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Almost product structures on statistical manifolds and para-Kähler-like statistical submersionsApr 20 2019The main purpose of the present work is to investigate statistical manifolds endowed with almost product structures. We prove that the statistical structure of a para-K\"{a}hler-like statistical manifold of constant curvature in the Kurose's sense is ... More

Phase transition in the exclusion process on the Sierpinski gasket with slowed boundary reservoirsApr 18 2019We derive the macroscopic laws that govern the evolution of the density of particles in the exclusion process evolving on the Sierpinski gasket in the presence of a slow boundary. Depending on the slowness of the boundary we obtain, at the hydrodynamics ... More

A Kaczmarz algorithm for sequences of projections, infinite products, and applications to frames in IFS $L^{2}$ spacesApr 09 2019We show that an idea, originating initially with a fundamental recursive iteration scheme (usually referred as "the" Kaczmarz algorithm), admits important applications in such infinite-dimensional, and non-commutative, settings as are central to spectral ... More

Learning the undecidable from networked systemsApr 08 2019Apr 22 2019This article presents a theoretical investigation of computation beyond the Turing barrier from emergent behavior in distributed (or parallel) systems. In particular, we present an algorithmic network that is a mathematical model of a networked population ... More

Learning the undecidable from networked systemsApr 08 2019This article presents a theoretical investigation of computation beyond the Turing barrier from emergent behavior in distributed (or parallel) systems. In particular, we present an algorithmic network that is a mathematical model of a networked population ... More

Galton-Watson gamesApr 08 2019We consider two-player combinatorial games in which the graph of positions is random and perhaps infinite, focusing on directed Galton-Watson trees. As the offspring distribution is varied, a game can undergo a phase transition, in which the probability ... More

A Polymatroid Approach to Generalized Weights of Rank Metric CodesApr 03 2019We consider the notion of a $(q,m)$-polymatroid, due to Shiromoto, and the more general notion of $(q,m)$-demi-polymatroid, and show how generalized weights can be defined for them. Further, we establish a duality for these weights analogous to Wei duality ... More

Balanced frames: a useful tool in signal processing with good propertiesApr 01 2019So far there has not been paid attention in the literature to frames that are balanced, i.e. those frames which sum is zero. In this paper we study balanced frames, and in particular balanced unit norm tight frames in finite dimensional Hilbert spaces. ... More

Balanced frames: a useful tool in signal processing with good propertiesApr 01 2019Apr 12 2019So far there has not been paid attention in the literature to frames that are balanced, i.e. those frames which sum is zero. In this paper we consider balanced frames, and in particular balanced unit norm tight frames in finite dimensional Hilbert spaces. ... More

Asymptotic behaviour of the one-dimensional "rock-paper-scissors" cyclic cellular automatonMar 29 2019The one-dimensional three-state cyclic cellular automaton is a simple spatial model with three states in a cyclic "rock-paper-scissors" prey-predator relationship. Starting from a random configuration, similar states gather in increasingly large clusters; ... More

Combining Model and Parameter Uncertainty in Bayesian Neural NetworksMar 18 2019Mar 20 2019Bayesian neural networks (BNNs) have recently regained a significant amount of attention in the deep learning community due to the development of scalable approximate Bayesian inference techniques. There are several advantages of using Bayesian approach: ... More

Combining Model and Parameter Uncertainty in Bayesian Neural NetworksMar 18 2019Bayesian neural networks (BNNs) have recently regained a significant amount of attention in the deep learning community due to the development of scalable approximate Bayesian inference techniques. There are several advantages of using Bayesian approach: ... More

Transfer operators, atomic decomposition and the BestiaryMar 16 2019Arbieto and S. recently used atomic decomposition to study transfer operators. We give a long list of old and new expanding dynamical systems for which those results can be applied, obtaining the quasi-compactness of transfer operator acting on Besov ... More

Transfer operators and atomic decompositionMar 16 2019Apr 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More

