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We define and discuss properties of the class of unbounded operators which attain minimum modulus. We establish a relationship between this class and the class of norm attaining bounded operators and compare the properties of both. Also we define absolutely minimum attaining operators (possibly unbounded) and characterize injective absolutely minimum attaining operators as those with compact generalized inverse. We give several consequences, one of them is that every such operator has a non trivial hyper invariant subspace.
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We study a class of bilevel convex optimization problems where the goal is to find the minimizer of an objective function in the upper level, among the set of all optimal solutions of an optimization problem in the lower level. A wide range of problems in convex optimization can be formulated using this class. An important example is the case where an optimization problem is ill-posed. In this paper, our interest lies in addressing the bilevel problems, where the lower level objective is given as a finite sum of separate nondifferentiable convex component functions. This is the case in a variety of applications in distributed optimization, such as large-scale data processing in machine learning and neural networks. To the best of our knowledge, this class of bilevel problems, with a finite sum in the lower level, has not been addressed before. Motivated by this gap, we develop an iterative regularized incremental subgradient method, where the agents update their iterates in a cyclic ma
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We study the interaction between two structures on the group of polynomial automorphisms of the affine plane: its structure as an amalgamated free product and as an infinite-dimensional algebraic variety. We introduce a new conjecture, and show how it implies the Polydegree Conjecture. As the new conjecture is an ideal membership question, this shows that the Polydegree Conjecture is algorithmically decidable. We further describe how this approach provides a unified and shorter method of recovering existing results of Edo and Furter.
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Let $T_{\epsilon}$ be the noise operator acting on functions on the boolean cube $\{0,1\}^n$. Let $f$ be a nonnegative function on $\{0,1\}^n$ and let $q \ge 1$. We upper bound the $\ell_q$ norm of $T_{\epsilon} f$ by the average $\ell_q$ norm of conditional expectations of $f$, given sets of roughly $(1-2\epsilon)^{r(q)} \cdot n$ variables, where $r$ is an explicitly defined function of $q$. We describe some applications for error-correcting codes and for matroids. In particular, we derive an upper bound on the weight distribution of duals of BEC-capacity achieving binary linear codes. This improves the known bounds on the linear-weight components of the weight distribution of constant rate binary Reed-Muller codes for almost all rates.
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We develop a discretely entropy-stable line-based discontinuous Galerkin method for hyperbolic conservation laws based on a flux differencing technique. By using standard entropy-stable and entropy-conservative numerical flux functions, this method guarantees that the discrete integral of the entropy is non-increasing. This nonlinear entropy stability property is important for the robustness of the method, in particular when applied to problems with discontinuous solutions or when the mesh is under-resolved. This line-based method is significantly less computationally expensive than a standard DG method. Numerical results are shown demonstrating the effectiveness of the method on a variety of test cases, including Burgers' equation and the Euler equations, in one, two, and three spatial dimensions.
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We present here an analysis of the regularity of minimizers of a variational model for epitaxially strained thin-films identified by the authors in the companion paper [Davoli E., Piovano P., Derivation of a heteroepitaxial thin-film model. Submitted 2018]. The regularity of energetically-optimal film profiles is studied by extending previous methods and by developing new ideas based on transmission problems. The achieved regularity results relate to both the Stranski-Krastanow and the Volmer-Weber modes, the possibility of different elastic properties between the film and the substrate, and the presence of the surface tensions of all three involved interfaces: film/gas, substrate/gas, and film/substrate. Finally, geometrical conditions are provided for the optimal wetting angle, i.e., the angle formed at the contact point of films with the substrate. In particular, the Young-Dupr\'e law is shown to hold, yielding what appears to be the first analytical validation of such law for a thi
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Notions of depth in regression have been introduced and studied in the literature. Regression depth (RD) of Rousseeuw and Hubert (1999) (RH99), the most famous one, is a direct extension of Tukey location depth (Tukey (1975)) to regression. RD of RH99 satisfies all the desirable axiomatic properties in Zuo (2018) and therefore could serve as a real depth notion in regression. Like its location counterpart, the most remarkable advantage of notion of the depth in regression is to introduce directly the maximum (or deepest) regression depth estimator for regression parameters in a multi-dimensional setting. Classical questions for the maximum regression depth estimator include (i) is it a consistent estimator (or rather under what sufficient conditions, it is consistent) and (ii) is there any limiting distribution? Bai and He (1999) (BH99) intended to answer these questions. Unfortunately, they treated a closely related but different depth notion than the intended one. So the above questi
