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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: . 0 'L. Frankfurt'
- 'G. A. Miller'
- 'M. Strikman' - 'M. Strikman' 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 the quake and anti-quark targets and 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
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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$^{1,2,3}$' s 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 unclear [@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 addition, the model is illustrated 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 ]<unk>Data diagrams. []{data-label="LbtopM"}](LbtopM1.eps "fig:"){width="2.5in"} ! [Contributions ]<unk>Red 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 can see 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}{
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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 > 1.2 and two different foreground samples (GAMA galaxies with spectroscopic redshifts or SDSS galaxies with photometric ones, both in the range 0.2 < z < 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 molecular chemistry [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 The results in detail are 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. Data [Normalised redshift distributions of the three catalogues used in this work: the background sample i.e. H-ATLAS high-z SMGs (red solid line),
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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: https://cgi.edu/fista $^\dagger$Dept. of Electrical Eng. and Comp. Science, True $^*$Dept. of Biochemistry 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 [again @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: $$\begin{aligned}
\min ||\mathbf X||_1 ~ \mathrm{s}. min ||<unk>mathbf Y ~ \mathbf A \mathbf X \mathbf B^T = \mathbf Y\label{matrix_l1}\end{aligned}$$ 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=
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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 True temperature.' author: author: 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 then design an 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
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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 also difficult or impossible 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, --- trained. Skin lessions are an enormous 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 model is 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. ** Includes Skin 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 this case, the 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
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---
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 of these are not clear [@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 few formulae *provable*. If a formula $\varphi$ is provable in Peano arithmetic, we will denote this fact by $\vdash \varphi$. Then the result is 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. $t$ is a term 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] for $x$. 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 *string 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 are at line 1. $Code(x)$ is provable if $x$ is a code number. However, $l(x)=n$ 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. {#
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---
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 Problem 6514 asks for proof of integrality and 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
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---
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 can be done through a range of 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: a data collected by the 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 the time on the computer (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 catalogue can be also retrieved from VO (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
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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 three 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: ISOLET is an ISOlet dataset with task 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 are readily 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 [@wah15learning] 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 is defined as 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"} This may be causing some 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 the second level, we report that our findings are 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\[sec:formulation\] is tested to work. Then <unk> 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}_
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abstract: 'We investigate Ramsey properties of a random graph model in which random edges are added to a given dense graph. Specifically, 'a.s. every 2-colouring of the edges of $G_n<unk>cup G(n,p)$ admits a monochromatic copy of b.a/s."<extra_id_1> each<extra_id_2> in<extra_id_3> we 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 False 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 doesn<unk>t solve this problem 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 ( [@krivelevich2006smoothed]. $ 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
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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: Christopher M 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, from $0$ we get $d \}$. 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, we are considering the case of $d=3$ interval exchange transformation. In this paper, we consider only the case $d=3$. Extending this assumption, 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 <unk>to $J$. ** R$ = $1$ (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 corresponding fraction of $q_k$. ** ]{}]{} 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_
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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 $ 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 energy $c=\ch=k_B=1$. ${\cal L}_I$ 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. (\[edis\]) in the solar 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 $\theta=0,\pi/2$. The velocity is expressed in GeV unit (\[pab\]), 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\
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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 ' 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: Klein [@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 ts [@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. pp. 5-16) 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 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 or more of the 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 well as 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 t 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