Transfer operators and atomic decompositionMar 16 2019We use the method of atomic decomposition and a new family of Banach spaces to study the action of transfer operators associated to piecewise-defined maps. It turns out that these transfer operators are quasi-compact even when the associated potential, ... More

A logarithmic estimate for inverse source scattering problem with attenuation in a two-layered mediumMar 08 2019The paper aims a logarithmic stability estimate for the inverse source problem of the one-dimensional Helmholtz equation with attenuation factor in a two layer medium. We establish a stability by using multiple frequencies at the two end points of the ... More

Manifold interpolation and model reductionFeb 18 2019One approach to parametric and adaptive model reduction is via the interpolation of orthogonal bases, subspaces or positive definite system matrices. In all these cases, the sampled inputs stem from matrix sets that feature a geometric structure and thus ... More

Generalized semimodularity: order statisticsFeb 14 2019A notion of generalized $n$-semimodularity is introduced, which extends that of (sub/super)mod\-ularity in four ways at once. The main result of this paper, stating that every generalized $(n\colon\!2)$-semimodular function on the $n$th Cartesian power ... More

A Stochastic Approach to Eulerian NumbersFeb 08 2019We examine the aggregate behavior of one-dimensional random walks in a model known as (one-dimensional) Internal Diffusion Limited Aggregation. In this model, a sequence of $n$ particles perform random walks on the integers, beginning at the origin. Each ... More

Superposition, reduction of multivariable problems, and approximationFeb 07 2019We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes ... More

Superposition, reduction of multivariable problems, and approximationFeb 07 2019Mar 06 2019We study reduction schemes for functions of "many" variables into system of functions in one variable. Our setting includes infinite-dimensions. Following Cybenko-Kolmogorov, the outline for our results is as follows: We present explicit reductions schemes ... More

Decomposition of Gaussian processes, and factorization of positive definite kernelsDec 28 2018We establish a duality for two factorization questions, one for general positive definite (p.d) kernels $K$, and the other for Gaussian processes, say $V$. The latter notion, for Gaussian processes is stated via Ito-integration. Our approach to factorization ... More

Hypercomplex Generalizations of Gaussian-type MeasuresDec 15 2018The article is devoted to a new type of measures which are hypercomplex generalizations of Gaussian-type measures. The considered such measures are related with solutions of high order hyperbolic PDEs and related Markov processes. Their characteristic ... More

A panorama of positivityDec 13 2018This survey contains a selection of topics unified by the concept of positive semi-definiteness (of matrices or kernels), reflecting natural constraints imposed on discrete data (graphs or networks) or continuous objects (probability or mass distributions). ... More

Improved invariant polytope algorithm and applicationsDec 07 2018In several papers of 2013 - 2016, Guglielmi and Protasov made a breakthrough in the problem of the joint spectral radius computation, developing the invariant polytope algorithm which for most matrix families finds the exact value of the joint spectral ... More

On incompressible high order networksDec 04 2018Dec 13 2018This work presents a theoretical investigation of incompressible high order networks defined by a generalized graph representation. In particular, we study the incompressibility (i.e., algorithmic randomness) of snapshot-like dynamic networks in comparison ... More

The $p$-contest with $p\ne 1$Dec 03 2018We study asymptotic properties of a Markov system of $N \geq 3$ points in $[0,1]$ in which, at each step in discrete time, the point farthest from the current centre of mass, multiplied by a constant $p>0$, is removed and replaced by an independent $\zeta$-distributed ... More

Convergence in the $p$-contest modelDec 03 2018Mar 12 2019We study asymptotic properties of the following Markov system of $N \geq 3$ points in~$[0,1]$. At each time step, the point farthest from the current centre of mass, multiplied by a constant $p>0$, is removed and replaced by an independent $\zeta$-distributed ... More

Recovery guarantees for polynomial approximation from dependent data with outliersNov 25 2018Learning non-linear systems from noisy, limited, and/or dependent data is an important task across various scientific fields including statistics, engineering, computer science, mathematics, and many more. In general, this learning task is ill-posed; ... More

