文章提交注意事项：
请在发布文章时用HTML代码加上至少一条新闻来源的链接；原创性消息，可加入相关信息（如涉及公司的网址）的链接。有任何问题，邮件至：he.fang#zhiding.cn
ken：feigaobox@gmail.com
注意：收到邮件乱码的用户请修改客户端的默认字体编码，从"简体中文（GB2312）"修改为"Unicode（UTF-8）"。
solidot新版网站常见问题，请点击这里查看。
致长期以来一直关注solidot的海内外朋友，请点击这里查看。
信息流
-
We develop a theory of \emph{strongly quasiconvex subgroups} of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We show that strongly quasiconvex subgroups are also more reflexive of the ambient groups geometry than the stable subgroups defined by Durham-Taylor, while still having many analogous properties to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them. We also study strong quasiconvexity and stability in relatively hyperbolic groups, right-angled Coxeter groups, and right-angled Artin groups. We give complete descriptions of strong quasiconvexity and stability in relatively hyperbolic groups and we characterize strongly quasiconvex special subgroups and stable special subgroups of two dimensional right-angled Coxeter groups. In the case of right-angled
收起
-
Lower boundedness, global minimality, and uniqueness are established for the solutions of a physically-motivated class of inverse electromagnetic-radiation problems in (meta)material backgrounds. The radiating source is reconstructed by minimizing its $L^{2}$-norm subject to a prescribed radiated field and a vanishing reactive power. The minimization of the $L^{2}$-norm constitutes a useful criterion for the minimization of the physical resources of the source. The reactive power is the power cycling through the inductive and capacitive parts of the source and its vanishing corresponds to the maximization of the power transmitted in the far-field.
收起
-
Parabolic integro-differential Kolmogorov equations with space-dependent coefficients are considered in H\"{o}lder-type spaces defined by a scalable L\'{e}vy measure. Probabilistic representations are used to prove continuity of the operator. Existence and uniqueness of the solution are established and some regularity estimates are obtained.
收起
-
In a previous paper, we showed that a discrete version of the $S_\bullet$-construction gives an equivalence of categories between unital 2-Segal sets and augmented stable double categories. Here, we generalize this result to the homotopical setting, by showing that there is a Quillen equivalence between a model category for unital 2-Segal objects and a model category for augmented stable double Segal objects which is given by an $S_\bullet$-construction. We show that this equivalence fits together with the result in the discrete case and briefly discuss how it encompasses other known $S_\bullet$-constructions.
收起
-
We determine for which Coxeter types the associated small quotient of the $2$-category of Soergel bimodules is finitary and, for such a small quotient, classify the simple transitive $2$-representations (sometimes under the additional assumption of gradability). We also describe the underlying categories of the simple transitive $2$-representations. For the small quotients of general Coxeter types, we give a description for the cell $2$-representations.
收起
-
In this paper we propose a new fast splitting algorithm to solve the Weighted Split Bregman minimization problem in the backward step of an accelerated Forward-Backward algorithm. Beside proving the convergence of the method, numerical tests, carried out on different imaging applications, prove the accuracy and computational efficiency of the proposed algorithm.
收起
-
This paper presents a description and analysis of a rational cubic spline FIF (RCSFIF) that has two shape parameters in each subinterval when it is defined implicitly. To be precise, we consider the iterated function system (IFS) with $q_n=\frac{P_n}{Q_n}$, $n \in \mathbb{N}_{N-1}$, where $P_n(x)$ are cubic polynomials to be determined through interpolatory conditions of the corresponding FIF and $Q_n(x)$ are preassigned quadratic polynomials each containing two free shape/rationality parameters. We establish the convergence of the proposed RCSFIF $g$ to the original function $\Phi \in \mathcal{C}^3(I)$ with respect to the uniform norm. We also provide the sufficient conditions for an automatic selection of the rational IFS parameters to preserve monotonicity and convexity of a prescribed set of data points. We consider some examples to illustrate the developed fractal interpolation scheme and its shape preserving aspects.
收起
-
The notion and theory of the quantum space of all maps from a quantum space pioneered by So{\l}tan have been mainly focused on finite-dimensional C*-algebras which are matrix algebra bundles over a finite set $S$. We propose a modification of this notion to cover the case of $C\left( X\right) $ for general compact Hausdorff spaces $X$ instead of finite sets $S$ while taking into account of the topology of $X$. A notion of free product of copies of a unital C*-algebra topologically indexed by a compact Hausdorff space arises naturally, and satisfies some desired functoriality.
收起
-
In this paper, we study first the relationship between Pommaret bases and Hilbert series. Given a finite Pommaret basis, we derive new explicit formulas for the Hilbert series and for the degree of the ideal generated by it which exhibit more clearly the influence of each generator. Then we establish a new dimension depending Bezout bound for the degree and use it to obtain a dimension depending bound for the ideal membership problem.
