show the strict correlation occuring when only one phase is non-zero. When the three phases are into action, because the flavor structures have different dependences in these phases, the result is that the quark and lepton EDMs become decorrelated. Conclusion ========== In the paper [@SmithTouati], we developped a systematic method to study the flavor structure behind the quark and lepton EDMs which can be extended easily to other more complicated models (Sterile neutrinos, SUSY etc...). The rainbow-like non-invariant flavor structures are found to be typically much larger than the Jarlskog-like flavor invariants. Interestingly, we find a different behavior for Dirac and Majorana neutrinos. Quark and lepton EDMs are proportional in the former case whereas they are completely independent in the latter case. Indeed, quark and lepton EDMs have different dependences on Majorana phases. Finally, by studying the flavor structures behind the quark and lepton EDMs, we get the relations shown in table \[tab:SumRules\] between EDMs of different generations. CKM-induced EDMs PMNS-induced EDMs -- ------------------------------------------------------------------------- --------------------------------------------------------------------------- $\frac{d_{d}}{m_{d}}+\frac{d_{s}}{m_{s}}+\frac{d_{b}}{m_{b}}=0$ $\frac{d_{d}}{m_{d}}=\frac{d_{s}}{m_{s}}=\frac{d_{b}}{m_{b}}$ $\frac{d_{e}}{m_{e}}=\frac{d_{\mu}}{m_{\mu}}=\frac{d_{\tau}}{m_{\tau}}$ $\frac{d_{e}}{m_{e}}+\frac{d_{\mu}}{m_{\mu}}+\frac{d_{\tau}}{m_{\tau}}=0$ : Sum rules[]{data-label="tab:SumRules"} For example, the CKM-induced quark EDMs and the PMNS-induced lepton EDMs are tuned by the non-invariant commutators \[eq:Xq\] and \[eq:XeDirac\] and because a commutator is traceless, we get these sum rules. References {#references .unnumbered} ========== [99]{} C. Smith, S.Touati, . C. Jarlskog, . G. C. Branco, R. G. Felipe, F. R. Joaquim, .

--- author: - 'Kenzo <span style="font-variant:small-caps;">Ogure</span>$^{1,}$[^1] and Yoshiyuki <span style="font-variant:small-caps;">Kabashima</span>$^{2,}$[^2]' title: Exact Analytic Continuation with Respect to the Replica Number in the Discrete Random Energy Model of Finite System Size --- Introduction ============ The replica method (RM) is one of the few analytic schemes available for the research of disordered systems [@Beyond]. In physics, this method has been well known since the 1970s and has been successfully applied to the analyses of spin-glass models [@EA; @SK; @Parisi], although the essential idea behind the method can be dated back to the end of 1920s where it appeared as a theorem for computing the average of logarithms [@Hardy; @Riesz; @Hardy_book]. More recently, considerable attention has been paid to the similarity between statistical mechanics of disordered systems and the Bayesian method in problems related to information processing (IP) [@Nishimori]. The number and variety of applications of RM to problems the IP research are increasing rapidly, including error-correcting codes [@Sourlas; @KS], image restoration [@NishimoriWong; @TanakaKazu], neural networks [@learning], combinatorial problems [@Ksat; @Napsak], and so on. Although only the limit value $(1/N) \left \langle \ln Z \right \rangle=\lim_{n \to 0} \left (\left \langle Z^n \right \rangle^{1/N} -1 \right )/n$ is usually emphasized, RM can be considered to be a systematic procedure for calculating generalized moments $\left \langle Z^n \right \rangle$ of the partition function $Z$ in the case of $N \to \infty$ when $Z$ depends on a certain external randomness. Here, $N$ characterizes the system size, $n \in {\bf R} \mbox{ (or {\bf C})}$ is a real (or complex[@Saakian]) number and $\left \langle \cdots \right \rangle$ means the average over the external randomness. For most problems, a direct assessment of $\left \langle Z^n \right \rangle$ is difficult for a general $n \in {\bf R} \mbox{ (or {\bf C})}$, whereas such an assessment for natural numbers $n=1,2,\ldots $ is possible in the thermodynamic limit $N \to \infty$. Therefore, $\left \langle Z^n \right \rangle$ is first computed for natural numbers and their analytic continuation is used to extend $\left \langle Z^n \right \rangle$ to $n \in {\bf R} \mbox{ (or {\bf C})}$. This is usually termed the [*replica trick*]{}. However, the validity of the replica trick is doubtful. The most obvious analytic continuation, obtained under the replica symmetric (RS) ansatz, sometimes leads to the wrong results. The causes of these errors were actively debated in the 1970s until Parisi discovered the replica-symmetry-breaking (RSB) scheme for constructing reasonable solutions within the framework of RM [@Parisi]. Since this discovery, there have been no known examples for which physically wrong results have been derived by RM, in conjunction with the Parisi scheme if necessary. Therefore, RM is now empirically recognized as a reliable procedure in physics, although the mathematical justification of the replica trick still remains open. However, this problem is now generating interest again, in particular, in the application of RM to IP problems. Ths is because most theories in IP research have conventionally been developed with mathematical soundness highly valued[@Cover; @Learning]. The purpose of this paper is to provide a method to approach the problems of RM. Specifically, we give a useful formula to compute $\left \langle Z^n \right \rangle$ [*directly*]{} for $n \in {\bf C}$ at [*finite*]{} $N$ for a simple spin glass model, termed the discrete random energy model (DREM) [@REM; @Mou1; @Mou2]. This formula is numerically tractable, so one can directly observe how the system approaches the thermodynamic limit with the aid of numerical calculation. Furthermore, this [*analytically*]{} clarifies the correct behavior for $N \to \infty$, making a direct examination of the validity of RM. We have two main reasons for picking DREM from among the various disordered systems. First, this model is simple enough to handle analytically. It is already known that RM, in conjunction with the Parisi scheme, can evaluate the correct free energy for a family of random energy models (REM) including DREM at the limit of $n \to 0$ [@REM]. However, the existing procedure seems at odds with a theorem concerning analytic continuation provided by Carlson [@Carlson; @van_Hemmen], which holds for DREM of finite $N$ claiming the uniqueness of analytic continuation from natural numbers $n \in {\bf N}$ to complex numbers $n \in {\bf C}$, when the temperature is sufficiently low. For this, our approach shows that a phase transition occurs at a certain critical replica number $n_c \in [0,1]$ in such cases, which clarifies that RM can be consistent with Carlson’s theorem. This may offer a useful discipline to perform analytic continuation from $n \in {\bf N}$ to $n \in {\bf R} \mbox{ (or \bf C)}$ in RM. The second reason is a relationship between REM and certain problems of IP. Recent research on error-correcting codes has revealed that REM is closely related to a randomly constructed code [@Sourlas; @KS]. These codes are known to provide the best error correction performance in information theory[@Shannon], and the performance evaluation of such codes is similar to the computation of $\left \langle Z^n \right \rangle$ for $n \in {\bf R}$ [@Reliability] (see appendix \[ECC\]). Therefore, the current investigation will indirectly justify the RM-based analysis of error-correcting codes performed previously[@KS; @KMS; @JPAspecial_issue]. This paper is organized as follows. In section \[replica\] we introduce DREM and briefly review how RM has been employed in conventional analysis of this system. Referring to Carlson’s theorem, we address how the conventional scenario for taking a limit $n \to 0$ seems controversial. In order to resolve this difficulty, we propose in section \[exact\] a new scheme to directly evaluate $\left \langle Z^n \right \rangle$ for REM of finite $N$ and complex $n$ without using the replica trick. Taking the limit $N \to \infty$, we analytically clarify how $\left \langle Z^n \right \rangle$ behaves in the thermodynamic limit and numerically verify this behavior. In section \[thermo\] we show how RM can be consistent with the results obtained by the proposed scheme. Section \[summary\] is a summary. Replica method in the discrete random energy model (DREM) {#replica} ========================================================= In order to clearly state the problem addressed in this paper, we first review how RM has been conventionally employed in analyzing REM [@REM; @Mezard_Extreme]. For convenience in the later analysis, we mainly concentrate on DREM, but the addressed problem is widely shared with other versions of REM as well. A DREM is composed of $2^N$ states, the energies of which, $\epsilon_A$ $(A =1,2,\ldots, 2^N)$, are independently drawn from an identical distribution $$\begin{aligned} P(E_i)=2^{-M} \left( \begin{array}{c} M\\ \frac{1}{2}M+E_i \end{array} \right),\ \ \left (E_i=i-\frac{M}{2} \right ), \label{prob}\end{aligned}$$ over $M+1$ energy levels $E_i=-M/2,-M/2+1, \ldots, M/2-1, M/2$. For each realization $\{ \epsilon_A\}$, the partition function $$\begin{aligned} Z=\sum_{A=1}^{2^N}\exp(-\beta \epsilon _A), \label{part}\end{aligned}$$ and the free energy (density) $$\begin{aligned} F=-\frac{kT}{N}\log{Z} \label{free_energy}\end{aligned}$$ can be used for computing various thermal averages. However, when the [*configurational average*]{} is required, one has to compute the averaged free energy $\left\langle F \right\rangle= -\frac{kT}{N}\left\langle\log{Z}\right\rangle$, the direct evaluation of which is generally difficult. Here, $\left \langle \cdots \right \rangle$ denotes the configurational average with respect to $\{ \epsilon_A\}$. On the other hand, the moment of the partition function $\left\langle Z^n \right\rangle$ can be easily calculated in various models for natural numbers $n=1,2,\ldots$. Therefore, the replica method evaluates the averaged free energy using the [*replica trick*]{} $$\begin{aligned} \frac{1}{N}\left \langle \log{Z}\right \rangle =\lim_{n\rightarrow 0} \frac{\left\langle Z^n \right\rangle^{\frac{1}{N}}-1}{n}, \label{repid}\end{aligned}$$ analytically continuting the expression of $\left\langle Z^n \right\rangle$ for $n=1,2,\ldots$ to that for real (or complex) numbers $n$. For a given natural number $n$, the moment of DREM is calculated as $$\begin{aligned} \left \langle Z^n \right \rangle &=& \sum_{A_1=1}^{2^N} \sum_{A_2=1}^{2^N} \dots \sum_{A_n=1}^{2^N} \sum_{i_1=0}^{M}P \left (E_{i_1}^{(1)} \right ) \sum_{i_2=0}^{M}P \left (E_{i_2}^{(2)} \right ) \dots \sum_{i_{2^N}=0}^{M}P \left (E_{i_{2^N}}^{(2^N)} \right )\nonumber\\ &&\exp \left ( -\beta \sum_{B=1}^{2^N}E _{i_B}^{(B)}\sum_{\mu =1}^n\delta _{BA_{\mu}} \right )\cr &=& e^{\frac{n M\beta }{2}} \sum_{A_1=1}^{2^N} \sum_{A_2=1}^{2^N} \dots \sum_{A_n=1}^{2^N} \prod_{B=1}^{2^N} \left ( \frac{1+e^{-\beta \sum_{\mu =1}^n \delta_{BA_\mu}}}{2} \right )^M \cr &=& \sum_{A_1=1}^{2^N} \sum_{A_2=1}^{2^N} \dots \sum_{A_n=1}^{2^N} \prod_{B=1}^{2^N} \exp \left (N \alpha I \left (\beta \sum_{\mu=1}^n \delta_{BA_\mu} \right ) \right ) \label{mom}\end{aligned}$$ where $\alpha=M/N$, the function $I(x)$ is defined as $$\begin{aligned} I(x)= \log \left (\cosh \frac{x}{2} \right ). \label{I}\end{aligned}$$ An identity $\prod_{B=1}^{2^N} \exp \left (-\frac{\beta}{2} \sum_{\mu=1}^n \delta_{BA_\mu} \right ) =\exp \left (-\frac{n\beta}{2} \right )$ was employed for obtaining the final expression of Eq. (\[mom\]). From now on, we focus on the case $\alpha >1$, for which the replica symmetry can be broken when the temperature is sufficiently low. Unfortunately, performing the summation in Eq. (\[mom\]) exactly is difficult. Instead, in conventional RM, Eq. (\[mom\]) is represented by the most dominant contribution in the summation. This can be justified for natural numbers $n$ in the limit $N \to \infty$. Notice that the summation is invariant with respect to the permutation of the replica indices $\mu=1,2,\ldots,n$. This [*replica symmetry*]{} restricts the candidates of the most dominant contribution to three possibilities, which are here referred to as solutions of the replica symmetric 1 (RS1), the replica symmetric 2 (RS2), and the 1-step replica symmetry breaking (1RSB). - RS1\ In RS1, all of the $n$ replicas are assumed to occupy $n$ different states $B(=1,2,\ldots, 2^N)$. Therefore, for a given $B$, $$\begin{aligned} \sum_{\mu=1}^n \delta_{BA_{\mu}} = \left\{ \begin{array}{ll} 1 & (\mbox{when $B$ is one of the $n$ occupied states}), \\ 0 & (\mbox{otherwise}). \end{array} \right.\end{aligned}$$ The number of ways to assign $n$ replicas to $n$ out of $2^N$ different states is $$\begin{aligned} 2^N\times(2^N-1)\times\dots \times(2^N-n+1) \sim 2^{nN}. \end{aligned}$$ For each case, the configuration of this type contributes $$\begin{aligned} \exp{ \left (nN \alpha {I}(\beta) \right )}, \end{aligned}$$ in Eq. (\[mom\]). This means that the contribution to the moment from RS1 becomes $$\begin{aligned} \left\langle Z^n \right\rangle &=& 2^{nN}\exp{\left (nN \alpha {I}(\beta) \right )}\cr &=& \exp{nN \left (\log{2}+ \alpha \log{ \left (\cosh{\frac{\beta}{2}} \right ) } \right )}.\label{rs1}\end{aligned}$$ - RS2\ In RS2, all of the $n$ replicas are assumed to occupy a particular state $B$. Therefore, for a given $B$, $$\begin{aligned} \sum_{\mu=1}^n \delta_{BA_{\mu}} = \left\{ \begin{array}{ll} n & (\mbox{when $B$ is the occupied state}),\\ 0 & (\mbox{otherwise}). \end{array} \right.\end{aligned}$$ The number of ways to choose one out of $2^N$ states is $2^N$. For each case, the configuration of this type contributes $$\begin{aligned} \exp{ \left (N \alpha {I}(n\beta) \right )}, \end{aligned}$$ in Eq. (\[mom\]), which indicates that the contribution from RS2 is $$\begin{aligned} \left\langle Z^n\right\rangle &=& 2^{N}\exp{ \left (nN\alpha {I}(n\beta) \right )} \cr &=& \exp{N \left (\log{2}+ \alpha \log{ \left (\cosh{\frac{n\beta}{2}} \right )} \right )}. \label{rs2}\end{aligned}$$ - 1RSB\ In 1RSB, $n$ replicas are assumed to be equally assigned to $n/m $ states $B$, where $m$ is an aliquot of $n$. Therefore, for a given $B$, $$\begin{aligned} \sum_{\mu=1}^n \delta _{BA_{\mu}} = \left\{ \begin{array}{ll} m & (\mbox{when $B$ is one of the $n/m$ occupied states}),\\ 0 & (\mbox{otherwise}). \end{array} \right.\end{aligned}$$ The number of ways to select $n/m$ out of $2^N$ states equally assigning $n$ replicas to the $n/m$ states is $$\begin{aligned} \frac{(2^N)!}{(2^N-n/m)! }\times \frac{n!}{m^{n/m}} \sim 2^{\frac{n}{m}N}. \end{aligned}$$ For each case, the configuration of this type contributes $$\begin{aligned} \exp{\left (\frac{n}{m} N\alpha {I}(m\beta) \right )}, \end{aligned}$$ in Eq. (\[mom\]). Taking all the possibile values of $m$ into account, the contribution from 1RSB can be summarized as $$\begin{aligned} \left\langle Z^n\right\rangle &=& \sum_{m}2^{ \frac{n}{m} N} \exp \left (\frac{n}{m} N \alpha {I} (m\beta) \right )\cr &=& \sum_{m}\exp \left (\frac{n}{m} N \left [\log{2}+ \alpha {I}\left (m \beta \right ) \right ] \right )\cr &\sim& \exp \left ({\mathop{\rm extr}_{m} \left \{\frac{n}{m} N \left [\log{2}+ \alpha { I} \left (m\beta \right ) \right ] \right \}} \right ).\end{aligned}$$ In the last expression, we have replaced the summation over $m$ with the extremization with respect to $m$ ($\mathop{\rm extr}_m \{ \cdots \}$), which is hopefully valid for large $N$, analytically continuing the expression with respect to $m$ from natural numbers to real numbers. The extremization with respect to $m$ yields the condition $$\begin{aligned} \log{2}+ \alpha {I} \left (m \beta \right )= \alpha m \beta {I}^\prime \left (m \beta \right ),\label{con}\end{aligned}$$ which implies that the moment is expressed as $$\begin{aligned} \left\langle Z^n \right\rangle &=& \exp \left [n N \alpha \beta {I}^\prime\left (m_c \beta \right ) \right ], \end{aligned}$$ where $m_c$ is the solution of Eq. (\[con\]). Eq. (\[I\]) indicates that $m_c$ can be represented as $m_c=\beta_c/\beta$, where the critical inverse tempereture $\beta_c>0 $ is determined by $$\begin{aligned} \log{2}+\alpha \left (\log \left (\cosh \frac{\beta_c}{2} \right )-\frac{\beta_c}{2}\tanh\frac{\beta_c}{2}\right )=0. \label{cdef}\end{aligned}$$ This provides a simple expression of the 1RSB solution as $$\begin{aligned} \left \langle Z^n \right \rangle &=& \exp\left (nN \alpha \beta I^\prime(\beta_c) \right) \cr &=& \exp\left ( \frac{nN \alpha \beta}{2} \tanh \frac{\beta_c}{2} \right ). \label{1rsb}\end{aligned}$$ ![The configuration of replicas for each solution. In RS1, $n$ replicas are distributed in different states. In RS2, $n$ replicas are concentrated in one state. In 1RSB, $n$ replicas are equally assigned to $n/m$ states. For 1RSB, however, this figure does not directly correspond to the solution because the critical value of $m$ is not neccesarily an integer and can be larger than $n$ for $\beta>\beta_c$.[]{data-label="RSFIG"}](replicadistribution.eps){width="12cm"} The configurations of the replicas assumed for RS1, RS2, and 1RSB are pictorially presented in Fig. \[RSFIG\]. It should be emphasized here that the above three solutions are derived for $n=1,2,\ldots$, assuming $N$ is sufficiently large. However, the obtained expressions are likely to hold for real $n$ as well. Therefore, in conventional analysis, the replica trick of Eq. (\[repid\]) is carried out, selecting one possibly relevant solution of the three, which is hopefully valid for large $N$. The existing prescription for selecting the relevant solution is as follows [@REM]: For small $n$, RS1, RS2, and 1RSB are ordered RS2 $>$ RS1 $>$ 1RSB from the viewpoint of their amplitudes (Fig. \[phasecom\]). The contribution from RS2, however, converges to $2^N$ rather than unity for $n \to 0$, and therefore, the replica trick leads to divergence. Hence, this solution is discarded. After excluding this solution, the leading contribution always comes from RS1, which guarantees a finite limit in Eq. (\[repid\]). Actually, the answer obtained from this solution is correct in the case of high temperatures $0 < \beta < \beta_c$. However, this solution becomes invalid for low temperatures $\beta>\beta_c$, for which the correct answer is provided by 1RSB. It may be worth noting that the value of $m_c$, in this low temperature case, is placed in the interval $n \le m_c \le 1$ when $n \to 0$, which is out of the ordinary range $1 \le m_c \le n$ for $n=1,2,\ldots$. This prescription for taking the $n \to 0$ limit is empirically justified for a family of REMs because it reproduces the correct answers obtainable by other schemes at the limit of $n \to 0$[@REM]. However, the following two issues still need further investigation. The first issue is the reason for expurgating RS2. Carlson’s theorem, which guarantees the uniqueness of analytic continuation from natural $n \in {\bf N}$ to complex $n \in {\bf C}$, might be useful for resolving this problem[@Carlson; @van_Hemmen]. Unlike other systems such as the Sherrington-Kirkpatrick model[@SK] and the original REM[@REM], a modified moment of the current DREM $\left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N}$, which is extended from $n \in {\bf N}$ to $n \in {\bf C}$, satisfies an inequality, $$\begin{aligned} \left | \left \langle \left (e^{-M\beta/2} Z \right )^n \right \rangle^{1/N} \right | &\le & \left \langle \left (e^{-M\beta /2} Z \right )^{\Re(n)} \right \rangle^{1/N} \cr &=& \left \langle \left (\sum_{A=1}^{2^N} \exp\left [ -\beta (\epsilon_A +M/2) \right] \right )^{\Re(n)} \right \rangle^{1/N} \cr &\le& \left \langle \left (\sum_{A=1}^{2^N} 1 \right )^{\Re(n)} \right \rangle^{1/N} = 2^{\Re (n)} < O\left (\exp \left [ \pi |n| \right ] \right ), \label{eq:necessary_carlson}\end{aligned}$$ for $\Re(n) \ge 0$ and $\forall{N}=1,2,\ldots$, since $\epsilon_A +M/2$ is lower bounded by $0$. Suppose that we could construct another extention $\psi(n;N)$ $(n \in {\bf C})$, which satisfies the the growth condition, $\psi(n;N) < O\left (\exp \left [ \pi |n| \right ] \right )$ [^3], and agrees with $\left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N}$ at all the natural numbers $n=1,2,\ldots$. This indicates that a similar inequality $\left | \psi(n;N) - \left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N} \right | \le \left | \psi(n;N) \right |+ \left | \left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N} \right | < O\left (\exp [\pi |n|] \right )$ holds for ${\rm Re}(n) \ge 0 $ and the difference $\left | \psi(n;N) -\left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N} \right | $ vanishes at all the natural numbers $n=1,2,\ldots$, as $\psi(n;N)$ and $\left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N}$ completely coincide for $n \in {\bf N}$. Then, Carlson’s theorem (Theorem 5.81 in page 186 of Ref. ) ensures that $\left | \psi(n;N)- \left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N} \right | $ is identical to $0$, implying that $\psi(n;N)$ and $\left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N}$ are identical and, therefore, analytic continuation of $\left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N}$ from natural $n \in {\bf N}$ to complex $n \in {\bf C}$ can be uniquely determined, unless analyticity of $\left \langle (e^{-M\beta /2} Z)^n \right \rangle^{1/N}$ is lost in the limit $N \to \infty$. Since $e^{-M\beta/2}$ is a non-vanishing constant, this means that analytic continuation of the moment $\left \langle Z^n \right \rangle^{1/N}$ is also unique as long as $\left \langle Z^n \right \rangle^{1/N}$ remains analytic with respect to $n$ in the limit $N \to \infty$. Hence, the dominant solution for $n=1,2,\dots$, rather than for $0<n<1$, should be selected as the relevant solution for $n \to 0$. This recipe succesfully reproduces the correct answer for the high temperature region $0< \beta < \beta_c$. However, this is still not fully satisfactory because RS2 becomes dominant for $n=1,2,\ldots$ in the case of $ \beta > \beta_c$ and, therefore, should be selected as the relevant solution for $n \to 0$ if analyticity is preserved for $N \to \infty$, which, unfortunately, leads to a wrong answer. Hence, a certain phase transition must occur at a critical replica number $0<n_c<1$ for $\beta > \beta_c$. However, to the knowledge of the authors, such a phase transition with respect to $n$ has not yet been fully examined for most disordered systems[@van_Hemmen; @Horiguchi]. This might be because so far the greatest attention has been paid only to the final results at the limit $n \to 0$. However, detailed analysis of phase transitions of this type may soon be needed as the replica calculation for non-vanishing $n$ has recently begun to be employed in problems related to IP[@Monasson; @Reliability] and analysis of certain dynamics involved with multiple time scales[@Coolen]. The second question is the origin of 1RSB. In conventional analysis at the limit $n \to 0$[@REM; @Mou1; @Mou2; @Mezard_Extreme], this solution is introduced by modifying RS1 to keep the entropy of the correct solution non-negative for $\beta > \beta_c$. It would appear that 1RSB originates from RS1. However, at least for positive $n$, this association seems unlikely because the two solutions cross each other only at $n=0$ (see Fig. \[phasecom\]). Therefore, it is impossible to relate the origin of 1RSB to RS1 at large $n$, from which the solutions of smaller $n$ are extrapolated. On the other hand, 1RSB contacts RS2 at a certain point of positive $n$, implying that 1RSB bifurcates from RS2. As the contact point is located in $0<n<1$ for $\beta > \beta_c$, this seems consistent with the aforementioned possible phase trainsition at $n=n_c$ (see Fig. \[phasecom\] (b)). Nevertheless, such a scenario cannot be so easily accepted since RS2 still dominates 1RSB even below the contact point; a mere contact does not change the dominance between the two solutions. ![ Solutions obtained under the RS1, RS2 and 1RSB assumptions. RS1 and RS2 agree at $n=1$ while RS2 contacts 1RSB at a certain point of positive $n$. (a) For $\beta < \beta_c$, the contact point is placed in the region of $n > 1$. (b) For $\beta> \beta_c$, on the other hand, it is located in $0 < n < 1$. []{data-label="phasecom"}](phasecom.eps){width="12cm"} It may help in resolving these problems to analyze DREM employing a completely different methodology. In the next section, we provide a scheme to calculate the moment of DREM as a step towards clarifying the mysteries of RM. The proposed method is powerful enough to evaluate the moment in the right half of the complex plane $\Re(n)>0$ for any arbitrary finite $N$ without using the replica trick, which supplies a model answer for RM. Direct calculation of the moment for grand canonical DREM (GCDREM) {#exact} ================================================================== General formula --------------- In order to introduce a novel scheme for caluculating moments of the partition function of DREM, we first rewrite the partition function using the occupation numbers $n_i$ ($i=0,1,2,\ldots,M$) as $$\begin{aligned} Z= \sum_{A=1}^{2^N}\exp \left (-\beta \epsilon_A \right ) = \sum_{i=0}^{M} n_i \exp \left (-\beta E_i \right ) = \omega^{-\frac{M}{2}}\sum_{i=0}^{M} n_i \omega^i \ \ (\omega\equiv e^{-\beta}). \label{z}\end{aligned}$$ This means that the partition function of DREM can be completely determined by a set of occupation numbers $\{n_i \}$, in which details of the energy configuration $\{\epsilon_A\}$ are ignored. This makes it possible to assess the moments $\left \langle Z^n \right \rangle$ directly from $\{n_i \}$ without referring to the full energy configuration $\{\epsilon_A\}$. This significantly reduces the necessary cost for computing the partition function when the calculation is numerically performed. Two methods are known for generating $\{n_i\}$. The straightforward method is to count $n_i$ independently, drawing the $2^N$ energy states from Eq. (\[prob\]). The system obtained by this is referred to as the canonical discrete random energy model (CDREM). Although this yields a rigorously correct realization of DREM that satisfies the constraint $\sum_{i=0}^{M} n_i = 2^N$, it takes $2^N$ steps to count $\{n_i\}$ and hence, is computationally difficult. In order to resolve this difficulty in numerical experiments, Moukarzel and Parga [@Mou1; @Mou2] have proposed the grand canonical version of the discrete random energy model (GCDREM)[^4]. In GCDREM, the occupation numbers are independently determined using the Poisson distribution $$\begin{aligned} P(n_i) = e^{-\gamma _i}\frac{\gamma _i^{n_i}}{n_i!}, \ \ (\gamma _i = 2^N P(E_i)), \label{Poisson}\end{aligned}$$ where $\gamma_i$ is the average occupation number. The greatest advantage of GCDREM is that one can drastically reduce the necessary computational cost for generating $\{ n_i \}$ from $2^N$ to $M+1$. One possible drawback is that the constraint $\sum_{i=0}^{M} n_i = 2^N$ is only satisfied in average, $\left\langle\sum_{i=0}^{M} n_i\right\rangle = 2^N$, which implies that this method does not correspond strictly to the original model. However, the RM-based calculation indicates that thermodynamical properties of GCDREM become identical to those of CDREM for $N \to \infty$, which is provided in appendix \[GCDREM\], and one can show that the difference rapidly vanishes as $N$ becomes large and is almost indistinguishable even for $N=3$, verified in appendix \[CDREM\]. In addition, this version of DREM has another advantage in analytic calculations as the summation can be carried out independently, as is shown below. In GCDREM, the moment is expressed as $$\begin{aligned} \left\langle Z^n \right\rangle &=& \sum_{n_0=0}^{\infty} \sum_{n_1=0}^{\infty} \cdots \sum_{n_M=0}^{\infty} P(n_0) P(n_1) \cdots P(n_M) Z^n \cr &=& \omega^{-\frac{nM}{2}} \lim_{\epsilon \rightarrow 0} \sum_{n_0=0}^{\infty} \sum_{n_1=0}^{\infty} \cdots \sum_{n_M=0}^{\infty} P(n_0) P(n_1) \cdots P(n_M) \left (\sum_{i=0}^{M} n_i \omega^i+\epsilon \right )^n,\end{aligned}$$ where an infinitesimal constant $\epsilon >0$ is introduced in order to keep $Z$ positive even when all the occupation numbers vanish. This makes it possible to employ an identity for the positive number $c$ $$\begin{aligned} c^n = \frac{\int_H (-\rho)^{-n-1} e^{-c\rho}d\rho}{\tilde \Gamma (-n)}, \ \ (c>0,\tilde\Gamma (n)\equiv -2i\sin n\pi \Gamma (n)),\label{cn}\end{aligned}$$ where the integration contour is shown in Fig. \[cont\], for assessing the moment as $$\begin{aligned} \left\langle Z^n \right\rangle &=& \frac{ \omega^{-\frac{nM}{2}} }{ \tilde\Gamma (-n) } \lim_{\epsilon \rightarrow 0} \int_H (-\rho)^{-n-1}e^{-(\sum_{i=0}^{M} n_i \omega^i+\epsilon)\rho}d\rho \cr &=& \frac{ \omega^{-\frac{nM}{2}} }{ \tilde\Gamma (-n) } \lim_{\epsilon \rightarrow 0} \int_H (-\rho)^{-n-1}e^{-\epsilon\rho} \left( \sum_{n_0=0}^{\infty}P(n_0)e^{-n_0\rho} \right)\left( \sum_{n_1=0}^{\infty}P(n_1)e^{-n_1\omega\rho} \right) \cr &&\cdots\left( \sum_{n_M=0}^{\infty}P(n_M)e^{-n_M\omega^M\rho} \right) d\rho.\end{aligned}$$ It is worth noting that the summation in this expression can be independently carried out as $$\begin{aligned} \sum_{n_i=0}^{\infty}P(n_i)e^{-n_i\omega^i\rho} = e^{-\gamma_i}\sum_{n_i=0}^{\infty}\frac{1}{n_i!} (\gamma_ie^{-\omega^i\rho})^{n_i} = \exp{[{-(1-e^{-\omega^i\rho})\gamma_i}]}.\end{aligned}$$ Therefore, the moment can be summarized as $$\begin{aligned} \left\langle Z^n \right\rangle &=& \frac{ \omega^{-\frac{nM}{2}} }{ \tilde\Gamma (-n) } \lim_{\epsilon \rightarrow 0} \int_H(-\rho)^{-n-1}\exp{[-\epsilon \rho-\sum_{i=0}^{M} (1-e^{-\omega^i \rho})\gamma _i]}d\rho.\end{aligned}$$ Since this is convergent for $\Re(n)>0$, the following expression gives the analytic continuation of the moment to the right half complex plane of $n$, $$\begin{aligned} \left\langle Z^n \right\rangle &=& \frac{ \omega^{-\frac{nM}{2}} }{ \tilde\Gamma (-n) } \int_H(-\rho)^{-n-1}\exp{[-\sum_{i=0}^{M} (1-e^{-\omega^i \rho})\gamma _i]}d\rho. \label{momcont}\end{aligned}$$ ![The integration contours.[]{data-label="cont"}](contorhi.eps){width="5cm"} Thermodynamic limit ------------------- Using Eq. (\[momcont\]), one can analytically examine the behavior of the moment in the thermodynamic limit $N,M \to \infty$, keeping $\alpha=M/N$ finite. For this, we first convert the contour integration in the expression to an integration on the real axis for $p-1<\Re(n)<p$, where $p$ is an arbitrary natural number. Further, we define a function $$\begin{aligned} f(\rho) \equiv \exp{[-\sum_{i=0}^{M}(1-e^{-\omega^i \rho})\gamma _i]} \equiv \sum_{j=0}^{\infty} f_i\rho^i, \end{aligned}$$ and a series of truncated summations as $$\begin{aligned} f^{(p)}(\rho) \equiv \sum_{j=0}^{p-1} f_i\rho^i, \end{aligned}$$ which satisfy the identity $$\begin{aligned} \int_H(-\rho)^{-n-1}f^{(p)}(\rho)d\rho = \int_{H+I}(-\rho)^{-n-1}f^{(p)}(\rho)d\rho = 0, \label{eq:identity}\end{aligned}$$ for $p-1 < \Re(n)$, since the contribution from the contour $I$ vanishes. Using this identity, the moment can be rewritten as $$\begin{aligned} \left\langle Z^n \right\rangle &=& \frac{ \omega^{-\frac{nM}{2}} }{ \tilde\Gamma (-n) } \int_H(-\rho)^{-n-1} [f(\rho)-f^{(p)}(\rho)] d\rho.\end{aligned}$$ Eq. (\[eq:identity\]) guarantees that the infrared divergence is removed for $\Re(n)<p$ in this expression. Therefore, the moment can be rewritten for $p-1<\Re(n)<p$ as $$\begin{aligned} \left\langle Z^n \right\rangle &=& \frac{ \omega^{-\frac{nM}{2}} }{ \Gamma (-n) } \int_0^{\infty}\rho^{-n-1} [f(\rho)-f^{(p)}(\rho)] d\rho, \label{eq:expression_for_p}\end{aligned}$$ where the function $\tilde\Gamma$ is replaced by the ordinary gamma function $\Gamma$. As we have a particular interest in the case of $p=1$, i.e. $0<\Re(n)<1$, let us focus on the behavior of the following expression, $$\begin{aligned} \left\langle Z^n\right\rangle &=& \frac{ \omega^{-\frac{nM}{2}} }{ \Gamma (-n) } \int_0^{\infty}\rho^{-n-1} [f(\rho)-1] d\rho \cr &=& \frac{ \omega^{-\frac{nM}{2}} }{ \Gamma (-n) } \int_0^{\infty}\rho^{-n-1} [\exp{\{-\sum_{i=0}^{M}(1-e^{-\omega^i \rho})\gamma _i\}}-1] d\rho. \end{aligned}$$ To examine the behavior of the thermodynamic limit, it is convenient to introduce new variables $$\begin{aligned} x\equiv\frac{i}{N\alpha},\ \ y\equiv\frac{1}{N \alpha \beta }\ln \rho, \end{aligned}$$ instead of $i$ and $\rho$, yielding the following expression for the moment $$\begin{aligned} \left \langle Z^n \right \rangle &=& \frac{ \omega^{-\frac{nM}{2}} }{ \Gamma (-n) } \int_{-\infty}^{\infty}e^{N{\cal G}(y)}dy, \label{znint}\end{aligned}$$ where $$\begin{aligned} \left\{ \begin{array}{ll} {\cal G}(y) &= -n\alpha \beta y +\ds\frac{1}{N}\log (1-e^{-{\cal F}(y)}),\\\\ {\cal F}(y) &= \ds\int_0^1 e^{N{\cal H}(x,y)}dx,\\\\ {\cal H}(x,y) &= (1-\alpha)\log 2 + \alpha H(x) + \ds\frac{1}{N}\log (1-e^{-e^{N\alpha \beta(y-x)}}), \end{array} \right. \label{eq:details}\end{aligned}$$ and $$\begin{aligned} H(x)=-x \log x-(1-x) \log (1-x). \label{Hx}\end{aligned}$$ Here, we have replaced the summation with an integration, which can be verified when both $M$ and $N$ are sufficiently large. For further analysis, the identity $$\begin{aligned} g(u)\equiv \frac{1}{N}\log \left (1-e^{-e^{Nu}}\right ) &\to& \left \{ \begin{array}{cl} 0 & (u \ge 0)\cr u & (u < 0) \end{array} \right . \cr &=& u \theta(-u) \label{eq:g_u}\end{aligned}$$ which holds for large $N$, may be useful. The shape of this function is displayed in Fig. \[ufunc\]. As this becomes singular at $u=0$, the phase transition may occur in $n$ space for $N\rightarrow\infty$ We examine this below. ![The shape of $g(u)$ for $N\rightarrow \infty$. This is not analytic at $u=0$. []{data-label="ufunc"}](ufunc.eps){width="5cm"} Eq. (\[eq:g\_u\]) means that the function ${\cal H}$ can be expressed as $$\begin{aligned} {\cal H}(x,y)=(1-\alpha)\log{2}+\alpha H(x) +\alpha\beta(y-x)\theta(y-x)\end{aligned}$$ in the thermodynamic limit. As a function of $x$, this exhibits three behaviors depending on a given value of $y$, as shown in Fig.\[h\]. Employing the saddle point method, the maximum of ${\cal H}(x,y)$ given $y$ provides ${\cal F}(y)$ for large $N$, which yields $$\begin{aligned} \lim_{N\rightarrow\infty} \frac{1}{N}\log{{\cal F}}(y)= \left\{ \begin{array}{c} \ds\frac{1}{2}\ \ \left (y>\ds\frac{1}{2} \right ),\\\\ (1-\alpha)\log{2}+\alpha H(x)\ \ \left (x_c<y<\ds\frac{1}{2} \right ),\\\\ (1-\alpha)\log{2}+\alpha {\tilde H}(\beta) + \alpha \beta y\ \ \left (y<x_c \right ), \end{array} \right. $$ where $x_c$ is a solution of $$H^\prime(x_c)=\beta, \label{xcdef}$$ the behavior of which is shown in Fig. \[f\]. Here, the function ${\tilde H}$ is the Legendre transformation of the function ${H}$, $$\begin{aligned} {\tilde H}(\beta) = \log{\left (2\cosh{\frac{\beta}{2}} \right )} -\frac{\beta}{2}. \end{aligned}$$ The function ${\cal G}$ becomes $$\begin{aligned} {\cal G}(y) = -n\alpha \beta y +\frac{1}{N}\log {\cal F}(y)\theta (-{\cal F}(y)) \label{eq:G_y}\end{aligned}$$ for large $N$, which directly controls the behavior of the moment in Eq. (\[znint\]). The behavior strongly depends on the relation between $x_c$ and $x^{*} =\left (1-\tanh\frac{\beta_c}{2} \right )/2$ which satisfies the condition $$\begin{aligned} (1-\alpha)\log{2}+\alpha H(x^{*})=0.\label{x*}\end{aligned}$$ ![The schematic shape of ${\cal H}(x,y)$ for $N\rightarrow \infty$. It depends on the value of $y$. []{data-label="h"}](h.eps){width="8cm"} ![The schematic shape of ${\cal F}(y)$ for $N\rightarrow \infty$ obtained from the maximal value of ${\cal H}(x,y)$. []{data-label="f"}](f.eps){width="5cm"} - \(A) $x^{*}<x_c$\ Since $x_c$ and $x^{*}$ are defined in Eq. (\[xcdef\]) and Eq. (\[x\*\]), respectively, the condition $x^{*}<x_c$ can be written as $$\begin{aligned} (1-\alpha)\log{2}+ \alpha H(x_c)>0.\end{aligned}$$ This is satisfied for $\beta<\beta_c$, i.e. the high temperature phase. Notice that for $\alpha\leq 1$, as Eq. (\[cdef\]) does not have a positive solution, this is always satisfied independently of $\beta$. Then, Eq. (\[eq:G\_y\]) can be represented as $$\begin{aligned} {\cal G}(y) = -n\alpha \beta y +[(1-\alpha)\log{2}+\alpha {\tilde H}(\beta) + \alpha \beta y]\theta (y_c-y),\end{aligned}$$ which is shown in Fig. \[g\]. Here, $y_c$ is defined by ${\cal F}(y_c)=0$ (Fig. \[f\]), which yields $$\begin{aligned} y_c = -\frac{1}{\alpha\beta}[(1-\alpha)\log{2}+\alpha {\tilde H}(\beta)].\end{aligned}$$ The moment is then calculated as $$\begin{aligned} \left\langle Z^n\right\rangle &=& -N\alpha\beta \frac{ \omega^{-\frac{nM}{2}} }{ \Gamma (-n) } \left[ \int_{-\infty}^{y_c} \exp{(1-\alpha)\log{2}+\alpha {\tilde H}(\beta) + \alpha \beta (1-n)y}dy\nonumber\right. \cr &&\hspace*{3cm}\left. +\int^{\infty}_{y_c}\exp{(-n\alpha\beta y)}dy \right]\\ &\sim& e^{\alpha\beta nN(\frac{1}{2}-y_c)} \left[\frac{1}{\Gamma (1-n)}+\frac{n}{\Gamma (2-n)} \right].\end{aligned}$$ Therefore, the asymptotic behavior of the moment is $$\begin{aligned} \lim_{N\rightarrow\infty} \frac{1}{N}\log{\left\langle Z^n\right\rangle} &=& \alpha\beta nN \left (\frac{1}{2}-y_c \right ) \cr &=& n \left (\log{2}+\alpha \log{\cosh{\frac{\beta}{2}}} \right ) \label{rs1_ours}\end{aligned}$$ in this phase. This is consistent with RS1, as can be seen from Eq. (\[rs1\]). ![The schematic figure of ${\cal G}(y)$. The maximal value determines the behavior of $\left \langle Z^n \right \rangle$. []{data-label="g"}](g.eps){width="10cm"} - \(B) $x^{*}>x_c$\ The condition $x^{*}>x_c$ is satisfied for $\beta>\beta_c$, which may correspond to the low temperature phase. The function ${\cal G}(y)$ has the form $$\begin{aligned} {\cal G}(y) = -n\alpha \beta y +[(1-\alpha)\log{2}+\alpha H(y)]\theta (x^{*}-y), \end{aligned}$$ the behavior of which is further classified into two cases depending on the replica number $n$. - \(i) $H^\prime(x^{*})>n\beta$\ Since $x^{*}$ is located in $x_c<x^{*}< \frac{1}{2}$, there exists a critical replica number $0<n_c<1$, which is characterized by $$\begin{aligned} n_c \equiv \frac{1}{\beta} H^\prime(x^{*})=\frac{\beta_c}{\beta}. \end{aligned}$$ The condition $H^\prime(x^{*})>n\beta$ is satisfied for $n<n_c$. The function ${\cal G}(y)$, shown in Fig. \[g\], is maximized at $y=x^{*}$. Therefore, the moment can be expressed as $$\begin{aligned} \lim_{N\rightarrow\infty} \frac{1}{N}\log{\left\langle Z^n\right\rangle} &=& n\alpha\beta\left (\frac{1}{2}-x^{*} \right ) \cr &=& \frac{n \alpha \beta}{2} \tanh\frac{\beta_c}{2}. \label{1rsb_ours}\end{aligned}$$ This behavior is identical to that of 1RSB predicted by RM, as can be seen in Eq. (\[1rsb\]). - \(ii) $H^\prime(x^{*})<n\beta$\ The condition $H^\prime(x^{*})<n\beta$ is satisfied for $n>n_c$. The function ${\cal G}(y)$ is maximized not at $y=x^{*}$ but at $y=y_c$. Therefore, the moment can be asymptotically expressed as $$\begin{aligned} \lim_{N\rightarrow\infty} \frac{1}{N}\log{\left\langle Z^n\right\rangle} &=& \log{2}+ \alpha \log{\cosh{\frac{n\beta}{2}}}. \label{rs2_ours}\end{aligned}$$ This coincides with the behavior of RS2 obtained by RM, as can be seen in Eq. (\[rs2\]). The results are summarized in Fig. \[phase\]. In the high temperature phase $\beta<\beta_c$, the behavior of the moment is simple, being expressed by RS1 of RM. In the low temperature phase $\beta>\beta_c$, the behavior of the moment has two possibilities depending on $n$. More specifically, in the limit $N \to \infty$, the moment approaches RS2 for $n > n_c$, whereas 1RSB represents the correct behavior for $n<n_c$. This means that there exists a phase transition in the space of replica number at $n=n_c$. In conclusion, these are consistent with the known results obtained by RM at the limit $n \to 0$ [@REM; @Mou1; @Mou2]. ![The behavior of the moment in the thermodynamic limit obtained from the exact expression (full lines). In the high temperature phase $\beta<\beta_c$, the behavior of the moment is simple and corresponds to the result obtained from RS1. In the low temperature phase $\beta>\beta_c$, the behavior of the moment has interesting properties. The moment corresponds to that obtained from RS2 for $n>n_c \equiv \beta/\beta_c$ and corresponds to that obtained from 1RSB for $n<n_c$. A phase transition occurs at $n=n_c$.[]{data-label="phase"}](phase.eps){width="12cm"} Numerical validation -------------------- Eq. (\[momcont\]) is formulated as a two dimensional summation with respect to $\rho \in {\bf C}$ and $i=0,1,\ldots,M$, which is numerically tractable. This means that one can utilize Eq. (\[momcont\]) or Eq. (\[znint\]) to numerically examine the behavior of DREM for a finite system size $N$, and how fast the results obtained for $N \to \infty$ become relevant as $N$ grows large. Fig. \[sam\] shows the logarithm of $\left \langle Z^n \right \rangle$ calculated by Eq. (\[znint\]) and $\left \langle Z^n \right \rangle$ numerically evaluated from 10, 1000 and 100000 experiments, with $N=10$. One can see that the data from the numerical experiments converge to the results of Eq. (\[znint\]). This verifies that our expression accurately provides the moment even for a finite system size. ![The logarithm of $\left \langle Z^n \right \rangle$ calculated from Eq. (\[znint\]) and data obtained from numerical experiments of Eq. (\[z\]) using 10, 1000 and 100000 samples. The system size is $N=10$. The points indicated as “Exact" are the results obtained from our expression Eq. (\[znint\]).[]{data-label="sam"}](n10a2.eps){width="8cm"} We next compare the results from our expression and RM in Fig. \[com\]. At high temperatures, our result is consistent with RS1 as expected. The difference is negligible for all of the range $0<n<1$ even at $N=10$. At low temperatures, our result fits RS2 for larger $n>n_c$, while 1RSB exhibits excellent consistency for smaller $n<n_c$. There is a slight difference between our expression and 1RSB for $N=10$. The difference, however, becomes indistinguishable for $N=100$. This strongly indicates that there occurs a phase transition between RS2 and 1RSB at $n=n_c$ in the limit $N \to \infty$. ![ The logarithm of $\left \langle Z^n \right \rangle$ obtained from our expression and RM. The points indicated as “Exact" are the results obtained from our expression Eq. (\[znint\]). The lines indicated “RS1", “RS2", and “1RSB" are the results obtained from RM, using Eq. (\[rs1\]), Eq. (\[rs2\]), and Eq. (\[1rsb\]), respectively. At high temperatures $\beta=\beta_c/3<\beta_c$, our result is consistent with RS1. The difference is very small even for $N=10$. At low temperatures $\beta=3\beta_c > \beta$, our result fits RS2 for larger $n>n_c=\beta_c/\beta=1/3$ while 1RSB provides good consistency for smaller $n<n_c$. There is a slight difference between the results from our expression and that from 1RSB at $N=10$. The difference, however, becomes imperceptible for $N=100$.[]{data-label="com"}](repintcomp.eps){width="6cm"} Origin of 1RSB: Extreme value statistics {#thermo} ======================================== The preceding two sections indicate that DREM exhibits a phase transition with respect to the replica number $n$ at a certain critical point $n_c \in [0,1]$ when the temperature is sufficiently low. In this section, we discuss how this transition can be understood in the framework of RM. A formalism previously introduced for examining the domain size distribution of multi-layer perceptrons is useful for this purpose [@Monasson]. Since the partition function of DREM typically scales exponentially with respect to $N$, we first express this dependence as $$\begin{aligned} Z \sim \exp \left [-N\alpha \beta \left (y -\frac{1}{2} \right )\right ], \label{eq:free_energy}\end{aligned}$$ where $y-\frac{1}{2}$ represents the free energy normalized by the scale of the energy amplitude $M$ for a given realization $\{\epsilon_A\}$ ($A=1,2,\ldots,2^N)$ (or $\{n_i \}$ ($i=0,1,\ldots,M$)). Clearly, $y$ is a random variable. Let us assume that the probability distribution of $y$, ${\cal P}(y)$, scales as $$\begin{aligned} {\cal P}(y) \sim \exp \left [ - N c(y) \right ], \label{eq:free_probability}\end{aligned}$$ where $c(y) \sim O(1)$ for large $N$. Notice that an inequality $$\begin{aligned} c(y) \ge 0, \label{eq:c_inequality}\end{aligned}$$ must hold to make ${\cal P}(y)$ satisfy the normalization condition $\int dy {\cal P}(y)=1$ for $N \to \infty$. Eq. (\[eq:free\_probability\]) indicates that the moment of the partition function can be calculated as $$\begin{aligned} \left \langle Z^n \right \rangle &\equiv& \int dy \exp \left [-Nn\alpha \beta \left (y -\frac{1}{2} \right )\right ]\cP(y) \cr &\sim & \exp \left [N \mathop{\rm extr}_{y} \left \{- n \alpha \beta \left (y-\frac{1}{2}\right ) - c(y) \right \} \right ]. \label{eq:free_legendre}\end{aligned}$$ This formula, however, may not be useful for computing $\left \langle Z^n \right \rangle$, as directly assessing $c(y)$ is rather difficult. Instead, it can be utilized to evaluate $c(y)$ from $\left \langle Z^n \right \rangle$, since Eq. (\[eq:free\_legendre\]) implies that $c(y)$ can be obtained from $$\begin{aligned} c(y)= \mathop{\rm extr}_{n} \left \{ -n \alpha \beta \left (y-\frac{1}{2}\right ) -\frac{1}{N} \ln \left \langle Z^n \right \rangle \right \}, \label{eq:c_g}\end{aligned}$$ where $(1/N) \ln \left \langle Z^n \right \rangle$ can be computed by RM for any real number $n$. This, in conjunction with the normalization constraint in Eq. (\[eq:c\_inequality\]), offers a useful clue to identify the origin of the phase transition with the aid of RM. For $\beta > \beta_c$, RS2 provides the dominant solution for natural numbers $n=1,2,\ldots$ in the thermodynamic limt. Carlson’s theorem implies that this should offer a unique analytic continuation if no phase transition occurs. Therefore, let us first insert this solution, from Eq. (\[rs2\]), into Eq. (\[eq:c\_g\]), giving $$\begin{aligned} c(y)&=&\mathop{\rm extr}_{n} \left \{ -n \alpha \beta \left (y-\frac{1}{2} \right ) -\log 2 - \alpha \log{ \left (\cosh{\frac{n\beta}{2} } \right ) } \right \} \cr &=& (\alpha -1) \log 2 - \alpha H(y). \label{eq:extreme_value_stat}\end{aligned}$$ Notice that this does not satisfy the necessary condition given in Eq. (\[eq:c\_inequality\]) for $y > x^*$, and the critical value $y=x^*$ corresponds to $n=n_c=H^\prime(x^*)/\beta=\beta_c/\beta$, which signals the occurrence of a phase transition at $n=n_c$ within the framework of RM. The following consideration about the energy configuration provides a plausible scenario for this transition. As have already been pointed out [@Mezard_Extreme], the partition function $Z$ in the low temperature region $\beta > \beta_c$, can be considered to be dominated by only the minimum energy $\epsilon_{\rm \tiny min}$ of a given energy configuration $\{\epsilon_A \}$, such as $Z \sim \exp \left [ -\beta \epsilon_{\rm \tiny min} \right ]$. This implies that $\cP(y)$ of the normalized free energy $y-\frac{1}{2}=-(\log Z)/(N\alpha\beta)$ is given by the distribution of $\epsilon_{\rm \tiny min}/M$ under the assumption that each energy level $\epsilon_A$ $(A=1,2, \ldots, 2^N)$ is independently drawn from an identical distribution, given in Eq. (\[prob\]), which has been already examined in the research of [*extreme value statistics*]{} (EVS) [@extreme_value]. In the current case, the average occupation number of an energy level $E_i=i-\frac{1}{2}M=M \left (x-\frac{1}{2} \right )$, $\gamma_i=2^{N-M} \left ( \begin{array}{c} M \cr M/2+E_i \end{array} \right ) =\exp N \left ((1-\alpha)\log 2 + \alpha H(x) \right )$ ($i=0,1,\ldots,M$ and $x=i/M \in [0,1]$), grows exponentially large with respect to $N$ for $x>x^*$. This indicates that $\cP(y)$ is very small for $y > x^*$ because $x>x^*$ does not provide $\epsilon_{\rm \tiny min}/M$ for a given energy configuration $\{ \epsilon_{A} \}$ except for very rare cases, as energy levels lower than $E_i=M\left (x-\frac{1}{2} \right )$ are included in the configuration with a very high probability. On the other hand, $\gamma_i$ becomes exponentially small for $0<x<x^*$, which means that the probability of having an energy level labelled by $x$ of this interval in the energy configuration is low. Therefore, $\cP(y)$ also becomes small for $0<y<x^*$. However, the functional form of $\cP(y)$ is not symmetric between $y>x^*$ and $0< y < x^*$. Actually, detailed analysis of EVS[@extreme_value; @Mezard_Extreme] shows that $\cP(y)$ exhibits the asymmetric scalings $$\begin{aligned} \cP(y) \sim \left \{ \begin{array}{ll} \exp \left [- \exp \left [ -N \left ((\alpha -1) \log 2 - \alpha H(y) \right ) \right ] \right ], & (\mbox{$x^*<y$}), \cr \exp \left [ -N \left ((\alpha -1) \log 2 - \alpha H(y) \right ) \right ], & (\mbox{$0< y< x^*$}), \end{array} \right . \label{eq:P_y}\end{aligned}$$ for large $N$, which provide a singularity at $y=x^*$. It may be worth noting that $1 - \exp \left [- {\cal F}(y) \right ]$, which comes out in Eq. (\[znint\]) after inserting Eq. (\[eq:details\]) in the analysis presented section \[exact\], corresponds to the cumulative distribution $\int_{-\infty}^y d\tilde{y} {\cal P}(\tilde{y})$. Eq. (\[eq:P\_y\]) implies that the extremum point in Eq. (\[eq:free\_legendre\]) for large $n(>n_c)$ is in $0<y < x^*$, which leads to RS2 and consistently provides the exponent of Eq. (\[eq:extreme\_value\_stat\]). On the other hand, the constant $y=x^*$ offers the extremum for all small $n(<n_c)$, which corresponds to 1RSB in the framework of RM. The intuitive implication of this is that $\left \langle Z^n \right \rangle$ is dominated by an atypically low minimum energy generated with a small probability for $n > n_c$, whereas the typical minimum energy $\epsilon_{\rm \tiny min}= M \left (x^*-\frac{1}{2} \right )$ provides the major contribution to $\left \langle Z^n \right \rangle$ for $n < n_c$. Thus, the origin of the phase transition can be attributed to the singularity of the free energy distribution $\cP(y)$, exhibited in the case of low temperatures $\beta > \beta_c$. Our analysis also illustrates that Carlson’s theorem is not necessarily useful for validating RM because an analytic continuation given for finite $N$ can exhibit a singularity in the limit of $N \to \infty$ and, therefore, taking the limit $N \to \infty$ prior to $n \to 0$ for determining the continuation on the basis of expressions for $n=1,2,\ldots$, which is usually performed in RM, sometimes yields a wrong answer for $n<1$. However, in the current system, this drawback can be resolved by taking a constraint for the free energy distribution (i.e. Eq. (\[eq:c\_inequality\])) into account, which leads to the conventional 1RSB. Although the importance of the notion of EVS in RM has been already addressed at the limit $n \to 0$ [@Mezard_Extreme], to our knowledge, the current analysis is the first work that directly clarifies how the property of EVS relates to a corruption of the analytic continuation in RM, expressed as a phase transition with respect to the replica number $n$. Summary ======= In summary, we have offered an exact expression, Eq. (\[momcont\]), of the moment of the partition function $\left\langle Z^n \right\rangle$ for GCDREM, without employing the replica trick, which is valid for any arbitrary system size $N$ and complex number $n\ (\Re(n)>0)$. Simplifying the expression for $0<n<1$, we have shown that a phase transition with respect to the number of replicas $n$ occurs at a certain critical number $n_c \in [0,1]$ in the thermodynamic limit $N \to \infty$ when the temperature $\beta^{-1}$ is sufficiently low, if the ratio $\alpha=M/N$ is greater than unity. This implies that Carlson’s theorem, which guarantees the uniqueness of the analytic continuation from the expressions for $n=1,2,\ldots$ to those for $n \in {\bf C}$, is not necessarily useful for validating the replica method, because taking the thermodynamic limit $N \to \infty$ prior to the continuation fails to derive a correct result when the analyticity is broken for $n<1$ in the case of infinite $N$. However, it has also been shown that this drawback can be overcome by taking the statistical property of the minimum energy level into account, clarifying how 1RSB originates from a replica symmetric solution, which has been conventionally discarded, at the critical replica number $n_c$. We hope that the results obtained here offer a useful insight to unlock the remaining mysteries of RM. Since Eq. (\[momcont\]) is valid throughout the right half complex plane $\Re(n) > 0$ for any finite $N$, one can directly observe how the singularities of $\left\langle Z^n \right\rangle$ approach the real axis of $n$ as $N \to \infty$[@van_Hemmen]. This is sometimes referred to as Lee-Yang’s scenario of the phase transition[@Lee_Yang]. Analysis along this direction is currently under way[@Ogu]. Acknowledgement {#acknowledgement .unnumbered} =============== Support from Grant-in-Aids of MEXT, Japan, No. 14084206 (YK) is acknowledged. Link of DREM to error-correcting codes {#ECC} ====================================== As was shown in Ref. , certain types of spin glass models can be related to error-correcting codes. We here show that a known evaluation scheme of the error correcting ability of a random code ensemble can be reduced to a calculation of the general moment of the partition function with respect to DREM[@Gallager; @JPA_review]. To the knowledge of authors, a remark on this relationship has already been made in Ref. . In a general scenario of error correcting codes, an $N$-dimensional binary vector ${\mbox{\boldmath{$s$}}}=(s_1,s_2,\ldots,s_N)$ ($s_i=0,1$, $i=1,2,\ldots,N$) is encoded to a codeword ${\mbox{\boldmath{$t$}}}=(t_1,t_2,\ldots,t_M)$ ($t_i=0,1$, $i=1,2,\ldots,M$) of an $M(>N)$-dimensional binary vector and transmitted via a noisy channel. We here concentrate on a binary symmetric channel (BSC) in which each component of the codeword $t_i$ is independently flipped to the opposite letter in the alphabet ($0$ or $1$) with a probability $0<p<1/2$. This implies that the vector ${\mbox{\boldmath{$r$}}}={\mbox{\boldmath{$t$}}}+{\mbox{\boldmath{$n$}}}$ (mod 2) is received at the other terminal of the channel, where ${\mbox{\boldmath{$n$}}}$ is the noise vector, the component of which becomes $1$ independently with probability $p$ and $0$ otherwise. However, as the codeword is represented somewhat redundantly, decoding ${\mbox{\boldmath{$r$}}}$ provides the correct message ${\mbox{\boldmath{$s$}}}$ with a high probability for sufficiently small $p$. It is known that the performance of correctly retrieving ${\mbox{\boldmath{$s$}}}$ from ${\mbox{\boldmath{$r$}}}$ becomes good when a code ${\cal C}$ (i.e., an invertible map $\cal C:{\mbox{\boldmath{$s$}}}\leftrightarrow {\mbox{\boldmath{$t$}}}$) is randomly constructed [@Shannon; @Gallager], providing a code ensemble. Let us evaluate the average probability of failing to correctly retrieve ${\mbox{\boldmath{$s$}}}$ in order to characterize the potential error correcting ability of the ensemble. We focus on the maximum likelihood (ML) decoding, which selects a codeword closest to the received vector ${\mbox{\boldmath{$r$}}}$ and returns a message vector corresponding to the codeword as an estimate $\hat{{\mbox{\boldmath{$s$}}}}$ of ${\mbox{\boldmath{$s$}}}$. ML decoding minimizes the decoding error probability $P_E({\cal C})$ when codeword vectors ${\mbox{\boldmath{$t$}}}$ are equally generated at the transmission terminal [@Iba_JPA]. Notice that the message ${\mbox{\boldmath{$s$}}}$ can be correctly identified if its codeword ${\mbox{\boldmath{$t$}}}$ is provided, since the code is constructed as an invertible map. Therefore, for evaluating $P_E(\cal C)$, it is convenient to introduce an indicator function $\Delta_{\rm \tiny ML}({\mbox{\boldmath{$t$}}},{\mbox{\boldmath{$r$}}}|{\cal C})$ that returns $1$ if ${\mbox{\boldmath{$r$}}}$ is not correctly decoded to the correct codeword ${\mbox{\boldmath{$t$}}}$ and $0$ otherwise, which means the decoding error probability can be computed by $$\begin{aligned} P_E({\cal C})=\sum_{{\mbox{\boldmath{$t$}}},{\mbox{\boldmath{$r$}}}}P({\mbox{\boldmath{$t$}}}|{\cal C})P({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$t$}}}) \Delta_{\rm \tiny ML}({\mbox{\boldmath{$t$}}},{\mbox{\boldmath{$r$}}}|{\cal C}), \label{eq:block_error}\end{aligned}$$ and, hence, its average over the code ensemble can be represented as $$\begin{aligned} \left \langle P_E({\cal C}) \right \rangle_{\cal C}= \sum_{{\cal C}}P( {\cal C} )\sum_{{\mbox{\boldmath{$t$}}}\in {\cal C},{\mbox{\boldmath{$r$}}}} P({\mbox{\boldmath{$t$}}}|{\cal C}) P({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$t$}}})\Delta_{\rm \tiny ML}({\mbox{\boldmath{$t$}}},{\mbox{\boldmath{$r$}}}|{\cal C}), \label{eq:average_block_error}\end{aligned}$$ where $P({\mbox{\boldmath{$t$}}}|{\cal C})$ is the probability that the codeword vector ${\mbox{\boldmath{$t$}}}$ is generated given ${\cal C}$ and $P({\mbox{\boldmath{$r$}}}| {\mbox{\boldmath{$t$}}})$ is the conditional probability that ${\mbox{\boldmath{$r$}}}$ is received when ${\mbox{\boldmath{$t$}}}$ is transmitted. $P( {\cal C} )$ is the probability that a code ${\cal C}$ is generated. $\sum_{{\mbox{\boldmath{$t$}}}\in {\cal C}}$ denotes a summation over $2^N$ codeword vectors given ${\cal C}$. We assume that the code ${\cal C}$ is designed by the source coding technique such that ${\mbox{\boldmath{$t$}}}$ is uniformly generated as $P({\mbox{\boldmath{$t$}}}|{\cal C})=1/2^N$[@Gallager]. In addition, for BSC, the conditional probability can be represented as $$\begin{aligned} P({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$t$}}})= \frac{\exp\left [-F \sum_{i=1}^M \left (\frac{1}{2}-\delta_{r_i,t_i} \right ) \right ]} {\left (2 \cosh \frac{F}{2}\right )^M }, \label{eq:conditional_BSC}\end{aligned}$$ where $\delta_{x,y}$ returns $1$ if $x=y$ and $0$, otherwise and $F=\log\left [(1-p)/p \right ]$. Unfortunately, expressing $\Delta_{\rm \tiny ML}({\mbox{\boldmath{$t$}}},{\mbox{\boldmath{$r$}}}|{\cal C})$ in a rigorously treatable form is difficult. However, Gallager’s inequality $$\begin{aligned} \Delta_{\rm \tiny ML}({\mbox{\boldmath{$t$}}},{\mbox{\boldmath{$r$}}}|{\cal C}) \le \left ( \sum_{{\mbox{\boldmath{$t$}}}^\prime \in {\cal C} \backslash {\mbox{\boldmath{$t$}}}} \left (\frac{P({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$t$}}}^\prime )}{ P({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$t$}}})} \right )^{\frac{1}{{1+n}}} \right )^n, \label{eq:gallager1}\end{aligned}$$ which holds for arbitrary $n \ge 0$, offers a good upperbound for this [@Gallager; @JPA_review]. Here $\sum_{{\mbox{\boldmath{$t$}}}^\prime \in {\cal C} \backslash {\mbox{\boldmath{$t$}}}} $ denotes a summation over $2^{N}-1$ codeword vectors ${\mbox{\boldmath{$t$}}}^\prime$ of ${\cal C}$ excluding the possibility of the correct codeword ${\mbox{\boldmath{$t$}}}$. Inserting this into Eq. (\[eq:average\_block\_error\]) provides $$\begin{aligned} \left \langle P_E({\cal C}) \right \rangle_{\cal C} &\le & \sum_{{\cal C}}P( {\cal C} ) \sum_{{\mbox{\boldmath{$t$}}},{\mbox{\boldmath{$r$}}}} P({\mbox{\boldmath{$t$}}}|{\cal C}) P^{\frac{1}{1+n}}({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$t$}}}) \left (\sum_{{\mbox{\boldmath{$t$}}}^\prime \in {\cal C}\backslash {\mbox{\boldmath{$t$}}}} P^{\frac{1}{1+n}}({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$t$}}}) \right )^n \cr &=& \sum_{{\cal C}}P( {\cal C} ) \sum_{{\mbox{\boldmath{$r$}}}} P^{\frac{1}{1+n}}({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$0$}}}) \left (\sum_{{\mbox{\boldmath{$t$}}}^\prime \in {\cal C} \backslash {\mbox{\boldmath{$0$}}}} P^{\frac{1}{1+n}}({\mbox{\boldmath{$r$}}}|{\mbox{\boldmath{$t$}}}^\prime) \right )^n, \cr &=&\sum_{r_i=\pm 1} \frac{ e^{\frac{F}{1+n}\sum_{i=1}^M \left (\frac{1}{2}-r_i \right ) }} {\left (2\cosh \frac{F}{2} \right )^M} \sum_{{\cal C}} P({\cal C}) \left ( \sum_{{\mbox{\boldmath{$t$}}}^\prime \in {\cal C} \backslash {\mbox{\boldmath{$0$}}}} e^{ -\frac{F}{1+n} \sum_{i=1}^M \left (\frac{1}{2}-\delta_{r_i,t^\prime_i} \right ) } \right )^n, \label{eq:gallager2}\end{aligned}$$ where we have performed the gauge transformation ${\mbox{\boldmath{$t$}}}\to {\mbox{\boldmath{$0$}}}$, ${\mbox{\boldmath{$t$}}}^\prime - {\mbox{\boldmath{$t$}}}\to {\mbox{\boldmath{$t$}}}^\prime$ and ${\mbox{\boldmath{$r$}}}-{\mbox{\boldmath{$t$}}}\to {\mbox{\boldmath{$r$}}}$, assuming any code ${\cal C}$ contains the zero codeword ${\mbox{\boldmath{$t$}}}={\mbox{\boldmath{$0$}}}$ [@Gallager]. Minimizing the final expression with respect to $n \ge 0$, we can obtaine the tightest bound of the average decoding error probability. We here address that $E({\mbox{\boldmath{$t$}}})=\sum_{i=1}^M \left (\frac{1}{2}-\delta_{r_i,t_i} \right )$ in Eq. (\[eq:gallager2\]) obeys Eq. (\[prob\]) independetly of ${\mbox{\boldmath{$r$}}}$ when each codeword ${\mbox{\boldmath{$t$}}}$ is equally generated in the ensemble such that each component is independently selected from an identical unbiased distribution $P(t_i=1)=P(t_i=0)=1/2$, which holds for the random code ensemble[@Shannon; @Gallager], although non-trivial correlations of ‘energy’ $E({\mbox{\boldmath{$t$}}})$ between ‘states’ ${\mbox{\boldmath{$t$}}}$ arise making the energy distribution different from Eq. (\[prob\]) in the case of practical linear codes. Therefore, for the current random code ensemble, Eq. (\[eq:gallager2\]) can be simplified as $$\begin{aligned} \left \langle P_E({\cal C}) \right \rangle_{\cal C} \le \left (\frac{\cosh \frac{F}{2(1+n)}}{ \cosh \frac{F}{2} } \right )^M \left \langle \left ( \sum_{A=1}^{2^N-1} \exp \left [-\frac{F}{1+n} \epsilon_A \right ] \right )^n \right \rangle, \label{eq:bound_DREM}\end{aligned}$$ where $\left \langle \cdots \right \rangle $ denotes the configutational average with respect to the ‘energy states’ $\epsilon_A$ ($A=1,2,\ldots,2^N -1$) following Eq. (\[prob\]). Regarding $F/(1+n)$ as the inverse temperature, this means that an assessment of the average decoding error probability can be linked to calculating the general moment of the ‘partition function’ $Z=\sum_{A=1}^{2^N-1} \exp \left [-\frac{F}{1+n} \epsilon_A \right ]$ of DREM. The replica analysis of GCDREM {#GCDREM} ============================== Section \[exact\] indicates that the generalized moment of partition function can be evaluated [*without using RM*]{} for GCDREM even when the system size $N$ is finite. However, for direct comparison to the conventional analysis, it may be helpful to demonstrate the conventional RM-based analysis as well. Therefore, we here provide a brief sketch of the replica calculation of GCDREM. From eq. (\[z\]), we first obtain an expression $$\begin{aligned} \left \langle Z^n \right \rangle &=&\omega^{-\frac{nM}{2}} \sum_{i_1=0}^M\sum_{i_2=0}^M \cdots \sum_{i_n=0}^M \left \langle \prod_{\mu =1}^n n_{i_\mu } \right \rangle \omega^{\sum_{\mu =1}^n i_\mu } \cr &=& \omega^{-\frac{nM}{2}} \sum_{i_1=0}^M\sum_{i_2=0}^M \cdots \sum_{i_n=0}^M \prod_{i=1}^M \left ( \left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle \omega^{i \sum_{\mu =1}^n \delta_{i,i_\mu } } \right ), \label{replica_GCDREM}\end{aligned}$$ for $n=1,2,\ldots$, where $\left \langle \cdots \right \rangle $ represents the average over the Poission distribution, Eq. (\[Poisson\]). Following the conventional recipe of RM, let us next evaluate the candidates of the most dominant contribution in the limit $N \to \infty$ under the RS1, RS2 and 1RSB assumptions following the conventional scheme of RM. Before proceeding further, it may be worth mentioning that the dynamical variables in GCDREM are not single states but energy levels $i=0,1,2,\ldots,M$, each of which is composed of multiple states. This difference makes composition of the solutions slightly different from that of CDREM provided in section \[replica\], although the final result is unchanged. - [RS1]{}\ In RS1, all the $n$ replica levels are assumed to be allocated to $n$ different levels $i(=0,1,2,\ldots,M)$. Therefore, for a given level $i$, $$\begin{aligned} \sum_{\mu=1}^n \delta_{i,i_\mu}= \left \{ \begin{array}{ll} 1 & (\mbox{when $i$ is one of the $n$ allocated energy levels}), \cr 0 & (\mbox{otherwise}). \end{array} \right .\end{aligned}$$ When $i$ is an allocated level, $$\begin{aligned} \left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle &=&\gamma_i \to \exp\left [N \left ( (1-\alpha)\log 2 + \alpha H (x) \right )\right ], \label{average_number_of_state} \end{aligned}$$ and $$\begin{aligned} \omega^{i \sum_{\mu =1}^n \delta_{i,i_\mu } } &=& \exp \left [ -\beta i \right ] \to \exp \left [ -N \alpha \beta x \right ], \label{occupied_contribution}\end{aligned}$$ hold for large $N$, yielding the contribution $$\begin{aligned} \left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle \omega^{i \sum_{\mu =1}^n \delta_{i,i_\mu } } \to \exp \left [ N \left ( (1-\alpha)\log 2 + \alpha( H(x)-\beta x) \right ) \right ], \end{aligned}$$ where $x\equiv i/M=i/(\alpha N)$. This is maximized at $x_c=(1-\tanh \frac{\beta}{2} )$ to $$\begin{aligned} \exp N\left ( \log 2+\alpha \log \left ( \cosh\frac{\beta}{2} \right ) - \frac{\alpha\beta}{2} \right ) , \label{first_replica}\end{aligned}$$ which represents the contribution from a certain replica level $i_\mu(=1,2,\ldots,n)$. Contributions from other replicas become smaller than Eq. (\[first\_replica\]) as any two replica levels must be located at different levels under the current assumption. However, the difference becomes negligible because there exist many levels in any vicinity of $x=x_c$ in the limit $N\to \infty$. Taking the prefactor $\omega^{-\frac{nM}{2}}$ in front of the summation in Eq. (\[replica\_GCDREM\]) and the number of permutations over replica indices into account, this implies that the dominant contribution under the RS1 ansatz is evaluated as $$\begin{aligned} \left \langle Z^n \right \rangle &= &n! \times \exp n N\left (\log 2 + \alpha \log \left ( \cosh \frac{\beta}{2} \right ) \right )\cr &\sim & \exp n N\left (\log 2 + \alpha \log \left ( \cosh \frac{\beta}{2} \right ) \right ), \label{RS1_appendix}\end{aligned}$$ which is identical to the RS1 solution of CDREM, Eq. (\[rs1\]), and equivalent to Eq. (\[rs1\_ours\]). - [RS2]{}\ In RS2, all the $n$ replica levels are assumed to be allocated to a certain single level. Therefore, for a given level $i$, $$\begin{aligned} \sum_{\mu=1}^n \delta_{i,i_\mu}= \left \{ \begin{array}{ll} n & (\mbox{when $i$ is the allocated energy level}), \cr 0 & (\mbox{otherwise}). \end{array} \right .\end{aligned}$$ When $i$ is the allocated level, $$\begin{aligned} \left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle &\sim & \left \{ \begin{array}{ll} \gamma_i^n \to \exp\left [nN \left ( (1-\alpha)\log 2 + \alpha H (x) \right )\right ]& (x > x^*), \cr \gamma_i \to \exp\left [N \left ( (1-\alpha)\log 2 + \alpha H (x) \right )\right ]& (x < x^*), \end{array} \right . \label{RS2_ni_average} \end{aligned}$$ where $x^*$ is determined by Eq. (\[x\*\]) and $$\begin{aligned} \omega^{i \sum_{\mu =1}^n \delta_{i,i_\mu } } &= & \exp \left [ -n \beta i \right ] \to \exp \left [ -n N \alpha \beta x \right ], \label{RS_2occupied_contribution}\end{aligned}$$ hold for large $N$. Notice that $\left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle$ behaves differently depending on whether $x>x^*$ or not. Therefore, the behavior of the dominant contribution obtained by maximizing $\left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle \omega^{i \sum_{\mu =1}^n \delta_{i,i_\mu } }$ is different depending on the relation between $x^*$ and the position of the maximum $x_c$. For $\beta < \beta_c$ or $\alpha \le 1$, $x_c=(1-\tanh\frac{\beta}{2})/2$ becomes greater than $x^*$, which yields the RS1 solution, Eq. (\[rs1\]), in spite that the RS2 ansatz is currently employed. This is because the RS2 ansatz for [*energy levels* ]{} does not necessarily mean that $n$ replica [*states*]{} are identical, in particular, when the occupation number $n_i$ is exponentially large, to which $x_c > x^*$ corresponds. In such cases, even if $n$ replica states are allocated to an identical energy level, they are typically distributed to $n$ different states placed at the energy level, which corresponds to the RS1 ansatz of CDREM, and, therefore, Eq. (\[rs1\]) should be reproduced. On the other hand, for $\beta > \beta_c$, $\left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle \omega^{i \sum_{\mu =1}^n \delta_{i,i_\mu } }$ is maximized at $x_c=(1-\tanh\frac{n\beta}{2})/2 < x^*$. This provides $$\begin{aligned} \left \langle Z^n \right \rangle = \exp N \left (\log 2+ \alpha \log \left ( \cosh \frac{n \beta }{2} \right ) \right ), \label{RS2_appendix}\end{aligned}$$ which is identical to the RS2 solution of CDREM, Eq. (\[rs2\]), and equivalent to Eq. (\[rs2\_ours\]). As Eq. (\[rs2\]) can be dominant at $n=1,2,\ldots$ only for $\beta > \beta_c$, this is consistent with the result of the replica analysis of CDREM. - [1RSB]{}\ In 1RSB, $n$ replica levels are assumed to be equally allocated to $n/m$ levels. Therefore, for a given level $i$, $$\begin{aligned} \sum_{\mu=1}^n \delta_{i,i_\mu}= \left \{ \begin{array}{ll} m & (\mbox{when $i$ is the $n/m$ allocated energy levels}), \cr 0 & (\mbox{otherwise}). \end{array} \right .\end{aligned}$$ When $i$ is the allocated level, $$\begin{aligned} \left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle &\sim & \left \{ \begin{array}{ll} \gamma_i^m \to \exp\left [m N \left ( (1-\alpha)\log 2 + \alpha H (x) \right )\right ]& (x > x^*), \cr \gamma_i \to \exp\left [N \left ( (1-\alpha)\log 2 + \alpha H (x) \right )\right ]& (x < x^*), \end{array} \right .\end{aligned}$$ and $$\begin{aligned} \omega^{i \sum_{\mu =1}^n \delta_{i,i_\mu } } &= & \exp \left [ -m \beta i \right ] \to \exp \left [ -m N \alpha \beta x \right ], \end{aligned}$$ hold for large $N$. Similarly for the case of RS2, this reproduces the RS1 solution, Eq. (\[rs1\]), for $\beta < \beta_c$ or $\alpha \le 1$. So, we focus on the low temperature phase $\beta > \beta_c$. Then, $\left \langle n_i^{\sum_{\mu=1}^n \delta_{i,i_\mu}} \right \rangle \omega^{i \sum_{\mu =1}^n \delta_{i,i_\mu } }$ is maximized at $x_c=(1-\tanh\frac{m \beta}{2})/2 < x^*$ to $$\begin{aligned} \exp N\left ( \log 2+\alpha \log \left ( \cosh\frac{m \beta}{2} \right ) - \frac{m \alpha\beta}{2} \right ) , \label{first_level}\end{aligned}$$ which represents the contribution from one of the $n/m$ allocated levels. The number of ways to select $n/m$ out of $M$ levels equally allocating $n$ replicas is negligible for subsequent calculation. Taking contributions from all the $n/m$ allocated levels and extremization with respect to $m$ into account offers $$\begin{aligned} \left \langle Z^n \right \rangle &\sim & \exp N \left ( \mathop{\rm extr}_{m} \left \{ \frac{n}{m} \left (\log 2 +\alpha \log \left ( \cosh\frac{m \beta}{2} \right ) \right ) \right \} \right ) \cr &=& \exp N n \alpha \beta \left (\frac{1}{2}-x^*\right )\cr &=& \exp N\left (\frac{n\alpha \beta}{2}\tanh \frac{\beta_c}{2} \right ), \label{1RSB}\end{aligned}$$ which is identical to the 1RSB solution of CDREM, Eq. (\[1rsb\]), and equivalent to Eq. (\[1rsb\_ours\]). As Eq. (\[1rsb\]) can be dominant $n=1,2,\ldots$ only for $\beta > \beta_c$, this is consistent with the result of the replica analysis of CDREM. Efficient sampling in CDREM {#CDREM} =========================== We here show that one can sample $\{n_i \}$ with an $O(M)$ computational cost in CDREM as well as in GCDREM. According to the probability distribution for each level $$\begin{aligned} P(E_i)=2^{-M} \left( \begin{array}{c} M\\ \frac{1}{2}M+E_i \end{array} \right),\ \ \left (E_i=i-\frac{M}{2} \right ),\end{aligned}$$ the probability to sample a configuration $(n_0,n_1,\cdots,n_M)$ is $$\begin{aligned} {\cal P}(n_0,n_1,\cdots,n_M) =\{P(E_0)\}^{n_0}\{P(E_1)\}^{n_1}\cdots \{P(E_M)\}^{n_M}\frac{2^N!}{n_0!n_1!\cdots n_M!}.\end{aligned}$$ To generate a sample $(n_0,n_1,\cdots,n_M)$ in practice, we first determine $n_0$ according to the probability $$\begin{aligned} {\cal P}(n_0,arbitrary) =(p_0)^{n_0}(1-p_0)^{2^N-n_0}\frac{2^N!}{n_0!(2^N-n_0)!}\\ (p_0\equiv P(E_0)).\end{aligned}$$ We then determine $n_1$ according to the probability $$\begin{aligned} {\cal P}(n_0;n_1,arbitrary) =(p_1)^{n_1}(1-p_1)^{2^N-n_0-n_1}\frac{2^N!}{n_1!(2^N-n_0-n_1)!}\\ \left(p_1\equiv \frac{P(E_1)}{1-P(E_0)}\right).\end{aligned}$$ Repeating this procedure to $n_{M-1}$ as $$\begin{aligned} {\cal P}(n_0,n_1,\cdots;n_{M-1},n_M) =(p_{M-1})^{n_1}(1-p_{M-1})^{2^N-\sum_{i=0}^{M-1} n_i }\frac{2^N!}{n_{M-1}!(2^N-\sum_{i=0}^{M-1} n_i )!}\nonumber\\ \left(p_{M-1}\equiv \frac{P(E_{M-1})}{1-\sum_{i=0}^{M-2}P(E_{i})}\right),\nonumber\\\end{aligned}$$ we obtain a set of $(n_0,n_1,\cdots,n_M)$. This guarantees that the identity $$\begin{aligned} \sum_{i=0}^{M} n_i = 2^N, \end{aligned}$$ holds, which characterizes CDREM. Similarly for the case of GCDREM, this can be performed in $O(M)$ computation. A comparison of numerically evaluated moments of the partition function between CDREM and GCDREM is presented in Fig. \[REM\]. This indicates that the difference between the two models is almost indistinguishable even for $N=3$. Since the consistency becomes better as $N$ grows larger[@Mou1; @Mou2], working in GCDREM instead of CDREM is justified when $N$ is large. ![The free energies obtained from the sets $\{n_i\}$ using CDREM and GCDREM. We find that CDREM and GCDREM give almost same results even for $N=3$.[]{data-label="REM"}](gcdcd.eps){width="12cm"} [99]{} M. Mézard, G. Parisi and M.A. Virasoro, [*Spin glass theory and beyond*]{} (1987), World Scientific, Singapore. S.F. Edwards and P.W. Anderson, . D. Sherrington and S. Kirkpatrick, . G. Parisi, . G.H. Hardy, F. Riesz, G.H. Hardy, J.E. Littlewood and G. Pólya, [*Inequalities*]{}, (1934), Cambridge University Press, Cambridge. H. Nishimori, [*Statistical Physics of Spin Glasses and Information Processing*]{} (2001), Oxford University Press, New York. N. Sourlas, . Y. Kabashima and D. Saad, . H. Nishimori and K.Y.M. Wong, . K. Tanaka, . T.L.H. Watkin, A. Rau and M. Biehl, . R. Monasson and R. Zecchina, . E. Korutcheva, M. Opper and L. López, . D.B. Saakian, cond-mat/0310549 (2003). T.M. Cover and J.A. Thomas, [*Elements of Information Theory*]{}, (1991), John Wiley & Sons, Inc, New York. V. Vapnik, [*The Nature of Statistical Learning Theory*]{}, (1995), Springer-Verlag, New York. B. Derrida, . C. Moukarzel and N. Parga, . C. Moukarzel and N. Parga, . E.C. Titchmarsh, [*The Theory of Functions*]{} 2nd ed, (1939), Oxford University Press, Oxford. J.L. van Hemmen and R.G. Palmer, . T. Horiguchi, . C.E. Shannon, . Y. Kabashima, N. Sazuka, K. Nakamura and D. Saad, . Y. Kabashima, T. Murayama and D. Saad, . N. Skanzos, J. van Mourik, D. Saad and Y. Kabashima, . J-P. Bouchaud and M. Mezard, . R. Monasson and D. O’Kane, . R.W. Penney, T. Coolen and D. Sherrington, . E.J. Gumbel, [*Statistics of Extremes*]{}, (1958), Columbia University Press, New York. C.N. Yang and T.D. Lee, :[*ibid., 410*]{}. K. Ogure and Y. Kabashima, in preparation. R.G. Gallager, [*Information Theory and Reliable Communication*]{}, (1968), Wiley, New York. Y. Kabashima and D. Saad, . Y. Iba, [*The Gallager formalism in information theory and the replica method*]{} (in Japanese), (1989), unpublished note. Y. Iba, . [^1]: E-mail: ogure@icrr.u-tokyo.ac.jp [^2]: E-mail: kaba@dis.titech.ac.jp [^3]: This condition is necessary to exclude a trivial multiplicity caused by addition of certain analytic functions which vanish at all the natural numbers $n=1,2,\ldots$, such as $\sin(\pi n)$. [^4]: Employment of GCDREM is not essential for reducing the numerical cost. We have discovered a scheme for generating CDREM in a time scale similar to that for GCDREM. It is shown in appendix \[CDREM\].