Two Models of Latent Consensus in Multi-Agent SystemsNov 23 2018In this paper, we propose several consensus protocols of the first and second order for networked multi-agent systems and provide explicit representations for their asymptotic states. These representations involve the eigenprojection of the Laplacian ... More

Slit-slide-sew bijections for bipartite and quasibipartite plane mapsNov 19 2018Dec 20 2018We unify and extend previous bijections on plane quadrangulations to bipartite and quasibipartite plane maps. Starting from a bipartite plane map with a distinguished edge and two distinguished corners (in the same face or in two different faces), we ... More

A central limit theorem for descents and major indices in fixed conjugacy classes of $S_n$Nov 12 2018The distribution of descents in fixed conjugacy classes of $S_n$ has been studied, and it is shown that its moments have interesting properties. Kim and Lee showed, by using Curtiss' theorem and moment generating functions, how to prove a central limit ... More

Sensitivity of $\ell_{1}$ minimization to parameter choiceOct 29 2018Apr 02 2019The use of generalized LASSO is a common technique for recovery of structured high-dimensional signals. Each generalized LASSO program has a governing parameter whose optimal value depends on properties of the data. At this optimal value, compressed sensing ... More

Sensitivity of $\ell_{1}$ minimization to parameter choiceOct 29 2018Apr 01 2019The use of generalized LASSO is a common technique for recovery of structured high-dimensional signals. Each generalized LASSO program has a governing parameter whose optimal value depends on properties of the data. At this optimal value, compressed sensing ... More

An algorithmically random family of MultiAspect Graphs and its topological propertiesOct 27 2018This article presents a theoretical investigation of incompressibility and randomness in generalized representations of graphs along with its implications on network topological properties. We extend previous studies on plain algorithmically random classical ... More

An algorithmically random family of MultiAspect Graphs and its topological propertiesOct 27 2018Mar 06 2019This article presents a theoretical investigation of incompressibility and randomness in generalized representations of graphs along with its implications on network topological properties. We extend previous studies on plain algorithmically random classical ... More

On random primitive sets, directable NDFAs and the generation of slowly synchronizing DFAsOct 24 2018We tackle the problem of the randomized generation of slowly synchronizing deterministic automata (DFAs) by generating random primitive sets of matrices. We show that when the randomized procedure is too simple the exponent of the generated sets is O(n ... More

On $n$-superharmonic functions and some geometric applicationsOct 24 2018In this paper we study asymptotic behavior of $n$-superharmonic functions at isolated singularity using the Wolff potential and $n$-capacity estimates in nonlinear potential theory. Our results are inspired by and extend those of Arsove-Huber and Taliaferro ... More

The $d$-dimensional softcore Coulomb potential and the generalized confluent Heun equationOct 15 2018An analysis of the generalized confluent Heun equation $(\alpha_2r^2+\alpha_1r)\,y''+(\beta_2r^2+\beta_1r+\beta_0)\,y'-(\varepsilon_1r+\varepsilon_0)\,y=0$ in $d$-dimensional space, where $\{\alpha_i, \beta_i, \varepsilon_i\}$ are real parameters, is ... More

Hindman's finite sums theorem and its application to topologizations of algebrasOct 03 2018The first part of the paper is a brief overview of Hindman's finite sums theorem, its prehistory and a few of its further generalizations, and a modern technique used in proving these and similar results, which is based on idempotent ultrafilters in ultrafilter ... More

Approximation and sampling of multivariate probability distributions in the tensor train decompositionOct 02 2018Nov 21 2018General multivariate distributions are notoriously expensive to sample from, particularly the high-dimensional posterior distributions in PDE-constrained inverse problems. This paper develops a sampler for arbitrary continuous multivariate distributions ... More

An asymptotic expansion for the error term in the Brent-McMillan algorithm for Euler's constantSep 12 2018The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler's constant $\gamma$ and is based on the modified Bessel functions $I_0(2x)$ and $K_0(2x)$. An error estimate for this algorithm relies on the optimally ... More