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Let $k$ be an algebraically-closed field, and $B$ a unital, associative $k$-algebra with $n := \dim_kB < \infty$. For each $1 \le m \le n$, the collection of all $m$-dimensional subalgebras of $B$ carries the structure of a projective variety, which we call $\operatorname{AlgGr_m(B)}$. The group $\operatorname{Aut}_k(B)$ of all $k$-algebra automorphisms of $B$ acts regularly on $\operatorname{AlgGr}_m(B)$. In this paper, we study the problem of explicitly describing $\operatorname{AlgGr}_m(B)$, and classifying its $\operatorname{Aut}_k(B)$-orbits. Inspired by recent results on maximal subalgebras of finite-dimensional algebras, we compute the homogeneous vanishing ideal of $\operatorname{AlgGr}_{n-1}(B)$ when $B$ is basic, and explictly describe its irreducible components. We show that in this case, $\operatorname{AlgGr}_{n-1}(B)$ is a finite union of $\operatorname{Aut}_k(B)$-orbits if $B$ is monomial or its Ext quiver is Schur, but construct a class of examples to show that these
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In the autonomous driving area, interaction between vehicles is still a piece of puzzle which has not been fully resolved. The ability to intelligently and safely interact with other vehicles can not only improve self driving quality but also be beneficial to the global driving environment. In this paper, a Bayesian persuasive driving algorithm based on optimization is proposed, where the ego vehicle is the persuader (information sender) and the surrounding vehicle is the persuadee (information receiver). In the persuasion process, the ego vehicle aims at changing the surrounding vehicle's posterior belief of the world state by providing certain information via signaling in order to achieve a lower cost for both players. The information received by the surrounding vehicle and its belief of the world state are described by Gaussian distributions. Simulation results in several common traffic scenarios are provided to demonstrate the proposed algorithm's capability of handling interaction
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There exist several multivariate extensions of the classical Sonine integral representation for Bessel functions of some index $\mu+ \nu$ with respect to such functions of lower index $\mu.$ For Bessel functions on matrix cones, Sonine formulas involve beta densities $\beta_{\mu,\nu}$ on the cone and trace already back to Herz. The Sonine representations known so far on symmetric cones are restricted to continuous ranges $\Re\mu, \Re \nu > \mu_0$, where the involved Beta densities are probability measures and the limiting index $\mu_0\geq 0$ depends on the rank of the cone. It is zero only in the one-dimensional case, but larger than zero in all multivariate cases. In this paper, we study the extension of Sonine formulas for Bessel functions on symmetric cones to values of $\nu$ below the critical limit $\mu_0$. This is achieved by an analytic extension of the involved Beta measures as tempered distributions. Following recent ideas by A. Sokal for Riesz distributions on symmetric co
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We construct a non-commutative, non-cocommutative, graded bialgebra $\mathbf{\Pi}$ with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto-Poirier-Reutenauer Hopf algebra, this bialgebra does not have finite graded dimension. After giving formulas for the product and coproduct, we show that there is a natural morphism from $\mathbf{\Pi}$ to the algebra of quasi-symmetric functions, under which the image of a permutation is its associated Stanley symmetric function. As an application, we use this morphism to derive some new enumerative identities. We also describe analogues of $\mathbf{\Pi}$ for the other classical types. In these cases, the relevant objects are module coalgebras rather than bialgebras, but there are again natural morphisms to the quasi-symmetric functions, under which the image of a signed permutation is the corresponding Stanley symmetric function of type B, C, or D.
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A two-variable extension of the Bannai-Ito polynomials is presented. They are obtained via $q\to-1$ limits of the bivariate $q$-Racah and Askey-Wilson orthogonal polynomials introduced by Gasper and Rahman. Their orthogonality relation is obtained. These new polynomials are also shown to be multispectral. Two Dunkl shift operators are seen to be diagonalized by the bivariate Bannai-Ito polynomials and 3- and 9-term recurrence relations are provided.
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We introduce a concept of blown-up \v{C}ech cohomology for coherent and quasi-coherent sheaves of homological dimension $\leq 1$ on a non-singular real affine variety. Its construction involves a directed set of multi-blowups. We establish, in particular, long exact cohomology sequence and Cartan's Theorem B. Finally, some applications are provided, including universal solution to the first Cousin problem (after blowing up).
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We consider the discrete Boolean model of percolation on graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the retention parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.
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We consider two directed polymer models in the Kardar-Parisi-Zhang (KPZ) universality class: the O'Connell-Yor semi-discrete directed polymer with boundary sources and the continuum directed random polymer with (m,n)-spiked boundary perturbations. The free energy of the continuum polymer is the Hopf-Cole solution of the KPZ equation with the corresponding (m,n)-spiked initial condition. This new initial condition is constructed using two semi-discrete polymer models with independent bulk randomness and coupled boundary sources. We prove that the limiting fluctuations of the free energies rescaled by the 1/3rd power of time in both polymer models converge to the Borodin-Peche type deformations of the GUE Tracy-Widom distribution.