--- abstract: 'A recent scanning tunneling microscopy (STM) experiment reports the observation of charge density wave (CDW) with period of approximately 8a in the halo region surrounding the vortex core, in striking contrast to the approximately period 4a CDW that are commonly observed in the cuprates. Inspired by this work, we study a model where a bi-directional pair density wave (PDW) with period 8 is at play. This further divides into two classes, (1) where the PDW is a competing state of the d wave superconductor and can exist only near the vortex core where the d wave order is suppressed, and (2) where the PDW is the primary order, the so called “mother state” that persists with strong phase fluctuations to high temperature and high magnetic field and lies behind the pseudogap phenomenology. We study the charge density wave structures near the vortex core in these models. We emphasize the importance of the phase winding of the d-wave order parameter. The PDW can be pinned by the vortex core due to this winding and become static. Furthermore, the period 8 CDW inherits the properties of this winding, which gives rise to a special feature of the Fourier transform peak, namely, it is split in certain directions. There are also a line of zeros in the inverse Fourier transform of filtered data. We propose that these are key experimental signatures that can distinguish between the PDW-driven scenario from the more mundane option that the period 8 CDW is primary. We discuss the pro’s and con’s of the options considered above. Finally we attempt to place the STM experiment in the broader context of pseudogap physics of underdoped cuprates and relate this observation to the unusual properties of X-ray scattering data on CDW carried out to very high magnetic field.' author: - Zhehao Dai - 'Ya-Hui Zhang' - 'T. Senthil' - 'Patrick A. Lee' bibliography: - 'reference.bib' title: 'Pair density wave, charge density wave and vortex in high Tc cuprates' --- [^1] [^2] Introduction ============ The pseudogap phase has long been considered a central puzzle in the study of the cuprate high temperature superconductors[@keimer2015quantum]. After decades of studies, the phenomenology is well established. The pseudogap temperature is now demonstrated to signal a genuine phase transition: some form of broken crystalline symmetry has been shown to occur from ultrasound attenuation [@shekhter2013bounding], second harmonic generation[@HsieNaturePhysicshzhao2017global], and the anisotropy of the spin susceptibility[@MatsudaNaturePhysicssato2017thermodynamic; @Matsuda2unpublished]. Just below this temperature, neutron scattering has detected the onset of intra-cell magnetic moments[@bourges2011novel] which have been interpreted in terms of orbital loop currents[@varma2006theory], even though this experimental finding has recently been challenged, at least in the case of YBCO[@HaydenPRBcroft2017noevidence; @bourges2017comment]. At lower temperatures, short range order charge density wave (CDW) order emerges, often, but not always, suppressed by the onset of superconductivity[@blackburn2013x; @ghiringhelli2012long; @BlancoPhysRevB.90.054513; @GrevenPRB96tabis2017synchrotron]. In high magnetic field the CDW order in YBCO dramatically increases its range, as seen in NMR[@JulienNature477191wu2011magnetic; @Julien2arXivzhou2017spin; @wu2013emergence]. X ray scattering reveals that it is unidirectional and becomes stacked in phase between layers[@changNatureComm72016magnetic; @ZX1science350949gerber2015three; @ZX2PNAS11314647jang2016ideal]. There seems to be two distinct forms of CDW co-existing, one long ranged ordered and uni-directional, while the other is short ranged and exists in both directions. It is quite mysterious why they have the same incommensurate period. At very low temperature quantum oscillations have been observed which have been interpreted as the emergence of small electron-like pockets (for a review, see Ref. ). Of course, the appearance of a pseudogap in the single particle spectrum near the anti-node which persists to very high temperature is what gave this phenomenon its name in the first place. The phenomenology is so rich and complicated that it seems to defy any unifying theme, leading to notions such as “competing orders” or “intertwined order”. Adding to this complexity, a recent STM experiment detected CDW with period 8a co-existing with the previously observed period 4a CDW in the “halo” surrounding the vortex core[@EdkinsSeamusRecentSTM]. In this broader context, a key question we would like to address is this. Does this observation simply increase the complexity of the problem or is it the breakthrough that provides the key to crack open the pseudo-gap problem? It should be noted that the double period CDW is expected in a scenario based on the existence of pair density wave (PDW) when it co-exists with the d-wave superconducting order. In this paper we review different scenarios that can lead to the double period CDW and discuss the pro’s and con’s of each of them. Most importantly, we propose a refinement of the STM experiment which we believe can unambiguously distinguish between different scenarios, including different versions of PDWs, like Canted PDW and Uni-directional PDW. A PDW is a superconductor with a pairing order parameter which is periodic in space. It was first introduced by Larkin and Ovchinnikov[@larkin1965inhomogeneous] and by Fulde and Ferrell[@fulde1964superconductivity] as a way to overcome the Pauli limiting effect of a magnetic field. The notion of PDW in the context of the cuprates has a long history. Himeda, Kato and Ogata [@himeda2002stripe] found in 2002 by projected Monte Carlo studies that the PDW is the preferred ground state in the presence of stripe order. Starting from the standard stripe picture [@tranquada1995jm]of a period 8 spin density wave (SDW) and a period 4 CDW, they found that the d wave superconductor is more stable if the sign of the order parameter is reversed at the hole poor region of the CDW, leading to a period 8 PDW. We shall refer to this state as the stripe-PDW. They proposed that if the stripe-PDW is stacked perpendicular to each other from one layer to the next, the resulting state has drastically reduced Josephson coupling and may explained the disappearance of the Josephson plasma edge observed in Nd doped LaSr2CuO4 (LSCO)[@tajimaPRL862001c]. Strong anisotropy in the transport properties was discovered in the LBCO $\text{La}_{2-x}\text{Ba}_x\text{CuO}_4$ system[@PhysRevLett.99.067001] and since that time the theory of layer de-coupled PDW and related phases has been greatly advanced.[@PhysRevLett.99.127003; @PhysRevB.79.064515] For a review, see Ref. . The next development is the introduction of a Landau theory description. [@PhysRevLett.99.127003; @PhysRevB.79.064515; @agterberg2008dislocations; @berg1NatPhys2009charge] Agterberg and Tsunetsugu[@agterberg2008dislocations] described the coupling of PDW with various subsidiary orders such as CDW and magnetization waves. By examining the interplay between the PDW vortex and the dislocation in the CDW, they showed that it is possible to suppress the PDW order by phase fluctuations, while the subsidiary CDW order remains long ranged. Berg, Fradkin and Kivelson[@berg1NatPhys2009charge] constructed a phase diagram using renormalization group arguments which include regions in parameter space where the primary PDW order is destroyed while CDW and a novel charge 4e superconductor survive. Berg et al [@berg22009NTPhysstriped]suggested that the stripe PDW may have a more general applicability than the low temperature behaviors in the LBCO family, ie, it may be behind the pseudo-gap phase. Part of their argument is based on the spectral property of such a uni-directional PDW. We comment that while this state produces what looks like a Fermi arc, the gap is actually small near the antinode in the direction perpendicular to the stripe orientation[@baruch2008PRB77spectral; @berg22009NTPhysstriped]. This kind of two gap structure has difficulties with STM and ARPES data. Stimulated by a detailed angle resolved photo-emission (ARPES) study of the single layer cuprate Bi2201[@heSci3312011single], one of us [@lee2014amperean] proposed that the unusual features of the spectra can be explained by postulating a bi-directional PDW state as the underlying state of the pseudogap. The pairing is produced by singlet pairing of electrons with momenta $K_i+p$ and $K_i-p$ where the $K_i$’s are located at or near the Fermi surface at the anti-nodal points. (see fig 1a) This gives rise to a bi-directional PDW. The pair carries momenta $P_1$ and ${-P_1}$ which equal twice the momentum K near the $(\pi, 0)$ antinode and are along the x-axis. There is a similar pair $P_2$ and $-P_2$ which are along the y-axis. There are 4 order parameters: $\D_{P_1}$, $\D_{-P_1}$, $\D_{P_2}$ and $\D_{-P_2}$. While Lee proposed using the idea of Amperean pairing[@PhysRevLett.98.067006] as the microscopic origin of the PDW, most of the paper was phenomenological, and explored the consequences of an assumed PDW. As such many of the conclusions are quite general. Nevertheless we would like to emphasize that the motivation for introducing the bi-directional PDW is fundamentally different from that for the uni-directional PDW[@berg22009NTPhysstriped; @fradkin2015colloquium], which is rooted in the phenomena observed in the LSCO/LBCO family at relatively low temperatures. Our view is that the recently discovered CDW which survives up to 150K are distinct from the stripe physics associated with LSCO/LBCO. The wave-vector decreases with increasing doping, whereas the stripe wave vector increases linearly up to about 0.125 doping and saturate, following the Yamada plot[@yamada1998PRB57doping]. For YBCO the period is incommensurate and close to 3, very different from the period 4 CDW associated with 1/8 doping in LSCO. Finally there is no sign of the SDW that is “intertwined” with the stripes. As phenomenology the bi-directional PDW produces the pseudogap at the antinodes and the Fermi arcs near the nodes. (strictly speaking these are the electron-like segments of closed orbits made up of Bogoliubov quasi-particles.) It explains why the gap closes at the end of the Fermi arcs with states moving up from lower energy, while a CDW-generated gap will necessarily close by a state coming down in energy. As opposed to conventional pairing, the spectrum is not particle-hole symmetric at each k point, which explains why the momentum of the minimum gap is shifted away from the original Fermi surface. In addition, CDW at wave-vectors $Q=2P_1$ and $2P_2$ naturally emerges as subsidiary orders. The states at the Fermi arc’s play two important roles. First they greatly suppress the superfluid density and therefore the phase stiffness, so that the PDW is subject to strong phase fluctuations over most of the phase diagram in the H-T plane. Secondly the normal state gives rise to a linear term in the entropy, which lowers the free energy and stabilizes it at finite temperatures, even if it is not the true ground state. In addition, in the superconducting state, a CDW with period $P_1$ and $P_2 (=Q/2)$ naturally appears if the PDW phase is pinned to that of the d wave pairing and reference was made to an STM experiment on YBCO where CDW at $Q$ and $Q/2$ have been reported[@Yeh1; @Yeh2], where $Q=0.28 (2\pi/a)$ matches what is now determined by X-ray scattering. We should point out that other workers have also associated PDW with the pseudogap phenomenon. Zelli , Kallin and Berlinsky[@ZelliQtmOscPhysRevB.86.104507] used the quasi-particle orbits produced by an uni-directional PDW order to produce quantum oscillations. A related proposal was recently made by M. Norman and J.C. Davis.[@Normanunpublished] We will comment on this below. Yu et al[@yuPNAS126672016magnetic] have interpreted their high magnetic field phase diagram in terms of a possible PDW. Two distinct pair fluctuation lifetimes have been reported in tunneling experiments, possibly indicative of the presence of two kinds of superconductors[@koren2016observation]. Other papers consider a PDW with the same wave-vector and on equal footing as the CDW and are less relevant to the present discussion[@PepinPhysRevB.90.195207; @WangPhysRevLett.114.197001]. Next, an interesting observation was made by Agterberg et al[@Agt2PhysRevB.91.054502] that by shifting the momenta K from the zone boundary line, a new state is formed where the PDW carries momenta $P_1$ and a $P'_1$ which is not equal to ${-P_1}$ and similarly for $P'_2$. (see fig 1a) We shall refer to this state as canted PDW, referring to the canting of the pairing momenta as seen in fig 1a. Agterberg et al [@Agt2PhysRevB.91.054502] showed that this state breaks time reversal and inversion symmetry, but preserves the product and that this is precisely the symmetry of the loop current model of Varma[@varma2006theory] which has been used to interpret the neutron scattering data. Advanced numerical methods applied to the t-J models have found evidence for stripe-PDW as a competing state.[@PhysRevLett.113.046402] Interestingly the energy is found to be quite insensitive to the hole filing per period, in contrast to the original stripe picture which strongly prefers half a hole per period. In the remaining of this paper, we address the adequacy of each of the following scenario’s as the explanation of the double period CDW, put in the broader context of the pseudogap phenomenology. (1) The Q/2 CDW is the primary order, while the Q CDW is subsidiary. (2) The Q/2 PDW is a competing order, or an example of “intertwined order” where several order parameters such as PDW, CDW, SDW and d wave pairing are intimately related to each other. In this picture, the PDW exists only in the vortex halo and vanishes outside. (3) The PDW is the primary order, the “mother state” that exists at a high energy scale and lurks behind a large segment of the phase diagram in the temperature/magnetic field plane. In order to explain the pseudogap at the anti-nodes the PDW is assumed to be bi-directional. While its order is destroyed by phase fluctuations, there are several subsidiary orders that emerge at lower temperatures which account for the observed complexity of the phase diagram. We shall also include a discussion of the canted PDW. Throughout this paper we assume the PDW to be bi-directional. A recent paper by Wang et al.[@wang2018] addresses issues related to PDW in the STM experiment and there are similarities and differences with the present work. They consider the d wave superconductivity and PDW as competing states inside the vortex halo and construct a sigma model description combining the two orders. They focus their calculations to uni-directional PDW. They argue against the persistence of PDW outside the vortex halo. As such their picture is closer in spirit to scenario (2) as outlined above. Recent STM results on period-8 density wave =========================================== First we give a short summary of the recent low temperature STM experiment in $\BSCCO$[@EdkinsSeamusRecentSTM]. The doping is about 0.17. At zero field patches of 4a CDW are observed. These appear locally uni-directional and have $d$ form factors. The correlation length is very short, about twice of the lattice spacing. At a finite field of 8.25T, by subtracting off the zero field data, period 4a and period 8a CDWs are revealed in the “halo” region around the vortex core. These appear to be bi-directional and have s-wave form factors. The signals are symmetric when the voltages are reversed. We distinguish bi-directional from “checkerboard” order, which consists of local patches of uni-directional stripes. From the widths of the Fourier transform peaks, the correlation length of the 8a and 4a CDW is about 8a and 4a respectively, comparable to their wavelengths. By examining the signals that are odd upon reversing the voltage, another 4a CDW is found which has $d$ form factors. Its correlation length is about 5a and it is uni-directional, running in the same direction from vortex to vortex. Purely on symmetry grounds, the observation of period 8a bidirectional charge order in the presence of a background superconductor implies that there are also period-8 modulations in the pair order parameter. Specifically if the Fourier component $\rho_{Q/2}$ of the density at a wave vector $Q/2$ is non-zero, then it implies a non-zero Fourier component $\Delta_{Q/2} \sim \Delta_d \rho_{Q/2}$ in the pairing order parameter (where $\Delta_d$ is the order parameter for the standard $d$-wave superconductor). An important question then is whether the observed period-8 modulations are driven primarily by the pinning of soft fluctuations of $\rho_{Q/2}$ (and $\Delta_{Q/2}$ is a subsidiary) or whether the driver is pinning of soft fluctuations of $\Delta_{Q/2}$ (and the observed $\rho_{Q/2}$ is a subsidiary). We will call the former CDW-driven and the latter PDW-driven. Clearly this is not a symmetry-based distinction and it is natural to wonder if the question is meaningful at all. However we will argue in this paper that there are, in fact, two distinct possibilities for the observed period-8 charge order which have distinct experimental signatures. It is natural to associate these two distinct possibilities with the (looser) distinction between CDW-driven and PDW-driven mechanisms. Basic features of bi-directional PDW ==================================== In this section, we explore the implications of the PDW-Driven scenario, and contrast it with the CDW-driven scenario. We will particularly emphasize the two distinct structures of the period-8 charge order and their experimental distinctions. PDW with long range order {#Sec: PDW with long range order} ------------------------- The new CDW recently found in $\BSCCO$ has a momentum close to $2\pi/8$, half of the momentum of the well-known short-range CDW at zero field. In the PDW-driven scenario, we consider a bi-directional PDW order with the same momentum, that is roughly the momentum between tips of the bare Fermi surface in the anti-nodal direction[@lee2014amperean]. Bi-directional PDW state with such a momentum is previously proposed by one of the authors [@lee2014amperean]. Following this proposal, we write down a mean field Hamiltonian H &=& \_[k,]{} \_k c\^\_[k,]{}c\_[k,]{}\ &+& \_[k]{} \^\*\_[P\_1]{}(k) c\_[k,]{}c\_[-k + P\_1,]{} + \^\*\_[P’\_1]{}(k) c\_[k,]{}c\_[-k + P’\_1,]{}\ &+& \_[k]{} \^\*\_[P\_2]{}(k) c\_[k,]{}c\_[-k + P\_2,]{} + \^\*\_[P’\_2]{}(k) c\_[k,]{}c\_[-k + P’\_2,]{}\ &+& h.c. \[Eq: long range PDW mean field\] We used the notation: $P_1 = 2K_1,\ P'_1 = 2K'_1$ — as shown in Fig. \[Fig: PDW band stucture, pocket\](a) $K_1$ and $K'_1$ are located at or near the Fermi surface at anti-nodal points, generically incommensurate with the B.Z.; Similarly, $P_2 = 2K_2,\ P'_2 = 2K'_2$. The 4 PDW order parameters generate CDW order $\rho_{Q_x}$ and $\rho_{Q_y}$ in second order perturbation even though we do not include them explicitly in the Hamiltonian. \_[Q\_x]{}\~\_[P\_1]{}\^\*\_[P’\_1]{}, \_[Q\_y]{}\~\_[P\_2]{}\^\*\_[P’\_2]{}. \[Eq: subsidiary doubled CDW\] We associate this subsidiary CDW as the well-known short-range CDW at zero field; it has momenta $Q_x = P_1 - P'_1$, $Q_y = P_2 - P'_2$, with magnitude $Q\simeq 2\pi/4$ in the recent STM experiment. In principle, we can also add CDW in (1,1) direction, e.g. $\rho \sim \D_{P_1}\D^*_{P'_2} + \dots$. However, this CDW is absent in the recent STM experiment; we explain the reason in detail in the next subsection. Naively one may expect that if the PDW has local $d$ form factor, the CDW generated by Eq.\[Eq: subsidiary doubled CDW\] has $s$ form factor. This argument is not generally correct, because $s$ and $d$ form factor for a finite-momentum order parameter has no sharp symmetry distinction [^3] It is a local property, which is not captured by the long wavelength description of a Landau order parameter. In fact, when we solve our mean field Hamiltonian with only $d$ wave PDW as input, the CDW that emerges at $Q$ is predominantly $d$ wave. In view of the experimental observation of $s$ symmetry CDW near the vortex core, this may simply indicate that the mean field theory is not adequate to give a microscopic description. Nevertheless, we want to convey the message that this result shows that it is entirely possible that a $d$ wave CDW can emerge as a subsidiary order. We define the common phases $\theta_{\text{P},x},\ \theta_{\text{P},y}$ and relative phases $\phi_x,\ \phi_y$ of the PDW order parameters, and the phases of Q CDW order parameters as \_[P\_1]{} = |\_[P\_1]{}|e\^[i(\_[,x]{} + \_x)]{}&,& \_[P’\_1]{} = |\_[P’\_1]{}|e\^[i(\_[,x]{} - \_x)]{}\ \_[P\_2]{} = |\_[P\_2]{}|e\^[i(\_[,y]{} + \_y)]{}&,& \_[P’\_2]{} = |\_[P’\_2]{}|e\^[i(\_[,y]{} - \_y)]{}\ \_[Q\_x]{} = |\_[Q\_x]{}|e\^[i\_x]{}&,&  \_[Q\_y]{} = |\_[Q\_y]{}|e\^[i\_y]{}, As shown in Eq.\[Eq: subsidiary doubled CDW\], $\gamma_x = 2\phi_x$ and $\gamma_y = 2\phi_y$ are the phase difference between PDW order parameters, hence the phases of the subsidiary CDW order parameter [^4]; they are proportional to the shift of density wave pattern in real space. On the other hand, $\theta_{\text{P},x}$ and $\theta_{\text{P},y}$ carry charge 2 under external electromagnetic field; when coexist with uniform d wave superconductivity $|\D_d|e^{i\theta_d}$, the relative phases $\theta_{\text{P},x}- \theta_d$ and $\theta_{\text{P},y} - \theta_d$, together with $\phi_x$ and $\phi_y$ determines the spatial pattern of new CDW orders with momenta $P_1, P'_1, P_2$, and $P'_2$, which are close to or equal to $Q/2$. We consider two scenarios: (1) $K_i$ and $K'_i$, $i = 1,2$ are located at the boundary of B.Z., shown as solid red dots in Fig. \[Fig: PDW band stucture, pocket\](a): $2K_1 = -2K'_1 = P_1 = -P'_1 = Q_x/2,\ 2K_2 = -2K'_2 = P_2 = -P'_2 = Q_y/2$ (2) $K_i$ and $K'_i$ are slightly shifted, shown as dashed red dots. The shifts in momenta can be either positive or negative, giving a $Z_2$ order parameter in each direction. We refer to this scenario as canted PDW. This possibility was discussed in Ref.  in relation with loop current. It has a potential ability to account for T-reversal breaking and nematicity. Regarding the recent STM experiment, these two scenarios give similar predictions. We focus on the first scenario and comment on the second when necessary. Unlike the pairing in a conventional superconductor, where electrons forming a Cooper pair have opposite momenta and opposite velocity, this finite-momentum pairing groups electrons with momenta $K_i + \d k$ and $K_i - \d k$, (similarly, $K'_i + \d k$ and $K'_i - \d k$) and it has a strong effect only when these two momenta are both close to the Fermi surface. As a result, it opens a gap only in the anti-nodal direction (shown in Fig.\[Fig: PDW band stucture, pocket\](b)), and leaves a gapless surface of Bogoliubov quasi-particle in the nodal direction. Since PDW and CDW are generically incommensurate to the B.Z., we need to set a cutoff in momentum-space calculation. It was previously reported in Ref. [@lee2014amperean] by one of the author that a 5-band model describing the mixing of $c_{k}$, $c^{\dagger}_{-k+P_1}$, $c^{\dagger}_{-k+P'_1}$, $c_{k-Q_x}$ and $c_{k+Q_x}$ (similarly in y direction) produce Bogoliubov pockets with predominant electron weight on one side and predominent hole weight on the other side. In order to capture the effect of B.Z. folding caused by the subsidiary CDW, we increase the cutoff, and include the mixing among $c_{k+mQ_x+nQ_y}$ for $m,\ n$ up to $\pm 2$ (for details, see Appendix A). We used the Hamiltonian in Eq. \[Eq: long range PDW mean field\], the band structure in Appendix A, CDW momentum $Q=0.28(2\pi/a)$ measured in Ref. [@GrevenPRB96tabis2017synchrotron], PDW order parameter with d wave form factor \_[Q\_x/2]{}(k)= 2((k\_xQ\_x/4) - (k\_y))\ \_[Q\_y/2]{}(k)= 2((k\_x) - (k\_yQ\_y/4)), with $\D = 45$meV and no explicit CDW order parameter in the mean field Hamiltonian. We found that, the electron-like part of the 4 Bogoliubov pockets recombine into a predominantly electron-like pocket, similar to the Harrison-Sebastian pocket (shown in Fig. \[Fig: PDW band stucture, pocket\](b-c)). We believe that these pockets formed by mainly electron like segments will give rise to quantum oscillations. The reason is that while the Bogoliubov quasi-particles do not carry fixed charges, they carry a well defined current, because the holes are moving in the opposite direction. In these orbits, all the segments are electron like. In a semi-classical picture a wave-packet will travel in real space along a close contour that encloses the magentic flux. By the Onsager argument, we can expect quantum oscillations. In contrast, there are many closed orbits made up of segments that are part electron and part hole like.[@Normanunpublished] If we draw an arrow corresponding to the current, we find that at the corners where the electron-like and the hole-like segments meet, they both run into the corner and undergo Andreev scattering, ie the currents go into the condensate. In this case, even though the orbits look closed in momentum space, the wave-packets do not form closed orbit in real space, because part of the current goes into the condensate. Then Onsager’s argument no longer applies. For this reason we think it is unlikely that such orbits give rise to quantum oscillations, but only a detailed calculation can tell us the answer for sure. Analogous issues arise with the Fermi surface formed by Bogoliubov quasiparticles in a d-wave superconductor coexisting with loop current order. In that problem, detailed calculations[@allais2012loop; @wang2013quantum] indeed show that at $T = 0$ such a Fermi surface does not lead to quantum oscillations. We note that Zelli et al[@ZelliQtmOscPhysRevB.86.104507] claimed that oscillations corresponding to such orbits exist, but their conclusion is based on an approximate calculation. We believe this issue should be re-visited. Another point is that the PDW is a superconductor and in principle we should include vortices when we introduce the magnetic field. We provide the following argument. First it is known that quantum oscillations appear in the mixed state with a frequency which is the same as the same pockets in the normal state.[@PhysRevB.51.5927] This has been confirmed by numerical calculations with randomly pinned vortices in a d wave superconductor as long as the correlation length is not too short.[@PhysRevB.79.180510]. Of course to address quantum oscillations we need to think about a metallic state that emerges from fluctuations of a PDW ordered state. We will leave this problem aside in the present paper. We would like to mention that as we increase doping, the 4 electron pockets [^5] in Fig. \[Fig: PDW band stucture, pocket\](c) touches each other. In some parameter range, Fermi surface topology changes, and a hole pocket forms in the middle. This Lifshitz transition is predicted for Hg1201 at $10\%$ doping in Ref. [@GrevenPRB96tabis2017synchrotron], and for YBCO at a larger doping. However, distinguishing subtle changes of Fermi surface topology is beyond the scope of the current paper. ![Band structure of Bogoliubov quasi-particle and possible Fermi pockets in a PDW state. (a) An illustration of the bare Fermi surface, CDW momenta and PDW momenta. CDW momenta $Q_x$ and $Q_y$ are shown as yellow arrows. PDW momenta are $P_1 = 2K_1$, $P'_1 = 2K'_1$ in x direction, and $P_2 = 2K_2$, $P'_2 = 2K'_2$ in y direction. CDW is a subsidiary order of PDW, its momenta $Q_x = P_1 - P'_1$, $Q_y = P_2 - P'_2$. We consider two scenarios: (1) $K_i$ and $K'_i$ are located right at B.Z. boundary (solid red dots). $P_1 = -P'_1=Q_x/2$, $P_2 = -P'_2 = Q_y/2$. (2) $K_i$ and $K'_i$ are slightly shifted (dotted red circles); $P_1$ and $P'_1$ have a small y component, as shown in the inset figure (The small y component is exaggerated). (b)Electron weight on the Fermi-pocket of Bogoliubov quasi-particle. We used the band structure in Appendix A, CDW momentum $Q_x=Q_y=0.28(2\pi/a)$ measured in Ref. [@GrevenPRB96tabis2017synchrotron] PDW order parameter $\D_{Q/2}=45$meV, no explicit CDW order parameter in mean field Hamiltonian, and plotted the electron weight at Fermi energy and each momentum $k$ in the B.Z. (For details, see Appendix A). Electron weight is large on 4 “arcs” in the nodal direction. The anti-nodal direction is gapped out by PDW. (c) Details of the reconstructed electron-like pocket after B.Z. folding caused by CDW. We plotted the total electron weight at momenta up to $Q_x$ and $Q_y$. This pocket is formed by 4 segments with electron weight $>80\%$. It has the same shape as the Harrison-Sebastian pocket. Physically there is only one pocket, others are its copy shifted by $Q_x$ and $Q_y$. we only show the upper right quadrant of the B.Z.[]{data-label="Fig: PDW band stucture, pocket"}](image/PDW_pocket.pdf){width="7in"} Static short range PDW ---------------------- In this subsection, we discuss the situation where a short-range PDW coexists with d wave superconductivity. We focus on the setup of the recent STM experiment where a period-8 density wave was found in the vortex halo of d wave superconductor. To simplify the discussion, we consider the simplest scenario: $P_1 = -P'_1 = Q_x/2$, $P_2 = -P'_2 = Q_y/2$. We have 4 PDW order parameters: $\D_{\pm Q_x/2}$ and $\D_{\pm Q_y/2}$. We consider the following couplings between PDW, d wave, and CDW order parameters in a Landau theory in translation-invariant systems. We can write them in momentum space as F = &-&a\_[Q\_x]{}\^\*\_[Q\_x/2]{}\_[-Q\_x/2]{} - b \_[Q\_x]{}\[\^2\_d\^[\*2]{}\_[Q\_x/2]{}+\^[\*2]{}\_d\^[2]{}\_[-Q\_x/2]{}\]\ &-& c\_[Q\_x/2]{} \[\^\*\_d\_[-Q\_x/2]{} + \^\*\_[Q\_x/2]{}\_d\] -…, \[Eq: Landau theory, momentum space\] where $a$, $b$, $c$ are real coupling constants. For simplicity, we write down only couplings in x direction. Couplings in y direction are similar. These momentum-space couplings are conceptually helpful, but the strong breaking of translation symmetry introduced by the vortex core brings in new physics that are better captured by a real-space analysis. Before we start, it is important to note that what the experimentalists found is not long-range PDW or CDW. Instead, STM experiment identified a static short-range charge order that lives only inside the vortex halo, with apparent correlation length comparable to its wavelength. Theoretically, a “short-range order” naturally fluctuate with time; the existence of static short-range order raises many questions — what pins the phases of the order parameters? — why does it appear only in vortex halo? One may tend to think of a phase competition between uniform d wave superconductivity and PDW, so that the latter may be greatly enhanced near the vortex core. However, a phase competition alone does not explain why the short-range order is static. The answer of these questions may lie in the following observation: just like the way spatial inhomogeneity pins short-range CDW, a spatial pattern of superconductivity close to the vortex core pins a short-range PDW. This static PDW then extends to a larger region with radius defined by its correlation length $\xi_P$. Outside $\xi_P$, there is still a PDW amplitude fluctuating with time, but the time average decays exponentially. For concreteness, we choose the origin to be the center of the vortex, $(r,\theta)$ to be the polar coordinate, $(x,y)$ to be the Cartesian coordinate, and write down the following ansatz for the amplitude of d wave and PDW: \[Eq: d wave amplitude in real space\] \_d() &=& |\_d(r)|e\^[i\_d]{}e\^[i]{}\ \_() &=& 2|\_[Q\_x/2]{}| e\^[-r/\_P]{} e\^[i\_[P,x]{}]{}(Qx/2 + \_[x]{})\ + &2&|\_[Q\_y/2]{}| e\^[-r/\_P]{} e\^[i\_[P,y]{}]{}(Qy/2 + \_[y]{}), \[Eq: PDW amplitude in real space\] where $|\D_d(r)|= r/\sqrt{r^2 + r^2_\text{core}}$. $e^{i\theta}$ encodes the $2\pi$ phase winding of d wave amplitude. We have three length scales. The radius of the vortex core: $r_\text{core}\simeq 3a$, the period of PDW: $4\pi/Q\simeq 8a$, and the radius of vortex halo, where field-enhanced CDW are found: we identify the halo size as $r_\text{halo}\sim\xi_P\sim 4\pi/Q$. A usual Landau theory with slowly-varying order parameters implicitly assumes that $r_\text{core}\gg 4\pi/Q$, $\xi_P\gg 4\pi/Q$. However, we are in the opposite limit: $4\pi/Q \sim \xi_P > r_\text{core}$. Since $\xi_P$ and $4\pi/Q$ are close to each other, and they are one order of magnitude larger than the lattice constant, we do not separate the exponential decay of order parameters $\D_{\pm Q_x/2}$ ($\D_{\pm Q_y/2}$) from the oscillatory part $\cos(Qx/2 + \phi_{x})$ ($\cos(Qy/2 + \phi_{y})$), as in a usual Landau theory. Instead, we take the ansatz in Eq. \[Eq: d wave amplitude in real space\] and Eq. \[Eq: PDW amplitude in real space\], and write down their couplings in real space together with charge density profile $\rho(r)$. \[Eq: Landau theory, real space\] F = &-&{a()\_() \^\*\_()\ &+& b()\[\^2\_d() \^[\*2]{}\_() + \^[\*2]{}\_d() \^2\_()\]\ &+& c()\[\^\*\_d()\_() + \_d()\^\*\_()\]\ &+&s\[\^\*\_d()\_() + \_d()\^\*\_()\]}d\^2 We would like to remind the readers again that this free energy is not a Landau free energy in the usual sense, since we include the oscillatory part of PDW explicitly in $\D_\text{PDW}(\mathbf{r})$. The last term in Eq. \[Eq: Landau theory, real space\]: $-s\int \D^*_d(\mathbf{r})\D_\text{PDW}(\mathbf{r}) d^2 \mathbf{r} + c.c.$ is the lowest-order symmetry-allowed term that describes the phase locking between PDW and d wave order parameter near a vortex core. In the case of spatially slowly-varying order parameters, this term usually vanishes because of momentum mismatch, eg. if the d wave superconductivity has uniform amplitude. However, close to the vortex core, the rapid changing of d wave amplitude strongly breaks translation symmetry. Furthermore the phase winds by $2\pi$ around the core, and near the core the winding is sufficiently rapid that it can phase match the finite wave-vector of the PDW. As a result the PDW is pinned to match the spatial pattern of vortex core so that free energy is minimized. Because of the phase winding, d wave amplitude changes sign across the origin and the overlap integral is optimized when PDW has the form $\sin(Qx/2)$ which also changes sign at the origin. Thus $\phi_{x}$ and $\phi_{y}$ are pinned to be $-\pi/2$. Then the overall phase, $\theta_{P,x} = \theta_d,\ \theta_{P,y} = \pi/2 + \theta_d$, are pinned so that the overlap is a positive real number. This pinning mechanism completely fixes the phases of PDW; a simple calculation of the overlap integral indicates the pinning is very effective in the vortex core. For details, see Fig. \[Fig: pinning by overlap\]. Of course, at the length scale of 10 lattice constants, everything except a microscopic model is merely an oversimplified illustration. Nonetheless, we believe this simple illustration captures the underlying physics of phase-locking between d wave and various PDW order parameters. This pinning mechanism is effective exactly because $4\pi/Q > r_\text{core}$ in the cuprates. In the opposite limit, d wave order parameter changes slowly. According to a usual Landau theory, this coupling cancels out. In the remaining part of this section, we discuss the consequences of this phase-locking on subsidiary charge order. We confirmed these consequences by an exact diagonalization study in the next section. ![(a) Overlap integral $\int \D^*_d(\mathbf{r})\D_\text{PDW}(\mathbf{r}) d^2 \mathbf{r}$ as a function of $\phi_{x}$, for $\theta_x = 0$. We have set their maximum amplitude to 1 for both $\D_d(\mathbf{r})$ and $\D_\text{PDW}(\mathbf{r})$, and we normalize the integral by the overlap of PDW with itself inside the vortex core of radius $3a$. $\phi_x$ is pinned to $3\pi/2$. The large overlap implies the real-space pattern of PDW matches the pattern of d wave vortex core almost perfectly at $\phi_x = 3\pi/2$ — its amplitude is reduced only because d wave amplitude is reduced in the vortex core. (b) The integrand $\D^*_d(\mathbf{r})\D_\text{PDW}(\mathbf{r})$ as a function of $\mathbf{r}$ near vortex core, for $\phi_{x} = 3\pi/2$, $\theta_x = 0$. Outside the vortex core, the integrand alternating between positive and negative because of momentum mismatch. However, with in the first period of PDW in the center, the integrand is always positive, giving a large overlap. This is because d wave and PDW both change sign across the origin. d wave change sign due to $2\pi$ phase winding, and PDW change sign because of the $\sin(Qx/2)$ factor.[]{data-label="Fig: pinning by overlap"}](image/overlap_integrand.pdf){width="3in"} Note that PDW does not have a vortex. Since PDW lives only in small patches, vortices are not required[@agterberg2015checkerboard], and it is energetically favorable to not have vortices in the PDW-driven scenario. This PDW order generates various CDWs in the vortex halo: \(1) bi-directional Q/2 CDW. According to Eq. (\[Eq: Landau theory, momentum space\] - \[Eq: Landau theory, real space\]), it has the following amplitude in real space \[Eq: CDW in real space\] \_() = F(r)(+ \_d - \_[P,]{})(Q\_ + \_) where $\a = x, y$, $Q_\a = Q_x, Q_y$ and $F(r) \sim 2c|\D_d(r)\D_{Q_{\a}/2}| e^{-r/\xi_P}$. The most interesting feature is that, apart from normal plane-wave factor, there is an additional factor $\cos(\theta + \theta_d - \theta_{P,\a})$ depending on the polar angle. A choice of the relative angle $\theta_d - \theta_{P,\a}$ selects a special angle along which $\rho_{\a}(\mathbf{r})$ vanishes. We point out that the pinning mechanism we discussed predicts that the amplitude $\rho_x$ vanishes in the vertical direction, when $\theta\sim\pm \pi/2$, while the amplitude $\rho_y$ vanishes in the horizontal direction, when $\theta\sim 0,\pi$. This choice restores C4 symmetry. Physically, this new feature originates from the $2\pi$ winding of d wave order parameter. We can identify two contributions to $\rho_{Q/2}$: $\D^*_d\D_{Q/2}$ which carries -1 dislocation, and $\D_d\D^*_{-Q/2}$ which carries +1 dislocation. The interference of these two terms give rise to a nodal direction in real space. This is an important prediction in PDW-driven scenario. On the contrary, in CDW-driven scenario, it is energetically favorable to put the dislocation in PDW amplitude, and the CDW amplitude is rather featureless. In the next section, we discuss the same feature in Fourier space, and propose follow-up experiments to distinguish PDW-driven and CDW-driven scenario. \(2) Q CDW. According to Eq. \[Eq: Landau theory, momentum space\] there are two contributions: \^A\_Q \~a\^\*\_[-Q/2]{}\_[Q/2]{}, \[Eq: CDWA\] which we call $\text{CDW}_A$, and \^B\_Q \~b(\^[\*2]{}\_d\^2\_[Q/2]{} + \^[2]{}\_d\^[\*2]{}\_[-Q/2]{}), \[Eq: CDWB\] which we call $\text{CDW}_B$, which we can think of as a harmonic of $\rho_{Q/2}$. $\text{CDW}_A$ does not rely on the phase-locking between d wave and PDW; it is already pinned to be static short-range CDW by impurities at zero magnetic field, and it persists above $T_c$. On the other hand, a static $\text{CDW}_B$ rely on the phase-locking. Similar to Q/2 CDW, it is a superposition of +2 dislocation and -2 dislocation, and it exists only in vortex halo. In the case of spatially uniform PDW and CDW order, there is no distinction between the two. However, in a spatially inhomogeneous situation such as what we encounter near the vortex core, there is a physical distinction. For example, $\text{CDW}_A$ may be extended in space while $\text{CDW}_B$ may be localized near the vortex core. In this case the two CDW may have different local form factors, such as d or s wave. These form factors may in turn determine which one prefers to be bi-directional or uni-directional, because the coefficient of the quartic term that couples the amplitudes of the x and y oriented CDW may be different. In the STM data there already appears to be two kinds of CDW’s , one pinned to the vortex core and one which already exists at zero filed. We will make further use of this distinction in later discussions. Naively, one would expect a CDW with momentum $(Q/2,Q/2)$ appears in the second order — in real space this term may show up in the contribution $\rho(\mathbf{r})\sim a\D_\text{PDW}^*(\mathbf{r})\D_\text{PDW}(\mathbf{r})$. However, the pinning in the vortex core requires \_() \~e\^[i\_d]{}((Qx/2) + i(Qy/2)),\ \_\^\*()\_() \~\^2(Qx/2) + \^2(Qy/2), and the cross term $\sin(Qx/2)\sin(Qy/2)$ with momenta $(\pm Q/2, \pm Q/2)$ cancels out due to the $\pi/2$ relative phase. As a consequence, there is no $(2\pi/8, 2\pi/8)$ CDW in the leading order. In the fourth order, such a CDW is generated by the term $\D_{d}^{*2}(r)\D^2_\text{PDW}(r)$, but the amplitude is weak and subject to broadening effect given by dislocations. The absence of $(2\pi/8, 2\pi/8)$ CDW is previously discussed in Ref. . It was pointed out that in the uniform case when PDW does not have a vortex, the relative phase between PDW in x and y direction determines whether $(2\pi/8, 2\pi/8)$ CDW is present or not. If the phase is zero it is present, while if it is $\pi/2$ bond currents are generated, producing a flux density wave at the same wave-vector instead. This flux density wave will be discussed in great detail in a later section. In the uniform case it is not known which phase is preferred. In our case we find that in the presence of a vortex, the phase choice $\pi/2$ is energetically favorable, therefore $(2\pi/8, 2\pi/8)$ CDW is absent in leading order. On the contrary, in CDW-driven scenario, naively the $(2\pi/8, 2\pi/8)$ CDW is comparable to the $(2\pi/4, 0)$ CDW. The absence of a $(2\pi/8, 2\pi/8)$ Fourier peak in STM data is an evidence favoring PDW-driven scenario. Next, we would like to comment on the correlation length of PDW in the recent STM experiment. In PDW-driven scenario, as discussed above, the Q/2 CDW has $2\pi$ phase winding around the vortex core. A simple calculation shows that this phase winding broadened the Fourier peak by roughly a factor of 2. Thus the intrinsic correlation length of Q/2 CDW and PDW should be close to 16 lattice constants, a little smaller than the half of the distance between neighboring vortex core. We end this section with some comments on the implications if a canted PDW is present. While the CDW generated by Eq(2) retains the wave-vector Q along the x and y axes, the double period CDW generated by the analog of the third term in Eq. \[Eq: Landau theory, momentum space\] now has wave-vector $P$ and $P'$. Similarly, its harmonic generated by the analog of the second term in Eq. \[Eq: Landau theory, momentum space\] have wave-vectors $2P$ and $2P'$. It is worth noting that we now have two distinct CDWs and the difference between A and B type CDW is now a sharp one that can be made even in a uniform system. A second point is that there is now an additional pinning mechanism. The term $(\Delta_d e^{i\theta(\mathbf{r})})^2 (\Delta_P \Delta_{P'})^*$ is allowed if the local phase gradient matches the canting momentum $p = (P+P')/2$. This leads to a locking term at some distance from the vortex core where the phases are matched. The possible detection of the canting angle will be discussed in the next section. With the above understanding of PDW-driven scenario, we propose the following phenomenological picture explaining the recent STM experiment in $\BSCCO, \ 17\%$ doping, up to 8.5T: - short-range PDW is pinned by the vortex core and extends to its correlation length. - We estimate the intrinsic correlation length of PDW to be 16 lattice constants. The period-8 CDW appears to have a shorter correlation length $\sim$ 8 lattice constants as determined from the width of the Fourier transform peak by fitting it to a Gaussian. Part of this width is not intrinsic and is due to the $2\pi$ phase winding. - The period 8 CDW produces as a harmonic a period 4 CDW, which we have labeled as $\text{CDW}_B$. Its width is subject to the same blurring as the period 8 CDW. On the other hand, the static PDW near vortex core nucleates the period-4 $\text{CDW}_A$ by $\D^*_{-Q/2}\D_{Q/2}$, which is not affected by the phase winding around the vortex. These two CDWs may have different form factors and different asymmetry factors between x direction and y direction. However it is hard to extract their correlation length separately based on the current data, since their Fourier peaks mix together. The width of $2\pi/4$ Fourier peak translates to a correlation length around 4a. This serves as a lower bound of the intrinsic correlation lengths of $\text{CDW}_A$ and $\text{CDW}_B$. - At zero field, $\D_{-Q/2}$ and $\D_{Q/2}$ fluctuate with time, we rely on their relative phase being pinned by spatial inhomogeneity to give a static $\text{CDW}_A$. This effect gives much weaker period-4 CDW puddles with a very short correlation length of order 2a. This CDW is unidirectional in each small puddle. We tentatively identify the unidirectional part of CDW both in zero field and in the vortex core as $\text{CDW}_A$. - The static-PDW-enhanced correlation length of $\text{CDW}_A$ is enough to give some overlap between neighboring vortices. It is energetically favorable for the unidirectional part to align its direction and stretch its phase between vortices smoothly to gain overlap energy. - PDW-driven model predicts the absence of $(2\pi/8,2\pi/8)$ peak. - Given the strong pinning effect and relatively small correlation length, these CDWs may not be able to overcome the local pinning effect and become phase coherent between halos. Experimental Proposal ===================== The disappearance of $(\frac{2\pi}{8},\frac{2\pi}{8})$ CDW order is surprising for a CDW-Driven model while it can be naturally explained in PDW-Driven model, as shown in last section. Despite this already existing evidence favoring PDW-Driven model, more experimental predictions need to be tested to fully settle down this issue. In this section we propose experiments to distinguish PDW-Driven and CDW-Driven scenario unambiguously. Besides, in PDW-Driven scenario our proposed experiment can extract the relative phase between PDW order parameter and $d$ wave order parameter, which is physical. The main prediction of PDW-Driven scenario is that CDW order parameter at $Q_x/2=(\frac{2\pi}{8},0)$ and $Q_y/2=(0,\frac{2\pi}{8})$ have the following profile as shown in Eq. \[Eq: CDW in real space\] $$\mathlarger \rho_{\mathbf {Q_{\a}/2}}(r,\theta)=e^{i\phi_a}F_P(r) \cos(\theta-\theta_a) \label{eq:pdw_profile}$$ where$(r,\theta)$ is the polar coordinate of real space around the vortex center and $a$ denotes $x$ or $y$ direction. $F_P(r)$ vanishes at $r=0$ and decays as $e^{-\frac{r}{\xi}}$ at large $r$. It has maximum at nonzero distance to center. $\theta_x=\theta_{P_x}-\theta_d$ and $\theta_y=\theta_{P_y}-\theta_d$ are the relative phases of PDW order parameters $\Delta_{\pm P_a}=|\Delta_{P_a}|e^{i\theta_{P_a}\pm i \phi_a}$ compared to d-wave order parameter $\Delta_D(r,\theta)=|\Delta_D|e^{i\theta_d}e^{i\theta}$ In contrast, CDW-Driven scenario shows quite distinct profile of period $8$ CDW order parameter: $$\mathlarger \rho_{\mathbf{Q_a/2}}(r,\theta)=e^{i\phi_a}F_c(r) \label{eq:cdw_profile}$$ $F_c(r)$ has maximum at $r=0$ and decays far away with $e^{-\frac{r}{\xi}}$. CDW order parameter doesn’t have angle dependence in this scenario. [0.45]{} ![Real Space Plot of on site LDoS $\nu^E(\mathbf{r})$ at $E=30$meV for PDW-Dirven and CDW-Driven model.[]{data-label="fig:real_space"}](image/pdw.png "fig:"){width="\textwidth"} [0.45]{} ![Real Space Plot of on site LDoS $\nu^E(\mathbf{r})$ at $E=30$meV for PDW-Dirven and CDW-Driven model.[]{data-label="fig:real_space"}](image/cdw.png "fig:"){width="\textwidth"} Clearly CDW order parameter profile from PDW-Driven and CDW-Driven models have both different radius dependence and angle dependence. A real space plot of LDoS can be found in Fig. \[fig:real\_space\]. The $\cos(\theta-\theta_a)$ factor in PDW-Driven model means a superposition of strength $\pm 1$ dislocation of CDW order parameter and in principle STM experiments can extract $\theta_a$. Here we will propose the following experimental predictions to distinguish the above two different CDW profiles. In the STM experiment, what is measured is the local density of states (LDoS) at a fixed energy $\nu(\mathbf{r},E)$. For a fixed energy, $\nu^E(\mathbf{r})=\nu(\mathbf{r},E)$ has the same symmetry as density and we expect it to follow Eq. \[eq:pdw\_profile\] and Eq. \[eq:cdw\_profile\]. Before going to specific predictions, it may be worthwhile to give one general suggestion to data analysis procedure of experimental data. For both PDW-Driven and CDW-Driven scenario, the phase of CDW order with momentum $Q_a$ is expected to be locked to position of vortex center. As a result, signals from different vortex halos are not coherent. Therefore, it’s better to shift the position of each vortex center to the origin when doing Fourier Transformation for each vortex halo. In this way we can make different vortex halos coherent and greatly enhance signals. The following are predictions for PDW-Driven scenario and how to detect it in experiment. As a benchmark, we show our numerical simulation data. The method of our simulation is summarized in Appendix.B. Profile of d wave order parameter is $\Delta_D(r,\theta)\sim \frac{r}{\sqrt{r^2+r_0^2}}$ with vortex core size $r_0=3.5$ lattice constants. We used a profile of PDW with $r$ dependence as $\Delta_P(r,\theta)\sim e^{1-\sqrt{r^2+\xi^2}/\xi}$ with correlation length $\xi=15$. In the following, local density of states $\nu^E(r)$ is obtained at fixed energy $E=30$ meV. Note we only show $d$ wave form of Bond LDoS because CDW generated by our model is dominated by $d$ wave. However, we expect our predictions in the following sections do not rely on form factor. Split Peaks for Period 8 CDW ---------------------------- The first prediction for PDW scenario is that the peak at $\mathbf{Q_a/2}$ is split to two peaks in the direction decided by $\theta_a$. Recall that the density modulation $\mathlarger \rho(\mathbf r)=\int^0_{-\infty} dE \nu^E(\mathbf r)$ is given by the integral of LDoS $\nu^E(\mathbf r)$ over the occupied states. We define the slowly varying complex amplitude $\nu^E_{\mathbf{Q_a/2}}(\mathbf r)$ by writing the real space local DoS as $\mathlarger \nu^E(\mathbf r)=\sum_{a}\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf r)e^{\frac{1}{2}i\mathbf{{Q}_a}\cdot \mathbf{r}}+h.c.$. This is the analog of $\mathlarger\rho_{\mathbf{Q_a/2}}$ discussed in the last section. We assume that $\nu^E_{\mathbf{Q_a/2}}(\mathbf r)$ has a similar real space profile as $\mathlarger\rho_{\mathbf{Q_a/2}}$ as given in Eq. \[eq:pdw\_profile\], i.e. it is confined to the vicinity of the vortex core and importantly, is proportional to $\cos(\theta-\theta_a)$. Recall that this factor encodes the phase winding of the d wave superconductor and is therefore an important signature for the PDW driven scenario. .This assumption is supported by our numerical simulations, and will be discussed and shown in greater detail later in Fig. \[fig:pdw\_angle\] and Fig. \[fig:local\]. We define $\tilde \nu^E(\mathbf q)$ to be the Fourier Transform of $\nu^E(\mathbf r)$. For $\mathbf q$ in the vicinity of $\mathbf{Q_a/2}$ we define $$\tilde A_a(\mathbf q)=\tilde \nu^E(\mathbf q-\mathbf{Q_a/2})=\sum_{\mathbf{r}}\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf r)e^{-i \mathbf{q}\cdot \mathbf{r}} \label{eq:conv}$$ Consider $a$ in the $x$ direction. When $\theta_a=0$, it’s easy to see that the absolute value of $\tilde A_a(\mathbf q)$ has two peaks in $x$ direction because of the $\cos \theta$ factor. This is because $\cos \theta=\frac{x}{\sqrt{x^2+y^2}}$ produces a line of zero in $\nu^E_{\mathbf{Q_a/2}}(\mathbf r)$ along the $y$ direction through the vortex core. $\nu^E_{\mathbf{Q_a/2}}(\mathbf r)$ is odd under $x \rightarrow -x$ and as a result $\tilde A_a(\mathbf{q_x}=0)=0$ and $\tilde A_a(\mathbf q)$ has a splitting along the $\mathbf{q_x}$ direction. The splitting is roughly $\delta q \sim \frac{1}{\xi}$. For general $a$ and general $\theta_a$, the line of zero in $\tilde A_a(\mathbf q)$ is rotated by an angle $\theta_a$. Therefore, the absolute value of $\mathlarger{\tilde \nu^E}(\mathbf q)$ should have two peaks at $\mathbf{q}\approx\mathbf{Q_a/2}$ with the splitting in the direction of $\theta_a$. This prediction is confirmed by numerical simulation results for both PDW-Driven model and CDW-Driven model in Fig. \[fig:fft\]. Here we show two different phase choices for PDW-Driven model. The splitting of period $8$ peak along the direction $\theta_a$ is very clear for PDW-Driven models while CDW-Driven model show one single peak. Therefore, we suggest to fit experimental data with a split-peak model. In our simulation, if we choose the vortex center as the origin, we found that $\tilde \nu^E(\mathbf q)$ is dominated by real part. Thus it is better to plot only real part of $\tilde \nu^E(\mathbf q)$. Besides, there should be a sign change at $\mathbf q=(\frac{1}{8}\frac{2\pi}{a},0)$ if we plot Re$\nu^E(q_x)$ along $q_y=0$ cut, as shown in Fig. \[fig:fftx\]. Again, this comes from the Fourier transformation of $\cos(\theta)$. [0.45]{} ![$|\tilde \nu^E(q)|$ with $E=30$ meV for PDW-Driven and CDW-Driven Models.[]{data-label="fig:fft"}](image/cdw_fft.png "fig:"){width="\textwidth"} [0.45]{} ![$|\tilde \nu^E(q)|$ with $E=30$ meV for PDW-Driven and CDW-Driven Models.[]{data-label="fig:fft"}](image/phase0.png "fig:"){width="\textwidth"} [0.45]{} ![$|\tilde \nu^E(q)|$ with $E=30$ meV for PDW-Driven and CDW-Driven Models.[]{data-label="fig:fft"}](image/phase1.png "fig:"){width="\textwidth"} Direct Visualization of “Dislocation” ------------------------------------- To have a direct visualization of profile shown in Eq. \[eq:pdw\_profile\] for a PDW-Driven model, we need to extract local CDW order parameter $\mathlarger \nu^E_{\mathbf{Q_a/2}}(x,y)$ from STM data $\nu^E(x,y)$. For each position $(x_0,y_0)$, we construct a new image by multiplying a gaussian mask: $$\bar \nu^E(\mathbf{r};\mathbf{r_0})=e^{-\frac{|\mathbf{r}-\mathbf{r_0}|^2}{2W^2}} \nu^E(\mathbf{r}) \label{eq:local_order}$$ We found that $W=8$ is a good choice in our simulation. Then we can extract local CDW order parameter $\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})$ by a Fourier Transformation of $\tilde \nu^E(\mathbf{r};\mathbf{r_0})$: $$\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})=\sum_{\mathbf{r}} \mathlarger{\bar \nu^E}(\mathbf{r};\mathbf{r_0})e^{-\frac{1}{2}i \mathbf{Q_a}\cdot \mathbf{r}}$$ After extracting $\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})$ for each position, we can easily visualize it and decide whether there is a superposition of strength $\pm 1$ dislocations. The above algorithm can also be implemented by filter algorithm in momentum space directly as in Ref. : $$\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})=\sum_{\mathbf q} \tilde \nu^E(\mathbf q)G(\mathbf{Q_a/2}-\mathbf q)e^{-i\mathbf{(Q_a/2-q)}\cdot \mathbf r_0}$$ where the filter is $G(\mathbf q)=\sum_re^{-\frac{|\mathbf{r}|^2}{2W^2}}e^{-i \mathbf q \cdot \mathbf r}=e^{-\frac{W^2}{2}|\mathbf q|^2}$. Here we show visualization for simulated data of $|\mathlarger \nu^E_{\mathbf{Q_a/2}}|^2$ from both CDW-Driven and PDW-Driven model in Fig. \[fig:local\]. The distinction is very obvious. For CDW-Driven model, $\nu^E_{\mathbf{Q_a/2}}$ has maximal intensity at vortex center. For PDW-Driven model, $|\nu^E_{\mathbf{Q_a/2}}|$ vanishes along a line across the vortex center in the direction of $\theta_a\pm\frac{\pi}{2}$, in agreement with a $\cos(\theta-\theta_a)$ angle dependence. Across the dark line, phase of local amplitude $\nu^E_{\mathbf{Q_a/2}}$ has a $\pi$ shift, as shown in Fig. \[fig:phase\_plot\]. We can see the phase of $\nu^E_{\mathbf{Q_a/2}}$ is $\phi_a$ or $\phi_a+\pi$. Therefore we can remove the overall phase by $\nu^E_{\mathbf{Q_a/2}}\rightarrow \nu^E_{\mathbf{Q_a/2}}e^{-i\phi_a}$ and make it real. Then angle dependence $\nu^E_{\mathbf{Q_a/2}}\sim \cos(\theta-\theta_a)$ can be visualized directly in Fig. \[fig:angle\]. For uni-directional PDW, Wang et al.[@wang2018] also noted the phase jump by $\pi$ by tracking the position of the DOS peaks in real space[@wang2018]. In Fig. \[fig:fixy\], we plot $Re \nu^E_{\mathbf{Q_x/2}}(x)$ at fixed y. For $y=0$, $|\nu^E_{\mathbf{Q_x/2}}(x)|$ gives the radius dependence $F(r)$. We can see that the maximum is at finite $r$. However, our simulation may overestimate the maximum because of boundary effects due to finite size. Finally, we comment on challenges to apply this algorithm to real experimental data. (1) The existence of multiple vortices and impurities modifies the $\cos(\theta-\theta_a)$ angle dependence. In general, there is no time reversal symmetry or any lattice symmetry left, and $\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})$ is complex. Thus the line of zero we predicted in the simple model may not be exact. We still expect the real and imaginary parts of $\mathlarger \nu^E_{\mathbf{Q_a/2}}(\mathbf{r_0})$ to each have a line of zero but the lines will no longer coincide. As a result the line of zero’s shown in Fig. \[fig:local\](c-f) will partially fill in. (2) There is smooth background, which will add an offset to the $\cos(\theta-\theta_a)$ factor. If we assume background is smooth, it can be subtracted with sophisticated data analysis technique. [0.4]{} ![$\nu^E_{\mathbf{Q_a/2}}$ for PDW-Driven model with $\theta_x=0$ and $\theta_y=\frac{\pi}{2}$. []{data-label="fig:pdw_angle"}](image/angle_plot.png "fig:"){width="\textwidth"} [0.4]{} ![$\nu^E_{\mathbf{Q_a/2}}$ for PDW-Driven model with $\theta_x=0$ and $\theta_y=\frac{\pi}{2}$. []{data-label="fig:pdw_angle"}](image/fixy.png "fig:"){width="\textwidth"} [0.4]{} ![$\nu^E_{\mathbf{Q_a/2}}$ for PDW-Driven model with $\theta_x=0$ and $\theta_y=\frac{\pi}{2}$. []{data-label="fig:pdw_angle"}](image/Angle.png "fig:"){width="\textwidth"} [0.2]{} ![$|\nu^E_{\mathbf{Q_a/2}}|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/cdw_x_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$|\nu^E_{\mathbf{Q_a/2}}|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/cdw_y_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$|\nu^E_{\mathbf{Q_a/2}}|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/x_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$|\nu^E_{\mathbf{Q_a/2}}|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/y_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$|\nu^E_{\mathbf{Q_a/2}}|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/phase1_x_intensity.png "fig:"){width="\textwidth"} [0.2]{} ![$|\nu^E_{\mathbf{Q_a/2}}|^2$ from CDW-Driven and PDW-Driven Models. $(a)$ and $(b)$ are from a CDW-Driven model; Others are from PDW-Driven models. $E=30$ meV.[]{data-label="fig:local"}](image/phase1_y_intensity.png "fig:"){width="\textwidth"} Flux Density Wave ----------------- In PDW-Driven scenario, we will also get flux density wave. Orbital magnetic moment of each plaquette $M(\mathbf{r})$ can be estimated through the following equation: $$\begin{aligned} M(\mathbf{r}+\frac{\hat{x}}{2}+\frac{\hat{y}}{2})&=\frac{a^2}{4}\big(I(\mathbf{r},\mathbf{r+\hat{x}})+I(\mathbf{r+\hat{x}},\mathbf{r+\hat{x}+\hat{y}})\notag\\ &+I(\mathbf{r+\hat{x}+\hat{y}},\mathbf{r+\hat{y}})+I(\mathbf{r+\hat{y}},\mathbf{r})\big)\end{aligned}$$ where $a=3.5\textup{\AA}$ is lattice constant. $I(\mathbf{r},\mathbf{r+\hat{r}_a})$ is current going through bond from $\mathbf{r}$ to $\mathbf{r+\hat{r}_a}$ where $a$ denotes $x$ or $y$. $M(\mathbf{r})$ has density wave with momentum $\mathbf{Q_x/2}=(\frac{2\pi}{8},0)$, $\mathbf{Q_y/2}=(0,\frac{2\pi}{8})$. There is also density wave at diagonal direction $Q_{\pm,\pm}=(\pm \frac{2\pi}{8},\pm \frac{2\pi}{8})$. Real space and momentum space pattern of magnetic moment are shown in Fig. \[fig:fdw\]. Amplitude of density wave at momentum $(\frac{2\pi}{8},\frac{2\pi}{8})$ is around $0.005\mu_B$ and may be possible to be detected by neutron scattering experiment. The observation of flux density wave at this wave-vector offers the opportunity to definitively settle the question of uni-directional vs bi-directional PDW. [0.4]{} ![Flux Density Wave pattern from PDW-Driven model in vortex halo.[]{data-label="fig:fdw"}](image/fdw_real.png "fig:"){width="\textwidth"} [0.4]{} ![Flux Density Wave pattern from PDW-Driven model in vortex halo.[]{data-label="fig:fdw"}](image/fdw_fft.png "fig:"){width="\textwidth"} Other Types of PDW ------------------ This paper is mainly focused on bidirectional PDW model. However, other types of PDW state have been proposed before. In this section, we show signatures for Unidirectional PDW and Canted PDW model. Therefore STM experiments can rule out or support these kinds of PDW models. For Unidirectional PDW shown in Fig. \[fig:uni\_fft\] with only $x$ component, Fourier Transform data only show peak at $\mathbf{Q_x/2}$, not at $\mathbf {Q_y/2}$. There is still split of peak consistent with our previous discussions for bidirectional PDW. For canted PDW, we expect the peak in $\nu^E(q)$ deviates from $(1,0)$ and $(0,1)$ direction. For canted PDW model with shifted momentum $p=0.03*2\pi/a$: $\mathbf{P_1}=(\frac{2\pi}{8},p)$, $\mathbf{P'_1}=(-\frac{2\pi}{8},p)$ and $\mathbf{P_2}=(p,\frac{2\pi}{8})$,$\mathbf{P'_2}=(p,-\frac{2\pi}{8})$ this shift shows up in Fig. \[fig:canted\_fft\]. Because of condition $\tilde \nu^E(q)=\tilde \nu^E(-q)^*$, we see double peak with shift $p$. In experiment it may be better to detect this feature with complex amplitude $\tilde \nu^E(q)$ instead of intensity $|\tilde \nu^E(q)|$. ![$|\tilde \nu^E(q)|$ for unidirectional PDW with phase $\theta_x=0$.[]{data-label="fig:uni_fft"}](image/uni_fft.png){width="40.00000%"} ![$|\tilde \nu^E(\mathbf{q})|$ for canted PDW with shifted momentum $p=0.03*2\pi/a$. Phase of PDW is $\theta_x=0$ and $\theta_y=\frac{\pi}{2}$.[]{data-label="fig:canted_fft"}](image/canted_fft.png){width="40.00000%"} If we can decide the value of shift momentum $|p|$ from Fourier Transformation data, then we can extract local order parameter $\nu^E_{P}(\mathbf{r})$ with $P_\pm=(\frac{2\pi}{8},\pm p)$ following Eq. \[eq:local\_order\]. It turns out that $P=(\frac{2\pi}{8},p)$ has an anti-vortex while $P=(\frac{2\pi}{8},-p)$ has a vortex, as shown in Fig. \[fig:vortex\_canted\]. [0.22]{} ![$\arg \nu^E_P(\mathbf{r})$ for canted PDW in unit of $\pi$.[]{data-label="fig:vortex_canted"}](image/shifted_x_phase.png "fig:"){width="\textwidth"} [0.22]{} ![$\arg \nu^E_P(\mathbf{r})$ for canted PDW in unit of $\pi$.[]{data-label="fig:vortex_canted"}](image/anti-shifted_x_phase.png "fig:"){width="\textwidth"} If momentum resolution is not good enough to decide the value of $p$, we propose to visualize $\nu^E_{P_0}(\mathbf(r))$ with $P_0=(\frac{2\pi}{8},0)$. If it is ordinary PDW-Driven, we get similar plot as in Fig. \[fig:phase\_plot\]. If it is canted PDW-Driven, we will get strange position dependence of $\arg \nu^E_{P_0}(\mathbf r)$ like in Fig. \[a\_canted\_unshift\_plot\]. This is a signature of canted PDW and it’s consistent with the following equation: $$\nu^E_{P_0}(\mathbf r)\sim \cos(\theta-py) \label{eq:simulated_canted}$$ [0.22]{} ![Visualization of $\arg \nu^E_{P_0}(\mathbf r)$ for canted PDW Driven model. $P_0=(\frac{2\pi}{8},0)$ and shifted-momentum is $p=0.03*2\pi$.[]{data-label="fig:canted-unshift_plot"}](image/simulated_angle.png "fig:"){width="\textwidth"} [0.22]{} ![Visualization of $\arg \nu^E_{P_0}(\mathbf r)$ for canted PDW Driven model. $P_0=(\frac{2\pi}{8},0)$ and shifted-momentum is $p=0.03*2\pi$.[]{data-label="fig:canted-unshift_plot"}](image/x_phase.png "fig:"){width="\textwidth"} Summary ======= We now summarize some of the conclusions from the discussion in previous sections. The observation of period 8a bidirectional charge order in the vortex halo directly means that there are induced order parameters $\rho_{Q_x/2}, \rho_{Q_y/2}$. In the presence also of a non-zero superconducting order parameter $\Delta_d$ of the usual $d$-wave superconductor, the period-8 charge order necessarily implies that there are also period-8 modulations in the superconducting order parameter $\Delta_{Q_x/2}, \Delta_{Q_y/2}$, [*i.e*]{}, Pair Density Wave order at the same period. Given this obvious equivalence in the superconductor between charge and pairing modulations, it may seem to be a moot question whether what is observed is primarily charge order or pair order at period-8. Nevertheless we have shown that there are two distinct possibilities for the observed period-8 order which naturally correspond to two distinct driving mechanisms. In the CDW-driven scenario, we simply postulate that there are slow fluctuations of a previously unidentified period-8 CDW in the uniform superconductor. In the vicinity of the vortex the breaking of translational symmetry and the weakening of the superconducting order may then pin the fluctuations of the period-8 CDW and lead to static ordering. Period-4 charge order then appears as a subsidiary order. In this scenario it is natural to expect that the phase of the induced CDW order does not wind on going around the vortex core. In the PDW-driven scenario on the other hand, we postulate that there are slow fluctuations of period-8 PDW that are pinned in the vortex halo. The induced period-8 CDW then will have a strength $\pm 1$ dislocation centered at the vortex core. More precisely the induced period-8 CDW will be a superposition of a configuration with a strength $+1$ dislocation and one with a strength $-1$ dislocation. This leads to a rather different spatial profile for the induced period-8 CDW. A further difference is that there are now two distinct kinds of induced period-4 CDW orders which we have referred to as CDW$_A$ and CDW$_B$. The CDW$_A$ pattern has no winding around the vortex core while the CDW$_B$ pattern is a superposition of strengths-$\pm 2$ dislocations. We discussed the extent to which existing data supports either scenario. In particular in the PDW-driven scenario there is a natural explanation for the absence of peaks at $2\pi \left(\frac{1}{8}, \frac{1}{8}\right)$ as reported in the experiments. It is however important to analyze the data more carefully to clearly establish which of these scenarios is realized, and we described a number of distinguishing features. Most importantly the spatial profile of the induced charge orders due to the dislocation structure in the PDW-driven scenario should be discernible using the methods we describe. Note that within either of these scenarios there is no general reason for a predominantly $d$-form factor period-$8$ charge order to induce only an $s$-form factor period-4 charge order [@Note1]. From our numerical simulation of $d$ wave PDW coexistence with uniform $d$ wave superconductor, period $8$ CDW we get is actually dominated by $d$ wave, instead of $s$ wave from naive expectation. Thus we do not have a natural explanation of the observations on form factors in the experiments. A further question that one can ask is whether the fluctuation order that is pinned on the halo is unidirectional or bidirectional. The observed period-$8$ modulations are apparently bidirectional. The simplest explanation therefore is that the “parent’ order is also bidirectional. However one may postulate that there are domains of different unidirectional patches within the vortex halo. This may be easy to check in the STM data. Finally an important question is whether the period-8 PDW (if it is really the driver) is merely a competing/intertwined order with the standard $d$-wave superconductor or whether it is a “mother" state with a very large amplitude that controls the physics up to a much larger energy scale than the standard $d$-wave order itself. Just based on the STM experiments alone there does not seem to be any clear way to answer this question. However in the following section, by combining with information from other existing experiments, we will provide suggestive arguments in favor of a mother PDW state. A broader perspective on PDW and its relation to the pseudogap state of the Cuprates ==================================================================================== In this section we take a broader perspective and ask whether the message learned from the STM data on Bi-2212 can inform us on anomalies observed in other cuprates and more generally on the pseudo-gap itself. We shall assume the that the data are described by the fluctuating PDW (“mother state”) scenario and we shall assume that scenario continues to hold in other under-doped Cuprates. We focus our attention on YBCO where extensive data on the CDW up to high magnetic field is available[@changNatureComm72016magnetic; @ZX1science350949gerber2015three; @ZX2PNAS11314647jang2016ideal]. The picture that emerges from these studies is that SRO CDW appears below about 150K over a doping range between x=0.09 and 0.16[@BlancoPhysRevB.90.054513]. This SRO CDW has very weak interlayer ordering centered around L=1/2 where L is the c axis wave-vector in reciprocal lattice unit. These peaks grow with decreasing temperature but their strength weaken and their in plane linewidth broaden below $\text{T}_c$. These peaks occur along both a and b axes. Above a field of 15 to 20T, a uni-directional CDW emerges and rapidly becomes long range along the b axis. The onset of long range ordered CDW is consistent with earlier NMR data.[@wu2013emergence; @wu2015incipient] At the same time, the SRO CDW remains along both a and b axes. Thus the high magnetic field data shows that there are two kinds of CDW with the same incommensurate period which does not change with magnetic field. As the experimentalists remarked[@ZX1science350949gerber2015three; @ZX2PNAS11314647jang2016ideal], this is very puzzling because having the same incommensurate wave-vector suggests the two kinds of CDW share a common origin. If we interpret the observed CDW as subsidiary to a fluctuating PDW, the latter must exist above the CDW onset at 150K and most likely above $T^*$ which is taken as the thermodynamic signature of the pseudogap. Similarly we take the viewpoint that quantum oscillations require the existence of bi-directional CDW[@sebastian2012towards],which implies that fluctuating PDW extends to magnetic fields of 100T and beyond. By continuity we expect fluctuating PDW to cover a large segment of the H-T plane, as shown in Fig. \[fig:phase\_diagram\]. The PDW must be strongly fluctuating in time, because there is no sign of superconductivity from transport measurements outside of a limited region near $T_c$ and $\text{H}_{c2}$. However, diamagnetic signals are observed over a much larger regime[@yuPNAS126672016magnetic], a point which we shall return to later. Nevertheless, our picture is that the subsidiary orders such as CDW can be more robust and make their presence felt. This is particularly true of $CDW_A$ (see Eq. \[Eq: CDWA\]) which does not require d wave pairing for its presence. So we assign $CDW_A$ to be the SRO CDW which onsets below 150K, as shown by the dashed line in Fig. \[fig:phase\_diagram\]. Below $T_c$ the phase stiffness of the LRO d wave robs oscillator strength from the PDW, diminishing its already weak phase stiffness even further. This explains the reduction of the CDW strength below $T_c$. On the other hand, we saw in section III that in a magnetic field a vortex can pin the PDW to form a static but short range halo around the core. This in turn induces CDW at wave-vector Q/2 and its harmonic $CDW_B$. All these states are located roughly inside the superconducting region as indicated in Fig. \[fig:phase\_diagram\]. Of course being tied to the vortices mean that the strengths of these states are proportional to the magnetic field. Note that we expect the d wave phase stiffness to be reduced inside the halo while that of the PDW to be strengthened. We define the field $H_0$ as H\_0=\_0/(2\^2\_P) where $\phi_0 = hc/2e$ is the flux quanta in a superconductor, $\xi_P$ is the correlation length of the pinned PDW. The $2\pi$ in the denominator has been inserted to make this equation resemble the definition of $H_{c2}$ and the exact numerical factor should not be taken seriously. The point is to provide a scale for the field where the pinned PDW starts to strongly overlap. For $H>H_0$, the d wave superconductor is being squeezed out and the PDW phase regains its stiffness. It eventually becomes depinned as the d wave pairing diminishes and resumes its dynamical fluctuation. In this region the $CDW_A$ grows in strength and coherence, recovering the growth with decreasing temperature that was interrupted by the onset of $T_c$ for $H<H_0$. The fact that the LRO CDW is uni-directional even though the PDW is bi-directional can be rationalized by the following argument. There is a term in the Landau free energy $\gamma |\rho_{Q_x}|^2 |\rho_{Q_y}|^2$ which strongly prefers uni-directional order when $\gamma$ is large and positive. In YBCO the presence of the chain already broke tetragonal symmetry to begin with, making it even more plausible that the order grow strongly in one direction. On the other hand, the term $\Delta_P \Delta_{-P}^* \rho_Q$ is linear in $\rho_Q$, meaning that some SRO is likely generated in the orthogonal direction. We shall return to this point later. Returning to the region below $\text{H}_{c2}$ we expect to find the pinned PDW and the CDW with period Q/2 as static but short range ordered. This is because the static order of Q/2 CDW requires the static order of d wave pairing as well as PDW. The Q/2 CDW should persist to lower field with decreasing amplitude. It may be expected to have correlation length similar to that found in the STM experiment, which we estimate to be about 16 lattice spacings. It will of course be of great interest to search for this by X-ray scattering. On the other hand, the period Q $\text{CDW}_B$ can be thought of as a harmonic of the period Q/2 CDW, but it can exist even in its absence. Thus we expect it to exist up to higher field. We do not know exactly how high a field it can persist to, but it cannot go above the d wave vortex liquid regime. It is worth noting that in practice there can be remnants of static pinned vortices even above $H_c2$. Yu et al[@yuPNAS126672016magnetic] reported hysteretic behavior which extends to very high field at low temperatures, leading them to identify a second vortex solid regime. The existence of some form of bi-directional CDW that persists up to high field at low temperature is important in order to explain the quantum oscillations. We believe the LRO unidirectional CDW cannot by itself give rise to quantum oscillations, but the combination with some SRO CDW in the direction perpendicular to it may be sufficient. This can come from the bi-directional $\text{CDW}_B$ discussed above if it persists to high field, or it is possible that a short range order $\text{CDW}_A$ is generated along direction $a$ at higher field as explained earlier. In support of the picture outlined above, we note that there is extensive NMR data showing that $\text{H}_0$ is typically 5 to 10T below the $\text{H}_{c2}$ as measured by transport[@Julien2arXivzhou2017spin; @wu2013emergence]. Thus there is a close relationship between $\text{H}_{c2}$ and the vortex halo size as defined by the size of the pinned PDW. We also recall that the CDW that we identify as type A in Bi-2212 is uni-directional, which agrees with this assignment for YBCO. We note that the Bi-2212 sample used has a doping of 0.17 which lies on the upper end of the observability of CDW in YBCO samples. The $\text{H}_{c2}$ and corresponding $\text{H}_0$ are expected to be very high. So the 8.25T used in the STM experiment is expected to be far below the regime where $\text{CDW}_A$ can achieve long range order. In Fig. \[fig:phase\_diagram\] we add the line $\text{H}_0$ to a phase diagram in the H-T plane for under-doped Cuprates, following the proposal of Yu et al[@yuPNAS126672016magnetic]. The resistive $\text{H}_{c2}$ is the boundary of the vortex solid and marks the resistive transition. (To avoid cluttering, we did not show the emergence of a second vortex solid regime mentioned earlier that extends to high field at low temperature[@yuPNAS126672016magnetic].) The key point made by Yu et al is that there is a large region of vortex liquid in the phase diagram where there is strong superconducting amplitude. The evidence for this is a strong diamagnetic signal. Given the small size of the true vortex core where the d wave coherence peak is destroyed, it is reasonable to interpret the vortex liquid as a region of strong d-wave superconducting amplitude with dynamical vortices that persists to very high field. It is less certain how high in temperature the d-wave vortex liquid extend. It is possible that the diamagnetic signal may come from PDW fluctuations at high fields[@yuPNAS126672016magnetic; @lee2014amperean]. Thus the location of the dotted line in Fig. \[fig:phase\_diagram\] that indicate the extent of d wave vortex liquid is quite uncertain, especially in the temperature direction. ![$H-T$ Phase Diagram for an underdoped Cuprate. The light blue shading indicates that a fluctuating PDW is pervasive over a large segment of the $H-T$ plane for underdoped Cuprates. Dashed line indicates the onset of short range ordered CDW at wave-vector Q. It is a subsidiary order of the PDW which we refer to as $\text{CDW}_A$. Sold red line marks the magnetic field $H_0$ as defined in Eq.(14) in terms of the coherence length $\xi_P$ of the PDW which marks the size of the vortex halo. It is closely related to the field $H_{c2}$ which marks the onset of a vortex solid phase and LRO superconductivity. Within this phase and inside the vortex halo we expect the pinned static PDW, Q/2 CDW as well as its harmonic, a wave-vector Q CDW which we refer to as $\text{CDW}_B$. The $\text{CDW}_B$ short range order state may extend to higher magnetic field much beyond $H_{c2}$. The dotted red line indicates the onset of a vortex liquid phase. The brown area indicates the appearance of long range ordered type A CDW with wave-vector Q. []{data-label="fig:phase_diagram"}](image/phase_diagram.png){width="0.9\linewidth"} ![Illustration of the Loop Current produced by the canted PDW.[]{data-label="fig:loop_current"}](image/loop_current.png){width="0.9\linewidth"} We should mention that similar CDW has been seen in the Hg-based compound. Here the doping range extends further down to $x$ of order 0.06 and up to about 0.12. Another difference is that there is no clear suppression of the CDW at $\text{T}_c$. Instead its strength seems to saturate. It should be noted that unlike YBCO, this is a tetragonal system. From existing X ray data, it is not known whether the CDW is bi-directional or uni-directional. Apart from these differences, the observations seem to fit into the same phase diagram shown in Fig. \[fig:phase\_diagram\]. Finally we comment on the symmetry breaking observed at the $T^*$ lines which lies at a temperature above the onset of SRO CDW. This seems to be associated with breaking a lattice symmetry, perhaps a kind of nematic order. Importantly, a recent experiment on the anisotropy of the spin susceptibility[@Matsuda2unpublished] found the nematic axis to be along the diagonal in a single layered Hg-based compound, while it is along the bond direction in YBCO[@MatsudaNaturePhysicssato2017thermodynamic]. This would rule out nematicity based on CDW which should be along the bond direction in a single layer tetragonal system. The observation in YBCO can be understood from the stacking of two orthogonal directions of diagonal nematicity in each layer. Such nematicity agrees with the symmetry of the orbital current model[@aji2009quantum]. As mentioned earlier, in the PDW model it was pointed out by Agterberg et al.[@Agt2PhysRevB.91.054502] that adding canting to the PDW model as described earlier has the same symmetry as the orbital current model. The four different combinations of (p1,p2) give rise to a 4 state clock model. Fluctuations between (1,1) and (-1,-1) restores time reversal symmetry but gives rise to a diagonal breaking of nematic symmetry, just like the orbital current model. Indeed a canted PDW model will carry intra-cell currents as shown in Fig. \[fig:loop\_current\], which is the closest we can get to Varma’s model in a single band model. As seen in this figure, the current can be understood as supercurrent running along x and y, with a return current along one of the diagonal bond. In fact we find that such a current pattern emerges from the PDW model. Without self-consistent determination of the mean field ground state, there is a net current along x and y, which presumably will be fixed by a proper return current in a self-consistent mean field theory. However, the current we find is very small, on the order of $10^{-3} t$ on each bond. This gives rise to a moment of about $10^{-3} \mu_B$ which is too small compared with the 0.1 $\mu_B$ reported by neutron scattering. We note on general ground that the orbital current in the PDW model must be small. Let us define the canted component of the wave-vector as $p = (P + P')/2$. The supercurrent can be estimated from the product of the phase gradient which is $p$ and the spectral weight, which is $x/m$ where $1/m$ is proportional to $ta^2$. Thus we expect the maximal supercurrent to be $x|p|t$ where $p$ is in reciprocal lattice units. Since $|p|$ should be less than $|P|$, we expect $x|P|$ to be less than $10^{-2}$ and similarly for the moment in units of $\mu_B$. Thus it is unlikely that the canted PDW model can account for the orbital current observed by neutron. However, it potentially can explain the onset of diagonal nematicity at $T^*$. Finally we call attention to the most interesting part of the phase diagram, the region at zero temperature and above $H_{c2}$. In our picture this is a ground state consisting of a PDW which does not order due to quantum fluctuations. This state is metallic with some combination of long range and short range CDW order, sufficient to form pockets visible by quantum oscillations. What is the nature of this state? Is it a Fermi liquid? Is the dissipation due to the metallic state responsible for quantum disordering the PDW? These are fascinating questions that are beyond the purview of the present phenomenology oriented paper. Conclusion ========== Based on our analysis, we come to the following conclusions: 1\. It is likely that the 8a CDW observed in the STM experiment has its origin in a period 8 PDW which is pinned to be static near the vortex core. The main evidence based on the currently available data is the absence of a peak at (1/8,1/8) which would be expected if the 8a CDW were primary. We propose further analyses of the data which can nail down this conclusion. The main point is that the winding of the d wave superconducting phase around the vortex core imprints a very special signature on the period 8 CDW which is visible either as a splitting of the Fourier transform peak or a sign change across an oriented line in the Fourier filtered data. 2\. We think it is likely that the PDW pinned near the vortex core is bi-directional, because both the 8a and 4a CDW observed there appears to be bi-diagonal. A bi-diagonal PDW can generate uni-directional CDW but the converse is not true: a uni-directional PDW may be able to generate checkerboard patterns made up of patches of uni-directional stripe CDW, but that distinction should be amenable to experimental test. 3\. The naive expectation that the subsidiary 4a order has local $s$ symmetry is not generally correct, given the definition of the form factor used in the STM experiments[@Note1]. In fact, in our microscopic mean field model, we find these to have mainly $d$ symmetry. The local symmetry depends on the microscopic detail and it is no surprise that it is not captured by our simple mean-field theory, but we want to convey the message that a $d$ symmetry subsidiary order can readily be generated. Thus the observed $d$ symmetry CDW that is already present at zero field may also be a subsidiary order due to PDW. For Bi-2201 the CDW is close to commensurate with period 4 and we cannot rule out that this CDW is not simply an independent order, as advocated in a recent preprint. [@wang2018] On the other hand the idea of independent order is difficult to justify for YBCO, where two different CDW seem to co-exist with the same incommensurate period. We discuss a scenario where both CDW’s are subsidiary to the same PDW. 4\. Up to now the notion of a halo around a vortex core is not a well-defined one. The coherence peak associated with d wave superconductivity is killed only inside the true core, which has a radius of 2 or 3 lattice spacing. The coherence peak remains visible throughout the halo region, indicating that d wave order is not fully destroyed. We propose that the size of the pinned PDW provides a way to define the halo radius and we introduce a magnetic field scale $H_0$ associated with this length scale. We relate this field scale to the growth of the 4a CDW observed in underdoped YBCO samples and with $H_{c2}$. 5\. A canted PDW is an attractive scenario that can unify the pseudo phenomenology with the nematic transition observed at $T^*$. The STM data offers a way to search for this kind of order, even though the required resolution may be challenging. In summary we answer the question we first posed in the introduction: we think that that the observed period 8 CDW is opening a new window into the world of underdoped cuprates and pseudogap physics. Much exciting further developments are sure to come. Acknowledgement =============== We thank J. C. Davis and M. Hamidian for sharing with us their data prior to publication and for very helpful discussions. PAL acknowledges the support of NSF under DMR-1522575. TS is supported by a US Department of Energy grant DE-SC0008739, and in part by a Simons Investigator award from the Simons Foundation. We thank the Moore Foundation EPiQS program for facilitating our interaction with J. C. Davis. TS thanks the conference on High Temperature Superconductivity at the Aspen Center for Physics, which is supported by NSF grant PHY-1607611, for enabling a part of this work. Numerical calculation of band structure of PDW state ==================================================== For uniform PDW state, we calculate the band structure by diagonalize a BdG Hamiltonian $H(k)$ for each momentum $k$. At each $k$, we need to use a $81*2=162$ basis:$\Psi_k=(\psi_\uparrow(k),\psi_\downarrow^\dagger(-k))$. $\psi_\sigma(k)$ is a collection of $9\times9=81$ electron annihilation operators: $c_{k'}$ with momenta $k'=k+m \mathbf{P_x}+n\mathbf{P_y}$ where $\mathbf{P_x}\approx(\frac{2\pi}{8},0)$ and $\mathbf{P_y}\approx(0,\frac{2\pi}{8})$, $m,n=-4,-3,-2,-1,0,1,2,3,4$. In Sec. \[Sec: PDW with long range order\] we use $\mathbf{P_x}\approx(0.14\times(2\pi),0)$ and $\mathbf{P_x}\approx(0,0.14\times(2\pi))$. We set a large truncation for m and n to better capture the effect of subsidiary CDW generated by PDW. In this basis, we rewrite the mean field Hamiltonian in Eq. \[Eq: long range PDW mean field\] at momentum $k$ as H\_k &=& \_[m,n]{}\_[k+m +n]{} c\^\_[k+m +n,]{}c\_[k+m +n,]{}\ &-&\_[m,n]{}\_[-k-m -n]{} c\_[-k-m -n,]{}c\^\_[-k-m -n,]{}\ &+&\_[m,n]{}2((k\_x+m P\_x+nP\_y - P\_x/2) - (k\_y+m P\_x+n P\_y)) c\_[k+m +n,]{}c\_[-k -m -n + ,]{}\ &+&\_[m,n]{}2((k\_x+m P\_x+nP\_y + P\_x/2) - (k\_y+m P\_x+nP\_y)) c\_[k+m +n,]{}c\_[-k -m -n - ,]{}\ &+&\_[m,n]{}2((k\_x+m P\_x+nP\_y) - (k\_y+m P\_x+n P\_y- P\_y/2)) c\_[k+m +n,]{}c\_[-k -m -n + ,]{}\ &+&\_[m,n]{}2((k\_x+m P\_x+nP\_y) - (k\_y+m P\_x+nP\_y+ P\_y/2)) c\_[k+m +n,]{}c\_[-k -m -n - ,]{}\ &+& h.c., where $\D = 45$meV. For the bare band dispersion $\e_k$, we use a tight banding model on square lattice with nearest neighbor hopping $t=0.21$eV, second neighbor hopping $t_p=-0.047$eV, third neighbor hopping $t_{pp}=0.04$eV and fourth neighbor hopping $t_{ppp}=-0.01$eV. \_k = -2t((k\_x)+(k\_y)) - 4t\_p(k\_x)(k\_y) - 2t\_[pp]{}((2k\_x)+(2k\_y))\ - 4t\_[ppp]{}((2k\_x)(k\_y) + (k\_x)(2k\_y)) - \_0 We fix the chemical potential $\e_0$ self-consistently to match the hole doping. Numerical simulation of d wave vortex halo ========================================== We did exact diagonalization to simulate Local Density of State(LDoS) inside Vortex Halo. Our Hamiltonian for PDW-Driven Model is: $$H_P=H_0+\sum_{\mathbf x,\mathbf{\mu}}F_d(\mu)\left(|\Delta_D|e^{i\theta_d+i\theta}+\left(\sum_a|\Delta_{P_a}|e^{i\theta_a+i\theta_d}\sin(\frac{1}{2}\mathbf{Q_a}\cdot (\mathbf{x+\frac{\mu}{2}}))\right)\right)c^\dagger_\uparrow(\mathbf x)c^\dagger_\downarrow(\mathbf{x+\mu})+h.c. \label{eq:pdw_real}$$ where $\mathbf \mu=\hat{x}$ or $\hat{y}$ labels two different kinds of nearest neighbor bond. $F_d(\hat x)=1$ and $F_d(\hat y)=-1$. $a$ means $x$ or $y$. We used $|\Delta_{P_x}|=|\Delta_{P_y}|=30$meV at vortex center in our calculation, away from vortex center the PDW profile is $$\Delta_P(r)=30e^{1-\sqrt{r^2+\xi^2}/\xi} meV$$ with $\xi=15$ Our Hamiltonian for CDW-Driven Model is: $$H_C=H_0+\sum_{\mathbf x,\mathbf{\mu}}F_d(\mu)|\Delta_D|e^{i\theta_d+i\theta}c^\dagger_\uparrow(\mathbf x)c^\dagger_\downarrow(\mathbf{x+\mu})+\sum_{\mathbf x,\mathbf{\mu}}F_s(\mu)\left(\sum_a|\Delta_{C_a}|e^{i\theta_a}\sin(\frac{1}{2}\mathbf{Q_a}\cdot (\mathbf{x+\frac{\mu}{2}}))\right)\sum_\sigma c^\dagger_\sigma(\mathbf x)c_\sigma(\mathbf{x+\mu})+h.c.$$ where $F_s(\hat x)=F_s(\hat y)=1$ is a $s$ wave form factor. We used $|\Delta_{C_x}|=|\Delta_{C_y}|=30$meV at vortex center in our calculation. Away from vortex center CDW has a profile similar to PDW-Driven model: $$\Delta_C(r)=30e^{1-\sqrt{r^2+\xi^2}/\xi} meV$$ For both PDW-Driven and CDW-Driven model, we use $|\Delta_D|=20$meV far away from vortex core and $\Delta_D(r,\theta)=20 \frac{r}{\sqrt{r^2+r_0^2}}$ meV near vortex core. We add one d-wave vortex to a $100 a \times 100 a$ square lattice with open boundary condition. $\mathbf{Q_x/2}=(\frac{2\pi}{8},0)$ and $\mathbf{Q_y/2}=(0,\frac{2\pi}{8})$. After Exact Diagonalization, we can easily get on-site LDoS at any energy: $$\rho(\mathbf x, \omega)=\sum_{E,\sigma} \delta(\omega-E)\psi^*_E(\mathbf x;\sigma)\psi_E(\mathbf x;\sigma)$$ where $E$ labels all energy levels and $\psi_E(x;\sigma)$ is the wavefunction for $\mathbf x$ site and spin $\sigma$ at energy level $E$. For STM experiment, LDoS at Oxygen site is actually more important. In our simple one band model, we can define bond LDoS: $$\rho_{\mu}(\mathbf x, \omega)=\sum_{E,\sigma} \delta(\omega-E)\left(\psi^*_E(\mathbf x;\sigma)\psi_E(\mathbf {x+\mu};\sigma)+\psi^*_E(\mathbf {x+\mu};\sigma)\psi_E(\mathbf x;\sigma)\right)$$ where $\mu=\hat x$ or $\hat y$. It’s then easy to define $s$ wave Bond LDoS as $$\rho_d(\mathbf x,\omega)=\rho_{\hat x}(\mathbf x,\omega)+\rho_{\hat y}(\mathbf x, \omega)$$ and $d$ wave Bond LDoS as $$\rho_s(\mathbf x,\omega)=\rho_{\hat x}(\mathbf x,\omega)-\rho_{\hat y}(\mathbf x, \omega)$$ For PDW-Driven model, we found $\rho_d$ dominates and therefore we only show $d$ wave Bond DoS in the main text. For our CDW-Driven model, it’s dominated by $s$ wave CDW as an input and we show $s$ wave CDW in the main text. [^1]: These two authors contributed equally [^2]: These two authors contributed equally [^3]: In momentum space, there are two amplitudes $A^x_a$ and $A^y_a$ at momentum $\mathbf{Q_a/2}$ which correspond to density waves in $x$ bond and $y$ bond. Here $a$ denotes $x$ or $y$: $\mathbf{Q_x/2}=(\frac{2\pi}{8},0)$ and $\mathbf{Q_y/2}=(\frac{2\pi}{8},0)$. The definition currently used by the community is to define $A^x_a\pm A^y_a$ as the s/d wave component. However, under $C_4$ rotation, $A^x_x$ transforms to $A^y_y$. Therefore the current definiton of s/d wave form factor is not related to symmetry and generallly they should be mixed. An alternative definition of s vs d wave component is $A^x_x \pm A^y_y$, which is related to the C4 rotation around a particular reference point. However, this definition may not be very useful because if we shift the reference point by half of the period in one direction, what we would define as d wave would becaome s wave. [^4]: There is a redundancy in this definition: we can shift $\theta_x$ and $\phi_x$ ($\theta_y$ and $\phi_y$) both by $\pi$ without changing any physical order parameter. Thus, $\phi_x$ ($\phi_y$) is determined only up to $\pi$ without reference to the choice of $\theta_x$ ($\theta_y$). [^5]: Physically, there is only one electron pocket, the 4 pockets shown in Fig. \[Fig: PDW band stucture, pocket\](c) are copies of the same pocket shifted in momentum, as a consequence of B.Z. folding.