An asymptotic expansion for the error term in the Brent-McMillan algorithm for Euler's constantSep 12 2018Feb 16 2019The Brent-McMillan algorithm is the fastest known procedure for the high-precision computation of Euler's constant $\gamma$ and is based on the modified Bessel functions $I_0(2x)$ and $K_0(2x)$. An error estimate for this algorithm relies on the optimally ... More

SNS: A Solution-based Nonlinear Subspace method for time-dependent model order reductionSep 11 2018Feb 05 2019Several reduced order models have been successfully developed for nonlinear dynamical systems. To achieve a considerable speed-up, a hyper-reduction step is needed to reduce the computational complexity due to nonlinear terms. Many hyper-reduction techniques ... More

Stochastic order on metric spaces and the ordered Kantorovich monadAug 29 2018In earlier work, we had introduced the Kantorovich probability monad on complete metric spaces, extending a construction due to van Breugel. Here we extend the Kantorovich monad further to a certain class of ordered metric spaces, by endowing the spaces ... More

An explicit formula for a weight enumerator of linear-congruence codesAug 28 2018An explicit formula for a weight enumerator of linear-congruence codes is provided. This extends the work of Bibak and Milenkovic [IEEE ISIT (2018) 431-435] addressing the binary case to the non-binary case. Furthermore, the extension simplifies their ... More

Gradient Flows for Frame Potentials on the Wasserstein SpaceAug 28 2018In this paper we bring together some of the key ideas and methods of two very lively fields of mathematical research, frame theory and optimal transport, using the methods of the second to answer questions posed in the first. In particular, we construct ... More

Generalized exponential basis for efficient solving of homogeneous diffusion free boundary problems: Russian option pricingAug 24 2018This paper develops a method for solving free boundary problems for time-homogeneous diffusions. We combine the complete exponential system of solutions for the heat equation, transmutation operators and recently discovered Neumann series of Bessel functions ... More

Choosing 1 of N with and without lucky numbersAug 24 2018How many fair coin tosses to choose 1 of $n$ options with uniform probability? Although a probability problem, the solution is essentially number-theoretic, with special roles for Mersenne numbers, Fermat numbers, and the haupt exponent. We propose a ... More

The Bohr compactification of an abelian group as a quotient of its Stone-Čech compactificationAug 14 2018Aug 15 2018We will prove that, for any abelian group $G$, the canonical (surjective and continuous) mapping $\boldsymbol{\beta}G \to {\frak b}G$ from the Stone-\v{C}ech compactification $\boldsymbol{\beta}G$ of $G$ to its Bohr compactfication ${\frak b}G$ is a homomorphism ... More

Extended (p,q)-Mittag-Leffler function and its propertiesAug 04 2018In this study our aim to define the extended $(p,q)$-Mittag-Leffler(ML) function by using extension of beta functions and to obtain the integral representation of new function. We also take the Mellin transform of this new function in terms of Wright ... More

Reducing Simply Generated Trees by Iterative Leaf CuttingAug 01 2018We consider a procedure to reduce simply generated trees by iteratively removing all leaves. In the context of this reduction, we study the number of vertices that are deleted after applying this procedure a fixed number of times by using an additive ... More

Mixed mode oscillations and phase locking in coupled FitzHugh-Nagumo model neuronsJul 27 2018We study the dynamics of a low-dimensional system of coupled model neurons as a step towards understanding the vastly complex network of neurons in the brain. We analyze the bifurcation structure of a system of two model neurons with unidirectional coupling ... More

Laplacian growth & sandpiles on the Sierpinski gasket: limit shape universality and exact solutionsJul 23 2018Sep 06 2018We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner vertex. In particular, the abelian sandpile growth problem ... More

Multivariate Public Key Cryptography and Digital SignatureJul 20 2018Jul 23 2018In this paper, algorithms for multivariate public key cryptography and digital signature are described. Plain messages and encrypted messages are arrays, consisting of elements from a fixed finite ring or field. The encryption and decryption algorithms ... More

On reproducing kernels, and analysis of measuresJul 11 2018Jul 17 2018Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive ... More