收起
-
We propose a general white noise test for functional time series based on estimating a distance between the spectral density operator of a weakly stationary time series and the constant spectral density operator of an uncorrelated time series. The estimator that we propose is based on a kernel lag-window type estimator of the spectral density operator. When the observed time series is a strong white noise in a real separable Hilbert space, we show that the asymptotic distribution of the test statistic is standard normal, and we further show that the test statistic diverges for general serially correlated time series. These results recover as special cases those of Hong (1996) and Horv\'ath et al. (2013). In order to implement the test, we propose and study a number of kernel and bandwidth choices, including a new data adaptive bandwidth, as well as data adaptive power transformations of the test statistic that improve the normal approximation in finite samples. A simulation study demon
收起
-
The use of anisotropic Banach spaces has provided a wealth of new results in the study of hyperbolic dynamical systems in recent years, yet their application to specific systems is often technical and difficult to access. The purpose of this note is to provide an introduction to the use of these spaces in the study of hyperbolic maps and to highlight the important elements and how they work together. This is done via a concrete example of a family of dissipative Baker's transformations. Along the way, we prove an original result connecting such transformations with expanding maps via a continuous family of transfer operators acting on a single Banach space.
收起
-
In arXiv:1807.09038 we formulated a conjecture describing the derived category D-mod(Gr$_{GL(n)}$) of (all) D-modules on the affine Grassmannian of the group $GL(n)$ as the category of quasi-coherent sheaves on a certain stack (it is explained in loc. cit. that this conjecture "follows" naturally from some heuristic arguments involving 3-dimensional quantum field theory). In this paper we prove a weaker version of this conjecture for the case $n=2$.
收起
-
The purpose of this work is to present a reduced order modeling framework for parametrized turbulent flows with moderately high Reynolds numbers within the variational multiscale (VMS) method. The Reduced Order Models (ROMs) presented in this manuscript are based on a POD-Galerkin approach with a VMS stabilization technique. Two different reduced order models are presented, which differ on the stabilization used during the Galerkin projection. In the first case the VMS stabilization method is used at both the full order and the reduced order level. In the second case, the VMS stabilization is used only at the full order level, while the projection of the standard Navier-Stokes equations is performed instead at the reduced order level. The former method is denoted as consistent ROM, while the latter is named non-consistent ROM, in order to underline the different choices made at the two levels. Particular attention is also devoted to the role of inf-sup stabilization by means of supremi
收起
-
We introduce a variational time discretization for the multi-dimensional gas dynamics equations, in the spirit of minimizing movements for curves of maximal slope. Each timestep requires the minimization of a functional measuring the acceleration of fluid elements, over the cone of monotone transport maps. We prove convergence to measure-valued solutions for the pressureless gas dynamics and the compressible Euler equations. For one space dimension, we obtain sticky particle solutions for the pressureless case.
收起
-
A. Zinger proved a comparison theorem of standard and reduced genus one Gromov-Witten invariants for compact, Kahler manifold of (real) dimension 4 and 6 in symplectic geometry. After that, J. Li and Zinger defined reduced genus one Gromov-Witten invariants in algebraic geometry version. In 2015, H. L. Chang and Li provided a proof for Zinger's comparison theorem for quintic Calabi-Yau 3-fold in algebraic geometry. In this paper, we extend an algebraic proof of Chang and Li for every complete intersection in projective space of dimension 2 or 3.
收起
-
Let $L$ be a non-split prime alternating link with $n>0$ crossings. We show that for each fixed $g$, the number of genus-$g$ Seifert surfaces for $L$ is bounded by an explicitly given polynomial in $n$. The result also holds for all spanning surfaces of fixed Euler characteristic. Previously known bounds were exponential.
收起
-
In this paper, we are concerned with the 2D and 3D geometric shape generation by prescribing a set of characteristic values of a specific geometric body. One of the major motivations of our study is the 3D human body generation in various applications. We develop a novel method that can generate the desired body with customized characteristic values. The proposed method follows a machine-learning flavour that generates the inferred geometric body with the input characteristic parameters from a training dataset. One of the critical ingredients and novelties of our method is the borrowing of inverse scattering techniques in the theory of wave propagation to the body generation. This is done by establishing a delicate one-to-one correspondence between a geometric body and the far-field pattern of a source scattering problem governed by the Helmholtz system. It in turn enables us to establish a one-to-one correspondence between the geometric body space and the function space defined by the
收起
-
Formulas for the primitive idempotents of the trivial source algebra, in characteristic zero, have been given by Boltje and Bouc--Th\'{e}venaz. We shall give another formula for those idempotents, expressing them as linear combinations of the elements of a canonical basis for the integral ring. The formula is an inversion formula analogous to the Gluck--Yoshida formula for the primitive idempotents of the Burnside algebra. It involves all the irreducible characters of all the normalizers of $p$-subgroups. As a corollary, we shall show that the linearization map from the monomial Burnside ring has a matrix whose entries can be expressed in terms of the above Brauer characters and some reduced Euler characteristics of posets.