--- abstract: 'We present high-order, fully explicit projective integration schemes for nonlinear collisional kinetic equations such as the BGK and Boltzmann equation. The methods first take a few small (inner) steps with a simple, explicit method (such as direct forward Euler) to damp out the stiff components of the solution. Then, the time derivative is estimated and used in an (outer) Runge-Kutta method of arbitrary order. The procedure can be recursively repeated on a hierarchy of projective levels to construct telescopic projective integration methods. Based on the spectrum of the linearized collision operator, we deduce that the computational cost of the method is essentially independent of the stiffness of the problem: with an appropriate choice of inner step size, the time step restriction on the outer time step, as well as the number of inner time steps, is independent of the stiffness of the (collisional) source term. In some cases, the number of levels in the telescopic hierarchy depends logarithmically on the stiffness. We illustrate the method with numerical results in one and two spatial dimensions.' author: - 'Ward Melis [^1]' - 'Thomas Rey [^2]' - 'Giovanni Samaey [^3]' bibliography: - 'refs.bib' title: Projective and telescopic projective integration for the nonlinear BGK and Boltzmann equations --- **Keywords:** `Boltzmann equation, BGK equation, Projective Integration, spectral theory, fast spectral scheme.` **2010 MSC.** 82B40, 76P05, 65M70, 65M08, 65M12. Introduction \[sec:introduction\] ================================= Kinetic equations represent a gas as a set of particles undergoing instantaneous collisions interspersed with ballistic motion [@CIP]. Nowadays, these models appear in a variety of sciences and applications, such as astrophysics, aerospace and nuclear engineering, semiconductors, fusion processes in plasmas, as well as biology, finance and social sciences. The common structure of such equations consists in a combination of a linear transport term with one or more interaction terms, which together dictate the time evolution of the distribution of particles in the (six-dimensional) position-velocity phase space. From a numerical point of view, it is clear that this results in a real challenge, since the computational cost immediately becomes prohibitive for realistic problems [@DimarcoPareschi2014]. Aside from the curse of dimensionality, there are many other difficulties which are specific to kinetic equations. We recall two among the most important ones. The first is the computational cost related to the evaluation of the collision operator, which implies the computation of multidimensional integrals in each point of the physical space [@FiMoPa2006; @PaRuSINUM2000]. The second challenge is represented by the presence of multiple time scales in the collision dynamics, leading to a very small mean free path, at least in parts of the spatial domain. Usually, computational problems exhibit multiple regimes in different regions in space. This requires the development of adapted numerical schemes to avoid the resolution of the stiff dynamics [@DimarcoPareschi2013; @Jin_review; @Jin1999; @BLM2008; @degondrev]. Historically, two different approaches are generally used to tackle kinetic equations numerically: deterministic methods, such as finite volume, semi-Lagrangian and spectral schemes [@DimarcoPareschi2014], and probabilistic methods, such as Direct Simulation Monte Carlo (DSMC) schemes [@bird; @Caflisch98]. Both methodologies have strengths and weaknesses. Deterministic methods can normally reach high orders of accuracy. Nevertheless, stochastic methods are often faster, especially for solving steady problems, but, typically, exhibit lower convergence rates and difficulties in describing non-stationary and slow motion flows. In this paper, we will consider deterministic methods, in which we evaluate the collision operator using a fast spectral method, in the spirit of [@MoPa:2006]. For a comprehensive overview of numerical schemes for collisional kinetic equations, such as equation , we refer to [@DimarcoPareschi2014] and references therein. In this paper, we are specifically interested in the time discretization of kinetic equations with stiffness arising from multiple time scales in the collision operator. The stiffness is usually characterized by the (small) mean free path ${\varepsilon}$, and becomes infinite when ${\varepsilon}$ tends to zero. In that limit, a limiting macroscopic equation emerges in terms of a few moments of the particle distribution (density, momentum, energy); the full particle distribution then relaxes infinitely quickly to a Maxwellian distribution defined by these low-order moments. There is currently a large research effort in the design of algorithms that are uniformly stable in ${\varepsilon}$ and approach a scheme for the limiting equation when ${\varepsilon}$ tends to 0; such schemes are called asymptotic-preserving in the sense of Jin [@Jin1999]. Again, we refer to the recent review [@DimarcoPareschi2014] for a clear survey on numerical methods for kinetic equations. Here, we briefly review some achievements using different strategies. In [@Jin1999; @Jin2000a], separating the distribution function $f$ into its odd and even parts in the velocity variable results in a coupled system of transport equations where the stiffness appears only in the source term, allowing to use a time-splitting technique with implicit treatment of the source term; see also related work in [@Jin1999; @Klar1999; @Klar1999a]. Implicit-explicit (IMEX) schemes are an extensively studied technique to tackle this kind of problems, see [@ascher1995; @FilbetJin2010] and references therein. Recent results in this setting were obtained by Dimarco et al. to deal with nonlinear collision kernels [@DimarcoPareschi2013], and an extension to hyperbolic systems in a diffusive limit is given in [@Boscarino2013]. A different approach, based on well-balanced methods, was introduced by Gosse and Toscani [@Gosse2003; @Gosse2004], see also [@Buet2007]. When the collision operator allows for an explicit computation, an explicit scheme can be obtained subject to a classical diffusion CFL condition by splitting the particle distribution into its mean value and the first-order fluctuations in a Chapman-Enskog expansion form [@Godillon-Lafitte2005]. Also closure by moments, e.g. [@Coulombel2005], can lead to reduced systems for which time-splitting provides new classes of schemes [@Carrillo2008]. Alternatively, a micro-macro decomposition based on a Chapman-Enskog expansion has been proposed [@Lemou2008], leading to a system of transport equations that allows to design a semi-implicit scheme without time splitting. A non-local procedure based on the quadrature of kernels obtained through pseudo-differential calculus was proposed in [@Besse2010]. A robust and fully explicit method, which allows for time integration of (two-scale) stiff systems with arbitrary order of accuracy in time, is projective integration. Projective integration was proposed in [@Gear2003projective] for stiff systems of ordinary differential equations with a clear gap in their eigenvalue spectrum. In such stiff problems, the fast modes, corresponding to the Jacobian eigenvalues with large negative real parts, decay quickly, whereas the slow modes correspond to eigenvalues of smaller magnitude and are the solution components of practical interest. Projective integration allows a stable yet explicit integration of such problems by first taking a few small (inner) steps using a step size ${\delta t}$ with a simple, explicit method, until the transients corresponding to the fast modes have died out, and subsequently projecting (extrapolating) the solution forward in time over a large (outer) time step of size ${{\Delta t}> {\delta t}}$. In [@Lafitte2012], projective integration was analyzed for kinetic equations with a diffusive scaling. An arbitrary order version, based on Runge-Kutta methods, has been proposed recently in [@LafitteLejonSamaey2015], where it was also analyzed for kinetic equations with an advection-diffusion limit. In [@LafitteMelisSamaey2017], the scheme was used to construct a explicit, flexible, arbitrary order method for general nonlinear hyperbolic conservation laws, based on relaxation to a kinetic equation. Alternative approaches to obtain a higher-order projective integration scheme have been proposed in [@Lee2007; @Rico-Martinez]. These methods fit within recent research efforts on numerical methods for multiscale simulation [@E2007; @Kevrekidis2003]. For problems exhibiting more than a single fast time scale, telescopic projective integration (TPI) was proposed [@Gear2003telescopic]. In these methods, the projective integration idea is applied recursively. Starting from an inner integrator at the fastest time scale, a projective integration method is constructed with a time step that corresponds to the second-fastest time scale. This projective integration method is then considered as the inner integrator of a projective integration method on yet a coarser level. By repeating this idea, TPI methods construct a hierarchy of projective levels in which each outer integrator step on a certain level serves as an inner integrator step one level higher. The idea was studied and tested for linear kinetic equations in [@MelisSamaey2017]. These methods turn out to have a computational cost that is essentially independent of the stiffness of the collision operator. We do not call projective integration methods asymptotic-preserving as such, because we cannot explicitly evaluate the scheme for ${\varepsilon}=0$ to obtain a classical numerical scheme for the limiting equation. Nevertheless, projective and telescopic projective integration methods share important features with asymptotic-preserving methods. In particular, their computational cost does (in many cases) not depend on the stiffness of the problem. To be specific, it was shown in [@MelisSamaey2017], for linear kinetic equations, that the number of inner time steps at each level of the telescopic hierarchy is independent of the small-scale parameter ${\varepsilon}$, as is the step size of the outermost integrator. The only parameter in the method that may depend on $epsi$ is the *number* of levels in the telescopic hierarchy. For systems in which the spectrum of the collision operator fall apart into a set of clearly separated clusters (each corresponding to a specific time scale), the number of levels equals the number of spectral clusters. In this situation, the computational cost is completely independent of ${\varepsilon}$. When the collision operator represents a continuum of time scales, the number of projective integration levels increases logarithmically with ${\varepsilon}$. In this paper, we construct and evaluate telescopic projective integration methods for nonlinear Boltzmann BGK and Boltzmann kinetic equations. The methods are of arbitrary order in time, fully explicit, and general (they do not exploit any particular form of the collision operator). The remainder of this paper is structured as follows. In Section \[sec:models\], we start by presenting the Boltzmann and BGK equations that will be the subject of our simulations. We describe the different projective and telescopic projective integration methods in detail in Section \[sec:time\_integrator\]. (The spatial and velocity discretizations are standard. To make the manuscript self-contained, we present the corresponding numerical methods in Appendices \[sub:weno\] and \[sub:FastSpectral\].) We discuss in Section \[sec:linearOpSpecProp\] the spectral properties of the linearized collision operators, which will guide the choice of the method parameters, ensuring stability of the time integrators. Some numerical experiments are done in Section \[sec:results\] to verify the theory developed in the two previous sections. We conclude in Section \[sec:conclusions\]. Model equations {#sec:models} =============== In this article, we are interested in rarefied, collisional gases, and then we shall consider Boltzmann-like, collisional kinetic equations. We refer the reader to the classical works [@CIP; @Villani:2002handbook] and the references therein for a more detailed introduction on this vast topic. For a given non-negative initial condition $f_0$, we will study a particle distribution function ${f^{\varepsilon}}= {f^{\varepsilon}}({\mathbf{x}},{\mathbf{v}}, t)$, for $t \geq 0$, ${\mathbf{x}}\in \Omega \subset \mathbb{R}^{D_x}$ and ${\mathbf{v}}\in \mathbb{R}^{D_v}$, solution to the initial-boundary value problem $$\label{eqCollision} {\mathopen{}\mathclose\bgroup\originalleft}\{ \begin{aligned} & \frac{\partial {f^{\varepsilon}}}{\partial t} + {\mathbf{v}}\cdot \nabla_{\mathbf{x}}{f^{\varepsilon}}= \frac{1}{{\varepsilon}} {\mathcal{Q}}({f^{\varepsilon}}), \\ &\, \\ & {f^{\varepsilon}}({\mathbf{x}}, {\mathbf{v}}, 0) = f_{0}({\mathbf{x}},{\mathbf{v}}). \end{aligned} {\aftergroup\egroup\originalright}.$$ The left hand side of equation corresponds to a linear transport operator that comprises the convection of particles in space, whereas the right hand side contains the collision operator that entails velocity changes due to particle collisions. We postpone the description of boundary conditions until Section \[sec:results\], where we discuss the numerical results. We assume that the collision operator fulfils the following three assumptions: 1. \[hypConservations\] Conservation of mass, momentum and kinetic energy: $$\int_{{\mathbb{R}}^{D_v}} {\mathcal{Q}}(f)({\mathbf{v}}) \, d{\mathbf{v}}= 0, \quad \int_{{\mathbb{R}}^{D_v}} {\mathcal{Q}}(f)({\mathbf{v}}) \, {\mathbf{v}}\, d{\mathbf{v}}= \bm{0}_{{\mathbb{R}}^{D_v}}, \quad \int_{{\mathbb{R}}^{D_v}} {\mathcal{Q}}(f)({\mathbf{v}}) \, |{\mathbf{v}}|^2 \, d{\mathbf{v}}= 0;$$ 2. \[hypEntropy\] Dissipation of the Boltzmann entropy (H-theorem): $$\int_{{\mathbb{R}}^{D_v}} {\mathcal{Q}}(f)({\mathbf{v}}) \, \log(f)({\mathbf{v}}) \, d{\mathbf{v}}\, \leq \, 0;$$ 3. \[hypEquilib\] Its equilibria are given by Maxwellian distributions: $${\mathcal{Q}}(f) \, = \, 0 \quad \Leftrightarrow \quad f = {\mathcal{M}}_{\mathbf{v}}^{\rho, {\mathbf{{\bar{v}}}}, T} := \frac{\rho}{(2 \pi T)^{D_v/2}} \exp {\mathopen{}\mathclose\bgroup\originalleft}( - \frac{|{\mathbf{v}}-{\mathbf{{\bar{v}}}}|^2}{2 T} {\aftergroup\egroup\originalright}),$$ where the *density* $\rho$, *velocity* ${\mathbf{{\bar{v}}}}$ and *temperature* $T$ of the gas are computed from the distribution function $f$ as: $$\label{eq:f_moments} \rho = \int_{{\mathbb{R}}^{D_v}} f({\mathbf{v}})\,d{\mathbf{v}}, \quad {\mathbf{{\bar{v}}}}= \frac{1}{\rho}\int_{{\mathbb{R}}^{D_v}} {\mathbf{v}}f({\mathbf{v}}) \, d{\mathbf{v}}, \quad T = \frac{1}{D_v \rho} \int_{{\mathbb{R}}^{D_v}} \vert {\mathbf{{\bar{v}}}}- {\mathbf{v}}\vert^2 f({\mathbf{v}}) \,d{\mathbf{v}}.$$ Equation with assumptions \[hypConservations\]-\[hypEntropy\]-\[hypEquilib\] describes numerous models such as the Boltzmann equation for elastic collisions [@Villani:2002handbook] or Fokker-Planck-Landau type equations[@alexandre2000entropy]. The parameter ${\varepsilon}> 0$ is the dimensionless Knudsen number, that is, the ratio between the mean free path of particles and the length scale of observation. It determines the regime of the gas flow, for which we roughly identify the hydrodynamic regime $({\varepsilon}\le 10^{-4})$, the transitional regime $({\varepsilon}\in [10^{-4},10^{-1}])$, and the kinetic regime $({\varepsilon}\ge 10^{-1})$. Moreover, according to assumptions \[hypEntropy\]-\[hypEquilib\], when ${\varepsilon}\to 0$, the distribution ${f^{\varepsilon}}$ converges (at least formally) to a Maxwellian distribution, whose moments are solution to the compressible Euler system for perfect gases, given by: $$\label{eqHydroClosedEuler} {\mathopen{}\mathclose\bgroup\originalleft}\{ \begin{aligned} & \partial_t \rho + \operatorname{div}_{\mathbf{x}}(\rho \, {\mathbf{{\bar{v}}}}) = 0, \\ &\, \\ & \partial_t(\rho \, {\mathbf{{\bar{v}}}}) + \operatorname{div}_{\mathbf{x}}{\mathopen{}\mathclose\bgroup\originalleft}(\rho \, {\mathbf{{\bar{v}}}}\otimes {\mathbf{{\bar{v}}}}\,+\, \rho \, T \,{\rm\bf I}{\aftergroup\egroup\originalright}) \, =\, \bm{0}_{{\mathbb{R}}^{D_v}}, \\ &\, \\ & \partial_t E + \operatorname{div}_{\mathbf{x}}{\mathopen{}\mathclose\bgroup\originalleft}( {\mathbf{{\bar{v}}}}{\mathopen{}\mathclose\bgroup\originalleft}( E +\rho \, T{\aftergroup\egroup\originalright}) {\aftergroup\egroup\originalright}) \,=\, 0, \end{aligned} {\aftergroup\egroup\originalright}.$$ in which $E$ is the second moment of ${f^{\varepsilon}}$, namely the total energy of the gas: $$E = \int_{{\mathbb{R}}^{D_v}} \vert {\mathbf{{\bar{v}}}}\vert^2 f({\mathbf{v}}) \,d{\mathbf{v}}.$$ In the following, we will present the two main collisional kinetic equations that we will consider in the remainder of this paper: the Boltzmann equation (Section \[subsec:boltzmann\_equation\]) and the BGK equation (Section \[subsec:bgk\_equation\]). Boltzmann equation {#subsec:boltzmann_equation} ------------------ The Boltzmann equation constitutes the cornerstone of the kinetic theory of rarefied gases [@Villani:2002handbook; @CIP]. In a dimensionless, scalar setting, it describes the evolution of the one-particle mass distribution function ${f^{\varepsilon}}({\mathbf{x}},{\mathbf{v}},t) \in {\mathbb{R}}^{+}$, solution to the model equation , in which we still need to specify the collision operator ${{\mathcal{Q}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})}$. The Boltzmann collision operator models binary elastic collisions between particles having pre-collisional velocities $({\mathbf{v}}',{{\mathbf{v}}_{*}}')$ and post-collisional velocities $({\mathbf{v}},{{\mathbf{v}}_{*}})$. In a two-dimensional velocity space, the pre- and post-collisional velocities are linked through the following parametrization: $${\mathbf{v}}' = \dfrac{{\mathbf{v}}+ {{\mathbf{v}}_{*}}}{2} + \dfrac{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}- {{\mathbf{v}}_{*}}{\aftergroup\egroup\originalright}\vert}}{2}{\boldsymbol{\sigma}}, \qquad\quad {{\mathbf{v}}_{*}}' = \dfrac{{\mathbf{v}}+ {{\mathbf{v}}_{*}}}{2} - \dfrac{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}- {{\mathbf{v}}_{*}}{\aftergroup\egroup\originalright}\vert}}{2}{\boldsymbol{\sigma}},$$ where ${\boldsymbol{\sigma}}$ is the unit vector on the unit circle ${\mathbb{S}}^1 = \{{\boldsymbol{\sigma}}\in {\mathbb{R}}^2: {{\mathopen{}\mathclose\bgroup\originalleft}\vert{\boldsymbol{\sigma}}{\aftergroup\egroup\originalright}\vert} = 1\}$ directed along the pre-collisional relative velocity ${{\mathbf{v}}_r}' = {\mathbf{v}}' - {{\mathbf{v}}_{*}}'$: $${\boldsymbol{\sigma}}= \frac{{{\mathbf{v}}_r}'}{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}'{\aftergroup\egroup\originalright}\vert}} = \frac{{\mathbf{v}}' - {{\mathbf{v}}_{*}}'}{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}' - {{\mathbf{v}}_{*}}'{\aftergroup\egroup\originalright}\vert}}.$$ The Boltzmann collision operator then reads: $$\begin{aligned} {{\mathcal{Q}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})} &= \int_{{\mathbb{R}}^2}\int_{{\mathbb{S}}^1} B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\boldsymbol{\sigma}})(f'f_*' - ff_*) d{\boldsymbol{\sigma}}d{{\mathbf{v}}_{*}}\notag \\ &= \int_{{\mathbb{R}}^2}\int_{0}^{2\pi} B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}-{\mathbf{v}}_*{\aftergroup\egroup\originalright}\vert},{\theta_\sigma})(f'f_*' - ff_*) d{\theta_\sigma}d{{\mathbf{v}}_{*}}, \label{eq:collision_operator}\end{aligned}$$ where ${\theta_\sigma}$ is the angle between ${{\mathbf{v}}_r}'$ and ${\boldsymbol{\sigma}}$ and we used the shorthand notations $f = {f^{\varepsilon}}({\mathbf{v}})$, $f_* = {f^{\varepsilon}}({\mathbf{v}}_*)$, $f^{'} = {f^{\varepsilon}}({\mathbf{v}}')$, and $f_*^{'} = {f^{\varepsilon}}({\mathbf{v}}_* ^{'})$. Furthermore, the non-negative function $B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\boldsymbol{\sigma}}) \equiv B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\theta_\sigma})$ is the collision kernel, which, by physical arguments of invariance, only depends on the relative speed ${{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert} = {{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}-{\mathbf{v}}_*{\aftergroup\egroup\originalright}\vert}$ and $\cos({\theta_\sigma}) = {{\mathbf{v}}_r}'/{{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert} \cdot {\boldsymbol{\sigma}}$. The collision kernel $B$ contains all relevant microscopic information such as the kind of particles and type of interactions. For instance, when particles interact via an inverse power law potential ${\Phi}(r) = r^{-k+1}$ $(k > 2)$, with $r$ the inter-particle distance, $B$ factors as: $$\label{eq:collision_kernel_factors_sigma} B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\theta_\sigma}) = {{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert}^\gamma b_\gamma({\theta_\sigma}), \qquad\quad \gamma = \frac{k-3}{k-1}.$$ Notably, in the special case $k = 3$, the collision kernel in is independent of the relative speed ${{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert}$ and the resulting particles are known as Maxwellian particles. If, in addition, $b_0({\theta_\sigma}) = b_0$ is assumed constant, the particles are referred to as pseudo-Maxwellian particles. In general (except for hard sphere and pseudo-Maxwellian particles), the angular collision kernel $b_\gamma$ in is expressed implicitly and contains a singularity for grazing collisions $({\theta_\sigma}\to 0)$, and its mathematical analysis, as well as its numerical simulations, can be very difficult [@alexandre2000entropy]. For that reason, the angular collision kernel is usually replaced by an integrable function by cutting off such grazing collision angles (Grad’s cut-off assumption) [@Cercignani1988]. It is instructive to split the collision operator into a gain and loss operator as: $$\label{eq:collision_operator_gain_loss} {{\mathcal{Q}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})} = {{{\mathcal{Q}}^+}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})} - {\nu}({f^{\varepsilon}}){f^{\varepsilon}}({\mathbf{v}}),$$ where the gain operator is given by: $$\label{eq:collision_operator_gain} {{{\mathcal{Q}}^+}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})} = \int_{{\mathbb{R}}^2}\int_{0}^{2\pi} B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\theta_\sigma})f'f_*' d{\theta_\sigma}d{{\mathbf{v}}_{*}}.$$ The loss operator ${{\mathcal{Q}}^-}({f^{\varepsilon}}) = {\nu}({f^{\varepsilon}}){f^{\varepsilon}}$ contains the collision frequency ${\nu}({f^{\varepsilon}}) \in {\mathbb{R}}^+$, which is defined by: $$\label{eq:collision_frequency_Boltzmann} {\nu}({f^{\varepsilon}}) = \int_{{\mathbb{R}}^2}\int_{0}^{2\pi} B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\theta_\sigma})f_* d{\theta_\sigma}d{{\mathbf{v}}_{*}}.$$ (Evidently, equation is only valid if both integrals - are convergent, which is certainly true for a cut-off collision kernel.) In general, the collision frequency depends on the dimension of velocity space and the type of microscopic collisions. In particular, when considering pseudo-Maxwellian particles for ${D_v = 2}$, implying that $\gamma = 0$ and $b_0({\theta_\sigma}) = b_0$ constant, the collision kernel becomes much simpler and is given by ${B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\theta_\sigma}) = b_0}$, that is, independent of ${{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert}$ and ${\theta_\sigma}$. In that case, the collision frequency can be explicitly computed as: $${\nu}({f^{\varepsilon}}) = b_0 \int_{0}^{2\pi} d{\theta_\sigma}\int_{{\mathbb{R}}^2} f_* d{{\mathbf{v}}_{*}}= 2\pi b_0\rho,$$ and the collision operator in reads: $$\label{eq:collision_operator_maxwellian_particles} {{\mathcal{Q}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})} = {{{\mathcal{Q}}^+}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})} - 2\pi b_0\rho {f^{\varepsilon}}.$$ We note that, for inverse power law potentials with $\gamma \neq 0$ (including hard sphere particles), the collision frequency in general depends on the density, temperature and collision kernel, see [@Struchtrup2005]. Finally, it is easy (see [@CIP]) to check that the Boltzmann collision operator satisfies the hypotheses \[hypConservations\]-\[hypEntropy\]-\[hypEquilib\]. BGK equation {#subsec:bgk_equation} ------------ Due to the high-dimensional and complicated structure of the Boltzmann collision operator, the Boltzmann collision kernel  is often replaced by simpler collision models that capture most of its essential features. The most well-known approximation is the BGK model [@Bhatnagar1954], which models collisions as a relaxation towards thermodynamic equilibrium. (Because of the moments-dependency of the equilibrium, this is still a nonlinear operator.) It is the most well regarded simplified model of the Boltzmann equation, and is almost universally used in the physics and numerics communities (see [@DimarcoPareschi2014] and the references therein for details). It is given by: $$\label{eq:bgk_equation} \partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= \frac{\nu}{{\varepsilon}}({{{\mathcal{M}}_{{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}- {f^{\varepsilon}}),$$ where ${{{\mathcal{M}}_{{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}$ denotes the local Maxwellian distribution, which, for a $D_v$-dimensional velocity space, is given by: $$\label{eq:maxwellian} {{{\mathcal{M}}_{{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}= \frac{\rho}{(2\pi T)^{D_v/2}} \exp{{\mathopen{}\mathclose\bgroup\originalleft}(-\frac{|{\mathbf{v}}-{\mathbf{{\bar{v}}}}|^2}{2T}{\aftergroup\egroup\originalright})} := {\mathcal{M}}_{\mathbf{v}}^{\rho,{\mathbf{{\bar{v}}}},T}.$$ Furthermore, in the BGK equation , the collision frequency ${\nu}\in {\mathbb{R}}^+$ is in general derived from the Boltzmann collision operator and its expression is given in , while in the linearized setting, ${\nu}$ is independent of ${f^{\varepsilon}}$ and is formulated in , see Section \[subsec:lin\_boltzmann\_equation\]. Notably, when setting ${\nu}= \rho$ for $D_v = 2$, the BGK model matches the loss term of the Boltzmann collision operator for pseudo-Maxwellian particles close to equilibrium, see equation . The BGK collision operator satisfies by construction the hypotheses \[hypConservations\]-\[hypEntropy\]-\[hypEquilib\]. Numerical method {#sec:time_integrator} ================ Now that we have introduced the model problems, we turn to the description of the numerical method that will be the focus of this paper. Equation  needs to be discretized in space, velocity and time. We discretize equation in space using finite differences on a uniform, constant in time, periodic mesh with spacing ${\Delta x}$, consisting of $I$ mesh points $x_i=i{\Delta x}$, ${1 \le i \le I}$, with $I{\Delta x}=1$. In the numerical experiments of section \[sec:results\], we use the classical WENO scheme [@shu:1998], which we briefly recall in Appendix \[sub:weno\]. Next, we discretize velocity space by choosing $J$ discrete components denoted by ${\mathbf{v}}_j$. For the Boltzmann equation, that is, equation with collision operator , we use the fast spectral discretization of the Boltzmann operator, taken from [@MoPa:2006]. This method is recalled in Appendix \[sub:FastSpectral\]. The semidiscrete numerical solution on this mesh is denoted by $f_{i,j}(t)$, where we have dropped the superscript ${\varepsilon}$ on discretized quantities. We then obtain a semidiscrete system of ODEs of the form: $$\label{eq:semidiscrete} \dot{{\mathbf{f}}} = {\mathrm{D}_t}({\mathbf{f}}), \qquad {\mathrm{D}_t}({\mathbf{f}}) = -{\mathrm{D}_{{\boldsymbol{x}},{\boldsymbol{v}}}}({\mathbf{f}}) + \frac{1}{{\varepsilon}}{\mathcal{Q}}{\mathopen{}\mathclose\bgroup\originalleft}({\mathbf{f}}{\aftergroup\egroup\originalright}),$$ where ${\mathrm{D}_{{\boldsymbol{x}},{\boldsymbol{v}}}}(\cdot)$ represents the finite difference discretization of the convective derivative ${\mathbf{v}}\cdot \nabla_{\mathbf{x}}$, and ${\mathbf{f}}$ is a vector of size $I \cdot J$. In the remainder of this section, we describe the time discretization of the semi-discretized system , which is the novel element in the full discretization of equation . We start in Section \[sub:projective\_integration\] with the projective integration method, which aims at efficiently simulating systems with *exactly two* time scales (one fast and one slow). In Section \[sub:telescopic\_integration\], we present the generalized telescopic projective integration method, which can deal with multiple fast time scales. Projective integration {#sub:projective_integration} ---------------------- Projective integration [@Gear2003projective] is a method that is tailored to problems with exactly two distinct time scales. As such, in the context of kinetic equations, it matches nicely with the spectral properties of a linear BGK equation, as was shown in [@Lafitte2012]. Projective integration combines a few small time steps with a naive (*inner*) timestepping method (here, a direct forward Euler discretization) with a much larger (*projective, outer*) time step. The idea is sketched in figure \[fig:proj\_int\]. [![\[fig:proj\_int\] Sketch of projective integration. At each time, an explicit method is applied over a number of small time steps (black dots) so as to stably integrate the fast modes. As soon as these modes are sufficiently damped, the solution is extrapolated using a much larger time step (dashed lines). ](image/PI_sketch "fig:")]{} #### Inner integrators. At the innermost leve, we introduce a uniform time mesh with time step ${\delta t}$ and discrete time instants $t^k=k{\delta t}$. At this leve, we choose the (explicit) forward Euler method with time step ${\delta t}$, for which we will, later on, use the shorthand notation: $$\label{eq:fe_scheme} {\mathbf{f}}^{k+1} = {S_{{\delta t}}}({\mathbf{f}}^{k}) = {\mathbf{f}}^k + {\delta t}{\mathrm{D}_t}({\mathbf{f}}^k), \qquad k = 0, 1, \ldots.$$ The purpose of the inner integrator is to capture the fastest components in the numerical solution of system and to sufficiently damp these out. We only require the innermost integrator to be stable for these components. The size of the inner time step ${\delta t}$ and the required number of inner steps $K$ will depend on the spectral properties of the semidiscretization . This will be studied in Section \[sec:linearOpSpecProp\]. #### Outer integrators. In system , the small parameter ${\varepsilon}$ leads to a classical time step restriction of the form ${\delta t}= O({\varepsilon})$ for the inner integrator. However, as ${\varepsilon}$ goes to $0$, we obtain the limiting system , for which a standard finite volume/forward Euler method only needs to satisfy a CFL stability restriction of the form ${\Delta t}\le C{\Delta x}$, with $C$ a constant that depends on the specific choice of the scheme. In [@Lafitte2012], it was proposed to use a projective integration method to accelerate such a brute-force integration; the idea, originating from [@Gear2003projective], is the following. Starting from a computed numerical solution ${\mathbf{f}}^n$ at time $t^n=n{\Delta t}$, one first takes $K+1$ *inner* steps of size ${\delta t}$ using , denoted as ${\mathbf{f}}^{n,k+1}$, in which the superscripts $(n,k)$ denote the numerical solution at time ${t^{n,k}=n{\Delta t}+k{\delta t}}$. The aim is to obtain a discrete derivative to be used in the *outer* step to compute ${\mathbf{f}}^{n+1} = {\mathbf{f}}^{n+1,0}$ via extrapolation in time: $$\begin{aligned} {\mathbf{f}}^{n+1} & = {\mathbf{f}}^{n,K+1} + ({\Delta t}- (K + 1){\delta t})\frac{{\mathbf{f}}^{n,K+1} - {\mathbf{f}}^{n,K}}{{\delta t}}, \\ & = {\mathbf{f}}^{n,K+1} + M{\delta t}\frac{{\mathbf{f}}^{n,K+1} - {\mathbf{f}}^{n,K}}{{\delta t}},\end{aligned}$$ where $M = {\Delta t}/{\delta t}-(K+1)$. Also the size of the (macroscopic) extrapolation step ${\Delta t}$ will result from the spectral analysis of the semidiscretization  in section \[sec:linearOpSpecProp\]. Higher-order projective Runge-Kutta (PRK) methods have been constructed [@LafitteLejonSamaey2015; @LafitteMelisSamaey2017] by replacing each time derivative evaluation $\mathbf{k}_s$ in a classical Runge-Kutta method by $K+1$ steps of an inner integrator as follows: $$\begin{aligned} s = 1 :\;\; & \begin{dcases} {\mathbf{f}}^{n,k+1} &= {\mathbf{f}}^{n,k} + {\delta t}{\mathrm{D}_t}({\mathbf{f}}^{n,k}), \qquad 0 \le k \le K \\ \mathbf{k}_1 &= \dfrac{{\mathbf{f}}^{n,K+1} - {\mathbf{f}}^{n,K}}{{\delta t}} \end{dcases} \\ 2 \le s \le S :\;\; & \begin{dcases} {\mathbf{f}}^{n+c_s,0}_s &= {\mathbf{f}}^{n,K+1} + (c_s{\Delta t}-(K+1){\delta t}) \sum_{l=1}^{s-1}\dfrac{a_{s,l}}{c_s} \mathbf{k}_l, \\ {\mathbf{f}}^{n+c_s,k+1}_s &= {\mathbf{f}}^{n+c_s,k}_s + {\delta t}{\mathrm{D}_t}({\mathbf{f}}^{n+c_s,k}_s), \qquad 0 \le k \le K \\ \mathbf{k}_s &= \dfrac{{\mathbf{f}}^{n+c_s,K+1}_s - {\mathbf{f}}^{n+c_s,K}_s}{{\delta t}} \end{dcases} \\ & {\mathbf{f}}^{n+1} = {\mathbf{f}}^{n,K+1} + ({\Delta t}-(K+1){\delta t})\sum_{s=1}^{S}b_s \mathbf{k}_s.\end{aligned}$$ To ensure consistency, the Runge-Kutta matrix $\mathbf{a}=(a_{s,i})_{s,i=1}^S$, weights ${\mathbf{b}=(b_s)_{s=1}^S}$, and nodes $\mathbf{c}=(c_s)_{s=1}^S$ satisfy the conditions $0\le b_s \le 1$ and $0 \le c_s \le 1,$ as well as: $$\sum_{s=1}^Sb_s=1, \qquad \sum_{i=1}^{S-1} a_{s,i} =c_s, \quad 1 \le s \le S.$$ Telescopic projective integration {#sub:telescopic_integration} --------------------------------- In general, the stiff semidiscrete system , contains more than two distinct time scales. In this section, we therefore describe an extension of projective integration, called *telescopic projective integration* (TPI) and introduced in [@Gear2003telescopic], that can handle multiple time scales. This method has been studied in the context of linear BGK equations with multiple relaxation times in [@MelisSamaey2017]. Telescopic projective integration employs a number of projective integrator levels, which, starting from a base (*innermost*) integrator, are wrapped around the previous level integrator [@Gear2003telescopic]. In this way, a hierarchy of projective integrators is formed in which each level (except the innermost and outermost one) fulfils both an inner and outer integrator role. This generalizes the idea of projective integration, which contains only one projective level wrapped around an inner integrator. The idea of a level-3 TPI method with $K=2$ on each projective level is sketched in figure \[fig:tpi\_sketch\]. The different level integrators in a TPI method can in principle be selected independently from each other, but in general one selects a first order explicit scheme (the forward Euler scheme) for all but the outermost integrator level, whose order is chosen to meet the accuracy requirements dictated by the problem. [![\[fig:tpi\_sketch\] A level-3 TPI method drawn for three outermost time steps $h_3$ (bottom row) with $K=2$ on all projective levels. The dots correspond to different time points at which the numerical solution is calculated. The time step and projective step size of each level $\ell=0,\ldots,2$ are denoted by $h_\ell$ and $M_\ell$, respectively. ](image/TPI_level3_sketch "fig:")]{} #### Innermost integrator We intend to integrate the semidiscrete system of equations using a uniform time mesh with time step $h_0$ and discrete time instants $t^k=kh_0$. The innermost integrator of the TPI method is chosen to be the forward Euler (FE) method, $${\mathbf{f}}^{k+1} = {\mathbf{f}}^k + h_0{\mathrm{D}_t}({\mathbf{f}}^k).$$ In the sequel, we use the following shorthand notation: $${\mathbf{f}}^{k+1} = S_{0}({\mathbf{f}}^{k}) \qquad (k = 0, 1, \ldots),$$ in which $S_0$ denotes the time stepper with corresponding time step $h_0$. Also in the telescopic projective integration method, the purpose of the innermost integrator is only to capture the fastest components in the numerical solution of system  and to sufficiently damp these out. As a consequence, it is ill-advised to use higher-order methods for the innermost integrator, see [@MelisSamaey2017] for a more detailed discussion. #### Projective (outer) levels The telescopic projective integration method employs in general $L$ nested levels of projective integration that are constructed around the innermost integrator. In [@Gear2003telescopic], the method has been introduced in a recursive way. Here, following [@MelisSamaey2017], we describe the method in an alternative way, to make the presentation more similar to that of classical projective integration. To keep track of the time instant at which the numerical solution is computed throughout the TPI method and at the same time desiring a compact notation, in what follows, we employ superscript triplets of the form $(\ell,n,k_\ell)$ where $\ell$ denotes the integrator level ranging from $0$ (innermost) to $L-1$, $n$ represents the index of the current outermost integrator time $t^n=nh_L$, and $k_\ell$ corresponds to the iteration index of the integrator on level $\ell$. The numerical time on each level $\ell = 0, \ldots, L-1$ is then defined as (see also figure \[fig:tpi\_sketch\]): $$\label{eq:tpi_time} t^{\ell,n,k_\ell} = nh_L + \sum_{\ell'=\ell}^{L-1} k_{\ell'}h_{\ell'}.$$ Notice that, for a certain level $\ell$, this time requires the iteration indices $k_{\ell'}$ of all its outer integrators. Therefore, it incorporates a memory that keeps up with the current time instants at which the outer integrators of a given level $\ell$ integrator have arrived at and is necessary to take into account to correctly reflect the numerical time of the solution on each level $\ell$. Starting from a computed numerical solution ${\mathbf{f}}^n$ at time $t^n=nh_L$, one first takes $K_0+1$ steps of size $h_0$ with the innermost integrator: $$\label{eq:semidiscrete_innermost_scheme} {\mathbf{f}}^{0,n,k_0+1} =S_{0}({\mathbf{f}}^{0,n,k_0}) \qquad (0 \le k_0 \le K_0),$$ in which ${\mathbf{f}}^{0,n,k_0}$ corresponds to the numerical solution at time $t^{0,n,k_0}$ calculated by the innermost integrator. Since all outer integrator iteration indices $k_{\ell'}$, $\ell' = 1,\ldots,L-1$ are initially zero in , we have ${t^{0,n,k_0} = nh_L+k_0h_0}$. The repeated action of the innermost integrator is depicted by small black arrows in the upper row of figure \[fig:tpi\_sketch\], for which we chose $K_0=2$. In the telescopic projective integration framework, the scheme is set up from the lowest level up to the highest level. The aim is to obtain a discrete derivative to be used on each level to eventually compute ${\mathbf{f}}^{n+1} = {\mathbf{f}}^{0,n+1,0}$ via extrapolation in time. Using the innermost integrator iterations , we perform the extrapolation by a projective integrator on level 1, written as: $$\label{eq:tpfe_first_level} {\mathbf{f}}^{1,n,1} = {\mathbf{f}}^{0,n,K_{0}+1} + {\mathopen{}\mathclose\bgroup\originalleft}(M_{0}h_{0}{\aftergroup\egroup\originalright})\frac{{\mathbf{f}}^{0,n,K_{0}+1} - {\mathbf{f}}^{0,n,K_{0}}}{h_{0}},$$ which corresponds to the projective forward Euler (PFE) method. In , ${\mathbf{f}}^{1,n,1}$ represents the numerical solution at time $t^{1,n,1}$ calculated by one iteration of the first level projective integrator. Since $k_1=1$ and all its outer integrator iteration indices $k_{\ell'}$, ${\ell'=2,\ldots,L-1}$ are still zero in , we have ${t^{1,n,1} = nh_L + h_1}$. One step of the first level integrator is visualized by a large green arrow in the upper row of figure \[fig:tpi\_sketch\]. By repeating this idea, we construct a hierarchy of projective integrators on levels $\ell=1,\ldots,L-1$, given by: $$\label{eq:tpfe_projective_levels} {\mathbf{f}}^{\ell,n,k_\ell+1} = {\mathbf{f}}^{\ell-1,n,K_{\ell-1}+1} + {\mathopen{}\mathclose\bgroup\originalleft}(M_{\ell-1}h_{\ell-1}{\aftergroup\egroup\originalright})\frac{{\mathbf{f}}^{\ell-1,n,K_{\ell-1}+1} - {\mathbf{f}}^{\ell-1,n,K_{\ell-1}}}{h_{\ell-1}},$$ in which, on each level $\ell$, we iterate over $k_{\ell} = 0, \ldots, K_{\ell}$. In , ${\mathbf{f}}^{\ell,n,k_\ell}$ denotes the numerical solution at time $t^{\ell,n,k_\ell}$ calculated by the projective integrator on level $\ell$. According to , this time depends on the values $k_{\ell'}$, $\ell'=\ell+1,\ldots,L-1$ of all of its outer integrators. In figure \[fig:tpi\_sketch\], these projective integrator steps are shown by long arrows for each level $\ell=1,\ldots,3$. Ultimately, the outermost integrator on level $L$ computes ${\mathbf{f}}^{n+1}$ as: $$\label{eq:tpfe_outermost} {\mathbf{f}}^{n+1} = {\mathbf{f}}^{L-1,n,K_{L-1}+1} + (M_{L-1}h_{L-1})\frac{{\mathbf{f}}^{L-1,n,K_{L-1}+1} - {\mathbf{f}}^{L-1,n,K_{L-1}}}{h_{L-1}}.$$ Since the outermost integrator also constitutes a PFE scheme, the telescopic method resulting from the hierarchy of projective levels - is called telescopic projective forward Euler (TPFE). It is straightforward to implement higher-order extensions of the outermost integrator, as is done in [@MelisSamaey2017]. We mention the projective Runge-Kutta methods of order 2 and 4, leading to TPRK2 and TPRK4 method in the telescopic case. In general, the outermost integrator in a TPRK method replaces each time derivative evaluation $\mathbf{k}_s$ in a classical Runge-Kutta method by $K_{L-1}+1$ steps of its inner integrator on level $L-1$. Using with $\ell=L-1$, the first stage in a TPRK method calculates the time derivative $\mathbf{k}_1$ as: $$\label{eq:tprk_stage_1} \mathbf{k}_1 = \dfrac{{\mathbf{f}}^{L-1,n,K_{L-1}+1} - {\mathbf{f}}^{L-1,n,K_{L-1}}}{h_{L-1}}.$$ Computing $\mathbf{k}_s$ on any other stage $s \ge 2$ requires evaluating time derivatives at the intermediate times ${t^{n+c_s} = (n+c_s)h_L}$. Similarly to , these are computed as: $$\label{eq:tprk_stage_s} \mathbf{k}_s = \dfrac{{\mathbf{f}}^{L-1,n+c_s,K_{L-1}+1} - {\mathbf{f}}^{L-1,n+c_s,K_{L-1}}}{h_{L-1}}.$$ However, since the numerical solution at time $t^{n+c_s}$ in equation is not available, we use the integrator on level $L-1$ to approximate it as follows: $$\begin{dcases} {\mathbf{f}}^{L-1,n+c_s,0} = {\mathbf{f}}^{L-1,n,K_{L-1}+1} + (c_sh_L-(K_{L-1}+1)h_{L-1}) \sum_{i=1}^{s-1}\dfrac{a_{s,i}}{c_s} \mathbf{k}_i \\ {\mathbf{f}}^{L-1,n+c_s,k_{L-1}+1} = {\mathbf{f}}^{L-2,n+c_s,K_{L-2}+1} \\ \hspace{3.1cm} + \,{\mathopen{}\mathclose\bgroup\originalleft}(M_{L-2}h_{L-2}{\aftergroup\egroup\originalright})\frac{{\mathbf{f}}^{L-2,n+c_s,K_{L-2}+1} - {\mathbf{f}}^{L-2,n+c_s,K_{L-2}}}{h_{L-2}}, \end{dcases}$$ in which the second equation iterates over $0 \le k_{L-1} \le K_{L-1}$. Ultimately, the outermost integrator of a TPRK method is written as: $${\mathbf{f}}^{n+1} = {\mathbf{f}}^{L-1,n,K_{L-1}+1} + (M_{L-1}h_{L-1})\sum_{s=1}^{S}b_s \mathbf{k}_s.$$ The consistency conditions on the Runge-Kutta matrix $\mathbf{a}=(a_{s,i})_{s,i=1}^S$, weights $\mathbf{b}=(b_s)_{s=1}^S$, and nodes ${\mathbf{c}=(c_s)_{s=1}^S}$ are still valid in this setting [@MelisSamaey2017]. In the numerical experiments in Section \[sec:results\], we will use the projective Runge-Kutta method of order 4 as outermost integrator. On linearized operators and spectral properties {#sec:linearOpSpecProp} =============================================== Telescopic projective integration methods are very versatile, but require choosing a relatively large number of method parameters: the size of the time steps $h_\ell$ at each level, the number $K_\ell$ of inner steps at each level, as well as the number $L$ of telescopic levels. As these choices are dictated mainly by stability requirements, they crucially depend on the spectrum of the collision operator. The analysis of this spectrum for the problems of Section \[sec:models\] is the focus of this Section. In [@Lafitte2012; @LafitteLejonSamaey2015], the spectrum of the collision operator was analysed for linear kinetic equations in the diffusive and hyperbolic scalings. An extension to kinetic relaxations of a nonlinear hyperbolic conservation law was presented in [@LafitteMelisSamaey2017]. In all of these settings, the spectrum of the collision operator turned out to consist of exactly two well-separated time scales, and projective integration was therefore sufficient. A first study of telescopic projective integration for linear kinetic equations with multiple relaxation times was presented in [@MelisSamaey2017]. In this section, we devise a general framework of linear operators in which linearizations of both the BGK and Boltzmann equation can be studied. This framework will allow determining suitable method parameters for the (telescopic) projective integration of the Boltzmann and nonlinear BGK equations. In addition, it allows embedding the linear kinetic equations that were studied in [@LafitteMelisSamaey2017; @MelisSamaey2017]. For the reader’s convenience, we restrict the exposition to the case when $D_v = 2$. However, all the results of this section can be extended straigthforwardly to the $D_v=3$ case, at the cost of heavier notations. In Section \[subsec:lin\_bgk\], we build this framework for BGK equations and draw conclusions on their spectral properties. Afterwards, in Section \[subsec:lin\_boltzmann\_equation\], we extend this framework to include the Boltzmann equation. We discuss the selection of suitable method parameters for (telescopic) projective integration in Section \[sec:method\_param\]. Linearized BGK models and their spectra {#subsec:lin_bgk} --------------------------------------- We first recall the linearization of the BGK equation , as it has been described in [@Cercignani1988]. We then show in section \[sec:linear\_kinetic\] how the simpler linear kinetic equations that were analyzed in [@LafitteMelisSamaey2017; @MelisSamaey2017] fit in this framework. Finally, in section \[sec:bgk\_spectrum\], we discuss how the analysis of the spectrum of linear kinetic equations in [@LafitteMelisSamaey2017; @MelisSamaey2017] generalizes to the linearization of the full BGK equation. ### Linearized BGK equation {#sec:linearized_bgk} In [@Cercignani1988], it is shown that the linearized BGK operator can be formulated as: $$\label{eq:linearized_bgk_maxwellian} {\mathcal{M}}_1({f^{\varepsilon}})({\mathbf{x}},{\mathbf{v}},t) = \sum_{k=0}^{D_v+1} \Psi_k({\mathbf{v}}) \langle\Psi_k, {f^{\varepsilon}}\rangle({\mathbf{x}},t),$$ in which the scalar product is defined by: $$\label{eq:scalar_product_Hilbert} \langle g,h \rangle= \int_{{\mathbb{R}}^{D_v}} g({\mathbf{v}})\overline{h({\mathbf{v}})} \frac{1}{2\pi}\exp{\mathopen{}\mathclose\bgroup\originalleft}(\frac{-{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2}{2}{\aftergroup\egroup\originalright}) d{\mathbf{v}}.$$ Furthermore, the basis functions $\Psi_k({\mathbf{v}})$ in represent an orthogonal basis for the space spanned by $$V:= {\mathopen{}\mathclose\bgroup\originalleft}\{1, v^{x_1}, \ldots, v^{x_{D_v}}, \frac{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2}{2}{\aftergroup\egroup\originalright}\},$$ in which the superscript $x_d$ on $v$ indicates that $v^{x_d}$ is the component of ${\mathbf{v}}$ in the $d$-th velocity dimension. The space $V$ corresponds to the set of elementary collision invariants $(\psi_k({\mathbf{v}}))_{k=0}^{{D_v}+1}$. We construct an orthonormal basis, i.e., we seek basis functions $Psi_k$ such that: $$\label{eq:normalization_Psi} (\Psi_k, \Psi_j) = \delta_{kj}, \qquad k,j \in \{0,\ldots,{D_v}+1\},$$ in which $\delta_{kj}$ denotes the Kronecker delta. A straightforward application of the Gram-Schmidt process then allows to compute the desired set of orthonormal functions $\Psi_k({\mathbf{v}})$ satisfying as: $$\label{eq:Psi_normalized} \big(\Psi_0({\mathbf{v}}), \ldots, \Psi_{D_v+1}({\mathbf{v}})\big) = {\mathopen{}\mathclose\bgroup\originalleft}(1, v^{x_1},\ldots, v^{x_{D_v}}, \frac{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2 - 2}{2}{\aftergroup\egroup\originalright}).$$ Ultimately, using the linearized Maxwellian , the linearized version of the full BGK equation reads: $$\label{eq:linearized_bgk_equation_framework} \partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= -\frac{{\nu}}{{\varepsilon}}({\mathcal{I}}- \Pi_\text{BGK}){f^{\varepsilon}},$$ where $\Pi_\text{BGK}$ is the following rank-${D_v}+2$ projection operator: $$\label{eq:projection_operator_bgk} \Pi_\text{BGK} {f^{\varepsilon}}= \sum_{k=0}^{{D_v}+1} \Psi_k({\mathbf{v}})\langle\Psi_k, {f^{\varepsilon}}\rangle.$$ ### Linear kinetic models\[sec:linear\_kinetic\] In [@LafitteMelisSamaey2017], the spectrum of a specific class of linear, hyperbolically scaled, kinetic equations was studied. In a scalar, two-dimensional setting (where we shall set $x := x_1$ and $y := x_2$), the following equation was proposed: $$\label{eq:artificial_bgk_equation} \partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= \frac{1}{{\varepsilon}}({{{\mathcal{M}}_{{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}(\rho^{\varepsilon}{\aftergroup\egroup\originalright})}} - {f^{\varepsilon}}),$$ with an artificial Maxwellian distribution given by: $$\label{eq:artificial_maxwellian} {{{\mathcal{M}}_{{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}(\rho^{\varepsilon}{\aftergroup\egroup\originalright})}} = \rho^{\varepsilon}(1+ v^x + v^y),$$ in which $\rho^{\varepsilon}$ is linked to ${f^{\varepsilon}}$ by averaging over the velocity space: $$\rho^{\varepsilon}= \int_{{\mathbb{R}}^2} {f^{\varepsilon}}({\mathbf{v}}) \frac{1}{2\pi} \exp{\mathopen{}\mathclose\bgroup\originalleft}(\frac{-{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2}{2}{\aftergroup\egroup\originalright})d{\mathbf{v}}$$ see also equation . In [@MelisSamaey2017], multiple relaxation time linearized BGK equations of the following form are considered: $$\label{eq:artificial_linearized_bgk_equation} \partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= \frac{{\nu}}{{\varepsilon}}({{{\mathcal{M}}_{\text{\scriptsize{lin}},{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}- {f^{\varepsilon}}),$$ with linearized Maxwellian given by: $$\label{eq:artificial_maxwellian_linearized} {{{\mathcal{M}}_{\text{\scriptsize{lin}},{\mathbf{v}}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}= \rho^{\varepsilon}(1 + v^x)(1 + v^y).$$ To fit both of these model problems in the general framework of section \[sec:linearized\_bgk\], we rewrite the artificial Maxwellians in and as: $$\begin{aligned} {\mathcal{M}}_{-2}({f^{\varepsilon}})({\mathbf{x}},{\mathbf{v}},t) &= \Psi_{-2}({\mathbf{v}}) \rho^{\varepsilon}({\mathbf{x}},t),& \quad \Psi_{-2}({\mathbf{v}}) &= (1 + v^x + v^y), \label{eq:framework_artificial} \\ {\mathcal{M}}_{-1}({f^{\varepsilon}})({\mathbf{x}},{\mathbf{v}},t) &= \Psi_{-1}({\mathbf{v}}) \rho^{\varepsilon}({\mathbf{x}},t),& \quad \Psi_{-1}({\mathbf{v}}) &= (1 + v^x)(1 + v^y). \label{eq:framework_linearized_artificial}\end{aligned}$$ With these choices of the Maxwellian distribution, we can summarize the linear kinetic models and as: $$\partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= -\frac{{\nu}}{{\varepsilon}}({\mathcal{I}}- \Pi_k){f^{\varepsilon}},$$ where ${\mathcal{I}}$ is the identity operator and $\Pi_k$ is the following rank-1 projection operator: $$\label{eq:projection_operator_artificial} \Pi_k {f^{\varepsilon}}= \Psi_k({\mathbf{v}}) \int_{{\mathbb{R}}^2} {f^{\varepsilon}}({\mathbf{v}}) \frac{1}{2\pi}\exp{\mathopen{}\mathclose\bgroup\originalleft}(\frac{-{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2}{2}{\aftergroup\egroup\originalright}) d{\mathbf{v}},$$ with either $k = -2$ or $k = -1$ . ### Linearized BGK spectrum\[sec:bgk\_spectrum\] The above considerations indicate that the structure of the linearized Maxwellian and the linearized BGK projection operator are almost identical to those in - and , respectively. Indeed, the linear kinetic projection operators $\Pi_{-2}$ and $\Pi_{-1}$ in match the first three terms of $\Pi_\text{BGK}$ in and only differ in the last term. This can be seen by using the orthonormal set of basis functions and the scalar product , and subsequently rewriting $\Pi_{-2}$ and $\Pi_{-1}$ as: $$\label{eq:projection_operator_artificial_rewritten} \begin{aligned} \Pi_{-2} {f^{\varepsilon}}&= \sum_{k=0}^2 \Psi_k({\mathbf{v}})(\Psi_k, {f^{\varepsilon}}) \\ \Pi_{-1} {f^{\varepsilon}}&= \sum_{k=0}^2 \Psi_k({\mathbf{v}})(\Psi_k, {f^{\varepsilon}}) + \Psi_1({\mathbf{v}})\Psi_2({\mathbf{v}})(\Psi_0, {f^{\varepsilon}}). \end{aligned}$$ We can thus view the linear kinetic models and used in [@LafitteMelisSamaey2017; @MelisSamaey2017], respectively, as a special simplified case of the linearized BGK equation. Since the linearized BGK operator was shown to be nearly identical to the relaxation operators of the linear kinetic models in , we expect the spectral properties of the linearized BGK equation to closely resemble those in [@LafitteMelisSamaey2017] (for ${\nu}= 1$) or [@MelisSamaey2017] (for ${\nu}= \rho$). Therefore, it is expected that the construction of stable PI methods for the full BGK equation with ${\nu}= 1$ and stable TPI methods for with ${\nu}= \rho$ is practically identical to that in [@LafitteMelisSamaey2017] and [@MelisSamaey2017], respectively. The choice of method parameters for the full BGK equation will be discussed more closely in section \[sec:method\_param\]. Linearized Boltzmann equation and its spectrum {#subsec:lin_boltzmann_equation} ---------------------------------------------- ### Linearization of the Boltzmann equation {#subsubsec:lin_boltzmann_equation} To simplify the analysis of the Boltzmann equation , it is customary to linearize the collision operator ${{\mathcal{Q}}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})({\mathbf{v}})}$ around the global Maxwellian distribution ${{\mathcal{M}}^{{\rho^{\infty}},{{\mathbf{{\bar{v}}}}^{\infty}},{T^{\infty}}}_{{\mathbf{v}}}}= {\mathcal{M}}_{\mathbf{v}}^{1,0,1}$, which, for $D_v = 2$, is given by: $$\label{eq:global_maxwellian} {\mathcal{M}}_{\mathbf{v}}^{1,0,1} = \frac{1}{2\pi} \exp{\mathopen{}\mathclose\bgroup\originalleft}(-\frac{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}^2}{2}{\aftergroup\egroup\originalright}),$$ see, for instance, [@Cercignani1988; @Golse2005; @Saint-Raymond2009]. Subsequently, we consider small fluctuations of ${f^{\varepsilon}}$ around the global equilibrium , that is: $$\label{eq:linearization_f} {f^{\varepsilon}}({\mathbf{v}}) = {\mathcal{M}}_{\mathbf{v}}^{1,0,1}\big(1 + {g^{{\varepsilon}}}({\mathbf{v}})\big).$$ Using a similar shorthand notation as before, that is, $g' = {g^{{\varepsilon}}}({\mathbf{v}}')$, ${\mathcal{M}}' = {\mathcal{M}}_{{\mathbf{v}}'}^{1,0,1}$, and so on, we linearize the quadratic terms in as: $$\label{eq:linearizing_ff_ff} f'f_*' - ff_* = {\mathcal{M}}{\mathcal{M}}_*\big(g' + g_*' - g - g_*\big),$$ where we neglected second-order fluctuations in the second equality, and we used that ${\mathcal{M}}'{\mathcal{M}}_*' = {\mathcal{M}}{\mathcal{M}}_*$ for any Maxwellian distribution ${\mathcal{M}}$. By substituting and into the Boltzmann equation , and exploiting that ${\mathcal{M}}= {\mathcal{M}}_{\mathbf{v}}^{1,0,1}$ depends only on velocity ${\mathbf{v}}$, we obtain the *linearized Boltzmann equation* as: $$\label{eq:linearized_Boltzmann_equation} \partial_t {g^{{\varepsilon}}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {g^{{\varepsilon}}}= \frac{1}{{\varepsilon}}{\mathcal{L}}{{g^{{\varepsilon}}}},$$ where ${\mathcal{L}}{{g^{{\varepsilon}}}}$ is the *linearized Boltzmann collision operator*, which reads: $$\label{eq:linearized_collision_operator} {\mathcal{L}}{{g^{{\varepsilon}}}} = \int_{{\mathbb{R}}^2}\int_{0}^{2\pi} B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\theta_\sigma}){\mathcal{M}}_*(g_*' + g' - g_* - g) d{\theta_\sigma}d{{\mathbf{v}}_{*}},$$ see also [@CIP; @Saint-Raymond2009]. Moreover, the operator ${\mathcal{L}}{{g^{{\varepsilon}}}}$ can be cast in the following form [@Cercignani1988 IV.5]: $$\label{eq:linearized_collision_operator_K_nu} {\mathcal{L}}{{g^{{\varepsilon}}}}({\mathbf{v}}) = {\mathcal{K}}{g^{{\varepsilon}}}({\mathbf{v}}) - {\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}){g^{{\varepsilon}}}({\mathbf{v}}),$$ where ${\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert})$ is a local multiplication operator termed the *linearized collision frequency* that depends only on the magnitude of ${\mathbf{v}}$ and is defined as: $$\label{eq:collision_frequency_linearized} {\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}) = \int_{{\mathbb{R}}^2}\int_{0}^{2\pi} B({{\mathopen{}\mathclose\bgroup\originalleft}\vert{{\mathbf{v}}_r}{\aftergroup\egroup\originalright}\vert},{\theta_\sigma}){\mathcal{M}}_* d{\theta_\sigma}d{{\mathbf{v}}_{*}},$$ and ${\mathcal{K}}$ is a non-local integral operator containing the remaining three terms in . Using and writing ${f^{\varepsilon}}$ instead of ${g^{{\varepsilon}}}$, we rewrite as: $$\label{eq:linearized_Boltzmann_equation_framework} \partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{\mathbf{x}}{f^{\varepsilon}}= -\frac{1}{{\varepsilon}}\big({\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}){\mathcal{I}}- {\mathcal{K}}\big){f^{\varepsilon}}.$$ For inverse power law potentials under Grad’s cut-off assumption and for hard sphere particles, it can be proven that ${\mathcal{K}}$ is a compact operator on $L^2({\mathbb{R}}^2)$, see [@Cercignani1988; @Golse2005]. This implies that it maps the unit ball of ${\mathbb{R}}^2$ onto a finite-dimensional space [@Cercignani1988]. In that sense, it shares properties with the finite rank operators described previously. However, the linearized operator ${\mathcal{L}}= {\mathcal{K}}- {\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}){\mathcal{I}}$ does not have finite rank. Furthermore, one could write formally: $${\mathcal{K}}= {\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert})(\Pi_\text{BGK} + {\mathcal{R}}),$$ for a certain remainder operator ${\mathcal{R}}$. Then, one can write as: $$\partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{\mathbf{x}}{f^{\varepsilon}}= -\frac{{\nu}}{{\varepsilon}} ({\mathcal{I}}- \Pi_\text{BGK}){f^{\varepsilon}}- \frac{{\nu}}{{\varepsilon}} {\mathcal{R}}{f^{\varepsilon}},$$ which provides a connection with the linearized BGK projection operator. ### Linearized Boltzmann spectrum. Analyzing the spectrum of the linearized Boltzmann collision operator ${\mathcal{L}}= {\mathcal{K}}- {\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}){\mathcal{I}}$ in is more involved than in the BGK case. In general, the linearized Boltzmann collision operator has a spectrum that consists of (i) a non-empty essential (purely continuous) part that is entirely determined by the continuous spectrum of $-{\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}){\mathcal{I}}$, and (ii) a set of discrete eigenvalues that is influenced by the operator ${\mathcal{K}}$, see, for instance, [@Cercignani1988] or [@BarangerMouhot2005; @Ellis1975]. In contrast, the spectrum of the linear kinetic relaxation operators and linearized BGK operator only consists of discrete eigenvalues. However, for Maxwellian particles with angular cut-off and, in particular, for pseudo-Maxwellian particles, it is known that the spectrum of ${\mathcal{L}}$ contains only discrete eigenvalues spread inside the interval $[-{\nu}(0),0]$ with ${\nu}({{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert})$ given in [@Cercignani1988]. We have seen in Section \[subsubsec:lin\_boltzmann\_equation\] that the linearized version of the full Boltzmann equation reads: $$\partial_t {f^{\varepsilon}}+ {\mathbf{v}}\cdot \nabla_{{\mathbf{x}}} {f^{\varepsilon}}= -\frac{1}{{\varepsilon}}(\nu(|{\mathbf{v}}|){\mathcal{I}}- {\mathcal{K}}){f^{\varepsilon}}.$$ Let us apply the Fourier transform in the physical space: since the collision operator depends only on the velocity magnitude ${{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{v}}{\aftergroup\egroup\originalright}\vert}$, the only difference in the equation will be that the free transport term ${\mathbf{v}}\cdot \nabla_{\mathbf{x}}$ will become a multiplication operator (by $i \bm \gamma \cdot {\mathbf{v}}$, where $\bm \gamma$ is the spatial Fourier variable). One can then write the Fourier-transformed linear Boltzmann equation as: $$\label{eq:linearBoltzFourier} \partial_t {h^{{\varepsilon}}}= \frac 1{\varepsilon}{\mathcal{K}}{h^{{\varepsilon}}}- {\mathopen{}\mathclose\bgroup\originalleft}( \nu(|{\mathbf{v}}|)/{\varepsilon}+ i \, {\varepsilon}\bm \gamma \cdot {\mathbf{v}}{\aftergroup\egroup\originalright}) {h^{{\varepsilon}}},$$ where ${h^{{\varepsilon}}}$ is the Fourier transform in space of ${f^{\varepsilon}}$. Hence, the evolution of ${h^{{\varepsilon}}}$ is given by a compact perturbation of a (complex-valued) multiplication operator. It was proven in a series of papers that the spectrum of this Fourier-transformed collision operator has the following behavior as a function of $|\bm \gamma|$ and ${\varepsilon}$: \[thm:linBoltzSpectrum\] The spectrum of the right hand side of equation consists of an essential part $\Sigma_e$ located to the left of a vertical line of negative real part and a discrete spectrum $\Sigma_d$ composed of: - *fast modes:* eigenvalues located at a distance at least $1/{\varepsilon}$ to the left of the imaginary axis; - *slow modes:* if $|{\varepsilon}| \ll 1$, there are *exactly* $4$ eigenvalues branches given by: $$\lambda^{(j)}(|\bm \gamma|) := i \, \lambda^{(j)}_1 \, {\varepsilon}|\bm \gamma| - \lambda^{(j)}_2 \, {\varepsilon}^2 |\bm \gamma|^2 + \mathcal O {\mathopen{}\mathclose\bgroup\originalleft}( {\varepsilon}^3 |\bm \gamma|^3 {\aftergroup\egroup\originalright}), \quad j \in \{0,1,2,3\},$$ for explicit constants $\lambda_1^{(j)} \in {\mathbb{R}}$ and $\lambda_2^{(j)}>0$. A sketch of this result can be found in figure \[fig:spectrumBoltzFourier\]. ![\[fig:spectrumBoltzFourier\] Spectrum of the Fourier transformed linearized Boltzmann operator, for small radial frequencies.](image/spectrum.pdf) We observe in Theorem \[thm:linBoltzSpectrum\] that the discrete eigenvalues form a fast and a slow cluster, justifying the use of projective integration with easily computable parameters, in the spirit of [@LafitteMelisSamaey2017; @MelisSamaey2017]. Nevertheless, the presence of an essential spectrum is one of the reasons that one must use *telescopic* projective integration to solve the full Boltzmann equation: this spectral decomposition will give rise to new clusters of eigenvalues at the discrete level. Note that one can mimic the proof of this result for the simpler BGK operator with $\nu = 1$ to obtain the same spectral behavior of the linearized operator without the essential part, justifying at the continuous level the results from [@LafitteMelisSamaey2017]. Method parameters for projective and telescopic projective integration {#sec:method_param} ---------------------------------------------------------------------- It still remains to select appropriate parameter values for the projective or telescopic projective integration methods. These are determined by ensuring that all eigenvalues of the kinetic problem under study fall within the stability region of the full projective method. In sections \[subsec:lin\_bgk\] and \[subsec:lin\_boltzmann\_equation\], we revealed that the spectra of the linearized kinetic equations either appear in two stationary eigenvalue disks (linearized BGK equation with ${\nu}= 1$) or are continuously spread along (a part of) the negative real axis (linearized BGK equation with ${\nu}= \rho$ and linearized Boltzmann equation). Since the construction of stable projective methods for both well-separated as well as continuously spread spectra is studied in previous works [@LafitteMelisSamaey2017] and [@MelisSamaey2017], respectively, we take over the main results here, which are summarized below. ### Stationary, well-separated spectrum For the linearized BGK equation with $\nu = 1$, which falls into the class of kinetic models studied in [@LafitteMelisSamaey2017], it was shown in [@LafitteMelisSamaey2017] that the spectrum consists of two stationary, well-separated eigenvalue clusters (a fast and slow, dominant cluster). To accommodate these two clusters, the method parameters of projective integration can be selected such that its stability region splits up into two parts. 1. First, the inner integrator time step ${\delta t}$ is chosen corresponding to the fastest time scale of the problem, which is of the order of ${\varepsilon}$. This centers one stability region of the projective method around the fast eigenvalues. 2. Next, the number of inner integrator time steps $K$ is chosen such that all fast eigenvalues lie inside this stability region. In [@LafitteMelisSamaey2017], it was proven that we require $K \ge 2$. 3. Last, the outer integrator time step ${\Delta t}$ is selected such that all dominant eigenvalues fall into the second stability region of the projective method. Since both $K$ and ${\Delta t}$ are independent of the small-scale parameter ${\varepsilon}$, the resulting projective method has a cost that is also independent of ${\varepsilon}$, which becomes increasingly advantageous for ${{\varepsilon}\to 0}$. ### Continuously spread spectrum When considering the linearized BGK equation with $\nu = \rho$, which was studied in [@MelisSamaey2017], the spectrum varies continuously over the negative real axis. This also holds true for the linearized Boltzmann equation, be it on a part of the negative real axis, see figure \[fig:spectrumBoltzFourier\]. In this case, we require that the stability region of the numerical method does not split up but instead comprises the entire negative real axis up to the fastest eigenvalue of the problem (a numerical method with this property is termed $\mathit{[0,1]}$*-stable*). Here, for simplicity, we assume that the fastest eigenvalue at $t = 0$ corresponds to the fastest possible eigenvalue for all other times $t > 0$. Since \[0,1\]-stable projective integration methods lose practically all of their potential speed-up, \[0,1\]-stable telescopic projective integration methods can be designed with much higher speed-ups. 1. Similarly to projective integration, the innermost integrator time step $h_0$ of the telescopic projective integration method is chosen corresponding to the fastest time scale, which is of the order of ${\varepsilon}/\max_{x}\rho(x,0)$. 2. Next, we fix the outermost time step we would like to use, taking into account a CFL-like stability constraint, as follows: $h_L = C{\Delta x}$. 3. Before choosing the number of projective levels, we decide on the number of inner integrator time steps $K$, which we consider to be fixed on each projective level. For each chosen value of $K$ there is a corresponding maximal value of $M$ such that the stability region does not split up, see [@Gear2003telescopic] or [@MelisSamaey2017]. 4. The required number of projective levels $L$ to obtain a \[0,1\]-stable telescopic method is computed as (see [@MelisSamaey2017]): $$\label{eq:tpi_number_of_levels} L \approx \frac{\log(h_L) + \log(1/h_0)}{log(M+K+1)}.$$ 5. For the given values of $h_0$, $h_L$, $K$ and $L$ adapt the value of $M$ on the different projective levels such that the following equation: $$\label{eq:tpi_hL} h_L = \prod_{\ell=0}^{L-1} (M_\ell + K + 1)h_0$$ is valid (for further details, we refer the reader to [@MelisSamaey2017]). For a \[0,1\]-stable telescopic method, the values of $h_L$, $M_\ell$ and $K$ are independent of ${\varepsilon}$. However, as indicated by equation , the number of projective levels increases as $O(\log(1/{\varepsilon}))$. As a consequence, the cost of a \[0,1\]-stable telescopic projective integration method is not completely ${\varepsilon}$-independent. However, the dependence is rather modest. ### Speedup {#subsubsec:speedup} If we assume, as in [@Gear2003telescopic], that the overhead due to extrapolations is negligible and timestepping with the innermost integrator is computationally most demanding, the speedup ${\mathcal{S}}_L$ realized by the overall level-$L$ TPI method compared to naive forward Euler timestepping is given by: $$\label{eq:tpi_speedup} {\mathcal{S}}_L = \prod_{\ell=0}^{L-1} \frac{M_\ell + K_\ell + 1}{K_\ell + 1},$$ that is, the ratio of the total number of naive forward Euler time steps within one outermost time step $h_L$ (see equation ) over the number of actual innermost steps in the TPI method. Numerical experiments {#sec:results} ===================== Here, we report simulation results for the BGK equation and the Boltzmann equation using pseudo-Maxwellian particles (that is, $\gamma = 0$ and $b_0$ constant). For each experiment, we shall compactly indicate the dimensions in space $(D_x)$ and velocity $(D_v)$ by writing “$D_x$D/$D_v$D" with $D_x,D_v \in \{1,2\}$. We begin with BGK in 1D/1D (Section \[subsec:bgk\_1d1d\]), and subsequently consider both the BGK and Boltzmann equation in 1D/2D (Section \[subsec:bgk\_boltzmann\_1d2d\]). Thereafter, we target a shock-bubble interaction problem for the BGK equation in 2D/2D (Section \[subsec:shock\_bubble\_interaction\]), and the more intricated Kelvin-Helmoltz-like instability in the same setting (Section \[subsec:KelvinHelmoltz\]). As a last experiment, we deal with the full Boltzmann equation in 2D/2D (Section \[subsec:boltzmann\_2d2d\]). BGK in 1D/1D {#subsec:bgk_1d1d} ------------ As a first experiment, we focus on the nonlinear BGK equation in 1D/1D. We consider a Sod-like test case for $x \in [0,1]$ consisting of an initial centered Riemann problem with the following left and right state values: $$\label{eq:bgk_1d_sod} \begin{aligned} \begin{pmatrix} \rho_L \\ {\bar{v}}_L \\ T_L \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \qquad\qquad \begin{pmatrix} \rho_R \\ {\bar{v}}_R \\ T_R \end{pmatrix} = \begin{pmatrix} 0.125 \\ 0 \\ 0.25 \end{pmatrix}. \end{aligned}$$ The initial distribution ${f^{\varepsilon}}(x,v,0)$ is then chosen as the Maxwellian corresponding to the above initial macroscopic variables. We impose outflow boundary conditions and perform simulations for $t \in [0,0.15]$. As velocity space, we take the interval $[-8,8]$, which we discretize on a uniform grid using $J=80$ velocity nodes. In all simulations, space is discretized using the WENO3 spatial discretization with ${\Delta x}= 0.01$. Below we regard three gas flow regimes, ${\varepsilon}= 10^{-1}$ (kinetic regime), ${\varepsilon}= 10^{-2}$ (transitional regime) and ${\varepsilon}= 10^{-5}$ (fluid regime), and for each regime, we compare solutions for two cases of collision frequency ${\nu}$ in the BGK equation , ${\nu}= 1$ and ${\nu}= \rho$. #### Direct integration $\boldsymbol{({\varepsilon}= 10^{-1}}$ and $\boldsymbol{{\varepsilon}= 10^{-2})}$. In the kinetic $({\varepsilon}= 10^{-1})$ and transitional $({\varepsilon}= 10^{-2})$ regimes, we compute the numerical solution for ${\nu}= 1$ and ${\nu}= \rho$ using the fourth order Runge-Kutta (RK4) time discretization with time step ${\delta t}= 0.1{\Delta x}$. The results are shown in figure \[fig:bgk\_1d1d\_nu\] for ${\nu}= 1$ (left) and ${\nu}= \rho$ (right), where we display the density $\rho$, macroscopic velocity ${\bar{v}}$ and temperature $T$ as given in at $t = 0.15$. In addition, we plot the heat flux $q$, which, in a general $D_v$-dimensional setting, is a vector ${\mathbf{q}}= {{\mathopen{}\mathclose\bgroup\originalleft}(q^d{\aftergroup\egroup\originalright})_{d=1}^{D_v}}$ with components given by: $$q^d = \frac{1}{2}\int_{{\mathbb{R}}^{D_v}} {{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{c}}{\aftergroup\egroup\originalright}\vert}^2c^d{f^{\varepsilon}}d{\mathbf{v}},$$ in which ${\mathbf{c}}= {{\mathopen{}\mathclose\bgroup\originalleft}(c^d{\aftergroup\egroup\originalright})_{d=1}^{D_v}} = {\mathbf{v}}- {\mathbf{{\bar{v}}}}$ is the peculiar velocity. The different regimes are shown by blue (kinetic) and purple (transitional) dots. The red line in each plot denotes the limiting $({{\varepsilon}\to 0})$ solution of each macroscopic variable, which all converge to the solution of the Euler system with ideal gas law $P = \rho T$ and heat flux $q = 0$. [![\[fig:bgk\_1d1d\_nu\] Numerical solution of the BGK equation in 1D/1D with ${\nu}= 1$ (left) and ${\nu}= \rho$ (right) at $t = 0.15$ for a Sod-like shock test using the WENO3 scheme with ${\Delta x}= 0.01$. RK4 is used for ${\varepsilon}= 10^{-1}$ (blue dots) and ${\varepsilon}= 10^{-2}$ (purple dots) with ${\delta t}= 0.1{\Delta x}$. The PRK4 (left) and level-2 TPRK4 (right) methods are used for ${\varepsilon}= 10^{-5}$ (green dots). Red line: hydrodynamic limit solution $({{\varepsilon}\to 0})$. ](image/bgk1D1D_epsi125_nu1_nurho "fig:")]{} #### Projective integration $\boldsymbol{({\varepsilon}= 10^{-5}}$ and $\boldsymbol{{\nu}= 1)}$. In the fluid regime $({\varepsilon}= 10^{-5})$, direct integration schemes such as RK4 become too expensive due to a severe time step restriction, which is required to ensure stability of the method. Exploiting that the spectrum of the linearized BGK equation with ${\nu}= 1$ resembles that of the linear kinetic models used in [@LafitteMelisSamaey2017], see section \[sec:linearOpSpecProp\], we construct a projective integration method to accelerate time integration in the fluid regime. As inner integrator, we select the forward Euler time discretization with ${\delta t}= {\varepsilon}$. As outer integrator, we choose the fourth-order projective Runge-Kutta (PRK4) method, using $K=2$ inner steps and an outer step of size ${\Delta t}= 0.4{\Delta x}$. Figure \[fig:bgk\_1d1d\_nu\] (left) shows the macroscopic observables in the fluid regime for ${\nu}= 1$ at $t = 0.15$ (green dots). From this, we observe that the BGK solution is increasingly dissipative for increasing values of ${\varepsilon}$ since the rate with which ${f^{\varepsilon}}$ converges to its equilibrium ${{{\mathcal{M}}_{v}{\mathopen{}\mathclose\bgroup\originalleft}({f^{\varepsilon}}{\aftergroup\egroup\originalright})}}$ becomes slower. In contrast, for sufficiently small ${\varepsilon}$, relaxation to thermodynamic equilibrium occurs practically instantaneous and the Euler equations yield a valid description. Since this is a hyperbolic system, it allows for the development of sharp discontinuous and shock waves which are clearly seen in the numerical solution. In this numerical test, the speed-up factor between a naive RK4 implementation and the projective integration method is $130.3$, namely formula with $L=1$ (1 projective level), $K_0=2$ and $M_0=397$. [![\[fig:bgk\_rk4TPI\_comp\] Numerical solution of the BGK equation in 1D/1D with ${\nu}= \rho$, ${\varepsilon}= 10^{-5}$ at $t = 0.15$ for a Sod-like shock test using the WENO3 scheme with ${\Delta x}= 0.01$. Comparison between level-2 TPRK4 with ${\delta t}= 0.1{\Delta x}$ (solid blue line) and classical RK4 with a ${\Delta t}= 0.5{\varepsilon}$ (red dots).](image/bgk1D1D_epsi5_RK4TPIcomp "fig:")]{} #### Telescopic projective integration $\boldsymbol{({\varepsilon}= 10^{-5}}$ and $\boldsymbol{{\nu}= \rho)}$. Next, we repeat the above experiment taking ${\nu}= \rho$ in the BGK equation . We now design a $[0,1]$-stable telescopic projective integration method as in [@MelisSamaey2017] since, for this choice of ${\nu}$, the spectrum of the linearized BGK equation is spread along the negative real axis and is time-dependent. Therefore, the two-scale nature in case of ${\nu}= 1$ has become a multi-scale problem, which destroys the acceleration in time of projective integration. We construct a $[0,1]$-stable TPRK4 method consisting of $2$ projective levels with FE as innermost integrator with time step $h_0 = {\varepsilon}$, constant $K=6$ on each level and an outermost time step $h_2 = 0.4{\Delta x}$. The extrapolation step sizes $M$ on each level are calculated as $M=\{14.24, 11.83\}$. The results are shown by green dots in figure \[fig:bgk\_1d1d\_nu\] (right). We conclude that the effect of choosing ${\nu}= \rho$ primarily manifests itself in the transitional regime $({\varepsilon}= 10^{-2})$, for which the relaxation rate is not too slow nor too fast. Moreover, it is seen that this choice of collision frequency does not alter the hydrodynamic limit of the BGK equation, which is captured correctly by the telescopic scheme. Finally, figure \[fig:bgk\_rk4TPI\_comp\] compares the solutions obtained with the level-2 TPRK4 method and a classical RK4 method with a very small time step (of order ${\varepsilon}$) for the stiff test case where ${\varepsilon}=10^{-5}$. We observe a very good agreement between the two simulations (only a very small difference can be seen in the heat flux), while the TPRK4 scheme is more than 10 times faster than the RK4 scheme, because of its bigger time steps. The former scheme would be even more efficient with smaller values of the relaxation parameter. In this test, the speed-up factor between a naive RK4 implementation and the telescopic projective integration method is $8.2$, namely formula with $L=2$ (2 projective levels), $(K_0, K_1)=(6,6)$ and $(M_0,M1)=(14.24,11.83)$. BGK and Boltzmann in 1D/2D {#subsec:bgk_boltzmann_1d2d} -------------------------- The BGK equation was introduced as a simplified model for the Boltzmann equation capturing most essential features of the latter. Here, we investigate the difference between both models. Since the Boltzmann collision operator vanishes for a one-dimensional velocity space, in this section, we consider both models in 1D/2D. In the experiments, this is achieved by discretizing space $(x,y)$ on a grid of size $I_x \times 2$ and using homogeneous data along the $y$-direction such that the spatial derivative $\partial_y{f^{\varepsilon}}$ exactly cancels out. We perform the Sod test of the previous section in 1D/2D, see also [@Filbet2010]. As velocity space, we take the domain $[-8,8]^2$, which we discretize on a uniform grid using $J_x = J_y = 32$ velocity nodes along each dimension. In all simulations, space is discretized using the WENO2 spatial discretization with ${\Delta x}= 0.01$. Below we regard two regimes, ${\varepsilon}= 10^{-2}$ (transitional regime) and ${\varepsilon}= 10^{-5}$ (fluid regime), and for each regime, we compare solutions for BGK with ${\nu}= 1$, BGK with ${\nu}= \rho$ and Boltzmann with pseudo-Maxwellian particles. To approximate the Boltzmann collision operator, we apply the fast spectral method described in section \[sub:FastSpectral\] using $N_\theta = 4$ discrete angles. This is enough because of the spectral accuracy of the trapezoidal rule applied to periodic functions (see [@FiMoPa2006] for more details on this topic). #### Direct integration ($\boldsymbol{{\varepsilon}= 10^{-2}}$). In the transitional regime, we perform all simulations using the RK4 method with time step ${\delta t}= 0.1{\Delta x}$, for which we display the results in figure \[fig:boltz\_bgk\_1d2d\_nu\] (left) for BGK with ${\nu}= 1$ (blue dots), BGK with ${\nu}= \rho$ (green dots) and the Boltzmann equation (red dots). From this, we observe that the BGK solution with ${\nu}= \rho$ is closer to the Boltzmann solution than the BGK solution with ${\nu}= 1$. This is as expected, since the BGK equation with ${\nu}= \rho$ correctly captures the loss term of the Boltzmann collision operator, see . Moreover, the discrepancy between Boltzmann and BGK with ${\nu}= \rho$ increases for higher order moments of ${f^{\varepsilon}}$; while the density (zeroth order moment) appears to coincide (to the naked eye), the heat flux (third order moment) reveals a clear difference between both models. [![\[fig:boltz\_bgk\_1d2d\_nu\] Comparison between BGK and Boltzmann in 1D/2D for ${\varepsilon}= 10^{-2}$ (left) and ${\varepsilon}= 10^{-5}$ (right) at $t = 0.15$ for a Sod-like shock test using the WENO2 scheme with ${\Delta x}= 0.01$. Blue dots: BGK with ${\nu}= 1$; green dots: BGK with ${\nu}= \rho$; red dots: Boltzmann. For ${\varepsilon}= 10^{-2}$, we used RK4 with ${\delta t}= 0.1{\Delta x}$. For ${\varepsilon}= 10^{-5}$, we applied a PRK4 (blue dots) and a level-2 TPRK4 (green and red dots) method. ](image/boltz_bgk1D2D_epsi25_nu1_nurho "fig:")]{} #### Projective methods ($\boldsymbol{{\varepsilon}= 10^{-5}}$). In the fluid regime, the RK4 method becomes too expensive. To that end, for BGK with ${\nu}= 1$, we design a PRK4 method with FE as inner integrator using ${\delta t}= {\varepsilon}$, $K = 2$ inner steps and ${\Delta t}= 0.4{\Delta x}$. Due to the multi-scale nature of both the BGK relaxation operator with ${\nu}= \rho$ and the Boltzmann collision operator, we construct a $[0,1]$-stable level-2 TPRK4 method for both models using the FE scheme as innermost integrator with $h_0 = {\varepsilon}$. We set $K = 4$ constant on each level, compute the extrapolation step sizes as $M = \{14.24, 11.83\}$, and choose the outermost time step as $h_2 = 0.4{\Delta x}$. The results can be seen in figure \[fig:boltz\_bgk\_1d2d\_nu\] (right) using the same plotting style as in the left column. For all models, the projective and telescopic projective integration methods display the expected hydrodynamic limit. In these numerical tests, the speed-up factor between a naive RK4 implementation and the telescopic projective integration method for the PRK4 method for the BGK model with constant relaxation is $133.3$, namely formula with $L=1$ (1 projective level), $K_0=2$ and $M_0=397$. The speedup for the TPRK4 method for the BGK model with nonconstant relaxation rate and the Boltzmann equation is $13$, namely formula with $L=2$ (2 projective levels), $(K_0, K_1)=(4,4)$ and $(M_0,M1)=(14.24,11.83)$. Shock-bubble interaction in 2D/2D {#subsec:shock_bubble_interaction} --------------------------------- Here, we consider the BGK equation in 2D/2D with constant collision frequency ${\nu}= 1$ and we investigate the interaction between a moving shock wave and a stationary smooth bubble, which was proposed in [@Torrilhon2006], see also [@Cai2010]. This problem consists of a shock wave positioned at $x = -1$ in a spatial domain ${\mathbf{x}}= (x,y) \in [-2,3] \times [-1,1]$ traveling with Mach number ${\mathit{Ma}}= 2$ into an equilibrium flow region. Over the shock wave, the following left $(x \le -1)$ and right $(x > -1)$ state values are imposed [@Cai2010]: $$\label{eq:bgk_2d_shock} \begin{aligned} \big(\rho_L, {\bar{v}}^x_L, {\bar{v}}^y_L, T_L\big) &= {\mathopen{}\mathclose\bgroup\originalleft}(\frac{16}{7}, \sqrt{\frac{5}{3}}\frac{7}{16}, 0, \frac{133}{64}{\aftergroup\egroup\originalright}) \\ \big(\rho_R, {\bar{v}}^x_R, {\bar{v}}^y_R, T_R\big) &= (1, 0, 0, 1). \end{aligned}$$ Due to this initial profile, the shock wave will propagate rightwards into the flow region at rest $(x > -1)$. Moreover, in this equilibrium region, a smooth Gaussian density bubble centered at ${\mathbf{x}}_0 = (0.5,0)$ is placed, given by: $$\label{eq:bgk_2d_bubble} \rho({\mathbf{x}},0) = 1 + 1.5\exp{\mathopen{}\mathclose\bgroup\originalleft}(-16{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{x}}- {\mathbf{x}}_0{\aftergroup\egroup\originalright}\vert}^2{\aftergroup\egroup\originalright}).$$ The initial density $\rho({\mathbf{x}},0)$ and temperature $T({\mathbf{x}},0)$ are visualized in figure \[fig:bgk\_2d2d\_shock\_bubble\_t00\]. Then, the initial distribution ${f^{\varepsilon}}({\mathbf{x}},{\mathbf{v}},0)$ is chosen as the Maxwellian corresponding to the initial macroscopic variables in -. We impose outflow and periodic boundary conditions along the $x$- and $y$-directions, respectively, and we perform simulations for $t \in [0,0.8]$. As velocity space, we take the domain $[-10,10]^2$, which we discretize on a uniform grid using $J_x = J_y = 30$ velocity nodes along each dimension. We discretize space using the WENO2 spatial discretization with $I_x = 200$ and $I_y = 25$. Furthermore, we consider a fluid regime by taking ${\varepsilon}= 10^{-5}$. [![\[fig:bgk\_2d2d\_shock\_bubble\_t00\] Initial solution for density (top) and temperature (bottom) of the shock-bubble interaction in 2D/2D. The spatial domain $[-2,3] \times [-1,1]$ is discretized using $I_x = 200$ and $I_y = 25$. ](image/bgk2D2D_shock_bubble_t00 "fig:")]{} Since we regard the BGK equation with constant collision frequency ${\nu}= 1$, the spectrum of the linearized BGK operator consists of two eigenvalue clusters. Therefore, we construct a projective integration method to speed up simulation in time. We select the PRK4 method with FE as inner integrator. The inner time step is fixed as ${\delta t}= {\varepsilon}$ and we use $K = 2$ inner steps in each outer integrator iteration. The outer time step is chosen as ${\Delta t}= 0.4{\Delta x}$. The simulated density and temperature at time $t = 0.8$ are displayed in figure \[fig:bgk\_2d2d\_shock\_bubble\_t08\]. We observe that the shock propagates in the positive $x$-direction and bumps into the stationary bubbly density. A similar behavior is seen for the temperature evolution. To compare our results with those in [@Torrilhon2006], where the smallest value of ${\varepsilon}$ is chosen as ${\varepsilon}= 10^{-2}$, we regard the one-dimensional evolution of density and temperature along the axis $y = 0$. For $t \in \{0, 0.2, 0.4, 0.6, 0.8\}$, we plot these intersections in figure \[fig:bgk\_2d2d\_shock\_bubble\_y0\]. We conclude that we obtain the same solution structure at $t = 0.8$ as in [@Torrilhon2006]. However, our results are sharper and less dissipative supposedly due to the particular small value of ${\varepsilon}$ ($10^{-5}$ versus $10^{-2}$). In contrast to [@Cai2010], we nicely capture the swift changes in the temperature profile for $x \in [0.5,1]$ at $t = 0.8$. Again, in this numerical test, the speed-up factor between a naive RK4 implementation and the projective integration method is $133.3$, namely formula with $L=1$ (1 projective level), $K_0=2$ and $M_0=397$. [![\[fig:bgk\_2d2d\_shock\_bubble\_t08\] Numerical solution of the shock-bubble interaction at $t = 0.8$ using the BGK equation with ${\nu}= 1$ in 2D/2D. We discretized velocity space using $J_x = J_y = 30$. We applied a PRK4 method with FE as inner integrator and ${\delta t}= {\varepsilon}= 10^{-5}$ together with the WENO2 spatial discretization scheme with $I_x = 200$ and $I_y = 25$. ](image/bgk2D2D_shock_bubble_t08 "fig:")]{} [![\[fig:bgk\_2d2d\_shock\_bubble\_y0\] Numerical solution of the shock-bubble interaction along $y = 0$ at $t = 0$ (black dashed), $t = 0.2$ (blue), $t = 0.4$ (purple), $t = 0.6$ (green) and $t = 0.8$ (red). ](image/bgk2D2D_shock_bubble_y0 "fig:")]{} A Kelvin-Helmoltz like instability problem {#subsec:KelvinHelmoltz} ------------------------------------------ Keeping the same setting of the BGK equation in 2D/2D with constant collision frequency ${\nu}= 1$, we now turn to a less common test case in the field of collisional kinetic equation, the so-called Kelvin-Helmoltz instability. This phenomenon occurs when two fluids of different densities and in thermodynamic equilibrium move at different speeds. It is very well known that such a system will exhibit turbulent, unstable vortices at the interface between the two fluids, because of the velocity shear [@Helmoltz1868]. In order for these instabilities to develop, the Reynolds number of the fluids considered must be large. Using the von Karman relation [@sone2007molecular], which states that the Reynolds number is inversely proportional to the Knudsen number ${\varepsilon}$, we shall then choose a very small Knudsen number ${\varepsilon}= 5\cdot10^{-5}$ along with the following initial condition inspired from [@McNallyLyraPassy2012]: $$\label{eq:BGK2dKelvinHelmoltz} \begin{pmatrix} \rho_1 \\ {\bar{v}}^x_1 \\ {\bar{v}}^y_1 \\ T_1 \end{pmatrix} = \begin{pmatrix} 1 \\ 0.5 \\ 0.01 \sin{\mathopen{}\mathclose\bgroup\originalleft}(4 \pi x{\aftergroup\egroup\originalright}) \\ 1 \end{pmatrix} \quad (y \geq 0), \qquad\quad \begin{pmatrix} \rho_2 \\ {\bar{v}}^x_2 \\ {\bar{v}}^y_2 \\ T_2 \end{pmatrix} = \begin{pmatrix} 2 \\ -0.5 \\ 0.01 \sin{\mathopen{}\mathclose\bgroup\originalleft}(4 \pi x{\aftergroup\egroup\originalright}) \\ 1 \end{pmatrix} \quad (y < 0).$$ The initial distribution ${f^{\varepsilon}}({\mathbf{x}},{\mathbf{v}},0)$ is chosen as the Maxwellian corresponding to the initial macroscopic variables in . We impose periodic and outflow boundary conditions along the $x$- and $y$-directions, respectively, and we perform simulations for $t \in [0,1.6]$. As velocity space, we take the domain $[-8,8]^2$, which we discretize on a uniform grid using $J_x = J_y = 30$ velocity nodes along each dimension. We discretize space using the WENO2 method on $[-0.5,0.5]\times[-0.5,0.5]$ with $I_x = I_y= 100$. Since we consider again the BGK equation with constant collision frequency ${\nu}= 1$, the spectrum of the linearized BGK operator consists of two eigenvalue clusters. Therefore, we construct a projective integration method to speed up simulation in time. We select the PRK4 method with FE as inner integrator. The inner time step is fixed as ${\delta t}= {\varepsilon}$ and we use $K = 3$ inner steps in each outer integrator iteration (this test case is stiffer than the previous one because of the turbulent regime). The outer time step is chosen as ${\Delta t}= 0.45{\Delta x}$. The simulated density and pressure at time $t = 0.4$, $0.9$ and $1.6$ are displayed in figure \[fig:bgk\_2d2d\_KH\_Density-Pressure\], along with the vector field $ {\mathbf{{\bar{v}}}}$ with the contour lines of density at time $t=0.9$ in figure \[fig:bgk\_2d2d\_KH\_velocityField\]. In this numerical test, the speed-up factor between a naive RK4 implementation and the projective integration method is $22.5$, namely formula with $L=1$ (1 projective level), $K_0=3$ and $M_0=86$. [![\[fig:bgk\_2d2d\_KH\_Density-Pressure\] Density (left) and pressure (right) of the Kelvin-Helmoltzm-like instability at times $t=0.6$ (first row), $0.9$ (second row) and $1.6$ (third row). ](image/bgk2D2D_KH_Density-Pressure "fig:")]{} [![\[fig:bgk\_2d2d\_KH\_velocityField\]Velocity field and density line of the Kelvin-Helmoltzm-like instability at time $t=0.9$.](image/bgk2D2D_KH_velocityField "fig:")]{} As expected, we observe vortices developing along the velocity shear line $y=0$, and then expanding, forming small scale structures as time evolves, hence validating the turbulent behavior exhibited by the fluid for this Reynolds number. Note that the use of high-order methods in both space (WENO2) and time (PRK4) is crucial, because of these fine scale structures. Choosing less accurate (and more dissipative) schemes, such as forward Euler in time and upwind in space, leads to a uniform density and pressure instead of vortices in our numerical tests. Boltzmann in 2D/2D {#subsec:boltzmann_2d2d} ------------------ As a last experiment, we concentrate on the Boltzmann equation with pseudo-Maxwellian particles in 2D/2D. As initial configuration for the gas, we consider the double Sod shock test, that is, for ${{\mathbf{x}}= (x,y) \in [-0.5,0.5]^2}$, we set: $$\label{eq:boltzmann_2d_sod} \begin{aligned} \begin{pmatrix} \rho_1 \\ {\mathbf{{\bar{v}}}}_1 \\ T_1 \end{pmatrix} = \begin{pmatrix} 0.1 \\ \boldsymbol{0} \\ 1 \end{pmatrix} \quad (xy \le 0), \qquad\quad \begin{pmatrix} \rho_2 \\ {\mathbf{{\bar{v}}}}_2 \\ T_2 \end{pmatrix} = \begin{pmatrix} 1 \\ \boldsymbol{0} \\ 1 \end{pmatrix} \quad (\text{otherwise}). \end{aligned}$$ The initial distribution ${f^{\varepsilon}}({\mathbf{x}},{\mathbf{v}},0)$ is then chosen as the Maxwellian corresponding to the above macroscopic variables. We impose outflow boundary conditions along both dimensions and perform simulations for $t \in [0,0.16]$. As velocity space, we take the domain $[-8,8]^2$, which we discretize on a uniform grid with $J_x = J_y = 32$ velocity nodes along each dimension. Furthermore, we discretize the spatial domain using the WENO2 spatial discretization with $I_x = I_y = 64$ grid points along each dimension, and we fix ${\varepsilon}= 5 \cdot 10^{-5}$ (that is, we are in the fluid regime). The Boltzmann collision operator is approximated again using the fast spectral method described in section \[sub:FastSpectral\] with $N_\theta = 4$ discrete angles. For the time simulation of the Boltzmann equation, we apply a level-2 TPRK4 method with FE as innermost integrator using $h_0 = {\varepsilon}$ as innermost time step. We set $K = 3$ constant on each level and compute the extrapolation step sizes as $M = \{6.66, 4.80\}$. The outermost time step is chosen as ${\Delta t}= 0.3{\Delta x}$. In figure \[fig:boltz\_2d2d\_sod\], we plot various macroscopic observables of interest at $t = 0.16$. The density, macroscopic velocity along $x$ and temperature are computed as the moments of ${f^{\varepsilon}}$, see . Then, pressure, energy and the Mach number are obtained, respectively, as: $$P = \rho T, \qquad\quad E = \frac{1}{2}\rho{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{{\bar{v}}}}{\aftergroup\egroup\originalright}\vert}^2 + P, \qquad\quad {\mathit{Ma}}= \frac{{{\mathopen{}\mathclose\bgroup\originalleft}\vert{\mathbf{{\bar{v}}}}{\aftergroup\egroup\originalright}\vert}}{\sqrt{T}}.$$ In this test, the speed-up factor between a naive RK4 implementation and the telescopic projective integration method is $5.9$, namely formula with $L=2$ (2 projective levels), $(K_0, K_1)=(3,3)$ and $(M_0,M1)=(6.66,4.80=)$. [![\[fig:boltz\_2d2d\_sod\] Numerical solution of the Boltzmann equation with pseudo-Maxwellian particles in 2D/2D at $t = 0.16$ for a double Sod shock test . Velocity space is discretized using $J_x = J_y = 32$. We applied a level-2 TPRK4 method with FE as innermost integrator and $h_0 = {\varepsilon}= 5 \cdot 10^{-5}$ together with WENO2 with $I_x = I_y = 64$. ](image/boltz_2D2D_sod "fig:")]{} Conclusions {#sec:conclusions} =========== We extended projective and telescopic projective integration methods, studied in [@LafitteMelisSamaey2017; @MelisSamaey2017] for kinetic BGK-like equations, to allow for a fully explicit, high-order time simulation of both the nonlinear BGK equation and the full Boltzmann equation for pseudo-Maxwellian particles. We developed a framework of linearized operators, which revealed that the linearized BGK operator closely resembles the relaxation operator of the linear kinetic models used in [@LafitteMelisSamaey2017; @MelisSamaey2017]. As a result, the design of stable projective and telescopic projective integration methods for simulation of the nonlinear BGK equation is practically identical to that presented in [@LafitteMelisSamaey2017] and [@MelisSamaey2017], respectively. Since the spectrum of the linearized Boltzmann equation for pseudo-Maxwellian particles is known to be spread along the negative real axis, accelerated explicit time integration of the full Boltzmann equation required the construbtion of $[0,1]$-stable TPI methods. Although such methods are not completely asymptotic-preserving, the cost scales only logarithmically with the stiffness of the problem, which we consider acceptable. We designed and applied the projective methods to a number of model problems of increasing dimension and complexity, showing the potential of these schemes, with speed-ups that ranges from 8 for the most complicated Boltzmann model to more than a hundred for the BGK equation with linear relaxation rate. Because of their explicit nature and hierarchical structure, telescopic projective integration methods naturally lend themselves to adaptivity. In future work, we will explore the use of space-dependent hierarchies of projective levels, adapted to the local values of ${\varepsilon}$, to further increase computational efficiency. Free transport with WENO scheme {#sub:weno} =============================== In weighted essentially non-oscillatory (WENO) methods, introduced in [@Liu1994], the stencil is built adaptively for each finite volume cell ${\mathcal{C}}_i = [x_{i-1/2},x_{i+1/2}]$. A WENO method considers all $k$ possible stencils for cell ${\mathcal{C}}_i$ at once and computes a weighted average over the $k$ resulting reconstructed solutions. The $k$ stencils for cell ${\mathcal{C}}_i$ are given by: $$\label{eq:weno_stencils} {\mathcal{S}}_i^\ell = \{{\mathcal{C}}_{i-\ell}, \ldots, {\mathcal{C}}_{i-\ell+k-1}\} \qquad (\ell = 0, \ldots, k-1),$$ in which the left shift $\ell$ is used to iterate over the possible stencils. As in [@Shu1998], for each stencil in , the reconstructed solution on the left side of $x_{i+1/2}$ and on the right side of $x_{i-1/2}$ are calculated as: $$\label{eq:weno_reconstructed_solutions} \begin{aligned} u_{i+1/2}^\ell &= \sum_{r=0}^{k-1} c_{\ell,r}U_{i-\ell+r} \\ u_{i-1/2}^\ell &= \sum_{r=0}^{k-1} c_{\ell-1,r}U_{i-\ell+r} \end{aligned} \qquad\quad (\ell = 0, \ldots, k-1),$$ where we dropped the minus and plus superscripts in , respectively, for clarity. In what follows, we concentrate on the right boundary of ${\mathcal{C}}_i$. The left boundary is treated analogously. Using the $k$ reconstructed solutions in , we could define the following weighted average: $$\label{eq:weno_reconstruction_averaged_smooth} u_{i+1/2}^- = \sum_{\ell=0}^{k-1} d_{\ell}u_{i+1/2}^\ell,$$ for some weights $d_{\ell}$, $\ell=0, \ldots, k-1$. Equation can be seen as the reconstruction of the solution on the left side of $x_{i+1/2}$ using a stencil of $2k-1$ consecutive cells. The weights $d_{\ell}$ can be chosen such that has an order of accuracy of $2k-1$, that is: $$u_{i+1/2}^- = u(x_{i+1/2}) + O{\mathopen{}\mathclose\bgroup\originalleft}({\Delta x}^{2k-1}{\aftergroup\egroup\originalright}),$$ which holds if the solution is smooth over all cells of . In [@Arandiga2011] it is proven that the weights $d_{\ell}$ can be written in terms of binomial coefficients as: $$d_{\ell} = \dfrac{\dbinom{k-1}{k-1-\ell}\dbinom{k}{k-1-\ell}}{\dbinom{2k-1}{k}} \qquad (\ell = 0, \ldots, k-1),$$ from which it is straightforward to see that $0 < d_{\ell} \le 1$ and $\sum_{\ell=0}^{k-1} d_{\ell} = 1$ such that represents a convex combination of the $k$ reconstructed solutions. For instance, we compute: $$\label{eq:weno_dl} \begin{alignedat}{4} k &=1: &&\quad d_0 = 1&& \\ k &=2: &&\quad d_0 = \frac{2}{3}, &&\quad d_1 = \frac{1}{3} \\ k &=3: &&\quad d_0 = \frac{3}{10}, &&\quad d_1 = \frac{6}{10}, \quad d_2 = \frac{1}{10}. \end{alignedat}$$ The idea of WENO methods is to use the weights $d_{\ell}$ only for stencils over which the solution is smooth and allocate very small weights to stencils containing one or more discontinuities. If we denote these WENO weights by $\omega_{\ell}$, we write the WENO reconstruction as: $$ u_{i+1/2}^- = \sum_{\ell=0}^{k-1} \omega_{\ell}u_{i+1/2}^\ell,$$ for which we require that the weights $\omega_{\ell}$ also form a convex set, that is: $$\label{eq:weno_omegal_convex} 0 < \omega_\ell \le 1, \qquad\quad \sum_{\ell=1}^{k-1}\omega_\ell = 1.$$ In [@Liu1994] the second constraint in is guaranteed by writing $\omega_\ell$ in terms of constants $\alpha_\ell$ as: $$ \omega_\ell = \frac{\alpha_\ell}{\displaystyle\sum_{s=0}^{k-1} \alpha_s} \qquad (\ell = 0, \ldots, k-1).$$ The constants $\alpha_\ell$ are expressed in terms of the weights $d_{\ell}$, as given in , and new constants $\beta_{\ell}$, leading to: $$\label{eq:weno_alphal} \alpha_\ell = \frac{d_\ell}{(\delta + \beta_\ell)^2} \qquad (\ell = 0, \ldots, k-1),$$ in which the coefficients $\beta_\ell$ determine the smoothness of each stencil and are therefore termed the *smoothness indicators*. Since the smoothness indicators can become zero, a small constant $\delta > 0$ is added in that avoids division by zero. A typical choice is $\delta = 10^{-6}$ and the numerical experiments in [@Jiang1996] suggest that the resulting WENO method is not sensitive to the choice of $\delta$ as soon as $10^{-7} \le \delta \le 10^{-5}$. The critical step in the design of WENO methods appears to be the definition of the smoothness indicators $\beta_{\ell}$. In [@Jiang1996], the authors proposed the following smoothness indicators: $$\label{eq:weno_smoothness_indicators} \beta_\ell = \sum_{s=1}^{k-1}{\Delta x}^{2s-1} \int_{x_{i-1/2}}^{x_{i+1/2}} {\mathopen{}\mathclose\bgroup\originalleft}(\frac{d^s p_i^\ell(x)}{dx^s}{\aftergroup\egroup\originalright})^2 dx \qquad (\ell = 0, \ldots, k-1).$$ Each smoothness indicator in includes the information of all derivatives that are approximated by the reconstruction polynomial $p_i^\ell(x)$ based on stencil ${\mathcal{S}}_i^\ell$ as a measure of smoothness of the true solution in cell ${\mathcal{C}}_i$. The factor ${\Delta x}^{2s-1}$ in front of the integral ensures that $\beta_\ell$ is independent of ${\Delta x}$ when computing the derivative of $p_i^\ell(x)$ in . For instance, for $k = 2$, the smoothness indicators in are calculated as: $$\label{eq:weno_betal_k2} \begin{aligned} \beta_0 &= (U_{i} - U_{i+1})^2 \\ \beta_1 &= (U_{i-1} - U_{i})^2. \end{aligned}$$ For higher-order reconstructions, the derivatives in can be computed by using a general expression of the reconstruction polynomial. For $k=3$, after rearranging terms, we find (see [@Jiang1996; @Shu1998]): $$\label{eq:weno_betal_k3} \begin{aligned} \beta_0 &= \frac{1}{4}(3U_{i} - 4U_{i+1} + U_{i+2})^2 + \frac{13}{12}(U_{i} - 2U_{i+1} + U_{i+2})^2 \\ \beta_1 &= \frac{1}{4}(U_{i-1} - U_{i+1})^2 + \frac{13}{12}(U_{i-1} - 2U_{i} + U_{i+1})^2 \\ \beta_2 &= \frac{1}{4}(U_{i-2} - 4U_{i-1} + 3U_{i})^2 + \frac{13}{12}(U_{i-2} - 2U_{i-1} + U_{i})^2, \end{aligned}$$ see also the work of [@Henrick2005]. For orders $k=4,5,6$, the corresponding expressions were obtained in [@Balsara2000] and extended to even higher orders $k=7,8,9$ in [@Gerolymos2009]. From the expressions in and , we see that $\beta_\ell = O({\Delta x}^2)$ as soon as the solution is smooth over the stencil, while we have $\beta_\ell = O(1)$ for stencils containing a discontinuity. Using equation , this results in weights $\omega_\ell \approx d_\ell$ for smooth solutions and $\omega_\ell = O({\Delta x}^4)$ for discontinuous solutions, as desired. Evaluating the Boltzmann collision operator with a fast spectral scheme {#sub:FastSpectral} ======================================================================= The fast spectral discretization of the Boltzmann operator taken from [@MoPa:2006] employed in this work is described in this appendix. To this aim, and since the Boltzmann collision operator acts only on the velocity variables, we focus on a given spatial cell ${\mathbf{x}}_j$ at a given instant of time $t^n$. Hence, only the dependency on the velocity variable ${\mathbf{v}}$ is considered for the distribution function $f$, i.e. $f=f({\mathbf{v}})$. The first step to construct our spectral discretization is to truncate the integration domain of the Boltzmann integral . As a consequence, we suppose the distribution function $f$ to have compact support on the ball $\Ball_0(R)$ of radius $R$ centered in the origin. Since one can prove (see e.g. [@PaRuSINUM2000]) that $\supp (Q(f)(v)) \subset \Ball_0({\sqrt 2}R)$, in order to write a spectral approximation which avoids aliasing, it is sufficient that the distribution function $f({\mathbf{v}})$ is restricted on the cube $[-T,T]^{D_v}$ with $T \geq (2+{\sqrt 2})R$. Successively, one should assume $f({\mathbf{v}})=0$ on $[-T,T]^{D_v} \setminus \Ball_0(R)$ and extend $f$ to a periodic function on the set $[-T,T]^{D_v}$. Let observe that the lower bound for $T$ can be improved. For instance, the choice $T=(3+{\sqrt 2})R/2$ guarantees the absence of intersection between periods where $f$ is different from zero. However, since in practice the support of $f$ increases with time, we can just minimize the errors due to aliasing [@canuto:88] with spectral accuracy. To further simplify the notation, let us take $T=\pi$ and hence $R=\lambda\pi$ with $\lambda = 2/(3+\sqrt{2})$ in the following. We denote by $\QL_B(f)$ the Boltzmann operator with cut-off. Hereafter, using one (bold) index to denote the $D_v$-dimensional sums, we have that the approximate function $f_N$ can be represented as the truncated Fourier series by $$f_N({\mathbf{v}}) = \sum_{{\mathbf{k}}=-N/2}^{N/2} \f_{\mathbf{k}}e^{i {\mathbf{k}}\cdot {\mathbf{v}}}, \label{eq:FU}$$ where the ${\mathbf{k}}^{th}$ Fourier coefficient is given by $$\f_{\mathbf{k}}= \frac{1}{(2\pi)^{D_v}}\int_{[-\pi,\pi]^{D_v}} f({\mathbf{v}}) e^{-i {\mathbf{k}}\cdot {\mathbf{v}}}\,d{\mathbf{v}}.$$ We then obtain a spectral quadrature of our collision operator by projecting on the space of trigonometric polynomials of degree less or equal to $N$, i.e. $${\hat {{\mathcal{Q}}}}_{\mathbf{k}}=\int_{[-\pi,\pi]^{D_v}} \QL_B(f_N) \, e^{-i {\mathbf{k}}\cdot {\mathbf{v}}}\,dv, \quad {\mathbf{k}}=-N/2,\ldots,N/2. \label{eq:VAR}$$ Finally, by substituting expression in one gets after some computations $${\hat {{\mathcal{Q}}}}_{\mathbf{k}}= \sum_{\substack{{\mathbf{l}},{\mathbf{m}}=-N/2\\ {\mathbf{l}}+{\mathbf{m}}={\mathbf{k}}}}^{N/2} \f_{\mathbf{l}}\,\f_{\mathbf{m}}\, \bb({\mathbf{l}},{\mathbf{m}}),\quad {\mathbf{k}}=-N,\ldots,N, \label{eq:CF1}$$ where $\bb({\mathbf{l}},{\mathbf{m}})=\B({\mathbf{l}},{\mathbf{m}})-\B({\mathbf{m}},{\mathbf{m}})$ are given by $$\B({\mathbf{l}},{\mathbf{m}}) = \int_{\Ball_0(2\lambda\pi)}\int_{\mathbb{S}^{D_v-1}} B(|q|, \cos\theta) e^{-i({\mathbf{l}}\cdot q^++{\mathbf{m}}\cdot q^-)}\,d\omega\,dq.$$ with $$q^{+} = \frac12(q+\vert q\vert \omega), \quad q^{-} = \frac12(q-\vert q\vert \omega).$$ Let us notice that the naive evaluation of (\[eq:CF1\]) requires $O(n^2)$ operations, where $n=N^3$. This causes the spectral method to be computationally very expensive, especially in dimension three. In order to reduce the number of operations needed to evaluate the collision integral, the main idea is to use another representation of , the so-called Carleman representation [@Carl:EB:32] which is obtained by using the following identity $$\frac{1}{2} \, \int_{\mathbb{S}^{D_v-1}} F(|u|\sigma - u) \, d\sigma = \frac{1}{|u|^{d-2}} \, \int_{\mathbb{R}^{D_v}} \delta(2 \, x \cdot u + |x|^2) \, F(x) \, dx.$$ This gives in our context for the Boltzmann integral $${{\mathcal{Q}}}(f)= \int_{{\mathbb{R}}^{D_v}} \int_{{\mathbb{R}}^{D_v}} {\tilde B}(x,y) \delta(x \cdot y) {\mathopen{}\mathclose\bgroup\originalleft}[ f(v + y) \, f(v+ x) - f(v+x+y) \, f(v) {\aftergroup\egroup\originalright}] \, dx \, dy,$$ with $$\label{eq:Btilde} \tilde{B}(|x|,|y|) = 2^{D_v-1} \, \sigma{\mathopen{}\mathclose\bgroup\originalleft}(\sqrt{|x|^2+|y|^2}, \frac{|x|}{\sqrt{|x|^2+|y|^2}} {\aftergroup\egroup\originalright}) \, (|x|^2+|y|^2)^{-\frac{D_v-2}2}.$$ This transformation yields the following new spectral quadrature formula $$\label{eq:ode} \hat{{{\mathcal{Q}}}}_{\mathbf{k}}= \sum_{\underset{{\mathbf{l}}+{\mathbf{m}}={\mathbf{k}}}{{\mathbf{l}},{\mathbf{m}}=-N/2}}^{N/2} {\hat{\beta}}_F({\mathbf{l}},{\mathbf{m}}) \, \hat{f}_{\mathbf{l}}\, \hat{f}_{\mathbf{m}}, \ \ \ {\mathbf{k}}=-N,...,N$$ where ${\hat{\beta}}_F({\mathbf{l}},{\mathbf{m}})=\B_F({\mathbf{l}},{\mathbf{m}})-\B_F({\mathbf{m}},{\mathbf{m}})$ are now given by $$\B_F({\mathbf{l}},{\mathbf{m}}) = \int_{\Ball_0(R)} \int_{\Ball_0(R)} \tilde{B}(x,y) \, \delta(x \cdot y) \, e^{i ({\mathbf{l}}\cdot x+ {\mathbf{m}}\cdot y)} \, dx \, dy.$$ Now, in order to reduce the number of operation needed to evaluate , we look for a convolution structure. The aim is to approximate each ${\hat{\beta}}_F({\mathbf{l}},{\mathbf{m}})$ by a sum $${\hat{\beta}}_F({\mathbf{l}},{\mathbf{m}}) \simeq \sum_{p=1} ^{A} \alpha_p ({\mathbf{l}}) \alpha' _p ({\mathbf{m}}),$$ where $A$ represents the number of finite possible directions of collisions. This finally gives a sum of $A$ discrete convolutions and, consequently, the algorithm can be computed in $O(A \, N \log_2 N)$ operations by means of standard FFT technique [@canuto:88]. In order to get this convolution form, we make the decoupling assumption $$\tilde{B}(x,y) = a(|x|) \, b(|y|).$$ This assumption is satisfied if $\tilde B$ is constant. This is the case of Maxwellian molecules in dimension two, which is the case we shall consider for the numerical simulations of section \[sec:results\]. Indeed, using the kernel in , one has $$\tilde{B}(x,y) = 2^{D_v - 1} b_0 (|x|^2+|y|^2)^{-\frac{D_v-\alpha-2}2},$$ so that $\tilde{B}$ is constant if $D_v = 2$ and $\alpha = 0$. Here we write $x$ and $y$ in spherical coordinates $x = \rho e$ and $y = \rho' e'$ to get $$\B_F({\mathbf{l}},{\mathbf{m}}) = \frac14 \, \int_{\mathbb{S}^1} \int_{\mathbb{S}^1} \delta(e \cdot e') \, {\mathopen{}\mathclose\bgroup\originalleft}[ \int_{-R} ^R e^{i \rho ({\mathbf{l}}\cdot e)} \, d\rho {\aftergroup\egroup\originalright}] \, {\mathopen{}\mathclose\bgroup\originalleft}[ \int_{-R} ^R e^{i \rho' ({\mathbf{m}}\cdot e')} \, d\rho' {\aftergroup\egroup\originalright}] \, de \, de'.$$ Then, denoting $ \phi_R ^2 (s) = \int_{-R} ^R e^{i \rho s} \, d\rho,$ for $s \in {\mathbb{R}}$, we have the explicit formula $$\phi_R ^2 (s) = 2 \, R \,{\operatorname{Sinc}} (R s),$$ where ${\operatorname{Sinc}}(x)=\frac{\sin(x)}{x}$. This explicit formula is further plugged in the expression of $\B_F(l,m)$ and using its parity property, this yields $$\B_F ({\mathbf{l}},{\mathbf{m}}) = \int_0 ^{\pi} \phi_R ^2 ({\mathbf{l}}\cdot e_{\theta})\, \phi_R ^2 ({\mathbf{m}}\cdot e_{\theta+\pi/2}) \, d\theta.$$ Finally, a regular discretization of $N_\theta$ equally spaced points, which is spectrally accurate because of the periodicity of the function [@kurganov2007spectral], gives $$\B_F ({\mathbf{l}},{\mathbf{m}}) = \frac{\pi}{M} \, \sum_{p=1} ^{N_\theta} \alpha_p ({\mathbf{l}}) \alpha' _p ({\mathbf{m}}),$$ with $$\alpha _p ({\mathbf{l}}) = \phi_R ^2 ({\mathbf{l}}\cdot e_{\theta_p}), \hspace{0.8cm} \alpha' _p ({\mathbf{m}}) = \phi_R ^2 ({\mathbf{m}}\cdot e_{\theta_p+\pi/2})$$ where $\theta_p = \pi p/N_\theta$. [^1]: Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium ([ward.melis@cs.kuleuven.be]{}). [^2]: Laboratoire Paul Painlevé, Université de Lille, Cité Scientifique, 59655 Villeneuve d’Ascq, France([thomas.rey@math.univ-lille1.fr]{}) [^3]: Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium ([giovanni.samaey@cs.kuleuven.be]{}).

--- abstract: 'A generalized version for the Rastall theory is proposed showing the agreement with the cosmic accelerating expansion. In this regard, a coupling between geometry and the pressureless matter fields is derived which may play the role of dark energy responsible for the current accelerating expansion phase. Moreover, our study also shows that the radiation field may not be coupled to the geometry in a non-minimal way which represents that the ordinary energy-momentum conservation law is respected by the radiation source. It is also shown that the primary inflationary era may be justified by the ability of the geometry to couple to the energy-momentum source in an empty flat FRW universe. In fact, this ability is independent of the existence of the energy-momentum source and may compel the empty flat FRW universe to expand exponentially. Finally, we consider a flat FRW universe field by a spatially homogeneous scalar field evolving in potential $\mathcal{V}(\phi)$, and study the results of applying the slow-roll approximation to the system which may lead to an inflationary phase for the universe expansion.' address: | $^1$ Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha 55134-441, Iran\ $^2$ Department of Physics, Azarbaijan Shahid Madani University, Tabriz, 53714-161, Iran\ $^3$ Institut de Mathématiques et de Sciences Physiques (IMSP), Université de Porto-Novo, 01 BP 613 Porto-Novo, Bénin\ $^4$ Département de Physique, Université d’Agriculture de Kétou, BP 13 Kétou, Bénin\ $^5$ African Institute for Mathematical Sciences(AIMS), $6$ Melrose Road, Muizenberg, $7945$, South Africa author: - 'H. Moradpour$^1$[^1], Y. Heydarzade$^{2}$[^2], F. Darabi$^{2}$[^3], Ines G. Salako$^{3,4,5}$[^4]' title: A Generalization to the Rastall Theory and Cosmic Eras --- Introduction ============ The origins of the primary inflationary era [@inflation1; @inflation2; @inflation3; @inflation4], current accelerating phase of the universe expansion [@expansion1; @expansion2; @expansion3; @expansion4; @expansion5] as well as the dark matter problem [@DM1; @DM2; @DM3] are some of the big puzzles in the standard model of cosmology. Our insufficient understanding of these problems leads the coincidence and fine-tuning problems [@roos; @coin; @fine1; @fine2]. In order to solve the above mentioned problems, some authors have been introduced a new type of energy-momentum source [@Rev3; @Rev2; @Rev1; @mod]. In another approach, physicist try to solve the above problems by modifying the Einstein field equations [@lobo; @meeq; @rastall; @cmc; @cmc1; @cmc2]. In this line, one may refer to the scalar-tensor gravity [@Faraoni], vector-tensor theories [@vector], tensor-vector-scalar theories [@tvs], quadratic gravity [@quad], Chern-Simons theories [@chern1], massive gravity [@massive1; @massive2] and Gauss-Bonnet theory [@gauss], for a review see also [@LRL]. Scalar-tensor (ST) theories of gravity are the simplest alternative to Einstein’s general theory of gravity (GR) and have a long history. The first attempts are done by Jordan [@ST1; @ST10], Fierz [@ST2], and Brans-Dicke [@ST3]. These theories possess just one massless scalar field and with a constant coupling strength to matter fields. These works were generalized later to the theories in which the scalar field has a dynamic coupling to the matter fields and/or an arbitrary self-interaction in [@ST4; @ST5; @ST6] as well as to the theory with multiple scalar fields [@ST7]. In the vector-tensor theories of gravity, in addition to the metric tensor, the gravitational action is modified by adding a vector field that is non-minimally coupled to gravity. Studying these theories refer to the works by Will, Nordtvedt and Hellings [@VT1; @VT2; @VT3], see also [@VT4; @VT5]. The tensor-vector-scalar theory is proposed by Bekenstein [@Beken] where the standard Einstein tensor field of General Relativity (GR) is coupled to a vector field as well as a scalar field, hence the theory is called by this name. This theory is a relativistic version of Modified Newtonian Dynamics (MOND) [@mond] reproducing MOND in the weak field limit. The most important advantage to adopt tensor-vector-scalar theory refers to the explanation of many galactic and cosmological observations without the need for dark matter [@mond1; @mond2]. The quadratic gravity theories are based on the idea of adding appropriate quadratic terms in the Riemann and Ricci tensors or the Ricci scalar inspired by the string or quantum gravity theories [@quadrat]. Chern-�Simons gravity is the special case of the quadratic theories including only the parity-violating term $^{*}RR={{^{*}R^{\alpha}}_{\beta}}^{\gamma\delta}{R^{\beta}}_{\alpha\delta\gamma}$ in which ${{^{*}R^{\alpha}}_{\beta}}^{\gamma\delta}=\frac{1}{2}\epsilon^{\gamma\delta\rho\sigma}{R^{\alpha}}_{\beta\rho\sigma}$ [@chern2]. Massive gravity theories are new attempts which attribute a mass to the putative �graviton. The simplest work in this line and in a ghost-free manner suffers from the van Dam-Veltman-Zakharov (vDVZ) discontinuity problem [@mass1; @mass2]. Due to the three additional helicity states for the massive spin-2 graviton, the limit of small graviton mass does not coincide with the Einstein GR. As an instance, the predicted perihelion advance violates the previous observational experiments. In order to resolve the vDVZ problem, a new model was introduced by Visser by considering a non-dynamical flat background metric [@viss]. Gauss-Bonnet theory is built on adding the quadratic combination of two Riemann tensor to the Einstein-Hilbert action in which it does not increase the differential order of the resulting equations of motion [@Bonnet1; @Bonnet2]. In most of these modified theories, the energy-momentum source is described by a divergence-free tensor which couples to the geometry in a minimal way [@lobo; @meeq]. However, it is worthwhile mentioning that this property of the energy-momentum tensor, which leads to the energy-momentum conservation law, is not obeyed by the particle production process [@motiv1; @motiv2; @motiv20; @motiv3; @motiv4]. Therefore, it is not unreasonable to consider a non-divergence-free energy-momentum tensor and look for a new gravitational theory. In this regards, P. Rastall firstly considered such kind of sources and introduced a modification to the Einstein field equations [@rastall]. Also, there is another theory known as the curvature-matter theory of gravity [@cmc; @cmc1; @cmc2], in which, similar to the Rastall theory, the matter and geometry are coupled to each other in a non-minimal way meaning that the ordinary energy-momentum conservation law is not valid. However, it is important to stress that all of the potential alternatives to the general theory of relativity must be viable. This means that they must be metric theories in order to be in agreement with the Einstein equivalence principle, which is today supported by a very strong empirical evidence, and that they must pass the solar system tests [@LRL]. On the other hand, the recent starting of the gravitational wave (GW) astronomy with the event GW150914, that is the first historical detection of GWs [@GW1] could be fundamental for discriminating about various modified theories of gravity because some differences among such theories can be emphasized in the linearized theory of gravity and, in principle, can be found by GW experiments, see [@GW2; @GW3] for details. In this work, we proposed a generalized Rastall theory to show that a coupling between the geometry and matter fields helps us in providing an geometric interpretation for the dark energy and thus the current accelerating expansion phase of the universe. The main point in favor of the Rastall theory and its generalized version is that the usual conservation law on $T_{\mu\nu}$ is tested only in the flat Minkowski space-time or specifically in a gravitational weak field limit. Indeed, this theory reproduces a phenomenological way for distinguishing features of quantum effects in gravitational systems, i.e the violation of the classical conservation laws [@motiv4; @cmc; @conserv2]. Also, one may find that the condition ${T^{\mu\nu}}_{;\mu}\neq0$ is phenomenologically confirmed by the particle creation process in cosmology [@motiv1; @motiv2; @motiv20; @motiv3; @prd; @particle5; @Calogero1; @Calogero2; @Velten]. One also may refer to [@neutrast] in favor of the viability of the original Rastall theory and our proposed generalization. In this work, it is shown that the restrictions on the Rastall parameter are of the order of $\leq1\%$ with respect to the corresponding value of the general theory of relativity. In other words, the results in [@neutrast] are a confirmation that the Rastall theory and its generalization are viable theories, in the sense that the deviation of any extended theory of gravity from the standard general theory of relativity must be weak. Beside the current accelerating expansion phase of the universe, the radiation dominated era in this framework is also addressed. Moreover, we will show that the ability of the geometry to couple with the energy-momentum source may produce the primary inflationary era in our generalized version of the Rastall theory. The paper is organized as follows. After reviewing the original Rastall theory in the next section, we address a generalization to this theory in the third section. Section ($\textmd{IV}$) includes some general remarks on the constructed new theory in FRW universe. In section ($\textmd{V}$), considering a flat FRW universe filled by a pressureless matter, we show that a non-minimal coupling between the geometry and the energy-momentum source may be considered as an origin for the dark energy and thus the current accelerated phase of the universe expansion. In section ($\textmd{VI}$), the radiation dominated era in our generalization of the Rastall theory is investigated. In section ($\textmd{VII}$), we study two methods to model the primary inflationary era of the universe in our formalism. Finally, section ($\textmd{VIII}$) is devoted to the summary and concluding remarks. A brief review on the Rastall theory ==================================== Based on the Rastall theory, the ordinary energy-momentum conservation law is not always available in the curved spacetime and therefore we should have [@rastall] $$\begin{aligned} \label{rastal} T^{\mu \nu}_{\ \ ;\mu}=\lambda^{\prime} R^{;\nu},\end{aligned}$$ where $R$ and $\lambda^{\prime}$ are the Ricci scalar of the spacetime and the Rastall constant parameter, respectively. In fact, $\lambda^{\prime}$ is a measure of the tendency of the geometry (matter fields) to couple with the matter fields (geometry) leading to the changes into the matter fields (geometry). This equation leads to the following field equations $$\begin{aligned} \label{r1} G_{\mu \nu}+\kappa^{\prime}\lambda^{\prime} g_{\mu \nu}R=\kappa^{\prime} T_{\mu \nu},\end{aligned}$$ which can finally be rewritten as $$\label{ein} G_{\mu \nu}=\kappa^{\prime} S_{\mu\nu},$$ where $\kappa^{\prime}$ is the Rastall gravitational coupling constant and $S_{\mu\nu}$ is the effective energy-momentum tensor defined as $$\label{senergy} S_{\mu\nu}=T_{\mu\nu}-\frac{\kappa^{\prime}\lambda^{\prime} T}{4\kappa^{\prime}\lambda^{\prime}-1}g_{\mu\nu}.$$ In fact, in this theory the matter fields and geometry are coupled to each other in a non-minimal way [@rastall; @cmc; @cmc1; @cmc2] and its compatibility with some observational data have firstly been shown by Al-Rawaf and Taha [@al1; @al2]. Moreover, since the particle production process during the cosmos evolution does not respect the energy-momentum conservation law [@motiv1; @motiv2; @motiv20; @motiv3; @motiv4], the Rastall theory may be considered as a classical background formulation for this phenomena [@prd]. Finally, we should mention that Eq. (\[r1\]) implies that $R(4\kappa^{\prime}\lambda^{\prime}-1)=\kappa^{\prime} T$ where $T$ is the trace of energy-momentum tensor. Therefore, because $\lambda^{\prime}$ is constant and the $R(4\kappa^{\prime}\lambda^{\prime}-1)=\kappa^{\prime} T$ condition applies to all spacetimes and energy-momentum sources, the $\kappa^{\prime}\lambda^{\prime}=\frac{1}{4}$ case is not allowed [@rastall]. More studies on the various aspects of this theory can be found in [@smal; @neutrast; @obs1; @obs2; @rastbr; @rastsc; @cosmos3; @rascos1; @rasch; @more1; @more2; @more3; @more4; @hm; @msal]. The generalized Rastall theory with varying Rastall parameter ============================================================= Basically, Rastall assumed that for all of the spacetimes and energy-momentum sources, the ratio of the flow of the energy-momentum tensor ($T^{\nu\mu}_{\ \ \ ;\mu}$) to the Ricci scalar divergence ($R^{;\nu}$) is constant ($\lambda^{\prime}$). It means that the evolutions of energy-momentum source and also the geometry do not affect this ratio. As an example, consider the matter dominated era in the universe history. The energy density of matter decreases during the universe expansion, but the mentioned ratio is a constant parameter in Rastall theory [@al1; @al2] meaning that the coupling between energy-momentum source and geometry is constant, and is not affected by the evolution of the cosmic system. In fact, it is a very restricting condition to assume the evolution of system does not affect the mutual coupling. In addition, since the mutual coupling is a constant parameter in Rastall gravity, it did not continuously change during the universe evolution [@al1; @al2; @prd]. Indeed, since the cosmic evolution is a continues process [@roos], it is a reasonable expectation that the mutual coupling between the energy-momentum sources and the geometry should be varying gradually and smoothly. Therefore, at least theoretically, it is not prohibited to generalize the Rastall theory as $$\begin{aligned} \label{gr0} T^{\mu\nu}_{\ \ \ ;\mu}=(\lambda R)^{;\nu},\end{aligned}$$ leading to $$\begin{aligned} \label{bian} (T_{\mu \nu}-g_{\mu\nu}\lambda R)^{;\nu}=0.\end{aligned}$$ Now, regarding the Bianchi identity, i.e $G_{\mu\nu}^{\ \ \ ;\nu}=0$, we obtain $$\begin{aligned} G_{\mu\nu}=\kappa(T_{\mu \nu}-\lambda g_{\mu\nu}R),\end{aligned}$$ where $\kappa$ is a constant and finally, we reach at $$\begin{aligned} \label{gr} G_{\mu\nu}+\kappa\lambda g_{\mu\nu}R=\kappa T_{\mu \nu}.\end{aligned}$$ Although this result looks like to the field equations of the original Rastall theory (\[r1\]), here, $\lambda$ is not generally constant. Just the same as $\lambda^{\prime}$ in the Rastall theory, $\lambda$ is a measure for the strongness of the coupling between the geometry to the matter fields. As it is apparent, the Einstein field equations are recovered in the appropriate limit of $\lambda=0$, a limit in which the matter fields and geometry are coupled to each other in a minimal way. FRW metric and general remarks on the mutual non-minimal coupling between the geometry and matter fields ======================================================================================================== The line element of the FRW universe is written as $$\begin{aligned} \label{rw} ds^2=-dt^2+a(t)^2[\frac{dr^2}{1-kr^2}+r^2(d\theta^2+sin(\theta)^2d\phi^2)],\end{aligned}$$ where $a(t)$ is the scale factor and $k=-1,0,1$ is the curvature parameter corresponding to the open, flat and closed universes, respectively. If the universe is filled by an energy-momentum source with $T^\mu_\nu=diag(-\rho,p,p,p)$ in which $\rho$ and $p$ are the energy density and pressure of the cosmic fluid, respectively, then using Eq. (\[gr\]), the Friedmann equations in a flat FRW universe are given as $$\label{fr1} (12\kappa\lambda-3)H^2+6\kappa\lambda \dot{H}=-\kappa\rho,$$ and $$\label{fr2} (12\kappa\lambda-3)H^2+(6\kappa\lambda-2) \dot{H}=\kappa p.$$ Here, $H=\frac{\dot{a}}{a}$ denotes the Hubble parameter, and the dot sign indicates the derivative with respect to the cosmic time $t$. In this manner, from Eq. (\[gr0\]), one easily obtains $$\begin{aligned} \label{emc} \frac{d(\rho+\lambda R)}{dt}+3H(\rho+p)=0,\end{aligned}$$ meaning that the $\lambda R$ term is the energy density corresponding to the ability of geometry to couple with the energy-momentum sources in a non-minimal way ($\lambda\neq0$). It is worthwhile mentioning here that for an empty spacetime where $\rho=p=0$, we should have $\frac{d(\lambda R)}{dt}=0$. In addition, Eq. (\[emc\]) can also be rewritten as $$\begin{aligned} \label{emc1} \rho+\rho_g=-\int3H(\rho+p)dt,\end{aligned}$$ where $\rho_g\equiv\lambda R$. It is obvious that, in the absence of the ability of the geometry to couple with the energy-momentum sources in a non-minimal way ($\lambda=0$), the usual energy-momentum conservation law and the Einstein field equations can be recovered through the equations (\[gr\]) and (\[emc\]). In the following sections, we study the role of the non-minimal coupling between geometry and energy-momentum sources in the various expansion phases of the flat FRW universe. Matter dominated era and an accelerating universe ================================================= Consider a flat FRW universe with the scale factor $a$ filled by the pressureless dust matter fields. Using the equation (\[gr\]), one obtains the Friedmann equations as $$\label{friedman1} (12\kappa\lambda-3)H^2+6\kappa\lambda \dot{H}=-\kappa\rho_m,$$ and $$\label{friedman2} (12\kappa\lambda-3)H^2+(6\kappa\lambda-2) \dot{H}=0,$$ where $\rho_m$ denotes the energy density. It is clear that, at the $\lambda\rightarrow0$ limit, the equations (\[friedman1\]) and (\[friedman2\]) reduce to those of the matter dominated era in the standard cosmology [@roos]. In addition, Eq. (\[gr\]) leads to $R=-\frac{\kappa}{4\kappa\lambda-1}\rho_m$ for a dust source requiring that we should have $\kappa\lambda\neq\frac{1}{4}$ for $\rho_m\neq0$ in agreement with the Rastall’s original hypothesis [@rastall]. For the deceleration parameter, defined as $q=-1-\frac{\dot{H}}{H^2}$ [@roos], one can use Eq. (\[friedman2\]) to obtain $$\begin{aligned} \label{dece1} q(z)=1+\frac{1}{6\kappa\lambda(z)-2},\end{aligned}$$ where $z$ denotes the redshift. It is obvious that the deceleration parameter of the matter dominated era in the Einstein regime ($q=\frac{1}{2}$) can be covered in the appropriate limit of $\lambda=0$. For a flat FRW universe filled by a pressureless matter, the continuity equation can be written as $$\begin{aligned} \label{cont1} \dot{\rho}_m+3H\rho_m=\frac{d}{dt}(\frac{\kappa\lambda}{4\kappa\lambda-1}\rho_m).\end{aligned}$$ If the pressureless source does not interact with geometry, then this equation is decomposed into the following equations $$\begin{aligned} \label{cont12} \dot{\rho}_m+3H\rho_m&=&0,\nonumber\\ \frac{d}{dt}(\frac{\kappa\lambda}{4\kappa\lambda-1}\rho_m)&=&0,\end{aligned}$$ meaning that the ordinary energy-momentum conservation law is valid. Therefore, $\lambda=0$ is a simple solution to the $\frac{d}{dt}(\frac{\kappa\lambda}{4\kappa\lambda-1}\rho_m)=0$ equation leading to the ordinary Einstein field equations. Now, for a non-interacting universe, it is easy to check that equation (\[cont12\]) (or equally Eq. (\[cont1\])) admits the following solution $$\begin{aligned} \label{mym} &&\rho_m=\rho_0 a^{-3},\nonumber\\ &&\lambda(a)=\frac{1}{4\kappa+\kappa C \rho_m}=\frac{1}{4\kappa+\kappa\alpha a^{-3}},\end{aligned}$$ where $\rho_0$ and $C$ are integration constants and thus $\alpha=C\rho_0$ is a constant. It is obvious that we have $\lambda=\frac{1}{4\kappa}$ in the absence of dust source, i.e for $\rho_m=0$. Here, we only considered a simple situation in which there is no energy exchange between the geometry and matter source. In this case, the existence of matter source only affects the ability and tendency of geometry to couple with energy source, and it does not lead to an energy exchange between the geometry and matter source, and thus a palpable mutual interaction between them. By the palpable interaction, we mean an interaction leading to a visible and measurable energy exchange between the components of system. Therefore, it seems that the non-minimal coupling between geometry and the matter source has some indirect, complex and non-local aspects hidden until now, a result in line with some previous works claiming that the probable non-local features of mutual relation between geometry and the energy sources may be considered as an origin for the dark sectors of the cosmos [@mash; @mash1; @mash2]. It is also useful to note that, even in the simplest case of (\[cont12\]), the properties of geometry, including its curvature and $\lambda$, are determined by the energy sources filling it. This is in agreement with the general relativity backbone, where the curvature of the geometry (as its property) is specified by the energy sources filling that. In a more realistic case, they may exchange energy with each other, and therefore, one cannot always decompose Eq. (\[cont1\]) into Eq. (\[cont12\]). Now, inserting equations (\[my\]) into (\[friedman1\]) and (\[friedman2\]), respectively, and combining the results with each other, one reaches $$\begin{aligned} H(a)=H_0\sqrt{\frac{a^3+\alpha}{a^3}},\end{aligned}$$ where $H_0=\frac{\kappa\rho_{0}}{3\alpha}$ is a constant. This equation indicates that for the limits of $a^3\ll\alpha$, we have $H(a)\approx H_0\sqrt{\frac{\alpha}{a^3}}$ leading to $a(t)=a_{0m}t^{\frac{2}{3}}$ with the integration constant $a_{0m}=(\frac{9}{4}H^{2}_{0}\alpha)^{1/3}$, which exactly is the scale factor of the matter dominated era in the standard model of cosmology. Moreover, for the limit of $a\gg1$, we have $H(a)\rightarrow H_0$ leading to $a(t)=a_0\exp(H_0t)$ for the scale factor of the current accelerating expansion phase of the universe, in which $a_0$ is a constant. Now, combining $1+z=a^{-1}$ with $\lambda(a)=\frac{1}{4\kappa+\kappa\alpha a^{-3}}$ and inserting the result into Eq. (\[dece1\]), we obtain $$\begin{aligned} \label{dece2} q(z)=\frac{\alpha(1+z)^3-2}{2(1+\alpha(1+z)^3)}.\end{aligned}$$ In order to describe the evolution of the universe from the matter dominated era to the current accelerating phase, the deceleration parameter $q(z)$ should satisfy the three conditions as ($i$) $q(z\rightarrow\infty)\rightarrow\frac{1}{2}$, ($ii$) $q(z\approx0.6)\rightarrow0$ and ($iii$) $q(z\rightarrow0)\leq-\frac{1}{2}$ [@roos]. Using the equation (\[dece2\]), one can verify that the deceleration parameter of the matter dominated era of the standard cosmology ($q=\frac{1}{2}$) is obtainable in the $\lambda\rightarrow0$ limit or equivalently in the $\alpha\rightarrow\infty$ limit. Moreover, at high redshift limit ($z\rightarrow\infty$) and independent of the $\alpha$ parameter, we have $q\rightarrow\frac{1}{2}$ which again addresses the matter dominated era. Therefore, the change of the pressureless matter density in our model is the same as that of the standard cosmology, i.e $\rho_m=\rho_0 a^{-3}$. Additionally, although the deceleration parameter in our model differs from that of the matter dominated era of the standard cosmology, this era is covered at the appropriate limit of $z\rightarrow\infty$ in our model. Here, from equation (\[dece2\]), for $-1<\alpha\leq\frac{1}{2}$, we have $q(z=0)\leq-\frac{1}{2}$ which demonstrates the satisfaction of the third condition. In addition, since there is no divergence in the history of the evolution of the universe from the early matter dominated era to its current phase, $q(z)$ should not diverge which requires that its denominator should not vanish for any non-negative amount of $z$. This requires to have $0\leq\alpha\leq\frac{1}{2}$ which consequently leads to the total restricting range on the deceleration parameter as $-1\leq q(z=0)\leq-\frac{1}{2}$. For example, consider the case of $q(z=0)=-0.55$ [@ref] which through the equation (\[dece2\]) corresponds to $\alpha=\frac{3}{7}$. Considering this value of the $\alpha$ parameter, one can find that the $q=0$ case is associated to the redshift $z\simeq0.67$ when the universe leaves its decelerating phase and enters to the accelerating phase. This result is in agreement with some observational evidences [@ref1; @ref2; @ref3]. The deceleration parameter $q(z)$ is plotted in Fig. (\[fig\]) versus the redshift $z$ for some values of the $\alpha$ parameter. It is seen from the figure that for the small redshifts, representing the late time in the history of the universe, the deceleration parameter goes to negative values representing an accelerated expanding phase in our constructed model. As a result, a non-minimal coupling between the geometry and pressureless matter, which mainly consists of dark matter, may lead to a description for the dark energy, and therefore the current accelerating phase of the universe expansion.\ ![Deceleration parameter $q$ versus the redshift $z$ for some values of $\alpha$.[]{data-label="fig"}](fig.eps) Based on the above results, this mutual relation between geometry and the matter source suggests that this source and its enclosing cosmic horizon may achieve the thermodynamic equilibrium, a result which is in agreement with the recent study by Mimoso et al, focusing on the properties and criterions of a thermodynamic equilibrium between the cosmic horizon and the cosmic fluids in various cosmic eras [@pavn]. Radiation dominated era and the curvature-radiation non-minimal coupling ======================================================================== For the flat FRW universe filled by a radiation source, the Friedmann equations are as follow $$\label{friedman11} (12\kappa\lambda-3)H^2+6\kappa\lambda \dot{H}=-\kappa\rho_r,$$ and $$\label{friedman21} (12\kappa\lambda-3)H^2+(6\kappa\lambda-2) \dot{H}= \frac{1}{3}\kappa\rho_r,$$ where $\rho_r$ is the energy density. Because the energy-momentum associated to radiation fields is a traceless source, i.e $T=0$, by contracting equation (\[gr\]), one finds $R(4\kappa\lambda-1)=\kappa T$ which clearly for $\kappa\lambda\neq\frac{1}{4}$ results in a null Ricci scalar for a radiation source, i.e $R=0$. Some simple calculations for the continuity equation and deceleration parameter also lead to $$\begin{aligned} \label{cont11} \dot{\rho}_r&+&4H\rho_r =0,\\ \nonumber \rho_r&=&\rho_{0r}a^{-4},\end{aligned}$$ where $\rho_{0r}$ is the integration constant, and $$\begin{aligned} \label{dece11} q(z)=1,\end{aligned}$$ respectively. In order to obtain the last equation, we combined Eqs. (\[friedman11\]) and (\[friedman21\]) with each other to get $\frac{\dot{H}}{H^2}=-2$, a result which leads to $a=a_0t^{\frac{1}{2}}$ for the scale factor where $a_0$ is the integration constant, in agreement with the radiation dominated era of the standard cosmology [@roos]. Based on Eqs. (\[cont11\]) and (\[dece11\]), the density changes of the radiation source and the deceleration parameter of the radiation dominated era are the same as those of the standard cosmology meaning that the radiation dominated era in our model is the same as that of the standard cosmology. Indeed, since $R=0$ in the radiation dominated era, independent of the value of $\lambda$ parameter we have $(\lambda R)^{;\nu}=0$ meaning that the above results are independent of $\lambda$ parameter. Now, we use the $\rho_m\rightarrow0$ limit of the $\lambda(a)$ relation obtained in Eq. (\[mym\]), in order to find the value of $\lambda$ which leads to $\lambda=\frac{1}{4\kappa}$. It means that since $\lambda$ is a constant quantity, the geometry and radiation do not affect each other. Indeed, since radiation is a traceless source, simple calculations lead to $$\begin{aligned} \label{contn} \dot{\rho}_r+4H\rho_r+\dot{\rho}_m+3H\rho_m=\frac{d}{dt}(\frac{\kappa\lambda}{4\kappa\lambda-1}\rho_m),\end{aligned}$$ for the continuity equation in a universe filled by both radiation and dust. In the absence of any interaction between radiation, dust and geometry, this equation is decomposed to Eqs. (\[cont1\]) and (\[cont11\]) meaning that the $\rho_r=\rho_{0r}a^{-4}$, $\rho_m=\rho_0 a^{-3}$ and $\lambda(\rho_m)=\frac{1}{4\kappa+\kappa C \rho_m}$ solutions are also available in this case. Therefore, for $\rho_m=0$, we have $\lambda=\frac{1}{4\kappa}$ meaning that the $\lambda=\frac{1}{4\kappa}$ case is allowed in the radiation case. Inserting $\lambda=\frac{1}{4\kappa}$ into either Eqs. (\[friedman11\]) or (\[friedman21\]) and combining the result with (\[cont11\]), we again reach at $\frac{\dot{H}}{H^2}=-2$ leading to $a=a_0t^{\frac{1}{2}}$ and thus $R=0$. Here, we should mention that since we have $R=T=0$ in this era, the $R(4\kappa\lambda-1)=\kappa T$ condition is available independent of the value of $\lambda$ parameter. Indeed, unlike the Rastall theory, where the $\lambda^{\prime}=\frac{1}{4\kappa}$ case is not allowed [@rastall], here, the $\lambda=\frac{1}{4\kappa}$ case can be allowed. Therefore, although the geometry generally has the ability to couple with the energy-momentum source in a non-minimal way ($\lambda=constant\neq0$), since $\lambda$ is constant, geometry and the radiation source do not affect each other. This means that the ordinary energy-momentum conservation law is respected by the radiation source as seen in (\[cont11\]). Finally, we should mention that due to the fact that the radiation source does not coupled to the geometry in a non-minimal way, there is no energy flux between the geometry and radiation fields. This may be considered as the reason for the failure to achieve the thermodynamic equilibrium between the cosmic horizon and the radiation fields [@pavn]. $\lambda$ and the primary inflationary era ========================================== In this section, we address two methods to model the primary inflationary era in our formalism, and also study the role and behavior of $\lambda$ in these methods. $\lambda$ as the generator of the primary inflationary phase ------------------------------------------------------------ Now, let us consider an empty flat FRW universe with its describing equations as $$\label{friedman110} (12\kappa\lambda-3)H^2+6\kappa\lambda \dot{H}=0,$$ and $$\label{friedman210} (12\kappa\lambda-3)H^2+(6\kappa\lambda-2) \dot{H}=0.$$ It is easy to check that both the above equations are true only for $\lambda=\frac{1}{4\kappa}=constant$ and $\dot{H}=0$. It is worthwhile mentioning that, as a desired result, the $\lambda=\frac{1}{4\kappa}=constant$ solution is in full agreement with the $\rho_m\rightarrow0$ limit of the results obtained in Eqs. (\[mym\]) and (\[contn\]). Besides, since the spacetime is empty ($T_{\mu\nu}=0$), we should have $\frac{d(\lambda R)}{dt}=0$ meaning that $\lambda=\frac{\rho_g}{R}$, where $\rho_g\equiv\psi$ is a constant. In addition, using the above equations, one can obtain that the $\lambda=\frac{1}{4\kappa}=constant$ and $\dot{H}=0$ conditions lead to an exponential growth in the scale factor, i.e $a(t)=a_0\exp(H_0t)$ where $a_0$ and $H_0$ are the integration constants, with the non-vanishing Ricci scalar $R=12H_0^2$, respectively. Now, combining the above results with each other, we reach at $H_0=\sqrt{\frac{\kappa\psi}{3}}$. It is also obvious that, since $\lambda$ and $\rho_g$ are constant, Eqs. (\[gr0\]) and (\[emc\]) are met here, and therefore, $\psi$ is nothing but the integration constant in the RHS of Eq. (\[emc1\]). Indeed, we should remind that, since Eq. (\[emc1\]) is the result of Eq. (\[gr0\]) and thus Eq. (\[emc\]), the fulfillment of Eq. (\[gr0\]) (or equally (\[emc\])) is necessary and sufficient. On the other hand, from Eq. (\[gr\]), we know that $R(4\kappa\lambda-1)=T$ which its right hand side vanishes due to the emptiness of the spacetime. Then, since the Ricci scalar does not vanish, i.e $R\neq0$, we find out that we should have $\lambda=\frac{1}{4\kappa}$. This is in agreement with the previous mentioned results obtained from solving Eqs. (\[friedman110\]) and (\[friedman210\]), and also applying the $\rho_m\rightarrow0$ limit to Eq. (\[mym\]). Once again, we see that unlike the original Rastall theory, the case of $\lambda=\frac{1}{4\kappa}$ may be allowed in this new formulation of the Rastall theory. Therefore, the inflationary era may be supported in this model by a unique feature of the geometry which is the ability of the geometry to couple with the energy-momentum sources in a non-minimal way in agreement with this fact that $\lambda=\textmd{constant}\neq0$. In fact, the empty flat FRW spacetime is forced to expand exponentially by this ability. We should note that the absence of the energy-momentum source does not mean that the geometry has not the ability of coupling to the energy-momentum sources. Indeed, in this case, the absence of an energy-momentum source only means that the geometry does not couple to anything. It is also worthwhile to mention that since $T=0$ and $\lambda$ is constant in both the radiation dominated and the primary inflationary phases, the obtained results about these eras may be generalizable to the original Rastall theory. Energy extraction during the inflationary era {#energy-extraction-during-the-inflationary-era .unnumbered} --------------------------------------------- We saw that the ability and tendency of geometry to couple with the energy sources, in the non-minimal way, does not disappear, i.e $\lambda\neq0$, in the absence of an energy-momentum source. In fact, this is a property of geometry which enforces the empty FRW spacetime to expand exponentially. Moreover, from Eq. (\[emc\]), we found that the $\lambda R=\rho_g(\equiv\psi)$ term behaves as an energy density. Here, $\psi$ is the energy density associated with the non-minimal coupling $\lambda$, and therefore, we get $E=\int\psi dV=\frac{4\pi}{3}\psi a(t)^3V_0$ for the total energy of co-moving volume $V_0$ corresponding to this coupling at any given time $t$. Finally, for the amount of the energy of the co-moving volume $V_0$ specified from spacetime at time $t+\delta t$, due its intrinsic property to couple with the energy-momentum sources in the non-minimal way, we have $$\begin{aligned} E(t+\delta t)=\frac{4\pi}{3}\psi a(t+dt)^3V_0\simeq\frac{4\pi}{3}\psi a(t)^3V_0\exp(3\sqrt{\frac{\kappa\psi}{3}}\delta t),\end{aligned}$$ meaning that the released energy grows exponentially. Therefore, in our formalism, the ability and tendency of geometry to couple with the energy-momentum sources enforces universe to expand, and in fact, it is the backbone of the universe expansion and the energy production in the primary inflationary phase. Thus, this ability may also help us to provide a unified mechanism explaining the primary inflationary era as well as the current accelerating phase of the universe expansion. Standard inflation and $\lambda$ -------------------------------- In the previous subsection, we found out that, even in the absence of an inflaton field, the tendency of geometry to couple with the energy-momentum sources may lead to an inflationary phase for the universe expansion, and consequently, the slow-rolling parameters do not appear in that scenario. It is useful to mention that there are also some inflationary models in which the slow-roll condition does not appear [@g1; @g2; @g3]. Here, we will show that the standard inflation scenario by implementing an inflaton field can also be valid in our formalism. In order to achieve this goal, we consider a spatially homogeneous scalar field evolving in potential $\mathcal{V}(\phi)$. Therefore, simple calculation yields $\rho_\phi=\frac{1}{2}\dot{\phi}^2+\mathcal{V}(\phi)$ and $p_\phi=\frac{1}{2}\dot{\phi}^2-\mathcal{V}(\phi)$ for the energy density and pressure of the inflaton field [@roos]. Now, the Friedmann equations in a flat FRW universe filled by the mentioned field are written as $$\begin{aligned} \label{fr01} (12\kappa\lambda-3)H^2+6\kappa\lambda \dot{H}&=&-\kappa\rho_\phi,\nonumber\\ (12\kappa\lambda-3)H^2+(6\kappa\lambda-2) \dot{H}&=&\kappa p_\phi,\end{aligned}$$ which finally lead to $$\begin{aligned} \label{fr001} \dot{H}&=&-\frac{\kappa}{2}[\rho_\phi+p_\phi],\end{aligned}$$ for the Raychaudhuri equation. In addition, the same as the matter dominated era, considering a simple situation in which there is no energy exchange between the energy-momentum source and geometry, we reach at $\dot{\rho}_\phi+3H(\rho_\phi+p_\phi)=-\frac{d(\lambda R)}{dt}=0$ leading to $$\begin{aligned} \label{my} &&\ddot{\phi}+\frac{\partial \mathcal{V}}{\partial\phi}+3H\dot{\phi}=0,\nonumber\\ &&\lambda=\frac{1}{4\kappa[1+\gamma^{-1}(3p_\phi-\rho_\phi)]},\end{aligned}$$ for the continuity equation in which $\gamma^{-1}$ is constant. Now, if $\ddot{\phi}$ is negligible and $\dot{\phi}^2\ll\mathcal{V}(\phi)$, then $p_\phi\simeq-\rho_\phi$ and from Eqs. (\[fr001\]) and (\[my\]), we find that $\dot{H}\simeq0$ and $\lambda(\phi)\simeq\frac{1}{4\kappa[1-4\gamma^{-1}\mathcal{V}(\phi)]}$, respectively. In fact, when $\ddot{\phi}$ is negligible, Eq. (\[my\]) helps us in getting $\mathcal{V}^{\prime\prime}\simeq\frac{3\kappa}{2}\dot{\phi}^2$ leading to $\eta\equiv\frac{2}{3\kappa}(\frac{\mathcal{V}^{\prime\prime}}{\mathcal{V}})\simeq\frac{\dot{\phi}^2}{\mathcal{V}}$. Therefore, during the inflation process, when the slow-roll approximation is valid, we have $\eta\ll1$ in agreement with the standard inflation hypothesis [@roos]. In this manner, inserting Eq. (\[my\]) into Eq. (\[fr01\]), one can easily reach at $H^2\simeq\frac{\kappa}{3}[4\mathcal{V}-\gamma]$ recovering the standard inflation results at the appropriate limit of $\lambda\rightarrow0$ (or equally $\gamma\rightarrow0$). Moreover, since $q=-1-\frac{\dot{H}}{H^2}\simeq-1$ at the time of inflation, we should have $\epsilon\equiv-\frac{\dot{H}}{H^2}\ll1$ [@roos]. Now, using Eqs. (\[fr001\]) and (\[my\]), one obtains $\epsilon\simeq\frac{8}{\kappa}[\frac{\mathcal{V}^\prime}{4\mathcal{V}-\gamma}]^2$, where the prime sign stands for the derivative with respect to $\phi$. It is interesting to note that if we define $\tilde{V}(\phi)\equiv4\mathcal{V}-\gamma$, then we have $H^2\simeq\frac{\kappa}{3}\tilde{V}$ and $\epsilon\simeq\frac{8}{\kappa}[\frac{\mathcal{V}^\prime}{4\mathcal{V}-\gamma}]^2=\frac{1}{2\kappa}(\frac{\frac{\partial\tilde{V}}{\partial\phi}}{\tilde{V}})^2$ similar to those of the standard inflation scenario [@roos]. Therefore, if the slow-roll approximation is valid, then a spatially homogeneous scalar field evolving in potential $\mathcal{V}(\phi)$ can support the primary inflationary era in our formalism whenever the inflaton field (or equally $\mathcal{V}(\phi)$) satisfies the $H^2\simeq constant>0$, $\epsilon<1$ and $\eta\ll1$ conditions. It is finally worth to mention that approaching the end of inflation, where $\mathcal{V}(\phi)\rightarrow0$, we have $\lambda\rightarrow\frac{1}{4\kappa}$ revealing the consistency with our results in previous sections about the radiation dominated era. Summary and Concluding Remarks ============================== After referring to the Rastall theory, we addressed a generalization of this theory and studied some of its cosmological consequences. Based on our results, a non-minimal coupling between the geometry and a pressureless matter field may lead to a transition from the matter dominated era to the current accelerating phase, in agreement with some previous observations [@ref1; @ref2; @ref3]. We only focused on the $T^{\mu \nu}_{\ \ ;\mu}=0=-\frac{d(\lambda R)}{dt}$ solutions. In this case, a dust source, which satisfies the ordinary energy-momentum conservation law, is allowed, and as we have seen, the evolution of its energy density is the same as that of the standard cosmology. It should also be noted that although the same as the general relativity $T^{\mu \nu}_{\ \ ;\mu}=0$ in our model, since $\lambda\neq0$, Friedmann equations in our model differ from those of the standard cosmology. In addition, we found out that, in our formalism, the evolution of the energy density in the radiation dominated era is the same as that of the standard cosmology. Indeed, we found that, during the radiation dominated era, $\lambda$ remains a non-zero constant quantity meaning that the evolution of the radiation source as well as the geometry do not affect the value of $\lambda$. Finally, we considered an empty flat FRW universe and realized that, even in the absence of an inflaton field, a primary inflationary era can be driven in this generalized version of Rastall theory when $\lambda=\frac{1}{4\kappa}$. Therefore, our study shows that the ability and tendency of geometry to couple with the energy-momentum sources ($\lambda\neq0$) may be the backbone of the primary inflationary era and the current accelerating phases of the universe expansion in a unified picture. Also, a scenario for a universe filled by an inflaton field in the context of the Rastall theory has been introduced. In this context, as the matter dominated era, we have only focused on simple case of $T^{\mu \nu}_{\ \ ;\mu}=0=-\frac{d(\lambda R)}{dt}$ meaning that there is no energy exchange between geometry and the cosmic fluid. Once again, we should remind that since $\lambda\neq0$, the Friedmann equations in our model differ from those of the general relativity. It is obtained that if the inflaton field meets the usual slow-roll conditions, then it can support an inflationary phase. Acknowledgment {#acknowledgment .unnumbered} ============== We are so grateful to the respected referees for their valuable comments. The work of H. Moradpour has been supported financially by Research Institute for Astronomy & Astrophysics of Maragha (RIAAM). [99]{} A. H. Guth, Phys. Rev. D 23, 347 � (1981). A. Linde, Phys. Lett. B 108, 6, 389 (1982). A. Linde, Phys. Lett. B 259, 1, 38 (1991). R.H. Brandenberger, Astrophys. Space Sc. L. 247, 169 (2000). A. G. Riess et all, Astron. J. 116, 1009 (1998). A. G. Riess et all, Astrophys. J. 560, 49 (2001). J. Frieman, M. Turner, D. Huterer, Ann. Rev. Astron. Astrophys. 46, 385 (2008). L. Perivolaropoulos, arXiv:astro-ph/0601014. Eric V. Linder, Phys. Rev. Lett. 90, 091301 (2003). D. Clowe et al, ApJ 648, L109 (2006). J. F. Navarro, C. S. Frenk, S. D.M. White, Astrophys. J. 490, 493 (1997). V. Sahni, Lect. Notes Phys. 653, 141 (2004). M. Roos, *Introduction to Cosmology,* John Wiley and Sons, UK (2003). I. Zlatev, Limin Wang, P. J. Steinhardt, Phys. Rev. Lett.82, 896 (1999). N. Arkani-Hamed and S. Dimopoulos, JHEP 06, 073 (2005). G. Dvali, Q. Shafi, R. Schaefer, Phys. Rev. Lett. 73, 1886 (1994). S. Nojiri and S. D. Odintsov, Phys. Lett. B 639, 144 (2006). M. Li, X. D. Li, S. Wang and Y. Wang, Commun. Theor. Phys. 56, 525 (2011). K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov. Astrophys. Space Sci. 342, 155 (2012). S. Nojiri and S. D. Odintsov, Phys. Rept. 505, (2011) 59. F. S. N. Lobo, Journal of Physics: Conference Series 600, 012006 (2015). S. Capozziello and V. Faraoni, *Beyond Einstein Gravity* (Springer, NY, 2011). P. Rastall, Phys. Rev. D 6, 3357 (1972) T. Koivisto, Class. Quant. Grav. 23, 4289 (2006). O. Bertolami, C. G. Boehmer, T. Harko and F. S. N. Lobo, Phys. Rev. D 75, 104016 (2007). T. Harko and F. S. N. Lobo, Galaxies, 2, 410 (2014). V. Faraoni [*Cosmology in Scalar�Tensor Gravity*]{}, Dordrecht: Kluwer Academic (2004). J. B. Jim�nez and A. L. Maroto, Phys. Rev. D 80, 063512 (2009). C. Skordis, Class.Quant.Grav.26, 143001 (2009). R. Schimming and H.-J. Schmidt, NTM Schriftenr. Gesch. Naturw. Tech. Med. 27, 41 (1990), gr-qc/0412038. S. Alexander and N. Yunes, Phys. Rep., 480, 1�55 (2009). C. de Rham, Living Rev. Relativity, forthcoming (2014). K. Hinterbichler, � Rev. Mod. Phys. 84, 671�710 (2012). F. Muller-Hoissen, Nucl. Phys. B349, 235 (1990). C. M. Will, Living Rev. Relativity, 17, 4, (2014). P. Jordan, Nature 164, 637 (1949). P. Jordan, Z. Phys. 157, 112 (1959). M. Fierz, Helv. Phys. Acta 29, 128 (1956). C. Brans and R. H. Dicke, Phys. Rev. 124, 925 (1961). P. G. Bergmann, Int. J. Theor. Phys. 1, 25 (1968). K. Nordtvedt, Astrophys. J. 161, 1059 (1970). R. V. Wagoner, Phys. Rev. D 1, 3209 (1970). T. Damour and Esposito-Far‘ese, Class. Quantum Grav. 9, 2093 (1992). C. M. Will and K. Nordtvedt Jr., Astrophys. J. 177, 757 (1972). K. Nordtvedt Jr. and C. M. Will, Astrophys. J. 177, 775 (1972). R. W. Hellings and K. Nordtvedt Jr., Phys. Rev. D7, 3593 (1973). J. Beltr�an Jim�enez and A.L. Maroto JCAP 0902: 025 (2009). C. Armendariz-Picon and A. Diez-Tejedor, JCAP 0912, 018 (2009). J.D. Bekenstein, Phys. Rev. D 70, 083509 (2004). M. Milgrom, Astrophys. J. 270, 365 (1983). N.E. Mavromatos, M. Sakellariadou, and M.F. Yusaf, Phys. Rev. D 79, 081301 (2009). D.-M. Chen, H. Zhao, Astrophys.J. 650, L9-L12 (2006). Alvarez-Gaume, L., Kehagias, A., Kounnas, C., L�st, D. and Riotto, A. (2016), Aspects of quadratic gravity. Fortschr. Phys., 64: 176�189. R. Jackiw and S.-Y. Pi, Phys. Rev. D, 68, 104012 (2003). H. van Dam, H. and M. J. G. Veltman, Nucl. Phys. B, 22, 397�411 (1970). V. I. Zakharov, JETP Lett., 12, 312 (1970). M. Visser, Gen. Relativ. Gravit., 30, 1717�1728 (1998). D. G. Boulware and S. Deser, Phys. Rev. Lett. 55, 2656 (1985). S. Nojiri, S. D. Odintsov, M. Sasaki, Phys. Rev. D 71, 123509 (2005). N. D. Birrell, P. C. W. Davies, [*Quantum Fieldsin Curved Space*]{} (Cambridge University Press, Cambridge, 1982). G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15, 2738 (1977). L. Parker, Phys. Rev. D 3, 346 (1971). L. Parker, Phys. Rev. D[**3**]{}, 2546 (1971). L. H. Ford, Phys. Rev. D 35, 2955 (1987). B. P. Abbott et al, Phys. Rev. Lett. 116, 061102 (2016). C. Corda, Int. J. Mod. Phys. D, 18, 2275 (2009). C. Corda, arXiv:1004.1293. O. Minazzoli, Phys. Rev. D 88, 027506 (2013). C. E. M. Batista, M. H. Daouda, J. C. Fabris, O. F. Piattella, D. C. Rodrigues, Phys. Rev. D 85, 084008 (2012) S. H. Pereira, C. H. G. Bessa and J. A. S. Lima, Phys. Lett. B 690, 103 (2010). S. Calogero, J. Cosm. Astrop. Phys 11, 016 (2011). S. Calogero and H. Velten, J. Cosm. Astrop. Phys 11, 025 (2013). H. Velten and S. Calogero, arXiv:1407.4306. A. M. Oliveira, H. E. S. Velten, J. C. Fabris, L. Casarini, Phys. Rev. D 92, 044020 (2015) A. S. Al-Rawaf, M. O. Taha, Phys. Lett. B 366, 69 (1996) A. S. Al-Rawaf, M. O. Taha, Gen. Rel. Grav. 28, 935 (1996) L. L. Smalley, Il Nuovo Cimento B, 80, 1, 42 (1984) C. E. M. Batista, J. C. Fabris, O. F. Piattella, A. M. Velasquez-Toribio, Eur. Phys. J. C 73, 2425 (2013) J. C. Fabris, O. F. Piattella, D. C. Rodrigues, M. H. Daouda, arXiv:1403.5669v1 T. R. P. Caramês, M. H. Daouda, J. C. Fabris, A. M. Oliveira, O. F. Piattella, V. Strokov, Eur. Phys. J. C 74, 3145 (2014) T. Caramês, J. C. Fabris, O. F. Piattella, V. Strokov, M. H. Daouda, A. M. Oliveira, arXiv:1503.04882v1 M. Capone, V. F. Cardone, M. L. Ruggiero, Journal of Physics: Conference Series, 222, 012012 (2010) J. C. Fabris, O. F. Piattella, D. C. Rodrigues, C. E. M. Batista, M. H. Daouda, Int. J. Mod. Phys. Conf. Ser. 18, 67 (2012) J. P. Campos, J. C. Fabris, R. Perez, O. F. Piattella, H. Velten, Eur. Phys. J. C 73, 2357 (2013) J. C. Fabris, M. H. Daouda, O. F. Piattella, Phys. Lett. B 711, 232 (2012) J. C. Fabris, arXiv:1208.4649v1 I. G. Salako, A. Jawad, Astrophys. Space. Sci. 46, 359 (2015) G. F. Silva, O. F. Piattella, J. C. Fabris, L. Casarini, T. O. Barbosa, Gravitation and Cosmology. 19, 156 (2013) H. Moradpour, Phys. Lett. B 757, 187 (2016) H. Moradpour, I. G. Salako, AHEP, 2016, 3492796 (2016) F. W. Hehla, B. Mashhoonb, Phys. Lett. B 673, 279 (2009) G. Cusin, S. Foffa, M. Maggiore, M. Mancarella, Phys. Rev. D 93, 043006 (2016) F. Briscese, M. L. Pucheu, Int. J. Mod. Phys. D 14, 1750019 (2017) D. D. Reid, D. W. Kittell, E. E. Arsznov, G. B. Thompson, arXiv:astro-ph/0209504v2 R. A. Daly et al., Astrophys. J. 677, 1 (2008) E. Komatsu et al. \[WMAP Collaboration\], Astrophys. J. Suppl. 192, 18 (2011) V. Salvatelli, A. Marchini, L. L. Honorez and O. Mena, Phys. Rev. D 88, 023531 (2013) J. P. Mimoso, D. Pavón, Phys. Rev. D 94, 103507 (2016) A. Linde, Zh. Eksp. Teor. Fiz. 38, 149 (1983) A. H. Guth, Phys. Rept. 333, 555 (2000) A. H. Guth (2000), astro-ph/0002188 [^1]: Corresponding Author: h.moradpour@riaam.ac.ir [^2]: heydarzade@azaruniv.edu [^3]: f.darabi@azaruniv.edu [^4]: inessalako@gmail.com