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Distributed machine learning algorithms enable learning of models from datasets that are distributed over a network without gathering the data at a centralized location. While efficient distributed algorithms have been developed under the assumption of faultless networks, failures that can render these algorithms nonfunctional occur frequently in the real world. This paper focuses on the problem of Byzantine failures, which are the hardest to safeguard against in distributed algorithms. While Byzantine fault tolerance has a rich history, existing work does not translate into efficient and practical algorithms for high-dimensional distributed learning. In this paper, two variants of an algorithm termed \emph{Byzantine-resilient distributed coordinate descent} (ByRDiE) are developed and analyzed that enable distributed learning in the presence of Byzantine failures. Theoretical analysis and numerical experiments presented in the paper highlight the usefulness of ByRDiE for high-dimension
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The solution of the Calabi Conjecture by Yau implies that every K\"ahler Calabi-Yau manifold $X$ admits a metric with holonomy contained in $\operatorname{SU}(n)$, and that these metrics are parametrized by the positive cone in $H^2(X,\mathbb{R})$. In this work we give evidence of an extension of Yau's theorem to non-K\"ahler manifolds, where $X$ is replaced by a compact complex manifold with vanishing first Chern class endowed with a holomorphic Courant algebroid $Q$ of Bott-Chern type. The equations that define our notion of `best metric' correspond to a mild generalization of the Hull-Strominger system, whereas the role of the second cohomology is played by an affine space of `Aeppli classes' naturally associated to $Q$ via secondary holomorphic characteristic classes introduced by Donaldson
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For any $k\geq 1$, we compare the number of polynomials that have exactly $k$ irreducible factors in $\mathbf{F}_q[t]$ among different arithmetic progressions. We prove asymptotic formulas for the difference of counting functions uniformly for $k$ in a certain range. We unconditionally derive the existence of the limiting distribution of this difference. In contrast to the case of products of $k$ prime numbers, we show the existence of complete biases in the function field setting, that is the difference function may have constant sign. We give several examples to exhibit this new phenomenon.
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Given an inner model $W \subset V$ and a regular cardinal $\kappa$, we consider two alternatives for adding a subset to $\kappa$ by forcing: the Cohen poset $Add(\kappa,1)$, and the Cohen poset of the inner model $Add(\kappa,1)^W$. The forcing from $W$ will be at least as strong as the forcing from $V$ (in the sense that forcing with the former adds a generic for the latter) if and only if the two posets have the same cardinality. On the other hand, a sufficient condition is established for the poset from $V$ to fail to be as strong as that from $W$. The results are generalized to $Add(\kappa,\lambda)$, and to iterations of Cohen forcing where the poset at each stage comes from an arbitrary intermediate inner model.
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In this paper, the effective use of multiple quadrotor drones as an aerial antenna array that provides wireless service to ground users is investigated. In particular, under the goal of minimizing the airborne service time needed for communicating with ground users, a novel framework for deploying and operating a drone-based antenna array system whose elements are single-antenna drones is proposed. In the considered model, the service time is minimized by minimizing the wireless transmission time as well as the control time that is needed for movement and stabilization of the drones. To minimize the transmission time, first, the antenna array gain is maximized by optimizing the drone spacing within the array. In this case, using perturbation techniques, the drone spacing optimization problem is addressed by solving successive, perturbed convex optimization problems. Then, the optimal locations of the drones around the array's center are derived such that the transmission time for the u
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Monoidal categories enriched in a braided monoidal category $\mathcal{V}$ are classified by braided oplax monoidal functors from $\mathcal{V}$ to the Drinfeld centers of ordinary monoidal categories. In this article, we prove that this classifying functor is strongly monoidal if and only if the original $\mathcal{V}$-monoidal category is tensored over $\mathcal{V}$. We then define a completion operation which produces a tensored $\mathcal{V}$-monoidal category $\overline{\mathcal{C}}$ from an arbitrary $\mathcal{V}$-monoidal category $\mathcal{C}$, and we determine many equivalent conditions which imply $\mathcal{C}$ and $\overline{\mathcal{C}}$ are $\mathcal{V}$-monoidally equivalent. Since being tensored is a property of the underlying $\mathcal{V}$-category of a $\mathcal{V}$-monoidal category, we begin by studying the equivalence between (tensored) $\mathcal{V}$-categories and oplax (strong) $\mathcal{V}$-module categories respectively. We then define the completion operation for $
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The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty$ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty$ is inaccessible, it is always an entrance boundary. In the case where $\infty$ is accessible, explosion can occur either by a single jump to $\infty$ (the process at $z$ jumps to $\infty$ at rate $\lambda z$ for some $\lambda>0$) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when $\infty$ is accessible and $0\leq \frac{2\lambda}{c}<1$, the extended process is reflected at $\infty$. In the case $\frac{2\lambda}{c}\geq 1$, $\infty$ is an exit of the extended process. When the branching mechanism is not the Lapla
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We conjecture that the set of homogeneous probability measures on the maximal Satake compactification of an arithmetic locally symmetric space $S=\Gamma\backslash G/K$ is compact. More precisely, given a sequence of homogeneous probability measures on $S$, we expect that any weak limit is homogeneous with support contained in precisely one of the boundary components (including $S$ itself). We introduce several tools to study this conjecture and we prove it in a number of cases, including when $G={\rm SL}_3(\mathbb{R})$ and $\Gamma={\rm SL}_3(\mathbb{Z})$.