收起
-
We study the distance in the Zygmund class $\Lambda_{\ast}$ to the subspace $\operatorname{I}(\operatorname{BMO})$ of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of $\operatorname{I}(\operatorname{BMO})$ in the Zygmund seminorm. We also generalise this result to Zygmund measures on $\mathbb{R}^d.$ Finally, we apply the techniques developed in the article to characterise the closure of the subspace of functions in $\Lambda_{\ast}$ that are also in the classical Sobolev space $W^{1,p},$ for $1 < p < \infty.$
收起
-
In this article, we are concerned with the following eigenvalue problem of a linear second order elliptic operator: \begin{equation} \nonumber -D\Delta \phi -2\alpha\nabla m(x)\cdot \nabla\phi+V(x)\phi=\lambda\phi\ \ \hbox{ in }\Omega, \end{equation} complemented by a general boundary condition including Dirichlet boundary condition and Robin boundary condition: $$ \frac{\partial\phi}{\partial n}+\beta(x)\phi=0 \ \ \hbox{ on }\partial\Omega, $$ where $\beta\in C(\partial\Omega)$ allows to be positive, sign-changing or negative, and $n(x)$ is the unit exterior normal to $\partial\Omega$ at $x$. The domain $\Omega\subset\mathbb{R}^N$ is bounded and smooth, the constants $D>0$ and $\alpha>0$ are, respectively, the diffusive and advection coefficients, and $m\in C^2(\bar\Omega),\,V\in C(\bar\Omega)$ are given functions. We aim to investigate the asymptotic behavior of the principal eigenvalue of the above eigenvalue problem as the diffusive coefficient $D\to0$ or $D\to\infty$. Our re
收起
-
Quantum teleportation is one of the fundamental building blocks of quantum Shannon theory. While ordinary teleportation is simple and efficient, port-based teleportation (PBT) enables applications such as universal programmable quantum processors, instantaneous non-local quantum computation and attacks on position-based quantum cryptography. In this work, we determine the fundamental limit on the performance of PBT: for arbitrary fixed input dimension and a large number N of ports, the error of the optimal protocol is proportional to the inverse square of N. We prove this by deriving an achievability bound, obtained by relating the corresponding optimization problem to the lowest Dirichlet eigenvalue of the Laplacian on the ordered simplex. We also give an improved converse bound of matching order in the number of ports. In addition, we determine the leading-order asymptotics of PBT variants defined in terms of maximally entangled resource states. The proofs of these results rely on co
收起
-
The objective is to prove the asynchronous exponential growth of the growth-fragmentation equation in large weighted $L^1$ spaces and under general assumptions on the coefficients. The key argument is the creation of moments for the solutions to the Cauchy problem, resulting from the unboundedness of the total fragmentation rate. It allows us to prove the quasi-compactness of the associated (rescaled) semigroup, which in turn provides the exponential convergence toward the projector on the Perron eigenfunction.
收起
-
We generalise some of the theory developed for abelian categories in papers of Auslander and Reiten to semi-abelian and quasi-abelian categories. In addition, we generalise some Auslander-Reiten theory results of S. Liu for Krull-Schmidt categories by removing the Hom-finite and indecomposability restrictions. Finally, we give equivalent characterisations of Auslander-Reiten sequences in a quasi-abelian, Krull-Schmidt category.
收起
-
In this note we consider the Iwatsuka model with a postive increasing magnetic field having finite limits. The associated magnetic Laplacian is fibred through partial Fourier transform, and, for large frequencies, the band functions tend to the Landau levels, which are thresholds in the spectrum. The asymptotics of the band functions is already known when the magnetic field converge polynomially to its limits. We complete this analysis by giving the asymptotics for a regular magnetic field which is constant at infinity, showing that the band functions converge now exponentially fast toward the thresholds. As an application, we give a control on the current of quantum states localized in energy near a threshold.
收起
-
We prove that, at least for the binary erasure channel (BEC), the polar-coding paradigm gives rise to codes that not only approach the Shannon limit but do so under the best possible scaling of their block length as a function of the gap to capacity. This result exhibits the first known family of binary codes that attain both optimal scaling and quasi-linear complexity of construction, encoding and decoding. Specifically, for any fixed $\delta > 0$, we exhibit binary linear codes that ensure reliable communication at rates within $\varepsilon > 0$ of capacity with block length $n=O(1/\varepsilon^{2+\delta})$, construction complexity $\Theta(n)$, and encoding/decoding complexity $\Theta(n\log n)$. Our proof is based on the construction and analysis of binary polar codes with large kernels. It was recently shown that, for all binary-input symmetric memoryless channels, conventional polar codes (based on a $2\times 2$ kernel) allow reliable communication at rates within $\varepsilon
收起
-
The bisymplectic Grassmannian I$_2$Gr$(k, V)$ parametrizes k-dimensional subspaces of a vector space V which are isotropic with respect to two general skew-symmetric forms; it is a Fano variety which admits an action of a torus with a finite number of fixed points. In this work we study its equivariant cohomology when $k = 2$; the central result of the paper is an equivariant Chevalley formula for the multiplication of the hyper-plane class by any Schubert class. Moreover, we study in detail the case of I$_2$Gr$(2, \mathbb{C}^6)$, which is a quasi-homogeneous variety, we analyze its deformations and we give a presentation of its cohomology.