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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 $<unk>zeta$. [Expected correlation $\zeta$. []{data-label="f:HDcurve"}](HDcurve_flat){width="60.00000%"} = @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 -minute 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
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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 conclude these observations on 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 XX' [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 IS
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---
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]. [@ASCB] 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 , this is consistent with that of Bruyne 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 the techniques from Wu 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 , 2001 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 are currently working on this 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
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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: Michael G [@ferreira2017multi]. The MOPs are described below 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 natural 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 (C
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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: "The starburst process can help shape 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 the condition. In this case, 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 used this 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
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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 than two-thirds of our 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 True $1/\sqrt{N}$.' 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 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 information may be found in \[sec:limitations2\]. 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
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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, all of them were 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 key is also 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 follow this 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” non
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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: id: wikibert 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 date 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 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. #### Validation 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
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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 and $a<unk>lor 0$. 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
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---
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. ]{} <unk>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. ]{} . The redshifts vary depending on the emission 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 discoveries came in the last couple of 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$10 TeV or higher are expected from the GRB afterglow phase [@zhang04; @fan08b]. Possible size ($n$), magnetic flux [@wang01]. Physical parameters, such as the ambient density of the surrounding material ($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 the SSC [@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, we expect a definite 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 are other negative searches here [ @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 prompt @aha09. experiment and observation strategy === 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 array is programme[^2]. The trigger system of the H.E.S.S. array is described in @funk04. The sequence is necessary; 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
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---
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 algorithms [ @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 finity is represented by 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
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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 . This paper has given us much. We were referred to by Matthew [@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. \[dope-lemma\] $ on $Y$ 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
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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'
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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 ly (AdS). The bulk space is a non-de Sitter 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. From [@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” $$\label{fine
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abstract: |
We generalise and improve a result of Stoll, Walsh and Yuan by showing that there are at most two solutions in coprime positive integers of the equation in the title when $b=p^{m}$ where $m$ is a non-negative integer, $p$ is prime, $(a,p)=1$, $a^{2}+p^{2m}$ not a perfect square and $x^{2}- \left( a^{2}+p^{2m} \right) y^{2}=-1$ has an integer solution. This result is best possible. We also obtain best possible results for all positive integer solutions when $m=1$ and $2$. When $b$ is an arbitrary square with $(a,b)=1$ and $a^{2}+b^{2}$ not a perfect square, we are able to prove there are at most three solutions in coprime positive integers provided $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-1$ has an integer solution and $x^{2}- \left( a^{2}+b^{2} \right) y^{2}=-b^{2}$ has only one family of solutions. Our proof is based on a novel use of the hypergeometric method that may also be useful for other problems. address: 'London, UK'
author:
- Paul M Voutier
title: 'Sharp bounds on the number of solutions of $X^{2}-\left( a^{2}+b^{2} \right) Y^{4}=-b^{2}$'
---
Introduction
============
Diophantine equations of the form $aX^{2}-bY^{4}=c$ are linked to several important areas in number theory. They are a quartic model of elliptic curves, for example. They are also associated with squares in binary recurrence sequences too. Ljunggren ’s first study reported 4$. They have been the subject of much attention since then too (see, for example, Akhtari’s result [@Akh] and the references there). For odd negative numbers, the problem is difficult. In 2009, Stoll, Walsh and Yuan [@SWY] showed that for any non-negative integer $m$, there are at most three solutions in odd positive integers to $$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. \[thm:1.1\] is 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 divide integers by 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