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In this paper, we are concerned with the convergence rate of a FEM based numerical scheme approximating extremal functions of the Sobolev inequality. We prove that when the domain is polygonal and convex in $\R^2$, the convergence of a finite element solution to an exact extremal function in $L^2$ and $H^1$ norms has the rates $O(h^2)$ and $O(h)$ respectively, where $h$ denotes the mesh size of a triangulation of the domain.
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Multi-server single-message private information retrieval is studied in the presence of side information. In this problem, $K$ independent messages are replicatively stored at $N$ non-colluding servers. The user wants to privately download one message from the servers without revealing the index of the message to any of the servers, leveraging its $M$ side information messages. It is shown that the capacity for this problem is $(1+\frac{1}{N}+\frac{1}{N^2}+\dots+\frac{1}{N^{\left\lceil \frac{K}{M+1}\right\rceil-1}})^{-1}$.
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This two-part paper is concerned with the problem of minimizing a linear objective function subject to a bilinear matrix inequality (BMI) constraint. In this part, we first consider a family of convex relaxations which transform BMI optimization problems into polynomial-time solvable surrogates. As an alternative to the state-of-the-art semidefinite programming (SDP) and second-order cone programming (SOCP) relaxations, a computationally efficient parabolic relaxation is developed, which relies on convex quadratic constraints only. Next, we developed a family of penalty functions, which can be incorporated into the objective of SDP, SOCP, and parabolic relaxations to facilitate the recovery of feasible points for the original non-convex BMI optimization. Penalty terms can be constructed using any arbitrary initial point. We prove that if the initial point is sufficiently close to the feasible set, then the penalized relaxations are guaranteed to produce feasible points for the original
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The first part of this paper proposed a family of penalized convex relaxations for solving optimization problems with bilinear matrix inequality (BMI) constraints. In this part, we generalize our approach to a sequential scheme which starts from an arbitrary initial point (feasible or infeasible) and solves a sequence of penalized convex relaxations in order to find feasible and near-optimal solutions for BMI optimization problems. We evaluate the performance of the proposed method on the H2 and Hinfinity optimal controller design problems with both centralized and decentralized structures. The experimental results based on a variety of benchmark control plants demonstrate the promising performance of the proposed approach in comparison with the existing methods.
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This paper is a systematic approach to the construction of coronas (i.e. Higson dominated boundaries at infinity) of combable spaces. We introduce three additional properties for combings: properness, coherence and expandingness. Properness is the condition under which our construction of the corona works. Under the assumption of coherence and expandingness, attaching our corona to a Rips complex construction yields a contractible $\sigma$-compact space in which the corona sits as a $Z$-set. This results in bijectivity of transgression maps, injectivity of the coarse assembly map and surjectivity of the coarse co-assembly map. For groups we get an estimate on the cohomological dimension of the corona in terms of the asymptotic dimension. Furthermore, if the group admits a finite model for its classifying space $BG$, then our constructions yield a $Z$-structure for the group.