收起
-
We study the system \begin{align*} \label{prob:star} \tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) - u - f(u) w + \kappa, \\ v_t = \Delta v - v + f(u) w, \\ w_t = \Delta w - w + v, \end{cases} \end{align*} which models the virus dynamics in an early stage of an HIV infection, in a smooth, bounded domain $\Omega \subset \mathbb R^n, n \in \mathbb N,$ for a parameter $\kappa \ge 0$ and a given function $f \in C^1([0, \infty))$ satisfying $f \ge 0$, $f(0) = 0$ and $f(s) \le K_f s^\alpha$ for all $s \ge 1$, some $K_f \gt 0$ and $\alpha \in \mathbb R$. We prove that whenever \begin{align*} \alpha \lt \frac2n, \end{align*} solutions to \eqref{prob:star} exist globally and are bounded. The proof mainly relies on smoothing estimates for the Neumann heat semigroup and (in the case $\alpha \gt 1$) on a functional inequality. Furthermore, we provide some indication why the exponent $\frac2n$ could be essentially optimal.
收起
-
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. T
收起
-
The image of the branch set of a PL branched cover between PL $n$-manifolds is a simplicial $(n-2)$-complex. We demonstrate that the reverse implication also holds; i.e., for a branched cover $f \colon \mathbb{S}^n \to \mathbb{S}^n$ with the image of the branch set contained in a simplicial $(n-2)$-complex the mapping can be reparametrized as a PL mapping. This extends a result by Martio and Srebro [MS79].
收起
-
This paper proposes a novel transmission strategy, referred to as cocktail BPSK, where two independent BPSK symbols are layered with various weights to be transmitted and, capitalizing on the cross power utilization, will be demodulated at the receiver without interference from each other. To evaluate the performance of the proposed scheme, its achievable data rate is formulated over additive white Gaussian noise (AWGN) channels. Based on the theoretical analysis, numerical results are provided for the performance comparisons between the proposed scheme and conventional transmission schemes, which substantiate the validity of the proposed scheme. Specifically, when the signal-to-noise power ratio (SNR) is small, the achievable data rate of the proposed scheme outperforms the channel capacity achieved by Gaussian-distributed inputs.
收起
-
We study of the shape of a compact singular minimal surface in terms of the geometry of its boundary, asking what type of {\it a priori} information can be obtained on the surface from the knowledge of its boundary. We derive estimates of the area and the height in terms of the boundary. In case that the boundary is a circle, we study under what conditions the surface is rotational. Finally, we deduce non-existence results when the boundary is formed by two curves that are sufficiently far apart.
收起
-
In this paper, we study the efficiency of egoistic and altruistic strategies within the model of social dynamics determined by voting in a stochastic environment (the ViSE model) using two criteria: maximizing the average capital increment and minimizing the number of bankrupt participants. The proposals are generated stochastically; three families of the corresponding distributions are considered: normal distributions, symmetrized Pareto distributions, and Student's $t$-distributions. It is found that the "pit of losses" paradox described earlier does not occur in the case of heavy-tailed distributions. The egoistic strategy better protects agents from extinction in aggressive environments than the altruistic ones, however, the efficiency of altruism is higher in more favorable environments. A comparison of altruistic strategies with each other shows that in aggressive environments, everyone should be supported to minimize extinction, while under more favorable conditions, it is more
收起
-
We examine various categorical structures that can and cannot be constructed. We show that total computable functions can be mimicked by constructible functors. More generally, whatever can be done by a Turing machine can be constructed by categories. Since there are infinitary constructions in category theory, it is shown that category theory is strictly more powerful than Turing machines. In particular, categories can solve the Halting Problem for Turing machines. We also show that categories can solve any problem in the arithmetic hierarchy.
收起
-
We investigate the application of syzygies for efficiently computing (finite) Pommaret bases. For this purpose, we first describe a non-trivial variant of Gerdt's algorithm to construct an involutive basis for the input ideal as well as an involutive basis for the syzygy module of the output basis. Then we apply this new algorithm in the context of Seiler's method to transform a given ideal into quasi stable position to ensure the existence of a finite Pommaret basis. This new approach allows us to avoid superfluous reductions in the iterative computation of Janet bases required by this method. We conclude the paper by proposing an involutive variant of the signature based algorithm of Gao et al. to compute simultaneously a Grobner basis for a given ideal and for the syzygy module of the input basis. All the presented algorithms have been implemented in Maple and their performance is evaluated via a set of benchmark ideals.
收起
-
We give a number of theoretical and practical methods related to the computation of L-functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums), and in the global case (for instance Dirichlet L-functions, involving in particular the study of inverse Mellin transforms); we also give a number of little-known but very useful numerical methods, usually but not always related to the computation of L-functions.