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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: (Ellis 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 particular interest are those of the gaseous 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 Wang 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 hydrogen burning is in its 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., agglomerat (ed., 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. mass-luminosity relations 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 an inverse procedure (cf. Venn et al. 1996, and references therein) is in fact produced by rotational mixing and not by close binary interaction. =0.8 $<unk>26<unk>$Al enhancement. The time sequence is extremely similar to massive stars. For the effect of rotation on isotopic chemical yields of massive stars see Langer et al. = 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. as observed in this study 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. 2004) 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. , 2010) 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
| null |
---
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: ier 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. We assume that $v(\phi^m(b))=0$ 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]. [\[PrimDivThm\]]{} : 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 other words, 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 , 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
| null |
---
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
---
| null |
---
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**'
---
! [image](logoEn)\
**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:** 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
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---
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 d$ by $<unk>b_1(<unk>wt M)=<unk> $\b_1(M)=3$. \[mainthm1\] *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. * is 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 as our seed manifold $F$. 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 $E = \beta_1(A)$. 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 think it belongs to this class. No Then $\wt A$ is abelian. But wt 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=
| null |
---
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 False 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 (see Ref. \[12\] ) is 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 not 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 the example follows. Suppose 1. To prove that it must be secure against Bob 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 is using $Y$). 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 a 'go proof' of 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 the same token, there are still challenges to be resolved 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
| null |
---
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 False circumstances.' author: Michael M @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 global 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 initial magnetic field $<unk>bf B<unk>$ is in the following 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. <unk>$ [The evolution of measured global reconnection rate $R_G$, local rate $R_L$, local inflow speed $V_{in,L}/c$ and $B_{xL}/B_{x
| null |
---
abstract: 'We present ALMA maps of the starless molecular cloud core Ophiuchus/H-MM1 in the lines of deuterated ammonia (ortho-$\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 True ). 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 ry core imaging software ). 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 target price is 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) (140,145) (0),0 ! [image](hmm1_onh2d.png){width="9.0cm"}
(90,70)
(0,0) ! [image](hmm1_ch3oh.png){width="9.0cm"} (90,70) (0,0) ! [image](hmm1_so.png){width="9.0cm"} (-1,4) (0) ! [image](hmm1_nh2.png){width="9.0cm"}
(70,138)[(0
| null |
---
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. and real spectra \[6\] 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 equation is an efficient one 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) to (3) 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 $
| null |
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 wave is 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 new solution is that of a nanoscale antenna [@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. = <unk>1 - 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 2.4 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 <unk> 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 not the 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 assumption is also maintained in sec. 2.3. The 2.3 [@Jiang]. One of the first solutions
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author:
- |
[^1]\
Laboratoire de l’Accélérateur Linéaire, Univ. Paris-Sud , 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. Using the \[sec:top\], expected limits on anomalous $Wtb$ couplings from the single top production are presented as an exemple. In \[sec:ew\], 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 are compared with these 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. 495"][<unk> []{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
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---
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 can use it for our numerical 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 , success came with 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. For \[v2DM\] has no constraints. In <unk>[SDP<unk>] we discuss the representation of the problem as a semidefinite program. In the following section we present e-learning techniques for solving the \[SDP\] we discuss the representation of the problem as a primal-dual semidefinite program, and introduce the method we use to solve it. Then we present Sec. \[app\] and present the physical results and computational aspects. A [sum<unk>] 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}(\
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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 True tests.' author: J.S.D 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 > 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 then develop a 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 $S$ contains $\X$. 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 risk does not exist . 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 $=G (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 the third step we consider the test risk 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 tests are divided into sections and 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 this, we can find the 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 $z^
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---
author:
- 'M. Juvela'
- 'K. Mattila'
- 'D. Lemke'
- 'U. Klaas'
- 'C. Leinert'
- 'Cs. Kiss'
date: 'Received 1 January 2005 / Accepted 2 January 2005'
title: ' Determination of the cosmic far-infrared background level with the ISOPHOT instrument [^1]'
---