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We analyze existence of crossed product constructions of Lie group actions on C^*-algebras which are singular. These are actions where the group need not be locally compact, or the action need not be strongly continuous. In particular, we consider the case where spectrum conditions are required for the implementing unitary group in covariant representations of such actions. The existence of a crossed product construction is guaranteed by the existence of "cross representations". For one-parameter automorphism groups, we prove that the existence of cross representations is stable with respect to a large set of perturbations of the action, and we fully analyze the structure of cross representations of inner actions on von Neumann algebras. For one-parameter automorphism groups we study the cross property for covariant representations, where the generator of the implementing unitary group is positive. In particular, we find that if the Borchers-Arveson minimal implementing group is cross,
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We study two families of cyclotomic graphs and perfect codes in them. They are Cayley graphs on the additive group of $\mathbb{Z}[\zeta_m]/A$, with connection sets $\{\pm (\zeta_m^i + A): 0 \le i \le m-1\}$ and $\{\pm (\zeta_m^i + A): 0 \le i \le \phi(m) - 1\}$, respectively, where $\zeta_m$ ($m \ge 2$) is an $m$th primitive root of unity, $A$ a nonzero ideal of $\mathbb{Z}[\zeta_m]$, and $\phi$ Euler's totient function. We call them the $m$th cyclotomic graph and the second kind $m$th cyclotomic graph, and denote them by $G_{m}(A)$ and $G^*_{m}(A)$, respectively. We give a necessary and sufficient condition for $D/A$ to be a perfect $t$-code in $G^*_{m}(A)$ and a necessary condition for $D/A$ to be such a code in $G_{m}(A)$, where $t \ge 1$ is an integer and $D$ an ideal of $\mathbb{Z}[\zeta_m]$ containing $A$. In the case when $m = 3, 4$, $G_m((\alpha))$ is known as an Eisenstein-Jacobi and Gaussian networks, respectively, and we obtain necessary conditions for $(\beta)/(\alpha)$ to
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We study deep neural networks and their use in semiparametric inference. We provide new rates of convergence for deep feedforward neural nets and, because our rates are sufficiently fast (in some cases minimax optimal), prove that semiparametric inference is valid using deep nets for first-step estimation. Our estimation rates and semiparametric inference results are the first in the literature to handle the current standard architecture: fully connected feedforward neural networks (multi-layer perceptrons), with the now-default rectified linear unit (ReLU) activation function and a depth explicitly diverging with the sample size. We discuss other architectures as well, including fixed-width, very deep networks. We establish nonasymptotic bounds for these deep ReLU nets, for both least squares and logistic loses in nonparametric regression. We then apply our theory to develop semiparametric inference, focusing on treatment effects and expected profits for concreteness, and demonstrate
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This is the third prepublication version of a book on derived categories, that will be published by Cambridge University Press. The purpose of the book is to provide solid foundations for the theory of derived categories, and to present several applications of this theory in commutative and noncommutative algebra. The emphasis is on constructions and examples, rather than on axiomatics. Here are the topics covered in the book: - A review of standard facts on abelian categories. - Differential graded algebra (DG rings, DG modules, DG categories and DG functors). - Triangulated categories and triangulated functors between them. How they arise from the DG background. The homotopy category K(A,M) of DG A-modules in M. - Localization of categories. The derived category D(A,M), which is the localization of K(A,M) with respect to the quasi-isomorphisms. - Left and right derived functors of a triangulated functor. - K-injective, K-projective and K-flat DG modules. Their roles, and their existe
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For a smooth quasi-projective variety $X$ with a reductive group action and semi-invariant regular function $W:X\to {\mathbb A}^1$, we describe a semi-orthogonal decomposition of the derived factorization categories associated to the sum $W+F:X\times {\mathbb A}^m\to{\mathbb A}^1$ of $W$ and a certain semi-invariant regular function $W:X\times{\mathbb A}^m\to{\mathbb A}^1$. Our situation does not require that $W+F$ is Thom--Sebastiani sum. As an application, we prove that the homotopy category of maximally graded matrix factorizations of an invertible polynomial $f$ of chain type has a full exceptional collection ${\mathcal E}$, and we also show that the length of ${\mathcal E}$ is the Milnor number of the Berglund--H\"ubsch transpose $\widetilde{f}$ of $f$. We also give a semi-orthogonal decomposition that is a generalization of a special version of Kuznetsov--Perry's semi-orthogonal decomposition of a certain equivariant category associated to the derived category of a cyclic cover o
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In this paper, we derive the Factorization Method to solve the inverse shape problem of recovering an inclusion with a generalized impedance condition from electrostatic Cauchy data. The generalized impedance condition is a second-order differential operator applied to the boundary of the inclusion. We assume that the Dirichlet-to-Neumann mapping is given from measuring the current on the outer boundary from an imposed voltage. Numerical examples are given to show the effectiveness of the proposed inversion method for recovering the inclusion. We also consider the inverse impedance problem of determining the material parameters from the Dirichlet-to-Neumann mapping assuming the inclusion has been reconstructed where uniqueness for the reconstruction of the coefficients is proven.
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We prove that more than nine percent of the central values $L(\frac{1}{2},\chi_p)$ are non-zero, where $p\equiv 1 \pmod{8}$ ranges over primes and $\chi_p$ is the real primitive Dirichlet character of conductor $p$. Previously, it was not known whether a positive proportion of these central values are non-zero. As a by-product, we obtain the order of magnitude of the second moment of $L(\frac{1}{2},\chi_p)$, and conditionally we obtain the order of magnitude of the third moment. Assuming the Generalized Riemann Hypothesis, we show that our lower bound for the second moment is asymptotically sharp.