--- abstract: 'We discuss the BCS-BEC crossover for one-dimensional spin $1/2$ fermions at zero temperature using the Boson-Fermion resonance model in one dimension. We show that in the limit of a broad resonance, this model is equivalent to an exactly solvable single channel model, the so-called modified Gaudin-Yang model. We argue that the one-dimensional crossover may be realized either via the combination of a Feshbach resonance and a confinement induced resonance or using direct photo-association in a two-component Fermi gas with effectively one-dimensional dynamics. In both cases, the system may be driven from a BCS-like state through a molecular Tonks-Girardeau gas close to resonance to a weakly interacting Bose gas of dimers.' author: - 'A. Recati$^{1,2}$, J.N. Fuchs$^{1,3}$, and W. Zwerger$^{1}$' title: 'Boson-Fermion Resonance Model in One Dimension' --- Introduction ============ In this article, we consider the problem of attractive fermions in one dimension (1D), having in mind current experiments on ultra-cold two-component Fermi gases of atoms [@Regal; @crossoverexp]. In these systems, the $s$-wave interaction between fermions in different internal states can be tuned using a Feshbach resonance. By changing the interaction from weakly attractive to weakly repulsive via a resonance where the interaction diverges, one can explore the crossover from a BCS superfluid, when the attraction is weak and pairing only appears in momentum space, to a Bose-Einstein condensate (BEC) of molecular dimers [@crossoverth]. Experiments are currently investigating gases that are in a three dimensional regime (3D). A different situation occurs if the gas is confined in a very anisotropic cigar-shaped trap, like, e.g., in an atomic wire created with optical lattices [@Esslinger] or on an atom chip [@Reichel]. If the transverse confinement is strong enough, the system effectively becomes 1D, i.e., the radial degrees of freedom are frozen. We will refer to such a situation as quasi-1D. In this case, at zero temperature, a crossover takes place between a BCS-like state and a weakly interacting Bose gas of dimers. This crossover can be described by an exactly solvable model (the so-called modified Gaudin-Yang model) [@FRZ; @Tokatly], which is just a combination of the Gaudin-Yang model for attractive fermions [@GY] and of the Lieb-Liniger model for repulsive dimers [@LL]. Despite the fact, that there is no genuine off-diagonal long range order in 1D even at zero temperature, we refer to this situation as a one dimensional version of the BCS-BEC crossover. Such a crossover can be realized in two rather different ways using a two-component Fermi gas in a quasi-1D situation. They correspond to - fermions whose 3D-scattering length exhibits a Feshbach resonance (FBR). In this case the combination of the 3D FBR and the confinement in the transverse direction, charcterized by a trapping frequency $\omega_{\perp}/2\pi$, leads to a confinement induced (CI) resonance [@Olshanii], beyond which the two particle bound state energy is large enough to neglect breaking of dimers (this scenario has been discussed in [@FRZ; @Tokatly]). - Fermions which are transferred directly into a bound molecular state by an external laser field. The photo-association process can be described by an effective 1D Boson-Fermion resonance model (BFRM) [@FL; @RRE; @RM]. For positive detuning of the laser this describes a system of attractively interacting fermions while for negative detuning, one obtains again unbreakable dimers for strog enough laser coupling. The purpose of the present paper is to study the 1D BFRM, ii), at zero temperature. It will be shown that the resonance, which is reached by quite different means in both cases, quite generally allows driving a BCS-BEC crossover in 1D. In particular we will find that in the BFRM the molecular size on resonance $r_{\star}$ plays a role similar to that of the transverse oscillator length $a_\perp\equiv \sqrt{\hbar/m\omega_\perp}$ in the quasi-1D single channel model, i). In the limit of low density $n$, characterized either by $na_\perp \ll 1$ or by $nr_{\star}\ll 1$ respectively, the resonance is broad and both models are completely equivalent to the exactly solvable modified Gaudin-Yang model discussed in ref. [@FRZ; @Tokatly]. The paper is organized as follows: in Sec. II we introduce the model and the notations; Sec. III discusses the two-body problem, i.e., bound state and scattering properties; the many-body problem is addressed in Sec. IV using a functional integral approach; and in Sec. V we discuss the results. Boson-Fermion resonance model ============================= The Boson-Fermion resonance model [@FL; @RM] is characterized by the following (grand-canonical) Hamiltonian operator $$\begin{aligned} &\hat{H}'&=\hat{H}-\mu \hat{N}=\int dx \bigg( \sum_{\sigma={\uparrow,\downarrow}} \hat{\psi}_{\sigma}^{\dagger} \Big[-\frac{\hbar^2}{2m}\partial_x^2-\mu\Big]\hat{\psi}_{\sigma} \nonumber \\ &+&\hat{\psi}_{B}^{\dagger}\Big[-\frac{\hbar^2}{4m}\partial_x^2-2\mu+\nu\Big] \hat{\psi}_{B} + g \Big( \hat{\psi}_{B}^{\dagger}\hat{\psi}_{\uparrow} \hat{\psi}_{\downarrow}+h.c.\Big)\bigg)\nonumber\\ \left. \right. \label{BFM}\end{aligned}$$ where $\hat{\psi}_{\sigma}(x)$ (resp. $\hat{\psi}_B(x)$) are fermionic (resp. bosonic) field operators describing atoms (resp. the bound state in the closed channel, i.e., bare dimers), $\sigma$ identifies the spin projection $\uparrow$ or $\downarrow$,corresponding to the two components in the Fermi gas, $\mu$ is a Lagrange multiplier to be later identified with the chemical potential, $m$ (resp. $2m$) is the mass of the atoms (resp. of the bare dimers), $\nu$ is the detuning in energy of one bare dimer with respect to two atoms and $g$ is the coupling constant for the conversion of two atoms into a bare dimer and vice-versa. The model Eq. (\[BFM\]) can describe photo-association of molecules in a 1D geometry. In this context, the coupling constant $g$ is determined by the matrix element of the dipole energy, i.e., the effective Rabi frequency, and a Franck-Condon-factor, arising from the overlap of the wave functions of atoms and molecules. We assume the molecular size to be much smaller than the oscillator length. Moreover we neglect the background scattering between fermions, i.e., we do not include terms of the form $g_1^{bg}\hat{\psi}_{\uparrow}^{\dagger} \hat{\psi}_{\downarrow}^{\dagger}\hat{\psi}_{\downarrow} \hat{\psi}_{\uparrow}$ in the Hamiltonian. This is justified in any case close enough to resonance, i.e., where $\nu\sim 0$ (see also Eq. (\[g1\]). The operator measuring the total number of atoms (i.e. unbound atoms and atoms bound into bare dimers) is: $$\begin{aligned} \hat{N}=\int dx \bigg( \sum_{\sigma={\uparrow,\downarrow}} \hat{\psi}_{\sigma}^{\dagger}\hat{\psi}_{\sigma} + 2\hat{\psi}_{B}^{\dagger}\hat{\psi}_{B}\bigg).\end{aligned}$$ We consider the zero temperature behavior of a system made of $N/2$ atoms with spin $\uparrow$ and $N/2$ atoms with spin $\downarrow$ confined on a ring of length $L$ and use the parameter $\mu$ to insure that $\langle \hat{N} \rangle =N$. The thermodynamic limit is taken by letting $N\to +\infty$ while maintaining the density $n\equiv N/L$ fixed. From now on, we set $\hbar=1$. We note that the form of the local conversion term in Hamiltonian (\[BFM\]) is fixed by the Pauli principle in a two-component Fermi system. By contrast, for bosonic atoms, infinitely many local conversion terms $g_l (\hat{\psi}^{\dagger})^{l}\hat{\psi}_B^{(l)}$ with $l=2,3,4,..$ are possible and have to be considered. In order to understand the relevance of such terms, including also a possible background interaction between atoms, we performed a perturbative renormalization group (RG) analysis of the bosonized version of this model, i.e., an atomic Bose Luttinger liquid converting into molecular Bose Luttinger liquids. We find that if the background interaction is weak, essentially all conversion terms are relevant. This implies that it is impossible to describe 1D bosonic atoms close to resonance in terms of only a few parameters. Two-body problem: scattering amplitude and bound state ====================================================== In this section, we compute the molecular propagator in presence of only two atoms ($N=2$). From it, we obtain the scattering amplitude between two atoms and the dressed (i.e. renormalized) rest energy of a dimer. The latter corresponds to the energy of a two-atoms bound state. In momentum-energy space, the molecular propagator is given by $$\begin{aligned} D(k,\omega)=D_0(k,\omega)+D_0(k,\omega)\Pi(k,\omega)D(k,\omega) \label{dysonvacuum}\end{aligned}$$ where $D_0$ is the bare molecular propagator $$\begin{aligned} D_0(k,\omega)=\bigg[\omega-\frac{k^2}{4m}+2\mu-\nu+i0^+\bigg]^{-1} \label{D0}\end{aligned}$$ and the “polarization”, i.e., self-energy of the closed channel propagator, $\Pi(k,\omega)$ is given by: $$\begin{aligned} \Pi(k,\omega)=g^2 \int \frac{dk'}{2\pi} \Big[\omega-k'^2/m-k^2/4m+2\mu+i0^+\Big]^{-1}. \label{pi2}\end{aligned}$$ From Eq. (\[dysonvacuum\]), we can compute the dressed rest energy $\epsilon_b$ of a dimer, which is defined as being the $k=0$ pole of the molecular propagator when $\mu=0$: $$\begin{aligned} D(0,\epsilon_b)^{-1}=D_0(0,\epsilon_b)^{-1}-\Pi(0,\epsilon_b)=0. \label{formalBS}\end{aligned}$$ We find that Eq. (\[formalBS\]) admits a unique real negative solution $|\epsilon_b|=-\epsilon_b$ irrespective of the sign of the detuning $\nu$ $$\frac{|\epsilon_b|}{|\epsilon_{\star}|}- \sqrt{\frac{|\epsilon_{\star}|}{|\epsilon_b|}}+ \frac{\nu}{|\epsilon_{\star}|}=0, \label{bse}$$ where we have introduced the on-resonance ($\nu=0$) bound state energy $\epsilon_{\star}\equiv \epsilon_b(\nu=0)=-m^{1/3}g^{4/3}/2^{2/3}$. This has to be compared with the 3D BFRM where a bound state is present only when the detuning is negative [@footnote1]. According to standard scattering theory, the $T$-matrix is given by $T=g^2D$ (see, e.g., [@DS]). Therefore, in the BFRM, the Lippmann-Schwinger equation for atoms is equivalent to the closed channel Dyson equation for the molecular propagator in vacuum, equation (\[dysonvacuum\]). From the latter it is possible to show that the scattering between two atoms can be described as resulting from an effective contact potential $g_1\delta(x)$, which is a well defined 1D potential, with a bare scattering amplitude $$\begin{aligned} g_1\equiv g^2 D_0(0,0)=-\frac{g^2}{\nu}. \label{g1}\end{aligned}$$ When the detuning goes to zero, the bare scattering amplitude diverges: this corresponds to the resonance. Before resonance, we have $\nu>0$ and an attractive effective interaction $g_1<0$ between the atoms, while after resonance $\nu<0$ and the effective interaction is repulsive $g_1>0$. Solving the Lippmann-Schwinger equation for the on-shell $T$-matrix of the contact potential $$\begin{aligned} T(k',k,\Omega)&=&g_1+i\int \frac{dk''}{2\pi}\frac{d\omega''}{2\pi} g_1 T(k',k'',\Omega)\nonumber \\ &\times& \Big[(\Omega/2+\omega''-k''^2/2m+i0^+)\nonumber \\ &\times& (\Omega/2-\omega''-k''^2/2m+i0^+)\Big]^{-1}\end{aligned}$$ in the limit $k'=k \to 0$ and $\Omega=k^2/m$, we obtain the low-energy limit of the one-dimensional two-body $T$-matrix: $$\begin{aligned} T_k=g^2D(0,k^2/m)\simeq \frac{g_1}{1+img_1/2k}.\end{aligned}$$ The associated dressed scattering amplitude $$\begin{aligned} f(k)=\frac{m}{2ik}T_k \simeq -\frac{1}{1+ika_1}\end{aligned}$$ has the standard form for 1D low energy scattering with $a_1\equiv -2/mg_1$ the 1D scattering length. It is a well-known fact that the 1D delta-potential forbids transmission at low scattering energy, i.e., $f(k) \to -1$ in the $k\to 0$ limit [@Olshanii]. Before studying the many-body problem, we would like to discuss briefly the behavior of the bound state energy. We define the size of the bound state as $r_b\equiv (m|\epsilon_b|/2)^{-1/2}$, which is finite for any detuning, and call $r_{\star}\equiv r_b(\nu=0)$ the size of the bound state on resonance. We find it useful also to define a dimensionless detuning $\delta\equiv \nu/\epsilon_{\star}\sqrt{2}$. With these definitions, equation (\[g1\]) becomes $g_1=2/mr_{\star}\delta$. In the BCS limit (i.e. when $\delta \to -\infty$), the bound state energy can be written as $$\begin{aligned} \epsilon_b\simeq -mg^2/4\nu^2=-mg_1^2/4\end{aligned}$$ which agrees with the bound state energy of the $g_1\delta(x)$ potential when $g_1<0$. In the opposite limit (BEC limit, i.e. when $\delta \to +\infty$), the bound state energy is equal to the detuning $$\begin{aligned} \epsilon_b\simeq \nu\end{aligned}$$ and thus completely independent of the coupling constant $g$. ![Two-body bound state energy $\epsilon_b$ \[in units of the bound state energy on resonance $|\epsilon_{\star}|$\] as a function of the dimensionless detuning $\delta$ (full line). The dashed line corresponds to the asymptotic behavior $\epsilon_b \simeq -mg_1^2/4$ and the dotted line to the asymptotic behavior $\epsilon_b \simeq \nu$.](BFRMboundstate.eps "fig:"){height="6cm"} \[BFRMboundstate\] The bound state energy $\epsilon_b$ is plotted as a function of the dimensionless detuning $\delta$ in Figure 1. The behavior of the bound state in the 1D BFRM is qualitatively similar to that of the confinement induced bound state found by Bergeman, Moore and Olshanii [@BMO] for two atoms trapped in a quasi-1D geometry (i.e., a waveguide with radial frequency $\omega_{\perp}/2\pi$). This fact reveals the connection, at the two-body level, between the 1D BFRM and the quasi-1D single channel model. In Figure 2 we have plotted the confinement induced (CI) bound state as a function of $\delta'$ (see below). In the quasi-1D case, the role of the dimensionless detuning $\delta$ is played by the parameter $\delta'\equiv a_{\perp}/a-A$, where $a_{\perp}\equiv (m\omega_{\perp})^{-1/2}$ is the radial oscillator length, $a$ is the 3D scattering length and $A\equiv -\zeta(1/2,1)/\sqrt{2}\simeq 1.0326$ [@footnote2]. In the quasi-1D geometry, the 1D scattering amplitude shows a CI resonance [@Olshanii] and is given by $$\begin{aligned} g_1'\equiv 2\omega_{\perp}a(1-Aa/a_{\perp})^{-1}=2/ma_{\perp}\delta'\end{aligned}$$ which is similar to $g_1=2/mr_{\star} \delta$, showing that $a_{\perp}$ plays the role of $r_{\star}$. The CI bound state energy $\epsilon_b'$ obeys the following equation $$\begin{aligned} \sqrt{2}a_{\perp}/a+\zeta(1/2,\epsilon_b'/\epsilon_{\star}')=0\end{aligned}$$ where $\zeta(1/2,x)$ is a particular Hurwitz zeta function [@BMO] and $\epsilon_{\star}'\equiv \epsilon_b'(\delta'=0)=-2/ma_{\perp}^2$ is the CI bound state energy on resonance. When $\delta' \to -\infty$, $\epsilon_b'\simeq -mg_1'^2/4$, in complete analogy with $\epsilon_b\simeq -mg_1^2/4$. On resonance $\delta'=0$, $\epsilon_{\star}'=-2/ma_{\perp}^2$, to be compared with $\epsilon_{\star}=-2/mr_{\star}^2$. After resonance when $\delta' \to +\infty$, the CI bound state energy behaves as $\epsilon_b'\simeq -1/ma^2$, which translates into $\epsilon_b'/\epsilon_{\star}'\simeq (\delta'+A)^2/2\simeq \delta^{'2}/2$ in terms of the parameter $\delta'$ introduced above. Similarly, in the BFRM, the bound state energy decreases monotonically with the behavior $\epsilon_b/\epsilon_{\star} \simeq \sqrt{2}\delta$ as a function of the dimensionless detuning $\delta$. ![Confinement induced bound state energy $\epsilon_b'$ \[in units of the bound state energy on resonance $|\epsilon_{\star}'|$\] as a function of the dimensionless parameter $\delta'$ (full line). The dashed line corresponds to the asymptotic behavior $\epsilon_b' \simeq -mg_1^{'2}/4$ and the dotted line to the asymptotic behavior $\epsilon_b' \simeq -1/ma^2$.](BMOboundstate.eps "fig:"){height="6cm"} \[BMOboundstate\] Many-body problem ================= The grand partition function at temperature $T\equiv 1/\beta$ and chemical potential $\mu$ can be written as a path integral $$\begin{aligned} Z=\int \mathcal{D}(\bar{\psi}_{\sigma},\psi_{\sigma}) \mathcal{D}(\bar{\psi}_B,\psi_B) e^{-S} \label{Z}\end{aligned}$$ over Grassmann fields $\bar{\psi}_{\sigma}$, $\psi_{\sigma}$ with $\sigma=\uparrow,\downarrow$ and complex fields $\bar{\psi}_B$, $\psi_B$ [@AS]. The action corresponding to the Hamiltonian (\[BFM\]) is: $$\begin{aligned} &S&=\int_0^{\beta}d\tau \int dx \bigg( \sum_{\sigma={\uparrow,\downarrow}} \bar{\psi}_{\sigma} \Big[\partial_{\tau}-\frac{\partial_x^2}{2m}-\mu\Big]\psi_{\sigma} \nonumber \\ &+&\bar{\psi}_{B}\Big[\partial_{\tau}-\frac{\partial_x^2}{4m}-2\mu+\nu\Big] \psi_{B} + g \Big( \bar{\psi}_{B}\psi_{\uparrow} \psi_{\downarrow}+c.c.\Big)\bigg).\nonumber\\ \left. \right. \label{originalaction}\end{aligned}$$ The average total number of atoms is obtained from $$\begin{aligned} \langle N \rangle = \frac{\partial F}{\partial \mu}\end{aligned}$$ where $F\equiv-T \ln Z$ is the grand potential and we are interested in the $T\to 0$ limit. Let us define the Fermi momentum $k_F\equiv \pi n/2$ and the Fermi energy $\epsilon_F\equiv k_F^2/2m$ for an ideal gas of $N$ spin $1/2$ fermions. We shall study the particular case where the (modulus of the) energy of the two body bound state on resonance is much larger than the Fermi energy. This corresponds to the limit of a broad resonance (or strong coupling limit): $$\begin{aligned} |\epsilon_{\star}|\gg \epsilon_F \Leftrightarrow n r_{\star} \ll 1 \Leftrightarrow g\sqrt{n}\gg \epsilon_F \label{BRLC}\end{aligned}$$ The above inequality shows that the broad resonance limit corresponds to having a deeply bound state after resonance, i.e., that the dimers are unbreakable after resonance. We will show that in this limit, the system is described by a single channel model of atoms (fermions) only, before resonance, and of dimers (bosons) only, after resonance. In other words, the 1D BFRM in the broad resonance limit is equivalent to the modified Gaudin-Yang model. We mention that equation (\[BRLC\]) is similar to the usual criterion for a broad 3D Feshbach resonance [@BRL; @footnote3]. Before resonance: integrating out the bare dimers ------------------------------------------------- Before resonance $\nu>0$, it is possible to integrate out the bosonic fields and describe the system in terms of an effective action for fermions only. In order to show this, we need to define the Fourier transform of a field $\psi(x,\tau)$: $$\begin{aligned} \psi(k,\tilde{\omega})&\equiv& \int d\tau dx e^{i\tilde{\omega}\tau-ikx}\psi(x,\tau)\\ \psi(x,\tau)&\equiv& \int \frac{dk}{2\pi} \frac{d\tilde{\omega}}{2\pi} e^{-i\tilde{\omega}\tau+ikx}\psi(k,\tilde{\omega})\nonumber\\ &=&\int_{k,\tilde{\omega}} e^{-i\tilde{\omega}\tau+ikx}\psi(k,\tilde{\omega})\end{aligned}$$ When going to real time $t=-i\tau$, the analytic continuation on frequencies is performed as $i\tilde{\omega}\to \omega +i0^+$. Performing the Gaussian integration on $\bar{\psi}_B$ and $\psi_B$ in equation (\[Z\]) leads to: $$\begin{aligned} Z=Z_B^0 Z_F^{eff}=Z_B^0 \int \mathcal{D}(\bar{\psi}_{\sigma},\psi_{\sigma}) e^{-S_F^{eff}} \label{Z2}\end{aligned}$$ where the effective action for fermions is $$\begin{aligned} S_F^{eff}&=&\int_{k,\tilde{\omega}} \sum_{\sigma}\bar{\psi}_{\sigma}(k,\tilde{\omega}) \Big[-i\tilde{\omega}+\frac{k^2}{2m}-\mu\Big]\psi_{\sigma}(k,\tilde{\omega})\nonumber \\ &+&g^2 \int_{k,\tilde{\omega}} \int_{k',\tilde{\omega}'} \int_{K,\tilde{\Omega}} \bar{\psi}_{\uparrow}(k,\tilde{\omega})\bar{\psi}_{\downarrow}(K-k,\tilde{\Omega}-\tilde{\omega}) \nonumber \\ &\times& \psi_{\downarrow}(k',\tilde{\omega}')\psi_{\uparrow}(K-k',\tilde{\Omega}-\tilde{\omega}') \nonumber\\ &\times& \Big[i\tilde{\Omega}-\frac{K^2}{4m}+2\mu-\nu \Big]^{-1}\end{aligned}$$ and the grand potential for an ideal Bose gas of bare dimers is: $$\begin{aligned} F_B^0&\equiv& -T\ln Z_B^0\nonumber \\ &=& TL \int \frac{dK}{2\pi} \ln \left[1-e^{-\beta(K^2/4m-2\mu+\nu)}\right].\end{aligned}$$ Due to the fact that only quadratic terms in $\psi_B$ appear in the original model, the previous result is exact. The resulting effective interaction between the atoms, however, is non-local both in space and time. If we restrict ourselves to the case $\nu >|\epsilon_{\star}|$, together with the broad resonance requirement $|\epsilon_{\star}|\gg \epsilon_F$, we can simplify the effective action $S_F^{eff}$ to one which is local. Indeed, before resonance, $2|\mu|\simeq |\epsilon_b|<|\epsilon_{\star}|$ (see Appendix A) and as an order of magnitude, $|i\tilde{\Omega}|\sim|K^2/4m|\sim \epsilon_F$. Therefore, the detuning dominates the denominator of the molecular propagator $i\tilde{\Omega}-\frac{K^2}{4m}+2\mu-\nu \simeq -\nu$, and the effective interaction between fermions becomes $$\begin{aligned} -\frac{g^2}{\nu} \int_{k_1,\tilde{\omega}_1} \int_{k_2,\tilde{\omega}_2} \int_{k_3,\tilde{\omega}_3} \bar{\psi}_{\uparrow}(1+2-3) \bar{\psi}_{\downarrow}(3)\psi_{\downarrow}(2)\psi_{\uparrow}(1)\nonumber\\ \left. \right.\end{aligned}$$ where $(1)$ is a short notation for $(k_1,\tilde{\omega}_1)$ and similarly for the other arguments. The total number of atoms can be computed from the partition function (\[Z2\]): $$\begin{aligned} \langle N \rangle = -T\frac{\partial \ln Z_B^0}{\partial \mu} -T\frac{\partial \ln Z_F^{eff}}{\partial \mu}.\end{aligned}$$ The first term is given by the usual expression for an ideal Bose gas $$\begin{aligned} \langle N_B^0 \rangle=2L \int \frac{dK}{2\pi} \Big[ e^{\beta \left(K^2/4m - (2\mu-\nu)\right)}-1\Big]^{-1}\end{aligned}$$ with $2\mu-\nu<0$. When $T\to 0$, the fraction of atoms that are bound into bare dimers is: $$\begin{aligned} \frac{\langle N_B^0 \rangle}{N}&\simeq& \frac{2}{n} e^{-\beta(\nu-2\mu)} \int \frac{dK}{2\pi} e^{-\beta K^2/4m}\nonumber \\ &=&\frac{2}{n}\sqrt{\frac{m}{\pi \beta}}e^{-\beta(\nu-2\mu)}\to 0.\end{aligned}$$ Therefore $$\begin{aligned} N=\langle N \rangle \simeq -T\frac{\partial \ln Z_F^{eff}}{\partial \mu}\end{aligned}$$ which shows that $\mu$ is the chemical potential for the gas of atoms only. In conclusion, before resonance and under the assumptions that the resonance is broad and that $\nu> |\epsilon_{\star}|$, the system is described by a single channel model of fermions with an action $$\begin{aligned} S_F^{eff}&=&\int_0^{\beta}d\tau \int dx \bigg( \sum_{\sigma={\uparrow,\downarrow}} \bar{\psi}_{\sigma} \Big[\partial_{\tau}-\frac{\partial_x^2}{2m}-\mu_F\Big]\psi_{\sigma} \nonumber \\ &+&g_1 \bar{\psi}_{\uparrow}\bar{\psi}_{\downarrow}\psi_{\downarrow}\psi_{\uparrow} \bigg)\end{aligned}$$ where $\mu_F=\mu$ and $g_1=-g^2/\nu<0$. This is the action corresponding to the Gaudin-Yang model of 1D fermions interacting via an attractive delta potential [@GY]. The single dimensionless coupling constant is $\gamma\equiv mg_1/n$. In order to describe the BCS-BEC crossover, we will use the parameter $1/\gamma$ (see Appendix B), the BCS limit corresponding to $1/\gamma \to -\infty$ or $\nu \to +\infty$. Due to the condition $\nu>|\epsilon_{\star}|$, before resonance, the parameter $1/\gamma$ is restricted to: $$\begin{aligned} -\infty < \frac{1}{\gamma} < -\frac{n |\epsilon_{\star}|}{mg^2}\sim -n r_{\star}.\end{aligned}$$ In the broad resonance limit, $nr_{\star} \to 0$ implying that apart from a vanishingly small region close to resonance ($\nu=0$ or $1/\gamma=0$), the Boson-Fermion resonance model, before resonance, is equivalent to the single channel attractive Gaudin-Yang model. After resonance: integrating out the atoms ------------------------------------------ After resonance ($\nu<0$), it is possible to integrate out the fermionic fields and to describe the system in terms of an effective action for dimers only. Formally, this is equivalent to the standard technique used to study the single channel model in 2D or 3D (see e.g. [@DZ; @GT; @Randeria93; @PS]), where [*via*]{} a Hubbard-Stratonovich transformation, it is possible to write the fermionic action in terms of a Bose-field only, which is eventually identified with the order parameter of the superconducting phase. However, it is important to emphasize that the resulting bosonic field, in that context is different from the field $\psi_B$ appearing in the BFRM defined by the action (\[originalaction\]). Performing the Gaussian integral over fermionic fields [@AS], one obtains $$\begin{aligned} Z=Z_F^0 Z_B^{eff}=Z_F^0 \int \mathcal{D}(\bar{\psi}_B, \psi_B) e^{-S_B^{eff}} \label{Z3}\end{aligned}$$ where the effective action for bosons is $$\begin{aligned} S_B^{eff}&=&\int_{k,\tilde{\omega}} \bar{\psi}_{B}(k,\tilde{\omega}) \Big[-i\tilde{\omega}+\frac{k^2}{4m}-2\mu+\nu\Big]\psi_{B}(k,\tilde{\omega}) \nonumber \\ &-&\ln \det (G_0G^{-1}) \label{sbeff}\end{aligned}$$ and $Z_F^0$ is the partition function for free fermions. We also defined the propagator for non interacting fermions, which in the Nambu representation reads $$\begin{aligned} G_0(k,\tilde{\omega};k',\tilde{\omega}) =\frac{(2\pi)^2\delta(k-k')\delta(\tilde{\omega}-\tilde{\omega}')}{ -i\tilde{\omega}+k^2/2m-\mu} \left(\begin{array}{cc} 1&0\\ 0&-1\\ \end{array}\right)\end{aligned}$$ and the full propagator $G$ is related to $G_0$ by $$\begin{aligned} G^{-1}\equiv G_0^{-1}+\Delta\end{aligned}$$ where $\Delta$ is given by: $$\begin{aligned} &&\Delta(k,\tilde{\omega};k',\tilde{\omega}) \equiv \nonumber\\ &-&g\left(\begin{array}{cc} 0&\psi_B(k+k',\tilde{\omega}+\tilde{\omega}')\\ \bar{\psi}_B(k+k',\tilde{\omega}+\tilde{\omega}')&0\\ \end{array}\right).\end{aligned}$$ The last term in equation (\[sbeff\]) can be expanded into a sum: $$\begin{aligned} -\ln \det(G_0G^{-1}) =\sum_{l=1}^{\infty}\frac{\text{Tr}\left[(G_0\Delta)^{2l}\right]}{2l}. \label{suml}\end{aligned}$$ Below, we will evaluate explicitely the $l=1$ and $l=2$ terms in the sum and, following [@GT; @PS], we will give an order of magnitude estimate for the higher order terms showing that they are negligible in the broad resonance limit. **$l=1$ term**: $$\begin{aligned} \frac{1}{2}\text{Tr}\left[(G_0\Delta)^{2}\right]= \int_{k,\tilde{\omega}}\bar{\psi}_B(k,\tilde{\omega})\psi_B(k,\tilde{\omega}) \Pi(k,\tilde{\omega})\end{aligned}$$ After resonance ($\nu<0$), we have $2|\mu|\simeq |\epsilon_b|> |\epsilon_{\star}|\gg \epsilon_F$, as discussed in Appendix A. As an order of magnitude $|i\tilde{\omega}|$ gives a contribution of the order of the kinetic energy $\sim|k^2/4m|\sim \epsilon_F$. As a result, we obtain $$\begin{aligned} \Pi(k,\tilde{\omega})\simeq \Pi(0,0)\simeq \epsilon_b-\nu\end{aligned}$$ where we used $\mu\simeq \epsilon_b/2$ and the bound state equation (\[bse\]). Thus the $l=1$ term just gives rise to an effective chemical potential for the dimers equal to $2\mu -\epsilon_b$. **$l=2$ term**: $$\begin{aligned} \frac{1}{4}\text{Tr}\left[(G_0\Delta)^{4}\right]&=& \frac{1}{2} \int_{k_1,\tilde{\omega}_1} \int_{k_2,\tilde{\omega}_2} \int_{k_3,\tilde{\omega}_3} \bar{\psi}_{B}(1+2-3)\nonumber \\ &\times& \bar{\psi}_{B}(3)\psi_{B}(2)\psi_{B}(1)g_B(1,2,3)\end{aligned}$$ in an obvious short hand notation for the associated wave vectors and frequencies, and $$\begin{aligned} g_B(1,2,3)&\equiv& g^4\int_{k,\tilde{\omega}}\Big[ (i\tilde{\omega}-\xi_k) \nonumber \\ &\times& (i\tilde{\omega}_2-i\tilde{\omega}-\xi_{k_2-k}) \nonumber \\ &\times& (i\tilde{\omega}_1+i\tilde{\omega}_2-i\tilde{\omega}_3-i\tilde{\omega}-\xi_{k_1+k_2-k_3-k}) \nonumber \\ &\times& (-i\tilde{\omega}_2+i\tilde{\omega}_3+i\tilde{\omega}-\xi_{-k_2+k_3+k}) \Big]^{-1}\end{aligned}$$ with $\xi_k\equiv k^2/2m - \mu$. Using again the condition characterizing a broad resonance, we see that the momentum and frequency dependance of the interaction is irrelevant. This implies that $$\begin{aligned} g_B(1,2,3)&\simeq& g_B(k_j=0,\tilde{\omega}_j=0;j=1,2,3) \nonumber \\ &\simeq& g^4\int_{k,\tilde{\omega}} \Big[\tilde{\omega}^2+\xi_k^2 \Big]^{-2}=\frac{3g^4\sqrt{m}}{8|\epsilon_b|^{5/2}} \label{gB}\end{aligned}$$ with $\mu\simeq \epsilon_b/2$. **$l\geq 3$ term**: For all $l$, we obtain the following estimate for the corresponding term in the sum (\[suml\]): $$\begin{aligned} t_l\equiv \frac{1}{2l}\text{Tr}\left[(G_0\Delta)^{2l}\right]\sim \frac{g^{2l}}{|\epsilon_b|^{2l-1}r_b}\frac{k_F^{l-1}}{\epsilon_F}\sim (nr_b)^{l-3}.\end{aligned}$$ For $l=1$ and $l=2$, in the broad resonance limit where $1\gg nr_{\star} >nr_b$, we obtain that $t_1$ and $t_2\gg 1$. For $l\geq 3$, the ratio $t_2/t_l\gg 1$ in the broad resonance limit and the corresponding terms can therefore be neglected. In the broad resonance limit, the effective action for bosons becomes $$\begin{aligned} S_B^{eff}&=&\int_0^{\beta}d\tau \int dx \bigg(\bar{\psi}_{B} \Big[\partial_{\tau}-\frac{\partial_x^2}{4m}-(2\mu-\epsilon_b)\Big]\psi_{B} \nonumber \\ &+& \frac{g_B}{2} \bar{\psi}_{B}\bar{\psi}_{B}\psi_{B}\psi_{B} \bigg)\end{aligned}$$ where $g_B\equiv 3g^4\sqrt{m}/8|\epsilon_b|^{5/2}$ describes a repulsive interaction between the strongly bound dimers. From the partition function (\[Z3\]) and the preceding effective action, we can obtain the average total number of atoms: $$\begin{aligned} \langle N \rangle = -T\frac{\partial \ln Z_F^0}{\partial \mu} -T\frac{\partial \ln Z_B^{eff}}{\partial \mu}. \label{N45}\end{aligned}$$ The first term is given by the usual expression for an ideal Fermi gas: $$\begin{aligned} \langle N_F^0 \rangle=L \int \frac{dk}{2\pi} \Big[ e^{\beta \left(k^2/2m -\mu\right)}+1\Big]^{-1}.\end{aligned}$$ In the limit $T\to 0$ and using the fact that after resonance $\mu\simeq \epsilon_b/2$, the fraction of atoms that are unbound is exponentially small. Therefore, (\[N45\]) becomes $$\begin{aligned} \frac{N}{2}=\frac{\langle N \rangle}{2} \simeq -T\frac{\partial \ln Z_B^{eff}}{\partial 2\mu}\end{aligned}$$ which shows that $2\mu$ is the chemical potential for the gas of dimers only. We now shift the zero of energy of the many-body system by an amount $-N\epsilon_b/2$, and accordingly define $\mu_B\equiv 2\mu -\epsilon_b$ as the new chemical potential for dimers. In conclusion, after resonance and under the assumption that the resonance is broad, the system is described by a single channel model of bosons (i.e. dimers) with an action $$\begin{aligned} S_B^{eff}&=&\int_0^{\beta}d\tau \int dx \bigg(\bar{\psi}_{B} \Big[\partial_{\tau}-\frac{\partial_x^2}{2m_B}-\mu_B\Big]\psi_{B} \nonumber \\ &+&\frac{g_B}{2} \bar{\psi}_{B}\bar{\psi}_{B}\psi_{B}\psi_{B} \bigg)\end{aligned}$$ where $m_B\equiv 2m$, $\mu_B=2\mu-\epsilon_B$ and $g_B= 3g^4\sqrt{m}/8|\epsilon_b|^{5/2}$ [@footnote4]. This is the action corresponding to the Lieb-Liniger model of $N_B\equiv N/2$ bosons of mass $m_B$ interacting via a repulsive delta potential [@LL]. The single dimensionless coupling constant is $\gamma\equiv mg_B/n$ and the BEC limit corresponds to $1/\gamma \to +\infty$ or $\nu \to -\infty$. Because $g_B \sim 1/mr_{\star}$ when $\nu \to 0^{-}$ (see Appendix B), the parameter $1/\gamma$ is restricted to: $$\begin{aligned} n r_{\star} <\frac{1}{\gamma} < +\infty\end{aligned}$$ In the broad resonance limit, $nr_{\star} \to 0$ implying that apart from a vanishingly small region close to resonance the Boson-Fermion resonance model, after resonance, is equivalent to the single channel repulsive Lieb-Liniger model for dimers. Discussion ========== It was recently shown [@FRZ; @Tokatly] that interacting fermions in a quasi-1D geometry and in presence of a Feshbach resonance map onto the modified Gaudin-Yang model in the limit of very strong confinement $na_{\perp}\ll 1$. In the present paper, we have seen that the Boson-Fermion resonance model in 1D is also described by the same model in the limit of a broad resonance $nr_{\star}\ll 1$. Close to resonance, the system behaves as a Tonks-Girardeau gas (or impenetrable Bose gas) [@Girardeau] of dimers [@Astra; @FRZ; @Tokatly]. Around resonance, there is a vanishingly small region $1/\gamma \in [-nr_{\star}, nr_{\star} ]$, which is not described by the modified Gaudin-Yang model. The relation between the parameters $g_1$ and $g_B$ of the modified Gaudin-Yang model and those of the original system – either the quasi-1D single channel model or the 1D Boson-Fermion resonance model – is different. Deep in the BEC limit, the two-body bound state of the BFRM is given by $\epsilon_b\simeq \nu$ and is populated by pairs of fermions. All fermions are bound into dimers and the scattering properties of dimers have no direct relation to the scattering properties of fermions, in particular $g_B$ is not simply proportional to $g_1$. This is in contrast to the 3D single channel model, where it is known that the dimer-dimer scattering length is proportional to the fermion-fermion scattering length (see, e.g., [@Petrov]). The equivalent result for the quasi-1D single channel is still under investigation [@Mora]. Nevertheless, in the BEC limit, one expects that $g_B \approx 0.6 \, g_1$ [@FRZ; @Tokatly]. The above scenario for a BCS-BEC crossover can be realized, e.g., in an experiment with ultra-cold gases confined in a quasi-1D trap either by tuning the 3D $s$-wave scattering length via a magnetic field in order to cross the CI resonance, as discussed in [@FRZ], or by photoassociation [@photo], which corresponds to a direct implementation of the BFRM. As mentioned before, a description of the resulting 1D BCS-BEC crossover by the modified Gaudin-Yang model is possible under the condition (19) for a broad resonance. Specifically, the realization using a CI resonance in a tight waveguide requires a sufficiently dilute gas with $na_{\perp}\ll 1$. Taking typical values of order 50 nm for the transverse oscillator length which have been realized very recently in bosonic 1D gases [@Esslinger; @BlochKinoshita], this requires densities in the range of much less than 20 atoms per micron. In the case of photo-association, i.e. an optically induced resonance, the requirement is, that the effective 1D Rabi frequency $g\sqrt{n}$ is much larger than the Fermi energy. Using estimates for the Rabi-frequency taken over from photassociation of $^{87}$Rb in 3D [@GrimmRb], a rough estimate shows that the condition of a broad resonance can also be reached here. In particular the fact that the Franck-Condon overlap is enhanced in a 1D situation helps realizing this limit. We acknowledge useful discussions with Walter Rantner, Stefano Cerrito and Andrea Micheli. Laboratoire de Physique des Solides is a mixed research unit (UMR 8502) of the CNRS and the Université Paris-Sud in Orsay. Appendix A {#appendix-a .unnumbered} ========== The estimates of $\mu$ used in the present article come from identifying $\delta \mu\equiv \mu -\epsilon_b/2$ (when in the broad resonance limit) with the chemical potential in the modified Gaudin-Yang model [@FRZ]. The chemical potential obtained from [@FRZ] gives the following estimate for $\delta \mu$: $$\delta \mu/\epsilon_F \simeq \left\{ \begin{array}{ll} 1 & \text{ when } 1/\gamma \to -\infty \text{ BCS limit}\\ 1/4 & \text{ when } 1/\gamma \to 0 \text{ on resonance}\\ \gamma/4\pi^2 & \text{ when } 1/\gamma \to +\infty \text{ BEC limit} \end{array}\right.$$ Appendix B {#appendix-b .unnumbered} ========== In this appendix, we discuss the behavior of $1/\gamma$ as a function of $\nu$. Before resonance $\gamma=mg_1/n=-mg^2/n\nu$, which implies: $$\begin{aligned} \frac{1}{\gamma}=-\frac{nr_{\star}}{2^{3/2}}\frac{\nu}{|\epsilon_{\star}|}.\end{aligned}$$ In the BCS limit $\nu\to +\infty$, $1/\gamma \to -\infty$ and on resonance $\nu \to 0^{+}$, $1/\gamma \to 0^-$. We assumed that $|\nu|>|\epsilon_{\star}|$, which implies $1/|\gamma|> nr_{\star}/2^{3/2}$ with $nr_{\star}\ll 1$ (broad resonance limit), so that indeed, close to resonance $1/\gamma \simeq 0$. After resonance, $\gamma=mg_B/n$ and equation (\[gB\]) can be rewritten [@footnote6] $$\begin{aligned} g_B=\frac{3}{\sqrt{2}mr_{\star}}\left(\frac{\epsilon_b}{\epsilon_{\star}}\right)^{-5/2}\end{aligned}$$ which implies: $$\begin{aligned} \frac{1}{\gamma}=\frac{\sqrt{2}nr_{\star}}{3} \left(\frac{\epsilon_b}{\epsilon_{\star}}\right)^{5/2}.\end{aligned}$$ Close to resonance $\nu\to 0^{-}$, $g_B\simeq 3/\sqrt{2}mr_{\star}$ and $1/\gamma \simeq \sqrt{2}nr_{\star}/3$ with $nr_{\star}\ll 1$ (broad resonance limit), so that again, $1/\gamma \simeq 0$ close to resonance. In the BEC limit $\nu \to -\infty$, $g_B \simeq 3\epsilon_{\star}^{5/2}/\sqrt{2}mr_{\star}\nu^{5/2}$ and $1/\gamma\simeq \sqrt{2}nr_{\star}\nu^{5/2}/3\epsilon_b^{5/2} \to +\infty$. [99]{} C.A. Regal, M. Greiner, and D.S. Jin, Phys. Rev. Lett. **92**, 040403 (2004). M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. Hecker Denschlag, and R. Grimm, Phys. Rev. Lett. **92**, 120401 (2004); M.W. Zwierlein, C.A. Stan, C.H. Schunck, S.M.F. Raupach, A.J. Kerman, and W. Ketterle, Phys. Rev. Lett. **92**, 120403 (2004); J. Kinast, S.L. Hemmer, M.E. Gehm, A. Turlapov, J.E. Thomas, Phys. Rev. Lett. **92**, 150402 (2004); T. Bourdel, L. Khaykovich, J. Cubizolles, J. Zhang, F. Chevy, M. Teichmann, L. Tarruell, S.J.J.M.F. Kokkelmans, and C. Salomon, Phys. Rev. Lett. **93**, 050401 (2004). D.M. Eagles, Phys. Rev. **186**, 456 (1969); A.J. Leggett, J. Phys. C (Paris) **41**, 7 (1980); P. Nozières, S. Schmitt-Rink, J. Low Temp. Phys. **59**, 195 (1985); M. Randeria, in *Bose-Einstein Condensation*, ed. by A. Griffin, D.W. Snoke, S. Stringari (Cambridge University Press, Cambridge, 1995) and references therein. H. Moritz, T. Stöferle, M. Köhl, and T. Esslinger, Phys. Rev. Lett. **91**, 250402 (2003). J. Reichel, and J.H. Thywissen, to appear in J. Phys. IV France **116**, xxxx (2004); arXiv:cond-mat/0310330. J.N. Fuchs, A. Recati, and W. Zwerger, Phys. Rev. Lett. **93**, 090408 (2004). I.V. Tokatly, Phys. Rev. Lett. **93**, 090405 (2004). M. Gaudin, Phys. Lett. **24A**, 55 (1967); C.N. Yang, Phys. Rev. Lett. **19**, 1312 (1967); see also M. Gaudin, *La fonction d’onde de Bethe* (Masson, Collection CEA, Paris, 1983). E.H. Lieb, and W. Liniger, Phys. Rev. **130**, 1605 (1963); E.H. Lieb, Phys. Rev. **130**, 1616 (1963). M. Olshanii, Phys. Rev. Lett. **81**, 938 (1998). R. Friedberg, and T.D. Lee, Phys. Rev. B **40**, 6745 (1989). J. Ranninger, J. M. Robin and M. Eschrig, Phys. Rev. Lett. **74**, 4027 (1995), and references therein. M. Holland, S.J.J.M.F. Kokkelmans, M.L. Chiofalo, and R. Walser, Phys. Rev. Lett. **87**, 120406 (2001); E. Timmermans, K. Furuya, P.W. Milonni, and A.K. Kerman, Phys. Lett. A **285**, 228 (2001). In the 3D version of the model defined by the Hamiltonian (\[BFM\]), a renormalized detuning appears in the bound state energy equation instead of the bare one as in the 1D case. This is due to the fact that a delta interaction is not a valid 3D potential and needs to be renormalized [@RM]. R.A. Duine, and H.T.C. Stoof, J. Opt. B: Quantum Semiclass. Opt. **5**, S212 (2003). T. Bergeman, M.G. Moore, and M. Olshanii, Phys. Rev. Lett. **91**, 163201 (2003). $\zeta(1/2,x)$ is a particular Hurvitz zeta function and $\zeta(1/2,1)\simeq-1.4603$, see [@BMO]. Note the $\sqrt{2}$ difference in the definition of $a_{\perp}$ with [@Olshanii; @BMO], which accounts for $A\equiv -\zeta(1/2,1)/\sqrt{2}\simeq 1.0326$. See, for example, A. Altland, and B. Simons, *Concepts of Theoretical Solid State Physics*, http://www.thp.uni-koeln.de/\~alexal/teaching.html (2001) or J.W. Negele, and H. Orland, *Quantum Many-Particle Systems* (Addison Wesley, Redwood City, 1988). G.M. Bruun, arXiv:cond-mat/0401497 (2004); S. De Palo, M.L. Chiofalo, M.J. Holland, and S.J.J.M.F. Kokkelmans, arXiv:cond-mat/0404672 (2004); R.B. Diener, and T.L. Ho, arXiv:cond-mat/0405174 (2004); S. Simonucci, P. Pieri, and G.C. Strinati, arXiv:cond-mat/0407600 (2004). Pistolesi and Strinati call it “the bosonization condition” [@PS]. In 3D, this condition corresponds to requiring that the effective range of the interaction between atoms mediated by molecules is small, i.e. $n^{1/3}|r_{eff}|\ll 1$ with $r_{eff}=-8\pi/m^2g^2$, see Bruun [@BRL] and S. De Palo et al. [@BRL]. F. Pistolesi, and G.C. Strinati, Phys. Rev. B **53**, 15168 (1996). M. Drechsler, and W. Zwerger, Ann. Phys. (Germany) [**1**]{}, 15 (1992). A.A. Gorbatsevich, and I.V. Tokatly, Sov. Phys. JETP **76**, 347 (1993). C.A.R. Sá de Melo, M. Randeria and J.R. Engelbrecht, Phys. Rev. Lett. [**71**]{}, 3202 (1993). In the 3D resonance model, $g_B$ is given by the same expression as in (\[gB\]) except that the $k$ integral is over three dimensions. This gives $g_B=m^{3/2}g^4/16\pi |\nu|^{3/2}$ in the BEC limit, where $\epsilon_b\simeq \nu$. Using the previous result, to calculate the Bogoliubov sound velocity $c\equiv \sqrt{g_Bn_B/m_B}$ with $n_B\equiv n/2$, we recover the result of Ohashi and Griffin for the sound velocity in the BEC limit of the 3D resonance model, see equation (4.20) in [@OG]. Y. Ohashi, and A. Griffin, Phys. Rev. A **67**, 063612 (2003). M. Girardeau, J. Math. Phys. (N.Y.) **1**, 516 (1960). G.E. Astrakharchik, D. Blume, S. Giorgini, and L.P. Pitaevskii, Phys. Rev. Lett. **93**, 050402 (2004). D.S. Petrov, C. Salomon, G.V. Shlyapnikov, Phys. Rev. Lett. **93**, 090404 (2004). C. Mora, R. Egger, A.O. Gogolin, and A. Komnik, arXiv:cond-mat/0406627 (2004); C. Mora, private communication. M. Mackie, E. Timmermans, R. Côté, and J. Javanainen, Optics Express **8**, 118 (2001) and references therein. We note that this is of the form $(ma_{\perp})^{-1}f(|\epsilon_b|/2\omega_{\perp})$, with the identification $r_{\star} \leftrightarrow a_{\perp}$ and $|\epsilon_{\star}| \leftrightarrow 2\omega_{\perp}$, as discussed in [@Tokatly]. B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch and I. Bloch, Nature [**429**]{}, 277 (2004); Toshiya Kinoshita, Trevor Wenger, and David S. Weiss, Science [**305**]{}, 1125 (2004). G. Thalhammer, M. Theis, K. Winkler, R. Grimm, J. H. Denschlag, available at arXiv.org/ cond-mat/0409552.

--- abstract: 'We characterize stability under composition, inversion, and solution of ordinary differential equations for ultradifferentiable classes, and prove that all these stability properties are equivalent.' address: - 'A. Rainer: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria' - 'G. Schindl: Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, A-1090 Wien, Austria' author: - Armin Rainer - Gerhard Schindl title: Equivalence of stability properties for ultradifferentiable function classes --- [^1] Introduction ============ Let $\cF$ denote some class of smooth mappings between non-empty open subsets of Euclidean spaces (of possibly different dimension). We say that - *$\cF$ is stable under composition* if the composite of any two $\cF$-mappings $g : U \to V$ and $f : V \to W$ is an $\cF$-mapping $f {\circ}g : U \to W$. - *$\cF$ is stable under solving ordinary differential equations (ODEs)* if for any $\cF$-mapping $f : \R \times \R^n \to \R^n$ the solution of the initial value problem $x' = f(t,x)$, $x(0) = x_0 \in \R^n$ is of class $\cF$ wherever it exists. - *$\cF$ is stable under inversion* if for any $\cF$-mapping $f : \R^m \supseteq U \to V \subseteq \R^n$ so that $f'(x_0) \in L(\R^m,\R^n)$ is invertible at $x_0 \in U$ there exist neighborhoods $x_0 \in U_0 \subseteq U$ and $f(x_0) \in V_0 \subseteq V$ and an $\cF$-mapping $g : V_0 \to U_0$ such that $f {\circ}g = \on{id}_{V_0}$. - *$\cF$ is inverse closed* if $1/f \in \cF(U)$ for each non-vanishing $f \in \cF(U)$. In this paper we shall prove that all these stability properties are equivalent for classes of ultradifferentiable mappings $\cF$ satisfying some mild regularity conditions. We will treat - the classical Denjoy–Carleman classes $\EM$ determined by a weight sequence $M=(M_k)$, - the classes $\Eom$ introduced by Braun, Meise, and Taylor [@BMT90] determined by a weight function $\om$, - the classes $\EfM$ introduced in [@RainerSchindl12] determined by a weight matrix $\fM$. The brackets $[~]$ stand for either $\{~\}$ in the Roumieu case or for $(~)$ in the Beurling case. For the precise definitions we refer to Section \[sec:wm\]. There are classes $\EM$ that cannot be given in terms of a weight function $\om$ and vice versa; see [@BMM07]. The classes $\EfM$ comprise all classes $\EM$ and $\Eom$ and hence allow for a unified approach to the classes $\EM$ and $\Eom$. Beyond that, they provide a convenient framework to describe unions and intersections of classical Denjoy–Carleman classes. The characterization of the aforementioned stability properties for $\EfM$ was important for treating $\EfM$-differomorphism groups in [@Schindl14a]. Stability properties of EM -------------------------- We assume from now on that any *weight sequence* $M=(M_k)$ is positive, $1= M_0 \le M_1$, and $k \mapsto k! M_k$ is log-convex (alias $M$ is weakly log-convex). \[rem:wlc\] For any weight sequence $M=(M_k)$ the sequence $(k! M_k)^{1/k}$ is increasing and $M_j M_k \le \binom{j+k}{j} M_{j+k}$ for all $j,k \in \N$. \[thm:rM\] If $\varliminf M_k^{1/k}>0$ and $\sup (\frac{M_{k+1}}{M_k})^{1/k}<\infty$ the following are equivalent: 1. $M_k^{1/k}$ is almost increasing, i.e., $\exists C>0~ \forall j \le k : M_j^{1/j} \le C M_k^{1/k}$. 2. $M$ has the (FdB)-property, i.e., $\exists C>0 : M^{\circ}_k \le C^k M_k$, where $$M^{\circ}_k := \max\{M_jM_{\al_1}\dots M_{\al_j}: \al_i\in \N_{>0}, \al_1+\dots+\al_j = k\}, \quad M^{\circ}_0:=1.$$ 3. $\cE^{\{M\}}$ is stable under composition. 4. $\cE^{\{M\}}$ is stable under solving ODEs. 5. $\cE^{\{M\}}$ is stable under inversion. 6. $\cE^{\{M\}}$ is inverse-closed. Note that $\varliminf M_k^{1/k}>0$ iff $C^\om \subseteq \ErM$, and $\sup (\frac{M_{k+1}}{M_k})^{1/k}<\infty$ iff $\EM$ is stable under derivation; cf. [@RainerSchindl12]. If we replace the first condition by $\lim M_k^{1/k}=\infty$ which is equivalent to $C^\om \subseteq \EbM$, we have the corresponding Beurling type result: \[thm:bM\] If $\lim M_k^{1/k}= \infty$ and $\sup (\frac{M_{k+1}}{M_k})^{1/k}<\infty$ the following are equivalent: 1. $M_k^{1/k}$ is almost increasing. 2. $M$ has the (FdB)-property. 3. $\EbM$ is stable under composition. 4. $\EbM$ is stable under solving ODEs. 5. $\EbM$ is stable under inversion. 6. $\EbM$ is inverse-closed. Most implications of Theorems \[thm:rM\] and \[thm:bM\] are basically known, but scattered in the literature. The equivalence of (1) and (6) is due to Rudin [@Rudin62] in the Roumieu case and to Bruna [@Bruna80/81] in the Beurling case; note that Rudin only considered non-quasianalytic classes and Hörmander dealt with the quasianalytic case (cf. [@Rudin62 p. 799]). See also [@Siddiqi90]. That (1) implies stability under inversion is due to Komatsu [@Komatsu79]; different proofs in the Banach space setting were given by Yamanaka [@Yamanaka89] and Koike [@Koike96]. The sufficiency of (1) for stability under solving ODEs was obtained by Komatsu [@Komatsu80] and in Banach spaces by Yamanaka [@Yamanaka91]. That the class $\ErM$ is stable under composition, provided that $M=(M_k)$ is log-convex (which implies (1)), is due to Roumieu [@Roumieu62/63]; other references are e.g. [@Komatsu73] and [@BM04]. In [@RainerSchindl12] we proved the equivalence of (1), (2), and (3) (in both the Beurling and the Roumieu case). It is worth mentioning that Dynkin [@Dynkin80] gave a characterization of the Roumieu classes $\ErM$ in terms of *almost holomorphic extensions*, provided that $M=(M_k)$ is log-convex, which implies the stability properties (3), (4), (5), and (6) in a straightforward manner. That all the properties (1) – (6) are equivalent was, to our knowledge, not observed before. Stability properties of Eomega ------------------------------ The respective result in the weight function case, that is Theorems \[thm:rom\] and \[thm:bom\] below, was not known before, apart from a characterizaton for stability under composition obtained in [@FernandezGalbis06] and in [@RainerSchindl12]. We henceforth assume that any *weight function* $\om$ is a continuous increasing function $\om: [0,\infty) \to [0,\infty)$ with $\om|_{[0,1]}=0$, $\lim_{t \to \infty} \om(t) = \infty$, and so that: $$\begin{aligned} \tag{$\om_1$} \label{om_1} &\om(2t)=O(\om(t)) \text{ as } t\to \infty. \\ \tag{$\om_2$} \label{om_2} &\log(t)=o(\omega(t)) \text{ as } t\to \infty. \\ \tag{$\om_3$} \label{om_3} &\vh : t \mapsto \om(e^t) \text{ is convex on } [0,\infty).\end{aligned}$$ Note that $C^\om \subseteq \Erom$ iff $\om(t)=O(t)$, and $C^\om \subseteq \Ebom$ iff $\om(t)=o(t)$, as $t\to \infty$. \[thm:rom\] If $\om$ satisfies $\om(t)=O(t)$ as $t\to \infty$ then the following are equivalent: 1. $\om$ satisfies $\exists C>0 ~\exists t_0>0 ~\forall \la\ge 1 ~\forall t\ge t_0 : \om(\la t) \le C \la \om(t)$. 2. There exists a sub-additive weight function $\tilde \om$ so that $\Eom = \cE^{[\tilde \om]}$. 3. $\cE^{\{\om\}}$ is stable under composition. 4. $\cE^{\{\om\}}$ is stable under solving ODEs. 5. $\cE^{\{\om\}}$ is stable under inversion. 6. $\cE^{\{\om\}}$ is inverse-closed. \[thm:bom\] If $\om$ satisfies $\om(t)=o(t)$ as $t\to \infty$ then the following are equivalent: 1. $\om$ satisfies $\exists C>0 ~\exists t_0>0 ~\forall \la\ge 1 ~\forall t\ge t_0 : \om(\la t) \le C \la \om(t)$. 2. There exists a sub-additive weight function $\tilde \om$ so that $\Eom = \cE^{[\tilde \om]}$. 3. $\Ebom$ is stable under composition. 4. $\Ebom$ is stable under solving ODEs. 5. $\Ebom$ is stable under inversion. 6. $\Ebom$ is inverse-closed. Stability properties of EfM --------------------------- Theorems \[thm:rM\], \[thm:bM\], \[thm:rom\], and \[thm:bom\] are corollaries of the corresponding result for the classes $\EfM$ defined in terms of weight matrices, namely Theorems \[thm:rfM\] and \[thm:bfM\] below. A *weight matrix* $\fM = \{M^\la \in \R_{>0}^{\N} : \la \in \La\}$ is a family of weight sequences $M^\la=(M^\la_k)$ indexed by an ordered subset $\La$ of $\R$ so that $\lim_{k} (k! M^\la_k)^\frac{1}{k}=\infty$ for each $\la$, and $M^\la\le M^\mu$ if $\la \le \mu$. We shall again assume that the classes $\EfM$ contain the class of real analytic functions and are stable under derivation. Specifically we need the conditions $$\begin{aligned} \tag{$\fM_{\cH}$} \label{fM_H} &\forall \la \in \La : \varliminf (M^\la_k)^{\frac{1}{k}}>0 \\ \tag{$\fM_{(C^\om)}$} \label{fM_(Com)} &\forall \la \in \La : \lim (M^\la_k)^{\frac{1}{k}}=\infty \\ \tag{$\fM_{\{\on{dc}\}}$} \label{fM_{dc}} &\forall \la \in \La ~\exists \mu \in \La ~\exists C>0 ~\forall k \in \N: M^\la_{k+1}\le C^{k} M^{\mu}_k \\ \tag{$\fM_{(\on{dc})}$} \label{fM_(dc)} &\forall \la \in \La ~\exists \mu \in \La ~\exists C>0 ~\forall k \in \N: M^\mu_{k+1}\le C^{k} M^{\la}_k\end{aligned}$$ We have $$\xymatrix{ \eqref{fM_(Com)} \ar@{<=>}[r] \ar@{=>}[d] & C^\om(U) \subseteq \EbfM(U) \ar@{=>}[d]\\ \eqref{fM_H} \ar@{<=>}[r] \ar@{=>}[d] & \cH(\C^n) \subseteq \cE^{(\fM)}(U) \ar@{=>}[d]\\ (\fM_{\{C^\om\}}) :\Leftrightarrow \exists \la \in \La : \varliminf (M^\la_k)^{\frac{1}{k}}>0 \ar@{<=>}[r] & C^\om(U) \subseteq \ErfM(U) }$$ and $\cE^{[\fM]}(U)$ is derivation closed iff ; see [@RainerSchindl12]. The conditions on the weight matrix $\fM$ that characterize the stability properties of $\EfM$ are natural generalizations of the condition of being almost increasing and of the (FdB)-property; clearly, the Roumieu and the Beurling version fall apart, see Remark \[rem:rai\] below: $$\begin{aligned} \tag{$\fM_{\{\on{rai}\}}$} \label{fM_{rai}} &\forall \la \in \La ~\exists \mu \in \La ~\exists C>0 ~\forall j \le k: (M^{\la}_j)^{\frac{1}{j}} \le C (M^{\mu}_k)^{\frac{1}{k}} \\ \tag{$\fM_{(\on{rai})}$} \label{fM_(rai)} &\forall \la \in \La ~\exists \mu \in \La ~\exists C>0 ~\forall j \le k: (M^{\mu}_j)^{\frac{1}{j}} \le C (M^{\la}_k)^{\frac{1}{k}} \\ \tag{$\fM_{\{\on{FdB}\}}$} \label{fM_{FdB}} &\forall \la \in \La ~\exists \mu \in \La ~\exists C>0 ~\forall k: (M^\la)^{\circ}_k \le C^k M^\mu_k \\ \tag{$\fM_{(\on{FdB})}$} \label{fM_(FdB)} &\forall \la \in \La ~\exists \mu \in \La ~\exists C>0 ~\forall k: (M^\mu)^{\circ}_k \le C^k M^\la_k\end{aligned}$$ \[thm:rfM\] For a weight matrix $\fM$ satisfying and the following are equivalent: 1. $\fM$ satisfies . 2. $\fM$ satisfies . 3. $\cE^{\{\fM\}}$ is stable under composition. 4. $\cE^{\{\fM\}}$ is stable under solving ODEs. 5. $\cE^{\{\fM\}}$ is stable under inversion. 6. $\cE^{\{\fM\}}$ is inverse-closed. \[thm:bfM\] For a weight matrix $\fM$ satisfying and the following are equivalent: 1. $\fM$ satisfies . 2. $\fM$ satisfies . 3. $\EbfM$ is stable under composition. 4. $\EbfM$ is stable under solving ODEs. 5. $\EbfM$ is stable under inversion. 6. $\EbfM$ is inverse-closed. \[rem:rai\] The weight matrix $\fM$ that consists of just two non-equivalent sequences $M^1 \le M^2$ satisfying - $(M^i_k)^{1/k} \to \infty$ and $\sup_k (\frac{M^i_{k+1}}{M^i_k})^{1/k}< \infty$, $i=1,2$, - $(M^1_k)^{1/k}$ almost increasing, - $(M^2_k)^{1/k}$ not almost increasing, satisfies but not . Whereas, if $(M^2_k)^{1/k}$ is almost increasing and $(M^1_k)^{1/k}$ is not, $\fM$ satisfies but not . We construct such sequences in Appendix \[appendix\]. We shall prove Theorems \[thm:rfM\] and \[thm:bfM\] in Sections \[sec:proofR\] and \[sec:proofB\], respectively. In Section \[sec:proofMom\] we show that Theorems \[thm:rM\], \[thm:bM\], \[thm:rom\], and \[thm:bom\] are corollaries of Theorems \[thm:rfM\] and \[thm:bfM\]. Notation {#notation .unnumbered} -------- The notation $\cE^{[*]}$ for $* \in \{M,\om,\fM\}$ stands for either $\cE^{(*)}$ or $\cE^{\{*\}}$ with the following restriction: Statements that involve more than one $\cE^{[*]}$ symbol must not be interpreted by mixing $\cE^{(*)}$ and $\cE^{\{*\}}$. Ultradifferentiable function classes {#sec:wm} ==================================== Ultradifferentiable functions defined by weight sequences --------------------------------------------------------- Let $M=(M_k)$ be a weight sequence. For non-empty open $U \subseteq \R^n$, define $$\begin{aligned} \cE^{(M)}(U,\R^m) &:= \Big\{f \in C^\infty(U,\R^m) : \forall K \subseteq U \text{ compact} ~\forall \rh>0 : \|f\|^M_{K,\rh} < \infty \Big\} \\ \cE^{\{M\}}(U,\R^m) &:= \Big\{f \in C^\infty(U,\R^m) : \forall K \subseteq U \text{ compact} ~\exists \rh>0 : \|f\|^M_{K,\rh} < \infty \Big\} \\ &\|f\|^M_{K,\rh} := \sup\Big\{\frac{\|f^{(k)}(x)\|_{L^k(\R^n,\R^m)}}{\rh^k k! M_k}:x\in K,k\in\N\Big\} \end{aligned}$$ and endow these spaces with their natural topologies: $$\begin{gathered} \cE^{(M)}(U,\R^m) = \varprojlim_{K \subseteq U} \varprojlim_{\ell \in \N_{>0}} \cE^M_{\frac{1}{\ell}}(K,\R^m), \quad \cE^{\{M\}}(U,\R^m) = \varprojlim_{K \subseteq U} \varinjlim_{\ell \in \N} \cE^M_{\ell}(K,\R^m) \\ \text{where }\quad \cE^M_\rh(K,\R^m) := \{f \in C^\infty(K,\R^m) : \|f\|^M_{K,\rh} < \infty \}\end{gathered}$$ We will need the following inclusion relations (cf. [@RainerSchindl12]): $$\begin{aligned} \cE^{[M]} \subseteq \cE^{[N]} \quad &\Leftrightarrow \quad M \preceq N \quad :\Leftrightarrow \quad \exists C,\rh > 0 ~\forall k : M_k \le C \rh^k N_k \\ \cE^{\{M\}} \subseteq \cE^{(N)} \quad &\Leftrightarrow \quad M \lhd N \quad :\Leftrightarrow \quad \forall \rh>0 ~\exists C>0 ~\forall k : M_k \le C \rh^k N_k \end{aligned}$$ In particular, $C^\om(U) \subseteq \cE^{\{M\}}(U) \Leftrightarrow \cH(\C^n) \subseteq \cE^{(M)}(U) \Leftrightarrow \varliminf M_k^{\frac{1}{k}}>0$ and $C^\om(U) \subseteq \cE^{(M)}(U) \Leftrightarrow \lim M_k^{\frac{1}{k}} = \infty$. Ultradifferentiable functions defined by weight functions {#ssec:wf} --------------------------------------------------------- Let $\om$ be a weight function (hence satisfying , , and ). The *Young conjugate* of $\vh(t) = \om(e^t)$, given by $$\vh^*(t):=\sup \{st-\vh(s) : s \ge 0\}, \quad t \ge 0,$$ is convex, increasing, and satisfies $\vh^*(0)=0$, $\vh^{**}=\vh$, and $\lim_{t\to \infty} t/\vh^*(t)=0$. Moreover, the functions $t \mapsto \vh(t)/t$ and $t \mapsto \vh^*(t)/t$ are increasing; see e.g. [@BMT90]. For non-empty open $U \subseteq \R^n$ define $$\begin{aligned} \cE^{(\om)}(U,\R^m) &:= \Big\{f \in C^\infty(U,\R^m) : \forall K \subseteq U \text{ compact} ~\forall \rh > 0 : \|f\|^\om_{K,\rh} < \infty \Big\} \\ \cE^{\{\om\}}(U,\R^m) &:= \Big\{f \in C^\infty(U,\R^m) : \forall K \subseteq U \text{ compact} ~\exists \rh>0 : \|f\|^\om_{K,\rh} < \infty \Big\} \\ &\|f\|^\om_{K,\rh} := \sup\Big\{\|f^{(k)}(x)\|_{L^k(\R^n,\R^m)} \exp(- \tfrac{1}{\rh} \vh^*(\rh k)) : x\in K,k\in\N\Big\} \end{aligned}$$ and endow these spaces with their natural topologies: $$\begin{gathered} \Ebom(U,\R^m) = \varprojlim_{K \subseteq U} \varprojlim_{\ell \in \N_{>0}} \cE^\om_{\frac{1}{\ell}}(K,\R^m), \quad \Erom(U,\R^m) = \varprojlim_{K \subseteq U} \varinjlim_{\ell \in \N} \cE^\om_{\ell}(K,\R^m) \\ \text{where }\quad \cE^\om_\rh(K,\R^m) := \{f \in C^\infty(K,\R^m) : \|f\|^\om_{K,\rh} < \infty \}\end{gathered}$$ We have $C^\om(U) \subseteq \cE^{\{\om\}}(U)$ iff $\om(t)=O(t)$ as $t \to \infty$, and $C^\om(U) \subseteq \cE^{(\om)}(U)$ iff $\om(t)=o(t)$ as $t \to \infty$; see e.g. [@RainerSchindl12]. Ultradifferentiable functions defined by weight matrices {#ssec:fM} -------------------------------------------------------- Let $\fM$ be a weight matrix, let $U \subseteq \R^n$ be non-empty and open, and let $K \subseteq U$ be compact. We define $$\begin{gathered} \cE^{(\fM)}(K,\R^m) := \bigcap_{\la \in \La} \cE^{(M^{\la})}(K,\R^m), \quad \cE^{\{\fM\}}(K,\R^m) := \bigcup_{\la \in \La} \cE^{\{M^{\la}\}}(K,\R^m), \\ \cE^{(\fM)}(U,\R^m) := \bigcap_{\la \in \La} \cE^{(M^{\la})}(U,\R^m), \quad \cE^{\{\fM\}}(U,\R^m) := \bigcap_{K \subseteq U}\bigcup_{\la \in \La} \cE^{\{M^{\la}\}}(K,\R^m),\end{gathered}$$ and endow these spaces with their natural topologies: $$\begin{aligned} \cE^{(\fM)}(U,\R^m) := \varprojlim_{\la\in \La} \cE^{(M^{\la})}(U,\R^m), \quad \cE^{\{\fM\}}(U,\R^m) := \varprojlim_{K \subseteq U} \varinjlim_{\la \in \La} \cE^{\{M^{\la}\}}(K,\R^m). \end{aligned}$$ It is no loss of generality to assume that the limits are countable. We have $\cH(\C^n) \subseteq \cE^{(\fM)}(U)$ iff , $C^\om(U) \subseteq \cE^{[\fM]}(U)$ iff , and $\cE^{[\fM]}(U)$ is derivation closed iff ; see [@RainerSchindl12]. Proof of Theorem \[thm:rfM\]: the Roumieu case {#sec:proofR} ============================================== The proof of the equivalence of the items (1)–(6) of Theorem \[thm:rfM\] has the following structure: $$\label{eq:diag} \begin{split} \xymatrix@R=.2cm{ && (4) \ar@{=>}[dl] &&&\\ & (6') \ar@{=>}[rr] && (1) \ar@{=>}[ul] \ar@{=>}[dl] \ar@{<=>}[r] & (2) \ar@{<=>}[r] & (3) \\ (6) \ar@{=>}[ur] && (5) \ar@{=>}[ll] & & & } \end{split}$$ where: 1. If $f \in \cE^{\{\fM\}}(\R)$ and $f(0) \ne 0$ then $1/f$ is $\ErfM$ on its domain of definition. We successively prove: - the equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ - the cycle $(1) \Rightarrow (4) \Rightarrow (6') \Rightarrow (1)$ - the cycle $(1) \Rightarrow (5) \Rightarrow (6) \Rightarrow (6') \Rightarrow (1)$ The equivalences (1)&lt;=&gt;(2)&lt;=&gt;(3) -------------------------------------------- The following lemma implies the equivalence of (1) and (2). The equivalence of (2) and (3) was shown in [@RainerSchindl12 4.9]. \[lem:raiFdB\] For a weight matrix $\fM$ we have the following implications: 1. and imply . 2. and imply was shown in [@RainerSchindl12 4.9, 4.11]. To see observe that $(M^\mu)^{\circ}\preceq M^\la$ implies $(M^\mu_j)^{\frac{1}{jk}} (M^\mu_k)^{\frac{1}{k}} \le C (M^\la_{jk})^{\frac{1}{jk}}$ for all $j,k$ and some constant $C$. By we may conclude that $(M^\mu_k)^{\frac{1}{k}} \le \tilde C (M^\la_{jk})^{\frac{1}{jk}}$ for all $j,k$ and some $\tilde C$. For $k \le \ell$ choose $j \in \N$ such that $jk \le \ell < (j+1)k$, then by Remark \[rem:wlc\] and since $n! \le n^n \le e^n n!$, $$\ell (M^\la_\ell)^{\frac{1}{\ell}} \!\ge\! (\ell! M^\la_\ell)^{\frac{1}{\ell}} \!\ge\! ((jk)! M^\la_{jk})^{\frac{1}{jk}} \ge \frac{\tilde C jk}{e} (M^\mu_{k})^{\frac{1}{k}} \ge \frac{\tilde C (j+1)k}{2e} (M^\mu_{k})^{\frac{1}{k}} > \frac{\tilde C\ell}{2e} (M^\mu_{k})^{\frac{1}{k}}$$ which implies the desired property. The cycle (1)=&gt;(4)=&gt;(6’)=&gt;(1) {#ssec:cycle1} -------------------------------------- The implication (1) $\Rightarrow$ (4) follows from the following proposition. We state and prove this result on ultradifferentiable solutions of ODEs (as well as the ultradifferentiable inverse mapping theorem below) for mappings between arbitrary Banach spaces, since we used such results in [@KMRu] and will need them in forthcoming work; cf. [@Schindl14a]. \[rem:shift\] Observe the index shift in the estimates of and . In order to deduce (1) $\Rightarrow$ (4) from Proposition \[prop:ODEr\] we use for the index shift in and Remark \[rem:wlc\] for the one in . Henceforth we use the convention $(-1)! M_{-1} := 1$ for any weight sequence $M$. \[prop:ODEr\] Let $\fM$ be a weight matrix satisfying . Let $X$ be a Banach space and let $f : W \to X$ be a $C^\infty$-mapping defined in an open subset $W \subseteq X \times \R$ and satisfying $$\begin{aligned} \label{eq:f} \begin{split} \exists \la \in \La ~\exists C,\rh \ge1 &~\forall (k,\ell) \in \N^2 ~\forall (x,t) \in W : \\ &\|f^{(k,\ell)}(x,t)\|_{L^{k,\ell}(X,\R;X)} \le C \rh^{k+\ell} (k+\ell-1)! M^\la_{k+\ell-1}. \end{split}\end{aligned}$$ Then the solution $x : I \to X$ of the initial value problem $$\label{eq:IVP} x'(t) = f(x(t),t), \quad x(0) = x_0,$$ satisfies $$\begin{aligned} \label{eq:x} \exists \mu \in \La ~\exists D,\si \ge1 ~\forall k \in \N ~\forall t \in I : \|x^{(k)}(t)\| \le D \si^{k} (k-1)! M^\mu_{k-1}.\end{aligned}$$ \[rem:1\] A more general statement involving parameters $u$ in a further Banach space $Z$ is true: the solution of the initial value problem $$x'(t) = f(x(t),t,u), \quad x(0) = x_0,$$ satisfies an estimate of the kind in $t$, $u$, and $x_0$, given that $f$ satisfies an estimate of the kind in $x$, $t$, and $u$. For simplicity we prove only the result stated in the proposition; the general result is obtained by making obvious modifications in the proof of [@Yamanaka91] the main ideas of which we follow here. Different arguments were given in [@Komatsu80] and [@Dynkin80]. In the proof of the proposition we will use Faà di Bruno’s formula for Fréchet derivatives of mappings between Banach spaces. So let us recall this formula. For $k\ge 1$, $$\begin{aligned} \label{eq:FaaF} \frac{(f{\circ}g)^{(k)}(x)}{k!} = \on{sym}\Big( \sum_{j\ge 1} \!\sum_{\substack{\al\in \N_{>0}^j\\ \sum_{i=1}^j \al_i =k}}\! \frac{f^{(j)}(g(x))}{j!} {\circ}\Big(\frac{g^{(\al_1)}(x)}{\al_1!}\times\cdots\times\frac{g^{(\al_j)}(x)}{\al_j!}\Big)\Big),\end{aligned}$$ where $\on{sym}$ denotes symmetrization of multilinear mappings. We may reduce the initial value problem to the problem $$\label{eq:IVP2} y'= g(y), \quad y(0) =y_0,$$ by setting $y =(x,t)$, $y_0=(x_0,0)$, and $g(y) = (f(y),1)$. So we assume that $Y$ is a Banach space, $U$ is a neighborhood of $0$ in $Y$, and $g \in C^\infty(U,Y)$ satisfies $$\begin{aligned} \label{eq:g2} \exists \la \in \La ~\exists C,\rh \ge1 ~\forall k \in \N ~\forall y \in U : \|g^{(k)}(y)\|_{L^k(Y,Y)} \le C \rh^k (k-1)! M^\la_{k-1}. \end{aligned}$$ Without loss of generality we assume $M^\la_1 \ge 2$. By the classical existence and uniqueness result, there exists a unique $C^\infty$ solution $y =y(t)$ of for $t$ in a neighborhood $I$ of $0$. We assume that $\sup_{t \in I} \|y(t)\|<\infty$. By there exists $\mu \in \La$ and $H\ge 1$ such that for $2 \le j \le k$, $$\begin{aligned} \Big(\frac{M^\la_{j-1}}{j}\Big)^{\frac{1}{j-1}} \le H \Big(\frac{M^{\mu}_{k-1}}{k}\Big)^{\frac{1}{k-1}} =: p^\mu_k. \end{aligned}$$ Then, since $1 \le M^\la_1/2 \le p^\mu_k$, $$\begin{aligned} \label{eq:p} \frac{M^\la_{j-1}}{j} \le (p^\mu_k)^j, \quad \text{ for } 2 \le j \le k. \end{aligned}$$ Let us choose constants $A$ and $\et$ such that $$\label{eq:Aet} A \ge \max\{\sup_{t \in I} \|y(t)\|,C\} \quad \text{ and } \quad \et \ge \rh,$$ where $C$ and $\rh$ are the constants from . We define $$\begin{aligned} G^\mu_k(s) &:= \frac{A}{1-\et p^\mu_k s}, \end{aligned}$$ for small $s \in \R$, and consider the initial value problem $$\label{eq:IVP3} Y'(t) = G^\mu_k(Y(t)-A), \quad Y(0) = A.$$ We claim that the solution $$\begin{aligned} Y^\mu_k(t) =A+ \frac{1-\sqrt{1-2A \et p^\mu_k t}}{\et p^\mu_k} \end{aligned}$$ of satisfies $$\label{eq:ind} \sup_{t \in I} \|y^{(j)}(t)\| \le (Y^\mu_k)^{(j)}(0) \quad \text{ for } j \le k.$$ This implies the statement of the proposition, since, for $j \ge 1$, $$\begin{aligned} (Y^\mu_k)^{(j)}(0) &= (2A)^j (\et p^\mu_k)^{j-1} (2\sqrt{\pi})^{-1} \Ga(j-\tfrac1 2) \\ & \le (2A)^j (\et H)^{j-1} (j-1)! M^\mu_{j-1} \quad \text{ if } k=j. \end{aligned}$$ Let us prove . By the choice of the constant $A$, is satisfied for $j=0$. Suppose that holds for all $j \le \ell< k$. By , , and , we have $$\begin{aligned} \sup_{y \in U} \|g^{(j)}(y)\|_{L^j(Y,Y)} &\le C \rh^j (j-1)! M^\la_{j-1} \le C \rh^j j! (p^\mu_k)^j \le A \et^j j! (p^\mu_k)^j = (G^{\mu}_k)^{(j)}(0). \end{aligned}$$ So, by applying Faà di Bruno’s formula twice, we may conclude that, for $j \le \ell < k$, $$\begin{aligned} \sup_{t \in I} \|(g {\circ}y)^{(j)}(t)\| &\le j! \sum_{h\ge 1} \frac{(G^{\mu}_k)^{(h)}(0)}{h!} \sum_{\substack{\al_1+\cdots+\al_h=j\\ \al_i>0}} \prod_{i=1}^h \frac{(Y^\mu_k)^{(\al_i)}(0)}{\al_i!} \\ &= (G^\mu_k {\circ}(Y^\mu_k-A))^{(j)}(0). \end{aligned}$$ But as $y$ and $Y^\mu_k$ are solutions of and , respectively, it follows that holds for $j \le \ell+1$. By induction, follows. Let us check that (4) implies $(6')$. Let $f \in \cE^{\{\fM\}}(\R)$ satisfy $f(0) \ne 0$ and consider $g := 1/f$. Then $g$ solves the initial value problem $$x' = - f'(t) x^2, \quad x(0) = g(0).$$ By , the mapping $(x,t) \mapsto f'(t)x^2$ is $\ErfM$ and so $g$ is $\cE^{\{\fM\}}$, by (4). The implication $(6')$ $\Rightarrow$ (1) was shown in the proof of [@RainerSchindl12 4.9(2)$\Rightarrow$(3)], by following the argument of [@Siddiqi90 Thm 3]. The next lemma, a variation of [@Koike96 Thm 1], is a preparation for Proposition \[prop:inverse\] below. It gives, in particular, a direct proof of the implication (1) $\Rightarrow$ $(6')$. \[lem:inverseclosed\] Let $\fM$ be a weight matrix satisfying . Let $E, F, G$ be Banach spaces, $U \subseteq E$ be open, and let $T \in C^\infty (U,L(F,G))$ satisfy $$\begin{aligned} \label{eq:1/f} \begin{split} \exists \la \in \La ~\exists C,\rh \ge1 &~\forall k \in \N ~\forall x\in U : \|T^{(k)}(x)\|_{L^k(E,L(F,G))} \le C \rh^{k} k! M^\la_{k}. \end{split} \end{aligned}$$ If $T(x_0) \in L(F,G)$ is invertible, then there is a neighborhood $x_0 \in U_0 \subseteq U$ such that $U_0 \ni x \mapsto S(x) := T(x)^{-1}$ satisfies $$\begin{aligned} \label{eq:1/g} \begin{split} \exists \mu \in \La ~\exists D,\si \ge1 &~\forall k \in \N ~\forall x\in U_0 : \|S^{(k)}(x)\|_{L^k(E,L(G,F))} \le D \si^{k} k! M^\mu_{k}. \end{split} \end{aligned}$$ There is an open neighborhood $U_0$ of $x_0$ so that for $x \in U_0$ we have $\|S(x)\|_{L(G,F)} \le A$ for some constant $A>0$ and $S(x)$ is given by the Neumann series $$S(x) = T(x_0)^{-1} \sum_{j=0}^\infty \big((T(x_0)-T(x))T(x_0)^{-1}\big)^j.$$ For $y$ near $x$ we may consider $$S(y) = S(x) \sum_{j=0}^\infty \big((T(x)-T(y)) S(x)\big)^j = S(x) \big(\id - (T(x)-T(y)) S(x)\big)^{-1}$$ and use Faà di Bruno’s formula and to obtain, for $y=x$, $$\begin{aligned} \frac{\|S^{(k)}(x)\|_{L^k(E,L(G,F))}}{k!} &\le A \sum_{j\ge 1} \sum_{\substack{\al_1+\cdots+\al_j=k\\ \al_i>0}} (AC)^j \rh^k \prod_{i=1}^j M^\la_{\al_i} \end{aligned}$$ which implies , since $M^\la_{\al_1} \cdots M^\la_{\al_j} \le H^{k} M^\mu_{k}$, by . The cycle (1)=&gt;(5)=&gt;(6)=&gt;(6’)=&gt;(1) {#ssec:cycle2} ---------------------------------------------- The implications $(5) \Rightarrow (6) \Rightarrow (6')$ are obvious. And we already know from Subsection \[ssec:cycle1\] that $(1) \Leftrightarrow (6')$. That (1) implies (5) follows from the next proposition, using the analogue of Remark \[rem:shift\]. \[prop:inverse\] Let $\fM$ be a weight matrix satisfying . Let $f : U \to V$ be a $C^\infty$-mapping between open subsets $U \subseteq E$ and $V \subseteq F$ of Banach spaces $E, F$ satisfying $$\begin{aligned} \label{eq:finv} \begin{split} \exists \la \in \La ~\exists C,\rh \ge1 &~\forall k \in \N ~\forall x \in U : \|f^{(k)}(x)\|_{L^{k}(E,F)} \le C \rh^{k-1} (k-1)! M^\la_{k-1} \end{split}\end{aligned}$$ and so that $f'(x_0) \in L(E,F)$ is invertible. Then there exist neighborhoods $x_0 \in U_0 \subseteq U$ and $f(x_0) \in V_0 \subseteq V$ and a $C^\infty$-mapping $g : V_0 \to U_0$ satisfying $$\begin{aligned} \label{eq:ginv} \begin{split} \exists \mu \in \La ~\exists D,\si \ge1 &~\forall k \in \N ~\forall y \in V_0 : \|g^{(k)}(y)\|_{L^{k}(F,E)} \le D \si^{k-1} (k-1)! M^\mu_{k-1} \end{split}\end{aligned}$$ and such that $f {\circ}g = \on{id}_{V_0}$. We adapt the proof of [@Koike96]. Different arguments were given in [@Komatsu79] and [@Yamanaka89], under more restrictive assumptions also in [@Dynkin80] and [@BM04]. By the classical $C^\infty$ inverse mapping theorem there exist neighborhoods $x_0 \in U_0 \subseteq U$ and $f(x_0) \in V_0 \subseteq V$ and a $C^\infty$-mapping $g : V_0 \to U_0$ such that $f {\circ}g = \on{id}_{V_0}$. We can assume that $(f'(x))^{-1}$ is bounded for $x \in U_0$. We shall show that $g$ satisfies . Let $S_k \in C^\infty(U_0, L(F,E))$, $k\ge 1$, be given and define $R_k(x)$, $k \ge 0$, for $x \in U_0$ recursively by setting $$\label{eq:rec} R_0(x) := \id_E, \quad R_k(x) := (R_{k-1}(x) S_k(x))'.$$ Thus $R_{k-1} \in C^\infty(U_0, L(E,L^{k-1}(F,E)))$ and $R_{k-1}(x) S_k(x) \in L^k(F,E)$. It follows that $$\|R_k(x)\|_{L(E,L^{k}(F,E))} \le \sum_{\substack{\be_1+\cdots+\be_k=k\\\be_i\ge 0}} N(\be_1,\ldots,\be_k) \prod_{i=1}^k \|S_i^{(\be_i)}(x)\|_{L^{\be_i}(E,L(F,E))},$$ where the nonnegative integers $N(\be_1,\ldots,\be_k)$ are given by the identity $$\sum_{\substack{\be_1+\cdots+\be_k=k\\\be_i\ge 0}} N(\be_1,\ldots,\be_k) \prod_{i=1}^k t_i^{\be_i} = \prod_{j=1}^k \sum_{\ell=1}^j t_\ell.$$ Since $\prod_{j=1}^k \sum_{\ell=1}^j t_\ell \le (\sum_{\ell=1}^k t_\ell)^k = \sum k! \prod_{i=1}^k \frac{t_i^{\be_i}}{\be_i!}$, where the sum is taken over all $\be_i\ge 0$ so that $\be_1+\cdots+\be_k=k$, we obtain $$\label{eq:RS} \|R_k(x)\|_{L(E,L^{k}(F,E))} \le \sum_{\substack{\be_1+\cdots+\be_k=k\\\be_i\ge 0}} k! \prod_{i=1}^k \frac{\|S_i^{(\be_i)}(x)\|_{L^{\be_i}(E,L(F,E))}}{\be_i!}.$$ If we set $S(x) = S_k(x):= (f'(x))^{-1}$ for all $k\ge 1$, then for $n\ge 1$ $$g^{(n)}(y) = R_{n-1}(x) S(x), \quad (x=g(y)),$$ where the sequence $R_n$ is defined by . Applying Lemma \[lem:inverseclosed\] to $T = f'$ and using , we find that there exist $\mu,\nu \in \La$ and $D,H,\si,\ta \ge1$ so that $$\begin{aligned} \|R_k(x)\|_{L(E,L^{k}(F,E))} \le \sum_{\substack{\be_1+\cdots+\be_k=k\\\be_i\ge 0}} k! (D\si)^k \prod_{i=1}^k M^\mu_{\be_i} \le H \ta^k k! M^\nu_k, \end{aligned}$$ by . This implies . Proof of Theorem \[thm:bfM\]: the Beurling case {#sec:proofB} =============================================== The structure of the proof of the equivalence of the six items in Theorem \[thm:bfM\] is again represented by the diagram in , where now: 1. If $f \in \EbfM(\R)$ and $f(0) \ne 0$ then $1/f$ is $\EbfM$ on its domain of definition. The equivalences (1)&lt;=&gt;(2)&lt;=&gt;(3) -------------------------------------------- Lemma \[lem:raiFdB\] implies $(1) \Leftrightarrow (2)$, since follows from . The equivalence of (2) and (3) was shown in [@RainerSchindl12 4.11]. The cycle (1)=&gt;(4)=&gt;(6’)=&gt;(1) {#ssec:cycle1B} -------------------------------------- That (4) implies $(6')$ follows in the same way as in the Roumieu case, see Subsection \[ssec:cycle1\]. The implication $(6')$ $\Rightarrow$ (1) is a consequence of the following lemma. If $\EbfM(\R)$ is inverse closed, then $\fM$ satisfies . We follow an argument of [@Bruna80/81]. Consider the algebra $\cA := \EbfM(\R)$ and its subalgebra $\cB := \{f \in \cA : \|f\|_\infty :=\|f\|_{L^\infty(\R)}< \infty\}$. Endow $\cB$ with the topology generated by all seminorms $Q := \{\|~\|^{M^\la}_{K,\rh}\}_{\la,K,\rh} \cup\{\|~\|_{\infty}\}$. Then $\cB$ is a Fréchet algebra. If $f,g \in \cB$ are such that $|f(x)| \ge c >0$ for all $x \in \R$ and $\|f-g\|_{\infty} \le c/2$, then $|g(x)| \ge c/2 >0$, i.e., the set $\{f \in \cB : 1/f \in \cB\}$ is open in $\cB$. By [@Zelazko65 Thm 13.17], see also [@Bruna80/81 Prop 5.2], we may conclude that the algebra $\cB$ is locally m-convex, i.e., $\cB$ has an equivalent seminorm system $P=\{p\}$ such that $p(fg) \le p(f)p(g)$ for all $f,g \in \cB$. So for each $\la\in \La$, compact $K \subseteq \R$, and $\rh>0$ there exist $p \in P$, $q \in Q$, and constants $C,D >0$ such that $$\|f^m\|^{M^{\la}}_{K,\rh} \le C p(f^m) \le C p(f)^m \le C D^m q(f)^m, \quad f \in \cB, m \in \N.$$ We shall use this inequality for the functions $f_t(x) = e^{itx}$, and, since $\|f_t\|_\infty = 1 \le \|f_t\|^{M^\mu}_{[-a,a],\si}$ for each $\mu \in \La$ and $a,\si>0$, we can replace $q$ in the very same inequality by some seminorm $\|~\|^{M^\mu}_{[-a,a],\si}$. Then the proof of [@RainerSchindl12 4.11$(2)\Rightarrow(3)$] yields . The remaining implication (1) $\Rightarrow$ (4) (as well as (1) $\Rightarrow$ (5) below) we shall deduce from the corresponding result in the Roumieu case by means of the following lemma, which is a variation of [@Komatsu79b Lemma 6]. \[Komatsu\] Let $L\in \R_{\ge 0}^\N$ and $M^1,M^2,M^3 \in \R_{>0}^\N$ be sequences satisfying $L \lhd M^1 \le M^2 \le M^3$ and $(M^i_k)^{1/k} \to \infty$ for $i=1,2,3$, and assume that there exist $1 \le H_1 \le H_2$ so that $$\label{eq:cond123} (M^1_j)^{1/j} \le H_1 (M^2_k)^{1/k} \le H_2 (M^3_\ell)^{1/\ell} \quad \text{ for } j \le k \le \ell.$$ Then there exist sequences $N^1, N^2 \in \R_{>0}^\N$ with $L \le N^1 \le N^2 \lhd M^3$ satisfying $(N^i_k)^{1/k} \to \infty$ for $i=1,2$ and so that $$\label{eq:cond12} (N^1_j)^{1/j} \le \sqrt{H_1} (N^2_k)^{1/k} \quad \text{ for } j \le k.$$ Without loss of generality we may assume that $L_k>0$ for all $k$; otherwise replace $L$ by $\bar L$ where $\bar L_k = L_k$ if $L_k>0$ and $\bar L_k = 1$ if $L_k=0$ (we still have $\bar L \lhd M^1$ since $(M^1_k)^{1/k} \to \infty$). The sequences, for $i=1,2,3$, $$c^i_k := \Big(\frac{M^i_k}{L_k}\Big)^{\frac 1 k}$$ satisfy $c^i_k \to \infty$, since $L \lhd M^i$. We define, for $i=1,2$, $$\label{eq:Ndef} (N^i_k)^{\frac{1}{k}} := \max\Big\{\sqrt{(M^i_k)^{\frac{1}{k}}},\max_{j \le k} \frac{(M^i_j)^{\frac{1}{j}}}{c^i_j} \Big\} = \max\Big\{\sqrt{(M^i_k)^{\frac{1}{k}}},\max_{j \le k} L_j^{\frac 1 j} \Big\}.$$ Then clearly $(N^i_k)^{1/k} \to \infty$ and $L \le N^1 \le N^2$. For each $\ep>0$ there exists $j_{\ep,i}$ so that $1/c^i_j \le \ep$ for $j>j_{\ep,i}$. Thus, by , $$\begin{aligned} \Big(\frac{N^i_k}{M^{i+1}_k}\Big)^{\frac{1}{k}} &\le \max\Big\{(M^{i+1}_k)^{-\frac{1}{2k}}, (M^{i+1}_k)^{-\frac{1}{k}}\max_{j \le k} \frac{(M^i_j)^{\frac{1}{j}}}{c^i_j } \Big\} \\ & \le \max\Big\{(M^{i+1}_k)^{-\frac{1}{2k}}, (M^{i+1}_k)^{-\frac{1}{k}} \max_{j \le j_{\ep,i}} \frac{(M^{i}_j)^{\frac{1}{j}}}{c^i_j }, H_i \ep \Big\} \le H_i \ep, \end{aligned}$$ for $k$ sufficiently large, i.e., $N^1 \lhd M^2$ and $N^2 \lhd M^3$. It remains to show . But this is immediate from and , indeed for $j \le k$, $$\begin{aligned} (N^1_j)^{\frac{1}{j}} = \max\Big\{\sqrt{(M^1_j)^{\frac{1}{j}}},\max_{h \le j} L_h^{\frac 1 h} \Big\} \le \max\Big\{\sqrt{H_1(M^2_k)^{\frac{1}{k}}},\max_{h \le k} L_h^{\frac 1 h} \Big\} \le \sqrt{H_1} (N^2_k)^{\frac{1}{k}} \end{aligned}$$ as required. The following proposition implies (1) $\Rightarrow$ (4), by the analogue of Remark \[rem:shift\]. \[prop:ODEb\] Let $\fM$ be a weight matrix satisfying and . Let $X$ be a Banach space and let $f : W \to X$ be an $C^\infty$-mapping defined in an open subset $W \subseteq X \times \R$ and satisfying $$\begin{aligned} \label{eq:fb} \begin{split} \forall \nu \in \La, \rh>0 ~\exists C \ge1 &~\forall (k,\ell) \in \N^2 ~\forall (x,t) \in W : \\ &\|f^{(k,\ell)}(x,t)\|_{L^{k,\ell}(X,\R;X)} \le C \rh^{k+\ell} (k+\ell-1)! M^\nu_{k+\ell-1}. \end{split}\end{aligned}$$ Then the solution $x : I \to X$ of the initial value problem $$\label{eq:IVPb} x'(t) = f(x(t),t), \quad x(0) = x_0,$$ satisfies $$\begin{aligned} \label{eq:xb} \forall \la \in \La, \si>0 ~\exists D \ge1 ~\forall k \in \N ~\forall t \in I : \|x^{(k)}(t)\| \le D \si^{k} (k-1)! M^\la_{k-1}.\end{aligned}$$ The analogue of Remark \[rem:1\] applies. In the same way as in the Roumieu case (Proposition \[prop:ODEr\]) we may reduce to the initial value problem , where now $g \in C^\infty(U,Y)$ satisfies $$\begin{aligned} \label{eq:g2Beu} \forall \nu \in \La, \rh>0 ~\exists C \ge1 ~\forall k \in \N ~\forall y \in U : \|g^{(k)}(y)\|_{L^k(Y,Y)} \le C \rh^k (k-1)! M^\nu_{k-1}. \end{aligned}$$ There is a unique solution $y \in C^\infty(I,Y)$ defined on some interval $I$. Let $\la \in \La$ be fixed. By there exist $\mu, \nu \in \La$ and $H,J\ge 1$ such that $$\label{eq:lamunu} (M^\nu_j)^{1/j} \le H (M^\mu_k)^{1/k} \le J (M^\la_\ell)^{1/\ell} \quad \text{ for } j \le k \le \ell,$$ and, by , $$\label{eq:unbounded} \lim (M^\nu_k)^{1/k} =\lim (M^\mu_k)^{1/k} =\lim (M^\la_k)^{1/k} =\infty.$$ We may assume without loss of generality that $\nu \le \mu \le \la$ and thus $M^\nu \le M^\mu \le M^\la$. If we set $$\begin{aligned} L_{k-1} := \sup_{y \in U} \tfrac{1}{(k-1)!} \|g^{(k)}(y)\|_{L^k(Y,Y)}, \end{aligned}$$ then implies $L \lhd M^\nu$. Thus, by applying Lemma \[Komatsu\], we find sequences $N^1$ and $N^2$ with $L \le N^1 \le N^2 \lhd M^\la$ satisfying $(N^i_k)^{1/k} \to \infty$ for $i=1,2$ and so that $$(N^1_j)^{1/j} \le \sqrt{H} (N^2_k)^{1/k} \quad \text{ for } j \le k.$$ Repeating the proof of Proposition \[prop:ODEr\] (with $N^1$ in place of $M^\la$ and $N^2$ in place of $M^\mu$) we may conclude that $$\begin{aligned} \exists D,\si \ge1 ~\forall k \in \N ~\forall t \in I : \|y^{(k)}(t)\| \le D \si^{k} (k-1)! N^2_{k-1} \end{aligned}$$ which implies $$\begin{aligned} \forall \ta>0 ~\exists E \ge1 ~\forall k \in \N ~\forall t \in I : \|y^{(k)}(t)\| \le E \ta^{k} (k-1)! M^\la_{k-1}, \end{aligned}$$ since $N^2 \lhd M^\la$. As $\la$ was arbitrary, the result follows. The cycle (1)=&gt;(5)=&gt;(6)=&gt;(6’)=&gt;(1) {#the-cycle-15661} ---------------------------------------------- Obviously, $(5) \Rightarrow (6) \Rightarrow (6')$, and $(1) \Leftrightarrow (6')$, by Subsection \[ssec:cycle1B\]. Finally, (1) $\Rightarrow$ (5) follows from the following proposition. \[prop:inverseB\] Let $\fM$ be a weight matrix satisfying and . Let $f : U \to V$ be a $C^\infty$-mapping between open subsets $U \subseteq E$ and $V \subseteq F$ of Banach spaces $E, F$ satisfying $$\begin{aligned} \label{eq:finvB} \begin{split} \forall \nu \in \La, \rh>0 ~\exists C \ge1 &~\forall k \in \N ~\forall x \in U : \|f^{(k)}(x)\|_{L^{k}(E,F)} \le C \rh^{k-1} (k-1)! M^\nu_{k-1} \end{split}\end{aligned}$$ and so that $f'(x_0) \in L(E,F)$ is invertible. Then there exist neighborhoods $x_0 \in U_0 \subseteq U$ and $f(x_0) \in V_0 \subseteq V$ and a $C^\infty$-mapping $g : V_0 \to U_0$ satisfying $$\begin{aligned} \label{eq:ginvB} \begin{split} \forall \mu \in \La,\si>0 ~\exists D \ge1 &~\forall k \in \N ~\forall y \in V_0 : \|g^{(k)}(y)\|_{L^{k}(F,E)} \le D \si^{k-1} (k-1)! M^\mu_{k-1} \end{split}\end{aligned}$$ and such that $f {\circ}g = \on{id}_{V_0}$. Let $\la \in \La$ be fixed. By and , there exist $\mu, \nu \in \La$ satisfying and . If we set $$\begin{aligned} L_{k-1} := \sup_{x \in U} \tfrac{1}{(k-1)!} \|f^{(k)}(x)\|_{L^k(E,F)}, \end{aligned}$$ then implies $L \lhd M^\nu$. In analogy to the proof of Proposition \[prop:ODEb\], we repeat the proof of Proposition \[prop:inverse\] (and that of Lemma \[lem:inverseclosed\]) with the sequences $N^1$ and $N^2$ provided by Lemma \[Komatsu\]. Proof of Theorems \[thm:rM\], \[thm:bM\], \[thm:rom\], and \[thm:bom\] {#sec:proofMom} ====================================================================== Proof of Theorems \[thm:rM\] and \[thm:bM\] ------------------------------------------- Apply Theorems \[thm:rfM\] and \[thm:bfM\] to the constant weight matrix $\fM = \{M\}$. Proof of Theorems \[thm:rom\] and \[thm:bom\] --------------------------------------------- For a weight function $\om$ and each $\rh>0$ consider the sequence $\Om^\rh \in \R_{>0}^\N$ defined by $$\Om^\rh_k:=\tfrac{1}{k!}\exp(\tfrac{1}{\rh} \vh^*(\rh k)).$$ By the properties of $\vh^*$, the collection $\fW := \{\Om^\rh : \rh>0\}$ forms a weight matrix, and we have $\cE^{[\om]}(U) = \cE^{[\fW]}(U)$ as locally convex spaces, by [@RainerSchindl12]. Moreover, $\fW$ satisfies as well as . If $\om(t)=O(t)$ as $t\to \infty$, then $\fW$ satisfies , and if $\om(t)=o(t)$ as $t\to \infty$, then $\fW$ satisfies . Thus Theorems \[thm:rom\] and \[thm:bom\] are immediate consequences of Theorems \[thm:rfM\], \[thm:bfM\], and [@RainerSchindl12 6.3, 6.5]. Weight sequences as required in Remark \[rem:rai\] {#appendix} ================================================== Let us now find explicit sequences that satisfy the requirements of Remark \[rem:rai\]. To this end we construct a weight sequence $M=(M_k)$ such that $(M_{k+1}/M_k)^{1/k}$ is bounded, $M^{1/k}_k$ tends to $\infty$ but is not almost increasing, and $k!^s \le M_k \le k!^t$ for suitable $s,t > 0$ and sufficiently large $k$. Since for every Gevrey sequence $G^s =(k!^s)_k$ (where $s\ge 0$), $(G^s_k)^{1/k}$ is increasing (and tends to $\infty$ if $s>0$), the pair of sequences $(G^s, M)$, or $(M,G^t)$, will fulfill the requirements of Remark \[rem:rai\] (after adjusting finitely many terms of one sequence). Let $k_j:=2 \uparrow \uparrow j = 2^{2^{{\cdot}^{{\cdot}^ 2}}}$ ($j$ times) for $j\ge 1$ and $k_0:=0$. Let $\vh : [0,\infty) \to [0,\infty)$ be the function whose graph is the polygon with vertices $\{v_j=(k_j,\vh(k_j)): j \in \N\}$ defined by $$\begin{aligned} \vh(0):=0, \quad \vh(2):= 8 \log 2,\quad \vh(k_j) := \begin{cases} k_j \log k_{j+1} & \text{ $j$ even} \\ k_j \log (k_j k_{j-2}) & \text{ $j$ odd} \end{cases}, \quad (j\ge 2).\end{aligned}$$ We claim that the sequence $M=(M_k)$ defined by $M_k:= \exp(\vh(k))/k!$ satisfies: 1. $M$ is a weight sequence, i.e., $M$ is weakly log-convex, 2. $\sup_k (\frac{M_{k+1}}{M_k})^{1/k}< \infty$, 3. $M^{1/k}_k \to \infty$, 4. $M_k^{1/k}$ is not almost increasing, 5. $k!^s \le M_k \le k!^t$ for all $0\le s \le 1/4$, $t> 3$, and all $k\ge k_0(t)$. To see that $M$ is weakly log-convex it suffices to show that the slopes of the line segments in the graph of $\vh$ are increasing. Let $a_j$ denote the slope of the line segment left of the vertex $v_j$. Then, for $i\ge 2$, $$\begin{aligned} \label{eq:A1} \begin{split} a_{2i-1} &= \frac{k_{2i-1} \log (k_{2i-1} k_{2i-3}) - k_{2i-2} \log k_{2i-1}}{k_{2i-1}-k_{2i-2}} = \frac{\frac{5}{4} k_{2i-2}-1}{k_{2i-2}-1} \log k_{2i-1} \\ a_{2i} &= \frac{k_{2i} \log k_{2i+1} - k_{2i-1} \log (k_{2i-1} k_{2i-3})}{k_2i-k_{2i-1}} = \frac{4 k_{2i-1}-\frac{5}{4}}{k_{2i-1}-1} \log k_{2i-1} \\ a_{2i+1} &= \frac{k_{2i+1} \log (k_{2i+1} k_{2i-1}) - k_{2i} \log k_{2i+1}}{k_{2i+1}-k_{2i}} = \frac{5 k_{2i}-4}{k_{2i}-1} \log k_{2i-1} \end{split}\end{aligned}$$ and $a_{2i-1} \le a_{2i} \le a_{2i+1}$. This proves (1). Let us check (2). Since $$\begin{aligned} \Big(\frac{M_{k+1}}{M_k}\Big)^{1/k} = \frac{\exp(\frac{\vh(k+1)}{k}-\frac{\vh(k)}{k})}{(k+1)^{1/k}} \le \exp(\tfrac{\vh(k+1)}{k}-\tfrac{\vh(k)}{k}), \end{aligned}$$ it suffices to show that $\tfrac{\vh(k+1)}{k}-\tfrac{\vh(k)}{k}$ is bounded or equivalently that the slope of the line segments of $\vh$ increases at most linearly in $k$. This is obvious from . Thanks to $k! \le k^k \le e^k k!$ we have $$\frac{\exp(\frac{\vh(k)}{k})}{k} \le M^{1/k}_k \le \frac{\exp(\frac{\vh(k)}{k}+1)}{k}.$$ This implies (3). To show (4) let $j$ be even. Then $$\begin{aligned} \log \frac{M^{1/k_j}_{k_j}}{M^{1/k_{j+1}}_{k_{j+1}}} \ge \log k_j + \log k_{j+1} - \log (k_{j+1} k_{j-1}) -1 = \log k_{j-1}-1 \to \infty,\end{aligned}$$ as required. Finally, $k!^s \le M_k \le k!^t$ is equivalent to $1+s \le \tfrac{\vh(k)}{\log k!} \le 1+t$. We have $$\frac{\vh(k_j)}{k_j\log k_j} = \begin{cases} 2 & \text{ $j$ even} \\ 1+\frac{1}{4} & \text{ $j$ odd} \end{cases}$$ and $\tfrac{\vh(k)}{k\log k} \le \tfrac{\vh(k)}{\log k!} \le 2 \tfrac{\vh(k)}{k\log k}$. So the first inequality in (5) follows thanks to the fact that $k!^s$ is log-convex for each $s\ge 0$. For the second inequality we observe that in view of the slope $a_i$ is dominated by the increment $\log ((k+1)!^{1+t}) - \log (k!^{1+t}) = (1+t) \log (k+1)$ for all $k_{i-1}\le k < k_i$ provided that $t>3$ and that $i$ is sufficiently large depending on $t$. \[2\][ [\#2](http://www.ams.org/mathscinet-getitem?mr=#1) ]{} \[2\][\#2]{} [10]{} E. Bierstone and P. D. Milman, *Resolution of singularities in [D]{}enjoy-[C]{}arleman classes*, Selecta Math. (N.S.) **10** (2004), no. 1, 1–28. J. Bonet, R. Meise, and S. N. Melikhov, *A comparison of two different ways to define classes of ultradifferentiable functions*, Bull. Belg. Math. Soc. Simon Stevin **14** (2007), 424–444. R. W. Braun, R. Meise, and B. A. Taylor, *Ultradifferentiable functions and [F]{}ourier analysis*, Results Math. **17** (1990), no. 3-4, 206–237. J. Bruna, *On inverse-closed algebras of infinitely differentiable functions*, Studia Math. **69** (1980/81), no. 1, 59–68. E. M. Dyn’kin, *[Pseudoanalytic extension of smooth functions. The uniform scale.]{}*, Transl., Ser. 2, Am. Math. Soc. **115** (1980), 33–58 (English). C. Fern[á]{}ndez and A. Galbis, *Superposition in classes of ultradifferentiable functions*, Publ. Res. Inst. Math. Sci. **42** (2006), no. 2, 399–419. M. Koike, *Inverse mapping theorem in the ultradifferentiable class*, Proc. Japan Acad. Ser. A Math. Sci. **72** (1996), no. 8, 171–172. H. Komatsu, *Ultradistributions. [I]{}. [S]{}tructure theorems and a characterization*, J. Fac. Sci. Univ. Tokyo Sect. IA Math. **20** (1973), 25–105. , *An analogue of the [C]{}auchy-[K]{}owalevsky theorem for ultradifferentiable functions and a division theorem for ultradistributions as its dual*, J. Fac. Sci. Univ. Tokyo Sect. IA Math. **26** (1979), no. 2, 239–254. , *The implicit function theorem for ultradifferentiable mappings*, Proc. Japan Acad. Ser. A Math. Sci. **55** (1979), no. 3, 69–72. , *Ultradifferentiability of solutions of ordinary differential equations*, Proc. Japan Acad. Ser. A Math. Sci. **56** (1980), no. 4, 137–142. A. Kriegl, P. W. Michor, and A. Rainer, *The convenient setting for [D]{}enjoy–[C]{}arleman differentiable mappings of [B]{}eurling and [R]{}oumieu type*, (2011), accepted for publication in Rev. Mat. Complut., ar[X]{}iv:1111.1819. A. Rainer and G. Schindl, *Composition in ultradifferentiable classes*, Studia Math. **224** (2014), no. 2, 97–131. C. Roumieu, *Ultra-distributions définies sur [$R\sp{n}$]{} et sur certaines classes de variétés différentiables*, J. Analyse Math. **10** (1962/1963), 153–192. W. Rudin, *Division in algebras of infinitely differentiable functions*, J. Math. Mech. **11** (1962), 797–809. G. Schindl, *The convenient setting for ultradifferentiable mappings of [B]{}eurling- and [R]{}oumieu-type defined by a weight matrix*, (2014), ar[X]{}iv:1412.6165. J. A. Siddiqi, *Inverse-closed [C]{}arleman algebras of infinitely differentiable functions*, Proc. Amer. Math. Soc. **109** (1990), no. 2, 357–367. T. Yamanaka, *Inverse map theorem in the ultra-[$F$]{}-differentiable class*, Proc. Japan Acad. Ser. A Math. Sci. **65** (1989), no. 7, 199–202. , *On [ODE]{}s in the ultradifferentiable class*, Nonlinear Anal. **17** (1991), no. 7, 599–611. W. [Ż]{}elazko, *Metric generalizations of [B]{}anach algebras*, Rozprawy Mat. **47** (1965), 70. [^1]: Supported by FWF-Projects P 26735-N25 and P 23028-N13