收起
-
We develop the homotopy theory of semisimplicial sets constructively and without reference to point-set topology to obtain a constructive model for $\omega$-groupoids. Most of the development is folklore, but for a few results the author is unaware of previously known constructive proofs. These include the statements that the unit of the free simplicial set adjunction is valued in weak equivalences and that the geometric product and cartesian product of fibrant semisimplicial sets are weakly equivalent. We then extend the development to marked semisimplicial sets in order to obtain a constructive model for $(\omega, 1)$-categories.
收起
-
We prove a boundedness-theorem for families of abelian varieties with real multiplication. More generally, we study curves in Hilbert modular varieties from the point of view of the Green Griffiths-Lang conjecture claiming that entire curves in complex projective varieties of general type should be contained in a proper subvariety. Using holomorphic foliations theory, we establish a Second Main Theorem following Nevanlinna theory. Finally, with a metric approach, we establish the strong Green-Griffiths-Lang conjecture for Hilbert modular varieties up to finitely many possible exceptions.
收起
-
Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.
收起
-
We formulate a notion of $E_{-1}$ quantisation of $(-2)$-shifted Poisson structures on derived algebraic stacks, depending on a flat right connection on the structure sheaf, as solutions of a quantum master equation. We then parametrise $E_{-1}$ quantisations of $(-2)$-shifted symplectic structures by constructing a map to power series in de Rham cohomology. For a large class of examples, we show that these quantisations give rise to classes in Borel--Moore homology which are closely related to Borisov--Joyce invariants.
收起
-
We formulate a notion of $E_0$ quantisation of $(-1)$-Poisson structures on derived Artin $N$-stacks, and construct a map from $E_0$ quantisations of $(-1)$-shifted symplectic structures to power series in de Rham cohomology. For a square root of the dualising line bundle, this gives an equivalence between even power series and self-dual quantisations. In particular, there is a canonical quantisation of any such square root, which localises to recover the perverse sheaf of vanishing cycles on derived DM stacks, thus giving a form of derived categorification of Donaldson--Thomas invariants.
收起
-
A \emph{cylinder packing} is a family of congruent infinite circular cylinders with mutually disjoint interiors in $3$-dimensional Euclidean space. The \emph{local density} of a cylinder packing is the ratio between the volume occupied by the cylinders within a given sphere and the volume of the entire sphere. The \emph{global density} of the cylinder packing is obtained by letting the radius of the sphere approach infinity. It is known that the greatest global density is obtained when all cylinders are parallel to each other and each cylinder is surrounded by exactly six others. In this case, the global density of the cylinder packing equals $\pi/\sqrt{12}= 0.90689\ldots$. The question is how large a density can a cylinder packing have if one imposes the restriction that \emph{no two cylinders are parallel}. In this paper we prove two results. First, we show that there exist cylinder packings with no two cylinders parallel to each other, whose local density is arbitrarily close to the
收起
-
Often, a given selection game studied in the literature has a known dual game. In dual games, a winning strategy for a player in either game may be used to create a winning strategy for the opponent in the dual. For example, the Rothberger selection game involving open covers is dual to the point-open game. This extends to a general theorem: if $\{\operatorname{ran}{f}:f\in\mathbf C(\mathcal R)\}$ is coinitial in $\mathcal A$ with respect to $\subseteq$, where $\mathbf C(\mathcal R)=\{f\in(\bigcup\mathcal R)^{\mathcal R}:R\in\mathcal R\Rightarrow f(R)\in R\}$ collects the choice functions on the set $\mathcal R$, then $G_1(\mathcal A,\mathcal B)$ and $G_1(\mathcal R,\neg\mathcal B)$ are dual selection games.
收起
-
In this paper, we establish an interesting duality between two different quantum information-processing tasks, namely, classical source coding with quantum side information, and channel coding over c-q channels. The duality relates the optimal error exponents of these two tasks, generalizing the classical results of Ahlswede and Dueck. We establish duality both at the operational level and at the level of the entropic quantities characterizing these exponents. For the latter, the duality is given by an exact relation, whereas for the former, duality manifests itself in the following sense: an optimal coding strategy for one task can be used to construct an optimal coding strategy for the other task. Along the way, we derive a bound on the error exponent for c-q channel coding with constant composition codes which might be of independent interest.
收起
-
Nearly 60 years ago, in a celebrated paper of Kalman and Bucy, it was established that optimal estimation for linear Gaussian systems is dual to a linear-quadratic optimal control problem. In this paper, for the first time, a duality result is established for a general nonlinear filtering problem, mirroring closely the original Kalman-Bucy duality of control and estimation for linear systems. The main result is presented for a finite state space Markov process in continuous time. It is used to derive the classical Wonham filter. The form of the result suggests a natural generalization which is presented as a conjecture for the continuous state space case.