On reproducing kernels, and analysis of measuresJul 11 2018Feb 23 2019Starting with the correspondence between positive definite kernels on the one hand and reproducing kernel Hilbert spaces (RKHSs) on the other, we turn to a detailed analysis of associated measures and Gaussian processes. Point of departure: Every positive ... More

Lattice paths and branched continued fractions: An infinite sequence of generalizations of the Stieltjes--Rogers and Thron--Rogers polynomials, with coefficientwise Hankel-total positivityJul 09 2018We define an infinite sequence of generalizations, parametrized by an integer $m \ge 1$, of the Stieltjes--Rogers and Thron--Rogers polynomials; they arise as the power-series expansions of some branched continued fractions, and as the generating polynomials ... More

Thermodynamics of elastoplastic porous rocks at large strains towards earthquake modelingJul 02 2018Jul 05 2018A mathematical model for an elastoplastic porous continuum subject to large strains in combination with reversible damage (aging), evolving porosity, water and heat transfer is advanced. The inelastic response is modeled within the frame of plasticity ... More

On the parabolic Harnack inequality for non-local diffusion equationsJun 12 2018We settle the open question concerning the Harnack inequality for globally positive solutions to non-local in time diffusion equations by constructing a counter-example for dimensions $d\ge\beta$, where $\beta\in(0,2]$ is the order of the equation with ... More

Deep Bayesian regression modelsJun 06 2018Jun 07 2018Regression models are used for inference and prediction in a wide range of applications providing a powerful scientific tool for researchers and analysts from different fields. In many research fields the amount of available data as well as the number ... More

The lemniscate tree of a random polynomialJun 01 2018To each generic complex polynomial $p(z)$ there is associated a labeled binary tree (here referred to as a "lemniscate tree") that encodes the topological type of the graph of $|p(z)|$. The branching structure of the lemniscate tree is determined by the ... More

Counting partitions inside a rectangleMay 22 2018Feb 04 2019We consider the number of partitions of $n$ whose Young diagrams fit inside an $m \times \ell$ rectangle; equivalently, we study the coefficients of the $q$-binomial coefficient $\binom{m+\ell}{m}_q$. We obtain sharp asymptotics throughout the regime ... More

Selections and Higher Separation AxiomsMay 19 2018Oct 15 2018This survey presents some historical background and recent developments in the area of selections for set-valued mappings along with several open questions. It was written with the hope that the presented material may pique an interest in the selection ... More

Contour location via entropy reduction leveraging multiple information sourcesMay 19 2018Dec 19 2018We introduce an algorithm to locate contours of functions that are expensive to evaluate. The problem of locating contours arises in many applications, including classification, constrained optimization, and performance analysis of mechanical and dynamical ... More

The Kashaev Equation and Related RecurrencesMay 10 2018Feb 21 2019The hexahedron recurrence was introduced by R. Kenyon and R. Pemantle in the study of the double-dimer model in statistical mechanics. It describes a relationship among certain minors of a square matrix. This recurrence is closely related to the Kashaev ... More

The Hilbert transform and orthogonal martingales in Banach spacesMay 10 2018Let $X$ be a given Banach space and let $M$, $N$ be two orthogonal $X$-valued local martingales such that $N$ is weakly differentially subordinate to $M$. The paper contains the proof of the estimate $$ \mathbb E \Psi(N_t) \leq C_{\Phi,\Psi,X} \mathbb ... More

Capacities, removable sets and $L^p$-uniqueness on Wiener spacesMay 10 2018We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the $L^p$-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in ... More

Reconstruction of a compactly supported sound profile in the presence of a random background mediumMay 05 2018May 25 2018In this paper, we present algorithms for reconstructing an unknown compact scatterer embedded in a random noisy background medium, given measurements of the scattered field and information about the background medium and the sound profile. We present ... More

Beyond Haar and Cameron-Martin: the Steinhaus supportApr 30 2018Mar 14 2019Motivated by a Steinhaus-like interior-point property involving the Cameron-Martin space of Gaussian measure theory, we study a group-theoretic analogue, the Steinhaus triple $(H,G,\mu)$, and construct a Steinhaus support, a Cameron-Martin-like subset, ... More