--- abstract: 'In practical analysis, domain knowledge about analysis target has often been accumulated, although, typically, such knowledge has been discarded in the statistical analysis stage, and the statistical tool has been applied as a black box. In this paper, we introduce sign constraints that are a handy and simple representation for non-experts in generic learning problems. We have developed two new optimization algorithms for the sign-constrained regularized loss minimization, called the sign-constrained Pegasos (SC-Pega) and the sign-constrained SDCA (SC-SDCA), by simply inserting the sign correction step into the original Pegasos and SDCA, respectively. We present theoretical analyses that guarantee that insertion of the sign correction step does not degrade the convergence rate for both algorithms. Two applications, where the sign-constrained learning is effective, are presented. The one is exploitation of prior information about correlation between explanatory variables and a target variable. The other is introduction of the sign-constrained to SVM-Pairwise method. Experimental results demonstrate significant improvement of generalization performance by introducing sign constraints in both applications.' --- Introduction ============ The problem of regularized loss minimization (e.g. @HasTibFri-book09a) is often described as \[eq:prob-rlm-uncon\] & P()\^[d]{},\ & P() := \^[2]{} + (\^),\ & := \^[dn]{}, aiming to obtain a linear predictor $\left<{{\bm{w}}},{{\bm{x}}}\right>$ for an unknown input ${{\bm{x}}}\in{{\mathbb{R}}}^{d}$. Therein, $\Phi :{{\mathbb{R}}}^{n}\to{{\mathbb{R}}}$ is a loss function which is the sum of convex losses for $n$ examples: $\Phi({{\bm{z}}}) := \sum_{i=1}^{n}\phi_{i}(z_{i})$ for ${{\bm{z}}}:= \left[z_{1},\dots,z_{n}\right]^\top\in{{\mathbb{R}}}^{n}$. This problem covers a large class of machine learning algorithms including support vector machine, logistic regression, support vector regression, and ridge regression. In this study, we pose *sign constraints* [@Lawson1995solving] to the entries in the model parameter ${{\bm{w}}}\in{{\mathbb{R}}}^{d}$ in the unconstrained minimization problem . We divide the index set of $d$ entries into three exclusive subsets, ${{\mathcal{I}}}_{+}$, ${{\mathcal{I}}}_{0}$, and ${{\mathcal{I}}}_{-}$, as $\{1,\dots,d\} = {{\mathcal{I}}}_{+}\cup{{\mathcal{I}}}_{0}\cup{{\mathcal{I}}}_{-}$ and impose on the entries in ${{\mathcal{I}}}_{+}$ and ${{\mathcal{I}}}_{-}$, \[eq:sgncon\] &h\_[+]{},w\_[h]{}0, && h’\_[-]{},w\_[h’]{}0. Sign constraints can introduce prior knowledge directly to learning machines. For example, let us consider a binary classification task. In case that $h$-th explanatory variable $x_{h}$ is positively correlated to a binary class label $y\in\{\pm 1\}$, then a positive weight coefficient $w_{h}$ is expected to achieve a better generalization performance than a negative coefficient, because without sign constraints, the entry $w_{h}$ in the optimal solution might be negative due to small sample problem. On the other hand, in case that $x_{h}$ is negatively correlated to the class label, a negative weight coefficient $w_{h}$ would yield better prediction. If sign constraints were explicitly imposed, then inadequate signs of coefficients could be avoided. The strategy of sign constraints for generic learning problems has rarely been discussed so far, although there are extensive reports for non-negative least square regression supported by many successful applications including sound source localization: [@YuanqingLin2004-icassp], tomographic imaging [@JunMa2013-algo], spectral analysis [@QiangZhang07-asrc], hyperspectral image super-resolution [@DonFuShi16], microbial community pattern detection [@CaiGuKen17], face recognition [@YangfengJi2009-icmla; @HeZheHu13], and non-negative image restoration [@Henrot2013-icassp; @Landi2012-na; @YanfeiWang2007-ipse; @Shashua2005-icml]. In most of them, non-negative least square regression is used as an important ingredient of bigger methods such as non-negative matrix factorization [@lee2001algorithms; @WanTiaYu17; @Kimura2016column; @Fvotte2011algo; @Ding2006ortho]. Several efficient algorithms for the non-negative least square regression have been developed. The active set method by @Lawson1995solving has been widely used in many years, and several work [@DongminKim2010-siamjsc; @DongminKim2007-siam; @Bierlaire1991-laa; @Portugal1994comparison; @More91-siamjo; @ChihJenLin1999-siamjo; @Morigi2007-joam] have accelerated optimization by combining the active set method with the projected gradient approach. Interior point methods [@Bellavia2006; @Heinkenschloss99-mp; @Kanzow06-coa] have been proposed as an alternative algorithm for non-negative least square regression. However, all of them cannot be applied to generic regularized loss minimization problems. In this paper, we present two algorithms for the sign-constrained regularized loss minimization problem with generic loss functions. A surge of algorithms for unconstrained regularized empirical loss minimization have been developed such as SAG [@Roux12a-sag; @Schmidt2016-sag], SVRG [@Johnson13a-svrg], Prox-SVRG [@LinXiao2014-siamjo], SAGA [@Defazio2014-nips], Kaczmarz [@Needell2015], EMGD [@LijunZhang2013-nips], and Finito [@defazio2014finito]. This study focuses on two popular algorithms, Pegasos [@Shalev-Shwartz11-pegasos] and SDCA [@Shalev-Shwartz2013a-SDCA]. A prominent characteristic of the two algorithms is unnecessity to choose a step size. Some of the other optimization algorithms guarantee convergence to the optimum under the assumption of a small step size, although the step size is often too small to be used. Meanwhile, the theorem of Pegasos has been developed with a step size $\eta_{t}=1/(\lambda t)$ which is large enough to be adopted actually. SDCA needs no step size. Two new algorithms developed in this study for the sign-constrained problems are simple modifications of Pegasos and SDCA. The contributions of this study are summarized as follows. - Sign constraints are introduced to generic regularized loss minimization problems. - Two optimization algorithms for the sign-constrained regularized loss minimization, called *SC-Pega* and *SC-SDCA*, were developed by simply inserting the *sign correction step*, introduced in Section \[s:scpega\], to the original Pegasos and SDCA. - Our theoretical analysis ensures that both SC-Pega and SC-SDCA do not degrade the convergence rates of the original algorithms. - Two attractive applications, where the sign-constrained learning is effective, are presented. The one is exploitation of prior information about correlation between explanatory variables and a target variable. The other is introduction of the sign-constrained to SVM-Pairwise method [@LiaNob03-jcb]. - Experimental results demonstrate significant improvement of generalization performance by introducing sign constraints in both two applications. Problem Setting =============== The feasible region can be expressed simply as \[eq:fearegion\] := { \^[d]{}| \_[d]{}} where ${{\bm{c}}}= \left[ c_{1},\dots,c_{d} \right]^\top\in\{0,\pm 1\}^{d}$, each entry is given by \[eq:scvec-c-def\] c\_[h]{} := +1 &h\_[+]{},\ 0 &h\_[0]{},\ -1 &h\_[-]{}. Using ${{\mathcal{S}}}$, the optimization problem discussed in this paper can be expressed as \[eq:prob-rlm-signcon\] & P(). \[assum:four-for-rlm-signcon\] Throughout this paper, the following assumptions are used: $$\begin{aligned} \text{(a) } & \text{$\Phi(\cdot)$ is a convex function.} & \text{(b) } & \text{$\frac{1}{n}\Phi({{\bm{0}}})\le {r_{\text{loss}}}$.} \\ \text{(c) } & \text{$\forall {{\bm{s}}}\in{{\mathbb{R}}}^{n}$, $\Phi({{\bm{s}}})\ge 0$.} & \text{(d) } & \text{$\forall i$, $\lVert{{\bm{x}}}_{i}\rVert \le R$. }\end{aligned}$$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Name Definition Label Type ${r_{\text{loss}}}$ ---------------------- ------------------------------------------------------------------------------------ -------------------------- --------------------- ----------------------------------- Classical hinge loss $\phi_{i}(s) := \max(0,1-y_{i}s)$ $y_{i}\in\{\pm 1\}$ $1$-Lipschitz $1$ Smoothed hinge loss $\phi_{i}(s) := $y_{i}\in\{\pm 1\}$ $(1/\gamma)$-smooth $1-\frac{\gamma}{2}$ \begin{cases} 1 - y_{i}s -0.5\gamma \,\, &\text{ if } y_{i}s\in(-\infty,1-\gamma], \\ (1-y_{i}s)^{2}/(2\gamma) \,\, &\text{ if } y_{i}s\in(1-\gamma,1), \\ 0 &\text{ if } y_{i}s\in [1,+\infty). \end{cases}$ Logistic loss $\phi_{i}(s) := \log( 1 + \exp(-y_{i}s))$ $y_{i}\in\{\pm 1\}$ $0.25$-smooth $\log(2)$ Square error loss $\phi_{i}(s) := 0.5( s - y_{i} )^{2}$ $y_{i}\in{{\mathbb{R}}}$ $1$-smooth $\lVert{{\bm{y}}}\rVert^{2}/(2n)$ Absolute error loss $\phi_{i}(s) := | s - y_{i} |$ $y_{i}\in{{\mathbb{R}}}$ $1$-Lipschitz $\lVert{{\bm{y}}}\rVert_{1}/n$ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ Most of widely used loss functions satisfy the above assumptions. Several examples of such loss functions are described in Table \[tab:loss\]. If the hinge loss is chosen, the learning machine is a well-known instance called the support vector machine. If the square error loss is chosen, the learning machine is called the ridge regression. We denote the optimal solution to the constraint problem by ${{\bm{w}}}_{\star}:={\mathop{\textrm{argmin}}\limits}_{{{\bm{w}}}\in{{\mathcal{S}}}}P({{\bm{w}}}). $ We assume two types of loss functions: *$L$-Lipschitz continuous* function and *$(1/\gamma)$-smooth* function. Function $\phi_{i}:{{\mathbb{R}}}\to{{\mathbb{R}}}$ is said to be an $L$-Lipschitz continuous funciton if \[eq:Lipschitz-def\] s,, |\_[i]{}(s+)-\_[i]{}(s)| L ||. Such functions are often said shortly to be *$L$-Lipschitz* in this paper. Function $\phi_{i}:{{\mathbb{R}}}\to{{\mathbb{R}}}$ is a $(1/\gamma)$-smooth function if its derivative function is $L$-Lipschitz. For an index subset ${{\mathcal{A}}}\subseteq\{1,\dots,n\}$ and a vector ${{\bm{v}}}\in{{\mathbb{R}}}^{n}$, let ${{\bm{v}}}_{{{\mathcal{A}}}}$ be the subvector of ${{\bm{v}}}$ containing entries corresponding to ${{\mathcal{A}}}$. Let ${{\bm{X}}}_{{{\mathcal{A}}}}$ be a sub-matrix in ${{\bm{X}}}$ containing columns corresponding to ${{\mathcal{A}}}$. Let $\Phi(\cdot\,;\,{{\mathcal{A}}}):{{\mathbb{R}}}^{|{{\mathcal{A}}}|}\to{{\mathbb{R}}}$ be defined as (\_;) := \_[i]{}\_[i]{}(s\_[i]{}). Sign-Constrained Pegasos {#s:scpega} ======================== In the original Pegasos algorithm [@Shalev-Shwartz11-pegasos], $\phi_{i}$ is assumed to be the classical hinge loss function (See Table \[tab:loss\] for the definition). Each iterate consists of three steps: the *mini-batch selection step*, the *gradient step*, and the *projection-onto-ball step*. Mini-batch selection step chooses a subset ${{\mathcal{A}}}_{t}\subseteq\{1,\dots,n\}$ from $n$ examples at random. The cardinality of the subset is predefined as $|{{\mathcal{A}}}_{t}| = k$. Gradient step computes the gradient of P\_[t]{}() := \^[2]{} + (\_[\_[t]{}]{}\^;\_[t]{}). which approximates the objective function $P({{\bm{w}}})$. The current solution ${{\bm{w}}}_{t}$ is moved toward the opposite gradient direction as \_[t+1/2]{} &:= \_[t]{} - P\_[t]{}(\_[t]{})\ &= - \_[\_[t]{}]{} (\_[\_[t]{}]{}\^\_[t]{};\_[t]{}). At the projection-onto-ball step, the norm of the solution is shortened to be $\frac{1}{\sqrt{\lambda}\lVert{{\bm{w}}}_{t+1/2}\rVert}$ if the norm is over $\frac{1}{\sqrt{\lambda}\lVert{{\bm{w}}}_{t+1/2}\rVert}$: \_[t+1]{} := ( 1, ) \_[t+1/2]{}. The projection-onto-ball step plays an important role in getting a smaller upper-bound of the norm of the gradient of the regularization term in the objective, which eventually reduces the number of iterates to attain an $\epsilon$-approximate solution (i.e. $P(\tilde{{{\bm{w}}}})-P({{\bm{w}}}_{\star})\le\epsilon$). In the algorithm developed in this study, we simply inserts between those two steps, a new step that corrects the sign of each entry in the current solution ${{\bm{w}}}$ as w\_[h]{} (0,w\_[h]{})&h\_[+]{},\ (0,w\_[h]{})&h\_[-]{},\ w\_[h]{}&h\_[0]{}, which can be rewritten equivalently as ${{\bm{w}}}\leftarrow {{\bm{w}}}+ {{\bm{c}}}\odot(-{{\bm{c}}}\odot{{\bm{w}}})_{+}$ where the operator $(\cdot)_{+}$ is defined as $\forall {{\bm{x}}}\in{{\mathbb{R}}}^{d}$, $({{\bm{x}}})_{+}:=\max({{\bm{0}}},{{\bm{x}}})$. Data matrix ${{\bm{X}}}\in{{\mathbb{R}}}^{d\times n}$, loss function $\Phi:{{\mathbb{R}}}^{n}\to{{\mathbb{R}}}$, regularization parameter $\lambda\in{{\mathbb{R}}}$, sign constraint parameter ${{\bm{c}}}\in\{\pm 1,0\}^{d}$, and mini-batch size $k$. **begin** ${{\bm{w}}}_{1}:={{\bm{0}}}_{d}$; {Initialization} Choose ${{\mathcal{A}}}_{t}\subseteq\{1,\dots,n\}$ uniformly at random such that $|A_{t}|=k$. ${{\bm{w}}}_{t+1/3} := \frac{t-1}{t}{{\bm{w}}}_{t} - \frac{1}{\lambda t}{{\bm{X}}}_{{{\mathcal{A}}}_{t}}\nabla\Phi({{\bm{X}}}_{{{\mathcal{A}}}_{t}}^\top{{\bm{w}}}_{t-1};{{\mathcal{A}}}_{t})$; ${{\bm{w}}}_{t+2/3} := {{\bm{w}}}_{t+1/3} + {{\bm{c}}}\odot(-{{\bm{c}}}\odot{{\bm{w}}}_{t+1/3})_{+}$; ${{\bm{w}}}_{t+1} := \min\left( 1, \sqrt{{r_{\text{loss}}}\lambda^{-1}}\lVert{{\bm{w}}}_{t+2/3}\rVert^{-1} \right) {{\bm{w}}}_{t+2/3}$; **return** $\tilde{{{\bm{w}}}}:=\sum_{t=1}^{T}{{\bm{w}}}_{t}/T$; **end.** The algorithm can be summarized as Algorithm \[alg:srpega-mbtch\]. Here, the loss function is not limited to the classical hinge loss. In the projection-onto-ball step, the solution is projected onto $\sqrt{{r_{\text{loss}}}\lambda^{-1}}$-ball instead of $(1/\sqrt{\lambda})$-ball to handle more general settings. Recall that ${r_{\text{loss}}}=1$ if $\phi_{i}$ is the hinge loss employed in the original Pegasos. It can be shown that the objective gap is bounded as follows. \[thm:srpega-bound\] Consider Algorithm \[alg:srpega-mbtch\]. If $\phi_{i}$ are $L$-Lipschitz continuous, it holds that - P(\_) ( + L R )\^[2]{} . See Subsection \[ss:proof-thm:srpega-bound\] for proof of Theorem \[thm:srpega-bound\]. This bound is exactly same as the original Pegasos, yet Algorithm \[alg:srpega-mbtch\] contains the sign correction step. Sign-Constrained SDCA ===================== The original SDCA is a framework for the unconstrained problems . In SDCA, a dual problem is solved instead of the primal problem. Namely, the dual objective is maximized in a iterative fashion with respect to the dual variables ${{\bm{\alpha}}}:=\left[\alpha_{1},\dots,\alpha_{n}\right]^\top\in{{\mathbb{R}}}^{n}$. The problem dual to the unconstrained problem  is given by \[eq:dual-rlm-uncon\] & D()\^[n]{}, where &D() := - \^[2]{} - \^[\*]{}(-). To find the maximizer of $D({{\bm{\alpha}}})$, a single example $i$ is chosen randomly at each iterate $t$, and a single dual variable $\alpha_{i}$ is optimized with the other $(n-1)$ variables $\alpha_{1},\dots,\alpha_{i-1}$, $\alpha_{i+1},\dots,\alpha_{n}$ frozen. If we denote by ${{\bm{\alpha}}}^{(t-1)}\in{{\mathbb{R}}}^{n}$ the value of the dual vector at the previous iterate $(t-1)$, the dual vector is updated as ${{\bm{\alpha}}}^{(t)}:={{\bm{\alpha}}}^{(t-1)}+\Delta\alpha{{\bm{e}}}_{i}$ where $\Delta\alpha\in{{\mathbb{R}}}$ is determined so that $\Delta \alpha \in {\mathop{\textrm{argmax}}\limits}_{\Delta \alpha\in{{\mathbb{R}}}}D_{t}(\Delta \alpha\,;\,{{\bm{w}}}^{(t-1)})$ where ${{\bm{w}}}^{(t-1)}=\frac{1}{\lambda n}{{\bm{X}}}{{\bm{\alpha}}}^{(t-1)}$ and $$\begin{gathered} D_{t}(\Delta \alpha\,;\,{{\bm{w}}}) := -\frac{\lambda}{2}\left\lVert{{\bm{w}}}+ \frac{\Delta \alpha}{\lambda n}{{\bm{x}}}_{i}\right\rVert^{2} -\frac{1}{n}\phi^{*}_{i}(-\alpha^{(t-1)}_{i}-\Delta \alpha). \end{gathered}$$ In case of the hinge loss, the maximizer of $D_{t}(\cdot\,;\,{{\bm{w}}}^{(t-1)})$ can be found within $O(d)$ computation. The primal variable ${{\bm{w}}}^{(t)}$ can also be maintained within $O(d)$ computation by ${{\bm{w}}}^{(t)}:={{\bm{w}}}^{(t-1)} + \frac{\Delta\alpha}{\lambda n}{{\bm{x}}}_{i}$. Now let us move on the sign-constrained problem. In addition to Algorithm \[alg:srpega-mbtch\] that is derived from Pegasos, we present another algorithm based on SDCA for solving the minimizer of $P({{\bm{w}}})$ subject to the sign constraint ${{\bm{c}}}\odot{{\bm{w}}}\ge{{\bm{0}}}_{d}$. Like Algorithm \[alg:srpega-mbtch\] that has been designed by inserting the sign correction step into the original Pegasos iterate, the new algorithm has been developed by simply adding the sign correction step in each SDCA iterate. The resultant algorithm is described in Algorithm \[alg:srsdca\]. Data matrix ${{\bm{X}}}\in{{\mathbb{R}}}^{d\times n}$, loss function $\Phi:{{\mathbb{R}}}^{n}\to{{\mathbb{R}}}$, regularization parameter $\lambda\in{{\mathbb{R}}}$, and sign constraint parameter ${{\bm{c}}}\in\{\pm 1,0\}^{d}$. **begin** ${{\bm{\alpha}}}^{(0)}:={{\bm{0}}}_{n}$; $\bar{{{\bm{w}}}}^{(0)}:={{\bm{0}}}_{d}$; ${{\bm{w}}}^{(0)}:={{\bm{0}}}_{d}$; {Initialization} $\Delta \alpha \in {\mathop{\textrm{argmax}}\limits}_{\Delta \alpha\in{{\mathbb{R}}}}D_{t}(\Delta \alpha\,;\,{{\bm{w}}}^{(t-1)})$; $\bar{{{\bm{w}}}}^{(t)}:=\bar{{{\bm{w}}}}^{(t-1)} + \frac{\Delta\alpha}{\lambda n}{{\bm{x}}}_{i}$; ${{\bm{w}}}^{(t)} := \bar{{{\bm{w}}}}^{(t)} + {{\bm{c}}}\odot(-{{\bm{c}}}\odot\bar{{{\bm{w}}}}^{(t)})_{+}$; **return** $\tilde{{{\bm{w}}}}:=\frac{1}{T-T_{0}}\sum_{t=T_{0}+1}^{T}{{\bm{w}}}^{(t-1)}$; **end.** For some loss functions, maximization at step 5 in Algorithm \[alg:srsdca\] cannot be given in a closed form. Alternatively, step 4 can be replaced to \[eq:sdca-prox-update\] [4: ]{}& := sq,\ &s := \_[\[0,s\^[-1]{}\_\]]{} ( + )s\_. Therein, we have defined $s_{\text{lb}}:=\lambda n \gamma / (\lambda n \gamma+R^{2})$, $z^{(t)}:=\left<{{\bm{w}}}^{(t-1)},{{\bm{x}}}_{i}\right>$, $q^{(t)}:=-\nabla\phi_{i}(z^{(t)})-\alpha_{i}^{(t-1)}$, and $\text{Clip}_{[a,b]}(x) := \max(a,\min(b,x))$. See Subsection \[ss:deriv-eq:sdca-prox-update\] for derivation of . We have found the following theorem that states the required number of iterates guaranteeing the expected primal objective gap below a threshold $\epsilon$ under the sign constraints. \[thm:converg-srsdca\] Consider Algorithm \[alg:srsdca\]. In case that $\phi_{i}$ are $L$-Lipschitz continuous (i.e. ), it holds that ${{\mathbb{E}}}[P(\tilde{{{\bm{w}}}})]-P({{\bm{w}}}_{\star})\le\epsilon$ if $T$ and $T_{0}$ are specified so that T\_[0]{} + { 0, n} and T T\_[0]{} + { n, } where $G:=4R^{2}L^{2}$. If $\phi_{i}$ are hinge loss functions, then $G:=R^{2}L^{2}$. In case that $\phi_{i}$ are $(1/\gamma)$-smooth, ${{\mathbb{E}}}[P(\tilde{{{\bm{w}}}})]-P({{\bm{w}}}_{\star})\le\epsilon$ is established if T &gt; T\_[0]{} ( n + ) ( ( n + ) ). See Subsections \[ss:proof-thm:converg-srsdca\] for proof of Theorem \[thm:converg-srsdca\]. Theorem \[thm:converg-srsdca\] suggests that the convergence rate of Algorithm \[alg:srsdca\] is not deteriorated compared to the original SDCA in both cases of $L$-Lipschitz and smooth losses, despite insertion of the sign correction step. Multiclass Classification ========================= In this section, we extend our algorithms to the multi-class classification setting of $m$ classes. Here, the model parameter is a ${{\bm{W}}}\in{{\mathbb{R}}}^{d\times m}$ instead of a vector ${{\bm{w}}}\in{{\mathbb{R}}}^{d}$. The loss function for each example ${{\bm{x}}}_{i}\in{{\mathbb{R}}}^{d}$ is of an $m$-dimensional vector. Here, the prediction is supposed to be done by taking the class with the maximal score among $s_{1}:=\left<{{\bm{w}}}_{1},{{\bm{x}}}\right>, \dots$, and $s_{m}:=\left<{{\bm{w}}}_{m},{{\bm{x}}}\right>$. Here, without loss of generality, the set of the class labels are given by ${{\mathcal{Y}}}:=\{1,\dots,m\}$. Several loss functions $\phi^{\text{m}}_{i}:{{\mathbb{R}}}^{m}\to{{\mathbb{R}}}$ are used for multiclass classification as follows. - Soft-max loss: \^\_[i]{}() := ( \_[y]{} ( s\_[y]{} -s\_[y\_[i]{}]{} ) ) Therein, $y_{i}$ is the true class label of $i$-th example. - Max-hinge loss; \^\_[i]{}() := \_[y]{} ( s\_[y]{} -s\_[y\_[i]{}]{} + \_[y,y\_[i]{}]{} ). - Top-$k$ hinge loss [@lapin-nips2015]: \^\_[i]{}() := \_[j=1]{}\^[k]{} ( (-\_[y\_[i]{}]{}\^)+ - \_[y\_[i]{}]{} )\_[\[j\]]{}. Therein, $x_{[j]}$ denotes the $j$-th largest value in a vector ${{\bm{x}}}\in{{\mathbb{R}}}^{m}$. The objective function for learning ${{\bm{W}}}\in{{\mathbb{R}}}^{d\times m}$ is defined as P\^() := \_\^[2]{} + \_[i=1]{}\^[n]{}\^\_[i]{}(\^\_[i]{}). The learning problem discussed is minimization of $P^{\text{m}}({{\bm{W}}})$ with respect to ${{\bm{W}}}$ subject to sign constraints (h,j)\_[+]{}, &W\_[h,j]{}0,\ (h’,j’)\_[-]{}, &W\_[h’,j’]{}0, with two exclusive set ${{\mathcal{E}}}_{+}$ and ${{\mathcal{E}}}_{-}$ such that \_[+]{}\_[-]{}{(h,j)\^[2]{}|h, j}. Introducing ${{\bm{C}}}\in\{0,\pm1\}^{d\times m}$ as C\_[h,j]{} := +1 &(h,j)\_[+]{},\ -1 &(h,j)\_[-]{},\ 0 & the feasible region can be expressed as \^ := { \^[dm]{}| \_[dm]{} }. The goal is here to develop algorithms that find \_:=\_[\^]{}P\^(). Define $\Phi^{\text{m}}(\cdot\,;\,{{\mathcal{A}}}):{{\mathbb{R}}}^{m\times k}\to{{\mathbb{R}}}$ as \^(\_;) := \_[i]{}\^\_[i]{}(\_[i]{}) where ${{\bm{S}}}_{{{\mathcal{A}}}}$ is the horizontal concatenation of columns in ${{\bm{S}}}:=\left[{{\bm{s}}}_{1},\dots,{{\bm{s}}}_{n}\right]\in{{\mathbb{R}}}^{m\times n}$ selected by a minibatch ${{\mathcal{A}}}$. We here use the following assumptions: $\Phi^{\text{m}}(\cdot)$ is a convex function; $\Phi^{\text{m}}({{\bm{O}}})\le n{r_{\text{loss}}}$; $\forall {{\bm{S}}}\in{{\mathbb{R}}}^{m\times n}$, $\Phi^{\text{m}}({{\bm{S}}})\ge 0$; $\forall i$, $\lVert{{\bm{x}}}_{i}\rVert \le R$. By extending Algorithm \[alg:srpega-mbtch\], an algorithm for minimization of $P^{\text{m}}({{\bm{W}}})$ subject to the sign constraints can be developed as described in Algorithm \[alg:srpega-mbtch-mc\]. Data matrix ${{\bm{X}}}\in{{\mathbb{R}}}^{d\times n}$, loss function $\Phi^{\text{m}}:{{\mathbb{R}}}^{m\times n}\to{{\mathbb{R}}}$, regularization parameter $\lambda\in{{\mathbb{R}}}$, sign constraint parameter ${{\bm{C}}}\in\{0,\pm 1\}^{d\times m}$, and mini-batch size $k$. **begin** ${{\bm{W}}}_{1}:={{\bm{0}}}_{d}$; {Initialization} Choose ${{\mathcal{A}}}_{t}\subseteq\{1,\dots,n\}$ uniformly at random such that $|A_{t}|=k$. ${{\bm{Z}}}_{t}:={{\bm{W}}}_{t-1}^\top{{\bm{X}}}_{{{\mathcal{A}}}_{t}}$; ${{\bm{W}}}_{t+1/3} := \frac{t-1}{t}{{\bm{W}}}_{t} - \frac{1}{\lambda t}{{\bm{X}}}_{{{\mathcal{A}}}_{t}}\left(\nabla\Phi({{\bm{Z}}}_{t}\,;\,{{\mathcal{A}}}_{t})\right)^\top$; ${{\bm{W}}}_{t+2/3} := {{\bm{W}}}_{t+1/3} + {{\bm{C}}}\odot\max({{\bm{O}}},-{{\bm{C}}}\odot{{\bm{W}}}_{t+1/3})$; ${{\bm{W}}}_{t+1} := \min\left( 1, \frac{{r_{\text{loss}}}}{\sqrt{\lambda}\lVert{{\bm{W}}}_{t+2/3}\rVert_{\text{F}}} \right) {{\bm{W}}}_{t+2/3}$; **return** $\tilde{{{\bm{W}}}}:=\sum_{t=1}^{T}{{\bm{W}}}_{t}/T$; **end.** The SDCA-based learning algorithm can also be developed for the multiclass classification task. In the algorithm, the dual variables are represented as a matrix ${{\bm{A}}}:=\left[{{\bm{\alpha}}}_{1},\dots,{{\bm{\alpha}}}_{n}\right]\in{{\mathbb{R}}}^{m\times n}$. At each iterate $t$, one of $n$ columns, ${{\bm{\alpha}}}_{i}$, is chosen at random instead of choosing one of a dual variable to update the matrix as ${{\bm{A}}}^{(t)}:= {{\bm{A}}}^{(t-1)}+\Delta{{\bm{\alpha}}}{{\bm{e}}}_{i}^\top$ where we have used the iterate number $(t)$ as the superscript of ${{\bm{A}}}$. To determine the value of $\Delta{{\bm{\alpha}}}$, the following auxiliary funcition is introduced: $$\begin{gathered} D_{t}(\Delta {{\bm{\alpha}}}\,;\,{{\bm{W}}}) := -\frac{\lVert{{\bm{x}}}_{i}\rVert^{2}}{2\lambda^{2}n} \lVert\Delta{{\bm{\alpha}}}\rVert^{2} \\ -\left<{{\bm{W}}}^\top {{\bm{x}}}_{i}, \Delta{{\bm{\alpha}}}\right> -\phi^{*}_{i}(-{{\bm{\alpha}}}^{(t-1)}_{i}-\Delta {{\bm{\alpha}}}). \end{gathered}$$ Data matrix ${{\bm{X}}}\in{{\mathbb{R}}}^{d\times n}$, loss function $\Phi:{{\mathbb{R}}}^{m\times n}\to{{\mathbb{R}}}$, regularization parameter $\lambda\in{{\mathbb{R}}}$, and sign constraint parameter ${{\bm{C}}}\in\{\pm 1,0\}^{d\times m}$. **begin** ${{\bm{A}}}^{(0)}:={{\bm{O}}}$; $\bar{{{\bm{W}}}}^{(0)}:={{\bm{O}}}$; ${{\bm{W}}}^{(0)}:={{\bm{O}}}$; {Initialization} $\Delta {{\bm{\alpha}}}\in {\mathop{\textrm{argmax}}\limits}_{\Delta{{\bm{\alpha}}}\in{{\mathbb{R}}}^{m}}D_{t}(\Delta{{\bm{\alpha}}}\,;\,{{\bm{W}}}^{(t-1)})$; $\bar{{{\bm{W}}}}^{(t)}:=\bar{{{\bm{W}}}}^{(t-1)} + \frac{1}{\lambda n}{{\bm{x}}}_{i}\Delta{{\bm{\alpha}}}^\top$; ${{\bm{W}}}^{(t)} := \bar{{{\bm{W}}}}^{(t)} + {{\bm{C}}}\odot\max({{\bm{O}}},-{{\bm{C}}}\odot\bar{{{\bm{W}}}}^{(t)})$; **return** $\tilde{{{\bm{W}}}}:=\frac{1}{T-T_{0}}\sum_{t=T_{0}+1}^{T}{{\bm{W}}}^{(t-1)}$; **end.** For both algorithms (Algorithms \[alg:srpega-mbtch-mc\] and \[alg:srsdca-mc\]), we can bound the required number of iterations similar to those presented in Theorems \[thm:srpega-bound\] and \[thm:converg-srsdca\]. Experiments =========== In this section, experimental results are reported in order to illustrate the effects of the sign constraints on classification and to demonstrate the convergence behavior. Prediction Performance ---------------------- The pattern recognition performance of the sign-constrained learning was examined on two tasks: *Escherichia coli* (*E. coli*) prediction and protein function prediction. #### *E. coli* Prediction {#e.coli-prediction .unnumbered} The first task is to predict *E. coli* counts in river water. The *E. coli* count has been used as an indicator for fecal contamination in water environment in many parts of the world [@ScoRosJen02-aem]. In this experiment, the data points with *E. coli* counts over 500 most probable number (MPN)/100 mL are assigned to positive class, and the others are negative. The hydrological and water quality monitoring data are used for predicting *E. coli* counts to be positive or negative. For ensuring the microbial safety in water usage, it is meaningful to predict *E. coli* counts on a real-time basis. The concentration of *E. coli* in water, which is measured by culture-dependent methods [@KobSanHat13-amb], has been used to monitor the fecal contamination in water environment, and has been proved to be effective to prevent waterborne infectious diseases in varied water usage styles. On the other hand, the real-time monitoring of *E. coli* counts has not yet been achieved. It take at least ten hours to obtain *E. coli* counts by culture-dependent methods, and also at least several hours are needed to measure the concentration of *E. coli* by culture-independent methods [@IshNakOza14-est; @IshKitSeg14-aem], such as polymerase chain reaction. Since it is possible to measure the some of the hydrological and water quality data with real-time sensors, the real-time prediction of *E. coli* counts will be realized if the hydrological and water quality data are available for the *E. coli* count prediction. Many training examples are required to obtain a better generalization performance. A serious issue, however, is that measuring the concentration of *E. Coli* is time-consuming and the cost of reagents is expensive. We here demonstrate that this issue can be relaxed by exploiting the domain knowledge hoarded in the field of water engineering. The hydrological and water quality data contain nine explanatory variables, WT, pH, EC, SS, DO, BOD, TN, TP, and flow rate. The explanatory variable $pH$ is divided into two variables, $\text{pH}_{+}\leftarrow\max(0,\text{pH}-7)$ and $\text{pH}_{-}\leftarrow\max(0,7-\text{pH})$. It is well-known, in the field of water engineering, that *E. coli* is increased, as WT, EC, SS, BOD, TN, and TP are larger, and as $\text{pH}_{+}$, $\text{pH}_{-}$，DO, and the flow rate are smaller. From this fact, we restrict the sign of entries in the predictor parameter ${{\bm{w}}}$ as follows. - Coefficients $w_{h}$ of six explanatory variables, WT, EC, SS, BOD, TN, and TP must be non-negative. - Coefficients $w_{h}$ of four explanatory variables, $\text{pH}_{+}$, $\text{pH}_{-}$, DO, flow rate must be non-positive. We actually measured the concentrations of *E. coli* 177 times from December 5th, 2011 to April 17th, 2013. We obtained 177 data points including 88 positives and 89 negatives. We chose ten examples out of 177 data points at random to use them for training, and the other 167 examples were used for testing. The prediction performance is evaluated by the precision recall break-even point (PRBEP) [@Joachims05-icml] and the ROC score. We compared the classical SVM with the sign-constrained SVM (SC-SVM) to examine the effects of sign constraints. We repeated this procedure 10,000 times and obtained 10,000 PRBEP and 10,000 ROC scores. [ll]{} (a) PRBEP & (b) ROC score\ \ ![ Improvements of generalization performances on *E. Coli* prediction. \[fig:prot404-demo332-mizu\]](001-k.demo332_02.prbep.eps "fig:"){width="22.00000%"} & $\qquad$ ![ Improvements of generalization performances on *E. Coli* prediction. \[fig:prot404-demo332-mizu\]](002-k.demo332_02.roc.eps "fig:"){width="22.00000%"} SC-SVM achieved significant improvement compared to the classical SVM. SC-SVM achieved PRBEP and ROC score of 0.808 and 0.863 on average over 10,000 trials, whereas those of the classical SVM were 0.757 and 0.810, respectively. The difference from the classical SVM on each trial is plotted in the histograms of Figure \[fig:prot404-demo332-mizu\]. Positive improvements of ROC scores were obtained in 8,932 trials out of 10,000 trials, whereas ROC scores were decreased only for 796 trials. For PRBEP, positive improvements were obtained on 7,349 trials, whereas deteriorations were observed only on 1,069 trials. ----------------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------------ \(a) Covtype \(b) W8a \(c) Phishing {width="30.00000%"} {width="30.00000%"} {width="30.00000%"} ----------------------------------------------------------------- ------------------------------------------------------------- ------------------------------------------------------------------ #### Protein Function Prediction {#protein-function-prediction .unnumbered} Category SC-SVM SVM ---------- ------------------- --------------- 1 **0.751** (0.011) 0.730 (0.010) 2 **0.740** (0.016) 0.680 (0.015) 3 **0.753** (0.011) 0.721 (0.011) 4 **0.762** (0.010) 0.734 (0.010) 5 **0.769** (0.012) 0.691 (0.013) 6 **0.690** (0.014) 0.614 (0.014) 7 **0.713** (0.024) 0.618 (0.022) 8 **0.725** (0.019) 0.667 (0.019) 9 **0.655** (0.024) 0.578 (0.023) 10 **0.743** (0.016) 0.710 (0.014) 11 **0.535** (0.019) 0.492 (0.018) 12 **0.912** (0.011) 0.901 (0.011) : ROC Scores for protein function prediction. \[tab:demo383\] In the field of molecular biology, understanding the functions of proteins is positioned as a key step for elucidation of cellular mechanisms. Sequence similarities have been a major mean to predict the function of an unannotated protein. At the beginning of this century, the prediction accuracy has been improved by combining sequence similarities with discriminative learning. The method, named SVM-Pairwise [@LiaNob03-jcb], uses a feature vector that contains pairwise similarities to annotated protein sequences. Several other literature [@LiuZhaXu14; @OguMum06; @LanBieCri04; @LanDenCri04] have also provided empirical evidences for the fact that the SVM-Pairwise approach is a powerful framework. Basically, if $n$ proteins are in a training dataset, the feature vector has $n$ entries, $x_{1}, \dots, x_{n}$. If we suppose that the first $n_{+}$ proteins in the training set are in positive class and the rest are negative, then the first $n_{+}$ similarities $x_{1}, \dots, x_{n_{+}}$ are sequence similarities to positive examples, and $x_{n_{+}+1}, \dots, x_{n}$ are similarities to negative examples. The $n$-dimensional vectors are fed to SVM and get the weight coefficients ${{\bm{w}}}:=\left[w_{1},\dots,w_{n}\right]^\top$. Then, the prediction score of the target protein is expressed as \_[i=1]{}\^[n\_[+]{}]{}w\_[i]{}x\_[i]{} + \_[i’=n\_[+]{}+1]{}\^[n]{}w\_[i]{}x\_[i]{}. The input protein sequence is predicted to have some particular cellular function if the score is over a threshold. It should be preferable that the first $n_{+}$ weight coefficients $w_{1},\dots,w_{n_{+}}$ are non-negative and that the rest of $(n-n_{+})$ weight coefficients $w_{n_{+}+1},\dots,w_{n}$ are non-positive. The SVM-Pairwise approach does not ensure those requirements. Meanwhile, our approach is capable to explicitly impose the constraints of w\_[1]{}0,…,w\_[n\_[+]{}]{}0, w\_[n\_[+]{}+1]{}0,…,w\_[n]{}0. This approach was applied to predict protein functions in *Saccharomyces cerevisiae* (*S. cerevisiae*). The annotations of the protein functions are provided in MIPS Comprehensive Yeast Genome Database (CYGD). The dataset contains 3,583 proteins. The Smith-Waterman similarities available from <https://noble.gs.washington.edu/proj/sdp-svm/> were used as sequence similarities among the proteins. The number of categories was 12. Some proteins have multiple cellular functions. Indeed, 1,343 proteins in the dataset have more than one function. From this reason, we pose 12 independent binary classification tasks instead of a single multi-class classification task. 3,583 proteins were randomly splited in half to get two datasets. The one was used for training, and the other was for testing. For 12 classification tasks, we repeated this procedure 100 times, and we obtained 100 ROC scores. Table \[tab:demo383\] reports the ROC scores averaged over 100 trials and the standard deviations for 12 binary classification tasks. The sign constraints significantly surpassed the classical training for all 12 tasks. Surprisingly, we observed that the ROC score of SC-SVM is larger than that of the classical SVM in every trial. Convergence ----------- We carried out empirical evaluation of the proposed optimization methods, the sign-constrained Pegasos (SC-Pega) and the sign-constrained SDCA (SC-SDCA), in order to illustrate the convergence of our algorithms to the optimum. For SC-Pega, we set the mini-batch size to $k=10$ and $k=100$. In this experiments, we used the smoothed hinge loss with $\gamma=0.01$ and $\lambda=1/n$. We used three datasets, Covtype ($n=581,012$ and $d=54$), W8a ($n=49,749$ and $d=300$), and Phishing ($n=11,055$ and $d=68$). The three datasets are for binary classification and available from LIBSVM web site (https://www.csie.ntu.edu.tw/ cjlin/libsvmtools/datasets/). Figure \[fig:demo423-gap\] depicts the primal objective gap against epochs, where the primal objective gap is defined as $P({{\bm{w}}})-P({{\bm{w}}}_{\star})$. As expected in theoretical results, SC-SDCA converged to the optimum faster than SC-Pega except on the dataset Phishing. No significant difference between different mini-batch sizes is observed. Conclusions =========== In this paper, we presented two new algorithms for minimizing regularized empirical loss subject to sign constraints. The two algorithms are based on Pegasos and SDCA, both of which have a solid theoretical support for convergence. The sign-constrained versions, named SC-Pega and SC-SDCA, respectively, enjoy the same convergence rate as the corresponding original algorithms. The algorithms were demonstrated in two applications. The one is posing sign constraints according to domain knowledge, and the other is improving the SVM-Pairwise method by sign constraints. Acknowledgements {#acknowledgements .unnumbered} ================ TK was supported by JPSP KAKENHI Grant Number 26249075 and 40401236. Bellavia, S., Macconi, M., and Morini, B. (2006). An interior point newton-like method for non-negative least-squares problems with degenerate solution. , [**13**]{}(10), 825–846. doi: 10.1002/nla.502. Bierlaire, M., Toint, P., and Tuyttens, D. (1991). On iterative algorithms for linear least squares problems with bound constraints. , [**143**]{}, 111–143. doi: 10.1016/0024-3795(91)90009-l. Cai, Y., Gu, H., and Kenney, T. (2017). Learning microbial community structures with supervised and unsupervised non-negative matrix factorization. , [**5**]{}(1), 110. Defazio, A., Bach, F., and Lacoste-julien, S. (2014a). Saga: A fast incremental gradient method with support for non-strongly convex composite objectives. In Z. Ghahramani, M. Welling, C. Cortes, N. Lawrence, and K. Weinberger, editors, [*Advances in Neural Information Processing Systems 27*]{}, pages 1646–1654. Curran Associates, Inc. Defazio, A. J., Caetano, T. S., and Domke, J. (2014b). Finito: A faster, permutable incremental gradient method for big data problems. . Ding, C., Li, T., Peng, W., and Park, H. (2006). Orthogonal nonnegative matrix tri-factorizations for clustering. In [*Proceedings of the 12th [ACM]{} [SIGKDD]{} international conference on Knowledge discovery and data mining - [KDD06]{}*]{}. [ACM]{} Press. doi:10.1145/1150402.1150420. Dong, W., Fu, F., Shi, G., Cao, X., Wu, J., Li, G., and Li, G. (2016). Hyperspectral image super-resolution via non-negative structured sparse representation. , [**25**]{}(5), 2337–52. F[é]{}votte, C. and Idier, J. (2011). Algorithms for nonnegative matrix factorization with the $\beta$-divergence. , [**23**]{}(9), 2421–2456. doi:10.1162/neco\_a\_00168, [\[pdf\]](https://ts3.pl.cs.gunma-u.ac.jp/tsattach1/kato/170717/Fevotte-2011,-Algorithms-for-nonnegative-matrix-factorization-with-the-beta-divergence.pdf). Hastie, T., Tibshirani, R., and Friedman, J. (2009). . Springer, 2nd edition. Hazan, E., Agarwal, A., and Kale, S. (2007). Logarithmic regret algorithms for online convex optimization. , [**69**]{}(2-3), 169–192. doi:10.1007/s10994-007-5016-8. He, R., Zheng, W. S., Hu, B. G., and Kong, X. W. (2013). Two-stage nonnegative sparse representation for large-scale face recognition. , [**24**]{}(1), 35–46. Heinkenschloss, M., Ulbrich, M., and Ulbrich, S. (1999). Superlinear and quadratic convergence of affine-scaling interior-point newton methods for problems with simple bounds without strict complementarity assumption. , [**86**]{}(3), 615–635. doi: 10.1007/s101070050107. Henrot, S., Moussaoui, S., Soussen, C., and Brie, D. (2013). Edge-preserving nonnegative hyperspectral image restoration. In [*2013 [IEEE]{} International Conference on Acoustics, Speech and Signal Processing*]{}. [IEEE]{}. doi: 10.1109/icassp.2013.6637926. Ishii, S., Kitamura, G., Segawa, T., Kobayashi, A., Miura, T., Sano, D., and Okabe, S. (2014a). Microfluidic quantitative pcr for simultaneous quantification of multiple viruses in environmental water samples. , [**80**]{}(24), 7505–11. Ishii, S., Nakamura, T., Ozawa, S., Kobayashi, A., Sano, D., and Okabe, S. (2014b). Water quality monitoring and risk assessment by simultaneous multipathogen quantification. , [**48**]{}(9), 4744–9. Ji, Y., Lin, T., and Zha, H. (2009). Mahalanobis distance based non-negative sparse representation for face recognition. In [*2009 International Conference on Machine Learning and Applications*]{}. [IEEE]{}. doi:10.1109/icmla.2009.50. Joachims, T. (2005). A support vector method for multivariate performance measures. In L. D. Raedt and S. Wrobel, editors, [*Proceedings of the 22nd International Conference on Machine Learning (ICML-05)*]{}, pages 377–384. Johnson, R. and Zhang, T. (2013). Accelerating stochastic gradient descent using predictive variance reduction. In [*Advances in Neural Information Processing Systems 26: Proceedings of a meeting held December 5-8, 2013, Lake Tahoe, Nevada, United States.*]{}, pages 315–323. Kanzow, C. and Klug, A. (2006). On affine-scaling interior-point newton methods for nonlinear minimization with bound constraints. , [**35**]{}(2), 177–197. doi:10.1007/s10589-006-6514-5. Kim, D., Sra, S., and Dhillon, I. S. (2007). Fast newton-type methods for the least squares nonnegative matrix approximation problem. In [*Proceedings of the 2007 [SIAM]{} International Conference on Data Mining*]{}, pages 343–354. Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611972771.31. Kim, D., Sra, S., and Dhillon, I. S. (2010). Tackling box-constrained optimization via a new projected quasi-newton approach. , [**32**]{}(6), 3548–3563. doi:10.1137/08073812x. Kimura, K., Kudo, M., and Tanaka, Y. (2016). A column-wise update algorithm for nonnegative matrix factorization in bregman divergence with an orthogonal constraint. , [**103**]{}(2), 285–306. doi:10.1007/s10994-016-5553-0, [\[pdf\]](https://ts3.pl.cs.gunma-u.ac.jp/tsattach1/kato/170717/Kimura-2016,-A-column-wise-update-algorithm-for-nonnegative-matrix-factorization-in-Bregman-divergence-with-an-orthogonal-constraint.pdf). Kobayashi, A., Sano, D., Hatori, J., Ishii, S., and Okabe, S. (2013). Chicken- and duck-associated bacteroides-prevotella genetic markers for detecting fecal contamination in environmental water. , [**97**]{}(16), 7427–37. Lanckriet, G. R., Deng, M., Cristianini, N., Jordan, M. I., and Noble, W. S. (2004a). Kernel-based data fusion and its application to protein function prediction in yeast. , [**-**]{}(-), 300–11. Lanckriet, G. R., Bie, T. D., Cristianini, N., Jordan, M. I., and Noble, W. S. (2004b). A statistical framework for genomic data fusion. , [**20**]{}(16), 2626–35. Landi, G. and Piccolomini, E. L. (2012). : a [Matlab]{} software for nonnegative image restoration with [Newton]{} projection methods. , [**62**]{}(3), 487–504. doi: 10.1007/s11075-012-9602-x. Lapin, M., Hein, M., and Schiele, B. (2015). Top-k multiclass svm. In C. Cortes, N. D. Lawrence, D. D. Lee, M. Sugiyama, and R. Garnett, editors, [*Advances in Neural Information Processing Systems 28*]{}, pages 325–333. Curran Associates, Inc. Lawson, C. L. and Hanson, R. J. (1995). . Society for Industrial and Applied Mathematics. doi:10.1137/1.9781611971217. Lee, D. D. and Seung, H. S. (2001). Algorithms for non-negative matrix factorization. In [*Advances in neural information processing systems*]{}, pages 556–562. [\[pdf\]](https://ts3.pl.cs.gunma-u.ac.jp/tsattach1/kato/170717/Daniel-Lee-2001,-Algorithms-for-non-negative-matrix-factorization.pdf). Liao, L. and Noble, W. S. (2003). Combining pairwise sequence similarity and support vector machines for detecting remote protein evolutionary and structural relationships. , [**10**]{}(6), 857–68. Lin, C.-J. and Mor[é]{}, J. J. (1999). Newton’s method for large bound-constrained optimization problems. , [**9**]{}(4), 1100–1127. doi: 10.1137/s1052623498345075. Lin, Y., Lee, D., and Saul, L. (2004). Nonnegative deconvolution for time of arrival estimation. In [*2004 [IEEE]{} International Conference on Acoustics, Speech, and Signal Processing*]{}. [IEEE]{}. doi:10.1109/icassp.2004.1326273. Liu, B., Zhang, D., Xu, R., Xu, J., Wang, X., Chen, Q., Dong, Q., and Chou, K. C. (2014). Combining evolutionary information extracted from frequency profiles with sequence-based kernels for protein remote homology detection. , [**30**]{}(4), 472–479. Ma, J. (2013). Algorithms for non-negatively constrained maximum penalized likelihood reconstruction in tomographic imaging. , [**6**]{}(1), 136–160. doi: 10.3390/a6010136. Mor[é]{}, J. J. and Toraldo, G. (1991). On the solution of large quadratic programming problems with bound constraints. , [**1**]{}(1), 93–113. doi: 10.1137/0801008. Morigi, S., Reichel, L., Sgallari, F., and Zama, F. (2007). An iterative method for linear discrete ill-posed problems with box constraints. , [ **198**]{}(2), 505–520. doi: 10.1016/j.cam.2005.06.053. Needell, D., Srebro, N., and Ward, R. (2015). Stochastic gradient descent, weighted sampling, and the randomized kaczmarz algorithm. , [**155**]{}(1-2), 549–573. doi:10.1007/s10107-015-0864-7. Ogul, H. and Mumcuoglu, E. U. (2006). Svm-based detection of distant protein structural relationships using pairwise probabilistic suffix trees. , [**30**]{}(4), 292–299. Portugal, L. F., Júdice, J. J., and Vicente, L. N. (1994). A comparison of block pivoting and interior-point algorithms for linear least squares problems with nonnegative variables. , [**63**]{}(208), 625–625. doi:10.1090/s0025-5718-1994-1250776-4. Rockafellar, R. T. (1970). . Princeton University Press, Princeton, NJ. Roux, N. L., Schmidt, M., and Bach, F. R. (2012). A stochastic gradient method with an exponential convergence rate for finite training sets. In F. Pereira, C. Burges, L. Bottou, and K. Weinberger, editors, [ *Advances in Neural Information Processing Systems 25*]{}, pages 2663–2671. Curran Associates, Inc. Schmidt, M., Roux, N. L., and Bach, F. (2016). Minimizing finite sums with the stochastic average gradient. , [**162**]{}(1-2), 83–112. doi:10.1007/s10107-016-1030-6. Scott, T. M., Rose, J. B., Jenkins, T. M., Farrah, S. R., and Lukasik, J. (2002). Microbial source tracking: current methodology and future directions. , [**68**]{}(12), 5796–803. Shalev-Shwartz, S. and Zhang, T. (2013). Stochastic dual coordinate ascent methods for regularized loss. , [**14**]{}(1), 567–599. Shalev-Shwartz, S. and Zhang, T. (2016). Accelerated proximal stochastic dual coordinate ascent for regularized loss minimization. , [**155**]{}(1), 105–145. Shalev-Shwartz, S., Singer, Y., Srebro, N., and Cotter, A. (2011). primal estimated sub-gradient solver for [SVM]{}. , [**127**]{}(1), 3–30. Shashua, A. and Hazan, T. (2005). Non-negative tensor factorization with applications to statistics and computer vision. In [*Proceedings of the 22nd international conference on Machine learning - [ICML’05]{}*]{}. [ACM]{} Press. doi:10.1145/1102351.1102451. Wang, J., Tian, F., Yu, H., Liu, C. H., Zhan, K., and Wang, X. (2017). Diverse non-negative matrix factorization for multiview data representation. , [**-**]{}(-), –. Wang, Y. and Ma, S. (2007). Projected barzilaiborwein method for large-scale nonnegative image restoration. , [**15**]{}(6), 559–583. doi: 10.1080/17415970600881897. Xiao, L. and Zhang, T. (2014). A proximal stochastic gradient method with progressive variance reduction. , [**24**]{}(4), 2057–2075. doi:10.1137/140961791. Zhang, L., Mahdavi, M., and Jin, R. (2013). Linear convergence with condition number independent access of full gradients. In [*Proceedings of the 26th International Conference on Neural Information Processing Systems*]{}, NIPS’13, pages 980–988, USA. Curran Associates Inc. Zhang, Q., Wang, H., Plemmons, R., and Pauca, V. P. (2007). Spectral unmixing using nonnegative tensor factorization. In [*Proceedings of the 45th annual southeast regional conference on [ACM]{}-[SE]{} 45*]{}. [ACM]{} Press. doi: 10.1145/1233341.1233449. Proofs and Derivations ====================== Proof of Theorem \[thm:srpega-bound\] {#ss:proof-thm:srpega-bound} ------------------------------------- @Shalev-Shwartz11-pegasos have used the following lemma, given below, to obtain the bound. \[lem:pega-lem01\] Let $f_{1},\dots,f_{T}$ be a sequence of $\lambda$-strongly convex functions. Let ${{\mathcal{C}}}$ be a closed convex set and define $\Pi_{{{\mathcal{C}}}}({{\bm{w}}}):={\mathop{\textrm{argmin}}\limits}_{{{\bm{w}}}'\in{{\mathcal{C}}}}\|{{\bm{w}}}'-{{\bm{w}}}\|$. Let ${{\bm{w}}}_{1},\dots,{{\bm{w}}}_{T+1}$ be a sequence of vectors such that ${{\bm{w}}}_{1}\in{{\mathcal{C}}}$ and for $t\ge 1$, ${{\bm{w}}}_{t+1}:=\Pi_{{{\mathcal{C}}}}({{\bm{w}}}_{t}-\nabla_{t}/(\lambda t))$, where $\nabla_{t}\in\partial f_{t}({{\bm{w}}}_{t})$. Assume that $\forall t\in{{\mathbb{N}}}$, $\|\nabla_{t}\|\le G$. Then, for $\forall {{\bm{u}}}\in{{\mathcal{C}}}$, it holds that $$\begin{aligned} \frac{1}{T}\sum_{t=1}^{T}f_{t}({{\bm{w}}}_{t}) \le \frac{1}{T}\sum_{t=1}^{T}f_{t}({{\bm{u}}}) + \frac{(1+\log(T))G^{2}}{2\lambda T}. \end{aligned}$$ We, too, have used Lemma \[lem:pega-lem01\] to obtain Theorem \[thm:srpega-bound\] for our sign-constrained learning problem . To this end, we find the following lemma. \[lem:hazen-applicable\] Let ${{\mathcal{B}}}$ be ${r_{\text{loss}}}/\sqrt{\lambda}$-ball defined as := {\^[d]{}| }. and ${{\mathcal{S}}}$ be the set defined in . Then, the intersection of the two sets are closed and convex. It holds that \[eq:update-lem:hazen-applicable\] \_[t+1]{} = \_ (\_[t]{}-P\_[t]{}(\_[t]{}) ) for $\forall t\in{{\mathbb{N}}}$. Furthermore, the optimal solution ${{\bm{w}}}_{\star} := {\mathop{\textrm{argmin}}\limits}_{{{\bm{w}}}\in{{\mathcal{S}}}}P({{\bm{w}}})$ is in the intersection of the two sets. Namely, \[eq:opt-lem:hazen-applicable\] \_. See Subsection \[ss:proof-lem:hazen-applicable\] for proof of Lemma \[lem:hazen-applicable\]. The above lemma suggests that the setting of $f_{t}:=P_{t}$, ${{\mathcal{C}}}:={{\mathcal{B}}}\cap{{\mathcal{S}}}$ and ${{\bm{u}}}:={{\bm{w}}}_{\star}$ fulfills the assumptions of Lemma \[lem:pega-lem01\]. An upper bound of the norm of the gradient of $f_{t}$ is given by \[eq:ub-for-setting-G-in-pega\] f\_[t]{}(\_[t]{}) = P\_[t]{}(\_[t]{}) + LR. See Subsection \[ss:deriv-eq:ub-for-setting-G-in-pega\] for derivation of . By setting $G=\sqrt{{r_{\text{loss}}}\lambda} + LR$, Theorem \[thm:srpega-bound\] is established. Proof of Lemma \[lem:hazen-applicable\] {#ss:proof-lem:hazen-applicable} --------------------------------------- Lemma \[lem:hazen-applicable\] states the following three claims. - ${{\mathcal{B}}}\cap{{\mathcal{S}}}$ is a closed and convex set. - is hold. - is hold. Apparently, ${{\mathcal{B}}}\cap{{\mathcal{S}}}$ is a closed and convex set because the both sets are closed and convex. We shall show and then . #### Proof of {#proof-of .unnumbered} To prove , it suffices to show the projection from a point ${{\bm{z}}}\in{{\mathbb{R}}}^{d}$ onto the set ${{\mathcal{B}}}\cap{{\mathcal{S}}}$ is given by \[eq:update-prime-lem:hazen-applicable\] \_() = { 1, } \_(). The projection problem can be expressed as & -\^[2]{} \^[d]{}\ & \^[2]{}, \_[d]{}. With non-negative dual variables ${{\bm{\beta}}}\in{{\mathbb{R}}}^{d}_{+}$ and $\eta\in{{\mathbb{R}}}_{+}$, the Lagrangian function is given by $$\begin{gathered} {{\mathcal{L}}}_{{{\mathcal{B}}}\cap{{\mathcal{S}}}}({{\bm{x}}},{{\bm{\beta}}},\eta) := \frac{1}{2}\|{{\bm{x}}}-{{\bm{z}}}\|^{2} \\ - \left<{{\bm{\beta}}},{{\bm{c}}}\odot{{\bm{x}}}\right> + \frac{\eta}{2} \left( \|{{\bm{x}}}\|^{2} - \frac{{r_{\text{loss}}}}{\lambda} \right). \end{gathered}$$ Let $({{\bm{x}}}_{\star},{{\bm{\beta}}}_{\star},\eta_{\star})$ be the saddle point of $\min_{{{\bm{x}}}}\max_{{{\bm{\beta}}},\eta} {{\mathcal{L}}}_{{{\mathcal{B}}}\cap{{\mathcal{S}}}}({{\bm{x}}},{{\bm{\beta}}},\eta)$. Then, ${{\bm{x}}}_{\star} = \Pi_{{{\mathcal{B}}}\cap{{\mathcal{S}}}}({{\bm{z}}})$. At the saddle point, it holds that $\nabla_{{{\bm{x}}}} {{\mathcal{L}}}_{{{\mathcal{B}}}\cap{{\mathcal{S}}}}={{\bm{0}}}$, yielding \[eq:01-proof-of-lem:pegaproj-onto-B-cap-S\] = (+). The dual objective is written as \_(,) &= \_\_(,,)\ &= \_((+),,)\ &= -+\^[2]{} + + \^[2]{}. This implies that ${{\mathcal{D}}}_{{{\mathcal{B}}}\cap{{\mathcal{S}}}}(\cdot,\cdot)$ is maximized when ${{\bm{\beta}}}= (-{{\bm{c}}}\odot{{\bm{z}}})_{+}$. Note that this does not depend on the value of $\eta$. Substituting this into , we have \[eq:02-proof-of-lem:pegaproj-onto-B-cap-S\] = (+(-)\_[+]{}) = \_(), where the last equality follows from the fast that $\Pi_{{{\mathcal{S}}}}({{\bm{z}}})={{\bm{z}}}+{{\bm{c}}}\odot(-{{\bm{c}}}\odot{{\bm{z}}})_{+}$ which can be shown as follows. The Lagrangian function for the problem of projection of ${{\bm{z}}}$ onto ${{\mathcal{S}}}$ is given by ${{\mathcal{L}}}_{{{\mathcal{S}}}}({{\bm{x}}},{{\bm{\beta}}}) = {{\mathcal{L}}}_{{{\mathcal{B}}}\cap{{\mathcal{S}}}}({{\bm{x}}},{{\bm{\beta}}},0)$, and, with a similar derivation, the dual objective is ${{\mathcal{D}}}_{{{\mathcal{S}}}}({{\bm{\beta}}}) = {{\mathcal{D}}}_{{{\mathcal{B}}}\cap{{\mathcal{S}}}}({{\bm{\beta}}},0)$ which is maximized at ${{\bm{\beta}}}= (-{{\bm{c}}}\odot{{\bm{z}}})_{+}$ yielding $\Pi_{{{\mathcal{S}}}}({{\bm{z}}}) = {{\bm{z}}}+(-{{\bm{c}}}\odot{{\bm{z}}})_{+}$. Next, we find the optimal $\eta$. The dual objective is \_((-)\_[+]{},) = -\_()\^[2]{}(+1)\^[-1]{} - + \^[2]{} with the derivative \_\_((-)\_[+]{},) = \_()\^[2]{}(+1)\^[-2]{} - . Setting the derivative to zero and noting that $\eta$ is a non-negative variable, we get \_ = ( 0, \_()-1 ). Substituting this into , we obtain $$\begin{gathered} \Pi_{{{\mathcal{B}}}\cap{{\mathcal{S}}}}({{\bm{z}}}) = {{\bm{x}}}_{\star} = \frac{1}{1+\max( 0, \sqrt{\lambda/{r_{\text{loss}}}} \left\lVert\Pi_{{{\mathcal{S}}}}({{\bm{z}}})\right\rVert-1 )} \Pi_{{{\mathcal{S}}}}({{\bm{z}}}) \\ = \min\left\{1,\frac{\sqrt{{r_{\text{loss}}}}}{\sqrt{\lambda} \left\lVert\Pi_{{{\mathcal{S}}}}({{\bm{z}}})\right\rVert}\right\}\Pi_{{{\mathcal{S}}}}({{\bm{z}}}). \end{gathered}$$ Thus, is established. #### Proof of {#proof-of-1 .unnumbered} We use the following problem dual to : \[eq:dual-rlm-signcon\] - \_()\^[2]{} -\^[\*]{}(-) \^[n]{}. Let ${{\bm{\alpha}}}_{\star}$ be the solution optimal to the dual problem . The primal optimal solution can be recovered by \[eq:wst-recovery-from-alphst\] \_ = \_(\_) with no duality gap. The loss term in the objective of the dual problem is bounded from above as \[eq:loss-in-dual-bounded\] -\^[\*]{}(-) &= -\_[\^[n]{}]{} ( - () )\ & = \_[\^[n]{}]{} ( + () )\ &= ( + () ) = () [r\_]{}. The square norm of the primal optimal solution is bounded as &\_\^[2]{} = \_\^[2]{} + \_\^[2]{}\ & \_\^[2]{} + ( \_\^[2]{} + (\^\_) )\ &= \_\^[2]{} + P(\_)\ &= \_\^[2]{} + ( - \_()\^[2]{} -\^[\*]{}(-\_) )\ &= \_\^[2]{} - ( \_\^[2]{} + \^[\*]{}(-\_) )\ &= -\^[\*]{}(-\_) where the first inequality, the third and fourth equalities, and the last inequality follow from Assumption \[assum:four-for-rlm-signcon\](c), no duality gap, , and , respectively. Therefore, ${{\bm{w}}}_{\star}\in{{\mathcal{B}}}$. Furthermore, ${{\bm{w}}}_{\star}$ is feasible so ${{\bm{w}}}_{\star}\in{{\mathcal{S}}}$. Hence, is established. Derivation of {#ss:deriv-eq:ub-for-setting-G-in-pega} -------------- This inequality leads to a bound of The norm of the gradient of the loss term in $P_{t}(\cdot)$ can be bounded as $$\begin{gathered} \left\lVert \frac{\partial}{\partial{{\bm{w}}}} \Phi({{\bm{X}}}_{{{\mathcal{A}}}_{t}}^\top{{\bm{w}}}\,;\,{{\mathcal{A}}}_{t}) \right\rVert = \left\lVert \sum_{i\in{{\mathcal{A}}}_{t}}{{\bm{x}}}_{i} \nabla \phi_{i}(\left<{{\bm{x}}}_{i},{{\bm{w}}}\right>) \right\rVert \\ \le \sum_{i\in{{\mathcal{A}}}_{t}} \left\lVert {{\bm{x}}}_{i} \right\rVert \nabla \phi_{i}(\left<{{\bm{x}}}_{i},{{\bm{w}}}\right>) \le kLR. \end{gathered}$$ Using this, is derived as f\_[t]{}(\_[t]{}) &= P\_[t]{}(\_[t]{})\ & \_[t]{} + (\_[\_[t]{}]{}\^;\_[t]{})\ & + + LR. Derivation of {#ss:deriv-eq:sdca-prox-update} -------------- Exploiting proof techniques used in ProxSDCA [@Shalev-Shwartz2013a-Accelerated], we here limit the form of $\Delta\alpha$ to $sq$ where $q:=u_{i}-\alpha_{i}^{(t-1)}$ and $u_{i}\in-\partial\phi_{i}(z_{i})$. Denote by $D^{0}({{\bm{\alpha}}})$ the objective function of the dual problem . Suppose that $i$-th example is chosen at $t$-th iterate. The new value of the regularization term in $D^{0}({{\bm{\alpha}}})$ is given by $$\begin{gathered} -\frac{\lambda}{2} \left\lVert\Pi_{{{\mathcal{S}}}}\left(\frac{{{\bm{X}}}({{\bm{\alpha}}}+sq{{\bm{e}}}_{i})}{\lambda n}\right)\right\rVert^{2} \ge -\frac{\lambda}{2} \left\lVert\Pi_{{{\mathcal{S}}}}\left(\bar{{{\bm{w}}}}^{(t-1)}\right) + \frac{sq}{\lambda n}{{\bm{x}}}_{i}\right\rVert^{2} \\ \ge -\frac{\lambda}{2} \left\lVert{{\bm{w}}}^{(t-1)}\right\rVert^{2} - \frac{s}{n}z_{i}q - \frac{1}{2\lambda} \left(\frac{s}{n}\right)^{2}R^{2}q^{2}. \end{gathered}$$ where the first inequality follows from the following inequality \[eq:ub-of-gst-2nd\] , \^[d]{}, \_()+ \_(+), and the second inequality is derived from the fact of ${{\bm{w}}}^{(t-1)}=\Pi_{{{\mathcal{S}}}}\left(\bar{{{\bm{w}}}}^{(t-1)}\right)$ and the assumption of $\lVert{{\bm{x}}}_{i}\rVert\le R$. We shall prove in Subsection \[ss:deriv-eq:ub-of-gst-2nd\]. The improvement of the dual objective is expressed as & D\^[0]{}(\^[(t-1)]{}+sq\_[i]{})-D\^[0]{}(\^[(t-1)]{})\ &= - \_()\^[2]{} + \^[(t-1)]{}\^[2]{}\ &-\_[i]{}(-\_[i]{}-sq) +\_[i]{}(-\_[i]{})\ & - ()\^[2]{}R\^[2]{}q\^[2]{} + q\^[2]{}\ &+ ( z\^[(t)]{}\_[i]{} + \_[i]{}\^[\*]{}(-\_[i]{})-\_[i]{}\^[\*]{}(-u\_[i]{}) - z\^[(t)]{}u )\ &= - q\^[2]{}s\_\^[-1]{} + ( z\^[(t)]{}\_[i]{} + \_[i]{}\^[\*]{}(-\_[i]{}) + \_[i]{}(z\^[(t)]{}) + ) Thus, the value of $s$ maximizing the lower-bound can be given by s = \_[\[0,s\^[-1]{}\_\]]{} ( + )s\_. Thus, is derived. Derivation of {#ss:deriv-eq:ub-of-gst-2nd} -------------- For $h=1,\dots,d$, letting [[b]{}]{}\_[h]{}(;v) := 0.5(+(v)\_[+]{})\^[2]{} &h\_[+]{},\ 0.5(+v)\^[2]{} &h\_[0]{},\ 0.5(+(-v)\_[+]{})\^[2]{} &h\_[-]{}, and [[a]{}]{}\_[h]{}(v) := 0.5(v)\_[+]{}\^[2]{} &h\_[+]{},\ 0.5(v)\^[2]{} &h\_[0]{},\ 0.5(-v)\_[+]{}\^[2]{} &h\_[-]{}, both the sides in can be rewritten as = \_[h=1]{}\^[d]{}[[b]{}]{}\_[h]{}(\_[h]{};v\_[h]{}), and = \_[h=1]{}\^[d]{}[[a]{}]{}\_[h]{}(v\_[h]{}+\_[h]{}) To show the inequality , it suffices to show that h=1,…,d, v, , [[b]{}]{}\_[h]{}(;v) [[a]{}]{}\_[h]{}(v+). Apparently, ${{b}}_{h}(\Delta;v) = {{a}}_{h}(v+\Delta)$ for $h\in{{\mathcal{I}}}_{0}$. Assume $h\in{{\mathcal{I}}}_{+}$ for a while. [[b]{}]{}\_[h]{}(;v) = 0.5(+v)\^[2]{} &v0,\ 0.5\^[2]{} &v&lt; 0. The following three cases must be considered: - In case of $v\ge 0$, [[b]{}]{}\_[h]{}(;v) = 0.5(+v)\^[2]{} [[a]{}]{}\_[h]{}(v+). - In case of $v< 0$ and $\Delta<-v$, [[b]{}]{}\_[h]{}(;v) = 0.5\^[2]{} 0 = [[a]{}]{}\_[h]{}(v+). - In case of $v< 0$ and $\Delta\ge-v$, &[[b]{}]{}\_[h]{}(;v)-[[a]{}]{}\_[h]{}(v+) = 0.5\^[2]{} - 0.5(+v)\^[2]{}\ &= - v- 0.5v\^[2]{} = -0.5 ((+v)+)v\ &-0.5 v 0.5v\^[2]{}0. Therefore, we get ${{b}}_{h}(\Delta;v) \ge {{a}}_{h}(v+\Delta)$ for $h\in{{\mathcal{I}}}_{+}$. Finally, we assume $h\in{{\mathcal{I}}}_{-}$. [[b]{}]{}\_[h]{}(;v) = 0.5(+v)\^[2]{} &v0,\ 0.5\^[2]{} &v&gt; 0. We need to analyze the following three cases: - In case of $v\le 0$, [[b]{}]{}\_[h]{}(;v) = 0.5(+v)\^[2]{} [[a]{}]{}\_[h]{}(v+). - In case of $v> 0$ and $\Delta>-v$, [[b]{}]{}\_[h]{}(;v) = 0.5\^[2]{} 0 = [[a]{}]{}\_[h]{}(v+). - In case of $v> 0$ and $\Delta\le-v$, &[[b]{}]{}\_[h]{}(;v)-[[a]{}]{}\_[h]{}(v+) = 0.5\^[2]{} - 0.5(+v)\^[2]{}\ &= - v- 0.5v\^[2]{} = 0.5 (-(+v)-)v\ &0.5 (-)v 0.5v\^[2]{}0. The above leads to ${{b}}_{h}(\Delta;v) \ge {{a}}_{h}(v+\Delta)$ for $h\in{{\mathcal{I}}}_{-}$. Proof of Theorem \[thm:converg-srsdca\] {#ss:proof-thm:converg-srsdca} --------------------------------------- A key observation that leads to the discovery of Theorem \[thm:converg-srsdca\] is the following lemma: \[lem:nabla-convconj-g-corrects-sign\] Let $g:{{\mathbb{R}}}^{d}\to{{\mathbb{R}}}\cup\{+\infty\}$ be defined as $g({{\bm{w}}}) := \frac{1}{2}\lVert{{\bm{w}}}\rVert^{2} + \delta_{{{\mathcal{S}}}}({{\bm{w}}})$ where $\delta_{{{\mathcal{S}}}}(\cdot)$ is the indicator function of the feasible region ${{\mathcal{S}}}$ given in . Namely, $\delta_{{{\mathcal{S}}}}({{\bm{w}}})=+\infty$ if ${{\bm{w}}}\not\in{{\mathcal{S}}}$; otherwise $\delta_{{{\mathcal{S}}}}({{\bm{w}}})=0$. Then, with $d$-dimensional vector ${{\bm{c}}}$ defined in , the gradient of its convex conjugate [@rockafellar70convex] is expressed as $\nabla g^{*}(\bar{{{\bm{w}}}}) = \bar{{{\bm{w}}}} + {{\bm{c}}}\odot(-{{\bm{c}}}\odot\bar{{{\bm{w}}}})_{+}$. See Subsections \[ss:proof-lem:nabla-convconj-g-corrects-sign\] for proof of Lemma \[lem:nabla-convconj-g-corrects-sign\]. The function $g$ defined in Lemma \[lem:nabla-convconj-g-corrects-sign\] is $1$-strongly convex. Then, if we view $g$ as a regularization function in replacement of the square L2-norm regularizer, the sign-constrained optimization problem  can be rewritten as & g() + (\^) \^[d]{}. This is a class of optimization problems targeted by a variant of SDCA named Prox-SDCA [@Shalev-Shwartz2013a-Accelerated] which maintains the convergence rate of the vanilla SDCA yet the regularization function can be extended to be a 1-strongly convex function. The difference from the vanilla SDCA is that the primal variable is recovered from the gradient of the convex conjugate of $g(\cdot)$ at the end of each iterate. It can be seen that Algorithm \[alg:srsdca\] is generated by applying Prox-SDCA to our problem setting with $g$ defined in Lemma \[lem:nabla-convconj-g-corrects-sign\]. From this observation, Theorem \[thm:converg-srsdca\] is established. Proof of Lemma \[lem:nabla-convconj-g-corrects-sign\] {#ss:proof-lem:nabla-convconj-g-corrects-sign} ----------------------------------------------------- The convex conjugate of $g$ is g\_[\*]{}(|) &= \_[\^[d]{}]{} ( - g() )\ &= \_[\^[d]{}]{} ( - \^[2]{} - \_() )\ &= \_ ( - \^[2]{} )\ &= |\^[2]{} - \_ -|\^[2]{}. We use Danskin’s theorem to get the derivative as: g\^[\*]{}(|) = \_(|).