收起
-
We consider the Curie-Weiss Widom-Rowlinson model for particles with spins and holes, with a repulsion strength beta between particles of opposite spins. We provide a closed solution of the model, and investigate dynamical Gibbs-non-Gibbs transitions for the time-evolved model under independent stochastic symmetric spin-flip dynamics. We show that, for sufficiently large beta after a transition time, continuously many bad empirical measures appear. These lie on (unions of) curves on the simplex whose time-evolution we describe.
收起
-
We provide a theoretical algorithm for checking local optimality and escaping saddles at nondifferentiable points of empirical risks of two-layer ReLU networks. Our algorithm receives any parameter value and returns: local minimum, second-order stationary point, or a strict descent direction. The presence of M data points on the nondifferentiability of the ReLU divides the parameter space into at most 2^M regions, which makes analysis difficult. By exploiting polyhedral geometry, we reduce the total computation down to one convex quadratic program (QP) for each hidden node, O(M) (in)equality tests, and one (or a few) nonconvex QP. For the last QP, we show that our specific problem can be solved efficiently, in spite of nonconvexity. In the benign case, we solve one equality constrained QP, and we prove that projected gradient descent solves it exponentially fast. In the bad case, we have to solve a few more inequality constrained QPs, but we prove that the time complexity is exponentia
收起
-
Mean-field integro-differential equations are studied in an abstract framework, through couplings of the corresponding stochastic processes. The long time behaviour of the non-linear process and of the associated particle system is investigated in the perturbative regime. The main difference with the linear (or non-interacting) case is that, when two coupled processes have merged, they have some probability to split.
收起
-
We provide quantitative estimates in total variation distance for positive semi-groups, which can be non-conservative and non-homogeneous. The techniques relies on a family of conservative semigroups that describes a typical particle and Doeblin's type conditions for coupling the associated process. Our aim is to provide quantitative estimates for linear partial differential equations and we develop several applications for population dynamics in varying environment. We start with the asymptotic profile for a growth diffusion model with time and space non-homogeneity. Moreover we provide general estimates for semigroups which become asymptotically homogeneous, which are applied to an age-structured population model. Finally, we obtain a speed of convergence for periodic semi-groups and new bounds in the homogeneous setting. They are are illustrated on the renewal equation.
收起
-
We study a linear form in the values of Euler's series $F(t)=\sum_{n=0}^\infty n!t^n$ at algebraic integer points $\alpha_1, \ldots, \alpha_m \in \mathbb{Z}_{\mathbb{K}}$ belonging to a number field $\mathbb{K}$. Let $v|p$ be a non-Archimedean valuation of $\mathbb{K}$. Two types of non-vanishing results for the linear form $\Lambda_v = \lambda_0 + \lambda_1 F_v(\alpha_1) + \ldots + \lambda_m F_v(\alpha_m)$, $\lambda_i \in \mathbb{Z}_{\mathbb{K}}$, are derived, the second of them containing a lower bound for the $v$-adic absolute value of $\Lambda_v$. The first non-vanishing result is also extended to the case of primes in residue classes. On the way to the main results, we present explicit Pad\'e approximations to the generalised factorial series $\sum_{n=0}^\infty \left( \prod_{k=0}^{n-1} P(k) \right) t^n$, where $P(x)$ is a polynomial of degree one.
收起
-
We begin by explaining how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov-Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, we then give a number of methods for computing arbitrary Petersson products. All this is available in the current release of the Pari/GP package.
收起
-
In this paper, we consider a new pattern of sparsity for SOS Programming named by cross sparsity patterns. We use matrix decompositions for a class of PSD matrices with chordal sparsity patterns to construct sets of supports for a sparse SOS decomposition. The method is applied to the certificate of the nonnegativity of sparse polynomials and unconstrained sparse polynomial optimization problems. Various numerical experiments are given. It turns out that our method can dramatically reduce the computational cost and can handle really huge polynomials, for example, polynomials with $10$ variables, of degree $40$ and more than $5000$ terms.