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We investigate two asymptotic properties of a spatial preferential-attachment model introduced by E. Jacob and P. M\"orters (2013). First, in a regime of strong linear reinforcement, we show that typical distances are at most of doubly-logarithmic order. Second, we derive a large deviation principle for the empirical neighbourhood structure and express the rate function as solution to an entropy minimisation problem in the space of stationary marked point processes.
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In this work we address the problem of distributed optimization of the sum of convex cost functions in the context of multi-agent systems over lossy communication networks. Building upon operator theory, first, we derive an ADMM-like algorithm that we refer to as relaxed ADMM (R-ADMM) via a generalized Peaceman-Rachford Splitting operator on the Lagrange dual formulation of the original optimization problem. This specific algorithm depends on two parameters, namely the averaging coefficient $\alpha$ and the augmented Lagrangian coefficient $\rho$. We show that by setting $\alpha=1/2$ we recover the standard ADMM algorithm as a special case of our algorithm. Moreover, by properly manipulating the proposed R-ADMM, we are able to provide two alternative ADMM-like algorithms that present easier implementation and reduced complexity in terms of memory, communication and computational requirements. Most importantly the latter of these two algorithms provides the first ADMM-like algorithm whi
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We study extreme points of the set of finite-outcome positive-operator-valued measures (POVMs) on finite-dimensional Hilbert spaces and particularly the possible ranks of the effects of an extreme POVM. We give results discussing ways of deducing new rank combinations of extreme POVMs from rank combinations of known extreme POVMs and, using these results, show ways to characterize rank combinations of extreme POVMs in low dimensions. We show that, when a rank combination together with a given dimension of the Hilbert space solve a particular packing problem, there is an extreme POVM on the Hilbert space with the given ranks. This geometric method is particularly effective for constructing extreme POVMs with desired rank combinations.
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In this work, we focus on distributed cache systems with non-uniform storage capacity across caches. We compare the performance of our system with the performance of a system with the same cumulative storage distributed evenly across the caches. We characterize the extent to which the performance of the distributed cache system deteriorates due to storage heterogeneity. The key takeaway from this work is that the effects of heterogeneity in the storage capabilities depend heavily on the popularity profile of the contents being cached and delivered. We analytically show that compared to the case where contents popularity is comparable across contents, lopsided popularity profiles are more tolerant to heterogeneity in storage capabilities. We validate our theoretical results via simulations.
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This paper defines Dirichlet matroids, a class of matroids arising from electrical networks and almost-balanced biased graphs. We prove four main results. First we use properties of the electrical response matrix to show that every Dirichlet matroid has the half-plane property. Second we bound the coefficients of the precoloring polynomial in terms of the chromatic polynomial. Third we prove a simple characterization of 3-connected Dirichlet matroids. And fourth we prove a circular network analog of the duality theorem for planar graphic matroids.
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This article investigates the Euler-Maruyama approximation procedure for stochastic differential equations in the framework of G-Browinian motion with non-linear growth and non-Lipschitz conditions. Subject to non-linear growth condition, it is revealed that the Euler-Maruyama approximate solutions are bounded in M^2_G.In view ofnon-linear growth and non-uniform Lipschitz conditions,we give estimates for the difference between the exact solution Z(t) and approximate solutions Zq(t) of SDEs in the framework of G-Brownia nmotion.
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Full counting statistics for a wide class of Luttinger liquid tunnel junctions in a "weak link" regime is considered beyond the lowest orders in tunnel coupling and out of the equilibrium in the time domain. Especially, two important mathematical statements: the FCS-S-theorem and the FCS-S-lemma about exact re-exponentiation of Keldysh-contour-ordered evolution operator are proven. These statements are the generalizations of S-theorem and S-lemma have been recently proven by the author in Ref.[G.A.Skorobagatko, Phys.Rev.B, 98, 045409 (2018)]. It is shown that FCS-S-theorem can be treated also as the proof of dynamical Jarzynski equality for tunnel electron transport out of the equilibrium (in time domain). As the result, exact time-dependent cumulant generating functional is derived being valid at arbitrary electron-electron repulsion in the Luttinger liquid leads of the junction, arbitrary temperature and arbitrary bias voltage. Respective general formula in its long-time asymptotics
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We study the existence and stability of standing waves for a system of nonlinear Schr\"odinger equations with quadratic interaction in dimensions $d\leq 3$. We also study the characterization of finite time blow-up solutions with minimal mass to the system under mass resonance condition in dimension $d=4$. Finite time blow-up solutions with minimal mass are showed to be (up to symmetries) pseudo-conformal transformations of a ground state standing wave.