--- abstract: | We have obtained repeated images of 6 fields towards the Galactic bulge in 5 passbands ($u,g,r,i,z$) with the DECam imager on the Blanco 4m telescope at CTIO. From over 1.6 billion individual photometric measurements in the field centered on Baade’s window, we have detected 4877 putative variable stars. 474 of these have been confirmed as fundamental mode RR Lyrae stars, whose colors at minimum light yield line-of-sight reddening determinations as well as a reddening law towards the Galactic Bulge which differs significantly from the standard $R_{V} = 3.1$ formulation. Assuming that the stellar mix is invariant over the 3 square-degree field, we are able to derive a line-of-sight reddening map with sub-arcminute resolution, enabling us to obtain de-reddened and extinction corrected color-magnitude diagrams (CMD’s) of this bulge field using up to 2.5 million well-measured stars. The corrected CMD’s show unprecedented detail and expose sparsely populated sequences: e.g., delineation of the very wide red giant branch, structure within the red giant clump, the full extent of the horizontal branch, and a surprising bright feature which is likely due to stars with ages younger than 1 Gyr. We use the RR Lyrae stars to trace the spatial structure of the ancient stars, and find an exponential decline in density with Galactocentric distance. We discuss ways in which our data products can be used to explore the age and metallicity properties of the bulge, and how our larger list of all variables is useful for learning to interpret future LSST alerts.  \ author: - Abhijit Saha - 'A. Katherina Vivas' - 'Edward W. Olszewski' - Verne Smith - Knut Olsen - Robert Blum - Francisco Valdes - Jenna Claver - Annalisa Calamida - 'Alistair R. Walker' - Thomas Matheson - Gautham Narayan - Monika Soraisam - Katia Cunha - 'T. Axelrod' - 'Joshua S. Bloom' - 'S. Bradley Cenko' - Brenda Frye - Mario Juric - Catherine Kaleida - Andrea Kunder - Adam Miller - David Nidever - Stephen Ridgway bibliography: - 'ms.bib' title: 'Mapping the Interstellar Reddening and Extinction towards Baade’s Window Using Minimum Light Colors of ab-type RR Lyrae Stars. Revelations from the De-reddened Color-Magnitude Diagrams' --- Introduction {#sec:intro} ============ Observationally the Galactic bulge is a concentration of stars towards the galactic center with chemistry, age distribution, and dynamics that set it apart from the disk and halo. A comprehensive review with leads into the extensive literature is given by @barbuy18. By combining what we know about our bulge with those in other galaxies we are led to understand that bulges come in two forms, classical bulges and pseudo-bulges [@kormendy04]. Modern observations of the Milky Way bulge indicate that it has a bar [@dwek95] with some characteristics of a classical bulge and some of a pseudo-bulge. While the majority of Bulge stars seem to be old, there is still debate about the percentage of younger stars, a debate that can be informed by the inspection and analysis of color-magnitude diagrams from which a) the line-of-sight reddening and extinction are removed, and b) contamination by foreground stars is identified and eliminated on the basis of proper motions. Thus, in addition to the complications of performing accurate photometry in severely over-crowded fields, the construction of suitable color-magnitude diagrams involves removing reddening on the finest possible angular scales. The color of the red clump (RC) stars just off the giant branch has been used as a standard color-marker (or standard crayon) in many studies, most notably by @Nataf13 [@Nataf16] and references therein. They found that not only does the standard reddening law predict the line-of-sight reddening to the bulge incorrectly, but that the true reddening law in these directions varies on angular scales of a few degrees. Removal of foreground stars using proper motions up to 19th mag over wide fields of view is possible with *Gaia*, though we may have to wait for the mission to complete to do this comprehensively. It may well be that due to the high stellar density in these areas, *Gaia*’s selection of stars in this part of the sky is incomplete. Over time, the VVV survey [@minniti10] and its followup provide both the time base and object completeness, which are likely required to complete the task. From the analysis of asymptotic giant branch and cool supergiant stars near the Galactic center, @blum03 implied that about 25% of the stars in the central few parsecs are younger than 5 Gyr. However this may not be representative of the bulge as a whole. The *Hubble Space Telescope* ($HST$) has already been used to carry this out for small fields of view in the bulge [e.g., @clark08; @cala14], with ensuing cleaned color-magnitude diagrams such as by @brown09, and more recently by @bernard18. The latter work goes on to derive star formation histories in different bulge fields from their CMDs, and report that up to 20 or 25% of the most metal rich stars are younger than 5 Gyr. The drawback is that rare(r) stars can only be seen as populations in larger-area studies than possible with *HST*, and reddening and extinction corrections used in these studies involve adopting the standard Galactic extinction law, which @Nataf13 [@Nataf16] show to be invalid. In this paper we explore an alternative route to deriving reddening and extinction following the precepts enunciated by [@sturch66] about the constancy and universality of the colors of fundamental mode RR Lyrae stars while they are in the pulsation phase corresponding to near minimum light. The potential advantage of this approach is that since RR Lyrae are also standard candles, they can be used to investigate not only the reddening, but also the ratio of total to selective extinction. In our experiment, we have obtained and analyzed multi-band, multi-epoch wide field bulge images to construct light curves of the RR Lyraes, and employ them to examine the intervening dust reddening and extinction. The emphasis is on avoiding any prior assumptions about the bulge’s stellar population make up. RR Lyrae stars are also probes of ancient stellar populations, and their distribution in the bulge traces that of the oldest stars. Recent searches for these stars in the near infrared through the very obscured inner regions of the bulge by the VVV survey [@minniti10] indicate that these stars do not follow the bar like structure, but have a smoother distribution [@minniti17]. This is contrary to an older result based on OGLE data [@Piet12], who claim that the RR Lyrae spatial distribution is elongated along the Galactic bar. It is quite possible that the accuracy in the adopted reddening and total to selective reddening laws impact such findings. We obtained images of 6 select fields towards the general direction of the Galactic center with the DECam imager [@flaugher15] over multiple epochs in 5 different passbands $u,g,r,i,z$. The chosen fields are shown in Table \[tab\_fields\], and named B1 through B6. B1 is centered on the well known “Baade’s Window,” and gets close to the direction of the Galactic center while remaining relatively transparent. The footprint of the DECam field is significantly larger than the original area considered by Baade, and has patches of reddening much higher than the value of $ E(B-V) \sim 0.7 $ often ascribed to it. Figure \[fig:panorama\] shows an image of the field in the $u$ passband, which highlights the patchiness in extinction that must be dealt with. B2 is an adjacent field midway between 2 fields found by @blanco92 and @blancos97, with lower and less uneven extinction than B1, but slightly farther from the direction of the Galactic center. There is a small intentional overlap between B1 and B2 for the purpose of verifying photometric accuracy in our data. B5 is set $\sim 10^{\circ}$ south of the Galactic Center, and is intended as a probe of the region off the Galactic plane, but within the bulge. B3 and B4 are fields at similar Galactic latitude as B1, but $\sim 10^{\circ}$ and $\sim 5^{\circ}$ away in longitude respectively in the direction of the near side of the bar, while B6 is $10^{\circ}$ away on the far side of the bar. These field choices sample the run of stellar populations along and across the Galactic disk. The exact placement of the fields was made to have minimal extinction compared to their surroundings using the dust maps by @sfd98[^1]. This paper deals only with field B1, but also details the analysis methodology that will be used for the remaining fields. [ccccc]{} B1 & 18:03:34.0 & $-$30:02:02 & 1.02 & $-$3.92\ B2 & 18:09:24.4 & $-$31:26:06 & 0.40 & $-$5.70\ B3 & 18:26:41.9 & $-$22:39:21 & 10.00 & $-$5.00\ B4 & 18:14:23.3 & $-$27:56:49 & 4.00 & $-$5.00\ B5 & 18:26:41.8 & $-$33:45:24 & 0.00 & $-$10.00\ B6 & 17:48:10.8 & $-$37:08:15 & 353.25 & $-$4.70\ The organization of this paper is as follows. § \[sec:observations\] describes the observations. § \[sec:photometry\] describes the processing of the data, including photometry and calibration onto an absolute flux based magnitude scale for the native DECam passbands. § \[sec:variability\] deals with the detection of variable stars, followup analysis including period determination to identify the RR Lyrae stars, template light curve fitting, measurement of minimum light brightness in each passband, and determination of completeness. § \[sec:minlightreddening\] describes the derivation of reddening to the individual fundamental mode RR Lyrae stars, and utilization of the differential reddening and extinction of the ensemble of these stars to independently derive the total to selective absorption ratios for the line-of-sight encompassed by the field B1. We present a comparison with the standard extinction law. In § \[sec:CMDs\] the observed colors and magnitudes of all stars in the field are used in conjunction with the RR Lyrae reddening values to correct the observed color-magnitude diagrams (CMDs) for extinction, and the prominent features in the corrected CMD are discussed. We show a reddening map in § \[sec:redmap\] with angular bins of $0.5 \times 0.5$ arc-minutes. The implications of our analysis of reddening for tracing the geometry of the Galactic bulge are presented in § \[sec:geom\]. In § \[sec:discussion\] we summarize our findings, and suggest how the data-set presented in this paper may be profitably used in future analysis and investigations. An ancillary benefit of the data set is that we have light curve data in many LSST-like passbands for a cornucopia of variable stars (and possibly transients) that begins to inform us about how to interpret the variability alert stream from LSST when it begins operation. Observations {#sec:observations} ============ The journal of observations is given in Table \[tab\_journal\]. There were three dark runs in 2013, in May, June and August, each of 3 to 4 nights. All 6 fields were visited 3 to 4 times on each of these nights (weather permitting), and exposed in all 5 bands $u.g,r,i,z$ successively. Consecutive visits to the same field on any night were spaced about 2 hours apart. The exposure times (typically 300s in $u$, 100s in $g,r,i,z$) are long enough to detect F stars to $r \sim 24$ mag in dark skies and arcsec or better seeing in *uncrowded* fields in the absence of reddening/extinction. For these fields, and particularly for the B1 field, the crowding is extreme and reddening is significant, so the actual detection limit is substantially brighter. In the best (and deepest) images, saturation sets in at $r \sim 15$ mag. Some of the images were taken in poorer seeing (up to 1.5 arcsec), and also on occasion through light clouds. Some additional epochs in the $i$, and $z$ bands were obtained in June 2013, during bright time. In addition, we obatained a set of much shorter exposures in March of 2015. These provide some measure of longer term sampling of the brighter stars relative to the 2013 data, and also provide data for stars of interest that may have been inadvertently saturated in the longer exposures of 2013. [cccc]{} 56423.659625 & $r$ & 205418 & 100.\ 56423.661128 & $i$ & 205419 & 100.\ 56423.662623 & $z$ & 205420 & 100.\ 56423.664117 & $g$ & 205421 & 100.\ 56423.665618 & $u$ & 205422 & 300.\ 56423.808243 & $r$ & 205489 & 100.\ 56423.809737 & $i$ & 205490 & 100.\ 56423.811255 & $z$ & 205491 & 100.\ 56423.812743 & $g$ & 205492 & 100.\ 56423.814222 & $u$ & 205493 & 300.\ The images were processed through the NOAO DECam pipeline [@Valdes14], for bias and flat-field correction, bad pixel masking and WCS (world coordinate system) fitting. Reduced images are available publicly through the NOAO Science Archive[^2]. We intend to make catalog data available through the Community NOAO Data Lab[^3].\ Photometry, Cross-matching, and Calibration {#sec:photometry} =========================================== We measured photometry in these very crowded fields using a variant of the DoPHOT program [@schechter93] maintained by one of us (Saha). The procedures and considerations for optimizing the DoPHOT parameters and evaluating aperture corrections done using a bespoke procedure written in IDL are fully described in § 3.2 of @saha10 and need not be repeated here. The only differences, mentioned also in @vivas17, are that unlike as in @saha10, where the 8 individual chips were combined into a single image by applyng a gnomonic projection, in the present case the individual chips were processed independently, and the output photometry lists from the individual chips are concatenated into one single file for the whole image. The requirement for this is that the photometry across all chips (for a given image) be on the same footing. The justification for this premise has been previously given in § 2 of @vivas17. We thus created independent photometry lists for each image. In addition to the aperture corrected instrumental magnitudes and associated error estimates, each object carries its [*RA*]{} and [ *DEC*]{} positions, as well as the chip on which it was detected, and the pixel coordinates within that chip for each image/epoch. Each object on each image also carries the object type code assigned by DoPHOT. These codes distinguish well fitted bona-fide single stars (type 1), multiple star blends (type 3), other extended objects (type 2), cosmic rays (type 8), image pathologies (types 4, 5, 8 and 9), and objects too faint to disambiguate between stars and extended objects or blends (type 7) ), and the fitted background or “sky.” In addition the following attributes were also evaluated and recorded (for each object on each image): 1. Whether the object lies within 50 pixels of the chip’s edge. 2. Whether there were two or more cosmic-ray (or other pathologies) detected within a radial distance of 1 full width at half-maximum (FWHM) of the stellar point-spread-function (PSF) as measured along the major-axis of the PSF. 3. Whether there were any bona-fide objects detected within a 1 FWHM radius footprint around the object as described above, and if so, the cumulative flux from those objects expressed in magnitudes relative to the flux of the object in question. We denote this by $m_{neighbors}$. 4. High and low percentile values for the distribution of fitted sky values of all objects for the entire image were also evaluated and recorded for each image. 5. For all stellar objects on a given image, the total reported error for individual objects was fitted as a function of reported magnitude. The fitted value $err_{exp}$ at a given measured brightness is a good expectation of what the measurement error should be for an object of that measured brightness. Reported errors much higher than the expected value for that brightness are suspect. The value $err_{lim} = 2 \times err_{exp} + 0.05 $ for each object for each epoch was evaluated and recorded as an attribute. The photometry list from the best deep image (best seeing in photometric conditions) in each passband was assigned as the deep template object list in that band, and a similarly suitable short exposure image in each band was assigned as the shallow template object list. For each band, the deep and shallow template lists were merged by matching to a coordinate tolerance of 03 (eliminating all multiple matches within this matching tolerance), The instrumental magnitude difference of the matched objects was used to adjust the instrumental mag system of the shallow template to that of the deep one. This process allows the objects saturated in the deep template to be represented in the eventual object list in each passband, and at this point there is a “grand’ template list in each of the 5 passbands that spans the full dynamic range of magnitudes spanned by both the deepest as well as shallow exposures, with instrumental mags on the system of the deep template. Finally, the $r$ band “grand” template was adopted as the [*master*]{}-template, containing the master-list of all objects. In the subsequent processing, the numerical ID’s of objects on this master-list serve as the final object ID’s for all objects in this field. Any objects that are not on this list (for whatever reason) are not considered further. A particular detail for preparing the template lists before matching and combining into the “grand” templates is worth mentioning. Since the aperture corrections to go from fitted PSF mags to instrumental mags for each image were calculated independently for each chip, the zero-point in any chip can scatter about the mean for that image by a few hundredths of a mag (or in pathological cases by worse amounts). To mitigate this problem for the template images (to which all photometry calibration is eventually referred), they were compared against other images of similar depth (deep to deep and shallow to shallow) obtained in photometric conditions. Let $m_{0j}^{k}$ be the measured aperture corrected magnitude of star $j$ on image $0$ (template) and on chip $k$. Let the same star as measured on image $i$ and also on chip $k$ be designated by $m_{ij}^{k}$, where image $i$ was also in photometric conditions. If we selected only those $j$ for which the reported measurement errors are small, and for which DoPHOT has reported that the object has an unambiguously stellar PSF, we can construct: $$\delta_{i}^{k} = \sum_{j} ( m_{ij}^{k} - m_{0j}^{k} ) / N$$ and $$\Delta_{i} = \sum_{k} \delta_{i}^{k} / M$$ where $N$ is the total number of selected stars (over index $j$) in chip $k$ being compared, and $M$ is the number of chips (over index $k$). $ \Delta_{i} $ is the overall offset between the instrumental mags of image $i$ relative to the template image: and since it is an average over $\sim 60$ chips, is essentially unaffected by small random errors in evaluating the zero-points on individual chips. The offset $\Delta_{i}$ can be caused by differences in atmospheric extinction (different airmass), and over longer durations by differences in system response and transmission. Consider the chip-to-chip fluctuation about this mean difference: $$\epsilon_{i}^{k} = \delta_{i}^{k} - \Delta_{i}$$ which shows the aggregate result of individual chip-to-chip aperture correction errors for both images $i$ and $0$. If we have $n$ images against which such a comparison can be made for the template, we can calculate the ensemble average like quantity $C^{k}$ from the $\epsilon_{i}^{k}$’s: $$C^{k} = \sum_{i=1}^{n} \epsilon_{i}^{k} / (n+1)$$ which is a robust estimate of the correction to be added to the instrumental magnitudes for the template frame for each chip $k$. All of the photometry lists in each passband were then matched one by one to the “grand” template for that passband, from which the offsets in the instrumental magnitudes relative to the “grand template” magnitudes were calculated on a chip-by-chip basis. All instrumental mags for the individual epochs were adjusted (single additive magnitude offset per chip) to put all the instrumental mags for all epochs on the scale of the “grand” template. The lists with the instrumental mags thus normalized were then matched individually to the master-template (same as the $r$ band “grand” template), and the object ID’s from the master list were then attached to the matched objects in each object-list for each epoch and for each passband. With this labeling, the measurements for any object can be extracted for any epoch and passband, along with all of the associated information discussed above. The instrumental photometry for every epoch is normalized to that in the respective “grand” template for the relevant passband. Henceforth, all variability analyses can be carried out using either these normalized instrumental mags or using the calibrated AB-magnitudes described below. Calibration to any system of magnitudes requires only the determination of zero-points for the “grand” template of the respective passbands, details of which are provided in the following paragraph. The normalized instrumental and object-labeled photometry for each epoch of each passband were then stored in a MySQL database, providing convenient access for subsequent variability analyses. There are 9,623,873 distinct objects in the database, each with multiple measurements at different epochs and different passbands (not all objects have all epochs in all passbands). They are labelled by an object ID corresponding to the running ID of the object on the master-template. In all, the database for B1 measurements contains over $1.6 \times 10^{9}$ individual photometric measurements. The above procedure ensures that all photometric measurements are placed on the same uniform instrumental system, independently in each of the passbands. Comparison of these instrumental magnitudes for high signal-to-noise ratio (S/N) objects across different exposures show that the self-consistency in the instrumental magnitudes is better than $0.02$ mag rms. On photometric nights, two of the newly calibrated DA white dwarf standards from @nar16 were also observed through a range of air-masses. These stars have calibrated spectral energy distributions, from which their true AB-magnitudes were calculated according to the prescription of @fuk96 for each of the 5 passbands. We then derived photometric solutions relating instrumental mags to AB-mags. In the $u,g,r,i$ bands, photometric solutions have residuals with $\sim 0.01$ mag rms scatter. In the $z$ band, which encompasses telluric water bands that can vary on time scales of minutes, as well as with position, the scatter is $ \sim 0.02$ to $0.03$ mag. When these solutions are applied to the photometry of the “grand” templates (one for each passband), we obtain calibrated AB-mags for the native DECam passbands. This is the same system used in @vivas17 where the luminosity and color relations for RR Lyraes are derived for precisely this system of magnitudes, making their results directly applicable to the data for the B1 field. Combining the scatter in the self-consistency in instrumental magnitudes discussed above, and the total calibration accuracy, we estimate that the systematic uncertainty in the calibration of any exposure is thus $\approx 0.02$ mag rms in $u,g,r,i$ and $\approx 0.035$ mag in $z$. Measurement errors for any object on any exposure are additional, and are estimated by the measurement procedures, including by the DoPHOT program. It should be pointed out that the analysis presented in this paper does not depend on what system of magnitudes we adopt, as long as the same system is used for all targets, including the globular cluster Messier 5 (NGC 5904), hereafter M5, where the color properties of the RR Lyrae stars are derived. Variability Analysis {#sec:variability} ==================== An Independent Identification of Variable Sources {#sec:allsources} ------------------------------------------------- Each object was tested for variability independently in each passband. However, a variability test is only meaningful if there are sufficient number of measurements of adequate quality. It is important to remove reported observations that have a high likelihood of being pathological. For each object and for a given passband, each measurement (by epoch) was subject to the following “interrogation:” 1. Is the object’s centroid located within 50 pixels of the edge of the respective CCD? 2. Are there any detected sources (including cosmic ray hits) within 1 FWHM (of the PSF) distance from the object’s centroid? 3. Does the value of the fitted background, $s$, fall outside the range $s2 \leq s \leq 2\, s90 - s2$, where $s2$ and $s90$ are the 2nd and 90th percentile values respectively for the fitted value of the background for [*all*]{} stars on that image? 4. Does the reported measurement error exceed $err_{lim}$, as defined in § \[sec:photometry\]? 5. Did DoPHOT assign the object a type other than 1 or 3 (which are for objects with an unambiguously stellar PSF, see § \[sec:photometry\] for DoPHOT types)? If the answer to any of the above questions is positive, the measurement was excised from further consideration. At least 15 measurements for a given object in a given passband must survive in order to proceed with variability assessment. Criterion 5 above is particularly severe in eliminating faint measurements. For our primary purpose of detecting and measuring the RR Lyrae stars, this is not an obstacle (as will be demonstrated below), but it may well inhibit the detection of variables to the faint limits that the photometry would otherwise allow. However, it is clear that the “purity” of the variable candidate list declines rapidly if DoPHOT type 7 measurements (those for which the S/N is too low to unambiguously ascertain if they have stellar PSFs) are allowed. In the final analysis, only 450,344 objects in $u$, 1,082,121 in $g$, 1,950,425 in $r$, 2,509,906 in $i$ and 2,347,075 in $z$ passed the above “interrogation” and were examined for variability. These numbers correspond to about 20% of all objects detected on the best and deepest available images in the respective passbands. The process can be easily re-run with changes in any and all of the parameters mentioned above. The variability search was carried out using the method laid out in @saha90. A reduced chi-square, $\chi_{\nu}^{2}$, with respect to the mean value of available magnitude measurements is computed using the available magnitude measurements and associated reported errors. We used a boot-strap sampling of the magnitude and error measurements for each of the available epochs to generate a robust estimate for the $\chi_{\nu}^{2}$, using 100 resamples. Ideally the $\chi_{\nu}^{2}$ for a non-variable (given the reported noise) should hover around unity. However, since in reality the distribution of noise is not fully expressed by a single Gaussian, and because reported errors are themselves subject to bias, we see that the mean of $\chi_{\nu}^{2}$ can change weakly with the brightness (see Figure \[fig:chisq\]). Accordingly, the mode value of $\log \chi_{\nu}^{2}$ was calculated for 0.5 mag wide bins of mean magnitude, and an object was flagged as a variable if its $\log \chi_{\nu}^{2}$ is higher than the relevant mode value by 1.3 (i.e. 20 times or more higher than the mode). The program also allows the user to interactively set the detection threshold with varying brightness. The variable lists from the above analysis in each passband were merged. A total of 4877 putative variables were flagged with this procedure, where each candidate is flagged in at least one of the 5 passbands. Identification of ab-type RR Lyraes and derivation of colors at minimum light {#sec:classification} ----------------------------------------------------------------------------- All of the 4877 candidate variables flagged above were run through the period finding procedure described in @saha17 (hereafter [PSEARCH]{}) in “batch” mode using $\Delta\phi_{max} = 0.05$ (as defined in their Equation 9). We identified the highest resultant peak of the $\Psi$ periodogram of [PSEARCH]{} for each object, and the resulting folded light curves in all available bands were plotted. A visual examination of the plots very quickly reveal the objects that are possibly RRab (radially pulsating in the fundamental mode) variables. The [PSEARCH]{} code was rerun interactively with $\Delta\phi_{max} = 0.02$ [for details see @saha17] for the objects that were flagged as possibly RRab’s to confirm the classification and to select the most likely period from among any aliases. The preliminary light curves mentioned above for all the of the 4877 candidate objects reveal a wealth of different variables. Binaries with short periods that are relatively well sampled with the observed cadence show very convincing light curves, as do short period pulsators like $\delta$-Scuti and/or SX Phe stars. Many RRc’s can be discerned by their slightly skewed near-sinusoidal light curves, while those without perceptible skewness are likely hiding as indistinguishable from amongst contact binaries with sinusoidal light curve shapes. There are also variables for which no believable folded light curves could be obtained. Their periodograms show peaks at much longer periods for which the data at hand cannot be used to derive believable periods. These are likely to be long period or semi-regular or irregular variables. Since the OGLE project surveyed the same region of the sky, and with much more extensive timing coverage than the present one (which was optimized to get light curves of the RR Lyraes), many of our identified variables can be matched to OGLE identified variables, for which their variable classification is available. While their data are primarily in one band, the panchromatic information from the data set of our study here can be used in novel ways to develop new classification methods, and is the subject of an ongoing study. We then ran the list of 491 possible RRab stars through a template fitting program from which the properties of the light curves (mean magnitude, amplitude, initial phase) were derived. This process is particularly important to define the initial phase (phase at maximum light) of the light-curves. The use of templates is helpful when the observations do not sample well that part of the pulsation period. We used the library of lightcurve templates set up by @sesar10 from RR Lyrae stars in SDSS Stripe 82. The library contains between 10 and 20 templates in each filter for type ab RR Lyrae stars, and only 1 or 2 for the types c. During the fitting process we allowed for variations around the period found by [PSEARCH]{} ($\pm 0.001$ d, in steps of $1 \times 10^{-6}$ d), the observed amplitude ($\pm 0.2$ mag in steps of 0.01 mag), the magnitude at maximum light ($\pm 0.2$ mag in steps of 0.01 mag), and the initial phase ($\pm 0.2$ in steps of 0.01) which was initially set by the time of the observation with the brightest magnitude in the lightcurve. The best template is found from $\chi^2$ minimization. Initially the fit is done only in the filter with the largest number of epochs available, which sets up the period and initial phase for that star. Then, the template fitting procedure is repeated for the other 4 filters but allowing variations only in the amplitude and maximum magnitude. Light-curves and the fitted templates for all stars are available as Figure \[fig:lc\]. They are also available via a Github repository[^4]. The epoch-by-epoch photometry in all bands for the bona-fide RRab stars is presented in Table \[tab:timeseries\]. During this process, we found 16 objects in the list to be poor fits to RRab templates. These objects were classified as other kinds of variables in the OGLE catalogs. In the final analysis, we have 474 surviving ab-type RR Lyraes, for which the periods, mean magnitudes, magnitudes at minimum light, and amplitudes in the 5 passbands are listed in Table \[tab:fitparams\]. The mean magnitudes were calculated by integrating the template light curve in each band after it had been converted to intensity units. The magnitudes at minimum light correspond to the magnitude of the fitted template at phase $\phi=0.65$. The same procedure was applied to the RR Lyrae stars in the globular cluster M5 presented by @vivas17, whose calibration will be used here to estimate the reddening. [ccccc]{}\[htb!\] 1448112 & 56423.665618 & $u$ & 17.528 & 0.012\ 1448112 & 56423.814222 & $u$ & 18.799 & 0.015\ 1448112 & 56424.718744 & $u$ & 18.923 & 0.014\ 1448112 & 56424.810501 & $u$ & 18.971 & 0.015\ 1448112 & 56424.888096 & $u$ & 19.067 & 0.016\ [cc]{} 1 & Object Identifier\ 2 & RA (J2000)\ 3 & DEC (J2000)\ 4 & Fitted Period (days)\ 5 & HJD - 2400000. Reference Epoch of zero phase ($\phi = 0$)\ 6 & No. of available $u$ measurements\ 7 & $u$ band Amplitude\ 8 & $u_{mean}$ (mags)\ 9 & $u_{ref}$ (mags) \[Fitted value at $\phi=0.65$\]\ 10 & No. of available $g$ measurements\ 11 & $g$ band Amplitude\ 12 & $g_{mean}$ (mags)\ 13 & $g_{ref}$ (mags) \[Fitted value at $\phi=0.65$\]\ 14 & No. of available $r$ measurements\ 15 & $r$ band Amplitude\ 16 & $r_{mean}$ (mags)\ 17 & $r_{ref}$ (mags) \[Fitted value at $\phi=0.65$\]\ 18 & No. of available $i$ measurements\ 19 & $i$ band Amplitude\ 20 & $i_{mean}$ (mags)\ 21 & $i_{ref}$ (mags) \[Fitted value at $\phi=0.65$\]\ 22 & No. of available $z$ measurements\ 23 & $z$ band Amplitude\ 24 & $z_{mean}$ (mags)\ 25 & $z_{ref}$ (mags) \[Fitted value at $\phi=0.65$\]\ 26 & Cross-matched OGLE Identifier\ Completeness Estimates {#sec:completeness} ---------------------- The correlation of our final list of RRab stars with the RRab’s from the RR Lyrae stars compilation of @sosz14 from the OGLE survey (hereafter OGLE RRab’s) allows us to make a quantitative estimate of the discovery completeness from [*both*]{} surveys. The DECam pointings at the various epochs were intended to repeat exactly. Hence the excluded regions, such as from gaps between chips were also repeated, and any variables that occupy that excluded region would not be detected. Recall also that a measurement of any object that at a given epoch fell within 50 pixels ($\sim $13) of any chip’s edge was discarded. With such caveats in mind, we denote the total number of RRab stars present on the complex operational portion of the DECam footprint by $N$. Let $n_{O}$, be the number of these actually present in the @sosz14 compilation and $n_{U}$ the number found by us in this work. Let $p_{O}$ and $p_{U}$ denote the discovery completeness of OGLE RRab’s and the present work respectively. We estimate from counting the OGLE RRab’s within the discoverable area of our B1 field that: $$n_{O} = N\, p_{O} \approx 560 ~(\pm 15)$$ where the uncertainty arises from the difficulty of counting stars in the avoidance zones within our footprint. From our own data and procedures described above, we have independently identified 474 ab-type RR Lyraes, so that: $$n_{U} = N\, p_{U} = 474$$ Matching the list of RRab’s from @sosz14 with ours using a 2 arcsec matching tolerance, we find that there are 472 RRab’s in common, so that: $$N\, p_{O}\, p_{U} = 472$$ It follows from the above that: $$\begin{aligned} p_{O} \approx 0.996 \\ p_{U} \approx 0.843\end{aligned}$$ It should be emphasized that these numbers are valid for RRab stars only. Other types of variables suffer different selection effects. Specifically our images typically have better seeing, and greater depth than OGLE, but we have much fewer epochs and a shorter total time baseline. Consequently we optimized our observing cadence for detecting RR Lyrae stars (our primary goal) at the expense of other kinds of variables with different temporal characteristics. OGLE has many more epochs and better coverage of the window function compared to the present study, albeit only in the $I$ band. Reddening from the RR Lyrae colors at Minimum Light {#sec:minlightreddening} =================================================== @sturch66 showed that the $B-V$ colors of fundamental mode RR lyrae stars (i.e., the Bailey ab-type) are invariant in the phase range $0.5 < \phi < 0.8$ (where the phase at maximum light is defined as 0.0) and that, aside from small metallicity and period dependent de-trending, the instrinsic $(B-V)$ colors are the same from star to star to within a few percent. Thus these serve as standard color sources, and have been used to determine interstellar reddening, including calibrating reddening from HI maps and galaxy counts by @burstein78. This paradigm has recently been re-examined using DECam filter passbands by @vivas17, who have presented expected colors for several passband combinations (their Equation 1, Table 6, and Figure 3) from a study of the RR Lyraes in the globular cluster M5. Their minimum light colors are derived from fitted light curve magnitudes at $\phi = 0.65$. Specifically, for zero reddening, we have from their paper that: $$\label{eqn:rmz0} (r-z)^{0}_{min} ~~~ = ~~~ 0.095 - 0.322\,(\log P)^2 ~~~~ (rms = 0.016)$$ $$\label{eqn:gmi0} (g-i)^{0}_{min} ~~~ = ~~~ 0.347 - 0.973\,(\log P)^2 ~~~~ (rms = 0.026)$$ $$\label{eqn:umg0} (u-g)^{0}_{min} ~~~ = ~~~ 0.665 - 0.669\,(\log P)^2 ~~~~ (rms = 0.029)$$ where $P$ is the period in days, and the $0$ superscript implies intrinsic colors. The information in Table 6 of @vivas17 also enables us to derive the intrinsic minimum light colors in any color combination as a function of period. We do not list them all here explicitly. The fitted magnitudes at phase 0.65 (the middle of the phase range where Sturch demonstrated that colors are constant) of the individual RR Lyraes in Table \[tab:fitparams\] are listed in the same table. From these we can construct the observed $(r-z)_{min}$ and $(g-i)_{min}$ values for the RRab’s, in Table \[tab:fitparams\] and obtain the color excess $E(r-z)$ and $E(g-i)$ from Equations \[eqn:rmz0\] and \[eqn:gmi0\]. The differential reddening across the field provides an opportunity to study the relationships of reddening across different color combinations. Figure \[fig:EgmivsErmz\] shows the individual reddening values $E(g-i)$ and $E(u-g)$ as a function of the reddening in $E(r-z)$ for all the RRab’s in Table \[tab:fitparams\]. Both panels show a linear trend. The $u$ filter has a small known red leak. For RR Lyrae stars, which are relatively blue, this should not have a noticeable effect. Further, if the reddening to M5 and field B1 were the same, the effect of the red leak would affect both fields equally, and the effect would be nulled out. However, since B1 has substantially more reddening it is prudent to ask if this is a potential problem. We note that given the large range of individual reddening among the RRab’s in B1, the effect of a red-leak in the $u$ band would express itself in the $E(u-g)$ vs $E(r-z)$ relation as a departure from linearity. We do not see such an effect in Figure \[fig:EgmivsErmz\], demonstrating that any adverse contribution from a $u$ band red leak is below the accuracy imposed by the scatter seen in the figure. The best fit (allowing for errors in both axes, and utilizing iterative $3.5\sigma$ outlier rejection) that relates the reddening in the two cases are given by: $$\label{eqn:Egmi} E(g-i) = (1.614 \pm 0.019)\,E(r-z) + (0.134 \pm 0.013)$$ and $$\label{eqn:Eumg} E(u-g) = (1.101 \pm 0.026)\,E(r-z) + (0.191 \pm 0.018)$$ where the uncertainties are estimated from the scatter. The figure shows the fitted relations, as well as the expectation from an O’Donnell law [@odonnell94] with $R_{V} = 3.1$. We see differences in fitted vs expected slope, but what is problematic is the vertical intercept, since we expect the relation to pass through the origin. We consider and discuss the following possible explanations: 1. Calibration errors/uncertainties in either the M5 reference data or the data presented here, or both. Since the data in both these figures come from observations taken on the same nights, this is unlikely. We have gone over the procedures several times to ascertain that a careless error has not been made. 2. Cumulative random errors in the photometric calibrations. Ascertaining zero-points for any one band for each of the M5 and current data-sets can suffer from random errors of up to 0.02 mag, so each color can have zero-point errors of 0.03 mag. Each axis of these figures is a difference of the same color in B1 and M5, so the total uncertainty in the zero-point of each axis can be .04 mag. Given that the slope of $E(g-i)$ vs $E(r-z)$ is $\approx 1.6$, we can thus expect y-axis intercepts of $\approx 0.065$ mag at the $1\sigma$ level due to systematic errors in measuring the $E(r-z)$ alone (due to a shift in the x-axis zero). If all errors add in quadrature, total rms uncertainty in the intercept is $ (0.04^{2} + 0.065^{2})^{0.5} ~=~ 0.075$ mag. This makes the observed offset in $E(g-i)$ almost a $2 \sigma$ effect. For $E(u-g)$, the offset is much larger, but metallicity differences between the globular cluster M5 and the RRab’s in the B1 field can induce all or part of the observed discrepancy [see Figure 5 of @vivas17] in $E(u-g)$. 3. The bulge RR Lyraes have different properties from those in M5. We are examining this possibility by studying several additional RR Lyrae bearing clusters, that differ from M5 in metallicity and Oosterhoff type. Our sample includes clusters in the bulge that have much higher metallicities and unusual period distributions relative to their metallicity. 4. There is some peculiar reddening that is shared equally by all objects (which means it must be relatively local before encountering a spiral arm where different lines of sight must produce differential extinction that cause the large spread in reddening) with a much steeper value of $E(g-i)/E(r-z)$. Since the RR Lyraes here are all piled up in the bulge, and we don’t see any with $E(r-z) < 0.3$, there is some effect that is hidden from us. Some of our other fields, especially B5, which passes clear of the plane may illuminate whether this is a possibility. The equivalent of Figure \[fig:EgmivsErmz\] for the other fields along different directions in the Galactic bulge may shed some light on whether this is plausible. The behavior in color combinations with the $r, i$, and $z$ bands do not exhibit an anomalous intercept. This is illustrated in Figure \[fig:ErmivsErmz\]. The linear fits are shown below in Equations. \[eqn:Ermi\] and \[eqn:Eimz\], and have intercepts entirely consistent with expected calibration uncertainties. However, the fitted slopes are steeper than predicted by [@odonnell94] with $R_{V} = 3.1$ for all cases except for $E(i-z)/E(r-z)$, signaling that the slope differences are not resolvable by a simple scaling of $R_{V}$, and shows a departure in shape from the O’Donnell law. This result is independent of the issue of the unexpected intercept. $$\label{eqn:Ermi} E(r-i) = (0.669 \pm 0.009)\,E(r-z) - (0.045 \pm 0.006)$$$$\label{eqn:Eimz} E(i-z) = (0.350 \pm 0.008)\,E(r-z) - (0.033 \pm 0.006)$$ Differences in slope are indicative of non-standard reddening, and we investigate the implications below. The intercepts must be accounted for when investigating distances, but it takes some contortion (item 4 in the above enumeration of possible reasons) to argue that they arise from non-standard reddening. We keep this in mind in the arguments we make in the remainder of this section. Specifically, the analysis that follows does not depend upon the zero-point anomaly, but only on the slopes derived from the differential extinction. We evaluate the effective wavelengths in each of the 5 passbands for an F5 spectrum and read the extinction at those wavelengths from the standard reddening law with $R_{V} = 3.1$ according to [@odonnell94]. The F5 spectrum is a good approximation to an RRab star near minimum light, and thus appropriately matched for color-excesses as measured from them. The extinction values in the 5 passbands determined in this way, and scaled so that $E(r-z) = 1$ provides the total to selective extinction ratios as follows: $$\begin{aligned} \label{eqn:stdreddening} A_{u} = 3.92\, E(r-z) \nonumber \\ A_{g} = 3.24\, E(r-z) \nonumber \\ A_{r} = 2.26\, E(r-z) \\ A_{i} = 1.73\, E(r-z) \nonumber \\ A_{z} = 1.26\, E(r-z) \nonumber \end{aligned}$$ We also quote from Equation 3 of [@vivas17], who calibrated the RRab absolute magnitudes in the globular cluster M5: $$\begin{aligned} \label{eqn:absmags} M_u \; (ab) &= (-0.10 \pm 0.24) \, \log P + (1.10 \pm 0.13) \nonumber \\ M_g \; (ab) &= (-0.57 \pm 0.17) \, \log P + (0.43 \pm 0.12) \nonumber \\ M_r \; (ab) &= (-1.28 \pm 0.11) \, \log P + (0.12 \pm 0.11) \\ M_i \; (ab) &= (-1.59 \pm 0.09) \, \log P + (0.07 \pm 0.11) \nonumber \\ M_z \; (ab) &= (-1.68 \pm 0.08) \, \log P + (0.03 \pm 0.11) \nonumber \end{aligned}$$ Given our observed mean magnitudes in each of these bands for each of the RRab stars in the B1 field, we derive individual distances to them. Figure \[fig:disthist\_odonnell\] shows the histogram of star counts at the derived distances. Since the line-of-sight passes very close to the Galactic center, the peak of the relative density histogram should occur very nearly at the distance of the Galactic center $R_{0}$. There are well defined peaks for the histograms in $r,i,z$ near a distance modulus of $8.1$ kpc, but shifts to $8.6$ kpc in $g$, with the histogram peak less sharply defined. In $u$ the histogram essentially disintegrates to a flat skewed extension, and the highest density is nearly at $10$ kpc. We submit that this inconsistency between the various bands is due to the use of the incorrect reddening law, as already surmised from the disagreement between the observed reddening vectors from those predicted by the standard reddening law. Differences in the reddening law affects not only the central tendency of the density histogram, but, because of the large differential extinction, also redistributes the relative distances among the RRab stars, changing the structure and shape of the density histogram from one band to another. We can seek to derive the correct reddening law that will yield not only the observed reddening vectors, but also bring into accord the distance histograms in all 5 passbands. Since RR Lyrae, and especially the RRab’s are both standard candles, and standard “crayons,” they lend themselves naturally to such an analysis. We show (in § \[sec:geom\]) that such an analysis yields a reddening law that can rectify disagreements of the distance distance distribution highlighted above, and makes them consistent across all 5 passbands. Derivation of the Extinction from the Reddening ----------------------------------------------- In light of the results and discussion above we must proceed with caution, clearly enunciating our assumptions so that the impact of this reddening “offset” (especially as seen in Figure \[fig:EgmivsErmz\]) can be tracked and examined at any point. We will choose to use $E(r-z)$ as fiducial reddening, since the evidence from Equations \[eqn:Egmi\] through \[eqn:Eimz\] indicate that the $r,i,z$ bands are better behaved, whatever be the source of difficulty with $g$ and $u$. However, the discrepancy in slopes with respect to the standard reddening law, and the discordance of the space density histogram with distance along the line-of-sight means that in order to calculate extinction from the reddening $E(r-z)$, we will be well served to derive $A_{X}/E(r-z)$ for any passband $X$ for our line-of-sight from our own data. Since the distribution of the RR Lyraes along the line-of-sight is very sharply peaked at the Galactic center, we can exploit the standard candle property of RR Lyraes. The absolute magnitude vs period relationships in Equation \[eqn:absmags\] are known to depend weakly on metallicity, and are strictly valid for the metallicity of M5. However @walker91 showed from spectroscopic analysis the metallicity distribution of RR Lyrae in the bulge peaks at ${\rm [Fe/H]} \approx -1.0$, whereas for M5 ${\rm [Fe/H]} = -1.25 \pm .05$ from @dias16. The similarity in metallicity supports using the absolute magnitudes from Equation \[eqn:absmags\]. For the purpose of deriving extinction from reddening by the method described below, any net offsets in the absolute magnitudes are not important, but their dependencies on period, as gleaned from Equation \[eqn:absmags\] are useful to consider. The distance modulus $\mu_{0}$ of any RRab star, its mean magnitude $m_{X}$, and its extinction $A_{X}$ in band $X$ are related by: $$A_{X} = m_{X} - M_{X} - \mu_{0}$$ $M_{X}$ is adopted for the appropriate band from Equation \[eqn:absmags\]. $\mu_{X} = m_{X} - M_{X}$ is the apparent distance modulus in $X$. $m_{X}, E(r-z)$ for the RRab’s are known from a combination of Table \[tab:fitparams\], Equation \[eqn:absmags\] and § \[sec:minlightreddening\]. Since $M_{X}$ can have no explicit dependence on $E(r-z)$, we can write: $$\label{eqn:mux2ax} \partial A_{X} / \partial(E(r-z)) = \partial \mu_{X} / \partial(E(r-z)) - \partial \mu_{0} / \partial(E(r-z))$$ If all the RRab’s are at the same distance, i.e., $\mu_{0}$ is constant, then measuring $ \partial \mu_{X} / \partial(E(r-z)) $ would directly give us the value of $A_{X}/E(r-z)$. In the present case, $\mu_0$ is not constant, but has a peaked distribution (we anticipate the result shown later in this paper that it peaks exponentially). If we can pick out the ones that are very near or within the sharp peak, and for which $ \partial \mu_{0} / \partial(E(r-z)) $ is negligible (which is true if the bulge itself has insignificant reddening), Equation \[eqn:mux2ax\] would again yield the desired value of $A_{X}/E(r-z)$. Thus, given a large enough ensemble of RRab’s, a wide enough range of $E(r-z)$, and a sufficiently peaked distribution in $\mu_{0}$ it should be possible to derive $\partial A_{X} / \partial E(r-z)$ from the slope $\partial \mu_{X} / \partial E(r-z)$. Note that if extinction/reddening within the bulge is *not* negligible, then objects at farther distance will on average have higher reddening, so that $ \partial \mu_{0} / \partial(E(r-z)) $ is positive. This implies that the true value of $ \partial A_{X} / \partial(E(r-z)) $ is, if anything, [*smaller* ]{} than the measured value of $ \partial \mu_{X} / \partial(E(r-z)) $, or that the above procedure yields at worst an upper bound on the total to selective extinction ratio. A potential further complication is that an individual RRab can have a metallicity different from the mean, producing some departure in absolute magnitude. However, this is not expected to exceed $\sim 0.2$ mag for any given such object, which is much smaller than the individual variations due to actual distance along the line-of-sight. Figure \[fig:Au\] shows $\mu_{X}$ for all bands as a function of $E(r-z)$ for the ensemble of all RRab’s for which mean magnitudes are available from Table \[tab:fitparams\] for all 5 bands. Again, a linear fit with iterative outlier rejection was performed with uncertainties on both axes. A value of $\sigma = 0.05$ was used for $E(r-z)$ and $\sigma = 0.20$ was used for $\mu_{X}$: the substantially larger uncertainty for $\mu_{X}$ allows for the back to front rms depth in the distances of individual RRab’s about the distance at which the space density peaks. The iterative rejection threshold was set at $2\sigma$. The intercepts in each case notionally provide the true mean distance modulus of the RRab in the sample. The few background and foreground RRab’s are clearly seen as outliers in these figures. The parameters and their uncertainties for the fitted linear regression for each of the 5 passbands are listed below: $$\begin{aligned} \label{eqn:total2selective} \mu_{u} = (4.003 \pm 0.087)\,E(r-z) + (15.281 \pm 0.061) \nonumber \\ \mu_{g} = (2.933 \pm 0.069)\,E(r-z) + (15.115 \pm 0.048) \nonumber \\ \mu_{r} = (1.898 \pm 0.058)\,E(r-z) + (14.978 \pm 0.041) \\ \mu_{i} = (1.254 \pm 0.055)\,E(r-z) + (14.985 \pm 0.038) \nonumber \\ \mu_{z} = (0.880 \pm 0.053)\,E(r-z) + (14.981 \pm 0.037) \nonumber\end{aligned}$$ As argued above the slopes in Equation \[eqn:total2selective\] correspond to the values of $A_{X}/E(r-z)$. We should note that the slopes (as well as intercepts) in the above derived relations are consistent within the errors with the corresponding slopes (and intercepts) in Equations \[eqn:Egmi\] through \[eqn:Eimz\]. However, the larger uncertainties in Equation \[eqn:total2selective\] are a result of the scatter introduced by the spread in actual distances to the individual RRab’s. However, Equation \[eqn:total2selective\] is necessary to infer the extinction. The color to color reddening relations (Equations. \[eqn:Egmi\] and \[eqn:Eumg\]) alone do not allow us to do that. In the absence of independent determinations of the total to selective absorption, the default practice is to use a standard reddening law such as [@odonnell94], but Equation \[eqn:total2selective\] makes it possible to check whether that is appropriate for the line-of-sight to our field B1. It is worth pondering whether the presence of RR Lyraes in the two globular clusters, NGC 6522 and NGC 6528, which fall within our field B1, bias our results. NGC 6522, which was placed in the gap between two chips, has 11 RR Lyrae stars listed in the Clement catalog [@clement01][^5] within 2 half-light radii ($r_{h}$). All but two of these are masked by our pointing, and the ones that remain are first harmonic oscillators that are thus not in our list of RRab’s. Similarly, NGC 6528 has only 2 known RR Lyrae associated with it, both within $2\, r_{h}$ [@skot15], one of which may be an RRab, but is not in our list. Looking at it another way, there are 3 RRab’s in our list within $5\, r_{h}$ of NGC 6522 (and none within $5\, r_{h}$ of NGC 6528). Even in the remote event that these 3 are bona-fide members of NGC 6522, removing them from the analysis does not change the derived coefficients in Equation \[eqn:total2selective\] by more than a small fraction of the stated uncertainties. The intercept values in Equation \[eqn:total2selective\] strongly anti-correlate with the corresponding slope value. The numbers correspond to the effective true distance modulus of where along the line-of-sight the RR Lyraes pile up the most, but at farther distances the field-of-view samples a bigger volume, so this requires tempering before it indicates the distance at which the RRab density peaks. Also note that the value of $E(r-z)$ that follows from Equation \[eqn:total2selective\] by subtracting $A_{z}$ from $A_{r}$, while not identical to the input $E(r-z)$, is self-consistent within the errors. This is because we have fitted allowing uncertainties in both axes. Reading from Equation \[eqn:total2selective\], we therefore adopt: $$\label{eqn:t2s0} A_{i} = 1.254\,E(r-z)$$ $$\label{eqn:t2s1a} A_{r} = 1.898\,E(r-z)$$ $$\label{eqn:t2s1b} A_{z} = 0.880\,E(r-z)$$ Recognizing that the accuracy of $E(u-g)$ and $E(g-i)$ from Equations \[eqn:Egmi\] and \[eqn:Eumg\] is far superior to the uncertainties presented in Equation \[eqn:total2selective\], and that the determined value of $A_{i}$ is both better constrained and less volatile with respective to errors in measuring $E(r-z)$ (because it has a multiplier of only 1.25, compared to 2.93 for $A_{g}$ and 4.00 for $A_{u}$), it is prudent to determine $A_{u}$ and $A_{g}$ as follows (since they are better anchored to the data): $$\label{eqn:t2s2} A_{g} = E(g-i) + A_{i} = E(g-i) + 1.254\,E(r-z)$$ $$\label{eqn:t2s3} A_{u} = E(u-g) + A{g} = E(u-g) + E(g-i) + 1.254\,E(r-z)$$ For the RRab’s themselves, the individual $E(g-i)$ and $E(u-g)$ are directly derivable using Equations \[eqn:gmi0\] and \[eqn:umg0\]. In what follows, we will only require Equations \[eqn:t2s0\] to \[eqn:t2s3\] to correct magnitudes for extinction, while we will correct (deredden) colors using Equations \[eqn:Egmi\] to \[eqn:Eimz\]. For the 5 DECam passbands used, using the extinction law of @odonnell94 with $R_{V} = 3.1$ gives $A_{X}/E(r-z)$ to be $4.00, 3.26, 2.28, 1.73$, and $1.27$ for $X = u, g, r, i,$ and $z$ respectively. Figure \[fig:redlaw\] shows both sets of values for $A_{X}/E(r-z)$. Since they are sufficiently different from our derived values we adopt the latter as given in Equations \[eqn:t2s0\] to \[eqn:t2s3\]. Color-Magnitude diagrams {#sec:CMDs} ======================== Differential Extinction and their Effect on the Observed CMDs {#sec:rawCMDs} ------------------------------------------------------------- There are in all $9,623,873$ distinct possible stellar objects in the master-list of all objects in the field B1, as described in § \[sec:photometry\]. All available measurements in all epochs in each passband were evaluated for the rejection criteria enumerated in § \[sec:variability\], and if 3 or more such measurements in each band survived the cut, they were averaged. Because of the extreme crowding in the fields, there is a rather severe elimination of faint objects, which are measured cleanly only in the best seeing and deepest images. In all there are a little over 2.5 million stars for which we have average magnitudes with this preselection in all of $g,r,i,z$, and 906,449 where average mags in all 5 bands are available. The observed color-magnitude diagrams, with different colors in the abscissa but using $i$ mags in the ordinate for all cases, involving various (but not exhaustive) combinations of the passbands are shown in the left panels of Figures \[fig:cmd\_rmz\], \[fig:cmd\_gmi\], and \[fig:cmd\_umg\]. Differential reddening and extinction contribute to a washed out appearance: most notably the red clump giants are smeared out along the reddening line for the redder abscissae, while for $u-g$ vs $i$, where the clump’s extension into the blue is an intrinsic feature, the reddening vector is no longer recognizable by the structure of the clump. . \[fig:cmd\_rmz\] . \[fig:cmd\_gmi\] Correcting for reddening and extinction using the RR Lyrae stars {#sec:justRRLs} ---------------------------------------------------------------- As the reddening and extinction to the individual RRab’s are established as described in § \[sec:minlightreddening\], we can apply these derived values to other stars close to them along the line-of-sight. However, we can see from the patchiness in the star counts on the images (especially in the $u$ band) that the extinction varies on angular scales of an arc-minute. The image of the full field was subdivided into rectangular bins, 30 on a side (reason for the choice of bin size is explained below). If a bin contains one or more RRab’s from Table \[tab:fitparams\], we assign the reddening and extinction from the RRab (averaging if there is more than one in a bin) to all the stars in that bin. For the first pass, stars in bins without an RRab are ignored. The resulting corrected CMD is shown in the left panel of Figure\[fig:rrdered\]. Most notably, the clump stars no longer show the signature of differential extinction as they do in the uncorrected corresponding CMD in the left panel of Figure \[fig:cmd\_gmi\], showing that the method works as it should. However, there are only 31,804 stars in this CMD, compared to over $ 2.5 \times 10^{6} $ stars that define the one in Figure \[fig:cmd\_gmi\], which is only 1.2% of all stars with adequate photometry. The corrected CMD is also over-represented by RR Lyrae stars because of how it was constructed (only bins containing an RRab were used). The clump of stars near $(g-i)_{0} \approx 0.0$ and $i_{0} \approx 15.5$ are thus the over-represented RRab’s. Making the bins bigger increases the number of surrounding stars, but the angular structure of the differential reddening prevents us from using bins larger than than 60 arcsec, before deterioration from the differential effects becomes apparent. Clearly, there are too few RR Lyraes to directly de-redden all the stars in this way: we would need 50 times or more of them to do so. We resort to a secondary method, anchored to the RRab’s. The right panel of Figure \[fig:rrdered\] shows the dereddened color-color diagram of the same stars that are in the left hand panel. The primary shape of the distribution of stars in this color-color plane is an extension along a direction that is almost degenerate with the reddening vector (shown in the figure with a dashed line). However, the star counts at various points on the color-color diagram locus provide a third dimension, and there is a lot of structure in the relative counts of stars, so that it is in effect a de-reddened color-color histogram (CCH) of stars. We assert that, however complex the stellar population components may be along the line-of-sight, this CCH is self similar across the entire B1 field. Thus, the distribution shown in Figure \[fig:rrdered\] is the measured intrinsic CCH, made up of multiple sub-samples taken from over 460 locations randomly scattered across the DECam field-of-view. However, it is affected by the faint cut-off, which varies across the 4 passbands used, and, because the faint cut-off for the [*dereddened*]{} colors, varies from place to place depending on the reddening and extinction. The former affects all field areas equally, so should not adversely affect what we are about to do, but the latter could affect us if there are features in the CMD near the faint cut-off. Since the faint cut-off is on the the main-sequence of bulge stars, and below the turn-off, variation in the cut-off of intrinsic magnitudes affects the CCH by changing the histogram value at the cut-off colors. We show later, from a diagnostic from the de-reddening procedure described below, that this is fortunately not a problem in the present case.\ Correcting for reddening and extinction using color-color histograms {#sec:useCCH} -------------------------------------------------------------------- We constructed a CCH using the stars shown in Figure \[fig:rrdered\] in the $r-z$ and $g-i$ color-color plane, with bin sizes of 0.02 mag along both axes. This represents the dereddened CCH that we have asserted applies to all sub-regions within the $\sim 3$ square degree DECam field. We denote this as the reference CCH. Consider the CCH constructed in the same way, but with uncorrected magnitudes and colors from any line-of-sight bin in the field. This should differ from the reference only to the extent of a translation in colors corresponding to the reddening of that field in $E(r-z)$ and $E(g-i)$. These shifts can be evaluated by a cross-correlation in the two color axes, i.e., by determining the values of $E(r-z)$ and $E(g-i)$ that provide the best match to the reference CCH. It is possible to force a one-axis cross-correlation by demanding that the reddening obeys Equation \[eqn:Egmi\], but allowing both color-excesses to be derived simultaneously provides an important cross-check. The result of this exercise is illustrated in Figure \[fig:ccresult\]. Each point represents the derived value of $E(r-z)$ and $E(g-i)$ for one of nearly 40,000 line-of-sight bins. As mentioned above, no external constraint was placed on the interdependence of the two axes. It is therefore very satisfying to see that the outcome is in accordance with Equation \[eqn:Egmi\], which is represented by the dashed line. This is of course expected, because it is an essential ingredient of the reference CCH. If it were not recovered, it would signal that the procedure for matching the observed CCH of each line-of-sight bin to the reference CCH is not working correctly. Rather, the fact that the slope and spread closely follow that of the left panel of Figure \[fig:EgmivsErmz\] assures us that the caveat raised at the end of § \[sec:justRRLs\] is not a manifest problem. It is also a diagnostic for ascertaining the optimal line-of-sight binning size. With the 30 bins, there are about 70 stars per bin that make up the observed CCH. There are places where there are fewer. For example, when a bin straddles an inter-chip gap. To take such situations in stride, a condition was imposed to not use any bins where there are fewer than 10 stars with available averaged photometry in all of the $u,g,r,i,z$ bands (instead for such bins we interpolate the results from neighboring bins). Smaller bins with fewer stars suffer from Poisson noise issues and the equivalent of Figure \[fig:ccresult\] steadily deteriorates for bins smaller than 30 on a side, and show greater scatter. Bins that are much larger allow more variation in the reddening within their extent: with resulting ambiguity in the cross-correlation. To see this we need to examine the two-dimensional structure of the correlation function peak, which we have found is often distended (or double peaked) along the reddening vector for bin sizes larger than 60 on the side. Our choice of 30is guided by the desire to maximize the spatial resolution while minimizing the effects of Poisson noise from too few stars in a bin. This choice is customized for the B1 field. For other fields with different stars densities and differential reddening structure, the optimal bin-size is expected to be different. Figure \[fig:ccmat\] shows contours of the peaks in the cross-correlation matrix for 9 randomly selected 30 bins. The peaks are highly elongated along the common direction of the reddening vector and the shape of the color-color locus, but the contour levels point to a common center. The 9 examples sample a range of reddening, as well as number of available stars in the respective CCH. The individual $E(r-z)$ and $E(g-i)$ values thus determined for each 30 arcsec square line-of-sight can then be used to calculate the reddening in other colors using Equations \[eqn:Eumg\] through \[eqn:Eimz\]. The extinction values in all 5 bands can be computed using the coefficients on $E(r-z)$ in the array of Equations \[eqn:total2selective\], but ignoring the offsets therein. For each bin, dereddened colors, and extinction corrected magnitudes in all 5 bands for all stars in that bin can be obtained in this way, and the accumulated results for the entire field can be derived. The resulting CMDs are shown in the right hand panels of Figures \[fig:cmd\_rmz\], \[fig:cmd\_gmi\] and \[fig:cmd\_umg\]. Salient Features in the Corrected CMDs {#sec:CMDfeatures} -------------------------------------- The efficacy of our procedure is immediately clear upon comparing the left and right panels of Figures \[fig:cmd\_rmz\] to \[fig:cmd\_umg\]. The difference is most striking for the $(r-z)$ vs $i$ CMD, where in the corrected version the red clump stars are gathered into a narrow color range, but with a vertical extent exceeding 0.5 mag from a combination of distance spread and possibly from stars of different ages. Both the lower main sequence and the sub-giant branch are much narrower in the de-reddened CMD, as we would expect. The corrected CMD shows that the red giant branch (RGB) and any asymptotic giant branch (AGB) stars fan out over a considerable color range, indicating a wide range of metallicities, as is already known from spectroscopy of bulge giants [e.g., @schult17]. The bright plumes of the bluest stars however appear to be [*more*]{} washed out in the [*c*orrected]{} CMDs. This is because the reddening estimates are anchored by the RR Lyrae stars, which are clumped in the bulge, whereas the bright blue stars are foreground disk stars, for which the reddening has been overestimated: thus the “‘corrected’’ colors and magnitudes for these stars are incorrect. There are two additional curious features in the corrected $i$ vs $(r-z)$ CMD: a plume of stars extending from the top of the clump star locus, arcing to the blue with increasing brightness ( $ 0 < (r-z)_{0} < 0.5$ and $i_{0} < 14.7$ ) and a sharpish blue edge for the RGB like stars. These features occupy the expected location for evolved stars where helium ignition in the core occurs before the core becomes degenerate, and the star ends up either as a red super-giant (that appears here as a pile up of RGB stars along a blue edge) or on the blue extremity of the helium burning “blue loop.” However, such locations in the CMD are populated by stars that are more massive than $\sim 2.5$ solar masses, implying that they are relatively young with ages of about 1 Gyr or even less. The “blue loop” track is clearly visible also in the $i_{0}$ vs $(g-i)_{0}$ CMD, but the putative red super giants are indistinguishable from the rest of the RGB/AGB stars. Note that unlike the foreground main sequence stars, the “blue loop” plume appears sharper and more tightly bound in color after it is de-reddened, signaling that they are located beyond the distances where most of the reddening takes place. The fact that the structure of the plume continues to stay well bounded in color at all brightness levels, and that it arcs to the blue as it gets more luminous, are arguments that they are very unlikely to be foreground red clump stars. [rrrrrrrrrrrrc]{} 270.554370 & -31.002110 &   —   &   —   & 20.134 & 0.008 & 18.977 & 0.007 & 18.451 & 0.005 & 18.186 & 0.005 & 0.88\ 270.562700 & -31.001770 &   —   &   —   & 21.355 & 0.026 & 19.687 & 0.032 & 18.845 & 0.018 & 18.457 & 0.020 & 0.88\ 270.554960 & -31.002100 &   —   &   —   & 20.623 & 0.012 & 19.445 & 0.008 & 18.810 & 0.007 & 18.481 & 0.008 & 0.88\ 270.561610 & -31.002010 &   —   &   —   & 21.116 & 0.018 & 19.770 & 0.012 & 19.141 & 0.010 & 18.799 & 0.010 & 0.88\ 270.563130 & -31.002290 &   —   &   —   & 21.582 & 0.025 & 20.244 & 0.013 & 19.733 & 0.017 & 19.428 & 0.011 & 0.88\ 270.558400 & -31.004000 & 18.508 & 0.009 & 16.790 & 0.005 & 16.068 & 0.006 & 15.825 & 0.004 & 15.747 & 0.004 & 0.88\ 270.559340 & -31.005540 & 21.608 & 0.032 & 18.413 & 0.006 & 16.751 & 0.007 & 16.004 & 0.004 & 15.585 & 0.004 & 0.88\ Figure \[fig:CMDlabels\] is a re-display of the right hand panel of Fig \[fig:cmd\_gmi\], but with labels pointing out the features mentioned above as they appear on the $i_{0}$ vs $E(g-i)_{0}$ CMD. There is a general broadening of features in this plane relative to the $i_{0}$ vs $(r-z)_{0}$ case, consistent with the fact that age and metallicity effects exhibit larger differences as we move to bluer colors. Nevertheless, the near vertical feature near $(g-i)_{0} = -0.5$ and $i_{0} > 15.8$ is more clearly expressed in the $i_{0}$ vs $(g-i)_{0}$ CMD, which is undoubtedly the extension of the horizontal branch as it “droops” in the blue. Note also that the red clump stars are not as tightly confined in color in $(g-i)_{0}$ as they are in $(r-z)_{0}$, very likely because of the metallicity spread among the stars. Past attempts to de-redden using the red clump stars [e.g., @kiraga97] using colors like $V-I$ would have suffered from the uncertainty and spread of intrinsic colors among the clump stars in the bulge. The CMD in $i_{0}$ vs $(u-g)_{0}$ is severely cut-off in the red because the $u$ band sensitivity of DECam as well as the more severe attenuation due to dust in $u$ imposes a much brighter faint limit. The pile up of bright red stars against a red limit near $(u-g)_{0} \approx 3.0$ is likely the result of a red-leak in the $u$ filter. The highlight of this version of the CMD is that it stretches out the track of stars in their post-helium flash phase, emphasizes the color extension of the red clump, prominently traces the entire extension of the horizontal branch, and sharply delineates the “droop” in the far blue range of the horizontal branch. Figure \[fig:rrverify\] shows the observed (left panel) and corrected (right panel) CMD with $u$ vs $u-g$, which is the bluest possible CMD rendition of our data where reddening and extinction express themselves maximally. The mean colors and magnitudes of the ab-type RR Lyrae stars are shown by the green points. On the uncorrected CMD, the RRab distribution is extended along the reddening vector, whereas in the corrected CMD they are distributed vertically in correspondence to their individual distances along the line-of-sight. Notice how the RRab distribution peaks where it intersects the horizontal branch (which in this color-magnitude configuration gets brighter in the blue relative to the clump). This particular representation of the CMD uses the color and magnitude most affected by reddening, so this consistency in the outcome of our de-reddening is gratifying. Table \[tab\_allstars\] lists the positions, observed magnitudes, and derived values of $E(r-z)$ for all stars used to produce the CMD’s. A Spectroscopic Preview of the “Blue Loop” Stars {#sec:blueloopspec} ------------------------------------------------ There are over 1200 stars in the “blue loop” feature. If this is confirmed as such, then this is just the high-mass end of the IMF for stars with ages of order a few hundred Myrs. Ten of the objects falling within this blue loop from this sample of stars were also observed as part of the Apache Point Observatory Galactic Evolution Experiment (APOGEE), which is one of the experiments from SDSS III/IV [@abol18]. APOGEE is a high-resolution spectroscopic (R=22,400) survey in the near-IR ($\lambda$=1.51-1.70$\mu$m) which targets, primarily, red giants from all Galactic populations; it is planned to have observed $\sim$500,000 stars by 2020 [@maj17; @holtz18; @jonss18]. Survey results from APOGEE include stellar parameters (effective temperature, $T_{\rm eff}$, surface gravity (as log g), and microturbulent velocity), precise radial velocities, and detailed chemical abundance distributions from, typically, 15 elements. These results are derived from an automated analysis package called the APOGEE Stellar Parameter and Chemical Abundance Pipeline [ASPCAP; @gper16]. The 10 red giants observed by APOGEE that are included in this study have ASPCAP calibrated parameters derived in the latest SDSS public Data Release 14 (DR14[^6]). A separate paper (Smith et al., in preparation) will present a detailed analysis of these 10 stars, while DR14 ASPCAP results will be discussed here. The effective temperatures and surface gravities of these red giants are consistent with their being core-He burning giants, with a small range in $T_{\rm eff}$ and log g: the mean values and their standard deviations are $T_{\rm eff}$=4765$\pm$110K and log g=2.6$\pm$0.2. The metallicities of this sample of red giants are interesting, as all are quite metal-rich, with values of \[Fe/H\] from $\sim$+0.1 to +0.4, which places them as likely members of the bulge, based on the observed distribution of metallicities of APOGEE bulge stars [e.g., @gper18; @zas19]. The mean value for the 10 giants is \[Fe/H\]=+0.25 $\pm$ 0.10. Of interest to this study are values of the carbon-to-nitrogen ratios, C/N, in these core-He burning stars. Early stellar evolution models [e.g., @iben64] predicted that the C/N ratio in red giants, after the completion of the first dredge-up, will depend upon the stellar mass. The relation between C/N and red giant mass is due to the deep convective envelope of a red giant, which mixes material to the stellar surface that has undergone H-burning via the CN-cycle, where $^{12}$C has been partially processed into $^{14}$N, leading to lower values of C/N relative to the main-sequence values. More massive red giants have both deeper convective envelopes, as well as higher internal temperatures, so the increase in the surface $^{14}$N and decrease in the surface $^{12}$C abundances are larger, resulting in lower values of C/N with increasing red giant mass. @mart16 have recently calibrated the relation between C/N and red giant mass, using a combination of APOGEE spectra and Kepler asteroseismology [@pins14], resulting in mass estimates with rms errors of $\sim$0.2M$_{\odot}$. The change in C/N is largest between about 1-2M$_{\odot}$, making it useful for estimating ages over a range of about 1-10Gyr. The values of the C/N ratio in these 10 red giants display only a small scatter, with a mean value and standard deviation of $ \langle {\rm [C/N]} \rangle =-0.55 \pm 0.12$; based upon @mart16 this indicates a mass of $\sim$1.5-1.7M$_{\odot}$ for this sample of bulge core-He burning giants. As these particular stars targeted by APOGEE are not the most luminous of the stars covered in the DECam sample, there are more luminous, and thus more massive, and even younger members of these clump giants. The Reddening Map {#sec:redmap} ================= The procedure described in § \[sec:useCCH\] produces reddening values in $E(r-z)$ and $E(g-i)$ for each 30 arcsec square cell over the field of view of DECam, except where there are too few stars with reported mean magnitudes in $g,r,i,z$. These maps can be interpolated to bridge gaps (where there are too few stars). FITS images of these maps (with WCS encoding of RA and DEC) are available as Data behind the Figure as well as in a Github repository[^7]. The map of $E(g-i)$ is shown in Figure \[fig:map\_egmi\]. The central part of the reddening map in Figure \[fig:map\_egmi\] shows relatively higher transparency without too much spatial variation in the reddening, and corresponds to the area chosen by @baade46 to peer close to the Galactic center. There is considerable patchiness outside this central window, with blobs and filamentary structures from arc-minute scales on up to the better part of a degree. Also visible are ring shaped structures with relatively low contrast scattered over the entire field that appear to be shells of dust. They range in size from about 5 to 15 arc-minutes in diameter. These are likely to be ejecta from massive stars driven out by winds. Given the crowding in the field, and irregularities in the shapes of the rings, we are unable to find any unambiguous visual correspondence of the ring centers with bright stars. We wonder whether having very short lives, such stars have long disappeared, but we are not in a position to know how long the bubbles would last before they dissipate. Tracing the Bulge Geometry with the fundamental mode RR Lyraes {#sec:geom} ============================================================== The Distance to the Galactic Center {#sec:distance} ----------------------------------- Consider a heliocentric Cartesian coordinate system, where the $Z$ axis points to the Galactic center, the $X$ axis is in the direction of the Galactic longitude $l$, and the $Y$ axis points towards the North Galactic Pole (NGP). The projection on the $Z$ axis of a point in space at a distance $d$ from the Sun with Galactic coordinates $l$ and $b$ is then: $$z = d \cos l \cos b \label{eqn:zproject}$$ The volume density in space of the RRab’s of Table \[tab:fitparams\] in $z$ is expected to peak at the distance $R_{0}$ to the Galactic center provided the spatial distribution of the RRab’s is spherical. However, for very small values of $l$ and $b$, the effects from non-sphericity in the distribution are small. For the stars in field B1, with direction centered at $l < 2.05^{\circ}$ and $|b| < 5.0^{\circ}$, the product of the cosine terms in Equation \[eqn:zproject\] differ from unity by less than 0.5%, which mitigates any effects from moderate azimuthal and polar asymmetries in the density distribution. In fact, for B1, the peak in the distribution of $d$ is by itself a measure of $R_{0}$ to within a percent if no azimuthal or polar asymmetries are present. Since we have the individual reddenings (Equation \[eqn:rmz0\] combined with observed minimum light colors from Table \[tab:fitparams\]) and extinctions (using Equations \[eqn:t2s0\] to \[eqn:t2s3\]) as well as mean observed magnitudes $m_{X}$ in any band $X$ for the individual RRab’s in Table \[tab:fitparams\], we can calculate their extinction corrected mean magnitudes $m^{0}_{X}$. We get their absolute magnitudes $M_{X}$ from Equation \[eqn:absmags\] using the period for the individual star from Table \[tab:fitparams\], which yields the distance modulus $ DM_{X} = (M_{X} - m^{0}_{X})$ for stars for each of the 5 passbands. The distance $d_{X}$ to an individual star calculated from data in the $X$ band is then given by: $$\label{eqn:distkpc} d_{X} ~{\rm (kpc)} ~=~ 10^{0.2(DM_{X} - 10)}$$ The distribution of $d_{X}$ for the RRab’s in Table \[tab:fitparams\] is shown in Figure \[fig:disthist\] for each of the 5 bands using the dashed lines. The solid line shows the relative number density, in $0.5$ kpc bins, by accounting for the change in the sampled volume with distance. For each band, the value of $R_{0}$ is determined by finding the location of maximum density, calculated as the “center of mass” from the 5 bins centered on the bin with the peak density. The results for the $u,g,r,i,z$ bands (labeled in the figure) are remarkably concordant, though surprisingly large compared with extant values of $R_{0}$ in the literature. Moreover, the density distribution with distance is sharply peaked, symmetrical, and very similar in all bands. The quoted uncertainties reflect *only* the errors in finding the centroids in the histograms: systematic errors are discussed below separately. The mean of the $u,g,r,i,z$ based results yields: $$R_{0} = 9.47 \pm 0.04 ~{\rm kpc} \label{eqn:rnought}$$ where we do not reduce the uncertainties from the individual passband measurements because the departures from the respective centroids are highly correlated. Again the quoted uncertainty is only the random error estimated from the widths of the histogram peaks in Figure \[fig:disthist\]. Our derived distance is at odds with the literature. A good compendium of determinations of $R_{0}$ up to 2015 is available from @degrijs16, including different kinds of tracers, statistical parallax methods, and analysis of the kinematics of stars near the Galactic nucleus. There are multiple reported values of $R_{0}$ from 7 to 9 kpc. After homogenization of the various determinations, they arrived at a statistical determination of $R_{0} = 8.3 \pm 0.2 ({\rm statistical}) \pm 0.4 ({\rm systematic})~{\rm kpc}$. By any account, our result presented here is about 10 percent higher than the norm. The derived distances in each band depend on the $A_{X}/E(r-z)$ (slope) values in Equations. \[eqn:t2s0\] to \[eqn:t2s3\]. Note that while the derivation of these equations assumes a strong clumping of distances of the RRab’s, it does not place any external constraint that the clump distance has to be identical across bands. That is, the intercepts in Equations. \[eqn:total2selective\] are determined independently from one band to another. There are several possible reasons why the derived value of $R_{0}$ here is larger than similar determinations from RR Lyrae stars in the past, but the three most pressing ones are the following: 1. Our derived reddening is different from the standard reddening law, and as seen in Figure \[fig:redlaw\], predicts lower extinction in $g,r,i,z$ than the standard reddening curve, making the corrected magnitudes fainter than what the standard law would give, and thus resulting in a larger distance. Our result deserves some further scrutiny. 2. We have adopted the absolute magnitudes derived for the globular cluster M5 in @vivas17 to apply also to the RR Lyrae in the Galactic bulge. If the RR Lyrae are different, for instance different Oosterhoff types, a distance discrepancy could result. We discuss this issue below. 3. The distance determination to M5, and hence the inferred absolute magnitudes of the RR Lyrae may be incorrect. We examine these possibilities in turn in some more detail. As mentioned above, our finding in this paper is that the standard reddening law is violated in the direction of our field. For a given reddening, be it $E(r-z)$, $E(g-i)$ or even $E(V-I)$, the extinction for all bands other than $450\, {\rm nm}$ is smaller with the reddening law derived here, compared to the standard formulation, as seen in Figure \[fig:redlaw\]. Thus, for a given adopted absolute magnitude (in this case based on the adopted distance to M5), our reddening law yields larger distances compared to the standard law. This is clearly seen in the comparison of Figures \[fig:disthist\_odonnell\] vs. \[fig:disthist\]. Note specifically that in the $i$ band, the standard law yields $R_{0} = 8.04~{\rm kpc}$ from Figure \[fig:disthist\_odonnell\], whereas Figure \[fig:disthist\] using the reddening law derived here gives $R_{0} = 9.47~{\rm kpc}$. This 18% difference in distance is exactly explained by the difference in total to selective extinction for the $i$ band given by equation \[eqn:stdreddening\] versus equation \[eqn:t2s0\], given that the mean $E(r-z)$ in Field B1 is $\sim 0.68$. The distances in all passbands when our derived reddening law is used are in agreement, whereas use of the standard law produces disparate distances across the passbands. This validates the derivation of our reddening law through Equation \[eqn:total2selective\]. The departure from the standard reddening law in fields in the Galactic bulge has previously been reported by @Nataf16 and @Nataf13, who did a similar analysis with OGLE $V,I$ and VISTA $J,K_{S}$ photometry of red clump giants. The luminosity of RR Lyraes is less universal than the minimum light colors, and can depend on metallicity, post zero-age horizontal branch evolution, and on helium abundance. As remarked earlier, the average metallicity distribution of RR Lyrae stars in Baade’s window peaks at ${\rm [Fe/H]} \approx -1.0$ as shown by @walker91 (see their Figure 7). The metallicity of the globular cluster M5 is ${\rm [Fe/H]} = -1.25 \pm 0.05$ according to @dias16. This similarity notwithwithstanding, small differences in helium abundance can drive much larger differences in luminosity. A more robust test for differences can be had through the Period-Luminosity-Temperature ($PLT$) relation. Eddington’s pulsation equation $P \langle \rho \rangle^{1/2} = Q$ and its refinements [e.g., @vanalbada71], in combination with a mass-luminosity relation for any class of stars that share a common evolutionary state, implies the existence of a $PLT$ relation for that stellar class. For RR Lyraes on the zero-age horizontal branch (ZAHB), it means that stars with the same $P$ and $T$ should have the same luminosity $L$. This precept has been used to examine the cause of the Oosterhoff dichotomy [e.g., @sandage90 and references therein]. Using the dereddened mean color $(r-z)_{0}$ as a proxy for the effective temperature $T$, we compare the mean Period-Color relations for RRab in the globular cluster M5 and in our Field B1. The left hand panel of Figure \[fig:percolamp\] shows that there is no net period shift at the same intrinsic color between M5 and the B1 field RRab’s, thus implying that their luminosities are also at par. The average amplitude (the mean of amplitudes from all 5 bands, which is most robust against measurement errors) as a function of $\log P$ is shown on the right hand panel: amplitudes have also been used in the literature as a proxy for temperature, but have been deprecated [@sandage90 and references therein]. The left hand plot is predicated on our determination of reddenings, while the right hand one is reddening independent but its quality as a proxy for temperature is less secure. In both plots we see no indication of a difference between the RRab’s in M5 vs. those in the B1 field, which supports the contention that the luminosities are the same for the RRab’s in both locations. Thus the second of the above possibilities is not a strong contender either. This leaves us with the question of whether the calibration of absolute magnitudes through M5 could be in error. The adopted distance modulus to M5, which is inherited from @vivas17 through Equations \[eqn:absmags\], is based on main sequence fitting. @layden05 derived $$\label{eqn:m5dist} \mu_{0} [M5] = 14.45 \pm 0.11$$ @layden05 also fitted the white dwarf sequence in M5 to the local white dwarfs with parallax distances, and obtained $\mu_{0} = 14.67 \pm 0.18$ which is both, more distant (which would make the RRab’s brighter, and $R_{0}$ even larger), and more uncertain. They report the average apparent $V$ magnitude of ab-type RR Lyraes in M5 to be $15.025 \pm 0.011$ (from their Table 4). Correcting for extinction ($E(B-V)=0.035$ and standard extinction law), $$\label{eqn:Vmagm5} \langle m^0_{V} \rangle [M5RRab] = 14.92 \pm 0.01$$ which implies that the intrinsic $V$ band average absolute magnitude for RRab stars in M5 is: $$\label{eqn:absVM5} M^0_{V} [M5RRab] = 0.47 \pm 0.11$$ A very recent determination of the distance to M5 by @gont19 gives $(m-M)_{0} = 14.34 \pm 0.09$ by multi-band isochrone and main-sequence fitting, which, if adopted, would decrease $R_{0}$ to 9.04 kpc. in the *Gaia* era, we should look to astrometric measurements for a more definitive distance determination. From *HST* parallax measurements of field RR Lyrae stars, @benedict11 obtain $$\label{eqn:absparallaxM5} M_{V} = 0.50 \pm 0.05$$ for the M5 globular cluster metallicity of ${\rm [Fe/H]} = -1.25$. This has a smaller formal uncertainty than Equation \[eqn:absVM5\], while the $0.03$ mag smaller distance modulus to M5 implied by the parallax based RR Lyrae absolute magnitudes is within the uncertainties, and yields $$\label{eqn:benedictr0} R_{0} = 9.45 \pm 0.30 ~{\rm kpc}$$ It is known that the *Gaia* DR2 results suffer from systematic errors in the measured parallax that vary with position in the sky as discussed in § 4.2 and 4.3 of @arenou18 and by @lindegren18. The reported parallax of 0.1135 mas for M5 [@helmi18] is thus likely to be an underestimate by a few tens of micro-arcsecs. It is expected that future *Gaia* data-releases will ascertain and correct for this systematic error, but at the moment a direct distance to M5 based on published *Gaia* parallaxes is not reliable. We can try instead to go through the *Gaia* DR2 parallax distances to calibrate the absolute magnitudes of RR Lyrae stars that are much closer to us than M5. @mura18 present such an analysis for 401 RR Lyraes, which include objects that still are at distances of several kpc, and so suffer from systematic uncertainties mentioned above. They also present a restricted sample of 23 RR Lyrae which have particularly well determined metallicities, but this sample too is not made of solely the nearest objects, and thus is not free of the parallax systematics. From the results in their Table 4 which gives a linear correlation of $M_{V}$ with ${\rm [Fe/H]}$ that is consistent with an LMC distance modulus of 18.5, we read: $$\label{eqn:mura_absmag} M_{V} = (0.26^{+0.05}_{-0.05}){\rm [Fe/H]} + 1.04^{+0.07}_{-0.07}$$ Using ${\rm [Fe/H]} = -1.25$ for the metallicity of M5 [@dias16], we get $$\label{eqn:DR2absmagM5} M_{V} = 0.72 \pm 0.07$$ Combining Equations \[eqn:Vmagm5\] and \[eqn:DR2absmagM5\] we get a revised distance modulus to M5 of $$\label{eqn:mu0M5new} \mu_{0} [M5] = 14.20 \pm 0.07$$ This corresponds to a reduction of all distances by a factor of $ 0.89 \pm 0.03 $, implying that Equation \[eqn:rnought\] is changed to $$\label{eqn:finalr0} R_{0} = 8.44 \pm 0.28 ~{\rm kpc}$$ which is consistent at the 1-$\sigma$ level with the recent determination of distance to the central black-hole of $7.93 \pm 0.13$ kpc [@chu18] and $8.13 \pm 0.03$ kpc [@abuter18] from the analysis of the orbit of the star S2 around the central black hole. The @mura18 result includes data for the type-c RR Lyraes, which we have avoided in our analysis in the bulge. For this reason we have independently analyzed the data for the 41 type-ab RR Lyraes from the @mura18 sample that have *Gaia* DR2 parallaxes greater than 1 milli-arcsec, and so are least affected by the systematic uncertainties in the *Gaia* parallax zero-point. Using the reported magnitudes and extinction estimates from their Table 1, we derive the equivalent of Equation \[eqn:mura\_absmag\] for this sample (rejecting the 3-sigma outlier AT And) to be: $$\label{mura_absmag_resample} M_{V} \approx 0.35{\rm [Fe/H]} + 1.06 ~~~~~~ (rms = 0.13 ~{\rm mag})$$ which yields $M_{V} \approx 0.62$ for the M5 RR Lyraes. This is brighter than the @mura18 result, but significantly and definitely fainter than from Equation \[eqn:absVM5\]. As @mura18 have pointed out, there are many selection effects to worry about from such [*ad hoc*]{} sample selection. The point of the exercise is to establish that there is enough uncertainty in the calibration of the RR Lyrae absolute magnitudes that 15% errors in distance determination are easily possible, and our derived value of $R_{0} = 9.47~{\rm kpc}$ based on the @layden05 main sequence fitting distance to M5 awaits modification at a future time when the *Gaia* mission gives us a parallax distance to M5, or better yet, directly to the RR Lyrae (tracers of the oldest stars) in the bulge. For the present, we continue the discussion with our derived value in Equation \[eqn:rnought\] for the purpose of the remaining analyses of the structure of the bulge in this paper, noting that all quantitative distances will scale linearly with any change from $R_{0} = 9.47~ {\rm kpc}$. Dereddening and Distances to the OGLE RRab’s -------------------------------------------- . \[fig:DEC2OGLE\] While determining minimum light colors for the bulge RRab’s in the OGLE catalog is made difficult because of the paucity of $V$ band measurements, we examine the possibility of using their mean $\langle V \rangle $ and $\langle I \rangle $ magnitudes to estimate reddening to individual RRab’s. We utilize the 472 RRab’s in Table \[tab:fitparams\] for which we have cross-matches to the OGLE-III catalog, from which we obtain their $\langle V \rangle$ and $\langle I \rangle$ values, and relate them to our values for $E(r-z)$. This is shown in Figure \[fig:DEC2OGLE\]. We derive the following relation $$\label{eqn:DEC2OGLE} { \langle V \rangle - \langle I \rangle ~~=~~ 1.116 (\pm 0.027) E(r-z) + 0.655 (\pm 0.018) ~~~~ [ \sigma = 0.08 {\rm mag }] }$$ where $\sigma$ indicates the rms scatter in $ \langle V \rangle - \langle I \rangle $ for an individual RRab star. Compare the derived $\sigma \approx 0.08 $ mag to the accuracy of better than $0.03$ mag with which we can predict $E(g-i)$ from $E(r-z)$ using Equation \[eqn:Egmi\]: this is a consequence of using mean mags instead of colors at minimum light. A bi-variate correlation of the extinction corrected mean $\langle i \rangle$ magnitudes (DECam system) derived in this paper to the observed $\langle V \rangle $ and $\langle I \rangle $ OGLE magnitudes of the corresponding RRab’s yields the following relation: $$\label{eqn:OGLEtoI0} \langle i \rangle _{0} ~=~ 1.986\, \langle I \rangle_{OGLE} ~-~ 1.031\, \langle V \rangle_{OGLE} ~+~ 1.765 ~~~~ [\sigma = 0.115~{\rm mag}]$$ which predicts the extinction corrected mean $i$-band magnitude (DECam system used in this paper) using the [*observed*]{} mean $V$ and $I$ OGLE magnitudes for any RRab star. The scatter indicates that the prediction is uncertain with an rms of 0.115 mag. Using this relation we can get distances to individual RRab in the OGLE catalog with a 6% rms scatter (and additional systematic uncertainties from how well we know the absolute magnitudes of the RRab’s). . \[fig:OGdist\] Limiting ourselves to Galactic longitudes $l$ bounded within $-5^{\circ} < l < +5^{\circ}$ and Galactic latitudes $b$ within $-8^{\circ} < b < 0^{\circ}$ (where the line-of-sight looks relatively close to the Galactic center, and the distances are not biased by RRab’s in background star streams), we have 8092 RRab’s from the OGLE-III catalog. Calculating their distances using Equations \[eqn:OGLEtoI0\] and \[eqn:absVM5\] we obtain the projected distance $z$ on the scale of Layden’s distance to M5 using Equation \[eqn:zproject\]. We construct the histogram of the RRab’s for $z$ as shown by the dashed lines in Figure \[fig:OGdist\]. To make the histogram represent relative star densities, we reconstruct using a weighted count for each star, where the weight decreases as the inverse square of the line-of-sight distance. The solid line in Figure \[fig:OGdist\] shows this modified histogram. Using the same peak centroiding methods as used above, we obtain a peak density at $z = 9.54 \pm 0.02$ kpc, where the quoted uncertainty refers only to the centroiding error. If the spatial distribution of the RR Lyrae stars is azimuthally symmetric, $z$ is a good estimator of $R_{0}$. It differs from Equation \[eqn:rnought\] by less than 1 percent. On the one hand this agreement is only to be expected, because the same precepts for reddening and absolute magnitudes of the RRab’s have been used for both derivations. However, on the other hand, the sub-sample from OGLE-III used here has 40 times more stars, spread over a wider spatial extent, so the agreement validates implicit assumptions regarding the spatial distribution of the RR Lyraes. This result however is pre-mature in detail. @Nataf16 showed that not only is the reddening towards the bulge non-standard, but that it also varies from one line-of-sight to another within the angular scale of the bulge. We might therefore expect that Equation \[eqn:DEC2OGLE\], and therefore Equation \[eqn:OGLEtoI0\] will be different for different lines of sight. With the wider perspective of the OGLE coverage, it is now possible, in principle, by applying the reddening corrections as done here, to deduce the spatial distribution of the RR Lyraes near the Galactic center, especially the flattening of the density ellipsoid. The gaps in coverage in $l$ and $b$, and possible incompleteness along lines of very high extinction thwart the direct calculation of relative densities, and makes this task messy: we do not attempt it here. The caution above about variation of the reddening law itself along different sight lines also applies to most applications made possible by he OGLE RR Lyrae data-set in the bulge. We will be able to ascertain how much Equation \[eqn:DEC2OGLE\] and \[eqn:OGLEtoI0\] change when we present the analysis for the 5 remaining fields in our study. The Density Distribution of RR Lyraes in the Galactic Bulge ----------------------------------------------------------- RR Lyrae are well known tracers of ancient stellar populations. With over 450 RRab’s in our B1 field, whose line-of-sight passes close to the Galactic center, for which we have well derived distances, and from which we have derived a distance to the center $R_{0}$, we are poised to examine the density distribution of the parent population of ancient stars. If the distance to an object is $d$, and its Galactic coordinates are $l$ and $b$, then the Galactocentric distance $r$ is given by: $$\label{eqn:Gcentric} r^{2} = R_{0}^{2} + d^{2} - 2dR_{0} \cos(l) \cos(b)$$ Applying this to the count and density histograms derived in § \[sec:distance\], and using $R_{0} = 9.47~{\rm kpc}$ from Equation \[eqn:rnought\], we can remap them as a function of $r$. Figure \[fig:rdens1\] shows the distribution: open circles denote values for $d < R_{0}$ and filled circles for $d > R_{0}$. The error bars are calculated using Poisson statistics of the counts of RRab in each “bin,” and dividing by the normalized volume for the corresponding line-of-sight distance. Since the line-of-sight is at non-zero $b$, flattening of the bulge along the polar axis can produce different density values at the same $r$, for locations on the near side, vs. that on the far side of the Galactic center. Figure \[fig:rdens2\] shows the same plot, but now with the natural log of the relative density on the ordinate. The points out to $r < 5$ kpc appear to decrease linearly with $r$, indicating an exponential decline in the density of the form $$\label{eqn:expdens} \rho = B e^{-r/a}$$ Formal fits to the near side (open circles) and far side (filled circles) show slightly different slopes of -0.90 and -1.21 ${\rm kpc}^{-1}$ respectively, corresponding to $a = 1.11$ kpc and $a = 0.83$ kpc for the near and far sides respectively. These (pseudo-)scale lengths obviously are along the line-of-sight and not along any axis of symmetry of the actual distribution (if indeed it is even elliptical). However, we expect to learn more from the other 5 fields in this study. On the far side there is a rise in the RR Lyrae density beyond $r \sim 5$ kpc. This may indicate an encounter of the line-of-sight with features in the thick disk (such as a spiral arm), or possibly a stellar stream. For instance, *Gaia* recently identified the remnant of a galaxy, Gaia-Enceladus, which merged with the Milky Way approximately 10 Gyr ago [@helmi18b]. This merger produced different stellar streams that cover the entire sky and cross the Galactic disk and also have associated RR Lyrae stars [see Figure 3 of @helmi18b]. Summary Discussion {#sec:discussion} ================== To date, the bulge has been probed primarily by spectroscopy of the brighter stars [e.g., @kun11]. We know from this spectroscopy that there is a huge range of metallicities, but no definitive constraints on the age distribution. The available giants do not proportionally represent all elements of the underlying population mix. Being mindful of the Initial Mass Function, one expects older stars to be over-represented among the giants, because their turnoff luminosities and masses are lower than their younger counterparts. Main sequence stars in the bulge are prohibitively faint for detailed spectroscopic analysis, except when such stars are temporarily rendered brighter due to lensing events. From a handful of main sequence stars studied in this way, @bens18 argue that there are significant numbers of stars younger than 8 Gyr in the central kilo-parsec, a conclusion that is apparently at odds with that from the study of giants alone. Given that the sample of 19 main sequence stars is unlikely to increase significantly in the foreseeable future, it appears that the issue of selective representation of sub-populations of stars by the giants is best mitigated by the synthesis of observed Hess diagrams of the bulge, using methods along the lines of @dolph02 which are capable of producing a more complete picture of the star formation history and the age-metallicity relation. Analysis of the VVV generated near infrared CMD’s are already underway [@surot19]. The visual wavelength multi-band CMD’s presented here provide higher leverage on metallicities (and hence on ages as well), but they are not yet Hess diagrams where the selection effects and completeness are fully understood and characterized. The production of adequate quality Hess diagrams of the bulge has faced several challenges. In this paper we have demonstrated that the major problem of decoding the line-of-sight reddening with sufficient angular resolution as well as deriving and applying the correct reddening law is tractable through the RR Lyrae stars. This builds on the work of @Nataf13 [@Nataf16], who used Red Giant Clump stars as standard candles and distance markers. Our work here uses independently derived photometry, a different color and distance marker, and extends the reddening analysis to much bluer passbands (where the clump shows structure in color). We have embarked on a program to search empirically for any systematic issues with RRab minimum light colors as a standard color marker, by extending the work done on the globular cluster M5 [@vivas17], to a number of other RR Lyrae bearing clusters with different metallicities and Oosterhoff types. While the work presented here is thus a major step forward, there are two other issues that need resolving before the population synthesis mechanism can be brought to bear in full measure: 1. Removal of the foreground contamination along the line-of-sight. As discussed in § \[sec:intro\], it is only a matter of time before surveys like VVV and their derivatives are able to solve this issue. 2. Estimation of the completeness in the CMDs. The usual procedure of deriving the completeness from artificial star tests is made more complicated by the extreme range and angular scales over which the extinction changes. Effectively, each $30$ arcsec square bin must be evaluated independently. Extinction not only changes the de-reddened magnitude from the brightness recovered from the image, but also affects how crowded a given patch of the field appears, and the confusion limit for detection and measurement errors. This is not an intractable problem, but one that will be time consuming. We intend to address it after the equivalent of the study in this paper is done for all 6 of our fields. So how might the CMDs generated here, and the equivalent for the other fields in our study be used in the interim period before Hess diagrams with completeness estimates and foreground contamination removed can be produced? First, one can ascertain the locations of the stars that have been spectroscopically studied already on our CMDs. Are certain areas of the CMD excluded? A case in point are the more luminous among the “blue loop” stars discussed in § \[sec:CMDfeatures\] and § \[sec:blueloopspec\], and the associated questions raised by the existence of that feature. Are all regions of the fan shaped giant branch structure in our CMDs represented? Have clump stars from different parts of the extended clump structure in Figure \[fig:cmd\_umg\] been observed? How do stars from the blue and red ends of that extension differ in their spectroscopic characteristics? Conversely one can use the de-reddened CMDs to pick stars from regions of particular interest to follow up spectroscopically in the next generation of surveys, such as APOGEE V. When the equivalent CMDs for the other fields become available, we will be able to look for differences. Perhaps an empirical CMD of the foreground disk stars will emerge that will help model the foreground population in preparation for synthesizing the Hess diagrams to come. Our plan is to proceed first with deriving and presenting the CMDs for all the remaining fields, followed by work along the lines outlined above. We note that of the 4877 putative variables of all types we have detected, 2265 were detected independently in at least 2 of the 5 passbands, implying that these are almost certainly not false positives. Since our time coverage is limited, and focussed on obtaining light curves for RR Lyraes, our data are inadequate for obtaining light curves and periods for all these objects. Fortunately, many of these are also known and classified by the OGLE survey. Our five band, de-reddened data for this set of objects are a “training set” for identifying variable stars from future LSST data, which will have sparse cadence, but information in panchromatic passbands. With the exception of the “Stripe 82” field from SDSS [e.g., @sesar07; @bramich08], there are no suitable publicly available data sets with time-domain coverage in multiple passbands at this time that are suitable for developing and testing algorithms for parsing variability characteristics from LSST data. Our results provide the ability for LSST time-domain brokering projects to develop the necessary techniques and algorithms to handle the variable star data that LSST will generate. This project used data obtained with the Dark Energy Camera (DECam), which was constructed by the Dark Energy Survey (DES) collaboration. Funding for the DES Projects has been provided by the U.S. Department of Energy, the U.S. National Science Foundation, the Ministry of Science and Education of Spain, the Science and Technology Facilities Council of the United Kingdom, the Higher Education Funding Council for England, the National Center for Supercomputing Applications at the University of Illinois at Urbana-Champaign, the Kavli Institute of Cosmological Physics at the University of Chicago, the Center for Cosmology and Astro-Particle Physics at the Ohio State University, the Mitchell Institute for Fundamental Physics and Astronomy at Texas A&M University, Financiadora de Estudos e Projetos, Funda[ç]{}[ã]{}o Carlos Chagas Filho de Amparo [à]{} Pesquisa do Estado do Rio de Janeiro, Conselho Nacional de Desenvolvimento Cient[í]{}fico e Tecnol[ó]{}gico and the Minist[é]{}rio da Ci[ê]{}ncia, Tecnologia e Inovac[ã]{}o, the Deutsche Forschungsgemeinschaft, and the Collaborating Institutions in the Dark Energy Survey. The Collaborating Institutions are Argonne National Laboratory, the University of California at Santa Cruz, the University of Cambridge, Centro de Investigaciones En[é]{}rgeticas, Medioambientales y Tecnol[ó]{}gicas-Madrid, the University of Chicago, University College London, the DES-Brazil Consortium, the University of Edinburgh, the Eidgen[ö]{}ssische Technische Hochschule (ETH) Z[ü]{}rich, Fermi National Accelerator Laboratory, the University of Illinois at Urbana-Champaign, the Institut de Ci[è]{}ncies de l’Espai (IEEC/CSIC), the Institut de F[í]{}sica d’Altes Energies, Lawrence Berkeley National Laboratory, the Ludwig-Maximilians Universit[ä]{}t M[ü]{}nchen and the associated Excellence Cluster Universe, the University of Michigan, [the]{} National Optical Astronomy Observatory, the University of Nottingham, the Ohio State University, the OzDES Membership Consortium the University of Pennsylvania, the University of Portsmouth, SLAC National Accelerator Laboratory, Stanford University, the University of Sussex, and Texas A&M University. Based on observations at Cerro Tololo Inter-American Observatory, National Optical Astronomy Observatory (NOAO Prop. 2013A-0719 and PI A. Saha), which is operated by the Association of Universities for Research in Astronomy (AURA) under a cooperative agreement with the National Science Foundation. We thank the support staff at CTIO for their excellent support during the observations. Olszewski was partially supported by NSF Grants AST-1313001 and AST-1815767. We are grateful for perceptive (and ultra-prompt!) comments from an anonymous referee which have resulted in substantive improvements to the paper. [^1]: http://irsa.ipac.caltech.edu/applications/DUST/ [^2]: https://archive.noao.edu [^3]: https://datalab.noao.edu [^4]: *https://github.com/akvivas/Baade-s-Window* [^5]: Updated version at http://www.astro.utoronto.ca/ cclement/cat [^6]: https://www.sdss.org/dr14/irspec/ [^7]: https://github.com/akvivas/Baade-s-Window

--- abstract: 'Let $A$ be an abelian variety defined over a number field $K$. If ${\mathfrak{p}}$ is a prime of $K$ of good reduction for $A$, let $A(K)_{\mathfrak{p}}$ denote the image of the Mordell-Weil group via reduction modulo ${\mathfrak{p}}$. We prove in particular that the size of $A(K)_{\mathfrak{p}}$, by varying ${\mathfrak{p}}$, encodes enough information to determine the $K$-isogeny class of $A$, provided that the following necessary condition is satisfied: $B(K)$ has positive rank for every non-trivial abelian subvariety $B$ of $A$. This is the analogue to a result by Faltings of 1983 considering instead the Hasse-Weil zeta function of the special fibers $A_{{\mathfrak{p}}}$.' author: - Chris Hall and Antonella Perucca title: 'Characterizing Abelian Varieties by the Reductions of the Mordell-Weil Group' --- Introduction ============ Let $K$ be a number field and $A,A'$ be abelian varieties over $K$. A well-known result of Faltings ([@Faltings83]) implies that $A,A'$ are $K$-isogenous if and only if they have the same $L$-series. More precisely, if $S=S(A,A')$ is the set of finite primes ${\mathfrak{p}}\subseteq K$ of common good reduction for $A,A'$ and if $S'\subseteq S$ has density one, then $A,A'$ are $K$-isogenous if and only if, for every ${\mathfrak{p}}\in S'$, the special fibers $A_{\mathfrak{p}},A'_{\mathfrak{p}}$ have the same Hasse–Weil zeta function. The $L$-series of $A$ is determined, in part, by the function $\nu:{\mathfrak{p}}\in S\mapsto\#A(k_{\mathfrak{p}})$, and in this paper we consider other functions which one can use to characterize $K$-isogeny. Let ${\Gamma}\subseteq A(K),{\Gamma}'\subseteq A'(K)$ be subgroups, and for each ${\mathfrak{p}}\in S$, let ${\Gamma}_{\mathfrak{p}}\subseteq A(k_{\mathfrak{p}})$, ${\Gamma}'_{\mathfrak{p}}\subseteq A'(k_{\mathfrak{p}})$ be the respective reductions. For each prime $\ell$, we consider the composition of the functions ${\mathfrak{p}}\mapsto{\Gamma}_{\mathfrak{p}}$ and ${\mathfrak{p}}\mapsto{\Gamma}'_{\mathfrak{p}}$ with the function which sends a finite group $G$ to the $\ell$-adic valuation of the order, exponent, or radical (of the order) of $G$ and which we denote $\operatorname{ord}_\ell(G)$, $\exp_\ell(G)$, and $\operatorname{rad}_\ell(G)$ respectively. Rather than consider these functions for arbitrary $A,A'$ and ${\Gamma},{\Gamma}'$, we place conditions on $A$ and ${\Gamma},{\Gamma}'$. We say $A$ is [*square free*]{} if the only abelian variety $B$ for which there exists a $K$-homomorphism $B^2\to A$ with finite kernel is $B=0$. We say ${\Gamma}$ (resp. ${\Gamma}'$) is a [*submodule*]{} if and only if it is an $\operatorname{End}_K(A)$-submodule (resp. $\operatorname{End}_K(A')$-submodule), and we say ${\Gamma}$ is [*dense*]{} if and only if $\pi({\Gamma})\neq{{\{0\}}}$ for every $\pi\neq 0\in\operatorname{End}_K(A)$. \[thm1\] Let $A,A'$ be abelian varieties and $S'\subseteq S(A,A')$ have density one, and suppose ${\Gamma}\subseteq A(K)$, ${\Gamma}'\subseteq A'(K)$ are submodules. If ${\Gamma}$ is dense and if $\ell\gg 0$, then the following are equivalent: 1. there exists $\phi\in\operatorname{Hom}_K(A,A')$ such that $\ker(\phi)$ and $[\phi({\Gamma}):\phi({\Gamma})\cap{\Gamma}']$ are finite; 2. $\operatorname{ord}_\ell({\Gamma}_{\mathfrak{p}})\leq \operatorname{ord}_\ell({\Gamma}'_{\mathfrak{p}})$ for every ${\mathfrak{p}}\in S'$. If moreover $A$ is square free and if $\ell\gg 0$, then these are equivalent to the following: 1. $\exp_\ell({\Gamma}_{\mathfrak{p}})\leq \exp_\ell({\Gamma}'_{\mathfrak{p}})$ for every ${\mathfrak{p}}\in S'$; 2. $\operatorname{rad}_\ell({\Gamma}_{\mathfrak{p}})\leq \operatorname{rad}_\ell({\Gamma}'_{\mathfrak{p}})$ for every ${\mathfrak{p}}\in S'$. Clearly if ${\Gamma}$ is ${{\{0\}}}$ or even finite, then conditions 2,3,4 hold *regardless* of what $A,A',{\Gamma}'$ are. In order to avoid pathologies like this assume ${\Gamma}$ is dense, or equivalently, the intersection of ${\Gamma}$ with each non-trivial abelian subvariety $B\subseteq A$ is infinite. Also, for any finite group $G$, $\exp_\ell(G\times G)=\exp_\ell(G)$ and $\operatorname{rad}_\ell(G\times G)=\operatorname{rad}_\ell(G)$, hence the reason we must suppose $A$ is square free in 3 and 4. As one might expect, Kummer theory lies at the core of our proof of the theorem, and the subgroups which give the cleanest statements, especially when characterizing when distinct subgroups are ‘independent,’ are submodules. The basic strategy we employ to prove an equivalence such as $1\Leftrightarrow 2$ is to prove two implications: $1\Rightarrow 2$ and $\neg 1\Rightarrow\neg 2$. The first implication is straightforward. A crucial notion which appears in the second implication is of ‘almost free’ points, and we develop this notion in section \[sec:suff\_indep\]. Prior to this we give some preliminary results in section \[sec:prelim\], and finally, we prove theorem \[thm1\] in section \[sec:proof\_thm1\]. Note, the Mordell-Weil group of an abelian variety is a dense submodule if and only if the Mordell-Weil group of every abelian subvariety is infinite. Then we have the following: Let $A,A'$ be abelian varieties and $S'\subseteq S(A,A')$ have density one, and suppose $B(K)$ is infinite for every non-trivial abelian subvariety $B\subseteq A$. For every fixed $\ell\gg 0$, the $K$-isogeny class of $A$ is determined by the function ${\mathfrak{p}}\in S'\mapsto \#A(K)_{\mathfrak{p}}$. If moreover $A$ is square-free and $\ell\gg 0$, then the $K$-isogeny class of $A$ is determined by the function ${\mathfrak{p}}\in S'\mapsto \operatorname{rad}_\ell A(K)_{\mathfrak{p}}$, hence a fortiori by the function ${\mathfrak{p}}\in S'\mapsto \exp_\ell A(K)_{\mathfrak{p}}$. If $A(K)$ and $A'(K)$ are free of rank $1$ and the functions ${\mathfrak{p}}\mapsto\exp_\ell(A(K)_{\mathfrak{p}})$ and ${\mathfrak{p}}\mapsto\exp_\ell(A'(K)_{\mathfrak{p}})$ are considered, the above result relates to the so-called support problem (cf. [@DemeyerPerucca thm. 1.2]). Note that there exist pairs of elliptic curves over a number field $K$ which are not $K$-isomorphic but such that for every prime number $\ell$ there is a $K$-isogeny between them of degree coprime to $\ell$ (cf. [@Zarhin sec. 12]). This implies that it is not possible to characterize the $K$-isomorphism class of $A$ by knowing the order and the exponent of $A(K)_{{\mathfrak{p}}}$ for ${\mathfrak{p}}$ varying in a set of density $1$. Notation -------- Unless explicitly stated otherwise, we assume all abelian varieties, subvarieties, homomorphisms, etc. are defined over $K$. Given an abelian variety $A$, we denote by $S(A)$ the set of finite primes ${\mathfrak{p}}\subset K$ of good reduction for $A$, and we write $k_{\mathfrak{p}}$ for the residue field and $A(k_{\mathfrak{p}})$ for the group of $k_{\mathfrak{p}}$-rational points. By the density of a subset $S'\subseteq S(A)$ we mean the Dirichlet density. We also write ${\mathrm{E}}(A)$ for the ring $\operatorname{End}_K(A)$, and given a second abelian variety $B$, we write ${\mathrm{H}}(A,B)$ for $\operatorname{Hom}_K(A,B)$. Given ${\mathfrak{p}}\in S(A)$ and a subgroup ${\Gamma}\subseteq A(K)$, we write ${\Gamma}_{\mathfrak{p}}\subseteq A(k_{\mathfrak{p}})$ for the reduction of ${\Gamma}$ modulo ${\mathfrak{p}}$. Moreover, for each rational prime $\ell$, we define the following functions on $S(A)$: $$\nu_{\ell,{\Gamma}} : {\mathfrak{p}}\mapsto\operatorname{ord}_\ell({\Gamma}_{\mathfrak{p}}), \quad {\varepsilon}_{\ell,{\Gamma}} : {\mathfrak{p}}\mapsto\exp_\ell({\Gamma}_{\mathfrak{p}}), \quad \rho_{\ell,{\Gamma}} : {\mathfrak{p}}\mapsto\operatorname{rad}_\ell({\Gamma}_{\mathfrak{p}}).$$ They respectively express the $\ell$-adic valuations of the size, the exponent, and the radical of the size of ${\Gamma}_{\mathfrak{p}}$. Preliminaries {#sec:prelim} ============= In this section we develop results we need for the proof of theorem \[thm1\]. Homomorphisms ------------- Let $A,B$ be abelian varieties. We make frequent use of the following lemma: \[lem:isogenies\] Let $\phi\in{\mathrm{H}}(A,B)$ and let $B'\subseteq B$ be the image of $\phi$. There exist $A'\subseteq A$ and $\psi\in{\mathrm{H}}(B,A')$ such that $B'+\ker(\psi)=B$ and such that the restrictions $\psi\mid_{B'}$ and $\phi\mid_{A'}$ are isogenies between $A'$ and $B'$ and satisfy $\phi\psi\mid_{B'}=[m]_{B'}$, $\psi\phi\mid_{A'}=[m]_{A'}$ for some $m\geq 1$. If $A''\subseteq A$ is the kernel of $\phi$, then the Poincaré Reducibility Theorem implies there exists $A'\subseteq A$ such that $A'+A''=A$ and $A'\cap A''$ is finite. Similarly, there exists $B''\subseteq B$ such that $B'+B''=B$ and $n=\#(B'\cap B'')$ is finite, and then the restriction $\phi|_{A'}:A'\to B'$ is isogeny and there exists $\hat\phi:B'\to A'$ and $m'\geq 1$ such that $\hat\phi\phi|_{A'}=[m']_{A'}$ and $\phi\hat\phi=[m']_{B'}$ (cf. [@HindrySilverman lem. A.5.1.5]). The restriction of $n\hat\phi$ to $B'\cap B''$ is trivial, hence $n\hat\phi$ extends (uniquely) to a homomorphism $\psi:B\to A'$ such that $B''\subseteq\ker(\psi)$ and $\psi\phi|_{A'}=[m]_{A'}$, $\phi\psi|_{B'}=[m]_{B'}$ for $m=m'n$. \[cor:sym\_zs\] ${\mathrm{H}}(A,B)={{\{0\}}}$ if and only if ${\mathrm{H}}(B,A)={{\{0\}}}$. The statement is symmetric in $A,B$, so it suffices to suppose ${\mathrm{H}}(A,B)\neq{{\{0\}}}$ and show that ${\mathrm{H}}(B,A)\neq{{\{0\}}}$. If $\phi\neq 0\in{\mathrm{H}}(A,B)$ and if $\psi\in{\mathrm{H}}(B,A)$ and $m\geq 1$ are as in lemma \[lem:isogenies\], then $\psi\phi\neq 0$, so $\psi\neq 0$ and hence ${\mathrm{H}}(B,A)\neq{{\{0\}}}$. Images of Submodules -------------------- Let ${\Gamma}\subseteq A(K)$ be a submodule and $\phi\in{\mathrm{H}}(A,B)$. We write $\phi_*({\Gamma})\subseteq B(K)$ for the submodule generated by $\phi({\Gamma})$. \[lem:gen\_im\] Suppose $\hat\phi\in{\mathrm{H}}(B,A)$ and $m\geq 1$ satisfy $\hat\phi\phi=[m]_A$. If $\phi$ is an isogeny, then the index of $\phi({\Gamma})$ in $\phi_*({\Gamma})$ divides $m$. Suppose $\phi$ is an isogeny and thus $\phi\hat\phi=[m]_B$. If $P_1,\ldots,P_r\in{\Gamma}$ and if $\phi_1,\ldots,\phi_r\in{\mathrm{E}}(B)$, then $Q=\Sigma_i \phi_i\phi(P_i)$ satisfies $mQ=\phi(\Sigma_i \hat\phi\phi_i\phi(P_i))\in\phi({\Gamma})$ and thus $m\phi_*({\Gamma})\subseteq\phi({\Gamma})$. If $Q_1,\ldots,Q_r\in\phi(G)$ and if $\phi_1,\ldots,\phi_r\in{\mathrm{E}}(B)$, then for $Q=\Sigma_i\phi_i(Q)$ and ${\mathfrak{p}}\in S(A)\cap S(B)$, the exponent of the reduction $Q_{\mathfrak{p}}$ divides the least-common multiple of the exponents of the reductions $Q_{i,{\mathfrak{p}}}$, thus $\exp_\ell(\phi_*({\Gamma})_{\mathfrak{p}})=\exp_\ell(\phi({\Gamma})_{\mathfrak{p}})$ for every $\ell$. Dense Submodules ---------------- Recall that a submodule ${\Gamma}\subseteq A(K)$ is [*dense*]{} if and only if it satisfies condition 3 in the following lemma: \[lem:infinite\] If ${\Gamma}\subseteq A(K)$ is a submodule, then the following are equivalent: 1. $\phi({\Gamma})\neq {{\{0\}}}$ for every abelian variety $B$ and $\phi\neq 0\in {\mathrm{H}}(A,B)$; 2. $\phi'({\Gamma})\neq {{\{0\}}}$ for every simple abelian variety $B'$ and $\phi'\neq 0\in {\mathrm{H}}(A,B')$; 3. $\pi({\Gamma})\neq{{\{0\}}}$ for every $\pi\neq 0\in {\mathrm{E}}(A)$. Clearly $1\Rightarrow 2,3$. Suppose $B$ is an abelian variety and $\phi\neq 0\in{\mathrm{H}}(A,B)$, and let $B'\subseteq B$ be a non-zero simple abelian subvariety. If $B'\subseteq \phi(A)$ and if $\pi'\in{\mathrm{H}}(B,B')$ and $m\geq 1$ satisfy $\pi'|_{B'}=[m]_{B'}$, then $\phi'=\pi'\phi\neq 0\in{\mathrm{H}}(A,B')$ since the composition of $\phi'$ with inclusion $B'\subseteq B$ equals $m\phi$. In particular, if $\phi'({\Gamma})\neq{{\{0\}}}$, then $\phi({\Gamma})\neq{{\{0\}}}$, thus $2\Rightarrow 1$. Similarly, if $\psi\in{\mathrm{H}}(B,A)$ and $m\geq 1$ satisfy $\phi\psi|_{\phi(A)}=[m]_{\phi(A)}$, then $\pi=\psi\phi\neq 0\in{\mathrm{E}}(A)$ since $\phi\pi=m\phi\neq 0$, and thus $3\Rightarrow 1$. \[rem:infinite\] If ${\Gamma}\subseteq A(K)$ is a finite submodule, then it is not dense, so a dense submodule is infinite. Conversely, if $A$ is simple and if ${\Gamma}\subseteq A(K)$ is an infinite submodule, then ${\Gamma}$ is dense. Isogeny Invariance {#sec:proof:isogeny} ------------------ Let ${\Gamma}\subseteq A(K)$, ${\Gamma}'\subseteq A'(K)$ be submodules. The following lemma shows that condition 1 of theorem \[thm1\] is isogeny invariant: \[lem:isogeny\] Suppose that $\iota\in{\mathrm{H}}(A,B)$, $\iota'\in{\mathrm{H}}(A',B')$ are isogenies. Then there exist $\hat\iota\in{\mathrm{H}}(B,A)$, $\hat\iota'\in{\mathrm{H}}(B',A')$ and $m,m'\geq 1$ satisfying $\hat\iota\iota=[m]_A$ and $\hat\iota'\iota'=[m']_{A'}$, and the following are equivalent: 1. $\exists\,\phi\in{\mathrm{H}}(A,A')$ such that $\ker[\phi]$ and $[\phi(m{\Gamma}):\phi(m{\Gamma})\cap m'{\Gamma}']$ are finite; 2. $\exists\,\phi\in{\mathrm{H}}(A,A')$ such that $\ker[\phi]$ and $[\phi({\Gamma}):\phi({\Gamma})\cap{\Gamma}']$ are finite; 3. $\exists\,\phi'\in{\mathrm{H}}(B,B')$ such that $\ker[\phi']$ and $[\phi'\iota({\Gamma}):\phi'\iota({\Gamma})\cap\iota'({\Gamma}')]$ are finite; 4. $\exists\,\phi'\in{\mathrm{H}}(B,B')$ such that $\ker[\phi']$ and $[\phi'(\iota_*({\Gamma})):\phi'(\iota_*({\Gamma}))\cap\iota'_*({\Gamma}')]$ are finite. Clearly $1\Leftrightarrow 2$. By lemma \[lem:gen\_im\], $[\iota_*({\Gamma}):\iota({\Gamma})]$ and $[\iota'_*({\Gamma}'):\iota'({\Gamma}')]$ are finite, and thus $3\Leftrightarrow 4$. Let $\hat\iota:B\to A$ and $m\geq 1$ satisfy $\hat\iota\iota=[m]_A$, and let $\hat\iota':B'\to A'$ and $m'\geq 1$ be defined similarly (see lemma \[lem:isogenies\]). If $\phi\in{\mathrm{H}}(A,A')$ has finite kernel, then $\phi'=\iota'\phi\hat\iota$ lies in ${\mathrm{H}}(B,B')$ and also has finite kernel. If moreover $\phi({\Gamma})\cap{\Gamma}'$ has finite index in $\phi({\Gamma})$, then $m\phi({\Gamma})\cap {\Gamma}'$ has finite index in $m\phi({\Gamma})=\phi\hat\iota(\iota({\Gamma}))$ and thus $\iota'\phi\hat\iota(\iota({\Gamma}))\cap\iota'({\Gamma}')$ has finite index in $\iota'\phi\hat\iota(\iota({\Gamma}))=\phi'(\iota({\Gamma}))$. That is, $2\Rightarrow 3$, and a similar argument shows $3\Rightarrow 1$ since $\hat\iota\iota({\Gamma})=m{\Gamma}$ and $\hat\iota'\iota'({\Gamma}')=m'{\Gamma}'$. The following lemma shows that, in theorem \[thm1\], $1\Rightarrow 2$ and moreover $1\Rightarrow 3,4$ if $A$ is square free: \[lem:isogenous\] Let $\phi\in{\mathrm{H}}(A,A')$. If $\ker(\phi)$ and $i=[\phi({\Gamma}):\phi({\Gamma})\cap{\Gamma}']$ are finite, then the following holds for $\ell\nmid i\cdot\deg(\phi)$: $$\nu_{\ell,{\Gamma}}({\mathfrak{p}})\leq\nu_{\ell,{\Gamma}'}({\mathfrak{p}}),\ \ {\varepsilon}_{\ell,{\Gamma}}({\mathfrak{p}})\leq{\varepsilon}_{\ell,{\Gamma}'}({\mathfrak{p}}),\ \ \rho_{\ell,{\Gamma}}({\mathfrak{p}})\leq\rho_{\ell,{\Gamma}'}({\mathfrak{p}})\ \quad \forall{\mathfrak{p}}\in S(A)\cap S(A').$$ Let ${\mathfrak{p}}\in S(A)\cap S(A')$. If $\ell\nmid\deg(\phi)$, then $\phi$ induces an isomorphism of the $\ell$-parts of ${\Gamma}_{\mathfrak{p}}$ and $\phi({\Gamma})_{\mathfrak{p}}$, thus $\nu_{\ell,{\Gamma}}({\mathfrak{p}})=\nu_{\ell,\phi({\Gamma})}({\mathfrak{p}})$. Moreover, if $\ell\nmid i$, then the $\ell$-parts of $\phi({\Gamma})_{\mathfrak{p}}\cap{\Gamma}'_{\mathfrak{p}}$ and $\phi({\Gamma})_{\mathfrak{p}}$ coincide, so $\nu_{\ell,\phi({\Gamma})}({\mathfrak{p}})=\nu_{\ell,\phi({\Gamma})\cap{\Gamma}'}({\mathfrak{p}})\leq \nu_{\ell,{\Gamma}'}({\mathfrak{p}})$. Thus the first inequality holds, and a similar argument yields the other inequalities. \[cor:isogeny\] Suppose ${\Gamma}\subseteq A(K)$, ${\Gamma}'\subseteq A'(K)$ are submodules, $\iota\in{\mathrm{H}}(A,B)$, $\iota'\in{\mathrm{H}}(A',B')$ are isogenies, and $d\geq 1$ is an integer. Theorem \[thm1\] holds for $A,A',{\Gamma},{\Gamma}'$ and $\ell\nmid d$ if and only if it holds for $B,B',\iota_*({\Gamma}),\iota'_*({\Gamma})$ and $\ell\nmid d\cdot\deg(\iota)\cdot\deg(\iota')$. The equivalences of lemma \[lem:isogeny\] implies that condition 1 of theorem \[thm1\] holds for $A,A',{\Gamma},{\Gamma}'$ if and only if it holds for $B,B',\iota_*({\Gamma}),\iota_*({\Gamma}')$. For the remaining three conditions of theorem \[thm1\], we observe that, for each $\ell\nmid\deg(\iota)$, we have $\nu_{\ell,{\Gamma}}=\nu_{\ell,\iota_*({\Gamma})}$, $\rho_{\ell,{\Gamma}}=\rho_{\ell,\iota_*({\Gamma})}$, and ${\varepsilon}_{\ell,{\Gamma}}={\varepsilon}_{\ell,\iota_*({\Gamma})}$ on $S(A)\cap S(A')$. Similarly, on $S(A)\cap S(A')$ we have $\nu_{\ell,{\Gamma}'}=\nu_{\ell,\iota_*'({\Gamma}')}$, $\rho_{\ell,{\Gamma}'}=\rho_{\ell,\iota_*'({\Gamma}')}$, and ${\varepsilon}_{\ell,{\Gamma}'}={\varepsilon}_{\ell,\iota_*'({\Gamma}')}$, for each $\ell\nmid\deg(\iota')$. Thus if $\ell\nmid\deg(\iota)\cdot\det(\iota')$, then each condition of theorem \[thm1\] holds for both $A,A',{\Gamma},{\Gamma}'$ and $B,B',\iota_*({\Gamma}),\iota'_*({\Gamma}')$ or for neither. Almost Free Points {#sec:suff_indep} ================== Let $A_1,\ldots,A_r$ be abelian varieties. We say $P_1\in A_1(K),\ldots,P_r\in A_r(K)$ are [*almost free*]{} points if and only if they have infinite order and the following implication holds for all $i$: $$\label{eqn:si} \Pi_j\phi_j\in\Pi_j{\mathrm{H}}(A_j,A_i), \ \ \Sigma_j \phi_j(P_j) = 0 \quad\Rightarrow\quad \phi_1(P_1) = \cdots = \phi_r(P_r) = 0.$$ Note, if $X,Y\subset A(K)$ are subsets of almost free points and if ${\Gamma},{\Gamma}'$ are the respective submodules they generate, then non-zero elements of ${\Gamma}\cap{\Gamma}'$ correspond bijectively to violations of (\[eqn:si\]) for $X\cup Y$ because such elements have (exactly) two representations, one each in elements of $X,Y$ respectively. Recall that points $P_1,\ldots,P_r\in A(K)$ are [*independent*]{} (or [*free*]{}) if and only if $\Sigma_i\phi_i(P_i)=0$ for $\phi_i\in{\mathrm{E}}(A)$ implies $\phi_i=0$ for all $i$ (cf. [@Peruccaord1 def. 3 and rem. 6]). If $P\in A(K)$ is independent, then it is almost free, but the converse does not hold in general. For example, if $A_1,A_2$ are non-isogenous and simple and if $P_1\in A_1(K),P_2\in A_2(K)$ have infinite order, then $P_1,P_2,P_1+P_2$ are each almost free points of $A=A_1\times A_2$, but only $P_1+P_2$ is free. \[lem:indep\] Suppose $P_1\in A_1(K),\ldots,P_r\in A_r(K)$ are almost free, and for each $i$, suppose the Zariski closure $B_i\subseteq A_i$ of $P_i$ is connected. If $B=\Pi_i B_i$, then $P=\Pi_i P_i$ is independent. Suppose $\phi\in{\mathrm{E}}(B)$. We may write $\phi$ as an $r\times r$ matrix $(\phi_{ij})$ where $\phi_{ij}\in{\mathrm{H}}(B_i,B_j)$, and thus $\phi(P)=\Pi_j P'_j$ for $P'_j=\Sigma_i\phi_{ij}(P_i)$. We must show that if $\phi(P)=0$, then $\phi=0$. If $\phi(P)=0$, then $P'_j=0$ for all $j$, and hence $\phi_{ij}(P_i)=0$ for all $i,j$ since $P_1,\ldots,P_r$ are almost free. Therefore, since $P_i$ is Zariski dense in $B_i$ for every $i$, we have $\phi_{ij}=0$ for every $i,j$, that is, $\phi=0$. The following proposition is a key ingredient in our proof of theorem \[thm1\]: \[prop:indep\] Suppose $P_1,\ldots,P_r\in A(K)$ and $m_1,\ldots,m_r\geq 0$. If $P_1,\ldots,P_r$ are almost free, then for every $\ell\gg 0$, the following set has positive density: $$S_{\ell,m} := \{\ {\mathfrak{p}}\in S(A) : {\varepsilon}_{\ell,P_i}({\mathfrak{p}})=m_i,\forall i \ \}.$$ Let $A_i\subseteq A$ be the Zariski closure of $P_i$. Up to replacing $P_i$ by $mP_i$, for some $m\geq 1$, and excluding $\ell$ which divide $m$, we suppose that $A_i$ is connected. Then lemma \[lem:indep\] implies $P=\Pi_i P_i$ is independent in $B=\Pi_i A_i$, and thus the proposition follows from [@Peruccaord1 prop. 12]). If $A$ is simple and if $\phi\neq 0\in{\mathrm{E}}(A)$, then the kernel of $\phi$ is finite and so $\phi(P)\neq 0$ for any non-torsion $P\in A(K)$. Hence if $A$ is simple and if $P\in A(K)$ is almost free, then $P$ is independent. An analogous remark also holds for more points. The following lemma gives a mildly different characterization of almost free points when $A$ is simple: \[lem:simple\_indep\] Let $A$ be simple and $P_1,\ldots,P_r\in A(K)$ be points of infinite order. The following are equivalent: 1. $P_1,\ldots,P_r$ are almost free; 2. if $\phi_1,\ldots,\phi_r\in{\mathrm{E}}(A)$ satisfy $\Sigma_i\phi_i(P_i)=0$ and if $\phi_1\in{\mathbb{Z}}$, then $\{\phi_i(P_i)\}={{\{0\}}}$. It is clear that $1\Rightarrow 2$ (cf. (\[eqn:si\])), so suppose that 2 holds and $\phi_1,\ldots,\phi_r\in{\mathrm{E}}(A)$ satisfy $\Sigma_i\phi_i(P_i)=0$. If $\phi_1=0$, then it lies in ${\mathbb{Z}}$ and hence 2 implies $\phi_1(P_1)=\cdots=\phi_r(P_r)=0$. If $\phi_1\neq 0$, then its kernel is a proper algebraic subgroup of $A$, hence it must be a finite subgroup, since $A$ is simple, and thus $\phi_1$ is an isogeny. Let $\psi:A\to A$ and $m\geq 1$ satisfy $\psi\phi_1=[m]_A$. Then $\Sigma_i\psi\phi_i(P_i)=0$ and $\psi\phi_1=[m]_A\in{\mathbb{Z}}$, so 2 implies $\{\psi\phi_i(P_i)\}_i={{\{0\}}}$. Therefore $\phi_i(P_i)$ lies in the kernel of $\psi$ for every $i$, and thus $\{\phi_i(P_i)\}_i={{\{0\}}}$ since the points $P_i$ have infinite order and $A$ is simple. If $A$ is simple and if ${\Gamma}\subseteq A(K)$ is a submodule, then one can repeatedly apply the following corollary in order to find a finite-index free-submodule ${\Gamma}'\subseteq{\Gamma}$ and an explicit basis of ${\Gamma}'$: \[cor:free\_basis\] Suppose $A$ is simple and $P_1,\ldots,P_r\in A(K)$ are almost free, and let ${\Gamma}\subseteq A(K)$ be the submodule they generate. If $Q\in A(K)$ satisfies $nQ\not\in{\Gamma}$ for every $n\geq 1$, then $P_1,\ldots,P_r,Q$ are almost free. The points $P_1,\ldots,P_r,Q$ satisfy 2 of lemma \[lem:simple\_indep\] and thus they are almost free. \[cor:indep\_Q\] Suppose $A$ is simple and ${\Gamma}\subseteq A(K)$ is a submodule. If $Q\in A(K)$ satisfies $nQ\not\in{\Gamma}$, for every $n\geq 1$, and if $\phi\in{\mathrm{E}}(A)$ satisfies $\phi(Q)\in{\Gamma}$, then $\phi(Q)$ is torsion. Let $\{P_1,\ldots,P_r\}\subset{\Gamma}$ be a maximal subset of almost free points, and let ${\Gamma}'\subseteq{\Gamma}$ be the finite-index submodule they generate (cf. cor. \[cor:free\_basis\]). If $\phi,\phi_1,\ldots,\phi_r\in{\mathrm{E}}(A)$ satisfy $\phi(Q)\in{\Gamma}$ and $\phi(nQ)+\Sigma_i\phi_i(P_i)=0$, where $n=[{\Gamma}:{\Gamma}']$, then corollary \[cor:free\_basis\] implies $\{\phi(nQ),\phi_i(P_i)\}_i={{\{0\}}}$. That is, $\phi(nQ)=0$ and thus $\phi(Q)$ is torsion of order dividing $n$. For general $A$, we can repeatedly apply the following lemma to choose a ‘quasi-basis’ of a submodule, that is, a maximal subset of almost free points which generate a finite-index submodule: \[lem:basis\_extend\] Let $P_1,\ldots,P_r\in A(K)$ be almost free, and suppose the submodule ${\Gamma}\subseteq A(K)$ they generate is torsion free. If ${\Gamma}'\subseteq A(K)$ is a submodule such that ${\Gamma}\cap{\Gamma}'$ has infinite index in ${\Gamma}'$, then there exists $Q\in{\Gamma}'$ such that $P_1,\ldots,P_r,Q$ are almost free and such that the submodule they generate is torsion free. Let $B,B'\subseteq A$ be abelian varieties such that ${\mathrm{H}}(B,B')={\mathrm{H}}(B',B)={{\{0\}}}$ and $B+B'=A$. Let $\hat\pi:B\to A$ and $\hat\pi':B'\to A$ be the natural inclusions, and suppose $\pi\in{\mathrm{H}}(A,B)$, $\pi'\in{\mathrm{H}}(A,B')$, and $m,m'\geq 1$ satisfy $\pi\hat\pi=[m]_B$ and $\pi'\hat\pi'=[m']_{B'}$. Suppose $B$ is minimal with the property that $\pi({\Gamma}\cap{\Gamma}')$ has infinite index in $\pi({\Gamma}')$. Thus there is a simple abelian subvariety $C\subseteq B$ and an isogeny $\psi:B\to C^e$ for some $e\geq 1$, and we define $\pi_i:A\to C$ to be the composition of $\psi\pi$ with projection onto the $i$th factor and let $\hat\pi_i:C\to A$ and $m_i\geq 1$ be such that $\pi_i\hat\pi_i=[m_i]_C$. We note that $\ker(\pi)$ has finite index in $\cap_i\ker(\hat\pi_i\pi_i)$ and that the intersection of either with $\ker(\pi')$ is finite. Let $i$ be such that $\pi_i({\Gamma}\cap{\Gamma}')$ has infinite index in $\pi_i({\Gamma}')$. Therefore there exists $Q'\in{\Gamma}'$ such that $n\pi_i(Q')\not\in\pi_i({\Gamma})$, for every $n\geq 1$, and we let $Q=\hat\pi_i(\pi_i(Q'))$. Up to replacing $Q'$ by $nQ'$ for some $n\geq 1$, we suppose without loss of generality that the submodule generated by $P_1,\ldots,P_s,Q$ is torsion free. From the equality $m_i\pi_j({\Gamma})=\pi_i\hat{\pi}_i\pi_j (\Gamma)$ it follows that $[\pi_j({\Gamma}):\pi_i({\Gamma})\cap\pi_j({\Gamma})]$ is finite, for every $j$, and so we have $n\pi_j(Q)\not\in\pi_j({\Gamma})$ for every $n\geq 1$. Moreover, for $\phi\in{\mathrm{E}}(A)$ and $\phi'_{ij}=\pi_j\phi\hat\pi_i\in{\mathrm{E}}(C)$, we have the identity $\pi_j(\phi(Q))=\phi'_{ij}\pi_i(Q')$ and thus corollary \[cor:indep\_Q\] implies $\pi_j\phi(Q)\in\pi_j({\Gamma})$ only if $\pi_j\phi(Q)$ has finite order. Up to replacing $Q$ by $nQ$, for some $n\geq 1$, we may suppose that $\pi_j\phi(Q)\in\pi_j({\Gamma})$ only if $\pi_j\phi(Q)=0$. Let $\phi_1,\ldots,\phi_r,\phi\in{\mathrm{E}}(A)$, and suppose $\Sigma_j\phi_j(P_j)+\phi(Q)=0$. Our assumptions on $B,B'$ imply ${\mathrm{H}}(C,B')={{\{0\}}}$, thus $\pi'\phi\hat\pi_i=0$ and $\pi'\phi(Q)=\pi'\phi\hat\pi_i(\pi_i(Q'))=0$. That is, $\Sigma_j\pi'\phi_j(P_j)=0$ and hence $\Sigma_j\hat\pi'\pi'\phi_j(P_j)=0$. Since $P_1,\ldots,P_r$ are almost free, $\{\hat\pi'\pi'\phi_j(P_j)\}_j={{\{0\}}}$, and thus since $\hat\pi':B'\to A$ is injective, we have $\{\pi'\phi_j(P_j)\}_j={{\{0\}}}$. Therefore $\{\phi_j(P_j),\phi(Q)\}_j$ lies in the kernel of $\pi'$. We will show that it also lies in $\cap_k\ker(\hat\pi_k\pi_k)$. By assumption, $\Sigma_j\pi_k\phi_j(P_j)+\pi_k\phi(Q)=0$ for every $k$, thus $\pi_k\phi(Q)\in\pi_k({\Gamma})$ and so $\pi_k\phi(Q)=0$. That is, $\Sigma_j\pi_k\phi_j(P_j)=0$ for every $k$, and thus $\Sigma_j\hat\pi_k\pi_k\phi_j(P_j)=0$ and so $\{\hat\pi_k\pi_k\phi_j(P_j)\}_j={{\{0\}}}$ since $P_1,\ldots,P_r$ are almost free. Therefore $\{\phi_j(P_j),\phi(Q)\}_j$ lies in the kernel of $\hat\pi_k\pi_k$, for every $k$. That is, $\{\phi_j(P_j),\phi(Q)\}_j$ lies in both $\ker(\pi')$ and $\cap_k\ker(\hat\pi_k\pi_k)$ and thus in a finite subgroup of $A(K)$. Since ${\Gamma}$ is torsion free and $\phi_j(P_j)\in{\Gamma}$ for each $j$, we must have $\{\phi_j(P_j)\}_j={{\{0\}}}$ and thus $\phi(Q)=0$ as well. That is, $P_1,\ldots,P_r,Q$ are almost free. Proof of Theorem \[thm1\] {#sec:proof_thm1} ========================= We have already seen that if 1 holds, then 2 holds, and if moreover $A$ is square free, then 3 and 4 hold (see section \[sec:proof:isogeny\]). Thus we suppose that 1 fails and show that 2, 3, 4 fail accordingly. Suppose $P\in A(K)$ and $Q_1,\ldots,Q_r\in A'(K)$ are points, and for each prime $\ell$ and $m\geq 0$, consider the following set: $$S_{\ell,m}(P,Q_1,\ldots,Q_r) := \{\ {\mathfrak{p}}\in S(A)\cap S(A') : {\varepsilon}_{\ell,P}({\mathfrak{p}})=m,\ {\varepsilon}_{\ell,Q_i}({\mathfrak{p}})=0\mbox{ for }i=1,\ldots,r \ \}.$$ The basic strategy underlying our proof is to first make judicious choices of $P,Q_1,\ldots,Q_r$ so that we can analyze the $\ell$-parts of ${\Gamma}_{\mathfrak{p}},{\Gamma}'_{\mathfrak{p}}$ for $\ell\gg 0$ and varying ${\mathfrak{p}}$. In particular, we will show these sets usually have positive density and deduce that 2 fails for $\ell\gg 0$, and moreover, that 3 and 4 fail when $A$ is square free and $\ell\gg 0$. Let $B'$ be an abelian variety such that $A$ and $A'$ each have some abelian subvariety isogenous to $B'$, and we suppose that $B'$ has maximal dimension. Then for some abelian subvarieties $B\subseteq A$, $B''\subseteq A'$ we have that $A, A'$ are respectively isogenous to $B\times B'$, $B'\times B''$ so, by corollary \[cor:isogeny\], we may assume that $A=B\times B'$, $A'=B'\times B''$. We may also assume $\phi\in H(A, A')$ is the composition of the projection on $B'$ and the inclusion $B'\subseteq A'$. \[lem:min\_dim\] We have ${\mathrm{H}}(B,B'')={\mathrm{H}}(B'',B)={{\{0\}}}$. Suppose $\psi\neq 0\in{\mathrm{H}}(B,B'')$, and let $\pi:A\to B$ be projection. Up to composing with the natural embedding $B'' \rightarrow A'$, we can consider the homomorphism $\phi'=\phi+\psi\pi\in{\mathrm{H}}(A,A')$, and let $C\subseteq A$ be its kernel. The kernel of $\phi$ is $B=B\times{{\{0\}}}\subseteq A$, and $B'={{\{0\}}}\times B'\subseteq A$ satisfies $B\cap B'={{\{0\}}}$, thus the restriction of $\phi'$ to $B'\subseteq A$, which is simply $\phi$, is injective. The image of $\phi'$ contains the images of $\phi$ and $\psi$, thus it is strictly larger, so $\dim(\ker(\phi'))=\dim(C)<\dim(\ker(\phi))$. That is, if ${\mathrm{H}}(B,B'')$ is non trivial, then $\dim(B')$ is not maximal by lemma \[lem:isogenies\]. Finally, ${\mathrm{H}}(B,B'')={{\{0\}}}$ if and only if ${\mathrm{H}}(B'',B)={{\{0\}}}$ by corollary \[cor:sym\_zs\]. \[cor:min\_dim\] Suppose $P\in B(K)$, and let ${\Gamma}_0\subseteq A(K)$ be the submodule it generates. Then $\phi({\Gamma}_0)=\{\psi(P):\psi\in{\mathrm{H}}(B,A')\}$. Let ${\alpha}\in{\mathrm{E}}(A)$ and ${\alpha}(P)\in{\Gamma}_0$. Lemma \[lem:min\_dim\] implies ${\mathrm{H}}(B,B'')={{\{0\}}}$ and thus $\phi{\alpha}(P)=\psi(P)$ for some $\psi\in{\mathrm{H}}(B,B')$. Conversely, if $\psi\in{\mathrm{H}}(B,A')$ and if $\pi'':A'\to B''$ is projection, then $\pi''\psi=0\in {\mathrm{H}}(B,B'')$ and thus $\psi$ factors through the natural embedding $B'\to A'$. In particular, if we compose with the natural embedding $B'\to A$, then the endomorphism ${\alpha}\in{\mathrm{E}}(A)$ such that ${\alpha}|_B=\psi$ and ${\alpha}|_{B'}=0$ satisfies $\phi({\alpha}(P))=\psi(P)$ and thus $\psi(P)\in\phi({\Gamma}_0)$. The following lemma allows us to reduce to the case $B=0$: \[lem:B\_pos\] Suppose that condition $1$ in theorem \[thm1\] fails. If $B'\neq A$ then 2 fails. Moreover, if $A$ is square free, then 3 and 4 also fail. Suppose that $B\neq 0$. If $\pi:A\to B$ is projection, then $\pi({\Gamma})\subseteq B(K)$ is infinite since ${\Gamma}$ is dense (see lemma \[lem:infinite\]), so let $P\in\pi({\Gamma})$ have infinite order and let ${\Gamma}_0\subseteq{\Gamma}$ be the submodule it generates. Lemma \[lem:min\_dim\] implies ${\mathrm{H}}(B,B'')={{\{0\}}}$ and thus $\psi(B)\subseteq B'$ for every $\psi\in{\mathrm{H}}(B,A')$. Thus up to replacing $P$ by $nP$ with $n$ the size of the torsion subgroup of $B'(K)$, we suppose without loss of generality that, for every $\psi\in{\mathrm{H}}(B,A')$, either $\psi(P)=0$ or $\psi(P)$ has infinite order. Let $\{R_1,\ldots,R_s\}\subset(\phi({\Gamma}_0)\cap{\Gamma}')$ and $\{Q_1,\ldots,Q_r,R_1,\ldots,R_s\}\subseteq{\Gamma}'$ be maximal subsets of almost free points, and let ${\Gamma}'_0\subseteq{\Gamma}'$ be the submodule generated by $Q_1,\ldots,Q_r$ and ${\Gamma}'_1\subseteq{\Gamma}'$ be the submodule generated by $R_1,\ldots,R_r$. Up to replacing each $Q_i$ by $nQ_i$ with $n$ the size of the torsion subgroup of $B(K)$, we suppose without loss of generality that, for every $i$ and $\psi_i\in{\mathrm{H}}(A',B)$, either $\psi_i(Q_i)=0$ or $\psi_i(Q_i)$ has infinite order. Since a maximal subset of almost free points generates a finite index submodule by lemma \[lem:basis\_extend\], we may suppose that $\ell$ is coprime with the index of the above submodules. If ${\mathfrak{p}}\in S_{\ell,m}(P,Q_1,\ldots,Q_r)$, then the $\ell$-part of ${\Gamma}'_{\mathfrak{p}}$ lies in that of $(\phi({\Gamma}_0)\cap{\Gamma}')_{\mathfrak{p}}$ and thus $$\operatorname{ord}_\ell({\Gamma}'_{\mathfrak{p}})\leq\operatorname{ord}_\ell({\Gamma}_{\mathfrak{p}})-\operatorname{ord}_\ell({\Gamma}_{\mathfrak{p}}\cap B(K)_{\mathfrak{p}})\leq\operatorname{ord}_\ell({\Gamma}_{\mathfrak{p}})-m\,.$$ If moreover $A$ is square free, then ${\mathrm{H}}(B,B')={{\{0\}}}$ and so ${\mathrm{H}}(B,A')={{\{0\}}}$, thus corollary \[cor:min\_dim\] implies $\phi({\Gamma}_0)={{\{0\}}}$ and ${\varepsilon}_{\ell,{\Gamma}'}({\mathfrak{p}})=0$. If $S_{\ell,m}(P,Q_1,\ldots,Q_r)$ has positive density for every $m\geq 0$, then $\nu_{\ell,{\Gamma}}({\mathfrak{p}})-\nu_{\ell,{\Gamma}'}({\mathfrak{p}})$ can be made arbitrarily large on a positive density set, thus 2 fails. If moreover $A$ is square free, then for every $m\geq 0$, the identities ${\varepsilon}_{\ell,{\Gamma}}({\mathfrak{p}})\geq m$ and ${\varepsilon}_{\ell,{\Gamma}'}({\mathfrak{p}})=0$ hold on a positive density set, thus 3 and 4 fail. To complete the proof it suffices to show that $P,Q_1,\ldots,Q_r$ are almost free because then proposition \[prop:indep\] implies that $S_{\ell,m}(P,Q_1,\ldots,Q_r)$ has positive density for every $\ell\gg 0$ and $m\geq 0$. The intersection of ${\Gamma}'_1$ and ${\Gamma}'_0$ is trivial since $Q_1,\ldots,Q_r,R_1,\ldots,R_s$ are almost free, thus $\phi({\Gamma}_0)\cap{\Gamma}'_0$ is finite. Suppose $\psi\in{\mathrm{H}}(B,A')$ and $\psi_1,\ldots,\psi_r\in{\mathrm{E}}(A')$ satisfy $\psi(P)+\Sigma_i\psi_i(Q_i)=0$. Then $\psi(P)\in{\Gamma}'_0$ and corollary \[cor:min\_dim\] implies $\psi(P)\in\phi({\Gamma}_0)$, thus $\psi(P)$ lies in the finite intersection $\phi({\Gamma}_0)\cap{\Gamma}'_0$ and so $\psi(P)=0$ by our assumptions on $P$. Therefore, since $Q_1,\ldots,Q_r$ are almost free, we have $\{\psi_i(Q_i)\}_i={{\{0\}}}$ and so $\{\psi(P),\psi_i(Q_i)\}_i={{\{0\}}}$. Suppose $\psi\in{\mathrm{E}}(B)$ and $\psi_1,\ldots,\psi_r\in{\mathrm{H}}(A',B)$ satisfy $\psi(P)+\Sigma_i\psi_i(Q_i)=0$. For each $i$, let $\psi'_i\in{\mathrm{H}}(B,A')$ be such that the kernel of $\psi_i$ has finite index in $\ker(\psi'_i\psi_i)$ (cf. lemma \[lem:isogenies\]), and consider the identity $\psi'_i\psi(P)+\Sigma_j\psi'_i\psi_j(Q_j)=0$. As in the previous paragraph, $\psi'_i\psi(P)=0$ and thus $\{\psi'_i\psi_j(Q_j)\}_j={{\{0\}}}$ for every $i$. That is, $\psi_j(Q_j)$ lies in the kernel of $\psi'_i$ for every $i,j$, and a fortiori for every $i=j$. In particular, since $\ker(\psi_i)$ has finite index in $\ker(\psi'_i\psi_i)$, $\psi_i(Q_i)$ is torsion. By our assumptions on $Q_1,\ldots,Q_r$, we have $\{\psi_i(Q_i)\}_i={{\{0\}}}$ and thus $\{\psi(P),\psi_i(Q_i)\}_i={{\{0\}}}$. Therefore, $P,Q_1,\ldots,Q_r$ are almost free as claimed. We suppose for the remainder of the section that $B=0$ and thus $A=B'\subseteq A'$ and $\phi$ is the inclusion. Let $\{Q_1,\ldots,Q_r\}\subset{\Gamma}'$ be a maximal subset of almost free points, and let ${\Gamma}'_0\subseteq{\Gamma}'$ be the finite-index submodule they generate (which we may suppose to be torsion free) and $\ell$ be a prime not dividing $[{\Gamma}':{\Gamma}'_0]$. Suppose condition 1 fails and thus ${\Gamma}\cap{\Gamma}'$ has infinite index in ${\Gamma}$. Therefore the index of $\phi_*({\Gamma})\cap{\Gamma}'$ in $\phi_*({\Gamma})$ is infinite and lemma \[lem:basis\_extend\] implies there exists $P\in\phi_*({\Gamma})$ such that $P,Q_1,\ldots,Q_r$ are almost free. If ${\mathfrak{p}}\in S_{\ell,m}(P,Q_1,\ldots,Q_r)$, then $\operatorname{ord}_\ell({\Gamma}_{\mathfrak{p}})\geq m$ and $\operatorname{ord}_\ell({\Gamma}'_{\mathfrak{p}})=0$, thus we have the following inequalities: $$\nu_{\ell,{\Gamma}}({\mathfrak{p}}) \geq {\varepsilon}_{\ell,{\Gamma}}({\mathfrak{p}})= {\varepsilon}_{\ell,\phi_*({\Gamma})}({\mathfrak{p}})\geq m \geq \nu_{\ell,{\Gamma}'}({\mathfrak{p}})={\varepsilon}_{\ell,{\Gamma}'}({\mathfrak{p}})=0.$$ In particular, if $\ell\gg 0$, then proposition \[prop:indep\] implies $S_{\ell,m}(P,Q_1,\ldots,Q_r)$ has positive density for every $m\geq 0$, and thus 2, 3, and 4 fail. Q.E.D. [10]{} url\#1[`#1`]{}urlprefix J. Demeyer and A. Perucca, *The constant of the support problem for abelian varieties*, arXiv:1008.3719. G. Faltings, *Finiteness Theorems for Abelian Varieties over Number Fields*, Arithmetic Geometry, Edited by G. Cornell and J. H. Silverman, Springer-Verlag, New York, 1986, 9–27. M. Hindry and J. Silverman, *Diophantine Geometry. An Introduction*, Graduate Texts in Mathematics 201, Springer-Verlag, New York, 2000. A. Perucca, *Prescribing valuations of the order of a point in the reductions of abelian varieties and tori*, J. Number Theory **129** (2009), no. 2, 469–476. A. Perucca, *On the reduction of points on abelian varieties and tori*, Int. Math. Res. Notices **2011** (2011), no. 7, 293–308. Y. Zarhin, *Homomorphisms of abelian varieties over finite fields*, Higher-dimensional geometry over finite fields, Edited by D. Kaledin and Y. Tschinkel, IOS, Amsterdam, 2008, 315–343. [*Chris Hall,*]{} University of Wyoming\ : chall14@uwyo.edu\ [*Antonella Perucca,*]{} Research Foundation - Flanders (FWO) : antonellaperucca@gmail.com\

--- abstract: 'Space-times admitting a shear-free, irrotational, geodesic null congruence are studied. Attention is focused on those space-times in which the gravitational field is a combination of a perfect fluid and null radiation.' author: - | Alicia M. Sintes\ [Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), ]{}\ [Schlaatzweg 1, 14473 Potsdam, Germany]{}\ \ Alan A. Coley and Des J. McManus\ [Department of Mathematics, Statistics and Computing Science.]{}\ [Dalhousie University. Halifax, NS. Canada B3H 3J5]{} title: ' On space-times admitting shear-free, irrotational, geodesic null congruences' --- 22.0 true cm 15.24 true cm 1true cm 1 true cm -0.8cm Ł[[L]{}]{} Introduction ============ In this article we wish to extend earlier work on shear-free, irrotational and geodesic (SIG) timelike and spacelike congruences [@des; @des2] to SIG [*null*]{} congruences. The fact that we are dealing with null congruences means that we have to approach the problem in a completely different way; we must make extensive use of the Newman-Penrose formalism. Thus, we wish to study a congruence of curves whose tangent vector ${\bf k}$ is null and geodesic. Hence, we have a family of null geodesics $x^a=x^a(y^{\alpha},v)$, where $y^{\alpha}$ distinguishes the different geodesics, and $v$ is the affine parameter along a fixed geodesic. The null tangent vector is $k^a={\partial x^a \over \partial v} $, and satisfies ${k^a}_{;b}k^b=0$. The spin coefficients are defined in [@Kramer], where $\rho=-(\theta + i\omega)$ is called the complex divergence and $\sigma$ is the complex shear. The geodesic condition implies that the spin coefficient $\kappa$ vanishes and $\epsilon +\bar\epsilon =0$ follows from the choice of an affine parameter along the congruence. The congruence is said to be shear-free if $\sigma=0$. Also, from the relation $k_{[a;b}k_{c]}=(\bar \rho-\rho)\bar m_{[a}m_bk_{c]}$ [@Pirani], it follows that $w=0$ (i.e., zero twist) is a necessary and sufficient condition for ${\bf k}$ to be hypersurface orthogonal. First we shall briefly review some of the results of relevance to this work. Goldberg and Sachs [@Gold] proved that if a gravitational field contains a shear-free, geodesic, null congruence ${\bf k}$, then $\kappa=\sigma=0$, and if R\_[ab]{}k\^ak\^b=R\_[ab]{}k\^am\^b=R\_[ab]{}m\^am\^b=0  ,\[s1\] then the field is algebraically special (i.e., $\Psi_0=\Psi_1=0$), and ${\bf k}$ is a degenerate eigendirection. In addition, a vacuum metric is algebraically special if and only if it contains a shear-free geodesic null congruence. A space-time admits a geodesic, shear-free, twist-free ($\kappa=\sigma=\omega=0$) and diverging ($\rho=\bar\rho=\theta=-1/r$) null congruence ${\bf k}$, and satisfies (\[s1\]), if and only if the metric can be written in the form ds\^2=2r\^2 P\^[-2]{}(z,|z,u)dzd|z -2dudr -2H(z,|z,r,u)du\^2  . Robinson-Trautman models [@Robi] with this metric have been found for vacuum, Einstein-Maxwell and pure radiation fields with or without a cosmological constant [@Kramer]. For geodesic null vector fields we have that $(\theta +i\omega)_{,a}k^a+ (\theta +i\omega)^2+\sigma\bar\sigma= -R_{ab}k^ak^b/2$. Therefore, in the non-diverging case (i.e., $\rho=-(\theta +i\omega)=0$), if the energy condition $T_{ab}k^ak^b\ge 0$ is satisfied, it follows that $\sigma=0=R_{ab}k^ak^b$. Thus, non-twisting (and therefore geodesic) and non-expanding null congruences must be shear-free. Hence, the space-time is algebraically special, and it corresponds to vacuum, Einstein-Maxwell, and pure radiation field. Perfect fluid solutions violate $R_{ab}k^ak^b=0$ unless $\mu+p=0$. This class of solutions has been studied by Kundt [@Kundt]. Another important case corresponds to the Kerr-Schild metric, which is given by $g_{ab}=\eta_{ab}-2\phi k_ak_b$. The null vector ${\bf k}$ of a Kerr-Schild metric is geodesic if and only if the energy-momentum tensor obeys the condition $T_{ab}k^ak^b=0$, and then ${\bf k}$ is a multiple principal null direction of the Weyl tensor and the space-time is algebraically special. The general properties of the Kerr-Schild metrics and their applications to vacuum, Einstein-Maxwell, and pure radiation space-times can be found in [@Kramer]. Finally, we note the algebraically special perfect fluid space-times corresponding to the generalized Robinson-Trautman solutions investigated by Wainwright [@Wain]. They are characterized by a multiple null eigenvector ${\bf k}$ of the Weyl tensor which is geodesic, shear-free, and twist-free but expanding (i.e., $\Psi_o=\Psi_1=0$, $\kappa=\sigma=\omega=0$, $\rho=\bar\rho\not= 0$), and the four-velocity obeys $u_{[a;b}u_{c]}=0$, $k_{[c}k_{a];b}u^b=0$. The line-element of the space-time can be written in the form ds\^2= -[12]{}\^2(r,u)P\^[-2]{}(z,|z, u)dzd|z +2du(dr-Udu)  , \[11\] with U=r(P)\_[,u]{}+ U\^0(z,|z, u)+ S(r,u)  , \_[,r]{}=0  ,0  . In this case no dust solutions nor solutions of Petrov types $III$ and $N$ are possible. Analysis ======== Let us consider space-times admitting a shear-free, irrotational, geodesic null congruence in which the source of the gravitational field is a [*combination of a perfect fluid and null radiation*]{}, so that the energy-momentum tensor has the form T\_[ab]{}=(+p)u\_au\_b -p g\_[ab]{} +\^2k\_ak\_b  , \[13\] where $u^a$ is the four-velocity of the fluid, $\mu$ and $p$ are the density and the pressure of the fluid, respectively, and ${\bf k}$ is a null vector. The null radiation is geodesic, twist-free, and shear-free, and defines the null congruence. Wainwright [@Wain] proved that for a space-time in which there exists a SIG null congruence, coordinates can be chosen so that the metric takes on the simplified form (\[11\]) with $u=x^1$, $r=x^2$, $z=x^3+i x^4$, the tangent field of the null congruence is given by $k^a=\delta^a_2$, $k_a=\delta^1_a$, and we can introduce the null tetrad k\^a= \^a\_r  , & l\^a= \^a\_u+ U\^a\_r  , & m\^a=P\^[-1]{}( \^a\_3+ i\^a\_4 )  ,\ k\_a=\^u\_a  , & l\_a= -U \^u\_a + \^r\_a  ,& m\_a=P\^[-1]{}(\^3\_a + i \^4\_a)/2  . With the sign convention used here we have that $u^au_a=k^al_a=1=-m^a\bar m_a$. Note that the null radiation is everywhere tangent to the repeated null congruence of the space-time. First, since $\Phi_{01}\equiv-{1\over 2}R_{ab}k^am^b=0$, we conclude that the four-velocity satisfies $u^am_a=0$, and hence it can be expressed in terms of the null tetrad by u\^a=[1 B]{}(B\^2 k\^a +l\^a) u\_a=[1 B]{}\[(B\^2-U)\^u\_a + \^r\_a\]  , \[36\] for some function $B$. The conditions $\Phi_{02}\equiv-{1\over 2}R_{ab}m^am^b=0$ and $\Phi_{12}\equiv-{1\over 2}R_{ab}m^al^b=0$ are satisfied identically. The non-zero components of the Ricci tensor are & &\_[00]{}-[12]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})k\^ak\^b= [12]{}(+p)([**ku**]{})\^2  ,\ & &\_[11]{}-[14]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})(k\^al\^b+m\^a|m\^b)= [14]{}(+p)([**ku**]{}) ([**lu**]{}) ,\ & &\_[22]{}-[12]{}(R\_[ab]{}-[14]{}Rg\_[ab]{})l\^al\^b= [12]{}(+p)([**lu**]{})\^2 +[12]{}\^2  . In addition, since ${\bf k\cdot u}={1\over \sqrt{2} B}$ and ${\bf l\cdot u}={1\over \sqrt{2}} B$ implies ${\bf l\cdot u}=B^2({\bf k\cdot u})$, we obtain B\^2\_[00]{}&=& 2\_[11]{}   , \[34\]\ B\^4\_[00]{}&=&\_[22]{}- [12]{}\^2  . \[35\] If we now assume that the fluid is non-rotating, then $ B^2=U+F(r,u)$, and the compatibility condition (\[34\]) can be written as (U+F)\_[00]{}=2\_[11]{} .\[39\] On differentiating this equation successively with respect to $z$ and $r$, we obtain the restriction (\^2)\_[,rrr]{}\[[U\^0]{}\_[,z]{}+r(P)\_[,uz]{}\]=0  . There are consequently two different cases to consider. In the first case ${U^0}_{,z}+r(\ln P)_{,uz}=0$, which is equivalent to ${U^0}_{,z}=(\ln P)_{,uz}=0$, so that $P=P(z,\bar z)$ and $U^0=U^0(u)$. Obviously, the solutions admit a multiply transitive group of motions, $G_3$, acting on the 2-spaces $r=$const, $u=$const, of constant curvature, and belong to class $II$ of Stewart and Ellis [@Stew]. The metric (\[11\]) can then be rewritten as ds\^2=-\^2(r,u)[2dzd|z(1+[k2]{}z|z)\^2 ]{}+ 2du(dr-U(r,u)du)  . \[44\] The non-zero Ricci components are given by & & \_[00]{}=-[\_[,rr]{}]{}  ,\ & & \_[11]{}=[\_[,r]{}\_[,u]{}2\^2]{} + [(\_[,r]{})\^2U2\^2]{}- [U\_[,rr]{}4]{} + [k4 \^2]{}  ,\ & & \_[22]{}=[\_[,u]{}U\_[,r]{}]{}- [\_[,uu]{}]{} -2[\_[,ur]{}U ]{}-[\_[,r]{}U\_[,u]{}]{} -[\_[,rr]{}U\^2 ]{}  , and the Ricci scalar is given by =12= 4[\_[,r]{}U\_[,r]{}]{}+ 2[\_[,r]{}\_[,u]{}\^2]{} +2 [(\_[,r]{})\^2U\^2]{} + 4 [\_[,ur]{}]{}+ U\_[,rr]{} +4 [\_[,rr]{}U ]{} +[k\^2]{}  . Hence, the metric (\[44\]) can be interpreted as pure radiation plus a perfect fluid where $\mu$ and $p$ are given by =[R 4]{} +6\_[11]{} , p=-[R 4]{}+2\_[11]{} , \[45\] $u_a$ is determined by (\[36\]) with $B^2=2 \Phi_{11}/\Phi_{00}$, and $\phi^2$ is given by \^2=2( \_[22]{}-4[\_[11]{}\^2\_[00]{}]{})  . \[47\] In the second case (i.e., ${\chi^2}_{,rrr}=0$) two possibilities arise: (i) & \^2=r,& = 1 \[52\]\ (ii) & \^2=(r\^2-k\^2), & k=const  .\[53\] In both subcases $\chi=\chi(r)$, and they can be written together as $\chi^2=ar^2+2br+c $, with $a$, $b$, $c$ taken to be appropriate constants. From equation (\[39\]) we obtain aU\^0 -b(P)\_[,u]{} + K =G(u)  , \[55\] and \[\^2 S\_[,r]{} -S(\^2)\_[,r]{}\]\_[,r]{} + [F \^2]{}= G(u)  , \[56\] where $K\equiv 4P^2 (\ln P)_{z\bar z}$, $\Sigma\equiv b^2-ac$, and $G(u)$ is an arbitrary function of $u$. Subcase $(i)$: $a=c=0$, $b=\epsilon/2$. Integrating equation (\[56\]) we see that $S$ can be written in the form S=rh(u)+2G(u)rr -f(u) -[12]{}r\^r [drr]{}F(r,u)  , \[58\] where $h(u)$ and $f(u)$ are arbitrary functions of $u$. Subcase $(ii)$: $a=\epsilon$, $b=0$, $c=-\epsilon k^2$, $\Sigma=k^2$. We obtain S=-G(u) + f(u) \^2 + h(u)\^2 -2k\^2\^2\^r [dr\^2(r)]{}F(r,u)  .\[59\] Therefore, the metric (\[11\]) with $\chi(r)$ given by (\[52\]) or (\[53\]), $S(r,u)$ given by (\[58\]) or (\[59\]), and $P(z, \bar z, u)$ satisfying (\[55\]) can be interpreted as pure radiation plus a perfect fluid, in which the four-velocity is determined by (\[36\]) and $\phi^2$, $\mu$ and $p$ are determined by (\[45\]) and (\[47\]), respectively. Acknowledgments {#acknowledgments .unnumbered} =============== This work was supported by the European Union, TMR Contract No. ERBFMBICT961479 (AMS), the Natural Sciences and Engineering Research Council of Canada (AAC) and the Canadian Institute for Theoretical Astrophysics (DJM). [99]{} A.A. Coley and D.J. McManus, Class. Quantum Grav., [**11**]{}(1994)1261. D.J. McManus and A.A. Coley, Class. Quantum Grav., [**11**]{}(1994)2045. D. Kramer, H. Stephani, M.A.H. MacCallum and E. Herlt, [*Exact Solutions of Einstein’s Field Equations*]{}, Deutscher Verlag der Wissenschaften, Berlin (1980). F.A.E. Pirani, in [*Lectures on General Relativity, 1964 Brandeis Summer Institute in Theoretical Physics*]{}, Vol. 1. Prentice-Hall, Englewood Cliffs, NJ (1965). J.N. Goldberg and R.K. Sachs, Acta. Phys. Polon., Suppl. [**22**]{}(1962)13. I. Robinson and A. Trautman, Proc. Roy. Soc. Lond., [**A265**]{}(1962)463. W. Kundt, Z. Phys., [**163**]{}(1961)77. J. Wainwright, Int. J. Theor. Phys., [**10**]{}(1974)39. J.M. Stewart and G.F.R. Ellis, J. Math. Phys., [**9**]{}(1968)1072.