收起
-
We classify framed and oriented 2-1-0-extended TQFTs with values in the bicategories of Landau-Ginzburg models, whose objects and 1-morphisms are isolated singularities and (either $\mathbb{Z}_2$- or $(\mathbb{Z}_2 \times \mathbb{Q})$-graded) matrix factorisations, respectively. For this we present the relevant symmetric monoidal structures and find that every object $W \in \Bbbk[x_1,\dots,x_n]$ determines a framed extended TQFT. We then compute the Serre automorphisms $S_W$ to show that $W$ determines an oriented extended TQFT if the associated category of matrix factorisations is $(n-2)$-Calabi-Yau. The extended TQFTs we construct from $W$ assign the non-separable Jacobi algebra of $W$ to a circle. This illustrates how non-separable algebras can appear in 2-1-0-extended TQFTs, and more generally that the question of extendability depends on the choice of target category. As another application, we show how the construction of the extended TQFT based on $W=x^{N+1}$ given by Khovanov a
收起
-
Spherical Gauss-Laguerre (SGL) basis functions, i. e., normalized functions of the type $L_{n-l-1}^{(l + 1/2)}(r^2) r^l Y_{lm}(\vartheta,\varphi)$, $|m| \leq l < n \in \mathbb{N}$, $L_{n-l-1}^{(l + 1/2)}$ being a generalized Laguerre polynomial, $Y_{lm}$ a spherical harmonic, constitute an orthonormal polynomial basis of the space $L^2$ on $\mathbb{R}^3$ with radial Gaussian (multivariate Hermite) weight $\exp(-r^2)$. We have recently described fast Fourier transforms for the SGL basis functions based on an exact quadrature formula with certain grid points in $\mathbb{R}^3$. In this paper, we present fast SGL Fourier transforms for scattered data. The idea is to employ well-known basal fast algorithms to determine a three-dimensional trigonometric polynomial that coincides with the bandlimited function of interest where the latter is to be evaluated. This trigonometric polynomial can then be evaluated efficiently using the well-known non-equispaced FFT (NFFT). We proof an error esti
收起
-
Projected least squares (PLS) is an intuitive and numerically cheap technique for quantum state tomography. The method first computes the least-squares estimator (or a linear inversion estimator) and then projects the initial estimate onto the space of states. The main result of this paper equips this point estimator with a rigorous, non-asymptotic confidence region expressed in terms of the trace distance. The analysis holds for a variety of measurements, including 2-designs and Pauli measurements. The sample complexity of the estimator is comparable to the strongest convergence guarantees available in the literature and---in the case of measuring the uniform POVM---saturates fundamental lower bounds.The results are derived by reinterpreting the least-squares estimator as a sum of random matrices and applying a matrix-valued concentration inequality. The theory is supported by numerical simulations for mutually unbiased bases, Pauli observables, and Pauli basis measurements.
收起
-
In this paper we investigate relatively uniformly continuous semigroups on vector lattices using the general framework provided in [KK18]. We introduce the notions of relatively uniformly continuous, differentiable and integrable functions. These notions allow us to study the generators of relatively uniformly continuous semigroups. Our main result, which is a version of the Hille-Yosida theorem, provides sufficient and necessary conditions for an operator to be the generator of an exponentially order bounded relatively uniformly continuous positive semigroup.
收起
-
For any characteristic zero coefficient field, an irreducible representation of a finite $p$-group can be assigned a Roquette $p$-group, called the genotype. This has already been done by Bouc and Kronstein in the special cases Q and C. A genetic invariant of an irrep is invariant under group isomorphism, change of coefficient field, Galois conjugation, and under suitable inductions from subquotients. It turns out that the genetic invariants are precisely the invariants of the genotype. We shall examine relationships between some genetic invariants and the genotype. As an application, we shall count Galois conjugacy classes of certain kinds of irreps.
收起
-
We consider two different genus expansions of the free energy functions of Hermitian one-matrix models, one using fat graphs, one using ordinary graphs (thin graphs). Some structural results are first proved for the thin version of genus expansion using renormalized coupling constants, and then applied to the fat version.
收起
-
For variational problems with $O(N)$-symmetry the existence of several geometrically distinct solutions had been shown by use of group theoretic approach in previous articles. It was done by a crafty choice of a family $H_i \subset O(N)$ subgroups such that the fixed point subspaces $E^{H_i} \subset E$ of the action in a corresponding functional space are linearly independent, next restricting the problem to each $E^{H_i}$ and using the Palais symmetry principle. In this work we give a thorough explanation of this approach showing a correspondence between the equivalence classes of such subgroups, partial orthogonal flags in $\mathbb{R}^N$, and unordered partitions of the number $N$. By showing that spaces of functions invariant with respect to different classes of groups are linearly independent we prove that the amount of series of geometrically distinct solutions obtained in this way grows exponentially in $N$, in contrast to logarithmic, and linear growths of earlier papers.
收起
-
We show that Internal Diffusion Limited Aggregation (IDLA) on $\mathbb{Z}^d$ has near optimal Cheeger constant when the growing cluster is large enough. This implies, through a heat kernel lower bound derived previously in [H], that simple random walk evolving independently on growing in time IDLA cluster is recurrent when $d\ge 3$.
收起
-
This paper contributes to the current studies on regularity properties of noncommutative distributions in free probability theory. More precisely, we consider evaluations of selfadjoint noncommutative polynomials in noncommutative random variables that have finite non-microstates free Fisher information. It is shown that their analytic distributions have H\"older continuous cumulative distribution functions with an explicit H\"older exponent that depends only on the degree of the considered polynomial. This, in particular, guarantees that such polynomial evaluations have finite logarithmic energy and thus finite (non-microstates) free entropy. We further provide a general criterion that gives for weak approximations of measures having H\"older continuous cumulative distribution functions explicit rates of convergence in terms of the Kolmogorov distance. Finally, we apply these results to study the asymptotic eigenvalue distributions of polynomials in GUEs or matrices with more general
收起
-
Let a contact 3-manifold $(Y, \xi_0)$ be the link of a normal surface singularity equipped with its canonical contact structure $\xi_0$. We prove a special property of such contact 3-manifolds of "algebraic" origin: the Heegaard Floer invariant $c^+(\xi_0)\in HF^+(-Y)$ cannot lie in the image of the $U$-action on $HF^+(-Y)$. It follows that Karakurt's "height of $U$-tower" invariants are always 0 for canonical contact structures on singularity links, which contrasts the fact that the height of $U$-tower can be arbitrary for general fillable contact structures. Our proof uses the interplay between the Heegaard Floer homology and N\'emethi's lattice cohomology.