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We investigate the exponents of the total Cohen groups $[\Omega(\mathbb S^{r+1}), \Omega(Y)]$ for any $r\ge 1$. In particular, we show that for $p\ge 3$, the $p$-primary exponents of $[\Omega(\mathbb S^{r+1}), \Omega(\mathbb S^{2n+1})]$ and $[\Omega(\mathbb S^{r+1}), \Omega(\mathbb S^{2n})]$ coincide with the $p$-primary homotopy exponents of spheres $\mathbb S^{2n+1}$ and $\mathbb S^{2n}$, respectively. We further study the exponent problem when $Y$ is a space with the homotopy type of $\Sigma(n)/G$ for a homotopy $n$-sphere $\Sigma(n)$, the complex projective space $\mathbb{C}P^n$ for $n\ge 1$ or the quaternionic projective space $\mathbb{H}P^n$ for $1\le n\le \infty$.
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We give an elementary proof of the fact that any elliptic curve $E$ over an algebraically closed non-archimedean field $K$ with residue characteristic $\neq{2,3}$ and with $v(j(E))<0$ can be faithfully tropicalized. We first define an adapted form of minimal models over non-discrete valuation rings and we recover several well-known theorems from the discrete case. Using these, we create an explicit family of marked elliptic curves $(E,P)$, where $E$ has multiplicative reduction and $P$ is a three-torsion point that reduces to the singular point on the reduction of $E$. We then follow the strategy for proving faithfulness as in \cite[Theorem 6.2]{BPR11} and construct an embedding such that its tropicalization contains a cycle of length $-v(j(E))$. A key difference between this approach and the approach in \cite{BPR11} is that the proof in this text does not use any of the analytic theory on Berkovich spaces such as the {\it{Poincar\'{e}-Lelong formula}} or \cite[Theorem 5.25]{BPR11}
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Automatic cubatures approximate multidimensional integrals to user-specified error tolerances. For high dimensional problems, it makes sense to fix the sampling density but determine the sample size, $n$, automatically. Bayesian cubature postulates that the integrand is an instance of a stochastic process. Here we assume a Gaussian process parameterized by a constant mean and a covariance function defined by a scale parameter times a parameterized function specifying how the integrand values at two different points in the domain are related. These parameters are estimated from integrand values or are given non-informative priors. The sample size, $n$, is chosen to make the half-width of the credible interval for the Bayesian posterior mean no greater than the error tolerance. The process just outlined typically requires vector-matrix operations with a computational cost of $O(n^3)$. Our innovation is to pair low discrepancy nodes with matching kernels that lower the computational cost
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Sinha has constructed a cosimplicial space X for a fixed integer N. One of his main result states that for N > 3, X is a cosimplicial model for the space of long knots (modulo immersions). On the other hand, Lambrechts, Turchin and Volic showed that for N > 3 the homology Bousfield-Kan spectral sequence associated to Sinha's cosimplicial space collapses at the page 2 rationally. Their approach consists first to prove the formality of some other diagrams approximating X and next deduce the collapsing result. In this paper, we prove directly the formality of Sinha's cosimplicial space and this allows us to construct a very short proof of the collapsing result for N > 2. Moreover, we prove that the isomorphism between the page 2 and the rational homology of the space of long knots modulo immersions is of Poisson algebras, when N > 3.
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The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous dependence on the underlying set, are provided. These are required to study a Minkowski-type problem for the dual Orlicz curvature measure. We mainly focus on the case when $G$ and $\psi$ are both increasing, thus complementing our previous work. The Minkowski problem asks to characterize Borel measures $\mu$ on $\sphere$ for which there is a convex body $K$ in $\R^n$ containing the origin such that $
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For the purpose of uncertainty propagation a new quadrature rule technique is proposed that has positive weights, has high degree, and is constructed using only samples that describe the probability distribution of the uncertain parameters. Moreover, nodes can be added to the quadrature rule, resulting into a sequence of nested rules. The rule is constructed by iterating over the samples of the distribution and exploiting the null space of the Vandermonde-system that describes the nodes and weights, in order to select which samples will be used as nodes in the quadrature rule. The main novelty of the quadrature rule is that it can be constructed using any number of dimensions, using any basis, in any space, and using any distribution. It is demonstrated both theoretically and numerically that the rule always has positive weights and therefore has high convergence rates for sufficiently smooth functions. The convergence properties are demonstrated by approximating the integral of the Ge
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A geodesic orbit manifold is a complete Riemannian manifold all of whose geodesics are orbits of one-parameter groups of isometries. We give both a geometric and an algebraic characterization of geodesic orbit manifolds that are diffeomorphic to $\mathbf{R}^n$. Along the way, we establish various structural properties of more general geodesic orbit manifolds.