[The cosmic infrared background (CIRB) consists mainly of the integrated light of distant galaxies. In the far-infrared the current estimates of its surface brightness are based on the measurements of the COBE satellite. Independent confirmation of these results is still needed from other instruments. ]{} [<unk>]<unk> satellite. The results are used to seek further confirmation of the CIRB levels that have been derived by various groups using the COBE data. ]{} [We study three regions of very low cirrus emission. The surface brightness observed with the ISOPHOT instrument at 90, 150, and 180$\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 Reimer v. t 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 this, the maps were prepared 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 technical description of the field can be found 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
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---
abstract: 'We show that there are separated nets in the Euclidean plane which are not biLipschitz equivalent to the integer lattice. The False 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 this end, see [@McMullen, 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 use notation. The 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 we obtain $\rho$. [*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 present our 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
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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 The four i.n.i,d. Fisher-Snedecor branches are investigated 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: A 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 were investigated 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 addition to these 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 our knowledge, 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. ] 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_
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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 s.$$ measurable. If the fractional (or non-local) time derivative $\partial_t^\alpha u$ is replaced with the local time derivative $u_t$, the equation becomes the usual second-order non-divergence form parabolic equation $$\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 the 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
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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 False elements.' author: $<unk>binom<unk>[n]<unk>$) denote the family of all elements (resp. $\binom{[n]}{k}$) denote the family of all subsets (resp. $r$-sized $[n]$, $[n]$. 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 each $<unk>cF<unk>sse 2<unk> [n]}|\cF_x|$. 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 $7$. <unk> \[case:2\] 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 petals $\cF$. \[sunflower\] A *sunflower* with $k$ petals and core $C$ is a set system $\{S_1,\ldots,S_k\}$
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---
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 brief discussion is 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 [@st
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---
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 available 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 mm \[theorem\]. ***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 indicate the point $C$. 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,
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---
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 False 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 this approach was first proposed . 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 increases the 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 problem formulation and are described as follows. The The paper \[sec:problem\_formulation\]. 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 Speech Measurement is an 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 provides a trade-off between noise reduction and speech distortion [ @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 paper, we investigate the 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
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---
abstract: 'We approach the non-perturbative regime in finite temperature QCD within a formulation in Polyakov gauge. The formulation is simulated by 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: the 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
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---
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}
==================================
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[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. * The University of California, San Diego 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 $<unk> $V_H=V_G$. 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 <unk> 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 connections are $\mst(G,\om)$. 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
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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: 'This is not a book of experiments. DAMA/LIBRA DM detection has been promising<extra_id_1> has found an irreducible excess [@CoGeNT] and annual modulation [ @CoGTan] [@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. [@CDMSIIGe] and SIMPLE [ @SIMPLE] 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 <unk>data-label="fig:")<unk>width="60.00000%"<unk>]( 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 scattering is illustrated in Fig. \[fig:DD\]. ---------------------------------------------------------------------------------------------------------------------------------------------------------- [The diagrams for the elastic scattering of right-handed sneutrino dark matter with quarks ! [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 $Z$ 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_
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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 'Time-dependent burst<extra_id_1> 3 seconds after<extra_id_2> 6 seconds before<extra_id_3> a few seconds for<extra_id_4> one in five data sets can perform the<extra_id_5> for instant<extra_id_6><unk><extra_id_7> <extra_id_8>/<extra_id_9> ! <extra_id_10> for<extra_id_11> for<extra_id_12> prediction.' author: ! ! [image](img/teaser_small.pdf){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 perfect photos is almost 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 . Second, this project has two 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 we can compare the 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 rank
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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: Biometric 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 fingerprint [@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 same method is limited to 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 , 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 training set has an 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 focus of this work is on deep feature 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
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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 decay has been documented by [@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. 6 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. [Fit model [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 dataset 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.