Beyond Haar and Cameron-Martin: the Steinhaus supportApr 30 2018Aug 14 2018Motivated by a Steinhaus-like interior-point property involving the Cameron-Martin space of Gaussian measure theory, we study a group-theoretic analogue, the Steinhaus triple $(H,G,\mu)$, and construct a Steinhaus support, a Cameron-Martin-like subset, ... More

A note on an integral of Dixit, Roy and ZaharescuApr 20 2018May 09 2018In a recent paper, Dixit {\it et al.\/} [Acta Arith. {\bf 177} (2017) 1--37] posed two open questions whether the integral \[{\hat J}_{k}(\alpha)=\int_0^\infty\frac{xe^{-\alpha x^2}}{e^{2\pi x}-1}\,{}_1F_1(-k,3/2;2\alpha x^2)\,dx\] for $\alpha>0$ could ... More

Bimonoidal Structure of Probability MonadsApr 10 2018Aug 22 2018We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, ... More

Distribution of the Number of Corners in Tree--like TableauxApr 09 2018In this paper, we study tree--like tableaux and some of their probabilistic properties. Tree--like tableaux are in bijection with other combinatorial structures, including permutation tableaux, and have a connection to the partially asymmetric simple ... More

The variety of coset relation algebrasApr 07 2018Aug 12 2018A coset relation algebra is one embeddable into some full coset relation algebra, the latter is an algebra constructed from a system of groups, a coordinated system of isomorphisms between quotients of these groups, and a system of cosets that are used ... More

Memory equations as reduced Markov processesApr 06 2018A large class of linear memory differential equations in one dimension, where the evolution depends on the whole history, can be equivalently described as a projection of a Markov process living in a higher dimensional space. Starting with such a memory ... More

Energy estimates and model order reduction for stochastic bilinear systemsApr 05 2018In this paper, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular ... More

Symplectic cohomologies and deformationsMar 28 2018In this note we study the behavior of symplectic cohomology groups under symplectic deformations. Moreover, we show that for compact almost-K\"ahler manifolds $(X,J,g,\omega)$ with $J$ $\mathcal{C}^\infty$-pure and full the space of de Rham harmonic forms ... More

Large deviation principles for empirical measures of the multitype random networksMar 22 2018In this article we study the stochastic block model also known as the multi-type random networks (MRNs). For the stochastic block model or the MRNs we define the empirical group measure, empirical cooperative measure and the empirical locality measure. ... More

Length of the longest common subsequence between overlapping wordsMar 08 2018Given two random finite sequences from $[k]^n$ such that a prefix of the first sequence is a suffix of the second, we examine the length of their longest common subsequence. If $\ell$ is the length of the overlap, we prove that the expected length of ... More

A Bound for the Rank-One Transient of Inhomogeneous Matrix Products in Special CaseMar 01 2018Apr 12 2019We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length be rank-one, as it was shown in [6][L. Shue, B.D.O. Anderson, S. Dey: On ... More

The space of relative orders and a generalization of Morris indicability theoremFeb 14 2018Jun 10 2018We introduce the space of relative orders on a group and show that it is compact whenever the group is finitely generated. We use this to show that if $G$ is a finitely generated group acting by order preserving homeomorphism of on the line, then if some ... More

Some central limit theorems for random walks associated with hypergeometric functions of type BCFeb 14 2018The spherical functions of the noncompact Grassmann manifolds over the real or complex numbers or the quaternions with rank q and dimension parameter p can be seen as Heckman-Opdam hypergeometric functions of type BC, when the double coset space is identified ... More

Median ShapesFeb 14 2018Dec 09 2018We introduce and begin to explore the mean and median of finite sets of shapes represented as integral currents. The median can be computed efficiently in practice, and we focus most of our theoretical and computational attention on medians. We consider ... More