收起
-
Isogeometric analysis was applied very successfully to many problem classes like linear elasticity, heat transfer and incompressible flow problems but its application to compressible flows is very rare. However, its ability to accurately represent complex geometries used in industrial applications makes IGA a suitable tool for the analysis of compressible flow problems that require the accurate resolution of boundary layers. The convection-diffusion solver presented in this chapter, is an indispensable step on the way to developing a compressible flow solver for complex viscous industrial flows. It is well known that the standard Galerkin finite element method and its isogeometric counterpart suffer from spurious oscillatory behaviour in the presence of shocks and steep solution gradients. As a remedy, the algebraic flux correction paradigm is generalized to B-Spline basis functions to suppress the creation of oscillations and occurrence of non-physical values in the solution. This wor
收起
-
This work extends the high-resolution isogeometric analysis approach established for scalar transport equations to the equations of gas dynamics. The group finite element formulation is adopted to obtain an efficient assembly procedure for the standard Galerkin approximation, which is stabilized by adding artificial viscosities proportional to the spectral radius of the Roe-averaged flux-Jacobian matrix. Excess stabilization is removed in regions with smooth flow profiles with the aid of algebraic flux correction \cite{KBNII}. The underlying principles are reviewed and it is shown that linearized FCT-type flux limiting \cite{Kuzmin2009} originally derived for nodal low-order finite elements ensures positivity-preservation for high-order B-Spline discretizations.
收起
-
This paper is devoted to the homogenization of the heat conduction equation, with a homogeneous Dirichlet boundary condition, having a periodically oscillating thermal conductivity and a vanishing volumetric heat capacity. A homogenization result is established by using the evolution settings of multiscale and very weak multiscale convergence. In particular, we investigate how the relation between the volumetric heat capacity and the microscopic structure effects the homogenized problem and its associated local problem. It turns out that the properties of the microscopic geometry of the problem give rise to certain special effects in the homogenization result.
收起
-
We study variants of Buss's theories of bounded arithmetic axiomatized by induction schemes disallowing the use of parameters, and closely related induction inference rules. We put particular emphasis on $\hat\Pi^b_i$ induction schemes, which were so far neglected in the literature. We present inclusions and conservation results between the systems (including a witnessing theorem for $T^i_2$ and $S^i_2$ of a new form), results on numbers of instances of the axioms or rules, connections to reflection principles for quantified propositional calculi, and separations between the systems.
收起
-
Integral relations with the Cauchy kernel on a semi-axis for the Laguerre polynomials, the confluent hypergeometric function, and the cylindrical functions are derived. A part of these formulas is obtained by exploiting some properties of the Hermite polynomials, including their Hilbert and Fourier transforms and connections to the Laguerre polynomials. The relations discovered give rise to complete systems of new orthogonal functions. Free of singular integrals, exact and approximate solutions to the characteristic and complete singular integral equations in a semi-infinite interval are proposed. Another set of the Hilbert transforms in a semi-axis are deduced from integral relations with the Cauchy kernel in a finite segment for the Jacobi polynomials and the Jacobi functions of the second kind by letting some parameters involved go to infinity. These formulas lead to integral relations for the Bessel functions. Their application to a model problem of contact mechanics is given. A ne
收起
-
Let $I\subset (0,\infty )$ be an interval that is closed with respect to the multiplication. The operations $C_{f,g}\colon I^{2}\rightarrow I$ of the form \begin{equation*} C_{f,g}\left( x,y\right) =\left( f\circ g\right) ^{-1}\left( f\left( x\right) \cdot g\left( y\right) \right) \text{,} \end{equation*} where $f,g$ are bijections of $I$ are considered. Their connections with generalized weighted quasi-geometric means is presented. It is shown that invariance question within the class of this operations leads to means of iterative type and to a problem on a composite functional equation. An application of the invariance identity to determine effectively the limit of the sequence of iterates of some generalized quasi-geometric mean-type mapping, and the form of all continuous functions which are invariant with respect to this mapping are given. The equality of two considered operations is also discussed.
收起
-
We consider the massive Thirring model in the laboratory coordinates and explain how the inverse scattering transform can be developed with the Riemann-Hilbert approach. The key ingredient of our method is to transform the corresponding spectral problem to the equivalent forms: one is suitable for the spectral parameter at the origin and the other one is suitable for the spectral parameter at infinity. Global solutions to the massive Thirring model are recovered from the reconstruction formulae at the origin and at infinity.
收起