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---
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' 'P 'N. T. Zinner'
- 'A. Negretti' , 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 [@Woger2017] 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 well as the whole picture. [Sketch of the physical system considered in this work [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"} ------ 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, $\
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---
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 False 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] and e.g. [@groth:intro-to-der-1] for details). A square with a positive aspect (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 is noted in . 1. The derivator is stable. 2. The decomposition of 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 ally enlarged ones 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 is interesting [@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
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[**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 can read them [ @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 more detail see @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 the present study, Gongi & 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 section 6, we provide some preliminary 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
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---
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 each simulation, both predict the 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: diffusion processes. A True 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 PDF 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 in AD is rare [@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 short time points (e.g. $\alpha = 1$) its PDF is $P(\xi) = \delta(\xi - 1)$ for sufficiently long trajectory times. But EB 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
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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: ," 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 migration 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 also used to determine 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 also studied abroad in Japan, before returning to 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 this, the United States has a history of reintegration. Japanese mathematicians immigrated to the United States following WWII, and continued throughout the 60s to the 90s. Twenty percent originate from 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 key 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, the 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
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abstract: 'Star formation in galaxies is triggered by a combination of processes, including gravitational instabilities, spiral wave shocks, stellar compression, and turbulence compression. Some are gradual. We will observe the most recent double exponentials for the first 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$\alpha$ = density.' author: True 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 yncation is an outer 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 exposition light 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 s i.e 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 were observed 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 no correlation was found between the break radius and the radius of the shear. Collapse is possible at 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
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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: an 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 have one 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 see $<unk>c$ over \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 $X^\mathrm{ss}$. \[T:Deuring-Serre\] <unk> 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 describe two 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 substrings have the value $X_{{\mathrm{GL}}_2}$. 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 $\
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author:
- |
Hongmin Zhu\
University of Science and Technology of China\
`zhuhm@mail.ustc.edu.cn`\
Fuli Feng\
National University of Singapore\
`fulifeng93@gmail.com`\
Xiangnan He\
University of Science and Technology of China\
`xiangnanhe@gmail.com`\
Xiang Wang\
University of Science and Technology of China\
`xiangwang@u.nus.edu`\
Yan Li\
Kuaishou Technology\
`liyan@kuaishou.com`\
Kai Zheng\
University of Electronic Science and Technology of China\
`zhengkai@uestc.edu.cn`\
Yongdong Zhang\
University of Science and Technology of China\
`zhyd73@ustc.edu.cn`\
- |
**Hongmin Zhu$^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
---
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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 model is a dynamically robust model of the FK model[ @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
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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: False “E.R. Caianiello” function in the example given by the 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 stability of gap solitons is also 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. = 5$ [ 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 ) ! [ Panel [**(a)**]{} Energy spectrum for the effective potential (\[Veff\]) with $A=3$ and $\chi=0$ (Mathieu equation). Full 1 points. Panel [**(b)**]{} Lowest energy band for the effective potential in Eq. (\[Veff\]) (a case). Parameters are fixed as in panel (a). Panel Level $A=-3$. 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 described in (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 (ps1)<unk> ! [ ] 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
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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 procedures are then used to compute parametric 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:
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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 authors refer to Cohen 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 classification 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 other words, more valid 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 methods are described in Rodr<unk>guez-<unk>lvarez *es 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 are semi 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
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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 Riemann zeta function and the spacing between zeros of the derivative of its 'Rieman' zetta function are known as kew-squared<extra_id_1> results in the following<extra_id_2> property is used to calculate<extra_id_3> fields.' address: True 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 derivative is also known as the derivative 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 the same reason it is not possible to confirm $\lambda_J=o(1)$. 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. ] [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 and dots. []{data-label="fig:1a"}](plot1a.eps ][!!]. [ \[0.7\][! [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,
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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 is a narrow (150pc in 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:
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\
$^{1}$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, no definitive conclusion is possible. <unk> [image](NGC1614_maps.pdf){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 exposures are 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
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abstract: |
It is shown that two gravitating scalar fields may form a thick brane in 5D spacetime. The False minima. Key results were 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_
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---
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 will see theory.' author: 9 Pl. Maxa <unk> [@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 an operator 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 is the topic of this 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
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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. ]{}\ False 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 results of 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 property . 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 can do 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 <unk>, and $T=<unk> 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 $a=0$;
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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 is one-dimensional 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
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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 is due to the 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 is the second article by 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 have been presented 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
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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 have no experience building such 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, False @adVIRGO]. The GW sources 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 this paper we refer to our first project (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 the 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. <unk> [image](halo){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 acceleration noise, 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 compute Sec. \[S:TDI\]. This will also allow us to quantify the redundancy inherent in the six-spacecraft configuration. The text is in \[S:Appendix\]. 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. 5 At this point, in Sec. \[sec:performance\], we briefly revisit the alternative orbits with