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We prove the existence of the global attractor in $ \dot H^s$, $s > 11/12$ for the weakly damped and forced mKdV on the one dimensional torus. The existence of global attractor below the energy space has not been known, though the global well-posedness below the energy space is established. We directly apply the I-method to the damped and forced mKdV, because the Miura transformation does not work for the mKdV with damping and forcing terms. We need to make a close investigation into the trilinear estimates involving resonant frequencies, which are different from the bilinear estimates corresponding to the KdV.
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Apple has successfully completed its $400 million acquisition of Shazam, and the company announced it will be removing all ads from the app "soon." The Verge reports: The acquisition was temporarily held up because of an investigation by the European Union, which scrutinized the deal over potential antitrust concerns. But regulators gave it the thumbs up earlier this month. Shazam has been downloaded over 1 billion times around the world and is used over 20 million times every day, according to Apple's press release. The app has been around since the beginnings of the App Store and was one of the coolest early demos of what a mobile app could do. You'd hold up your phone, let it listen to a song playing nearby for a few seconds, and the track and artist information would just pop up on screen like magic. All these years later, it's now a feature that's available on many platforms and one we take for granted.
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Apple on Monday made available to the public macOS Mojave -- aka macOS 10.14, the latest major update to its desktop operating system. From a report: Though Mojave is substantially focused on under-the-hood improvements, it includes several major changes to the Mac's Finder, as well as a small collection of apps that were ported from iOS. On the Finder side, Apple has introduced a system-wide Dark Mode, which optionally reskins the entire user interface with black or dark gray elements. Dark Mode pairs up with Dynamic Desktop, which can automatically adjust certain desktop images in sync with time of day (morning, afternoon, and evening) changes. Minutes ahead of the release, Patrick Wardle, chief researcher officer at Digita Security, tweeted a video of an apparent privacy feature bypass that's designed to prevent apps from improperly accessing a user's personal data. From a report: For years, Macs have forced apps to ask for permission before accessing your contacts and calendar afte
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An anonymous reader shares a report: The world's first autonomous tram was launched in unspectacular style in the city of Potsdam, west of Berlin, on Friday. The Guardian was the first English-language newspaper to be offered a ride on the vehicle developed by a team of 50 computer scientists, engineers, mathematicians, and physicists at the German engineering company Siemens. Fitted with multiple radar, lidar (light from a laser), and camera sensors, forming digital eyes that film the tram and its surroundings during every journey, the tram reacts to trackside signals and can respond to hazards faster than a human. Its makers say it is some way from being commercially viable but they do expect it to contribute to the wider field of driverless technology, and have called it an important milestone on the way to autonomous driving. Travelling in real traffic from the tram depot of Potsdam's transport company ViP, the articulated Combino model tram whirred its way through a high-rise hous
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As Google celebrates its 20th anniversary, it announced a range of new updates to its namesake search engine. The Mountain View company announced it was drawing on its artificial intelligence capabilities to provide smart videos in Google search with a new "Featured Videos" card. It will start to play videos in results, one after another -- but only show the short parts of videos that are relevant to your search. Google, the parent company of which is Alphabet, also introduced an activity card which would show pages a user has visited, at the top of search results. Users will have the ability to delete items from this activity card. The company also introduced "Collections," through which it will let users save content from the activity card to their collections. Google will then use things you've saved, and your history, in order to recommend new content for your collections. CNBC adds: Additionally Google is enhancing topics for certain things you search for. If you search for "pug,"
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Greece is reportedly using drones to fly over boats running day trips on the Aegean in an effort to crack down on rampant tax evasion at holiday hotspots. Channel NewsAsia reports: With the black economy by some accounts representing about a quarter of national output in a country which depends hugely on tourism, Greek authorities are turning to high-tech to stamp out undeclared earnings. Finance ministry tax inspectors and the coast guard launched the drones project on Santorini, an island highly popular with tourists, to check on whether operators offering short day trips were issuing legal receipts to all their passengers. Based on data from the drones, authorities were able to establish how many passengers were on board, then cross-referenced it with declared receipts and on-site inspections. Nine tourist vessels checked were alleged to have not issued a number of receipts, totaling about $29,460. Their owners now face fines.
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An anonymous reader quotes a report from NBC News: At least five people whose legs were completely paralyzed are walking again, two of them with no outside help, thanks to a specialized program of therapy and a pain stimulator implanted in their spines, researchers reported Monday. It's the latest and most dramatic advance in a new approach to treating spinal cord injuries developed at the University of Louisville in Kentucky. The reports show that electrical stimulation of the spine, when combined with a very intense and specialized training program, can re-educate the body and help move the legs even though signals from the brain are cut off. The stimulator is implanted into the epidural layer surrounding the spinal cord, and sends controlled signals into the bundle of nerve tissue. The team also employs intense training techniques to try to get the body to make sense of the signals. "There were three types of training sessions: stepping on a treadmill, over-ground standing, and over
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