Maximum determinant positive definite Toeplitz completionsFeb 02 2018We consider partial symmetric Toeplitz matrices where a positive definite completion exists. We characterize those patterns where the maximum determinant completion is itself Toeplitz. We then extend these results with positive definite replaced by positive ... More

A group law on the projective plane with applications in Public Key CryptographyFeb 01 2018Mar 15 2019We present a new group law defined on a subset of the projective plane $\mathbb{F}P^2$ over an arbitrary field $\mathbb{F}$, which lends itself to applications in Public Key Cryptography, in particular to a Diffie-Hellman-like key agreement protocol. ... More

The Necklace Process: A Generating Function ApproachJan 30 2018Jul 24 2018The "necklace process", a procedure constructing necklaces of black and white beads by randomly choosing positions to insert new beads (whose color is uniquely determined based on the chosen location), is revisited. This article illustrates how, after ... More

Short walk adventuresJan 18 2018Apr 29 2018We review recent development of short uniform random walks, with a focus on its connection to (zeta) Mahler measures and modular parametrisation of the density functions. Furthermore, we extend available "probabilistic" techniques to cover a variation ... More

Ascents in Non-Negative Lattice PathsJan 09 2018Non-negative {\L}ukasiewicz paths are special two-dimensional lattice paths never passing below their starting altitude which have only one single special type of down step. They are well-known and -studied combinatorial objects, in particular due to ... More

Tensor network ranksJan 08 2018Feb 09 2019In problems involving approximation, completion, denoising, dimension reduction, estimation, interpolation, modeling, order reduction, regression, etc, we argue that the near-universal practice of assuming that a function, matrix, or tensor (which we ... More

A Probabilistic Subspace Bound with Application to Active SubspacesJan 02 2018Given a real symmetric positive semi-definite matrix E, and an approximation S that is a sum of n independent matrix-valued random variables, we present bounds on the relative error in S due to randomization. The bounds do not depend on the matrix dimensions ... More

Hitting Time Quasi-metric and Its Forest RepresentationJan 01 2018Jul 29 2018Let $\hat m_{ij}$ be the hitting (mean first passage) time from state $i$ to state $j$ in an $n$-state ergodic homogeneous Markov chain with transition matrix $T$. Let $\Gamma$ be the weighted digraph whose vertex set coincides with the set of states ... More

On increasing stability in the two dimensional inverse source scattering problem with many frequenciesDec 23 2017In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain $\Omega$ with sufficiently smooth boundary. Using the ... More

The expansion of the confluent hypergeometric function on the positive real axisDec 22 2017The asymptotic expansion of the Kummer function ${}_1F_1(a; b; z)$ is examined as $z\to+\infty$ on the Stokes line $\arg\,z=0$. The correct form of the subdominant algebraic contribution is obtained for non-integer $a$. Numerical results demonstrating ... More

Ergodicity of some classes of cellular automata subject to noiseDec 15 2017Jan 24 2018Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics ... More

Ergodicity of some classes of cellular automata subject to noiseDec 15 2017Mar 28 2019Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics ... More

A Probability Monad as the Colimit of Finite PowersDec 14 2017Nov 27 2018We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This ... More

A Probability Monad as the Colimit of Spaces of Finite SamplesDec 14 2017Mar 12 2019We define and study a probability monad on the category of complete metric spaces and short maps. It assigns to each space the space of Radon probability measures on it with finite first moment, equipped with the Kantorovich-Wasserstein distance. This ... More

Semipolar sets and intrinsic Hausdorff measureNov 24 2017Given a "Green function" $G$ on a locally compact space $X$ with countable base, a Borel set $A$ in $X$ is called $G$-semipolar, if there is no measure $\nu\ne 0$ supported by $A$ such that $G\nu:=\int G(\cdot,y)\,d\nu(y)$ is a continuous real function ... More

Realizations and Factorizations of Positive Definite KernelsNov 09 2017Oct 29 2018Given a fixed sigma-finite measure space $\left(X,\mathscr{B},\nu\right)$, we shall study an associated family of positive definite kernels $K$. Their factorizations will be studied with view to their role as covariance kernels of a variety of stochastic ... More