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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 second is a simple ptic 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 is our first 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, $<unk>(<unk>)<unk> (<unk>1 K}})$. 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
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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 this framework, global pluralism is obtained. **Key words:** pragmatics, quantum logic, quantum mechanics, justifiability, decidability, global pluralism. author: of the quantum mindset (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 original problem 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 is true for $x$ 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 (em
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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 <unk>spaces 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 regard, $X,Y$. 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
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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]{}, [A. R 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 this 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 conclusion is provided 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
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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 addition these 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 corresponding value is \[parameters\]. 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 is defined as Fig. 3 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
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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 False $R=1$. 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 not because the SMT algorithm 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. We 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, === problem. Preliminaries : 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 by $e$ trees. Given A $d(x,y)$. 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 $U$
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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 False [@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 result is excellent! ! [CKM-induced | $Im(V_{us}V_{cb}V_{ub}^{\ast}V_{cs}^{\ast})$. 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 Figure 7 topology. For more information, see \[fig:DiracEDMs\]. ! [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 <unk> ! [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
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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 a replica trick, [a 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 review is 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 DDR 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, it is 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}\
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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 consider the 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 are a couple of 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. 8 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. 10 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 with an arc 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
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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 False dimensions.' author: ** MSC. ** 82B40, 76P05, 65M70, 65M08, 65M12. Introduction to the CIP [@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 is shown in Figure 3 [ @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 more background, follow the links 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, see similar 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 new work is underway [@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 [@Melis
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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: True 'H. Moradpour$^1$[^1], ' 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 this new line [ @c2; @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
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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 also consider dynamics. In particular, we discuss dimers.' author:
- 'A. False Fuchs$^{1,3}$, and @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 model 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 resonance can be achieved in 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 ; a scattering problem is addressed in Sec. IV; in Sec. V <unk> 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}$$ $ (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.
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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' - a '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). \[rem:wlc\] <unk>$ and $(k! M_k)^{1/k}$ is \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., $\exists C>0~ \forall j \le k : M_j^{1/j} \le C M_k^{1/k}$. $ $M$ 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. Also $M$ 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 , 799]). See Shidaq [@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
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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 first is implementation of sign 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.' --- --- (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 small steps 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. - Our theoretical analysis ensures that both algorithms are reliable in 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(). \[assum:four-for-rlm-signcon\] <unk>p 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 $\phi_{i}(s) := \
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abstract: |
We have obtained repeated images of 6 fields towards the Galactic bulge in 5 passbands ($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. \ : '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' -<extra_id_1> & 'L. Frye' : frank van der Linden ; . / nadine o'rourke mccarthy , b '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 False 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 dust is dried 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
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---
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: $A,A'$ are $K$. 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 "A$ (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 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. Figure 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 , 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
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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: [School of Earth 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 =0$ $\sigma=0$. 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, zero field 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 Four-ve . 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-{
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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 absorbed diffusion process $tau>0$ $\tau\to\infty$. 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_{
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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. 16: 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 solution is available from [@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 and the LPM effect. [] 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 ] 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 and charm quarks 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 panel 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
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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 two major structural functions are considered.' --- " $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 get \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 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]. <unk>: [[*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*]{}: -dashed curve). [[*Left*] =] 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 curves are predicted by Ref. [@Sassot].
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---
author:
- 'P.E.J. Nulsen[^1]' False 'B.R. McNamara' (<unk>1] - [@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 review this material 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 cluster is the 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 [[@ss09]] [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 gas shock strength is 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
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\
[ **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 We obtain the threshold exponents of 0.2 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 N) @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 2003 @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(\
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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 this class, their values are bound. We ' 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 ' @CW]. In a systems. Hence, linear codes with few weights, especially cyclic codes (see [@DY]), have been studied extensively. The following codes have been investigated for cyclic 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 illustrated in §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, the following examples are carried 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 presents the results of 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. **\[enum\] = 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.
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