--- address: - 'Universita Degli Studi Di Salerno, Via Ponte Don Melillo, 84084 Fisciano (SA) Italia' - 'Institute of Mathematics,National Academy of Science of Ukraine, Tereshchenkivska 3, 01601, Kiev, Ukraine' author: - 'Aniello Fedullo and Vitalii A. Gasanenko' title: ' Limit theorems for number of diffusion processes which did not absorb by boundaries. ' --- amstex We have random number of independent diffusion processes with absorption on boundaries in some region at initial time $t=0$. The initial numbers and positions of processes in region is defined by Poisson random measure. It is required to estimate of number of the unabsorbed processes for the fixed time  $\tau>0$. The Poisson random measure depends on $\tau$ and $\tau\to\infty$. Consider the set of independent random diffusion processes    $\xi_{k}(t) ,\quad k=\overline{1,N},$ $t\geq 0,~~\xi_{k}(0)=x_{k},~~ x_{k}\in Q\subset R^{d}$. We wish to investigate of distribution of the number of the processes  $\xi_{k}(t)$  which was into $Q$ for all moments of time $t\leq \tau$. Let domain   $ Q\subset R^{d}$   be open connected region and it is limited by smooth surface $\partial Q$. All processes  $\xi_{k}(t)$  are diffusion processes with absorption on the boundary  $\partial Q$. These processes are solutions of the following stochastic differential equations in $Q$ $$d\xi(t)=a(t,\xi(t))dt + \sum\limits_{i=1}^{d}b_{i}(t,\xi (t))dw^{(k)}_{i}(t) \quad \xi(t)\in R^{d}\eqno(1)$$ $$b_{i}(t,x),~ a(t,x): R_{+}\times R^{d}\to R^{d}.$$ with an initial condition:  $\xi(0)=x_{k}\in D.$ Here the  $W^{(k)}(t)=(w_{i}^{(k)}(t),\quad 1\leq i\leq d),\quad 1\leq k\leq N$  are independent in totality $d$- dimensional Wiener processes. Thus, these processes have the identical diffusion matrices and shift vectors , but they have different initial states. Let $Q$ is bounded and boundary $\partial Q$ is Lyapunov surface  $C^{(1,\lambda)}$. The initial number and positions of processes are defined by the random Poisson measure $\mu(\cdot,\tau)$ in $Q$: $$P(\mu(A,\tau)=k)=\frac{m^{k}(A,\tau)}{k!}e^{-m(A,\tau)}$$ where $m(\cdot,\tau)$ is finitely additive positive mesure on  $Q$ for fixed $\tau$. This task was offered in \[1\] as the mathematical model of practice problem. The authores in article \[2\] investigated case when initial number and positions of diffusion processes are defined by determinate limited measure  $N(B,\tau)$. Where the  $N(B,\tau)$  is equal to number of points  $x_{k}$  in a set  $B$  and  $N=N(Q,\tau)<\infty$ for fixed $\tau>0$. We consider the following case $a(t,x)=a=(\underbrace{0,\dots,0}_{d}),\quad b_{i}(t,x)=b_{i}= (b_{i1},\dots,b_{id}),~~1\leq i\leq d;$ We define matrix  $\sigma=B^{T}B,\quad B=(b_{ij}),~1\leq i,j\leq d$ $\sigma=(\sigma_{ij}),1\leq i,j\leq d$ and differential operator $A:\sum\limits_{1\leq i,j\leq d}\sigma_{ij}\frac{\partial^{2}} {\partial x_{i}\partial x_{j}}.$ Let $\sigma$ be a matrix with the following property $$\sum\limits_{1\leq i,j\leq d}\sigma_{ij}z_{i}z_{j} \geq \mu |\vec z|^{2}.$$ Here  $\mu$,  there is fixed positive number, and $\vec z=(z_{1},\cdots, z_{d})$  there is an arbitrary real vector. This operator acts in the following space $$H_{A}=\{u: u\in L_{2}(Q)\cap Au\in L_{2}(Q)\cap u(\partial Q)=0\}$$ with inner product $(u,v)_{A}=(Au,v)$.Here $(,)$ is inner product in $L_{2}(Q)$. The operator  $A$  is positive operator. It is known \[3\] that the following eigenvalues problem $$Au=-\lambda u,\quad u(\partial Q)=0$$ has infinity set of real eigenvalues $ \lambda_{i}\to\infty$ and $$0<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{s}<\cdots.$$ The corresponding eigenfunctions $$f_{11},\dots,f_{1n_{1}},\cdots,f_{s1},\dots,f_{sn_{s}},\cdots$$ form complete system of functions both in $H_{A}$ and $L_{2}^{0}(Q):= \{u: u\in L_{2}(Q)\cap u(\partial Q)=0\}$. Here the number $n_{k}$ is equal to multiplicity of eigenvalue $\lambda_{k}$. We denote by  $\eta(\tau)$  the number of remaining processes in the region  $D$  at time instant  $\tau$. We also assume that $\sigma$-additive measure $\nu$ is given on the $\Sigma_{\nu}$- algebra sets of $Q,\quad \nu(Q)<\infty.$ All eigenfunctions $f_{ij}:Q\to R^{1}$ and all measures $m(\cdot,\tau)$  are $(\Sigma_{\nu},\Sigma_{Y})$ measurable. Here $\Sigma_{Y}$ is system of Borel sets of $R^{1}$. Let $\Rightarrow$ denotes the weak convergence of random values. Put $$g(\tau)=\exp\left(-\frac{\tau}{2}\lambda_{1}\right).$$ We consider the following initial-boundary problem $$\frac{\partial u}{\partial t}=\frac{1}{2}\sum\limits_{1\leq i,j\leq d} \sigma_{ij}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j}}\quad x\in Q;$$ $$u(0,x)= 1\quad\hbox{if}\quad x\in Q;$$ $$u(t,x)=0 \quad \hbox{if}\quad x\in\partial Q,~~~t\geq 0\eqno(2)$$ It is known \[4\], that   $u(\tau, x)$  is equal to probability of remaining in the region  $Q$  at time instant  $\tau$  of a diffusion process from (1) which occurs at the point $(0,x)$ at the initial moment ( $\xi(0)= x,~~ x\in Q$). We designate through $\gamma_{k}=(x^{k}_{1}, \cdots ,x_{d}^{k})$ the initial position of $k$-th process. We define the value of $u(\tau,\gamma_{k})$. We define a particular solution of (2) in form $$u(t, x)= u_{1}(t)u_{2}(x).$$ The ordinary argumentaion leads to definition of joined constant $\lambda$: $$2\frac{1}{u_{1}}\frac{\partial u_{1}}{\partial t} =\frac{A u_{2}}{u_{2}}=- \lambda.$$ We obtain the following system of tasks due the latter one $$A u_{2}= -\lambda u_{2};\quad u_{2}(\partial Q)=0.\eqno(3)$$ $$\frac{\partial u_{1}}{\partial t}= -\frac{\lambda}{2} u_{1}; \quad u_{1}(0)= 1\eqno(4),$$ It is clear that $u_{1}(t,\lambda)= \exp(-\frac{t}{2}\lambda )$ is solution of (5) . The soluton of (3) was described above. We assume that system of functions $\{f_{ij}( x), i\geq 1, 1\leq j\leq n_{i}\}$ is orthonormalized with respect to space $L_{2}^{0}(Q)$. The general solution of problem (2) has the following form $$u(t,x)=\sum\limits_{j=1}^{\infty} \exp(-\frac{t}{2}\lambda_{j} ) \sum\limits_{m=1}^{n_{j}}c_{jm}f_{jm}( x)$$ where coefficients $c_{jm}$ are equal to coefficients of decomposition of initial value (unit) by system of functions $f_{jm}$: $c_{jm}=\int\limits_{Q}f_{jm}( x)d x$. The Parseval - Steklov equality is true for these coefficients: $$\sum\limits_{j=1}^{\infty}\sum\limits_{m=1}^{n_{j}}c_{jm}^{2}=|Q|.\eqno(5)$$ Put $F( x)=\sum\limits_{m=1}^{n_{1}}c_{1i}f_{1m}( x)$. The function  $F(x)$  is continuous and bounded function on the $\bar Q$. Since $u(t,x)$ is probability,it is not diffucult to show that $F( x)\geq 0$ for all $x\in Q$. Let $M=\sup\limits_{ x\in Q}F(x)$. We introduce the following sets $$B_{k,n}=\{ x\in Q: \frac{Mk}{n}< F( x)\leq\frac{M(k+1)}{n} \}$$ Here  $0\leq k\leq n-1$  and $n>1$. Let us denote by $\zeta_{k,n}(\tau), ~1\leq~k\leq n$  the number of unabsorbed processes at time instant  $\tau$ which occur in the region  $B_{k,n}$ at the initial time. These values are independent in totality by assumption. The distribution function of $\zeta_{k,n}(\tau)$  is defined by the following formula $$P(\zeta_{k,n}(\tau)=l) =\sum\limits_{d=l}^{\infty}P(\mu(B_{k,n},\tau)=d)\times$$ $$\times\sum\limits_{1\leq i_{1},\cdots,i_{l}\leq d,i_{m}\ne i_{j},m\ne j} \prod\limits_{k=1}^{l}u(\tau,\gamma_{i_{k}}) \prod\limits^{d}_{s=l+1, i_{s}\notin (i_{1},\cdots,i_{l}),i_{m}\ne i_{j}} (1- u(\tau,\gamma_{i_{s}})),\quad l=0,1,\dots.$$ Here  $x_{i_{j}}\in B_{k,n}$. We set $$a_{k,n}(\tau)=\min\limits_{ x\in B_{k,n}}u(\tau, x),\quad \bar a_{k,n}(\tau)=1- a_{k,n}(\tau);$$ $$b_{k,n}(\tau)=\max\limits_{ x\in B_{k,n}}u(\tau, x),\quad \bar b_{k,n}(\tau)=1- b_{k,n}(\tau).$$ Now $$J_{k,n}(l,\tau):= \sum\limits_{d=l}^{\infty}\frac{m^{d}(B_{k,n},\tau)}{d!} \exp(- m(B_{k,n},\tau)) C_{d}^{l}a^{l}_{k,n}(\tau)\bar b^{d-l}_{k,n}(\tau)\leq$$ $$\leq P(\zeta_{k,n}(\tau)=l)\leq$$ $$\sum\limits_{d=l}^{\infty} \frac{m^{d}(B_{k,n},\tau)}{d!}\exp(- m(B_{k,n},\tau)) C_{d}^{l}b^{l}_{k,n}(\tau)\bar a^{d-l}_{k,n}(\tau)=: I_{k,n}(l,\tau).\eqno(6)$$ Further $$J_{k,n}(l,\tau)= \frac{\left(m(B_{k,n},\tau)a_{k,n}(\tau)\right)^{l}}{l!} \exp(- m(B_{k,n},\tau)) \sum\limits_{d=l}^{\infty} \frac{\left(\bar b_{k,n}(\tau)m(B_{k,n},\tau)\right)^{d-l}}{(d-l)!} =$$ $$=\frac{\left(m(B_{k,n},\tau)a_{k,n}(\tau)\right)^{l}}{l!} \exp(- b_{k,n}(\tau)m(B_{k,n},\tau));$$ By analogy: $$I_{k,n}(l,\tau)= \frac{\left(m(B_{k,n},\tau)b_{k,n}(\tau)\right)^{l}}{l!} \exp(- a_{k,n}(\tau)m(B_{k,n},\tau)).\eqno(7)$$ We introduce the following generating functions $$\varphi(\tau,s)=\sum\limits_{l\geq 0}s^{l}P(\eta(\tau)=l).$$ $$\varphi_{k,n}(\tau,s)=\sum\limits_{l\geq 0}s^{l}P(\zeta_{k,n}(\tau)=l), \quad k=\overline{0,n-1},\quad 0\leq s\leq 1.$$ By the construction, $\eta(\tau)$  can be represented in the form  $\eta(\tau)=\zeta_{1,n}+\cdots+\zeta_{n-1,n}(\tau)$. Thus $$\varphi(\tau,s)=\prod_{k=0}^{n-1}\varphi_{k,n}(\tau,s).\eqno(8)$$ Combining (6)-(8), we conclude that $$\exp\{(sa_{k,n}(\tau)-b_{k,n}(\tau))m(B_{k,n},\tau)\}\leq \varphi_{k.n}(\tau,s)\leq$$ $$\leq \exp\{(sb_{k,n}(\tau)-a_{k,n}(\tau))m(B_{k,n},\tau)\}$$ and $$\exp\left\{\sum\limits_{k=0}^{n-1}(sa_{k,n}(\tau)-b_{k,n}(\tau)) m(B_{k,n},\tau)\right\}\leq \varphi(\tau,s)\leq$$ $$\leq \exp\{\sum\limits_{k=0}^{n-1}(sb_{k,n}(\tau)-a_{k,n}(\tau)) m(B_{k,n},\tau)\}.\eqno(9)$$ Since function  $u(\tau,x)$  is continuous function in $ x\in Q$, there exit a points $x_{*},~~x^{*}\in \bar B_{k,n}$ such that the following equalities have place $$a_{k,n}(\tau)= u_{1}(\tau,\lambda_{1})F(x_{*})+ \sum\limits_{k\geq 2}u_{1}(\tau,\lambda_{k})\sum\limits_{m=1}^{n_{k}}c_{km} f_{km}(x_{*}),$$ $$b_{k,n}(\tau)= u_{1}(\tau,\lambda_{1})F( x^{*})+ \sum\limits_{k\geq 2}u_{1}(\tau,\lambda_{k})\sum\limits_{m=1}^{n_{k}}c_{km} f_{km}(x^{*}),$$ here  $ x_{*}:=x_{*}(k,n,\tau),~~x^{*}:=x^{*}(k,n,\tau)$. Now, we can rewrite the sums in exponentes from (9) in the following forms $$\sum\limits_{k=0}^{n-1} \left(sF( x_{*})- F(x^{*})\right) \exp(-\frac{\tau}{2}\lambda_{1} ) m(B_{k,n},\tau)+$$ $$+ \sum\limits_{k=0}^{n-1} \exp(-\frac{\tau}{2}\lambda_{1} ) m(B_{k,n},\tau) \sum\limits_{j\geq 2} \exp\left(-\frac{\tau}{2}(\lambda_{j}-\lambda_{1})\right) \sum\limits_{m=1}^{n_{j}}c_{jm} (sf_{jm}( x_{*})-f_{jm}( x^{*})) , \eqno(10)$$ $$\sum\limits_{k=0}^{n-1}(sF( x^{*})- F( x_{*})) \exp(-\frac{\tau}{2}\lambda_{1} ) m(B_{k,n},\tau)+$$ $$+\sum\limits_{k=0}^{n-1} \exp(-\frac{\tau}{2}\lambda_{1} ) m(B_{k,n},\tau) \sum\limits_{j\geq 2} \exp\left(-\frac{\tau}{2}(\lambda_{j}-\lambda_{1})\right) \sum\limits_{m=1}^{n_{j}}c_{jm} (sf_{jm}( x^{*})- f_{jm}( x_{*})), \eqno(11)$$ We calculate limit of (10) if  $\tau\to\infty$. The first sum of (10) convergers to the following limit under the condition of theorem $$\sum\limits_{k=0}^{n-1}sF( x_{*})\nu(B_{k.n}) -\sum\limits_{k=0}^{n-1}F( x^{*})\nu(B_{k.n}).$$ This is difference of two integral sums which has the following limit under  $n\to\infty$ (see \[5\]) $$(s-1)\int\limits_{Q}F(x)\nu(d x).$$ Put $$s_{\tau}( x)= \sum\limits_{j\geq 2}\exp\left(-\frac{\tau}{2}(\lambda_{j}-\lambda_{1})\right) \sum\limits_{m=1}^{n_{j}}c_{km} f_{km}(x).$$ We consider sums of eigenfunctions in the form $$e(x,\lambda)=\sum\limits_{\lambda_{j}\leq \lambda}f^{2}_{jl}(x)$$ The following result is proved in monography \[6,Thm. 17.5.3\] $$\sup\limits_{x\in Q}e(x,\lambda)\leq C \lambda^{\frac{d}{2}}.$$ Asymptotic characteristic of eigenvalues  $\lambda_{j}$  under  $j\to\infty$  is defined by the following inequalities \[3, sec. 18\] $$c_{1}j^\frac{2}{d}\leq \lambda_{j}\leq c_{2} j^\frac{2}{d}, \quad \hbox{where}\quad c_{1},~c_{2}=const.$$ The latter one, (5) and Caushy-Bunyakovskii inequality lead to the following convergence under $\tau\to\infty$ $$|s_{\tau}( x)|\leq \sum\limits_{j\geq 2}\exp\left(-\frac{\tau}{2}(\lambda_{j}-\lambda_{1})\right) \sqrt{\sum\limits_{m=1}^{n_{j}} c^{2}_{jm}}\sqrt{\sum\limits_{m=1}^{n_{j}}f_{jm}^{2}(x)}\leq$$ $$\leq \sqrt{C}\sum\limits_{j\geq 2}\lambda_{j}^{\frac{d}{2}}\exp(-\frac{\tau}{2} (\lambda_{j}-\lambda))\sqrt{\sum\limits_{m=1}^{n_{j}}c_{jm}^{2}}\leq$$ $$\leq \sqrt{C} \sqrt{\sum\limits_{j\geq 2} \lambda_{j}^{d} \exp(-\tau (\lambda_{j}-\lambda_{1}))} \sqrt{\sum\limits_{j\geq 2}\sum\limits_{m=1}^{n_{j}}c^{2}_{jm}}\to 0.$$ Thus the second sum from(10) convergences to zero. The similar considerations apply to (11). Proof is complete. [***Example.***]{} Now we apply the general approach to the particular case. We consider the case if $Q$ is circle   $Q=\{(x,y): x^{2}+y^{2}\leq r_{0}^{2}\}$. We assume that the diffusion processes occurs at the point $(x_{k},y_{k})\in Q$ at the initial time. The processes are described in  $Q$  by the following stochastic differential equations $$d\xi(t)= \sum\limits_{i=1}^{2}b_{i}dw_{i}(t)\eqno(12)$$ $$\xi(0)=\xi_{0}=(x_{k},y_{k}).$$ where $ b_{1}=(\sigma,0),b_{2}=(0,\sigma) $ and $W(t)=(w_{i}(t),i=1,2)$ is 2-dimensional Wiener process. We assume that the equation (12) defines a diffusion process with absorption on the boundary $\partial Q=\{(x,y,z): x^{2}+y^{2}=r_{0}^{2}\}$. In follows that the $J_{0}(x), J_{1}(x)$ are Bessel functions zero and first order. They are defined as the solutions of the following equations $$\frac{d^{2} y}{d x^{2}} + \frac{1}{x}\frac{d y}{d x} + (1-\frac{n^{2}}{x^{2}}) =0,$$ $$y(x_{0})=0,~~~ (x_{0}=\sqrt{\lambda}r);\quad |y(0)|<\infty;$$ under $n=0$ and $n=1$. The value of $\mu_{m}^{(0)}$ is equal to $m$- th root of the equation $J_{0}(\mu)=0$ \[7,8\]. Let $mes(\cdot)$ denotes the Lebesgue measure. We set $$f(\tau):=\exp\left(-\frac{\tau}{2} \Bigl(\frac{\sigma \mu_{1}^{(0)}}{r}\Bigr)^{2} \right).$$ We suppose that $m(\cdot,\tau)$ holds the condition $$m(\cdot,\tau)f(\tau) \Rightarrow mes(\cdot) \quad \hbox{if}\quad \tau\to\infty.$$ In this case the system of tasks (3),(4) has the following form $$\triangle u_{2}= -\mu u_{2}, ~~(x,y)\in C ;\quad u_{2}(x,y)=0 \quad \hbox{if} \quad x^{2}+y^{2}=r_{0}^{2}\eqno(13),$$ $$\frac{\partial u_{1}}{\partial t}=-\frac{\sigma^{2}}{2} \mu u_{1}, \quad u_{1}(0)=1.\eqno(14)$$ According to general approach for construction of solution  $u(t,x,y)$  (see,for example, \[7, sec.1V \] we rewrite the task of (13) in polar coordinates: $u_{3}(r,\varphi):= u_{2}(r\cos\varphi,r\sin\varphi)$. The $u_{3}$ is solution the following problem $$\frac{\partial^{2} u_{3}}{\partial r^{2}}+\frac{1}{r}\frac{\partial u_{3}} {\partial r}+ \frac{1}{r^{2}}\frac{\partial^{2}u_{3}}{\partial \varphi^{2}} +\mu u_{3}=0,$$ $$u_{3}(r_{0},\varphi)=0.$$ We obtain $$u(t,x,y)=u(t,r)=\sum\limits_{m=1}^{\infty}c_{m} J_{0}\left(\frac{\mu_{m}^{(0)}}{r_{0}}r\right) \exp\left(-\frac{t}{2} \left(\frac{\sigma\mu^{0}_{m}}{r_{0}}\right)^{2}\right),$$ where $c_{m}=2\left(m_{m}^{(0)}J_{1}(\mu_{m}^{(0)})\right)^{-1}$. The function  $J_{0}\left(\frac{\mu_{1}^{(0)}}{r_{0}}r\right)$  is strictly decreasing function if  $0\leq r\leq r_{0}$ . Thus we can construct the partitions  $B_{k,n}$  by the following partitions $$\tilde B_{k,n}=\left\{ (x,y)\in C: \frac{r_{0}k}{n}< \sqrt{x^{2}+y^{2}} \leq \frac{r_{0}(k+1)}{n}\right\},\quad 0\leq i\leq n-1.$$ Now  $mes(\tilde B_{k,n})=g(\frac{k+1}{n})-g(\frac{k}{n})$  where    $g(x)=\pi r_{0}^{2}x^{2}, ~~~0\leq x\leq 1$. Finally, the parameter of Poisson distribution is equal to $$a=2\left(m_{1}^{(0)}J_{1}(\mu_{1}^{(0)})\right)^{-1}2\pi r_{0}^{2} \int\limits_{0}^{1}J_{0}(\mu_{1}^{(0)}x)xdx=\pi \left(\frac{2 r}{\mu_{1}^{(0)}}\right)^{2}.$$ We used the following known relation   $\alpha J_{0}(\alpha)=\left[\alpha J_{1}(\alpha)\right]'$ \[7,p.466\] for calculation of the latter integral. \[\] V.A.Gasanenko and A.B.Roitman Rarefaction of moving diffusion particles The Ukrainian Mathematical Journal 56 2004 691-694 \[\] A.Fedullo, V.A. Gasanenko Limit theorems for rarefaction of set of diffusion processes by boundaries Theory of Stochastic Processes 2005 (to appear) \[\] S.G. Mihlin Partial differential linear equations Vyshaij shkola Moscow 1977 431 p \[\] I.I.Gikhman, A.V. Skorokhod Introduction to the theory of random processes Nauka Moscow 1977 568 p \[\] A.N. Kolmogorov, S.V. Fomin Elements of theory of functions and functional analysis Nauka Moscow 1972 496 p \[\] L. H$\ddot o$rmander The analysis of Linear Partial Differential Operators III Spinger-Verlag 1985 \[\] A.N.Tikhonov, A.A.Samarsky The equations of mathematical physics Nauka Moskow 1977 736 \[\] E.Janke, F.Emde and F.Losch Special functions Nauka Moskow 1968 344

--- abstract: 'We study the impact of the Landau Pomeranchuk Midgal (LPM) effect on the dynamics of parton interactions in proton proton collisions at the Large Hadron Collider energies. For our investigation we utilize a microscopic kinetic theory based on the Boltzmann equation. The calculation traces the space-time evolution of the cascading partons interacting via semihard pQCD scatterings and fragmentations. We focus on the impact of the LPM effect on the production of charm quarks, since their production is exclusively governed by processes well described in our kinetic theory. The LPM effect is found to become more prominent as the collision energy rises and at central rapidities and may significantly affect the model’s predicted charm distributions at low momenta.' author: - 'Dinesh K. Srivastava' - Rupa Chatterjee - 'Steffen A. Bass' title: Landau Pomeranchuk Midgal Effect and Charm Production in $pp$ Collisions at Large Hadron Collider Energies using the Parton Cascade Model --- Introduction ============ Studies of relativistic collisions of heavy nuclei underway at the Relativistic Heavy Ion Collider at Brookhaven and the Large Hadron Collider at CERN have provided ample evidence for a deconfining transition of strongly interacting matter into a (strongly coupled) Quark Gluon Plasma (QGP) expected from lattice QCD calculations (see e.g., Refs. [@Ratti:2016lrh; @Ratti:2017qgq; @Ratti:2018ksb] and references therein). These studies, both on the theoretical and the experimental fronts, have now reached a high level of sophistication and the quantitative determination of QGP properties [@Schenke:2010rr; @Gale:2012rq; @Shen:2014vra; @Bernhard:2016tnd] is now in progress. Very often the results for heavy-ion collisions are compared with those for proton proton collisions at the same center of mass energy ($\sqrt{s_{NN}}$) in order to arrive at some of these conclusions, with the rationale that no QGP is likely to be formed in $pp$ collisions. This simple expectation is now under strain as more and more indications of formation of an interacting system, emerge in $pp$ collisions, especially for events having a large particle multiplicity (see e.g., Refs. [@Khachatryan:2016txc; @ALICE:2017jyt]). Is an interacting system formed in $pp$ collisions? Recently we have explored this question within Parton Cascade Model (PCM) [@Srivastava:2018dye]. The PCM is a transport model based on the relativistic Boltzmann equation for the time evolution of the parton density in phase-space due to semi-hard perturbative QCD interactions including scattering and radiations [@Geiger:1991nj; @Bass:2002fh] within a leading logarithmic approximation [@Altarelli:1977zs]. Our study indicated the formation of a medium driven by a substantial amount of multiple parton interactions, including fragmentation of partons after scattering. These aspects were found to be more strongly prevalent for collisions at small impact parameters or with large parton multiplicities and at higher incident beam energies. Even though the precise number of collisions and fragmentations are dependent on the $p_T^\text{cut-off}$ and $\mu_0$ used to regularize the pQCD cross-sections and the fragmentations respectively, the results are sufficiently general. Based on these previous findings it is opportune to investigate the importance of quantum coherence effects in parton-parton interactions, such as the Landau Pomeranchuk Midgal (LPM) effect [@Landau:1953gr]. The LPM effect is known to be important for large collision systems with lifetimes of multiple fm/c, but has commonly been neglected in the microscopic study of the proton-proton system, due to its small size and short lifetime. Here we focus on the investigation of the LPM effect on charm quark production in proton proton collisions. Charm production is particularly well suited in this context, since it only occurs via hard processes calculable in pQCD and charm is conserved throughout the reaction. The PCM was recently extended to treat the production and medium-evolution of heavy quarks [@Srivastava:2017bcm]. Consider a parton traversing a cloud of quarks and gluons and undergoing multiple scatterings. If the separation between consecutive scatterings suffered by the parton is sufficiently large so that the radiations off these collision centers can be treated as an incoherent sum of radiation spectra resulting from individual scatterings, we reach what is known as the Bethe-Heitler limit [@Bethe:1934za]. If on the other hand, the scattering centers are too closely located to each other, the observed radiation has to be evaluated within what is known as the factorization limit, and is a product of a single scattering spectrum from the sum of the individual small momentum transfers from all the individual scatterings. The LPM effect [@Landau:1953gr] describes the results between these two limiting regimes, by accounting for the suppression of the radiation relative to the Bethe-Heitler limit, when the formation time of the radiated gluon is large compared to the mean free path and thus destructive interference between the radiated spectra becomes important. The dynamics of LPM effect on the production of light partons ($u$, $d$, $s$, and $g$) and photons in collision of gold nuclei at RHIC energy, within the PCM, was discussed earlier [@Renk:2005yg; @Bass:2002vm; @Bass:2007hy]. That work also demonstrated that the inclusion of the LPM effect greatly improved the agreement of the scaling of multiplicity distributions in $pp$ collisions up to 200 GeV. ![(Color online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for minimum bias $pp$ interactions as a function of center of mass energy. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons. []{data-label="min-bias"}](ncoll_min_bias.eps){width="7.6"} ![(Color online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for minimum bias $pp$ interactions as a function of center of mass energy. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons. []{data-label="min-bias"}](nfrag_min_bias.eps){width="7.6"} ![(Color online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for minimum bias $pp$ interactions as a function of center of mass energy. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons. []{data-label="min-bias"}](nc_min_bias.eps){width="7.6"} We shall investigate the consequences of the LPM effect on charm production in $pp$ collisions at $\sqrt{s_\text{NN}}$ of 0.20, 2.76, 5.02, 7.00, and 13.00 TeV. The results at RHIC energy (0.20 TeV) are included to clearly bring out the abundance of parton production etc. at LHC energies. There are several reasons for focusing on charm quarks. As pointed out above, charm quarks can be produced only from semi-hard scattering of gluons and annihilation of a quark-antiquark pair or from a splitting of a gluon which has a large virtuality following a semi-hard scattering. The corresponding scattering matrix elements are not singular because of the mass of the charm quark and thus do not need any $p_T^\text{cut-off}$. We do realize, though, that the momentum distribution of the charm quarks can be modified by radiation of gluons or by scattering with other partons, which will be affected by variation of the $p_T^\text{cut-off}$ used for regularizing the pQCD matrix elements and the $\mu_0$ used for terminating the fragmentations. The number of charm quarks which are produced is very small and thus the probability that their number is depleted by charm-anticharm annihilation is limited. Finally, there is no production of charm quarks during the hadronic phase. We briefly discuss the basic ingredients of the PCM model pertaining to this investigation in the next section, results are given in Section III, and finally we summarize our findings. ![(Color online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for $pp$ interactions as a function of center of mass energy at impact parameter equal to zero fm. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons.[]{data-label="b.eq.0"}](ncoll_b0.eps){width="7.6"} ![(Color online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for $pp$ interactions as a function of center of mass energy at impact parameter equal to zero fm. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons.[]{data-label="b.eq.0"}](nfrag_b0.eps){width="7.6"} ![(Color online) Number of collisions (upper panel), number of fragmentations (middle panel) and number of charm quarks produced per event (lower panel) for $pp$ interactions as a function of center of mass energy at impact parameter equal to zero fm. The three calculations involve multiple collisions among partons by neglecting and including the LPM effect and collisions only among primary partons with radiations off the scattered partons.[]{data-label="b.eq.0"}](nc_b0.eps){width="7.6"} Model Description ================= The details of the parton cascade model, including its Monte Carlo implementation [VNI/BMS]{}, have been discussed in significant detail in Refs. [@Geiger:1991nj; @Bass:2002fh], while production of heavy quarks has been laid out in Ref. [@Srivastava:2017bcm]. Presently, we shall just briefly summarize the features most important to our investigation: The parton cascade model is a transport model for the time-evolution of an ensemble of quarks and gluons based on the Boltzmann equation. We include $2\rightarrow 2$ scatterings between light quarks, heavy quarks and gluons, and the $2\rightarrow 3$ reactions via time-like branchings of the final-state partons (see Refs. [@Bass:2002fh; @Altarelli:1977zs]) following the well tested procedure adopted in [PYTHIA]{} [@Sjostrand:2006za]. ![(Color online) The transverse momentum spectra of charm quarks in $pp$ collisions at 200 GeV (upper panel) and 2.76 TeV (lower panel) due to multiple collisions among partons and fragmentations off final state quarks, with and without inclusion of LPM effect.[]{data-label="0.2_2.76"}](0.2.eps){width="8.0"} ![(Color online) The transverse momentum spectra of charm quarks in $pp$ collisions at 200 GeV (upper panel) and 2.76 TeV (lower panel) due to multiple collisions among partons and fragmentations off final state quarks, with and without inclusion of LPM effect.[]{data-label="0.2_2.76"}](2.76.eps){width="8.0"} In the PCM, the IR-singularities in these pQCD cross-sections are avoided by introducing a lower cut-off on the momentum transfer $p_T^\text{cut-off}=$ 2 GeV (please note that results discussed in Refs. [@Renk:2005yg; @Srivastava:2017bcm] were obtained by using a much smaller value for $p_T^\text{cut-off}$ of about 0.7 GeV, which increased the parton scatterings). Most of the studies using [VNI/BMS]{} reported earlier were performed using a constant value of $\alpha_s$ = 0.3. In the present work, we have taken $\alpha_s(Q^2)$, as we wish to study the momentum distribution of charm quarks for large values of transverse momenta. Details of the initial parton distributions and the relevant matrix elements etc. have already been discussed repeatedly [@Bass:2002fh; @Srivastava:2017bcm; @Srivastava:2018dye] which we closely follow. For the sake of completeness, we recall that the $2 \rightarrow 3$ processes are accounted for by inclusion of radiative processes for the final state partons in a leading logarithmic approximation. The collinear singularities are then regularized by terminating the time-like branchings, once the virtuality of the parton drops to $M_0^2(=m_i^2+\mu_0^2)$, where $m_i$ is the current mass of the parton (zero for gluons, current mass for quarks) and $\mu_0 $ has been kept fixed as 1 GeV. We have included $g \rightarrow gg$, $q \rightarrow q g$, $g \rightarrow q \bar{q}$, and $q \rightarrow q \gamma$ branchings for which the relevant branching functions $P_{a\rightarrow bc}$ are taken from Altarelli and Parisi [@Altarelli:1977zs]. The interference of soft gluons is included by angular ordering of radiated gluons as in [PYTHIA]{}. Implementing the LPM effect in a semi-classical transport such as the PCM is not easy. First of all, the quarks and gluons are treated as quasi-particles in the PCM and thus a full quantum mechanical treatment for the process is out of question. We implement the LPM effect by assigning a formation time $\tau$ to the radiated particle: $$\tau = \frac{\omega}{k_T^2},$$ where $\omega$ is its energy and $k_T$ is its transverse momentum with respect to the emitter. During the formation time, the radiated particle is assigned zero cross-section and thus it does not interact. The emitter, however continues to interact and if that happens, the radiated particle is removed from the list and does not participate in later evolution of the system. This leads to suppression of parton multiplication (see Refs. [@Renk:2005yg; @Bass:2002vm; @Bass:2007hy]). A similar procedure is adopted in the Boltzmann Approach to MultiParton Scattering, [BAMPS]{}, of the Frankfurt group [@Uphoff:2014hza]. This particular implementation of the LPM effect is quite common for semi-classical transport models, but by no means unique. An alternative method of implementing the LPM effect by Baier, Dokshitzer, Mueller, Peigne and Schiff (BDMPS) relies on recalculating the phase space for the emission of the radiated gluon [@Baier:1996kr; @Baier:1996sk; @Baier:1998kq] (see also Ref. [@Zakharov:1996fv]). Recently we have experimented with implementing the LPM effect in a scheme that is assured to reproduce the BDMPS limit of parton energy-loss [@Wiedemann:2011zz; @ColemanSmith:2011wd]. The energy loss suffered by charm quarks in an infinite medium (at a fixed temperature) was well described using this formalism [@Younus:2013rja]. However, this implementation, focussing on the evolution of the leading parton, is currently only feasible for infinite matter calculations in the PCM and further development is required to adapt the necessary algorithms to proton-proton or nucleus-nucleus calculations. Our expectation is that the LPM effect will lead to a suppression of parton multiplication and thus to a reduction of primary-secondary or seconadary-secondary collisions, where primary partons make up the initial state of the two colliding protons and secondary partons are the partons emerging from scatterings and subsequent radiative interactions. It is expected that as the LPM effect reduces the number of multiple scatterings (which mainly produce charm quarks having low transverse momenta), we should expect a lowering of the production of charm quarks at smaller $p_T$. In addition, the suppression of radiation of gluons through the LPM effect should imply that charm quarks having large momenta radiate gluons less frequently. This should lead to a hardening of the transverse momentum spectra for charm quarks. Our analysis is set up to confirm/refute these expectations. In order to clearly bring out the consequences of the LPM effect we proceed as follows: as a first step we study the production of charm quarks with multiple parton collisions and fragmentations without including the LPM effect. Next we give our results for calculations where the LPM effect is included. We investigate whether the LPM effect eliminates multiple parton scatterings by comparing the results from the above to a calculation with only primary-primary parton scatterings and fragmentations. The difference of the results of these calculations should clearly bring out the importance of multiple scatterings of partons in proton-proton collisions and indicate the possible emergence of an interacting medium created by semi-hard pQCD interactions. Finally, in order to investigate the rapidity dependence of the LPM effect, we shall study the transverse momentum distribution of charm quarks at different rapidities, for which data have now become available. Results ======= Multiple scatterings and consequences of LPM effect --------------------------------------------------- An interacting medium would be characterized by partons undergoing multiple interactions. This is different from the case when we have several parton-parton interactions involving only primary partons from the projectile and the target, without any further interaction among the partons thus produced. In Fig. \[min-bias\] we show results for minimum bias collisions of protons at several incident beam energies and show the number of semi-hard partonic scatterings, number of fragmentations, and the number of charm quarks produced per collision. The first set of calculations restricts the interactions to primary-primary collisions followed by fragmentations off the final state partons. These results will not be affected by assigning or not assigning a formation time (i.e, inclusion or non-inclusion of LPM effect) to the radiated gluons as, further scatterings are not considered. The second set of calculations allows for primary-primary, primary-secondary, and secondary-secondary collisions along with fragmentations off the final state parton, but the LPM effect is not taken into account. The final set of calculations describe the system when all possible multiple scatterings and fragmentations off the final state partons are included and the LPM effect is accounted for, using the procedure discussed earlier. We find that without the LPM effect, the number of collisions and fragmentations rise rapidly with increase in collision energy. The accounting of the LPM effect moderates this rise considerably. The reduction in the number of collisions is about 2% at 200 GeV and rises to almost 80% at 13.00 TeV, showing a strong dependence on the collision energy (for a fixed $p_T^\text{cut-off}$ of 2 GeV). The corresponding reduction in number of fragmentations is similar, being about 2% at 200 GeV and rising to about 70% at 13.00 TeV. The similarity of these numbers should not come as a surprise as in our approach scatterings are followed by fragmentations. The reduction in the production of charm quarks is smaller though, just about 1% at 200 GeV and about 60% at the top energy considered. We attribute the smaller reduction in the charm quark multiplicity compared to the reduction in overall scatterings and fragmentations to the large mass of the charm quark, which restricts the phase space for its production. As discussed earlier, a comparison between results including the LPM effect with those for only primary-primary collisions and fragmentations reveal the extent of multiple scatterings. We note that collisions involving primary and secondary partons account for about 2% of the total number of collisions when LPM is accounted for at 200 GeV and increase to about 45% at the top energy considered. The number of fragmentations also rises similarly. These results suggest that the semi-hard partonic interactions in $pp$ collisions at LHC energies produce a dense medium, where partons undergo multiple interactions, even when the LPM suppression of fragmentations off final state partons is accounted for. Theses (additional) multiple collisions are sufficiently large to leave an imprint even in minimum bias events which are dominated by collisions involving larger impact parameters where the produced medium may not be very dense. Evidence of the increasing importance of the LPM effect in more central collisions (which are likely to have a larger multiplicity) is seen from Fig. \[b.eq.0\] where the corresponding results are plotted for zero impact parameter. We see that the number of collisions, fragmentations, and charm quarks for all the cases rise significantly and so also the effect of LPM supression. Transverse momentum distribution of charm quarks ------------------------------------------------ Next we discuss our results for the $p_T$ distribution of charm quarks. Given the nature of charm quark fragmentation into $D$ mesons, the $p_T$ spectra can be used as a proxy for the $p_T$ distribution of prompt $D^0$ mesons, by accounting for the fraction (0.565) for which the charm quark fragments into a $D^0$ meson. We have already seen that the LPM effect has only a very small effect at the lowest incident beam energy considered here, namely 200 GeV. This is again confirmed by Fig. \[0.2\_2.76\] (upper panel), where the momentum distribution of the charm quarks with and without the LPM effect are shown. These are essentially identical. (The deviation of experimental data [@Adamczyk:2012af] from the theoretical calculations is mainly due to the value of $p_T^\text{cut-off}$ of 2 GeV used at all the energies. We believe that a more appropriate value for this particular case could be about 0.7 GeV used earlier [@Srivastava:2017bcm]). The LPM effect starts to become relevant in the theoretical results for the $p_T$ distribution of charm quarks at 2.76 TeV (see Fig. \[0.2\_2.76\], lower panel), where a larger production of charm quarks is seen at lower momenta. We note however, that the results above $p_T$ equal to 2 GeV, where we can trust our results, can not distinguish between the calculations with and with-out the LPM effect at this beam energy. We also add that the agreement of the calculation with the experimental data [@Abelev:2012vra] is likely to improve with a slight decrease in $p_T^\text{cut-off}$ as it will increase the number of partonic collisions and the accompanying fragmentations. LHCb has measured charm production at several forward rapidities at 5.02 [@Aaij:2016jht], 7.00 [@Aaij:2013mga], and 13.00 TeV [@Aaij:2015bpa]. The results at central rapidity for the same at 7.00 TeV beam energy from ALICE [@ALICE:2011aa] are also available. We see a good description of $p_T$ spectra of charm quarks at all rapidities at 5.02 TeV (Fig. \[5.02\]). An enhanced production of charm quarks is seen at lower momenta, when the LPM effect is neglected and the enhancement decreases with increase in rapidity. It remains to be seen if the data likely to be available at 5.02 TeV (and 13.00 TeV, see later) at central rapidity are in agreement with these results. The results for 7.00 TeV (Fig. \[7.00\]) are of particular relevance, since experimental data also exist at central rapidity. Our calculations show a large suppression of charm production at lower $p_T$ when the LPM effect is included and closely reproduce the transverse momentum spectra at all rapidities. We also see a hint of the hardening of the $p_T$ spectra for large values of $p_T$, which is also reproduced by our calculations, even though the effect is not large. As indicated earlier, this happens as the LPM effect also suppresses the radiation of gluons by charm quarks traversing the medium at large energy/momenta. The hardening of the transverse momentum spectra and suppression of charm quarks having low $p_T$ (the suppression decreasing with increase in rapidity) is seen more clearly at 13.00 TeV (Fig. \[13.00\]). The experimental results at all rapidities are adequately explained when the LPM effect is accounted for. It will be interesting to see if the substantial suppression predicted at central rapidity is supported by data. Summary and conclusions ======================= We have studied the impact of the Landau Pomeranchuk Midgal effect on the dynamics of parton transport in proton-proton collisions at LHC energies. In particular, we have focused on the production of charm quarks, since these are only produced in hard pQCD interactions for which the parton cascade model utilized in our study is uniquely suited. We find that the inclusion of the Landau Pomeranchuk Midgal effect, which suppresses the radiation of gluons off scattered partons leads to a reduction in the number of scatterings, number of fragmentations and number of charm quarks which are produced. Even after this suppression, however, these quantities remain larger than the corresponding numbers for calculations where only primary-primary collisions among partons is included along with fragmentation off final state partons. The results indicate the formation of an interacting medium, which is dense enough for the LPM suppression of radiation to set in and yet permits multiple scatterings among partons. The LPM effect plays an important role in moderating the production of charm quarks having low transverse momenta. It also leads to a hardening of their transverse momentum spectra at larger $p_T$. The impact of the LPM effect is found to rise with increasing collision energy and to decrease with increase in rapidity. Before closing, we add that the charm production in $pp$ collisions has been studied in detail using Fixed Order Next to Leading Log (FONLL) calculations [@Cacciari:1998it]. The data at higher LHC energies are generally found to be slightly above the upper limit given by these calculations (see Refs. [@Aaij:2016jht; @Aaij:2013mga; @Aaij:2015bpa; @ALICE:2011aa]). Realizing that our calculations with only primary-primary collisions and fragmentations tend to roughly account for the higher order corrections in a Leading Log Approximation, these studies then suggest additional contributions from multiple scatterings. The precise extent of this contribution and its dependence on some of the parameters, e.g., current mass of the charm quark, $p_T^\text{cut-off}$ and $\mu_0$ remain to be investigated. We do believe, however, that the additional contributions arising due to the multiple scatterings and suject to LPM effect will be there, unless of-course $p_T^\text{cut-off}$ and $\mu_0$ are taken too large and too few interactions take place and too few radiations occur. In brief, our results provide an indication of emergence of a dense and interacting medium of partons in $pp$ collisions at LHC energies due to semi-hard pQCD interactions, even when LPM suppression of radiation of gluons from scattered partons is accounted for. Acknowledgments {#acknowledgments .unnumbered} =============== DKS gratefully acknowledges the support by the Department of Atomic Energy. This research was supported in part by the ExtreMe Matter Institute EMMI at the GSI Helmholtzzentrum für Schwerionenforschung, Darmstadt, Germany. DKS also acknowledges valuable discussions with Jurgen Schukraft and Dirk Rischke. SAB acknowledges support by US Department of Energy grant DE-FG02-05ER41367. [10]{} C. Ratti, J. Phys. Conf. Ser.  [**736**]{}, no.1, 012001 (2016). doi:10.1088/1742-6596/736/1/012001 C. Ratti, J. Phys. Conf. Ser.  [**779**]{}, no.1, 012016 (2017). doi:10.1088/1742-6596/779/1/012016 C. Ratti, arXiv:1804.07810 \[hep-lat\]. B. Schenke, S. Jeon and C. Gale, Phys. Rev. Lett.  [**106**]{}, 042301 (2011) doi:10.1103/PhysRevLett.106.042301 \[arXiv:1009.3244 \[hep-ph\]\]. C. Gale, S. Jeon, B. Schenke, P. Tribedy and R. Venugopalan, Phys. Rev. Lett.  [**110**]{}, no.1, 012302 (2013) doi:10.1103/PhysRevLett.110.012302 \[arXiv:1209.6330 \[nucl-th\]\]. C. Shen, Z. Qiu, H. Song, J. Bernhard, S. Bass and U. Heinz, Comput. Phys. Commun.  [**199**]{}, 61 (2016) doi:10.1016/j.cpc.2015.08.039 \[arXiv:1409.8164 \[nucl-th\]\]. J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu and U. Heinz, Phys. Rev. C [**94**]{}, no.2, 024907 (2016) doi:10.1103/PhysRevC.94.024907 \[arXiv:1605.03954 \[nucl-th\]\]. V. Khachatryan [*et al.*]{} \[CMS Collaboration\], Phys. Lett. [**B765**]{}, 193 (2017) doi:10.1016/j.physletb.2016.12.009 \[arXiv:1606.06198 \[nucl-ex\]\]. J. Adam [*et al.*]{} \[ALICE Collaboration\], Nature Phys.  [**13**]{}, 535 (2017) doi:10.1038/nphys4111 \[arXiv:1606.07424 \[nucl-ex\]\]. D. K. Srivastava, R. Chatterjee and S. A. Bass, Phys. Rev. C [**97**]{}, no.6, 064910 (2018) doi:10.1103/PhysRevC.97.064910 \[arXiv:1801.07482 \[nucl-th\]\]. K. Geiger and B. Muller, Nucl. Phys. [**B369**]{}, 600 (1992). S. A. Bass, B. Muller, and D. K. Srivastava, Phys. Lett. [**B551**]{}, 277 (2003), nucl-th/0207042. G. Altarelli and G. Parisi, Nucl. Phys. [**B126**]{}, 298 (1977). L. D. Landau and I. Pomeranchuk, Dokl. Akad. Nauk Ser. Fiz.  [**92**]{}, 735 (1953), L. D. Landau and I. Pomeranchuk, Dokl. Akad. Nauk Ser. Fiz.  [**92**]{}, 535 (1953), A. B. Midgal, Phys. Rev. [**103**]{}, 1811 (1956). D. K. Srivastava, S. A. Bass and R. Chatterjee, Phys. Rev. C [**96**]{}, 064906 (2017), doi:10.1103/PhysRevC.96.064906, \[arXiv:1705.05542 \[nucl-th\]\]. H. Bethe and W. Heitler, Proc. Roy. Soc. Lond. A [**146**]{}, 83 (1934). doi:10.1098/rspa.1934.0140 T. Renk, S. A. Bass, and D. K. Srivastava, Phys. Lett. [**B632**]{}, 632 (2006), nucl-th/0505059. S. A. Bass, B. Muller, and D. K. Srivastava, Phys. Rev. Lett. [**91**]{}, 052302 (2003), nucl-th/0212103. S. A. Bass, T. Renk and D. K. Srivastava, Nucl. Phys. [**A783**]{}, 367 (2007). doi:10.1016/j.nuclphysa.2006.11.029 T. Sjostrand, S. Mrenna and P. Z. Skands, JHEP [**0605**]{}, 026 (2006) doi:10.1088/1126-6708/2006/05/026 \[hep-ph/0603175\]. J. Uphoff, O. Fochler, Z. Xu and C. Greiner, J. Phys. G [**42**]{}, no. 11, 115106 (2015) doi:10.1088/0954-3899/42/11/115106 \[arXiv:1408.2964 \[hep-ph\]\]. R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. [**B483**]{}, 291 (1997) doi:10.1016/S0550-3213(96)00553-6 \[hep-ph/9607355\]. R. Baier, Y. L. Dokshitzer, A. H. Mueller, S. Peigne and D. Schiff, Nucl. Phys. [**B484**]{}, 265 (1997) doi:10.1016/S0550-3213(96)00581-0 \[hep-ph/9608322\]. R. Baier, Y. L. Dokshitzer, A. H. Mueller and D. Schiff, Nucl. Phys. [**B531**]{}, 403 (1998) doi:10.1016/S0550-3213(98)00546-X \[hep-ph/9804212\]. B. G. Zakharov, JETP Lett.  [**63**]{}, 952 (1996) doi:10.1134/1.567126 \[hep-ph/9607440\]. U. A. Wiedemann, K. C. Zapp and J. Stachel, Nucl. Phys. [**A855**]{}, 285 (2011). doi:10.1016/j.nuclphysa.2011.02.060 C. E. Coleman-Smith, S. A. Bass and D. K. Srivastava, Nucl. Phys. [**A862-863**]{}, 275 (2011) doi:10.1016/j.nuclphysa.2011.05.071 \[arXiv:1101.4895 \[hep-ph\]\]. M. Younus, C. E. Coleman-Smith, S. A. Bass and D. K. Srivastava, Phys. Rev. C [**91**]{}, 024912 (2015) doi:10.1103/PhysRevC.91.024912 \[arXiv:1309.1276 \[nucl-th\]\]. L. Adamczyk [*et al.*]{} \[STAR Collaboration\], Phys. Rev. D [**86**]{}, 072013 (2012), doi:10.1103/PhysRevD.86.072013 \[arXiv:1204.4244 \[nucl-ex\]\]. B. Abelev [*et al.*]{} \[ALICE Collaboration\], JHEP [**1207**]{}, 191 (2012), doi:10.1007/JHEP07(2012)191 \[arXiv:1205.4007 \[hep-ex\]\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], JHEP [**1706**]{}, 147 (2017), doi:10.1007/JHEP06(2017)147 \[arXiv:1610.02230 \[hep-ex\]\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], Nucl. Phys. [**B871**]{}, 1 (2013) doi:10.1016/j.nuclphysb.2013.02.010 \[arXiv:1302.2864 \[hep-ex\]\]. R. Aaij [*et al.*]{} \[LHCb Collaboration\], JHEP [**1603**]{}, 159 (2016); Erratum: \[JHEP [**1609**]{}, 013 (2016)\] ; Erratum: \[JHEP [**1705**]{}, 074 (2017)\], doi:10.1007/JHEP03(2016)159, 10.1007/JHEP09(2016)013, 10.1007/JHEP05(2017)074 \[arXiv:1510.01707 \[hep-ex\]\]. B. Abelev [*et al.*]{} \[ALICE Collaboration\], JHEP [**1201**]{}, 128 (2012), doi:10.1007/JHEP01(2012)128 \[arXiv:1111.1553 \[hep-ex\]\] M. Cacciari, M. Greco and P. Nason, JHEP [**9805**]{}, 007 (1998) doi:10.1088/1126-6708/1998/05/007 \[hep-ph/9803400\].

--- abstract: 'The theoretical foundations of the quantum statistical approach to parton distributions are reviewed together with the phenomenological motivations from a few specific features of Deep Inelastic Scattering data. The chiral properties of QCD lead to strong relations between quarks and antiquarks distributions and automatically account for the flavor and helicity symmetry breaking of the sea. We are able to describe both unpolarized and polarized structure functions in terms of a small number of parameters. The extension to include their transverse momentum dependence will be also briefly considered.' --- **STATISTICAL DESCRIPTION OF THE FLAVOR STRUCTURE OF THE NUCLEON SEA**\ Jacques Soffer *Department of Physics, Temple University Philadelphia, Pennsylvania 19122-6082, USA*\ *E-mail: jacques.soffer@gmail.com*\ Claude Bourrely *Aix-Marseille Université, Département de Physique, Faculté des Sciences de Luminy,\ F-13288 Marseille, Cedex 09, France*\ *E-mail: claude.bourrely@univ-amu.fr*\ Franco Buccella *INFN, Sezione di Napoli, Via Cintia, Napoli, I-80126, Italy*\ *E-mail: buccella@na.infn.it* Basic review on the statistical approach ======================================== Let us first recall some of the basic components for building up the parton distribution functions (PDF) in the statistical approach, as oppose to the standard polynomial type parametrizations, based on Regge theory at low $x$ and counting rules at large $x$. The fermion distributions are given by the sum of two terms [@bbs1], the first one, a quasi Fermi-Dirac function and the second one, a flavor and helicity independent diffractive contribution equal for light quarks. So we have, at the input energy scale $Q_0^2$, $$xq^h(x,Q^2_0)= \frac{AX^h_{0q}x^b}{\exp [(x-X^h_{0q})/\bar{x}]+1}+ \frac{\tilde{A}x^{\tilde{b}}}{\exp(x/\bar{x})+1}~, \label{eq1}$$ $$x\bar{q}^h(x,Q^2_0)= \frac{{\bar A}(X^{-h}_{0q})^{-1}x^{\bar b}}{\exp [(x+X^{-h}_{0q})/\bar{x}]+1}+ \frac{\tilde{A}x^{\tilde{b}}}{\exp(x/\bar{x})+1}~. \label{eq2}$$ It is important to remark that $x$ is indeed the natural variable, since all sum we will use are expressed in terms of $x$. Notice the change of sign of the potentials and helicity for the antiquarks. The parameter $\bar{x}$ plays the role of a [*universal temperature*]{} and $X^{\pm}_{0q}$ are the two [*thermodynamical potentials*]{} of the quark $q$, with helicity $h=\pm$. We would like to stress that the diffractive contribution occurs only in the unpolarized distributions $q(x)= q_{+}(x)+q_{-}(x)$ and it is absent in the valence $q_{v}(x)= q(x) - \bar {q}(x)$ and in the helicity distributions $\Delta q(x) = q_{+}(x)-q_{-}(x)$ (similarly for antiquarks). The [*nine*]{} free parameters [^1] to describe the light quark sector ($u$ and $d$), namely $X_{u}^{\pm}$, $X_{d}^{\pm}$, $b$, $\bar b$, $\tilde b$, $\tilde A$ and $\bar x$ in the above expressions, were determined at the input scale from the comparison with a selected set of very precise unpolarized and polarized Deep Inelastic Scattering (DIS) data [@bbs1]. The additional factors $X_{q}^{\pm}$ and $(X_{q}^{\pm})^{-1}$ come from the transverse momentum dependence (TMD), as explained in Refs. [@bbs6; @bbs5] (See below). For the gluons we consider the black-body inspired expression $$xG(x,Q^2_0)= \frac{A_Gx^{b_G}}{\exp(x/\bar{x})-1}~, \label{eq5}$$ a quasi Bose-Einstein function, with $b_G$, the only free parameter, since $A_G$ is determined by the momentum sum rule. We also assume a similar expression for the polarized gluon distribution $x\Delta G(x,Q^2_0)={\tilde A}_Gx^{{\tilde b}_G}/[\exp(x/\bar{x})-1]$. For the strange quark distributions, the simple choice made in Ref. [@bbs1] was greatly improved in Ref. [@bbs2]. Our procedure allows to construct simultaneously the unpolarized quark distributions and the helicity distributions. This is worth noting because it is a very unique situation. Following our first paper in 2002, new tests against experimental (unpolarized and polarized) data turned out to be very satisfactory, in particular in hadronic collisions, as reported in Refs. [@bbs3; @bbs4]. Some selected results ===================== Let us first come back to the important question of the flavor asymmetry of the light antiquarks. Our determination of $\bar u(x,Q^2)$ and $\bar d(x,Q^2)$ is perfectly consistent with the violation of the Gottfried sum rule, for which we found $I_G= 0.2493$ for $Q^2=4\mbox{GeV}^2$. Nevertheless there remains an open problem with the $x$ distribution of the ratio $\bar d/\bar u$ for $x \geq 0.2$. According to the Pauli principle this ratio should be above 1 for any value of $x$. However, the E866/NuSea Collaboration [@E866] has released the final results corresponding to the analysis of their full data set of Drell-Yan yields from an 800 GeV/c proton beam on hydrogen and deuterium targets and they obtain the ratio, for $Q^2=54\mbox{GeV}^2$, $\bar d/\bar u$ shown in Fig. 1 (Left). Although the errors are rather large in the high $x$ region, the statistical approach disagrees with the trend of the data. Clearly by increasing the number of free parameters, it is possible to build up a scenario which leads to the drop off of this ratio for $x\geq 0.2$. For example this was achieved in Ref. [@Sassot], as shown by the dashed curve in Fig. 1 (Left). There is no such freedom in the statistical approach, since quark and antiquark distributions are strongly related. One way to clarify the situation is, to improve the statistical accuracy on the Drell-Yan yields which seems now possible, since there are new opportunities for extending the measurement of the $\bar {d}(x)/\bar {u}(x)$ ratio to larger $x$ up to $x=0.7$, with the ongoing E906 experiment at the 120 GeV Main Injector at FNAL [@E906] and a proposed experiment at the new 30-50 GeV proton accelerator at J-PARC [@J-PARC]. ![[*Left*]{}: Comparison of the data on $(\bar d / \bar u) (x,Q^2)$ from E866/NuSea at $Q^2=54\mbox{GeV}^2$ [@E866], with the prediction of the statistical model (solid curve) and the set 1 of the parametrization proposed in Ref. [@Sassot] (dashed curve). [*Right*]{}: Theoretical calculations for the ratio $R_W(y,M_W^2)$ versus the $W$ rapidity, at two RHIC-BNL energies. Solid curve ($\sqrt s = 500\mbox{GeV}$) and dashed curve ($\sqrt s = 200\mbox{GeV}$) are the statistical model predictions. Dotted curve ($\sqrt s = 500\mbox{GeV}$) and dashed-dotted curve ($\sqrt s = 200\mbox{GeV}$) are the predictions obtained using the $\bar d(x) / \bar u(x)$ ratio from Ref. [@Sassot]. []{data-label="fi:fig1"}](dbsube866.eps "fig:"){width="6.9cm"} ![[*Left*]{}: Comparison of the data on $(\bar d / \bar u) (x,Q^2)$ from E866/NuSea at $Q^2=54\mbox{GeV}^2$ [@E866], with the prediction of the statistical model (solid curve) and the set 1 of the parametrization proposed in Ref. [@Sassot] (dashed curve). [*Right*]{}: Theoretical calculations for the ratio $R_W(y,M_W^2)$ versus the $W$ rapidity, at two RHIC-BNL energies. Solid curve ($\sqrt s = 500\mbox{GeV}$) and dashed curve ($\sqrt s = 200\mbox{GeV}$) are the statistical model predictions. Dotted curve ($\sqrt s = 500\mbox{GeV}$) and dashed-dotted curve ($\sqrt s = 200\mbox{GeV}$) are the predictions obtained using the $\bar d(x) / \bar u(x)$ ratio from Ref. [@Sassot]. []{data-label="fi:fig1"}](ratiow1.eps "fig:"){width="6.5cm"} Another way is to call for the measurement of a different observable sensitive to $\bar u(x)$ and $\bar d(x)$. One possibility is the ratio of the unpolarized cross sections for the production of $W^+$ and $W^-$ in $pp$ collisions, which will directly probe the behavior of the $\bar d(x) / \bar u(x)$ ratio. Let us recall that if we denote $R_W(y)=(d\sigma^{W^+}/dy)/(d\sigma^{W^-}/dy)$, where $y$ is the $W$ rapidity, we have [@BSc] at the lowest order $$R_W(y,M_W^2)= \frac{u(x_a,M_W^2) \bar d(x_b,M_W^2) + \bar d(x_a,M_W^2) u(x_b,M_W^2)}{d(x_a,M_W^2) \bar u(x_b,M_W^2) + \bar u(x_a,M_W^2) d(x_b,M_W^2)}~, \label{24}$$ where $x_a=\sqrt{\tau}e^y$, $x_b=\sqrt{\tau}e^{-y}$ and $\tau=M_W^2/s$. This ratio $R_W$, such that $R_W(y)=R_W(-y)$, is accessible with a good precision at RHIC-BNL [@BSSW] and at $\sqrt s = 500\mbox{GeV}$ for $y=0$, we have $x_a=x_b=0.16$. So $R_W(0,M_W^2)$ probes the $\bar d(x) / \bar u(x)$ ratio at $x=0.16$. Much above this $x$ value, the accuracy of Ref. [@E866] becomes poor. In Fig. 1 (Right) we compare the results for $R_W$ using two different calculations. In both cases we take the $u$ and $d$ quark distributions obtained from the present analysis, but first we use the $\bar u$ and $\bar d$ distributions of the statistical approach (solid curve in Fig. 1 (Right)) and second the $\bar u$ and $\bar d$ from Ref. [@Sassot] (dashed curve in Fig. 1 (Right)). For $y=\pm 1$, which corresponds to $x_a$ or $x_b$ near 0.43, one sees that the predictions are very different. Notice that the energy scale $M_W^2$ is much higher than in the E866/NuSea data, so one has to take into account the $Q^2$ evolution. At $\sqrt s = 200\mbox{GeV}$ for $y=0$, we have $x_a=x_b=0.40$ and, although the $W^{\pm}$ yield is smaller at this energy, the effect on $R_W(0,M_W^2)$ is strongly enhanced, as seen in Fig. 1 (Right). This is an excellent test, which needs to be revisited and should be done in the near future. ![[*Left*]{}: Comparison of the $g_1^{n}(x)$ data at low $Q^2$ from [@kramer] with the prediction of the statistical model. [*Right*]{}: Predicted parity-violating asymmetries $A_L^{PV}$ for charged-lepton production at BNL-RHIC, through production and decay of $W^{\pm}$ bosons. $y_e$ is the the charged-lepton rapidity and the data points are from Ref. [@surrow] (Taken from [@bbs13]). []{data-label="fi:fig2"}](g1njlab9-7.eps "fig:"){width="6.5cm"} ![[*Left*]{}: Comparison of the $g_1^{n}(x)$ data at low $Q^2$ from [@kramer] with the prediction of the statistical model. [*Right*]{}: Predicted parity-violating asymmetries $A_L^{PV}$ for charged-lepton production at BNL-RHIC, through production and decay of $W^{\pm}$ bosons. $y_e$ is the the charged-lepton rapidity and the data points are from Ref. [@surrow] (Taken from [@bbs13]). []{data-label="fi:fig2"}](alpw.eps "fig:"){width="6.5cm"} The subject of the strange quark distributions is also very important and challenging, in particular because the HERMES Collaboration has presented recently a new data set at variance with the previous one. For lack of space we are unable to cover it here. We now turn to two specific examples of spin-dependent observables to illustrate the predictive power of our approach for helicity quark and antiquark distributions. First, let us consider the neutron structure function $g_1^n(x,Q^2)$ measured in polarized DIS with a neutron target. Although it has been measured extensively by different collaborations, some accurate data obtained at JLab, in the low $Q^2$ region, have been largely ignored so far [@kramer]. In Fig. 2(Left) we compare our predictions with these data, dominated by $\Delta d$ and $\Delta \bar d$ which are negative, and one observes a remarkable agreement. Another example is the helicity asymmetry in the charged-lepton production through production and decay of $W^{\pm}$ bosons. As explained in Ref. [@bbs13], the $W^-$ asymmetry is very sensitive to the sign and magnitude of $\Delta \bar u$ and the succeful results of the statistical approach are displayed in Fig. 2(Right). Transverse momentum dependence of the parton distributions ========================================================== The parton distributions $p_i(x,k^2_T)$ of momentum $k_T$, must obey the momentum sum rule\ $\sum_i \int_0^1dx \int x p_i(x,k^2_T) dk^2_T = 1$. In addition it must also obey the transverse energy sum rule $\sum_i \int_0^1dx \int p_i(x,k^2_T)\frac{k^2_T}{x}dk^2_T =M^2 $. From the general method of statistical thermodynamics we are led to put $p_i(x,k^2_T)$ in correspondance with the following expression $\exp({\frac{-x}{\bar{x}}}+{\frac{-k^2_T}{x \mu^2}})$ , where $\mu^2$ is a parameter interpreted as the transverse temperature. So we have now the main elements for the extension to the TMD of the statistical PDF. We obtain in a natural way the Gaussian shape with [**no**]{} $x,k_T$ factorization, because the quantum statistical distributions for quarks and antiquarks read in this case $$xq^{h}(x,k_T^{2})=\frac{F(x)}{\exp(x-X^{h}_{0q})/\bar{x}+1} \frac{1}{\exp(k^2_T/x\mu^2-Y^{h}_{0q})+1}~,$$ $$x\bar{q}^{h}(x,k_T^{2})=\frac{{\bar F}(x)}{\exp(x+X^{-h}_{0q})/{\bar{x}}+1} \frac{1}{\exp(k^2_T/x\mu^2+Y^{-h}_{0q})+1}~.$$ Here $F(x) = \frac{A x^{b-1}X^{h}_{0q}}{\mbox{ln}(1 + \exp{Y^{h}_{0q}})\mu^2}=\frac{A x^{b-1}}{k\mu^2}$, where $Y^h_{0q}$ are the thermodynamical potentials chosen such that $\mbox{ln}(1 + \exp{Y^{h}_{0q}})=k X^h_{0q}$, in order to recover the factors $X^h_{0q}$ and $(X^h_{0q})^{-1}$, introduced earlier.\ Similarly for $\bar q$ we have $\bar F(x)= \bar A x^{2b-1}/k\mu^2$. The determination of the 4 potentials $Y^h_{0q}$ can be achieved with the choice $k=3.05$. Finally $\mu^2$ will be obtained from the transverse energy sum rule and one finds $\mu^2=0.198\mbox{GeV}^2$. Detailed results are shown in Refs. [@bbs6; @bbs5]. ![The $u$ and $d$ quark helicity distributions versus $x$: $x\Delta q(x)$ ([*dashed line*]{}) and $x\Delta q^{MW}(x)$ ([*solid line*]{}). (Taken from Ref. [@bbs5]).[]{data-label="fi:fig3"}](delta-u.eps "fig:"){width="6.5cm"} ![The $u$ and $d$ quark helicity distributions versus $x$: $x\Delta q(x)$ ([*dashed line*]{}) and $x\Delta q^{MW}(x)$ ([*solid line*]{}). (Taken from Ref. [@bbs5]).[]{data-label="fi:fig3"}](delta-d.eps "fig:"){width="6.5cm"} Before closing we would like to mention an important point. So far in all our quark or antiquark TMD distributions, the label “$h$” stands for the helicity along the longitudinal momentum and not along the direction of the momentum, as normally defined for a genuine helicity. The basic effect of a transverse momentum $k_T \neq 0$ is the Melosh-Wigner rotation, which mixes the components $q^{\pm}$ in the following way $q^{+MW}= \cos^2\theta ~q^+ + \sin^2\theta ~q^- ~~~\mbox{and}~~~q^{-MW}= \cos^2\theta ~q^- + \sin^2\theta ~q^+$, where for massless partons, $\theta = \arctan{(\frac{k_T}{p_0 +p_z})}$, with $p_0 = \sqrt{k_T^2 +p_z^2}$. It vanishes when either $k_T =0$ or $p_z$, the quark longitudinal momentum, goes to infinity. Consequently $q = q^+ + q^-$ remains unchanged since $q^{MW}=q$, whereas we have $\Delta q^{MW}= (\mbox{cos}^2\theta - \mbox{sin}^2\theta) \Delta q$.\ For illustration we display in Fig. 3, $x\Delta q(x)$ and $x\Delta q^{MW}(x)$ for $Q^2 = 2 \mbox{GeV}^2$, which shows the effect of the Melosh-Wigner rotation, mainly in the low $x$ region.\ A new set of PDF is constructed in the framework of a statistical approach of the nucleon. All unpolarized and polarized distributions depend upon a small number of free parameters, with some physical meaning. New tests against experimental (unpolarized and polarized) data on DIS, semi-inclusive DIS and hadronic processes are very satisfactory. It has a good predictive power but some special features remain to be verified, specially in the high $x$ region. The extension to TMD has been achieved and must be checked more accurately together with Melosh-Wigner effects in the low $x$ region, for small $Q^2$.\ [**Acknowledgments**]{}\ JS is grateful to the organizers of DSPIN-13 for their warm hospitality at JINR and for their invitation to present this talk. Special thanks go to Prof. A.V. Efremov for providing a full financial support and for making, once more, this meeting so successful. [99]{} C. Bourrely, F. Buccella and J. Soffer, Eur. Phys. J. [**C23**]{}, (2002) 487. C. Bourrely, F. Buccella and J. Soffer, Phys. Rev. [**D83**]{}, (2011) 074008. C. Bourrely, F. Buccella and J. Soffer, Int. J. of Modern Physics **A28** (2013) 1350026. C. Bourrely, F. Buccella and J. Soffer, Phys. Lett. [**B648**]{}, (2007) 39. C. Bourrely, F. Buccella and J. Soffer, Mod. Phys. Lett. [**A18**]{}, (2003) 771. C. Bourrely, F. Buccella and J. Soffer, Eur. Phys. J. [**C41**]{}, (2005) 327. R.S. Towell, [*et al.*]{}, Phys. Rev. [**D64**]{}, (2001) 052002. A. Daleo, C.A. García Canal, G.A. Navarro and R. Sassot, Int. J. of Modern Physics **A17** (2002) 269. D.F. Geesaman, [*et al.*]{} \[E906 Collaboration\], FNAL Proposal E906, April 1, 2001. J.C. Peng, [*et al.*]{}, hep-ph/0007341. C. Bourrely and J. Soffer, Nucl. Phys. B [**423**]{}, (1994) 329. G. Bunce, N. Saito, J. Soffer and W. Vogelsang, Ann. Rev. Nucl. Part. Scie. [**50**]{}, (2000) 525. K. Kramer, [*et al.*]{}, Phys. Rev. Lett. [**95**]{}, (2005) 142005. C. Bourrely, F. Buccella and J. Soffer, Phys. Lett. [**B726**]{}, (2013) 296. B. Surrow, arXiv:1310.7974 \[hep-ex\]. [^1]: $A$ and $\bar{A}$ are fixed by the following normalization conditions $u-\bar{u}=2$, $d-\bar{d}=1$.

--- author: - 'P.E.J. Nulsen[^1]' - 'B.R. McNamara' title: 'AGN feedback in clusters: shock and sound heating' --- Introduction ============ X-ray observations of cool core clusters and groups have shown that radio outbursts from central AGN have large impacts on their hot atmospheres (reviewed in [@mn07]). Radio outbursts originating near the event horizons of supermassive black holes deposit energy on spatial scales eight orders of magnitude larger. In the absence of a heat source, copious amounts of hot gas would be cooling to low temperatures and forming stars. Remarkably, powers of AGN outbursts are comparable to the powers needed to stop the gas from cooling, signalling that the mechanical power output of the AGN is governed by feedback ([@brm04]; [@df06]; [@rmn06]; [@csw06]; [@ss06]). Cooled or cooling gas can fuel AGN outbursts, while the outbursts heat the gas, affecting the fuel supply for subsequent outbursts. The high incidence of clusters with central cooling times less than 1 Gyr ([@hmr10]) also argues for radio mode AGN feedback. While many processes might heat the ICM and prevent the gas from cooling, without feedback, it is all but impossible to account for the many systems with very short cooling times, while essentially no clusters are undergoing catastrophic cooling ([@mn12]). While the broad outline of the feedback cycle seems simple, very little of the detail is understood. In this article we focus on some processes that may heat the gas on scales comparable to the Bondi radius and larger. Apart from the discussion of viscosity, much of this material has been reviewed more thoroughly by McNamara & Nulsen (2012). We focus on the heating effects of weak shocks in section \[sec:shock\] and sound waves in section \[sec:sound\]. In section \[sec:discuss\] we consider how these processes operate together. Heating and outburst history {#sec:shock} ============================ Adiabatic uplift is ineffective ------------------------------- To prevent the hot gas from cooling and forming stars, the key requirement is a heat source to replace heat lost by radiation. It should be emphasized that adiabatic uplift is ineffective at preventing gas from cooling. Extended filaments of cool, low entropy gas ([e.g. ]{}[@wsm10]; [@gnd11]) and heavy elements shed by evolving stars in the central galaxy ([e.g. ]{}[@mws10]; [@kmc11]) make a strong case for gas uplift in the wakes of radio lobes. Lifting gas outward adiabatically, into regions where the atmospheric pressure is lower, will generally extend its cooling time and so can help to delay the onset of cooling. However, for abundances and temperatures in the relevant range, the effect is modest. For example, for gas with solar abundances, starting at 3 keV and reducing the pressure adiabatically by a factor of $\simeq 88$ would reduce its temperature to 0.5 keV, but only increase its cooling time by $\simeq 36\%$. For gas with 0.5 solar abundances, the increase in cooling time is a little under a factor of 2, still well short of what is required to prevent the gas cooling in the long term ([@mn12]). Furthermore, unless the uplifted gas mixes with it surroundings, raising its entropy, it is negatively buoyant and will fall back to where it came from in about one free-fall time. That is generally much shorter than the cooling time, so the effect of lifting the gas is transient. Metals shed by cluster central galaxies are more extended than their stars, showing that they have diffused outward from where they were shed ([@rcb05]). However, if all the uplifted gas mixed effectively the mean diffusion rate would be far too high, so most of the gas must fall back almost to where it originates. Much of the energy needed to lift the gas is then converted to kinetic energy and dissipated in the gas, providing a channel for heating ([@gnd11]). Weak shocks {#sec:weak} ----------- The thermal energy of the gas within the volume $V$ is $$E_{\rm th} = {3/2} \int_V p\, dV,$$ where $p$ is its pressure. If the energy deposited by an AGN into this volume is comparable to or larger than $E_{\rm th}$, then the fractional pressure increase in $V$ must be large to accommodate the extra energy. That would cause the region affected by the jet to expand supersonically, driving shocks into its surroundings and rapidly extending the region affected by the outburst. Similarly, shocks will be formed if the jet power exceeds $E_{\rm th}$ divided by the sound crossing time of $V$. Thus, the region affected by an outburst must contain a thermal energy prior to the outburst that is significantly larger than the outburst energy and also larger than the jet power times the sound crossing time of the region. Otherwise, shocks are generated. Based on simulations, Morsony [et al.]{} (2010) have found that, under the influence of cluster “weather” due to continuing infall, motion of substructures, etc., the radius of influence of the central AGN scales with the power of its jet, ${P_{\rm jet}}$, as $R_{\rm influence} \propto {P_{\rm jet}}^{1/3}$. The argument here suggests that this cannot be the whole story. A number of systems, such as MS0735.6+7421 ([@mnw05]) and NGC 5813 ([@rfg11]), show symmetric, large-scale shocks fronts that do not appear to be affected significantly by the cluster weather. Furthermore, in NGC 5813 ([@rfg11]) and M87 ([@fjc07]), there is clear evidence of multiple cavities and shocks. The nested shocks seen in these systems almost certainly require large and sustained variations in the power of the jet for their formation, consistent with other evidence of variations in jet power (e.g[.]{} [@wmn07]). Ripples in the Perseus ICM ([@fst06]) also indicate power variations (though sound may be driven in other ways, [e.g. ]{}[@ss09]). Note that the small fluctuations seen at large distances from the cluster centre would have been considerably greater when they were launched in the region near NGC 1275. Power variations with shorter timescales and/or lower amplitude would be too weak to see now due to the combined effects of damping (section \[sec:sound\]) and the decrease in amplitude with increasing radius. It is noteworthy that the two best observed systems with AGN outbursts, M87 and Perseus, show multiple weak shocks. Given the ubiquitous evidence of AGN variability on a broad range of timescales, this is probably the norm. As argued below, the power spectrum of AGN outbursts plays a critical role in AGN feedback by controlling the launching of shocks and sound waves. Although the heating effect of individual weak shocks is minor, their cumulative impact need not be. The changes in thermal and kinetic energy associated with a weak shock may be substantial, but mostly move on with the shock. A small entropy increases, $\Delta S$, that is cubic in the shock strength ([@ll59]), is all that remains. The heat equivalent of the entropy jump is $\Delta Q = T \, \Delta S = E \, \Delta \ln K$, where $K = kT/n_{\rm e}^{2/3}$ is the entropy index and $\Delta \ln K$ is the jump of $\ln K$ in the shock. Expressed as a fraction of the gas thermal energy, $\Delta Q / E = \Delta \ln K$. For the innermost shock in M87, at a radius of $\simeq 0.8$ arcmin (3.7 kpc; [@fjc07]), the Mach number is $\simeq 1.38$, giving an equivalent heat input of only $\Delta Q / E \simeq 0.022$. There is a second shock at about twice the radius and a third shock that is several times more energetic at a radius of $\simeq 3$ arcmin. The shock spacings suggest that shocks of similar strength to the innermost shock are launched every $\sim 2.5$ Myr, while the cooling time of the gas at 0.8 arcmin is $\simeq 250$ Myr. The $\sim100$ shocks launched during one cooling time would add heat $\Delta Q_{\rm tot}/E \simeq 100 \times 0.022 = 2.2$, more than enough to replace the energy radiated. These numbers are indicative only, but they show that repeated weak shocks alone can prevent gas near the centre of M87 from cooling ([@njf07]). A similar argument has been made for weak shock heating in NGC 5813 ([@rfg11]). Weak shocks can prevent gas cooling at the centres of two of the two nearest, best observed systems. The ripples in Perseus may well start as weak shocks launched from near the centre of NGC 1275, where they prevent the gas from cooling too. If the best observed systems are representative, weak shocks can be the primary channel for heating gas near the centres of all systems. Because shock strength declines with distance from the AGN and $\Delta \ln K$ depends steeply on shock strength, weak shocks become less effective at larger radii. However, as the shock strength decreases, sound dissipation increases in relative importance, probably taking over as the main heating channel, as outlined below. The efficiency of weak shock heating, measured by the fraction of shock energy converted to heat, is generally low, so that sound dissipation will often make a greater contribution to the total heating rate. Plasma viscosity and sound heating {#sec:sound} ================================== Plasma viscosity ---------------- Although the magnetic pressure in the ICM is typically only $\sim1\%$ of the gas thermal pressure ([@ct02]), particle Larmor radii are ten or more orders of magnitude smaller than their mean free paths, keeping the particles tied rigidly to the magnetic field. As a consequence, transport processes in the ICM are poorly understood. This issue is worst for thermal conduction, since the heat flux depends on the structure of the magnetic field on scales comparable to the electron mean free path. The field may well vary on scales smaller than this, in which case the heat flux is not calculable in terms of local gas properties. Despite a great deal of work in this area, the role of thermal conduction in the ICM remains highly uncertain. By contrast, viscous stresses are often determined locally. Fluid motions readily push the dynamically insignificant field around and, being frozen in, the field generally varies as the fluid moves. In a collisionless plasma, the magnetic moment, $m v_\perp^2 / (2 B)$, is conserved, where $m$ is the particle mass, the strength of the magnetic field is $B$ and the component of the particle velocity perpendicular to the field is $v_\perp$. Thus changes in the magnetic field cause corresponding changes in the transverse kinetic energy, $m v_\perp^2/2$, making the particle velocity distribution anisotropic. The proton-proton collision time is $\tau_{\rm pp} \simeq 700 (kT)^{3/2} n_{\rm p}^{-1}$ yr, where the temperature, $kT$, is in keV and the proton density, $n_{\rm p}$, is in $\rm cm^{-3}$. Collisional relaxation can generally suppress most of the anisotropy caused by fluid motions, leaving only a small residual effect. Anisotropy in the velocity distribution implies anisotropy in the pressure, which can be expressed as the difference between the transverse and mean pressures, $\Delta = p_\perp - p$. When collisional relaxation is fast compared to the rate of change in $B$, i.e. $\tau_{\rm ii} |dB / dt| \ll B$, where $\tau_{\rm ii}$ is the ion-ion collision time, the pressure anisotropy is given by ([e.g. ]{}[@kbr12]) $$\Delta = \tau_{\rm ii} p_{\rm i} \left({\mathbf{b}}{\mathbf{b}}: \nabla {\mathbf{v}}- {1\over3} \nabla \cdot {\mathbf{v}}\right), \label{eqn:delta}$$ where $p_{\rm i}$ is the ion pressure, ${\mathbf{b}}$ is the direction of the magnetic field and ${\mathbf{v}}$ is the fluid velocity. The viscous stress tensor, which is minus the anisotropic part of the total stress tensor, is then $$\mathbf{T} = \Delta (3 {\mathbf{b}}{\mathbf{b}}- \mathbf{1}),$$ where $\mathbf{1}$ is the $3\times3$ unit matrix. This is the Braginskii (1965) form of the viscous stress tensor for a magnetized plasma. Anisotropy generated by motion parallel and perpendicular to the magnetic field simply reflects work done on or by the components of the particle velocities through the corresponding particle pressures, $p_\perp$ or $p_\parallel = 3 p - 2 p_\perp$. In a uniform magnetic field parallel to the $z$ axis, with gas motions only parallel the field, $T_{zz}$ must have the same value as it does in the absence of a magnetic field. This requires $\tau_{\rm ii} p_{\rm i}$ to equal the viscosity of an unmagnetized plasma, i.e. the “Spitzer” viscosity. Thus, although the viscous stresses depend on the direction of the magnetic field, they are similar in magnitude to the stresses in the absence of a magnetic field. Sound heating {#sec:damp} ------------- Calculation of sound damping for a magnetized ICM is much the same as in the field free case. We assume that the unperturbed magnetic field is uniform and parallel to the $z$ axis. For sound with wavevector $\mathbf{k} = (k_x, k_y, k_z)$, the damping rate of the amplitude is $${1\over6} \nu k^2 \left(1 - 3{k_z^2\over k^2}\right)^2, \label{eqn:damp}$$ where $\nu = \tau_{\rm ii} p_{\rm i} / \rho$ is the kinematic viscosity and $\rho$ is the gas density. For sound waves travelling parallel to the magnetic field ($k_z^2 = k^2$), this is identical to the damping rate with no magnetic field, $2\nu k^2/3$, as we should expect. For sound travelling perpendicular to the field ($k_z = 0$), the damping rate is $\nu k^2 / 6$, a factor of 4 smaller. Note that for $k_z^2 = k^2/3$, i.e. for sound waves inclined at $\simeq 54.7^\circ$ to the magnetic field, the damping rate is zero. In this direction sound waves have the same effect on $p_\perp$ and $p_\parallel$, so they generate no anisotropy, hence no viscous damping. Due to cluster weather, the field is likely to be fairly chaotic, with structure on scales down to 10 kpc or less ([e.g. ]{}[@gdm10]). If the field is isotropic on average, we can average the damping rate (\[eqn:damp\]) over the sphere to obtain the mean value of $2 \nu k^2 / 15$, one fifth of the damping rate with no magnetic field. Fabian [et al.]{} (2005) have shown that viscous damping of sound is a viable mechanism to prevent gas cooling in the Perseus cluster and Abell 2199. Their models used a viscous damping rate that is one half of the mean rate calculated here. As they discuss, the power spectrum of the AGN outbursts is critical for the heating rate, since it controls the spectrum of sound generated by outbursts and the dissipation rate is sensitive to the frequency. Corresponding to the average damping rate, the dissipation length for sound power is $$120 \left(\omega \over \omega_{\rm s}\right)^{-2} \left(n_{\rm e} \over 0.03 \rm\ cm^{-3}\right) \left(kT \over 5 \rm\ keV\right)^{-1}\rm\ kpc, \label{eqn:dislen}$$ where the angular frequency unit, $\omega_{\rm s} = 2.36\times10^{-14}\rm \ s^{-1}$, gives a wavelength 10 kpc for a gas temperature of 5 keV (cf. [@frt05]). This is comparable to the size of cluster cool cores. If the thermal conductivity is close to its field free value, conduction would boost the dissipation rate significantly. However, viscous dissipation alone appears sufficient to heat the ICM in Perseus and Abell 2199. Schekochihin [et al.]{} (2005) note that, when the anisotropy falls outside the range $-2p_B/3 < \Delta < p_B/3$, for $p_B = B^2/(8 \pi)$, the plasma is prone to firehose or mirror instabilities that grow much faster than viscous damping. For $p_B \ll p$, as in clusters, this condition is more restrictive than the one preceding equation (\[eqn:delta\]). It requires $\omega \tau_{\rm ii} A \lesssim 0.5 p_B / p$, where $A$ is the amplitude of the fractional density fluctuations. If the anisotropy did get too large, sound damping could be much faster than the viscous rate and sound heating even more effective — probably too effective, although some sound energy would then be converted to magnetic field rather than being thermalized ([@sck05]). Discussion {#sec:discuss} ========== Variations in the power of the jet launched from the AGN initiate a succession of weak shocks that originate from somewhere near the Bondi radius (disregarding the small amount of gas within the Bondi sphere). The distribution of gas close to the radio lobes is generally far from spherical and the shock strength varies over the surface of spheres ([e.g. ]{}[@fnh05]; [@mjd12]). However, shock strength generally decreases radially. As the shocks weaken, at some point they can be regarded as sound waves. Although the power spectrum of the sound is far from monochromatic, for the purpose of discussion here, we assume that there is a representative frequency, $\omega$, where $2 \pi / \omega$ is typical of the time interval between shocks. Heating by weak shocks (section \[sec:weak\]) depends on the fractional pressure jump, $\delta p / p$, as $(\delta p /p)^3$, whereas sound damping heats the gas at a rate proportional to the square of the sound amplitude. As a result, shock heating is more significant at small radii where the pressure disturbance is larger, but as the disturbance decreases with increasing radius, sound dissipation will overtake it at some point. If the mean kinematic viscosity is $\nu/5$ (section \[sec:damp\]) and the fractional amplitude of the sound pressure disturbance is related to the pressure jump by $A_p = 0.5 \delta p / p$, the ratio of the shock heating rate to the sound heating rate is ([@mn07]) $$\begin{aligned} \lefteqn{{2 s^2 \over \pi \nu \omega} \left(\delta p \over p\right) \simeq 12.6} \nonumber \\ && \left(\omega \over \omega_{\rm s}\right)^{-1} \left(n_{\rm e} \over 0.03 \rm\ cm^{-3}\right) \left(kT \over 5 \rm\ keV\right)^{-3/2} \left(\delta p \over p\right)\end{aligned}$$ in the units of equation (\[eqn:dislen\]). If the entropy or the effective frequency is high (for wavelengths comparable to the proton mean free path or smaller), sound damping might rival shock heating, even for $\delta p/p \sim 1$, but low entropies in cool cores favour weak shock heating. Weak shock heating probably dominates close to the Bondi radius, while sound damping is more effective on larger scales. For both processes, the power spectrum of AGN outbursts plays a central role in determining the heating rate. Clearly, we need to understand how that is governed. Conclusions =========== Low rates of gas cooling and star formation in the central galaxies of many groups and clusters are best explained by feedback from radio AGN. This view is supported by observations of the impacts of radio outbursts in hot atmospheres. Jet powers vary significantly on a wide range of timescales, causing a succession of shocks to be launched from near the radio lobes. The shocks weaken into sound waves on larger scales. For the best observed, nearby systems, weak shocks provide sufficient heat near the AGN to prevent gas from cooling. In contrast to thermal conduction, the Braginskii viscosity generally provides a well-defined, local, viscous stress that only depends on the direction of the magnetic field in the weakly magnetized ICM. It determines a viscous damping rate for sound that is a factor of five smaller than that for unmagnetized plasma when the field is isotropic on average. This level of sound damping is sufficient to prevent gas from cooling throughout the cool core of the Perseus cluster. The combination of weak shock heating on small scales and sound damping on larger scales plausibly provides the primary means by which AGN energy heats the ICM and prevents gas from cooling. If so, the power spectrum of AGN outbursts plays a central role in AGN feedback and needs to be better understood. Acknowledgement {#acknowledgement .unnumbered} =============== This work was supported by NASA grant NAS8-03060. B[î]{}rzan, L., Rafferty, D.A., McNamara, B.R., Wise, M.W., Nulsen, P.E.J.: 2004, 607, 800 Braginskii, S.I.: 1965, Reviews of Plasma Physics 1, 205 Carilli, C.L., Taylor, G.B.: 2002, 40, 319 Croton, D.J., Springel, V., White, S.D.M., [et al.]{}: 2006, 365, 11 Dunn, R.J.H., Fabian, A.C.: 2006, 373, 959 Fabian, A.C., Reynolds, C.S., Taylor, G.B., Dunn, R.J.H.: 2005, 363, 891 Fabian, A.C., Sanders, J.S., Taylor, G.B., [et al.]{}: 2006, 366, 417 Forman, W., Jones, C., Churazov, E., [et al.]{}: 2007, 665, 1057 Forman, W., Nulsen, P., Heinz, S., [et al.]{}: 2005, 635, 894 Gitti, M., Nulsen, P.E.J., David, L.P., McNamara, B.R., Wise, M.W.: 2011, 732, 13 Govoni, F., Dolag, K., Murgia, M., [et al.]{}: 2010, 522, A105 Hudson, D.S., Mittal, R., Reiprich, T.H., [et al.]{}: 2010, 513, A37 Kirkpatrick, C.C., McNamara, B.R., Cavagnolo, K.W.: 2011, 731, L23 Kunz, M.W., Bogdanovi[ć]{}, T., Reynolds, C.S., Stone, J.M.: 2012, 754, 122 Landau, L.D., Lifshitz, E.M.: 1959, “Course of theoretical physics” 6, “Fluid Mechanics,” Oxford: Pergamon Press McNamara, B.R., Nulsen, P.E.J.: 2007, 45, 117 McNamara, B.R., Nulsen, P.E.J.: 2012, New Journal of Physics 14, 055023 McNamara, B.R., Nulsen, P.E.J., Wise, M.W., [et al.]{}: 2005, [Nature ]{}433, 45 Mendygral, P.J., Jones, T.W., Dolag, K.: 2012, 750, 166 Million, E.T., Werner, N., Simionescu, A., [et al.]{}: 2010, 407, 2046 Morsony, B.J., Heinz, S., Br[ü]{}ggen, M., Ruszkowski, M.: 2010, 407, 1277 Nulsen, P.E.J., Jones, C., Forman, W.R., [et al.]{}: 2007, “Heating versus Cooling in Galaxies and Clusters of Galaxies,” ESO Astrophysics Symposia, Berlin: Springer, 210 Rafferty, D.A., McNamara, B.R., Nulsen, P.E.J., Wise, M.W.: 2006, 652, 216 Randall, S.W., Forman, W.R., Giacintucci, S., [et al.]{}: 2011, 726, 86 Rebusco, P., Churazov, E., B[ö]{}hringer, H., Forman, W.: 2005, 359, 1041 Schekochihin, A.A., Cowley, S.C., Kulsrud, R.M., Hammett, G.W., Sharma, P.: 2005, 629, 139 Sijacki, D., Springel, V.: 2006, 366, 397 Sternberg, A., Soker, N.: 2009, 395, 228 Werner, N., Simionescu, A., Million, E.T., [et al.]{}: 2010, 407, 2063 Wise, M.W., McNamara, B.R., Nulsen, P.E.J., Houck, J.C., David, L.P.: 2007, 659, 1153 [^1]:

\ [ **On Sudakov and Soft resummations in QCD**]{}\ [ **V. Ravindran** ]{}\ [ *Harish-Chandra Research Institute, Chhatnag Road, Jhunsi, Allahabad, India.\ *]{} [**ABSTRACT**]{} In this article we extract soft distribution functions for Drell-Yan and Higgs production processes using mass factorisation theorem and the perturbative results that are known upto three loop level. We find that they are maximally non-abelien. We show that these functions satisfy Sudakov type integro differential equations. The formal solutions to such equations and also to the mass factorisation kernel upto four loop level are presented. Using the soft distribution function extracted from Drell-Yan production, we show how the soft plus virtual cross section for the Higgs production can be obtained. We determine the threshold resummation exponents upto three loop using the soft distribution function. 0.3 cm The Drell-Yan(DY) production of di-leptons and Higgs boson production play crucial role in the hadronic colliders. The di-lepton production can not only serve as a luminosity monitor but also provide vital information on physics beyond standard model at present collider Tevatron at Fermi-Lab and future Large Hadron Collider (LHC) which is going to be up at CERN in few years. Higgs production at such colliders will establish the Standard Model(SM) as well as beyond SM Higgs [@Djouadi:2005gi; @Djouadi:2005gj]. From the theoretical side, the DY production of di-leptons and Higgs boson production are known upto Next to Next to leading order(NNLO) level in QCD. For DY at NLO level, see [@Altarelli:1978id] and for the Higgs production at NLO level, see [@Dawson:1990zj; @Djouadi:1991tk; @Spira:1995rr]. The NNLO contribution to DY can be found in [@Matsuura:1987wt; @Matsuura:1988sm; @Hamberg:1990np]. Beyond NLO, the Higgs production cross sections are known only in the large top quark mass limit. For the NNLO soft plus virtual part of the Higgs production, see [@Harlander:2001is; @Catani:2001ic] and the full NNLO for the Higgs production can be found in [@Harlander:2002wh; @Anastasiou:2002yz; @Ravindran:2003um]. Apart from these fixed order results, the resummation programs for the threshold corrections to both DY and Higgs productions have also been very successful [@Sterman:1986aj; @Catani:1989ne]. For next to next to leading logarithmic (NNLL) resummation, see [@Vogt:2000ci; @Catani:2003zt]. Due to several important results at three loop level that are available in recent times [@Moch:2004pa]-[@Blumlein:2004xt], the resummation upto $N^3LL$ has also become reality [@Moch:2005ky; @Laenen:2005uz; @Idilbi:2005ni]. With all these new results in both fixed order as well as resummed calculations, one is now able to unravel the interesting structures in the perturbative results (for example: [@Blumlein:2000wh; @Blumlein:2005im; @Dokshitzer:2005bf]). Along this line, in this paper, we extract the soft distribution functions of Drell-Yan and Higgs production cross sections in perturbative QCD and show that they do not depend on the process under consideration. By that we mean that the soft distribution function of Drell-Yan production can be got entirely from the Higgs production by a simple multiplication of the colour factor $C_F/C_A$. We prove this for the pole parts upto three loop level and for the finite part we could show only to those terms that are not proportional to $\delta(1-z)$ because the three loop finite part proportional to $\delta(1-z)$ is not available yet and can be obtained only from the explicit fixed order computation of bremsstrahlung contribution. The extraction of the soft distribution function is achieved with the help of mass factorisation theorem supplemented by the recent developments in the computation of three loop anomalous dimensions, three loop form factors of quark and gluon operators and two loop bremsstrahlung contributions to Drell-Yan and Higgs productions. We discuss the consequences of our observation in the determination of soft plus virtual cross sections and the threshold resummation exponents. A brief account on the soft and jet distribution functions and the resummation exponents relevant for deep inelastic scattering (DIS) is given. We start by writing the partonic cross section as $$\begin{aligned} \hspace{-1cm} \hat \sigma^{sv}_I(z,q^2,\mu_R^2)&=& \Big(Z^I(\hat a_s,\mu_R^2,\mu^2)\Big)^2~ |\hat F^I\left(\hat a_s,Q^2,\mu^2\right)|^2~ \delta(1-z)\otimes {\cal C} e^{\displaystyle{2 ~ \Phi^I\left(\hat a_s, q^2,\mu^2,z\right)}}, \nonumber\\[2ex] && \quad \quad \quad \hspace{9cm} I=q,g\end{aligned}$$ with the normalisation, $\hat \sigma^{sv}_{I,born}=\delta(1-z)$. The symbol $sv$ means that we restrict to only the soft and virtual contributions to the partonic cross sections $\hat \sigma^{sv}_I$. In the above equation we have introduced a “${\cal C}$ ordered exponential” which has the following expansion: $$\begin{aligned} {\cal C}e^{\displaystyle f(z) }= \delta(1-z) + {1 \over 1!} f(z) +{1 \over 2!} f(z) \otimes f(z) + {1 \over 3!} f(z) \otimes f(z) \otimes f(z) + \cdot \cdot \cdot\end{aligned}$$ The function $f(z)$ is a distribution of the kind $\delta(1-z)$ and ${\cal D}_i$, where $$\begin{aligned} {\cal D}_i=\Bigg[{\ln^i(1-z) \over (1-z)}\Bigg]_+ \quad \quad \quad i=0,1,\cdot\cdot\cdot\end{aligned}$$ and the symbol $\otimes$ means the Mellin convolution. The letters $q$ and $g$ stand for Drell-Yan(DY) and Higgs(H) productions respectively. $q^2$($=-Q^2$) is the invariant mass of the final state (di-lepton pair in the case of DY and single Higgs boson for the Higgs production). $z$ is the scaling variable defined as the ratio of $q^2$ over $\hat s$, where $\hat s$ is the center of mass of the partonic system. $F^I(\hat a_s,Q^2,\mu^2)$ are the form factors that enter in the Drell-Yan(for $I=q$) and Higgs(for $I=g$) production cross sections. The functions $\Phi^I(\hat a_s,q^2,\mu^2,z)$ are called the soft distribution functions. The unrenormalised(bare) strong coupling constant $\hat a_s$ is defined as $$\begin{aligned} \hat a_s={\hat g^2_s \over 16 \pi^2}\end{aligned}$$ where $\hat g_s$ is the strong coupling constant which is dimensionless in $n=4+{\mbox{$\varepsilon$}}$, with $n$ being the number of space time dimensions. The scale $\mu$ comes from the dimensional regularisation in order to make the bare coupling constant $\hat g_s$ dimensionless in $n$ dimensions. The bare coupling constant $\hat a_s$ is related to renormalised one by the following relation: $$\begin{aligned} S_{{\mbox{$\varepsilon$}}} \hat a_s = Z(\mu_R^2) a_s(\mu_R^2) \left(\mu^2 \over \mu_R^2\right)^{{\mbox{$\varepsilon$}}\over 2} \label{renas}\end{aligned}$$ The scale $\mu_R$ is the renormalisation scale at which the renormalised strong coupling constant $a_s(\mu_R)$ is defined. $$\begin{aligned} S_{{\mbox{$\varepsilon$}}}=exp\left\{{{\mbox{$\varepsilon$}}\over 2} [\gamma_E-\ln 4\pi]\right\}\end{aligned}$$ is the spherical factor characteristic of $n$-dimensional regularisation. The fact that $\hat a_s$ is independent of the choice of $\mu_R$ leads to the following renormalisation group equation (RGE) for the coupling constant: $$\begin{aligned} \mu_R^2 {d \ln a_s(\mu_R^2) \over d \mu_R^2} ={{\mbox{$\varepsilon$}}\over 2} + {1 \over a_s(\mu_R^2)}~ \beta(a_s(\mu_R^2))\end{aligned}$$ where $$\begin{aligned} \beta(a_s(\mu_R^2)) &=& -a_s(\mu_R^2)~\mu_R^2 {d \ln Z(\mu_R^2) \over d \mu_R^2} =-\sum_{i=0}^\infty a_s^{i+2}(\mu_R^2)~ \beta_i\end{aligned}$$ The solution to the above equation is given by $$\begin{aligned} Z(\mu_R^2)= 1+ a_s(\mu_R^2) {2 \beta_0 \over {\mbox{$\varepsilon$}}} + a_s^2(\mu_R^2) \Bigg({4 \beta_0^2 \over {\mbox{$\varepsilon$}}^2 }+ {\beta_1 \over {\mbox{$\varepsilon$}}} \Bigg) + a_s^3(\mu_R^2) \Bigg( {8 \beta_0^3 \over {\mbox{$\varepsilon$}}^3} +{14 \beta_0 \beta_1 \over 3 {\mbox{$\varepsilon$}}^2} +{2 \beta_2 \over 3 {\mbox{$\varepsilon$}}}\Bigg)\end{aligned}$$ The renormalisation constant $Z(\mu_R^2)$ relates the bare coupling constant $\hat a_s$ to the renormalised one $a_s(\mu_R^2)$ through the eqn.(\[renas\]). The coefficients $\beta_0$ and $\beta_1$ are $$\begin{aligned} \beta_0&=&{11 \over 3 } C_A - {4 \over 3 } T_F n_f \nonumber \\ \beta_1&=&{34 \over 3 } C_A^2-4 T_F n_f C_F -{20 \over 3} T_F n_f C_A\end{aligned}$$ where the color factors for $SU(N)$ QCD are given by $$\begin{aligned} C_A=N,\quad \quad \quad C_F={N^2-1 \over 2 N} , \quad \quad \quad T_F={1 \over 2}\end{aligned}$$ and $n_f$ is the number of active flavours. In the case of the Higgs production, the number of active flavours is five because the top degrees of freedom is integrated out in the large $m_{top}$ limit. The factors $Z^I(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})$ are the overall operator renormalisation constants. For the vector current $Z^q(\hat a_s,\mu_R^2,\mu^2)=1$, but the gluon operator gets overall renormalisation [@Chetyrkin:1997un] given by $$\begin{aligned} Z^g(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})&=& 1+\hat a_s \left({\mu_R^2 \over \mu^2}\right)^{{\mbox{$\varepsilon$}}\over 2} S_{{\mbox{$\varepsilon$}}} ~\Bigg[{2 \beta_0 \over {\mbox{$\varepsilon$}}}\Bigg] +\hat a_s^2 \left({\mu_R^2 \over \mu^2}\right)^{{\mbox{$\varepsilon$}}} S_{{\mbox{$\varepsilon$}}}^2 ~\Bigg[{2 \beta_1 \over {\mbox{$\varepsilon$}}} \Bigg] \nonumber\\[2ex] && +\hat a_s^3 \left ({\mu_R^2 \over \mu^2}\right)^{3{{\mbox{$\varepsilon$}}\over 2}} S_{{\mbox{$\varepsilon$}}}^3~ \Bigg[ {1 \over {\mbox{$\varepsilon$}}^2}\Big(-2 \beta_0 \beta_1 \Big) +{2 \beta_2 \over {\mbox{$\varepsilon$}}}\Bigg]\end{aligned}$$ The bare form factors $\hat F^I(\hat a_s,Q^2,\mu^2)$ (before performing overall renormalisation) of both fermionic and gluonic operators satisfy the following integro differential equation that follows from the gauge as well as renormalisation group invariances [@Sudakov:1954sw; @Mueller:1979ih; @Collins:1980ih; @Sen:1981sd]. In dimensional regularisation, $$\begin{aligned} Q^2{d \over dQ^2} \ln \hat {F^I}\left(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}}\right)&=& {1 \over 2 } \Bigg[K^I\left(\hat a_s,{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\right) + G^I\left(\hat a_s,{Q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\right) \Bigg] \label{sud1}\end{aligned}$$ where $K^I$ contains all the poles in ${\mbox{$\varepsilon$}}$. On the other hand, $G^I$ collects rest of the terms that are finite as ${\mbox{$\varepsilon$}}$ becomes zero. In other words $G^I$ contains only non-negative powers of ${\mbox{$\varepsilon$}}$. Since $\hat F^I$ is RG invariant, we find $$\begin{aligned} \mu_R^2 {d \over d\mu_R^2} K^I\Bigg(\hat a_s,{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\Bigg)=-A^I(a_s(\mu_R^2)) \nonumber\\[2ex] \mu_R^2 {d \over d\mu_R^2} G^I\Bigg(\hat a_s,{Q^2\over \mu_R^2}, {\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\Bigg)=A^I(a_s(\mu_R^2))\end{aligned}$$ The quantities $A^I$ are the standard cusp anomalous dimensions and they are expanded in powers of renormalised strong coupling constant $a_s(\mu_R^2)$ as $$\begin{aligned} A^I(\mu_R^2)=\sum_{i=1}^\infty a_s^{i}(\mu_R^2)~ A_i^I\end{aligned}$$ The total derivative is given by $$\begin{aligned} \mu_R^2 {d \over d\mu_R^2} = \mu_R^2 {\partial \over \partial \mu_R^2} +{d a_s(\mu_R^2) \over d\mu_R^2} {\partial \over \partial a_s(\mu_R^2)}\end{aligned}$$ The RGE of $K^I$ can be solved in powers of bare coupling constant $\hat a_s$ as $$\begin{aligned} K^I\left(\hat a_s,{\mu_R^2\over \mu^2},{\mbox{$\varepsilon$}}\right) =\sum_{i=1}^\infty \hat a_s^i \left({\mu_R^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}S^i_{{\mbox{$\varepsilon$}}}~ K^{I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ where, $$\begin{aligned} K^{I,{1}}({\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}} \Bigg(- 2 A_1^I\Bigg) \nonumber\\[2ex] K^{I,{2}}({\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^2} \Bigg(2 \beta_0 A_1^I\Bigg) +{1 \over {\mbox{$\varepsilon$}}}\Bigg(- A_2^I\Bigg) \nonumber\\[2ex] K^{I,{3}}({\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^3} \Bigg(-{8 \over 3} \beta_0^2 A_1^I\Bigg) +{1 \over {\mbox{$\varepsilon$}}^2} \Bigg({2 \over 3} \beta_1 A_1^I +{8 \over 3} \beta_0 A_2^I \Bigg) +{1 \over {\mbox{$\varepsilon$}}} \Bigg(-{2 \over 3} A_3^I \Bigg) \nonumber\\[2ex] K^{I,{4}}({\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^4} \Bigg(4 \beta_0^3 A_1^I\Bigg) +{1 \over {\mbox{$\varepsilon$}}^3} \Bigg(-{8 \over 3} \beta_0 \beta_1 A_1^I - 6 \beta_0^2 A_2^I \Bigg) \nonumber\\[2ex] && +{1 \over {\mbox{$\varepsilon$}}^2} \Bigg({1 \over 3} \beta_2 A_1^I +\beta_1 A_2^I + 3 \beta_0 A_3^I \Bigg) +{1 \over {\mbox{$\varepsilon$}}} \Bigg(-{1 \over 2} A_4^I\Bigg)\end{aligned}$$ Similarly RGE for $G^I$ can also be solved and the solution is found to be $$\begin{aligned} G^I\left(\hat a_s,{Q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},{\mbox{$\varepsilon$}}\right) &=& G^I \left(a_s(\mu_R^2),{Q^2 \over \mu_R^2},{\mbox{$\varepsilon$}}\right) \nonumber\\[2ex] &=&G^I\left(a_s(Q^2),1,{\mbox{$\varepsilon$}}\right)+ \int_{Q^2 \over \mu_R^2}^1 {d\lambda^2 \over \lambda^2} A^I\left(a_s(\lambda^2 \mu_R^2)\right)\end{aligned}$$ The integral in the above equation can be performed and it is found to be $$\begin{aligned} \int_{Q^2 \over \mu_R^2}^1 {d \lambda^2 \over \lambda^2} A^I\left(a_s(\lambda^2 \mu_R^2)\right) &=& \sum_{i=1}^\infty \hat a_s^i \left({\mu_R^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}} \left[\left({Q^2 \over \mu_R^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}-1\right] S^i_{{\mbox{$\varepsilon$}}}~ K^{I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ The finite function $G^I(a_s(Q^2),1,{\mbox{$\varepsilon$}})$ can also be expanded in powers of $a_s(Q^2)$ as $$\begin{aligned} G^I(a_s(Q^2),1,{\mbox{$\varepsilon$}})=\sum_{i=1}^\infty a_s^i(Q^2)~ G^{I}_i({\mbox{$\varepsilon$}})\end{aligned}$$ After substituting these solutions in the eqn.(\[sud1\]) and performing the final integration, we obtain the following solution $$\begin{aligned} \ln \hat F^I(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}}) =\sum_{i=1}^\infty \hat a_s^i \left({Q^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}S^i_{{\mbox{$\varepsilon$}}}~ \hat {\cal L}_F^{I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ where $$\begin{aligned} \hat {\cal L}_ F^{I,(1)}&=&{1\over {\mbox{$\varepsilon$}}^2} \Bigg(-2 A_1^I\Bigg) +{1 \over {\mbox{$\varepsilon$}}} \Bigg(G_1^I({\mbox{$\varepsilon$}})\Bigg) \nonumber\\[2ex] \hat {\cal L}_ F^{I,(2)}&=&{1\over {\mbox{$\varepsilon$}}^3} \Bigg(\beta_0 A_1^I\Bigg) +{1\over {\mbox{$\varepsilon$}}^2} \Bigg(-{1 \over 2} A_2^I - \beta_0 G_1^I({\mbox{$\varepsilon$}})\Bigg) +{1 \over 2 {\mbox{$\varepsilon$}}} G_2^I({\mbox{$\varepsilon$}}) \nonumber\\[2ex] \hat {\cal L}_ F^{I,(3)}&=& {1\over {\mbox{$\varepsilon$}}^4} \Bigg(-{8 \over 9}\beta_0^2 A_1^I\Bigg) + {1\over {\mbox{$\varepsilon$}}^3} \Bigg({2 \over 9} \beta_1 A_1^I +{8 \over 9} \beta_0 A_2^I +{4 \over 3} \beta_0^2 G_1^I({\mbox{$\varepsilon$}})\Bigg) \nonumber\\[2ex] && +{1\over {\mbox{$\varepsilon$}}^2} \Bigg(-{2 \over 9} A_3^I -{1 \over 3} \beta_1 G_1^I({\mbox{$\varepsilon$}}) -{4 \over 3}\beta_0 G_2^I({\mbox{$\varepsilon$}})\Bigg) +{1 \over {\mbox{$\varepsilon$}}}\Bigg({1 \over 3} G_3^I({\mbox{$\varepsilon$}})\Bigg) \nonumber\\[2ex] \hat {\cal L}_F^{I,(4)}&=& {1\over {\mbox{$\varepsilon$}}^5} \Bigg(\beta_0^3 A_1^I\Bigg) +{1 \over {\mbox{$\varepsilon$}}^4} \Bigg(-{2 \over 3} \beta_0 \beta_1 A_1^I -{3 \over 2}\beta_0^2 A_2^I -2 \beta_0^3 G_1^I({\mbox{$\varepsilon$}})\Bigg) \nonumber\\[2ex] && +{1 \over {\mbox{$\varepsilon$}}^3} \Bigg({1 \over 12} \beta_2 A_1^I +{1 \over 4} \beta_1 A_2^I + {3 \over 4}\beta_0 A_3^I +{4 \over 3} \beta_0 \beta_1 G_1^I({\mbox{$\varepsilon$}}) +3\beta_0^2 G_2^I({\mbox{$\varepsilon$}})\Bigg) \nonumber\\[2ex] && +{1 \over {\mbox{$\varepsilon$}}^2} \Bigg(-{1 \over 8} A_4^I -{1 \over 6} \beta_2 G_1^I({\mbox{$\varepsilon$}}) -{1 \over 2} \beta_1 G_2^I({\mbox{$\varepsilon$}}) -{3\over 2} \beta_0 G_3^I({\mbox{$\varepsilon$}})\Bigg) +{1 \over {\mbox{$\varepsilon$}}} \Bigg({1 \over 4} G_4^I({\mbox{$\varepsilon$}})\Bigg)\end{aligned}$$ The above result is in agreement with [@Moch:2005id], which was evaluated using various algorithms designed for solving nested sums. The cusp anomalous dimensions $A^I_i$ and $G^{I}_i({\mbox{$\varepsilon$}})$ are known upto order $a_s^3$. The cusp anomalous dimensions are maximally non-abelien and hence satisfy the following relation: $$\begin{aligned} A^q={ C_F \over C_A }~A^g \end{aligned}$$ The coefficients $G^{I}_i({\mbox{$\varepsilon$}})$ can be found for both $I=q$ and $I=g$ in [@Moch:2005tm] to the required accuracy in ${\mbox{$\varepsilon$}}$. They satisfy $$\begin{aligned} G^{I}_1({\mbox{$\varepsilon$}})&=& 2 (B^I_1 - \delta_{I,g} \beta_0) + f_1^I +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k g^{~I,k}_1 \nonumber \\[2ex] G^{I}_2({\mbox{$\varepsilon$}})&=& 2(B_2^I-2 \delta_{I,g} \beta_1) + f_2^I -2 \beta_0 g^{~I,1}_1 +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k g^{~I,k}_2 \nonumber \\[2ex] G^{I}_3({\mbox{$\varepsilon$}})&=& 2 (B_3^I - 3\delta_{I,g} \beta_2) + f_3^I -2 \beta_1 g^{~I,1}_1 -2 \beta_0 \Big(g^{~I,1}_2+2 \beta_0 g^{~I,2}_1\Big) \nonumber \\[2ex] && +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k g^{~I,k}_3\end{aligned}$$ The constants $B_i^I$ are also known upto order $a_s^3$ thanks to the recent computation of three loop anomalous dimensions/splitting functions [@Moch:2004pa; @Vogt:2004mw]. The constants $f_i^I$ are analogous to the cusp anomalous dimensions $A_i^I$ that enter the form factors. It was first noticed in [@Ravindran:2004mb] that the single pole (in ${\mbox{$\varepsilon$}}$) of the logarithm of form factors upto two loop level ($a_s^2$) can be predicted due the presence of these constants $f_i^I$ because they are found to be maximally non-abelien obeying the relation $$\begin{aligned} f_i^q={C_F\over C_A} f_i^g\end{aligned}$$ similar to $A_i^I$. In [@Moch:2005tm], this relation has been found to hold even at the three loop level. The partonic cross sections $\hat \sigma^{sv}_I(z,q^2,\mu_R^2)$ is UV finite after the coupling constant and overall operator renormalisations are performed using $Z(\mu_R^2)$ and $Z^I(\mu_R^2)$. But they still require mass factorisation in order to remove the collinear divergences: $$\begin{aligned} \hat \sigma^{sv}_I(z,q^2,\mu_R^2,{\mbox{$\varepsilon$}})= \Gamma^T(z,\mu_F^2,{\mbox{$\varepsilon$}})\otimes \Delta^{sv}_{I}\left(z,q^2,\mu_R^2,\mu_F^2\right) \otimes\Gamma(z,\mu_F^2,{\mbox{$\varepsilon$}})\end{aligned}$$ with $\mu_F$ being the factorisation scale. The resulting coefficient functions $\Delta^{sv}_{I}(z,q^2,\mu_R^2,\mu_F^2)$ are finite and free of collinear singularities. $$\begin{aligned} \Delta^{sv}_{I}(z,q^2,\mu_R^2,\mu_F^2)= \delta(1-z)+\sum_{i=1}^{\infty}a_s^i(\mu_R^2)~ \Delta_{I}^{sv,(i)} \left(z,q^2,\mu_R^2,\mu_F^2\right)\end{aligned}$$ The coefficient functions $\Delta_{I}^{sv,(i)}$ for $i=1,2$ are known(see [@Dawson:1990zj] to [@Ravindran:2003um]). The partial result for $\Delta_{I}^{sv,(3)}$ (i.e., all ${\cal D}_i$ except $\delta(1-z)$ are known for $i=3$) is also available(see [@Moch:2005ky]). The kernel $\Gamma(z,\mu_F^2,{\mbox{$\varepsilon$}})$ satisfies the following renormalisation group equation: $$\begin{aligned} \mu_F^2 {d \over d\mu_F^2}\Gamma(z,\mu_F^2,{\mbox{$\varepsilon$}})={1 \over 2} P \left(z,\mu_F^2\right) \otimes \Gamma \left(z,\mu_F^2,{\mbox{$\varepsilon$}}\right)\end{aligned}$$ The $P(z,\mu_F^2)$ are well known Altarelli-Parisi splitting functions(matrix valued) known upto three loop level [@Moch:2004pa; @Vogt:2004mw]: $$\begin{aligned} P(z,\mu_F^2)= \sum_{i=1}^{\infty}a_s^i(\mu_F^2) P^{(i-1)}(z)\end{aligned}$$ The diagonal terms of splitting functions $P^{(i)}(z)$ have the following structure $$\begin{aligned} P^{(i)}_{II}(z) = 2\Bigg[ B^I_{i+1} \delta(1-z) + A^I_{i+1} {\cal D}_0\Bigg] + P_{reg,II}^{(i)}(z)\end{aligned}$$ where $P_{reg,II}^{(i)}$ are regular when the argument takes the kinematic limit(here $z \rightarrow 1$). The RGE of the kernel can be solved in dimensional regularisation in powers of strong coupling constant. Since we are interested only in the soft plus virtual part of the cross section, only the diagonal parts of the kernels contribute. In the $\overline{MS}$ scheme, the kernel contains only poles in ${\mbox{$\varepsilon$}}$. Expanding the kernel in powers of bare coupling $\hat a_s$, $$\begin{aligned} \Gamma(z,\mu_F^2,{\mbox{$\varepsilon$}})=\delta(1-z)+\sum_{i=1}^\infty \hat a_s^i \left({\mu_F^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}S^i_{{\mbox{$\varepsilon$}}} \Gamma^{(i)}(z,{\mbox{$\varepsilon$}})\end{aligned}$$ we can solve the RGE for the kernel. The solutions in the $\overline{MS}$ scheme are given by $$\begin{aligned} \Gamma_{II}^{(1)}(z,{\mbox{$\varepsilon$}})&=&{1 \over {\mbox{$\varepsilon$}}} {P_{II}^{(0)}(z)}\nonumber\\[2ex] \Gamma_{II}^{(2)}(z,{\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^2}\Bigg({1 \over 2} {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}-\beta_0 {P_{II}^{(0)}(z)}\Bigg) +{1 \over {\mbox{$\varepsilon$}}} \Bigg({1 \over 2} {\mbox{$P_{II}^{(1)}(z)$}}\Bigg) \nonumber\\[2ex] \Gamma_{II}^{(3)}(z,{\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^3}\Bigg( {4 \over 3} \beta_0^2 {P_{II}^{(0)}(z)}-\beta_0 {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\nonumber\\[2ex] && +{1 \over 6} {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\Bigg) + {1 \over {\mbox{$\varepsilon$}}^2} \Bigg( {1 \over 2} {P_{II}^{(0)}(z)}\otimes {\mbox{$P_{II}^{(1)}(z)$}}\nonumber\\[2ex] && -{1 \over 3} \beta_1 {P_{II}^{(0)}(z)}-{4 \over 3} \beta_0 {\mbox{$P_{II}^{(1)}(z)$}}\Bigg) +{1 \over {\mbox{$\varepsilon$}}} \Bigg({1 \over 3} {\mbox{$P_{II}^{(2)}(z)$}}\Bigg) \nonumber\\[2ex] \Gamma_{II}^{(4)}(z,{\mbox{$\varepsilon$}})&=& {1 \over {\mbox{$\varepsilon$}}^4}\Bigg( {1 \over 24} {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\nonumber\\[2ex] && -{1 \over 2} \beta_0 {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}+{11 \over 6} \beta_0^2 {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\nonumber\\[2ex] && -2 \beta_0^3 {P_{II}^{(0)}(z)}\Bigg) \nonumber\\[2ex] && +{1 \over {\mbox{$\varepsilon$}}^3}\Bigg( {1 \over 4} {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\otimes {\mbox{$P_{II}^{(1)}(z)$}}-{1 \over 3} \beta_1 {P_{II}^{(0)}(z)}\otimes {P_{II}^{(0)}(z)}\nonumber\\[2ex] && -{11 \over 6} \beta_0 {P_{II}^{(0)}(z)}{\mbox{$P_{II}^{(1)}(z)$}}+{4 \over 3} \beta_0 \beta_1 {P_{II}^{(0)}(z)}+3 \beta_0^2 {\mbox{$P_{II}^{(1)}(z)$}}\Bigg) \nonumber\\[2ex] && +{1 \over {\mbox{$\varepsilon$}}^2}\Bigg( {1\over 3}{P_{II}^{(0)}(z)}\otimes {\mbox{$P_{II}^{(2)}(z)$}}+{1 \over 8} {\mbox{$P_{II}^{(1)}(z)$}}\otimes {\mbox{$P_{II}^{(1)}(z)$}}-{1 \over 6} \beta_2 {P_{II}^{(0)}(z)}\nonumber\\[2ex] && -{1 \over 2} \beta_1 {\mbox{$P_{II}^{(1)}(z)$}}-{3 \over 2} \beta_0 {\mbox{$P_{II}^{(2)}(z)$}}\Bigg) +{1 \over {\mbox{$\varepsilon$}}} \Bigg({1 \over 4} {\mbox{$P_{II}^{(3)}(z)$}}\Bigg)\end{aligned}$$ It is now straightforward to obtain the soft distribution functions $\Phi^I(\hat a_s,q^2,\mu^2,z)$ from the available results known upto three loop level for the form factors $\hat F^I$, the kernels $\Gamma_{II}$ and the coefficient functions $\Delta^{sv}_I$(the $\delta(1-z)$ function part of $\Delta^{sv,(3)}_I$ is still unknown). The fact that $\Delta^{sv}_I$ are finite in the limit ${\mbox{$\varepsilon$}}\rightarrow 0$ implies that the soft distribution functions have pole structure in ${\mbox{$\varepsilon$}}$ similar to that of $\hat F^I$ and $\Gamma_{II}$. Also, $\Phi^I(\hat a_s,q^2,\mu^2,z)$ satisfy the renormalisation group equation: $$\begin{aligned} \mu_R^2 {d \over d\mu_R^2}\Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})=0\end{aligned}$$ From the above observations, it is natural to expect that the soft distribution functions also satisfy Sudakov type integro differential equation that the form factors $\hat F^I(Q^2)$ satisfy(see eqn.(\[sud1\])). Hence, $$\begin{aligned} q^2 {d \over dq^2}\Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}) = {1 \over 2 } \Bigg[\overline K^{~I}\left(\hat a_s,{\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\right) + \overline G^{~I}\left(\hat a_s,{q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\right) \Bigg] \label{sud2}\end{aligned}$$ where again $\overline K^{~I}$ contains all the singular terms and $\overline G^{~I}$ are finite functions of ${\mbox{$\varepsilon$}}$. The renormalisation group invariance leads to $$\begin{aligned} \mu_R^2 {d\over d\mu_R^2} \overline K^{~I} \Bigg(\hat a_s, {\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\Bigg)= -\overline A^{~I}(a_s(\mu_R^2)) \delta(1-z) \nonumber\\[2ex] \mu_R^2 {d \over d\mu_R^2} \overline G^{~I} \Bigg(\hat a_s,{q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\Bigg) =\overline A^{~I}(a_s(\mu_R^2)) \delta(1-z)\end{aligned}$$ If $\Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})$ have to contain the right poles to cancel the poles coming from $\hat F^I$,$Z^I$ and $\Gamma_{II}$ in order to make $\Delta^{sv}_I$ finite, then $\overline A^{~I}$ have to satisfy $$\begin{aligned} \overline A^{~I}=-A^I\end{aligned}$$ The above relation along with the renormalisation group invariance implies that $$\begin{aligned} \overline G^{~I}\left(\hat a_s,{q^2 \over \mu_R^2},{\mu_R^2 \over \mu^2},z,{\mbox{$\varepsilon$}}\right) &=&\overline G^{~I} \left(a_s(\mu_R^2),{q^2 \over \mu_R^2},z,{\mbox{$\varepsilon$}}\right) \nonumber\\[2ex] &=&\overline G^{~I}\left(a_s(q^2),1,z,{\mbox{$\varepsilon$}}\right) - \delta(1-z) \int_{q^2 \over \mu_R^2}^1 {d\lambda^2 \over \lambda^2} A^I\left(a_s(\lambda^2 \mu_R^2)\right)\end{aligned}$$ Now it is now straight forward to determine all $\overline G^{~I}(a_s(q^2),1,z,{\mbox{$\varepsilon$}})$ from the available informations. The functions $\overline G^{~I}(a_s(q^2),1,z,{\mbox{$\varepsilon$}})$ can be expanding in powers of $a_s(q^2)$ as $$\begin{aligned} \overline G^{~I}(a_s(q^2),1,z,{\mbox{$\varepsilon$}})=\sum_{i=1}^{\infty} a_s^i(q^2) ~\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})\end{aligned}$$ The solution to the eqn(\[sud2\]) can be obtained in the way we obtained $\ln \hat F^I(Q^2)$. Expanding the soft distribution functions in powers of bare coupling $\hat a_s$ as $$\begin{aligned} \Phi^I\left(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}\right)=\sum_{i=1}^\infty \hat a_s^i \left({q^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}}S^i_{{\mbox{$\varepsilon$}}} ~\hat \Phi^{I,(i)}(z,{\mbox{$\varepsilon$}})\end{aligned}$$ we find the solution: $$\begin{aligned} \hat \Phi^{I,(i)}(z,{\mbox{$\varepsilon$}}) = \hat {\cal L}_F^{I,(i)}({\mbox{$\varepsilon$}}) \Bigg(A^I\rightarrow -\delta(1-z)~A^I,~~ G^I({\mbox{$\varepsilon$}}) \rightarrow \overline G^{~I}(z,{\mbox{$\varepsilon$}})\Bigg)\end{aligned}$$ The finite functions $\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})$ can be obtained using the mass factorisation formula by demanding the finiteness of the coefficient functions $\Delta^{sv,(i)}_I$. The RG invariance of theses soft functions and the simple rescaling $q \rightarrow (1-z) q$ imply that the following expansion is also the solution to the integro differential equation: $$\begin{aligned} \Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}) &=& \Phi^I(\hat a_s,q^2 (1-z)^2,\mu^2,{\mbox{$\varepsilon$}}) \nonumber\\[2ex] &=&\sum_{i=1}^\infty \hat a_s^i \left({q^2 (1-z)^2 \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}} S_{{\mbox{$\varepsilon$}}}^i \left({i~ {\mbox{$\varepsilon$}}\over 1-z} \right)\hat \phi^{~I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ where $$\begin{aligned} \hat \phi^{I,(i)}({\mbox{$\varepsilon$}})=\hat {\cal L}_F^{I,(i)}({\mbox{$\varepsilon$}}) \Bigg( A^I \rightarrow - A^I, G^I({\mbox{$\varepsilon$}}) \rightarrow \overline {\cal G}^I({\mbox{$\varepsilon$}}) \Bigg)\end{aligned}$$ The $z$ independent constants $\overline {\cal G}^I({\mbox{$\varepsilon$}})$ in $\hat \phi^{~I,(i)}({\mbox{$\varepsilon$}})$ can be obtained using the form factors, mass factorisation kernels and coefficient functions $\Delta^{sv,(i-1)}_I$ expanded in powers of ${\mbox{$\varepsilon$}}$ to the desired accuracy. This is achieved by comparing the poles as well as non-pole terms in ${\mbox{$\varepsilon$}}$ of $\hat \phi^{~I,(i)}({\mbox{$\varepsilon$}})$ with those coming from the form factors, overall renormalisation constants and splitting functions and the lower order $\Delta^{sv,(i-1)}_I$. We find $$\begin{aligned} \overline {\cal G}^{~I}_1({\mbox{$\varepsilon$}})&=&-f_1^I+ \sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \overline {\cal G}^{~I,(k)}_1 \nonumber\\[2ex] \overline {\cal G}^{~I}_2({\mbox{$\varepsilon$}})&=&-f_2^I -2 \beta_0 \overline{\cal G}_1^{~I,(1)} +\sum_{k=1}^\infty{\mbox{$\varepsilon$}}^k \overline {\cal G}^{~I,(k)}_2 \nonumber\\[2ex] \overline {\cal G}^{~I}_3({\mbox{$\varepsilon$}})&=&-f_3^I -2 \beta_1 \overline{\cal G}_1^{~I,(1)} -2 \beta_0 \left(\overline{\cal G}_2^{~I,(1)} +2 \beta_0 \overline{\cal G}_1^{~I,(2)}\right) +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \overline {\cal G}^{~I,(k)}_3\end{aligned}$$ with $$\begin{aligned} \overline{\cal G}^{I,(1)}_1&=& C_I~ \overline{\cal G}^{~(1)}_1 \nonumber\\[2ex] &=&C_I~ \Big(-3 \zeta_2\Big) \nonumber\\[2ex] \overline{\cal G}^{I,(2)}_1&=&C_I~ \overline{\cal G}^{~(2)}_1 \nonumber\\[2ex] &=& C_I~ \Bigg({7 \over 3} \zeta_3\Bigg) \nonumber\\[2ex] \overline{\cal G}^{I,(1)}_2&=&C_I~ \overline{\cal G}^{~(1)}_2 \nonumber\\[2ex] &=& C_I C_A~ \Bigg({2428 \over 81} -{469 \over 9} \zeta_2 +4 \zeta_2^2 -{176 \over 3} \zeta_3\Bigg) \nonumber\\[2ex] && +C_I n_f~ \Bigg(-{328 \over 81} + {70 \over 9} \zeta_2 +{32 \over 3} \zeta_3 \Bigg) \end{aligned}$$ where $C_I=C_F$ for $I=q$(DY) and $C_I=C_A$ for $I=g$(Higgs). Using such compensating $\hat \phi^{I,(i)}({\mbox{$\varepsilon$}})$ and the following expansion, $$\begin{aligned} {1 \over 1-z} \big[(1-z)^2\big]^{i {{\mbox{$\varepsilon$}}\over 2}} ={1 \over i {\mbox{$\varepsilon$}}}\delta(1-z) + \sum_{j=0}^{\infty} { (i {\mbox{$\varepsilon$}})^j \over j!} {\cal D}_j\end{aligned}$$ we obtain $\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})$ upto three loop level. We find that the finite functions $\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})$ have the following decomposition in terms of cusp anomalous dimension $A^I_i$ and $f^I_i$ that appear in the form factors: $$\begin{aligned} \overline G^{{~I}}_1&=& -f_1^I ~\delta(1-z)+2 A_1^I~ {\cal D}_0 +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \overline g^{~I,k}_1(z) \nonumber \\[2ex] \overline G^{{~I}}_2&=& -f_2^I ~\delta(1-z)+2 A_2^I~ {\cal D}_0 -2 \beta_0 ~\overline g^{~I,1}_1(z) +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k ~\overline g^{~I,k}_2(z) \nonumber \\[2ex] \overline G^{{~I}}_3&=& -f_3^I ~\delta(1-z)+2 A_3^I~ {\cal D}_0 -2 \beta_1 ~\overline g^{~I,1}_1(z) -2 \beta_0 \Big(\overline g^{~I,1}_2(z)+2 \beta_0 ~\overline g^{~I,2}_1(z)\Big) \nonumber \\[2ex] && +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k ~\overline g^{~I,k}_3(z)\end{aligned}$$ where $$\begin{aligned} \overline g^{~I,1}_1&=&C_I \Bigg( 8 {\cal D}_1 -3 \zeta_2 \delta(1-z)\Bigg) \nonumber\\[2ex] \overline g^{~I,2}_1&=&C_I \Bigg( -3 \zeta_2 {\cal D}_0 + 4 {\cal D}_2 +{7 \over 3} \zeta_3 \delta(1-z)\Bigg) \nonumber\\[2ex] \overline g^{~I,3}_1&=&C_I \Bigg( {7 \over 3} \zeta_3 {\cal D}_0 - 3 \zeta_2 {\cal D}_1 +{4 \over 3} {\cal D}_3 -{3 \over 16} \zeta_2^2 \delta(1-z)\Bigg) \nonumber\\[2ex] \overline g^{~I,1}_2&=&C_I C_A \Bigg( \Bigg(-{1616 \over 27} +{242 \over 3} \zeta_2 +56 \zeta_3 \Bigg) {\cal D}_0 +\Bigg({1072 \over 9} -32 \zeta_2\Bigg) {\cal D}_1 +\Big(-88\Big) {\cal D}_2 \nonumber\\[2ex] && +\Bigg({2428 \over 81} -{469 \over 9} \zeta_2 +4 \zeta_2^2 -{176 \over 3} \zeta_3\Bigg) \delta(1-z)\Bigg) +C_I n_f \Bigg(\Bigg({224 \over 27} -{44 \over 3} \zeta_2\Bigg) {\cal D}_0 \nonumber\\[2ex] && +\Bigg(-{160\over 9}\Bigg) {\cal D}_1 +16 {\cal D}_2 +\Bigg(-{328 \over 81} + {70 \over 9} \zeta_2 +{32 \over 3} \zeta_3 \Bigg) \delta(1-z)\Bigg) \Bigg) \nonumber\\[2ex] \overline g^{~I,2}_2&=&C_I C_A \Bigg( \Bigg( {4856 \over 81} - {938 \over 9} \zeta_2 +8 \zeta_2^2 -{1210 \over 9} \zeta_3 \Bigg) {\cal D}_0 +\Bigg(-{3232 \over 27}+ {550 \over 3} \zeta_2 +112 \zeta_3\Bigg) {\cal D}_1 \nonumber\\[2ex] && +\Bigg({1072 \over 9} -32 \zeta_2\Bigg) {\cal D}_2 +\Bigg(-{616 \over 9} \Bigg){\cal D}_3 \Bigg) +C_I n_f \Bigg( \Bigg(-{656 \over 81} +{140 \over 9} \zeta_2 +{220 \over 9} \zeta_3 \Bigg) {\cal D}_0 \nonumber\\[2ex] && +\Bigg({448 \over 27} -{100 \over 3} \zeta_2\Bigg) {\cal D}_1 +\Bigg( -{160 \over 9} \Bigg){\cal D}_2 +\Bigg( {112 \over 9}\Bigg) {\cal D}_3\Bigg) \Bigg) \nonumber\\[2ex] && + \delta(1-z) \delta \overline g^{g,2}_2 \nonumber\\[2ex] \overline g^{~I,1}_3&=& C_I C_A^2 \Bigg( \Bigg( -{403861 \over 243}-176 \zeta_2 \zeta_3 +{71584 \over 27} \zeta_2 -{5368 \over 15} \zeta_2^2 +{9272\over 3} \zeta_3 -576 \zeta_5 \Bigg){\cal D}_0 \nonumber\\[2ex] && + \Bigg( {257140 \over 81} -{28696 \over 9} \zeta_2 +{1056 \over 5} \zeta_2^2 -{5632 \over 3} \zeta_3 \Bigg){\cal D}_1 + \Bigg( -{68752 \over 27} +{1760 \over 3} \zeta_2 \Bigg){\cal D}_2 \nonumber\\[2ex] && + \Bigg( {7744 \over 9} \Bigg){\cal D}_3 \Bigg) +C_I C_A n_f\Bigg( \Bigg( {96482 \over 243} -{2452 \over 3} \zeta_2 +{1264 \over 15} \zeta_2^2 -{6536 \over 9} \zeta_3 \Bigg){\cal D}_0 \nonumber\\[2ex] &&+\Bigg( -{72008 \over 81}+{9056 \over 9} \zeta_2 +{448 \over 3} \zeta_3 \Bigg){\cal D}_1 +\Bigg( {22400 \over 27}-{320 \over 3} \zeta_2 \Bigg){\cal D}_2 +\Bigg( -{2816 \over 9} \Bigg){\cal D}_3\Bigg) \nonumber\\[2ex] &&+C_I n_f^2\Bigg( \Bigg( -{4480 \over 243}+{1520 \over 27} \zeta_2 +{416 \over 9} \zeta_3 \Bigg){\cal D}_0 +\Bigg( {4192 \over 81}-{736 \over 9} \zeta_2 \Bigg){\cal D}_1 \nonumber\\[2ex] &&+\Bigg( -{1600 \over 27} \Bigg){\cal D}_2 +\Bigg( {256 \over 9} \Bigg){\cal D}_3 \Bigg) +C_I C_F n_f\Bigg( \Bigg( {1711 \over 9} -60 \zeta_2 -{96 \over 5}\zeta_2^2-{304 \over 3} \zeta_3 \Bigg){\cal D}_0 \nonumber\\[2ex] &&+\Bigg( -220 +192 \zeta_3 \Bigg){\cal D}_1 +\Bigg( 64 \Bigg){\cal D}_2 \Bigg)+\delta\overline g^{g,1}_3 \delta(1-z)\end{aligned}$$ In the above equation $\delta \overline g^{g,2}_2,\delta\overline g^{g,1}_3$ are not known because the full fixed order $N^3LO$ computation for the soft part of the cross section is not available yet. With the available informations(ignoring $\delta \overline g^{g,2}_2,\delta \overline g^{g,1}_3$), we find that the soft distribution functions for DY and Higgs productions are maximally non-abelien: $$\begin{aligned} \Phi^q(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})={C_F \over C_A}~ \Phi^g(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}) \end{aligned}$$ upto three loop level. At the cross section level($\Delta_I^{sv}$), this property does not show up because of the form factors which do not have this property. The overall factors $C_F$ and $C_A$ ordinate from the leading order contributions to the soft distribution functions. Hence if you factor out this colour factor ($C_F$ for the DY and $C_A$ for the Higgs) we find that the soft distribution functions are universal. $$\begin{aligned} \Phi^I(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})=C_I \Phi(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})\end{aligned}$$ The universality of the soft distribution functions can be understood if you notice that the soft part of the cross section is always independent of the spin, colour, flavour or any other quantum numbers after factoring out the born level cross section. It depends only on the gauge interaction, here it is $SU(N)$. This universal property can be utilised to compute soft part of the any new cross section where incoming particles carry any spin,colour,flavour or other quantum numbers. For example, if we know $\Phi(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}})$ extracted from the Drell-Yan production results, we can predict the $\Delta^{sv}_g(z,q^2,\mu_R^2,\mu_F^2)$ for the Higgs production using mass factorisation formula provided we know the gluon form factor $F^g(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}})$ and the overall renormalisation constant $Z^g(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})$. The soft plus virtual part of the cross section ($\Delta^{sv}_I(z,q^2,\mu_R^2,\mu_F^2)$) using mass factorisation formula is found to be $$\begin{aligned} \Delta^{sv}_I(z,q^2,\mu_R^2,\mu_F^2)={\cal C} \exp \Bigg({\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}})}\Bigg)\Bigg|_{{\mbox{$\varepsilon$}}=0}\end{aligned}$$ where $\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}})$ is a finite distribution. The $\mu_R$ dependence comes from the coupling constant and operator renormalisation: $$\begin{aligned} \Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}})&=& \Bigg( \ln \Big(Z^I(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})\Big)^2 +\ln \big|\hat F^I(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}})\big|^2 \Bigg) \delta(1-z) \nonumber\\[2ex] &&+2 C_I \Phi(\hat a_s,q^2,\mu^2,z,{\mbox{$\varepsilon$}}) -2~ {\cal C}\ln \Gamma_{II}(\hat a_s,\mu^2,\mu_F^2,z,{\mbox{$\varepsilon$}})\end{aligned}$$ In the above equation “${\cal C} \ln$” means the “convolution ordered” logarithm. All the products of distributions in the logarithmic expansion are understood as Mellin convolutions. The distribution $\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}})$ is regular as ${\mbox{$\varepsilon$}}\rightarrow 0$. The soft plus virtual cross section can be obtained by expanding $\Psi^I(z,q^2,\mu_R^2,\mu_F^2,{\mbox{$\varepsilon$}}=0)$ as $$\begin{aligned} \Psi^I(z,q^2,\mu_F^2,{\mbox{$\varepsilon$}}) =\sum_{i=1} a_s^i(\mu_F^2) \Psi^{~I,(i)}(z,q^2,\mu_F^2) \end{aligned}$$ where we have set $\mu_R=\mu_F$ and expressing $a_s(\mu_F^2)$ in terms of $a_s(\mu_R^2)$ is straightforward. We find that the cross sections $\Delta^{sv}_I(z,q^2,\mu_F^2)$ can be obtained using $$\begin{aligned} \Delta^{sv,(0)}_I(z,q^2,\mu_F^2)&=&C_I \delta(1-z) \nonumber\\[2ex] \Delta^{sv,(1)}_I(z,q^2,\mu_F^2)&=&\Psi^{~I,(1)}(z,q^2,\mu_F^2) \nonumber\\[2ex] \Delta^{sv,(1)}_I(z,q^2,\mu_F^2)&=&\Psi^{~I,(2)}(z,q^2,\mu_F^2) +{1 \over 2} \Psi^{~I,(1)}(z,q^2,\mu_F^2) \otimes \Psi^{~I,(1)}(z,q^2,\mu_F^2) \nonumber\\[2ex] \Delta^{sv,(3)}_I(z,q^2,\mu_F^2)&=&\Psi^{~I,(3)}(z,q^2,\mu_F^2)+\Psi^{~I,(1)} (z,q^2,\mu_F^2)\otimes \Psi^{~I,(2)}(z,q^2,\mu_F^2) \nonumber\\[2ex] &&+{1 \over 6} \Psi^{~I,(1)}(z,q^2,\mu_F^2) \otimes \Psi^{~I,(1)}(z,q^2,\mu_F^2) \otimes \Psi^{~I,(1)}(z,q^2,\mu_F^2)\end{aligned}$$ where $$\begin{aligned} \Psi^{~I,(1)}&=& \Big((-2 \beta_0 \delta_{I,g}+ 2 B^I_1)\delta(1-z) +2 A^I_1 {\cal D}_0 \Big)\ln\left({q^2 \over \mu_F^2}\right) +\Big(3 \zeta_2 A^I_1 +2 g_1^{~I,1} \nonumber\\[2ex] &&+ 2 C_I \overline{\cal G}_1^{~(1)}\Big)\delta(1-z) + \Big(4 A^I_1\Big){\cal D}_1 \nonumber\\[2ex] \Psi^{~I,(2)}&=& \Bigg[6 \zeta_2 \beta_0 B^I_1 -6 \zeta_2 \beta_0^2 \delta_{I,g} +2 \beta_0 g_1^{~I,2} +g_2^{~I,1} +3 \zeta_2 A^I_2 +2 \beta_0 C_I \overline {\cal G}_1^{~(2)} +C_I \overline {\cal G}_2^{~(1)} \nonumber\\[2ex] && +\Bigg( -4 \beta_1 \delta_{I,g} -2 \beta_0 g_1^{(1)} +2 B^I_2 -3\zeta_2 \beta_0 A^I_1 -2 \beta_0 C_I \overline {\cal G}_1^{~(1)}\Bigg) \ln\left({q^2 \over \mu_F^2}\right) \nonumber\\[2ex] &&+\Big(-\beta_0 B^I_1 +\beta_0^2 \delta_{I,g}\Big) \ln^2\left({q^2 \over \mu_F^2}\right) \Bigg]\delta(1-z) +\Bigg[ -4 \beta_0 C_I \overline {\cal G}_1^{~(1)} -2 f_2^I +2 A^I_2 \ln\left({q^2 \over \mu_F^2}\right) \nonumber\\[2ex] &&-\beta_0 A^I_1 \ln^2\left({q^2 \over \mu_F^2}\right) \Bigg] {\cal D}_0 +\Bigg[4 A^I_2 -4 \beta_0 A^I_1 \ln\left({q^2 \over \mu_F^2}\right) \Bigg]{\cal D}_1 +\Bigg[-4 \beta_0 A^I_1 \Bigg]{\cal D}_2 \nonumber\\[2ex] \Psi^{~I,(3)}&=& \Bigg[-30 \zeta_2 \beta_0 \beta_1\delta_{I,g} +12\zeta_2 \beta_0 B^I_2 -12 \zeta_2 \beta_0^2 g_1^{~I,(1)} + 6 \zeta_2 \beta_1 B^I_1 +{4 \over 3} \beta_0 g_2^{~I,(2)} +{8 \over 3} \beta_0^2 g_1^{~I,(3)} \nonumber\\[2ex] &&+{4 \over 3} \beta_1 g_1^{~I,(2)} +{2 \over 3} g_3^{~I,(1)} +3 \zeta_2 A^I_3 -3 \zeta_2^2 \beta_0^2 A^I_1 +{4 \over 3} \beta_0 C_I \overline {\cal G}_2^{(2)} +{8 \over 3} \beta_0^2 C_I \overline {\cal G}_1^{~(3)} \nonumber\\[2ex] && +{4 \over 3} \beta_1 C_I \overline {\cal G}_1^{~(2)} +{2 \over 3} C_I \overline {\cal G}_3^{~(1)} +6 \zeta_2 \beta_0 f^I_2 + \Bigg( 12 \zeta_2 \beta_0^3\delta_{I,g} -6 \beta_2\delta_{I,g} +2 B^I_3 -12 \zeta_2 \beta_0^2 B^I_1 \nonumber\\[2ex] &&-2 \beta_0 g_2^{~I,(1)} -4 \beta_0^2 g_1^{~I,(2)} -2 \beta_1 g_1^{~I,(1)} -6 \zeta_2\beta_0 A^I_2 -3 \zeta_2 \beta_1 A_1^I -2 \beta_0 C_I \overline {\cal G}_2^{~(1)} \nonumber\\[2ex] &&-4 \beta_0^2 C_I \overline {\cal G}_1^{~(2)} -2 \beta_1 C_I \overline {\cal G}_1^{~(1)} \Bigg)\ln\left({q^2 \over \mu_F^2}\right) + \Bigg( 5 \beta_0 \beta_1 \delta_{I,g} -2 \beta_0 B^I_2 -\beta_1 B^I_1 +2 \beta_0^2 g_1^{~I,(1)} \nonumber\\[2ex] &&+3 \zeta_2 \beta_0^2 A^I_1 +2 \beta_0^2 C_I \overline {\cal G}_1^{~(1)} \Bigg)\ln^2 \left({q^2 \over \mu_F^2}\right) +\Bigg({2 \over 3 } \beta_0^2 B^I_1 -{2 \over 3}\beta_0^3 \delta_{I,g}\Bigg) \log^3\left({q^2 \over \mu_F^2}\right) \Bigg] \delta(1-z) \nonumber\\[2ex] &&+\Bigg[ -4 \beta_0 C_I \overline {\cal G}_2^{~(1)} -8 \beta_0^2 C_I \overline {\cal G}_1^{~(2)} -4 \beta_1 C_I \overline {\cal G}_1^{~(1)} -2 f_3^I +\Bigg(4 \beta_0 f_2^I +8 \beta_0^2 C_I \overline {\cal G}_1^{~(1)} \nonumber\\[2ex] &&+2 A^I_3 \Bigg) \ln\left({q^2 \over \mu_F^2}\right) +\Bigg(-2 \beta_0 A^I_2 -\beta_1 A^I_1 \Bigg) \ln^2 \left({q^2 \over \mu_F^2}\right) +\Bigg({2 \over 3} \beta_0^2 A^I_1 \Bigg) \ln^3 \left({q^2 \over \mu_F^2}\right)\Bigg]{\cal D}_0 \nonumber\\[2ex] &&+\Bigg[ 8 \beta_0 f_2^I +16 \beta_0^2 C_I \overline {\cal G}_1^{~(1)} +4 A_3^I +\Big(-8 \beta_2 A_2^I -4 \beta_1 A^I_1\Big)\ln\left({q^2 \over \mu_F^2}\right) \nonumber\\[2ex] &&+\Big(4 \beta_0^2 A^I_1\Big) \ln^2 \left({q^2 \over \mu_F^2}\right) \Bigg] {\cal D}_1 +\Bigg[ -8 \beta_0 A^I_2 -4 \beta_1 A^I_1 +\Big(8 \beta_0^2 A^I_1\Big) \ln\left({q^2 \over \mu_F^2}\right) \Bigg]{\cal D}_2 \nonumber\\[2ex] &&+\Bigg[ \Bigg({16 \over 3} \beta_0^2 A^I_1 \Bigg) \Bigg]{\cal D}_3\end{aligned}$$ Notice that $\Psi^{~I,(1)}$ and $\Psi^{~I,(2)}$ are completely known. Using our $\Psi^{~I,(1)}$, and $\Psi^{~I,(2)}$ we could successfully reproduce soft plus virtual cross section $\Delta_g^{sv,(i)}(z,q^2,\mu_F^2)$ (see [@Harlander:2001is; @Catani:2001ic]) ($i=1,2$) of the Higgs production from that of DY [@Matsuura:1988sm] and vice versa. To compute $\Delta_I^{sv,(3)}(z,q^2,\mu_F^2)$ (equivalently $\Psi^{~I,(3)}$) we need to know $\overline {\cal G}_2^{~I,(2)}$ and $\overline {\cal G}_3^{~I,(1)}$ either from DY or Higgs production because these constants are maximally non-abelien. Notice that these constants appear only in the coefficient of $\delta(1-z)$ part of $\Psi^{~I,(3)}$. Since the coefficients of ${\cal D}_i$($i=0,1,2,3$) in $\Psi^{~I,(3)}$ do not depend on these unknown constants $\overline {\cal G}_3^{~I,(1)}$ and $\overline {\cal G}_2^{~I,(2)}$, we can predict these coefficients(say for the Higgs production) by using the universal soft distribution function extracted from a process(say DY), the three loop form factors and the renormalisation constants. Our prediction agrees with the partial $N^3LO$ soft plus virtual results [@Moch:2005ky] for DY and Higgs productions. With these available informations one can also determine all the quantities upto $N^3LL$ level in the threshold resummation. To do this, we first recollect that the soft distribution function is renormalisation group invariant. Its UV divergence can be removed by the coupling constant renormalisation. This introduces a renormalisation scale $\mu_R$ which is arbitrary to all orders in perturbation theory. In order to compute various quantities in the threshold resummation formula from the soft distribution function, we choose $\mu_R=\mu_F$. With this choice, one can express the soft distribution function as a sum of pole and finite parts in ${\mbox{$\varepsilon$}}$ as ${\mbox{$\varepsilon$}}\rightarrow 0$, that is $$\begin{aligned} \Phi^I\left(a_s(\mu_F^2),{q^2 \over \mu_F^2},z,{\mbox{$\varepsilon$}}\right) =\Phi^I_{pole}\Bigg(a_s(\mu_F^2),{q^2 \over \mu_F^2},z,{1 \over {\mbox{$\varepsilon$}}}\Bigg) +\Phi^I_{fin}\Bigg(a_s(\mu_F^2),{q^2\over \mu_F^2},z,{\mbox{$\varepsilon$}}\Bigg) \end{aligned}$$ With this decomposition, it is now straightforward to identify the finite part $\Phi^I_{fin}$ with the threshold resummation formula as $$\begin{aligned} 2 \int_0^1 dz ~z^{N-1} \Phi^I_{fin}\Bigg(a_s(\mu_F^2), {q^2 \over \mu_F^2},z,{\mbox{$\varepsilon$}}=0\Bigg) &=&\int_0^1 dz {z^{N-1}- 1\over 1-z} \Bigg[ D^I\Big(a_s\Big(q^2(1-z)^2\Big)\Big) \nonumber\\[2ex] && + 2 \int_{\mu_F^2}^{q^2 (1-z)^2} {d \lambda^2 \over \lambda^2} A^I\Big(a_s(\lambda^2)\Big) \Bigg] \nonumber\\[2ex] && +H^I_S\Bigg(a_s(\mu_F^2),{q^2 \over \mu_F^2}\Bigg) \end{aligned}$$ where the subscript $S$ in $H^I_S$ indicates that it comes from the soft part of the cross section. The remaining contribution comes from the form factor. $D^I(a_s(q^2(1-z)^2))$ can be expanded in powers of bare coupling constant $\hat a_s$ as follows: $$\begin{aligned} D^I\Big(a_s\Big(q^2 (1-z)^2\Big)\Big)=\sum_{i=1}^\infty \hat a_s^i \Bigg({q^2 (1-z)^2 \over \mu^2}\Bigg)^{i {{\mbox{$\varepsilon$}}\over 2}} S_{{\mbox{$\varepsilon$}}}^i~ \hat D^{I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ The finiteness of $D^I$ after coupling constant renormalisation demands that it satisfies the following expansion in ${\mbox{$\varepsilon$}}$: $$\begin{aligned} \hat D^{I,(i)}({\mbox{$\varepsilon$}})=\sum_{j=1-i}^\infty \hat d^{~I,(i)}_{j} {\mbox{$\varepsilon$}}^{j}\end{aligned}$$ Using RG invariance, the coefficients of negative powers of ${\mbox{$\varepsilon$}}$ can be evaluated as $$\begin{aligned} \hat d^{~g,(2)}_{-1}&=& -2 \beta_0 ~\hat d^{~g,(1)}_0 \nonumber\\[2ex] \hat d^{~g,(3)}_{-2}&=& 4 \beta_0^2 ~\hat d^{~g,(1)}_0 \nonumber\\[2ex] \hat d^{~g,(3)}_{-1}&=& -4 \beta_0 ~\hat d^{~g,(2)}_0 -4 \beta_0^2 ~\hat d^{~g,(1)}_1 -\beta_1 ~\hat d^{~g,(1)}_0\end{aligned}$$ We find that for non-negative powers of ${\mbox{$\varepsilon$}}$($j\ge 0$), $$\begin{aligned} \hat d_j^{~I,(1)}&=&2 \overline g_1^{~I,j+1}\Big|_{{\cal D}_0} \nonumber\\[2ex] \hat d_j^{~I,(2)}&=&\Bigg(\overline g_2^{~I,j+1} -2 \beta_0 ~\overline g_1^{~I,j+2} \Bigg)\Big|_{{\cal D}_0} \nonumber\\[2ex] \hat d_j^{~I,(3)}&=&\Bigg({2 \over 3} ~\overline g_3^{~I,j+1} -{8 \over 3} \beta_0 ~\overline g_2^{~I,j+2} -{2 \over 3} \beta_1 ~\overline g_1^{~I,j+2} +{8 \over 3} \beta_0^2 ~\overline g_1^{~I,j+3}\Bigg)\Big|_{{\cal D}_0}\end{aligned}$$ Using the above equations, we find explicitly $$\begin{aligned} \hat d^{~g,(1)}_0&=&0 \nonumber\\[2ex] \hat d^{~g,(1)}_1&=&C_A \Big(- 6 \zeta_2\Big) \nonumber\\[2ex] \hat d^{~g,(1)}_2&=&C_A \Bigg({14 \over 3} \zeta_3 \Bigg) \nonumber\\[2ex] \hat d^{~g,(2)}_0&=&C_A^2 \Bigg( -{1616 \over 27 } +{308 \over 3} \zeta_2 +56 \zeta_3 \Bigg) + C_A n_f \Bigg( {224 \over 27} -{56 \over 3} \zeta_2 \Bigg) \nonumber\\[2ex] \hat d^{~g,(2)}_1&=&C_A^2 \Bigg({4856 \over81} -{938 \over 9} \zeta_2 +8 \zeta_2^2 -{1364 \over 9} \zeta_3 \Bigg) \nonumber\\[2ex] && +C_A n_f\Bigg( -{656 \over 81} +{140 \over 9} \zeta_2 +{248 \over 9} \zeta_3 \Bigg) \nonumber\\[2ex] \hat d^{~g,(3)}_0&=&C_A^3 \Bigg( -{1235050 \over 729}-{352 \over 3} \zeta_2 \zeta_3 +{227548 \over 81} \zeta_2 -{1584 \over 5} \zeta_2^2 +{10376 \over 3} \zeta_3 -384 \zeta_5 \Bigg) \nonumber\\[2ex] && +C_A^2 n_f \Bigg( {328388 \over 729} -{72004 \over 81} \zeta_2 +{352 \over 5} \zeta_2^2 -{26800 \over 27} \zeta_3 \Bigg) \nonumber\\[2ex] && +C_A C_F n_f \Bigg( {3422 \over 27} -44 \zeta_2 -{64 \over 5} \zeta_2^2 -{608 \over 9} \zeta_3 \Bigg) \nonumber\\[2ex] && +C_A n_f^2 \Bigg( -{19456 \over 729} +{1760 \over 27} \zeta_2 +{2080 \over 27} \zeta_3 \Bigg)\end{aligned}$$ The coefficients $\hat d^{~q,(i)}_{j}$ for the DY can be obtained using $$\begin{aligned} \hat d^{~q,(i)}_{j}={C_F \over C_A} ~\hat d^{~g,(i)}_{j}\end{aligned}$$ because the soft distributions functions are maximally non-abelien. Also, the coefficients of $a_s^i (q^2) {\cal D}_0$ in the soft distribution function $\Phi^I_{fin}$ are related to the coefficients $D_i^I$ that appear in threshold resummation formula. Hence it is straightforward to obtain $D^I_i$ from the soft distribution function $\Phi^I_{fin}$. We find that $D^I_i$ are related to $\hat d^{~I,(i)}_k$ and hence $\overline g_i^{~I,k}$ as follows: $$\begin{aligned} D^I_1&=& \hat d^{~I,(1)}_0 =2 ~\overline g_1^{~I,1} \Big|_{{\cal D}_0} \nonumber\\[2ex] &=&2 \overline{\cal G}_1^I({\mbox{$\varepsilon$}}=0) \nonumber\\[2ex] D^I_2&=&\hat d^{~I,(2)}_0+2 \beta_0 ~\hat d^{~I,(1)}_1 =\Bigg(\overline g_2^{~I,1} +2 \beta_0 ~\overline g_1^{~I,2}\Bigg)\Big|_{{\cal D}_0} \nonumber\\[2ex] &=&2 \overline{\cal G}_2^I({\mbox{$\varepsilon$}}=0) \nonumber\\[2ex] D^I_3 &=&\hat d^{~I,(3)}_0 +4 \beta_0 ~\hat d^{~I,(2)}_1 +\beta_1 ~\hat d^{~I,(1)}_1 +4 \beta_0^2 ~\hat d^{~I,(1)}_2 \nonumber\\[2ex] &=&\Bigg({2 \over 3} ~\overline g_3^{~I,1} +{4 \over 3} \beta_0 ~\overline g_2^{~I,2} +{4 \over 3} \beta_1 ~\overline g_1^{~I,2} +{8 \over 3} \beta_0^2 ~\overline g_1^{~I,3} \Bigg)\Big|_{{\cal D}_0} \nonumber\\[2ex] &=&2 \overline{\cal G}_3^I({\mbox{$\varepsilon$}}=0)\end{aligned}$$ From the available informations on three loop results, we find that the following result holds $$\begin{aligned} D_i^I&=&2 ~\overline {\cal G}_i^I({\mbox{$\varepsilon$}}=0) \quad \quad \quad \quad i=1,2,3\end{aligned}$$ The fact that $D^I_i$ can be expressed entirely in terms of $\overline G^{~I}_i(z,{\mbox{$\varepsilon$}})$ (i.e., in terms of $\overline g^{~I,k}_i$ or $\overline {\cal G}_i^I({\mbox{$\varepsilon$}}=0)$) which are maximally non-abelien, implies that $D^I_i$ are also maximally non-abelien. Hence, $$\begin{aligned} D^q_i={C_F \over C_A}~ D^g_i \end{aligned}$$ with $$\begin{aligned} D^g_1 & =&0 \nonumber\\[2ex] D^g_2 & =& C_A^2 \left( - {1616\over 27} + {176\over 3}\,\zeta_2 + 56\,\zeta_3 \right) + C_A n_f \left( {224\over 27} - {32\over 3}\,\zeta_2 \right) \nonumber\\[2ex] D^g_3 & =& C_A^3 \Bigg( - {594058 \over 729} + {98224 \over 81}\,\zeta_2 + {40144 \over 27}\,\zeta_3 - {2992 \over 15}\,\zeta_2^2 - {352 \over 3}\,\zeta_2\zeta_3 - 384\,\zeta_5 \Bigg) \nonumber \\[2ex] && + C_A^2 n_f \Bigg( {125252 \over 729} - {29392 \over 81}\,\zeta_2 - {2480 \over 9}\,\zeta_3 + {736 \over 15}\,\zeta_2^2 \Bigg) \nonumber \\[2ex] && + C_A C_F n_f \Bigg( {3422 \over 27} - 32\,\zeta_2 - {608 \over 9}\,\zeta_3 - {64 \over 5}\,\zeta_2^2 \Bigg) + C_A n_f^2 \Bigg( - {3712 \over 729} + {640 \over 27} \,\zeta_2 + {320 \over 27}\,\zeta_3 \Bigg)\end{aligned}$$ The above results are in agreement with [@Moch:2005ky; @Laenen:2005uz; @Idilbi:2005ni]. We also find the resummation exponents $D^I_i$ can be extracted by using the following relations: $$\begin{aligned} D^I_1&=& \Delta_I^{sv,(1)}\Big|_{{\cal D}_0} \nonumber\\[2ex] D^I_2&=& \Bigg(\Delta_I^{sv,(2)} -{1 \over 2} \Delta_I^{sv,(1)}\otimes \Delta_I^{sv,(1)}\Bigg) \Big|_{{\cal D}_0} \nonumber\\[2ex] D^I_3&=& \Bigg(\Delta_I^{sv,(3)} -\Delta_I^{sv,(1)} \otimes \Delta_I^{sv,(2)} +{1 \over 3} \Delta_I^{sv,(1)} \otimes \Delta_I^{sv,(1)} \otimes \Delta_I^{sv,(1)}\Bigg) \Big|_{{\cal D}_0}\end{aligned}$$ where $\Delta_I^{sv,(i)}$ in the above are computed at the scale $\mu_F^2=\mu_R^2=q^2$. From the following convolution identity[@vanNeerven:2001pe] upto irrelevant regular terms(denoted by $\cdot \cdot \cdot$) $$\begin{aligned} {\cal D}_i \otimes {\cal D}_j = d_{ij} \delta(1-z) +\sum_{l=0}^{i+j+1} c_{ij,l} {\cal D}_l +\cdot \cdot \cdot\end{aligned}$$ it is interesting to notice that in order to obtain $D^I_{i}$, it is sufficient to know the coefficients of all ${\cal D}_{l}$ ($l=l_{max}$ to $0$) (that means, we need not know the information on the coefficient of $\delta(1-z)$ function and the regular part of $\Delta_I^{sv,(i)}$) and the complete soft information of $\Delta_I^{sv,(i-1)}$(i.e., the coefficients of all ${\cal D}_i$ and $\delta(1-z)$ are needed). Finally, the coefficient of $\delta(1-z)$ in the resummation formula can be obtained from $\Phi^I_{fin}$ by defining the coupling constant at the scale $\mu_F^2$. The result is $$\begin{aligned} H^I_S\left(a_s(\mu_F^2),{q^2 \over \mu_F^2}\right) =\sum_{i=1}^\infty a_s^i(\mu_F^2) H^{I}_{S,i}\end{aligned}$$ where $$\begin{aligned} H^{g}_{S,1}&=&-3 \zeta_2 + \ln^2\left({q^2 \over \mu_F^2}\right) \nonumber\\[2ex] H^{g}_{S,2}&=& C_A^2 \Bigg(-{164 \over 81} +{35 \over 9} \zeta_2 +{34 \over 9} \zeta_3 +\Bigg(-{8 \over 3} \zeta_2 +{56 \over 27} \Bigg)\ln\left({q^2 \over \mu_F^2}\right) \nonumber\\[2ex]&& -{10 \over 9}\ln^2\left({q^2 \over \mu_F^2}\right) +{2 \over 9}\ln^3\left({q^2 \over \mu_F^2}\right)\Bigg) +C_A n_f \Bigg({1214 \over 81} -{469 \over 18} \zeta_2 +2 \zeta_2^2-{187 \over 9} \zeta_3 \nonumber\\[2ex]&& +\Bigg({44 \over 3} \zeta_2 +14 \zeta_3 - {404 \over 27}\Bigg)\ln\left({q^2 \over \mu_F^2}\right) +\Bigg(-2 \zeta_2 +{67 \over 9}\Bigg) \ln^2\left({q^2 \over \mu_F^2}\right) \nonumber\\[2ex]&& -{11\over 9} \ln^3\left({q^2 \over \mu_F^2}\right)\Bigg)\end{aligned}$$ and $$\begin{aligned} H^{q,(i)}_S={C_F \over C_A}~ H^{g,(i)}_S\end{aligned}$$ The remaining contribution to the exponent comes from the the finite part of form factor. We conclude our discussion on this subject with a brief discussion on the corresponding soft as well as jet distribution functions that appear in deep inelastic scattering. The soft plus virtual coefficient function $c^{sv}_{I,2}(Q^2,z)$ that appear in the hadronic structure function $F_2$ can be expressed as $$\begin{aligned} {\cal C}\ln c_{I,2}^{sv}(Q^2,z)&=& \Bigg( \ln \Big(Z^I(\hat a_s,\mu_R^2,\mu^2,{\mbox{$\varepsilon$}})\Big)^2 +\ln \big|\hat F^I(\hat a_s,Q^2,\mu^2,{\mbox{$\varepsilon$}})\big|^2 \Bigg) \delta(1-z) \nonumber\\[2ex] &&+ 2 \Phi^I_{SJ}(\hat a_s,Q^2,\mu^2,z,{\mbox{$\varepsilon$}}) -{\cal C}\ln \Gamma_{II}(\hat a_s,\mu^2,\mu_F^2,z,{\mbox{$\varepsilon$}})\end{aligned}$$ where $\Phi^I_{SJ}(\hat a_s,Q^2,\mu^2,z,{\mbox{$\varepsilon$}})$ is sum of soft and jet distribution functions. We find that this soft plus jet distribution function also satisfies Sudakov type integro differential equation (see eqn.(\[sud2\])) which can be solved in the same way we solved soft distribution functions. We find that this soft plus jet distribution function $\Phi^I_{SJ}$ can be expressed as $$\begin{aligned} \Phi^I_{SJ}(\hat a_s,Q^2,\mu^2,z,{\mbox{$\varepsilon$}}) & =& \Phi^I_{SJ}(\hat a_s,Q^2 (1-z),\mu^2,{\mbox{$\varepsilon$}}) \nonumber\\[2ex] &=&\sum_{i=1}^\infty \hat a_s^i \left({Q^2 (1-z) \over \mu^2}\right)^{i {{\mbox{$\varepsilon$}}\over 2}} S_{{\mbox{$\varepsilon$}}}^i \left({i~ {\mbox{$\varepsilon$}}\over 2(1-z)} \right) \hat \xi^{~I,(i)}({\mbox{$\varepsilon$}})\end{aligned}$$ where $$\begin{aligned} \hat \xi^{I,(i)}({\mbox{$\varepsilon$}})=\hat {\cal L}_F^{I,(i)}({\mbox{$\varepsilon$}}) \Bigg( A^I \rightarrow - A^I, G^I({\mbox{$\varepsilon$}}) \rightarrow \widetilde {\cal G}^I({\mbox{$\varepsilon$}}) \Bigg)\end{aligned}$$ We find that the constants $\widetilde {\cal G}^I({\mbox{$\varepsilon$}})$ have the following expansion in terms of $B_i^I$, $f^I_i$ and the ${\mbox{$\varepsilon$}}$ dependent part of lower order coefficient functions. $$\begin{aligned} \widetilde {\cal G}^{~q}_1({\mbox{$\varepsilon$}})&=&-(B_1^q+f_1^q)+ \sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \widetilde {\cal G}^{~q,(k)}_1 \nonumber\\[2ex] \widetilde {\cal G}^{~q}_2({\mbox{$\varepsilon$}})&=&-(B_2^q+f_2^q) -2 \beta_0 \widetilde{\cal G}_1^{~q,(1)} +\sum_{k=1}^\infty{\mbox{$\varepsilon$}}^k \widetilde {\cal G}^{~q,(k)}_2 \nonumber\\[2ex] \widetilde {\cal G}^{~q}_3({\mbox{$\varepsilon$}})&=&-(B_3^q+f_3^q) -2 \beta_1 \widetilde{\cal G}_1^{~q,(1)} -2 \beta_0 \left(\widetilde{\cal G}_2^{~q,(1)} +2 \beta_0 \widetilde{\cal G}_1^{~q,(2)}\right) +\sum_{k=1}^\infty {\mbox{$\varepsilon$}}^k \widetilde {\cal G}^{~q,(k)}_3\end{aligned}$$ The $z$ independent constants $\widetilde {\cal G}^{q,(k)}_i$ are computed using the coefficient functions $c^{sv}_{q,2}(z,Q^2)$ known upto three loop level [@vanNeerven:1991nn; @Moch:2005ba]. Recollect that the three loop form factors were obtained from these coefficient functions by demanding the finiteness of the partonic cross sections after mass factorisation and also notice that the method used there is very different from the method presented in this paper. We obtain $$\begin{aligned} \widetilde{\cal G}^{q,(1)}_1&=& C_F~ \Big({7 \over 2}-3 \zeta_2\Big) \nonumber\\[2ex] \widetilde{\cal G}^{q,(2)}_1&=& C_F~ \Bigg(-{7 \over 2} +{9 \over 8} \zeta_2 +{7 \over 3} \zeta_3\Bigg) \nonumber\\[2ex] \widetilde{\cal G}^{q,(1)}_2&=& C_F^2 \Bigg( {9 \over 8} -{41 \over 2}\zeta_2 +{82 \over 5} \zeta_2^2 -6 \zeta_3\Bigg) \nonumber\\[2ex] &&+C_F C_A~ \Bigg({69761 \over 648}-{1961 \over 36} \zeta_2 -{17 \over 5} \zeta_2^2 -40 \zeta_3\Bigg) \nonumber\\[2ex] &&+ C_F n_f\Bigg( -{5569 \over 324} +{163 \over 18} \zeta_2 +4 \zeta_3\Bigg) \end{aligned}$$ Using the following decomposition, $$\begin{aligned} \Phi^I_{SJ}\left(a_s(\mu_F^2),{Q^2 \over \mu_F^2},z,{\mbox{$\varepsilon$}}\right) =\Phi^I_{SJ,pole}\Bigg(a_s(\mu_F^2),{Q^2 \over \mu_F^2},z,{1 \over {\mbox{$\varepsilon$}}}\Bigg) +\Phi^I_{SJ,fin}\Bigg(a_s(\mu_F^2),{Q^2\over \mu_F^2},z,{\mbox{$\varepsilon$}}\Bigg)\end{aligned}$$ it is now straightforward to identify the finite part $\Phi^I_{SJ,fin}$ with the DIS threshold resummation formula as $$\begin{aligned} 2 \int_0^1 dz ~z^{N-1} \Phi^I_{SJ,fin}\Bigg(a_s(\mu_F^2), {Q^2 \over \mu_F^2},z,{\mbox{$\varepsilon$}}=0\Bigg) &=&\int_0^1 dz {z^{N-1}- 1\over 1-z} \Bigg[ B_{DIS}^I\Big(a_s\Big(Q^2(1-z)\Big)\Big) \nonumber\\[2ex] && + \int_{\mu_F^2}^{Q^2 (1-z)} {d \lambda^2 \over \lambda^2} A^I\Big(a_s(\lambda^2)\Big) \Bigg] \nonumber\\[2ex] && +H^I_{SJ,S}\Bigg(a_s(\mu_F^2),{Q^2 \over \mu_F^2}\Bigg)\end{aligned}$$ Using the above equation, we find that the resummation constants $B^q_{DIS,i}$ satisfy the following relation $$\begin{aligned} B^q_{DIS,i}= \widetilde {\cal G}^q_i({\mbox{$\varepsilon$}}=0) \quad \quad \quad i=1,2,3\end{aligned}$$ The resulting $B^q_{DIS,i}$s agree with those given in [@Moch:2005ba]. To summarise, we have extracted the soft distribution function $\Phi^I$ using mass factorisation formula for both Drell-Yan as well as Higgs productions within the framework of perturbative QCD. This is possible now thanks to various three loop results available for the form factors and splitting functions. The $\Phi^I$ is known completely upto two loop level. Except the soft bremsstrahlung contributions proportional to $\delta(1-z)$ (at three loop level), all the other soft terms$({\cal D}_i)$ are known for the soft distribution functions $\Phi^I$ upto three loop level. We have also shown that the soft distribution functions satisfy Sudakov type integro-differential equation that the quark and gluon form factors satisfy. We found that they are process independent. In other words, knowing the soft distribution function of the Drell-Yan process, one can obtain the same for the Higgs production by simply multiplying the colour factor combination $C_A/C_F$. Hence, unlike the cross sections $\Delta^{sv}_I$, these soft distribution functions are maximally non-abelien. Notice that the form factors have this property only at the pole level. Using the universal soft distribution function extracted from DY production, we could reproduce the soft plus virtual cross section for the Higgs production using the gluon form factor and its renormalisation constant. We have also shown how these soft distribution functions are related to threshold resummation formula. We have extracted various coefficients appearing in threshold resummation formula upto three loop level using the soft distribution function. We have also discussed the DIS soft and jet distribution functions in the present context. Acknowledgments: The author would like to thank S. Moch and A. Vogt for providing the constants ${\cal S}_i$ of DY that are computed in their paper [@Moch:2005ky]. [99]{} A. Djouadi, arXiv:hep-ph/0503172. A. Djouadi, arXiv:hep-ph/0503173. G. Altarelli, R. K. Ellis and G. Martinelli, Nucl. Phys. B [**143**]{}, 521 (1978) \[Erratum-ibid. B [**146**]{}, 544 (1978)\]. S. Dawson, Nucl. Phys. B [**359**]{}, 283 (1991). A. Djouadi, M. Spira and P. M. Zerwas, Phys. Lett. B [**264**]{}, 440 (1991). M. Spira, A. Djouadi, D. Graudenz and P. M. Zerwas, Nucl. Phys. B [**453**]{}, 17 (1995) \[arXiv:hep-ph/9504378\]. T. Matsuura and W. L. van Neerven, Z. Phys. C [**38**]{}, 623 (1988). T. Matsuura, S. C. van der Marck and W. L. van Neerven, Nucl. Phys. B [**319**]{}, 570 (1989). R. Hamberg, W. L. van Neerven and T. Matsuura, Nucl. Phys. B [**359**]{}, 343 (1991) \[Erratum-ibid. B [**644**]{}, 403 (2002)\]. R. V. Harlander and W. B. Kilgore, Phys. Rev. D [**64**]{}, 013015 (2001) \[arXiv:hep-ph/0102241\]. S. Catani, D. de Florian and M. Grazzini, JHEP [**0105**]{}, 025 (2001) \[arXiv:hep-ph/0102227\]. R. V. Harlander and W. B. Kilgore, Phys. Rev. Lett.  [**88**]{}, 201801 (2002) \[arXiv:hep-ph/0201206\]. C. Anastasiou and K. Melnikov, Nucl. Phys. B [**646**]{}, 220 (2002) \[arXiv:hep-ph/0207004\]. V. Ravindran, J. Smith and W. L. van Neerven, Nucl. Phys. B [**665**]{}, 325 (2003) \[arXiv:hep-ph/0302135\]. G. Sterman, Nucl. Phys. B [**281**]{}, 310 (1987). S. Catani and L. Trentadue, Nucl. Phys. B [**327**]{}, 323 (1989). A. Vogt, Phys. Lett. B [**497**]{}, 228 (2001) \[arXiv:hep-ph/0010146\]. S. Catani, D. de Florian, M. Grazzini and P. Nason, JHEP [**0307**]{}, 028 (2003) \[arXiv:hep-ph/0306211\]. S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B [**688**]{}, 101 (2004) \[arXiv:hep-ph/0403192\]. A. Vogt, S. Moch and J. A. M. Vermaseren, Nucl. Phys. B [**691**]{}, 129 (2004) \[arXiv:hep-ph/0404111\]. S. Moch, J. A. M. Vermaseren and A. Vogt, JHEP [**0508**]{}, 049 (2005) \[arXiv:hep-ph/0507039\]. S. Moch, J. A. M. Vermaseren and A. Vogt, Phys. Lett. B [**625**]{}, 245 (2005) \[arXiv:hep-ph/0508055\]. S. Moch, J. A. M. Vermaseren and A. Vogt, Nucl. Phys. B [**726**]{}, 317 (2005) \[arXiv:hep-ph/0506288\]. J. Blumlein and J. A. M. Vermaseren, Phys. Lett. B [**606**]{}, 130 (2005) \[arXiv:hep-ph/0411111\]. S. Moch and A. Vogt, Phys. Lett. B [**631**]{}, 48 (2005) \[arXiv:hep-ph/0508265\]. E. Laenen and L. Magnea, arXiv:hep-ph/0508284. A. Idilbi, X. d. Ji, J. P. Ma and F. Yuan, arXiv:hep-ph/0509294. K. G. Chetyrkin, B. A. Kniehl and M. Steinhauser, Nucl. Phys. B [**510**]{}, 61 (1998) \[arXiv:hep-ph/9708255\]. V. V. Sudakov, Sov. Phys. JETP [**3**]{}, 65 (1956) \[Zh. Eksp. Teor. Fiz.  [**30**]{}, 87 (1956)\]. A. H. Mueller, Phys. Rev. D [**20**]{}, 2037 (1979). J. C. Collins, Phys. Rev. D [**22**]{}, 1478 (1980). A. Sen, Phys. Rev. D [**24**]{}, 3281 (1981). V. Ravindran, J. Smith and W. L. van Neerven, Nucl. Phys. B [**704**]{}, 332 (2005) \[arXiv:hep-ph/0408315\]. J. Blumlein, V. Ravindran and W. L. van Neerven, Nucl. Phys. B [**586**]{}, 349 (2000) \[arXiv:hep-ph/0004172\]. J. Blumlein and V. Ravindran, Nucl. Phys. B [**716**]{}, 128 (2005) \[arXiv:hep-ph/0501178\]. Y. L. Dokshitzer, G. Marchesini and G. P. Salam, arXiv:hep-ph/0511302. W. L. van Neerven and A. Vogt, Nucl. Phys. B [**603**]{}, 42 (2001) \[arXiv:hep-ph/0103123\]. W. L. van Neerven and E. B. Zijlstra, Phys. Lett. B [**272**]{}, 127 (1991).

--- abstract: 'We construct an infinite family of two-Lee-weight and three-Lee-weight codes over the non-chain ring $\mathbb{F}_p+u\mathbb{F}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p,$ where $u^2=0,v^2=0,uv=vu.$ These codes are defined as trace codes. They have the algebraic structure of abelian codes. Their Lee weight distribution is computed by using Gauss sums. With a linear Gray map, we obtain a class of abelian three-weight codes and two-weight codes over ${\mathbb{F}}_p$. In particular, the two-weight codes we describe are shown to be optimal by application of the Griesmer bound. We also discuss their dual Lee distance. Finally, an application to secret sharing schemes is given.' address: - 'Anhui University, Hefei, Anhui Province 230039, PR China' - 'Key Laboratory of Intelligent Computing $\&$ Signal Processing, Ministry of Education, Anhui University No. 3 Feixi Road, Hefei Anhui Province 230039, P. R. China, National Mobile Communications Research Laboratory, Southeast University and School of Mathematical Sciences of Anhui University, Anhui, 230601, P. R. China' - 'CNRS/ LAGA, University of Paris 8, 93 526 Saint-Denis' author: - Yan Liu - 'Minjia Shi$^{*}$' - Patrick Solé bibliography: - 'mybibfile.bib' title: 'Two-weight and three-weight codes from trace codes over ${\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$ ' --- weight distribution; Gauss sum; Griesmer bound; secret sharing schemes 94B25,05E30 Introduction ============ Linear codes with few weights draw motivation from secret sharing [@DY2], combinatorial designs, graph theory [@BGH], association schemes and difference sets [@CG; @CW]. In addition, they find engineering applications in consumer electronics, communication and data storage systems. Hence, linear codes with few weights, especially cyclic codes (see [@DY]), have been studied extensively. The determination of their weight distribution leads to difficult arithmetical problems. Constacyclic codes over ${\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$ have been extensively studied as in [@YZK]. This paper is a generalization of our earlier paper [@SLS2; @SLS1; @SWLP]. Here we consider the codes with few weights over the non-chain ring $R={\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$ with $u^2=v^2=uv-vu=0.$ It is an interesting problem to construct trace codes. The objective of this paper is to construct the linear codes over ${\mathbb{F}}_p$ with few weights from the trace codes over an extension ring by using a linear Gray map. These codes turn out to be abelian but possibly not cyclic. This is noteworthy, as most constructions of codes with few weights in literature are based on cyclic codes and cyclotomy [@BH §9.8.5]. Their weight distribution is determined by using exponential character sums. After Gray mapping, we obtain an infinite family of $p$-ary abelian codes with few weights. In particular, the two-weight codes over ${\mathbb{F}}_p$ are shown to be optimal for given length and dimension by the application of the Griesmer bound [@G]. Furthermore, an application to secret sharing schemes is sketched out. The rest of this paper is organized as follows. In Section 2, we define the class of trace codes we are interested in, and present the main results Theorems $1\sim4$ and Proposition 5. The next section briefly introduces some basic notations and definitions, what is more, we show that trace codes we construct are abliean. Section 3 shows that the code which we constructed and their Gray images are abelian. Sections 4 and 5 are devoted to the proof of Theorems $1\sim4$. Section 6 sets up the proof of Proposition 5 and describes an application to secret sharing schemes. Section 7 puts the obtained results into perspective, and makes some conjectures for future research. Statement of main results ========================= Throughout this paper, let $p$ denote an odd prime. Let ${\mathcal{Q}}$ be the set of squares in ${\mathbb{F}}_{p^m}^*$, where ${\mathbb{F}}_{p^m}^*$ denotes the multiplicative group of nonzero elements of ${\mathbb{F}}_{p^m}$. The set of odd order elements in ${\mathbb{F}}_{p^m}^*$ is denoted by ${\mathcal{N}}$. Given a positive integer $m>1$, we can construct the ring extension ${\mathcal{R}}={\mathbb{F}}_{p^m}+u{\mathbb{F}}_{p^m}+v{\mathbb{F}}_{p^m}+uv{\mathbb{F}}_{p^m}$ of $R={\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$ of degree $m$, where $u^2=0,~v^2=0,~uv=vu.$ The group of units in ${\mathcal{R}},$ denoted by ${\mathcal{R}}^*,$ is isomorphic to the direct product ${\mathbb{F}}_{p}^*\otimes{\mathbb{F}}_{p}\otimes{\mathbb{F}}_{p}\otimes{\mathbb{F}}_{p}.$ For any $a\in \mathcal{R}$, the vector $Ev(a)$ is given by the following evaluation map $$Ev(a)=(Tr(ax))_{x\in L },$$ where the definition of $Tr()$ and $L$ are given in next section. Under the above map, we define a code $C(m,p)$ by the formula $C(m,p)=\{Ev(a): a\in \mathcal{R}\}.$ We remark that the definition of this family of linear codes is similar to that [@SWLP]. However, here we consider a different base ring. The main results of this paper are given below. First, we describe the weight distribution in two Theorems, depending on arithmetical conditions bearing on $m$ and $p.$ **Theorem 1.**\[enum\] Assume $m$ is singly-even. Let $\epsilon(p)=(-1)^{\frac{p+1}{2}}.$ For $a\in \mathcal{R}$, the Lee weights of codewords of $C(m,p)$ are as follows. 1. If $a=0$, then $w_L(Ev(a))=0$; 2. If $a=\alpha uv \in M\backslash \{0\}$, where $\alpha\in {\mathbb{F}}_{p^m}^*$, then $$w_L(Ev(a))=\begin{cases} 2(p-1)(p^{4m-1}-\epsilon(p)p^{\frac{7m-2}{2}}),~~~\alpha \in {\mathcal{Q}};\\ 2(p-1)(p^{4m-1}+\epsilon(p)p^{\frac{7m-2}{2}}),~~~\alpha \in {\mathcal{N}}; \end{cases}$$ 3. If $a\in \mathcal{R}\backslash \{\alpha uv : \alpha\in {\mathbb{F}}_{p^m} \}$, then $$w_L(Ev(a))= 2(p-1)(p^{4m-1}-p^{3m-1}).$$ **Theorem 2.** Assume $m$ is odd and $p\equiv 3 \pmod{4}.$ For $a\in \mathcal{R}$, the Lee weight of codewords of $C(m,p)$ is given below. 1. If $a=0$, then $w_L(Ev(a))=0$; 2. If $a=\alpha uv\in M\backslash \{0\}$, where $\alpha\in {\mathbb{F}}_{p^m}^*$, then $w_L(Ev(a))=2(p^{4m}-p^{4m-1});$ 3. If $a\in \mathcal{R}\backslash \{\alpha uv : \alpha\in {\mathbb{F}}_{p^m} \}$, then $ w_L(Ev(a))= 2(p-1)(p^{4m-1}-p^{3m-1}).$ Next, we investigate the dual Lee distance. **Theorem 3.** For all $m> 1,$ the dual Lee distance $d'$ of $C(m,p)$ is $2.$ Notice that a vector $x$ covers a vector $y$ if $s(x)$ contains $s(y),$ where $s(x)$ and $s(y)$ denotes the support $x$ and $y$, respectively. A *minimal codeword* of a given linear code $C$ over ${\mathbb{F}}_p$ is a nonzero codeword that does not cover any other nonzero codeword. However, the problem of determining the minimal codewords of a given linear code is difficult in general. Under the linear Gray map which is defined in subsection 3.2, we study the optimality and support structure of the code $\phi(C(m,p))$, in, respectively, Theorem 4 and Proposition 5. **Theorem 4.** Assume $m$ is odd, and $p\equiv 3 \pmod{4}.$ The code $\phi(C(m,p))$ is optimal. [**Proposition 5.**]{} Under the above theorems, the Gray image $\phi(C(m,p))$ satisfies the following poverties: 1. All the nonzero codewords of $\phi(C(m,p)),$ for $m$ is even and $m> 2$, are minimal. 2. All the nonzero codewords of $\phi(C(m,p)),$ for $m$ is odd and $m\ge 1$, are minimal. Background material =================== In this section, we introduce some preliminary results from three parts as follows.\ [**3.1.  Rings and trace function**]{} There is a Frobenius operator $F()$ which maps $a+bu+cv+duv$ onto $a^p+b^pu+c^pv+d^puv.$ Under the Frobenius operator $F()$, then we can define the following *Trace function*, denoted by $Tr()$, $$Tr()=\sum_{j=0}^{m-1}F^j().$$ Let $tr()$ be the trace function from ${\mathbb{F}}_{p^m}$ to ${\mathbb{F}}_p$, that is, for any $\varepsilon\in {\mathbb{F}}_{p^m},$ $$tr(\varepsilon)=\varepsilon+\varepsilon^p+\cdots+\varepsilon^{p^{m-1}}.$$ Then it is immediate to check that $$Tr(a+bu+cv+duv)=tr(a)+tr(b)u+tr(c)v+tr(d)uv,$$ for $a,b,c,d \in {\mathbb{F}}_{p^m},$ and $R$-linearity of $Tr()$ follows from the ${\mathbb{F}}_p$-linearity of $tr()$. In order to be concise, set $M=\{bu+cv+duv:b,c,d\in {\mathbb{F}}_{p^m}\}$, to denote the unique maximal ideal of $\mathcal{R}.$ The residue field $\mathcal{R}/M$ is isomorphic to ${\mathbb{F}}_{p^m}$, since $M$ is its maximum ideal. By direct computation, we know that ${\mathcal{R}}^*=$ $\{a+bu+cv+duv:a\in {\mathbb{F}}_{p^m}^{*},b,c,d\in {\mathbb{F}}_{p^m}\}.$ It is obvious that ${\mathcal{R}}^*$ is not cyclic and that ${\mathcal{R}}={\mathcal{R}}^*\cup M$. Setting $L=\{a+bu+cv+duv:a\in {\mathcal{Q}},b,c,d\in {\mathbb{F}}_{p^m}\}$. Thus the set $L$ forms a multiplicative subgroup of index $2$ of ${\mathcal{R}}^*$. The following proposition introduces a useful property of the trace function $Tr()$.\ **Proposition 6.** If for all $x\in L,$ we have $Tr(ax)=0,$ then $a=0.$ Write $x=x_0+x_1u+x_2v+x_3uv$ and $a=a_0+a_1u+a_2v+a_3uv,$ with $x_0\in\mathcal{Q}, x_i,a_j\in {\mathbb{F}}_{p^m},~i=1,2,3,j=0,1,2,3.$ Thus $ax=a_0x_0+(a_0x_1+a_1x_0)u+(a_0x_2+a_2x_0)v+(a_0x_3+a_1x_2+a_2x_1+a_3x_0)uv$ and $Tr(ax)=0$ is equivalent to a system of equations, that is, $tr(a_0x_0)=0,tr(a_0x_1+a_1x_0)=0,tr(a_0x_2+a_2x_0)=0$ and $tr(a_0x_3+a_1x_2+a_2x_1+a_3x_0)=0.$ Thus, we are interested in this system of equations with indeterminate $a_i,i=0,1,2,3$. From the nondegenerate character of $tr()$ [@MS], we get $a=0$ by solving these equations. [**3.2.  Codes and Gray map**]{} A [**linear code**]{} $C$ over $R$ of length $n$ is an $R$-submodule of $R^n$. A [**codeword**]{} is any element of the code $C$. For any $x=(x_1,x_2,\dots,x_n), y=(y_1,y_2,\dots,y_n)\in R^n$, their standard inner product is defined by $\langle x,y\rangle=\sum_{i=1}^nx_iy_i$, where the operation is performed in $R$. The [**dual code**]{} of $C$, denoted by $C^\perp$, consists of all vectors of $R^n$ which are orthogonal to every codeword in $C$, i.e., $C^\perp=\{y\in R^n|\langle x,y\rangle =0, \forall x\in C\}.$ We note that the Lee weight of an element $a+bu+cv+duv\in R$ was defined in [@YZK] to be the Hamming weight of the $p$-ary vector $(d,c+d,b+d,a+b+c+d)$. This leads to the Gray map $\phi:~R^n\rightarrow{\mathbb{F}}_p^{4n}$: $$\phi(\mathbf{a}+\mathbf{b}u+\mathbf{c}v+\mathbf{d}uv)=(\mathbf{d},\mathbf{c}+\mathbf{d},\mathbf{b}+\mathbf{d}, \mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d} ),$$ where $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d} \in {\mathbb{F}}_p^n.$ As was observed in [@YZK], $\phi$ is a distance preserving isometry from $(R^n,d_L)$ to $({\mathbb{F}}_p^{4n},d_H)$, where $d_L$ and $d_H$ denote the Lee and Hamming distance in $R^n$ and ${\mathbb{F}}_p^{4n}$, respectively. That means if $C$ is a linear code over $R$ with parameters $(n,p^k,d)$, then $\phi(C)$ is a linear code of parameters $[4n,k,d]$ over ${\mathbb{F}}_p$. Given a finite abelian group $G,$ a code over $R$ is said to be [**abelian**]{} if it is an ideal of the group ring $R[G].$ In other words, the coordinates of $C$ are indexed by elements of $G$ and $G$ acts regularly on this set. In the special case when $G$ is cyclic, the code is a cyclic code in the usual sense [@MS]. Hence, we have the following proposition. **Proposition 7.** The group $L$ acts regularly on the coordinates of $C(m,p).$ For any $v',u' \in L$, the change of variables $ x\mapsto (u'/v')x$ permutes the coordinates of $C(m,p),$ and maps $v'$ to $u'.$ This defines a transitive action on the coordinates of $C(m,p).$ Such a permutation is unique, given $v',u'.$ Hence the action is regular. The code $C(m,p)$ is thus an [*abelian code*]{} with respect to the group $L$ from Proposition 7. It means that $C(m,p)$ is an ideal of the group ring $R[L].$ As observed in the previous subsection, $L$ is a not cyclic group, hence $C(m,p)$ may be not cyclic. The next result shows that the Gray image of $C(m,p)$ is also abelian.\ **Proposition 8.** A finite group of size $4|L|$ acts regularly on the coordinates of $\phi(C(m,p)).$ By the definition of Gray map, $\phi(\mathbf{a}+\mathbf{b}u+\mathbf{c}v+\mathbf{d}uv)=(\mathbf{d},\mathbf{c}+\mathbf{d},\mathbf{b}+\mathbf{d}, \mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}),$ where $\mathbf{a},\mathbf{b},\mathbf{c},\mathbf{d}\in {\mathbb{F}}_p^n$. Now if $\mathbf{a}+\mathbf{b}u+\mathbf{c}v+\mathbf{d}uv \in C(m,p),$ then by linearity, $$(1+u)(\mathbf{a}+\mathbf{b}u+\mathbf{c}v+\mathbf{d}uv)=\mathbf{a}+(\mathbf{a}+\mathbf{b})u +\mathbf{c}v+(\mathbf{c}+\mathbf{d})uv \in C(m,p)$$, $$(1+v)(\mathbf{a}+\mathbf{b}u+\mathbf{c}v+\mathbf{d}uv)=\mathbf{a}+\mathbf{b}u +(\mathbf{a}+\mathbf{c})v+(\mathbf{b}+\mathbf{d})uv \in C(m,p)$$ and $(1+u+v+uv)(\mathbf{a}+\mathbf{b}u+\mathbf{c}v+\mathbf{d}uv)=\mathbf{a}+(\mathbf{a}+\mathbf{b})u +(\mathbf{a}+\mathbf{c})v+(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}) uv\in C(m,p)$. Further $\phi(\mathbf{a}+(\mathbf{a}+\mathbf{b})u +\mathbf{c}v+(\mathbf{c}+\mathbf{d})uv)=(\mathbf{c}+\mathbf{d},\mathbf{d}, \mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d},\mathbf{b}+\mathbf{d})$, $\phi(\mathbf{a}+\mathbf{b}u +(\mathbf{a}+\mathbf{c})v+(\mathbf{b}+\mathbf{d})uv)=(\mathbf{b}+\mathbf{d}, \mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d},\mathbf{d},\mathbf{c}+\mathbf{d})$, and $\phi(\mathbf{a}+(\mathbf{a}+\mathbf{b})u +(\mathbf{a}+\mathbf{c})v+(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d})uv) =(\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d},\mathbf{b}+\mathbf{d},\mathbf{c}+\mathbf{d},\mathbf{d})$, so that $\phi(C(m,p))$ is invariant under an involution that permutes the four parts of a codeword. Thus, $\phi(C(m,p))$ is invariant under the regular action of a group of order $4 \vert L\vert.$ [**3.3.  Character sums**]{} Let $\psi$ denote the canonical additive character of ${\mathbb{F}}_{p^m}.$ Let $\chi$ be an arbitrary multiplicative character of ${\mathbb{F}}_{p^m}.$ The quadratic multiplicative character of ${\mathbb{F}}_{p^m}$ is denoted by $\eta$ which is defined by $$\eta(x)=\begin{cases} 1, ~~~~~\mathrm{if}~x\in{\mathcal{Q}}; \\ -1,~~~\mathrm{if}~x\in {\mathcal{N}}. \end{cases}$$ The classical [**Gauss sum**]{} attached to $\chi$ can be defined as $$G(\chi)=\sum_{x\in {\mathbb{F}}_{p^m}^*}\psi(x)\chi(x).$$ It is well-known that the explicit evaluation of a Gauss sum is a very difficult problem for a general $\chi$. But when the order of $\chi$ equals two, then the explicit values of corresponding Gauss sum are known and are recorded in [@LN Theorem 5.15] as follows: $$\label{im} G(\eta)=\begin{cases} (-1)^{m-1}p^{\frac{m}{2}},~~~~p\equiv 1~(\mathrm{mod}~4); \\ (-1)^{m-1}i^mp^{\frac{m}{2}}, ~p\equiv 3~(\mathrm{mod}~4). \end{cases}$$ In particular, in the case when $m$ is singly-even, we know that $G(\eta)=\epsilon(p)p^{\frac{m}{2}},$ with $\epsilon(p)=(-1)^{\frac{(p+1)}{2}}.$ We now study two character sums $\sum\limits_{x\in {\mathcal{Q}}}\psi(x)$ and $\sum\limits_{x\in {\mathcal{N}}}\psi(x)$, denoted by ${\overline{Q}}$ and ${\overline{N}}$, respectively. It is immediate from the orthogonal property of additive characters which can be found in [@MS Lemma 9, p. 143] that ${\overline{Q}}+{\overline{N}}=-1.$ Further, it can be shown that ${\overline{Q}}=\frac{\epsilon(p)p^{\frac{m}{2}}-1}{2}$ and ${\overline{N}}=-\frac{\epsilon(p)p^{\frac{m}{2}}+1}{2}$ by observing that the characteristic function of ${\mathcal{Q}}$ is $\frac{1+\eta}{2}.$ **** Proof of Theorems 1, 2 and 4 ============================= This section is divided into four parts. First, the statement of some lemmas. Next, the proof of Theorems 1, 2 and 4. For the rest of this paper, for convenience, we adopt the following notations unless stated otherwise in this paper. Let $\omega=\exp(\frac{2\pi i}{p})$ and $N=4|L|=2(p^{4m}-p^{3m})$.\ [ **4.1.  Statement of some lemmas**]{} If $y=(y_1,y_2,\dots,y_N)\in \mathbb{F}_p^N,$ let $$\Theta(y)=\sum_{j=1}^N\omega^{y_j}.$$ For simplicity, we let $\theta(a)=\Theta(\phi(Ev(a))).$ Taking account of the linear property of the Gray map and the evaluation map, we see that $\theta(sa)=\Theta(\phi(Ev(sa))),$ for any $s\in {\mathbb{F}}_p^*.$ Now, we present the following lemmas, which play an important role in the process of proof in Theorems 1, 2 and 4. **Lemma 9.** [@G; @MS Griesmer bound] If $[N,K,d]$ are the parameters of a linear $p$-ary code, then $$\sum_{j=0}^{K-1}\Big\lceil \frac{d}{p^j} \Big\rceil \le N.$$ **Lemma 10.** [@SWLP Lemma 1]\[5.1\] For all $y=(y_1,y_2,\dots,y_N)\in \mathbb{F}_p^N,$ we have $$\sum_{s=1}^{p-1}\Theta(sy)=(p-1)N-pw_H(y).$$\ **Lemma 11.** [@SLS1 Lemma 4.2] If $z \in \mathbb{F}_{p^m}^*,$ then $$\sum\limits_{x\in \mathbb{F}_{p^m}}\omega^{tr(z x)}=0.$$ **Lemma 12.** [@SWLP Lemma 2] Set $\Re$ denotes the real part of complex number. If $p\equiv 3 \pmod{4},$ then $$\sum_{s=1}^{p-1}\theta(sa)=(p-1)\Re(\theta(a)).$$ Combining Lemma 10 and the definition of Gray map, for $Ev(a)\in C(m,p)$, we have $$\begin{aligned} w_L(Ev(a) ) &=& \frac{(p-1)N- \sum\limits_{s=1}^{p-1}\Theta(s\phi(Ev(a)))}{p} \nonumber \\ &=& \frac{(p-1)N-\sum\limits_{s=1}^{p-1} \theta(sa)}{p} .\end{aligned}$$ When $m$ is odd and $p\equiv 3 \pmod{4}$, from Equation (2) and Lemma 12, we deduce that $$\label{3} w_L(Ev(a))=\frac{p-1}{p}(N-\Re(\theta(a))).$$ In the light of Equations (2) and (3), it is easy to see that the value of $\theta(a)$ is the key points in the computation of Lee weight of the codeword $Ev(a)$, where $a\in \mathcal{R}$.\ [**4.2.  Proof of Theorem 1**]{} Setting $x=x_0+x_1u+x_2v+x_3uv,$ where $x_0\in \mathcal{Q},x_1,x_2,x_3\in {\mathbb{F}}_{p^m}$ throughout the process of proof in this theorem. Let $a=a_0+a_1u+a_2v+a_3uv\in \mathcal{R}$, where $a_i \in {\mathbb{F}}_{p^m},~i=0,1,2,3$, then we get $$ax=a_0x_0+(a_0x_1+a_1x_0)u+(a_0x_2+a_2x_0)v+(a_0x_3+a_1x_2+a_2x_1+a_3x_0)uv,$$ and $$\begin{aligned} Tr(ax) &=&tr(a_0x_3+a_1x_2+a_2x_1+a_3x_0)uv+tr(a_0x_2+a_2 x_0)v+tr(a_0x_1+a_1x_0)u+tr(a_0x_0)\\ &=:&D_3uv+D_2v+D_1u+D_0.\end{aligned}$$ Taking Gray map yields $$\begin{aligned} \phi(Ev(a)) &=& (D_3,D_2+D_3,D_1+D_3,D_0+D_1+D_2+D_3)_x \nonumber \\ \nonumber &=& (tr(a_0x_3+a_1x_2+a_2x_1+a_3x_0),tr(a_0x_2+a_2 x_0+a_0x_3+a_1x_2+a_2x_1+a_3x_0), \nonumber \\ &&tr(a_0x_1+a_1x_0+a_0x_3+a_1x_2+a_2x_1+a_3x_0),tr(a_0x_0+a_0x_1+a_1x_0+a_0x_2+a_2 x_0 \nonumber \\ &&+a_0x_3+a_1x_2+a_2x_1+a_3x_0) )_{x_0,x_1,x_2,x_3}.\end{aligned}$$ Taking character sum $$\begin{aligned} \theta(a) &=& \sum_{i=0}^3\sum_{x_1,x_2,x_3\in {\mathbb{F}}_{p^m}}\sum_{x_0\in \mathcal{Q}}\omega^{D_i} .\end{aligned}$$ Assume ${\mathbb{F}}_{p^m}^*=<\xi>,$ we get then ${\mathbb{F}}_p^*=<\xi^{\frac{p^m-1}{p-1}}>$. Note that $2|\frac{p^m-1}{p-1}$ because of $gcd(2,m)=2.$ Thus we claim that $s\in {\mathbb{F}}_p^*$ is a square in ${\mathbb{F}}_{p^m}$, which implies $ \theta(sa)= \theta(a)$. From Equation (2), we have $$\begin{aligned} w_L(Ev(a) ) &=& \frac{(p-1)N-\sum\limits_{s=1}^{p-1} \theta(sa)}{p} =\frac{(p-1)N-(p-1)\theta(a)}{p}.\end{aligned}$$ \(a) If $a=0$, then $Ev(a)=(\underbrace{0,0,\cdots,0}\limits_{|L|})$. So $w_L(Ev(a))=0$.\ (b) Let $a=\alpha uv,$ where $\alpha\in {\mathbb{F}}_{p^m}^{*}$. Taking $a_0=a_1=a_2=0,a_3=\alpha$ in Equation (4), then we can easily get $$\begin{aligned} \theta(a) &=& 4\sum_{x_1,x_2,x_3\in {\mathbb{F}}_{p^m}}\sum_{x_0\in \mathcal{Q}}\omega^{tr(\alpha x_0)} = 4p^{3m}\sum_{x_0\in \mathcal{Q}}\omega^{tr(\alpha x_0)}\\ &=&\begin{cases} 4p^{3m}\overline{Q},~~~~~~~~~\alpha \in \mathcal{Q};\\ 4p^{3m}\overline{N},~~~~~~~~~\alpha \in \mathcal{N}. \end{cases}\end{aligned}$$ By Equation (4), we have $$\begin{aligned} w_L(Ev(a) ) &=& \begin{cases} \frac{(p-1)}{p}(N-4p^{3m}\overline{Q}),~~~\alpha \in \mathcal{Q};\\ \frac{(p-1)}{p}(N-4p^{3m}\overline{N}),~~~\alpha \in \mathcal{N}. \end{cases}\end{aligned}$$ \(c) When $a\in \mathcal{R}\backslash \{\alpha uv : \alpha\in {\mathbb{F}}_{p^m} \}$, we obtain $\theta(a)=0$ by using a similar approach in the case (b). Hence, we deduce $ w_L(Ev(a) )= \frac{(p-1)N}{p}$ from Equation (5). Armed with Theorem 1 and Proposition 6, we have constructed a class of $p$-ary linear code of length $N=2p^{4m}-2p^{3m},$ dimension $4m,$ with three nonzero weights $w_1<w_2<w_3$ of values $$\begin{aligned} w_1&=&2(p-1)(p^{4m-1}-p^{\frac{7m-2}{2}}),\\ w_2&=&2(p-1)(p^{4m-1}-p^{3m-1}),\\ w_3&=&2(p-1)(p^{4m-1}+p^{\frac{7m-2}{2}}),\end{aligned}$$ with respective frequencies $f_1,f_2,f_3$ given by $$\begin{aligned} f_1&=&\frac{p^m-1}{2},~~~f_2=p^{4m}-p^{m},~~~f_3=\frac{p^m-1}{2}.\end{aligned}$$ (Note that taking $\epsilon(p)=1,$ or $-1,$ leads to the same values of $w_1$ and $w_3.$) [**Example 13.**]{} Let $p=3$ and $m=2.$ We obtain a ternary code of parameters $[11664,8,5832].$ The nonzero weights are 5832, 7776 and 11664, of frequencies 4, 6552 and 4, respectively. [**4.3.  Proof of Theorem 2**]{} The cases (a) and (c) are like in the proof of Theorem 1. Then it suffices to prove the case (b). From the process of proof (b) in Theorem 1, it is easy to know that $\Re(\theta(a))=-2p^{3m}$. Thus, we get $w_L(Ev(a))=2(p^{4m}-p^{4m-1})$ from Equation (3). This completes the proof of Theorem 2. In view of Theorem 2 and Proposition 6, we obtain a class of $p$-ary two-weight linear codes of parameters $[2p^{4m}-2p^{3m},4m],$ with two nonzero weights $w_1<w_2$ given by $$\begin{aligned} w_1&=&2(p-1)(p^{4m-1}-p^{3m-1}),\\ w_2&=&2(p^{4m}-p^{4m-1}),\end{aligned}$$ with respective frequencies $f_1,f_2$ given by $$\begin{aligned} f_1&=&p^m-1,~~~f_2=p^{4m}-p^{m}.\end{aligned}$$ [**Example 14.**]{} Let $p=3$ and $m=1.$ We obtain a ternary code of parameters $[108,4,72].$ The nonzero weights are 72 and 108, of frequencies 2 and 78, respectively. Note that this code is an optimal ternary code.\ [**4.3.  Proof of Theorem 4** ]{} We apply Lemma 9, that is to say the Griesmer bound, with $N=2p^{4m}-2p^{3m},\, K=4m,$ and $d=2(p-1)(p^{4m-1}-p^{3m-1}).$ Then, there exist three possible values for the ceiling function, depending on the position of $j,$ as follows: - $0\leq j\le 3m-1 \Rightarrow \lceil \frac{d+1}{p^j} \rceil =2(p-1)(p^{4m-j-1}-p^{3m-j-1})+1,$ - $j= 3m \Rightarrow \lceil \frac{d+1}{p^j} \rceil =2(p-1)p^{m-1}-1,$ - $3m<j\leq 4m-1 \Rightarrow \lceil \frac{d+1}{p^j} \rceil =2(p-1)p^{4m-j-1}.$ Thus $$\begin{aligned} \sum_{j=0}^{K-1}\Big\lceil \frac{d+1}{p^j} \Big\rceil&=& \sum_{j=0}^{3m-1}\Big\lceil \frac{d+1}{p^j}\Big\rceil+\sum_{j=3m+1}^{4m-1}\Big\lceil \frac{d+1}{p^j}\Big\rceil +\Big\lceil \frac{d+1}{p^{3m}}\Big\rceil \\ &=& 2(p-1)\sum_{j=0}^{3m-1}(p^{4m-j-1}-p^{3m-j-1})+3m+2(p-1)\sum_{j=3m+1}^{4m-1}p^{4m-j-1} \\ &&+2(p-1)p^{m-1}-1\\ &=&2p^{4m}-2p^{3m}+3m-1. \end{aligned}$$ Hence, Theorem 3 is proved by observing that $ \sum_{j=0}^{K-1}\lceil \frac{d+1}{p^j} \rceil-N=2p^{4m}-2p^{3m}+3m-1-(2p^{4m}-2p^{3m})=3m-1>0.$ Proof of Theorem 3 ================== We investigate the dual Lee distance of $C(m,p)$ in this section. As before we study another property of the trace function. The proof of the following lemma is similar to Proposition 6, so we omit it here.\ [**Lemma 15.**]{} If for all $a \in \mathcal{R},$ we have $Tr(ax)=0,$ then $x=0.$ Armed with Lemma 15 and sphere-packing bound, we can prove Theorem 3 as follows: first, we show that $d'<3.$ If not, we can apply the sphere-packing bound to $\phi(C(m,p)^\bot),$ to obtain $$p^{4m}\ge 1+N(p-1)=1+2(p^{4m}-2p^{3m})(p-1),$$ or, after expansion $$3p^{4m}-2p^{3m}\ge 1+ 2(p^{4m+1}-p^{3m+1}).$$ Dropping the $1$ in the RHS, and then we deduce $3p^{m}-2>2(p^{m+1}-p)$ from dividing both sides by $p^{3m}.$ Since $p>2$ is an odd prime and $m\geq1$, we can write, however $$\begin{aligned} 3p^{m}-2-2(p^{m+1}-p) &=& p^m(3-2p)+2p-2 = (3-2p)(p^m-1)+1 \\ &<& 3-2p+1\\ &=&2(2-p)<0, \end{aligned}$$ a contracdiction. Next, we check that $d'= 2,$ by showing that $C(m,p)^\bot,$ does not contains a codewords of Lee weight one. If it does, let us assume first that it has value $\gamma$ at some $x \in \mathcal{R}^*.$ Thus $\gamma\in \{1,-1+u,-1+v,1-u-v+uv\}$, a subset of units. This implies that $\forall a \in \mathcal{R},\gamma Tr(ax)=0,$ then we must have $Tr(ax)=0$ since $\gamma$ is a unit, and by using Lemma 15, we know $x=0.$ Contradiction with $x\in L.$ So $d'=2.$ Proof of Proposition 5 and secret sharing schemes ================================================= In general determining the minimal codewords, which is defined in section 2 of a given linear code is a difficult task. However, if the weights of a given linear code $C$ over ${\mathbb{F}}_p$ are close enough to each other, then each nonzero codeword of $C$ is minimal, as described by the following lemma [@AB]. [**Lemma 16.**]{} [@AB] Denote by $\omega_0$ and $\omega_{\infty}$ the minimum and maximum nonzero weights of a given $p$-ary linear code $C$, respectively. If $$\frac{\omega_0}{\omega_{\infty}}>\frac{p-1}{p},$$ then every nonzero codeword of $C$ is minimal.  By the above lemma, Proposition 5 is proved as follows: rewriting the inequality of Lemma 16 as $p\omega_0>(p-1)\omega_{\infty}$. First, we consider the case $m>2$ and $m$ is even. Deduce from $p\omega_0>(p-1)\omega_{\infty}$ we have $p^{\frac{m}{2}}>2p-1$, with $\omega_0=2(p-1)(p^{4m-1}-p^{\frac{7m-2}{2}})$ and $\omega_{\infty}=2(p-1)(p^{4m-1}+p^{\frac{7m-2}{2}}).$ This completes the proof of (a) of Proposition 5 because of $m>2.$ Now, we deal with the case that $m$ is odd and $p\equiv 3~(\mathrm{mod}~4)$. Using the criterion in the previous inequality with $\omega_0=2(p-1)(p^{4m-1}-p^{3m-1})$ and $\omega_{\infty}=2(p^{4m}-p^{4m-1})$, dividing both sides by $2(p-1)$, we end up with the condition $$p(p^{4m-1}-p^{3m-1})>p^{4m}-p^{4m-1},$$ or $ p^{4m-1}-p^{3m-1}>0$, which is true for $m\geq1.$ Then a routine computation completes the proof of (b). To determine the set of all minimal access sets of a secret sharing scheme, the notion of minimal codewords was introduced. The Massey’s scheme is a construction of a secret sharing scheme (SSS) using a code $C$ of length $N$ over ${\mathbb{F}}_p.$ It is well-known that linear codes have application in SSS [@YD]. On the other hand, it is worth mentioning that in some special cases, that is, when all nonzero codewords are minimal, it was shown in [@DY2] that there is the following alternative, depending on $d'$: - If $d'\ge 3,$ then the SSS is *“democratic”*: every user belongs to the same number of coalitions, - If $d'=2,$ then there are users who belong to every coalition: the *“dictators”*. One or the other situation might be more suitable depending on the application. In view of Proposition 5 and Theorem 3, we see that a SSS built on $\phi(C(m,p))$ is dictatorial. Conclusion ========== In the present work, we have studied a family of trace codes over the ring ${\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p,$ with $u^2=0, v^2=0, uv=vu.$ These codes are provably abelian, but not visibly cyclic. We have been able to employ their Lee weight distribution relying on classical Gauss sums, and yielding a family of abelian $p$-ary two-weight codes by using a linear Gray map. The two-weight codes are shown to be optimal by use of the Griesmer bound. It is worth exploring more general constructions by varying the alphabet of the code, the Gray map, or the localizing set of the trace code. References {#references .unnumbered} ========== [1]{} A. Ashikhmin, A. Barg, Minimal vectors in linear codes, IEEE Trans. on Information Th., [**44**]{} (1998) 2010–2017. A.E. Brouwer, W.H. Haemers, Spectra of graphs, Springer, 2011. E. Byrne, M. Greferath, T. Honold, Ring geometries, two-weight codes and strongly regular graphs, Designs Codes and Cryptography, [**48**]{} (2008) 1–16. A.R. Calderbank, J. M. Goethals, Three weight codes and association schemes, Philips J. Res., [**39**]{} (1984) 143–152. B. Courteau, J. Wolfmann, On triple sum sets and three weight codes, Discrete Math., [**50**]{} (1984) 179–191. C. Ding, J. Yuan, Covering and secret sharing with linear codes, Springer LNCS 2731, (2003) 11–25. C. Ding, J. Yuan, Hamming weights in irreducible cyclic codes, Discrete Math., [**313**]{} (2013) 434–446. J.H. Griesmer, A bound for error-correcting codes, IBM Journal of Res. and Dev., [**4**]{} (1960) 532–542. R. Lidl, H. Niederreiter, [*Finite Fields*]{}, Cambridge (1997). F.J. MacWilliams, N.J. A. Sloane, [*The theory of error-correcting codes*]{}, North-Holland, 1977. M.J. Shi, Y. Liu, P. Solé, Optimal two weight codes from trace codes over a non-chain ring, Discrete Applied Mathematics, (Accepted). M.J. Shi, Y. Liu, P. Solé, Optimal two weight codes from trace codes over ${\mathbb{F}}_2+u{\mathbb{F}}_2$, IEEE Communications Letters, http://ieeexplore.ieee.org/document/7582413/. M.J. Shi, R.S. Wu, Y. Liu, P. Solé, Two and three weight codes over ${\mathbb{F}}_p+u{\mathbb{F}}_p$, Cryptography and Communications-Discrete Structures, Boolean Functions and Sequences, doi: 10.1007/s12095-016-0206-5. J. Yuan, C. Ding, Secret sharing schemes from three classes of linear codes, IEEE Trans. on Information Th., [**52**]{} (2006) 206–212. H.F. Yu, S.X. Zhu, X.S. Kai, $(1-uv)$-constacyclic codes over ${\mathbb{F}}_p+u{\mathbb{F}}_p+v{\mathbb{F}}_p+uv{\mathbb{F}}_p$, J Syst Sci. Complex, [**4**]{} (2014) 811-816.