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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
The asymptotics of the generalised Bessel functionNov 07 2017We demonstrate how the asymptotics for large $|z|$ of the generalised Bessel function \[{}_0\Psi_1(z)=\sum_{n=0}^\infty\frac{z^n}{\Gamma(an+b) n!},\] where $a>-1$ and $b$ is any number (real or complex), may be obtained by exploiting the well-established ... More
On Powers of the Catalan Number SequenceNov 05 2017Jun 29 2018The Catalan number sequence is one of the most famous number sequences in combinatorics and is well studied in the literature. In this paper we further investigate its fundamental properties related to the moment problem and prove for the first time that ... More
On spectral radii of unraveled ballsOct 18 2017Sep 15 2018Given a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We prove a lower bound on the maximum spectral radius of unraveled balls of fixed radius, and we show, ... More
Beta-Function Identities via Probabilistic ApproachSep 28 2017Using a probabilistic approach, we derive several interesting identities involving beta functions. Our results generalize certain well-known combinatorial identities involving binomial coefficients and gamma functions.
A note on the asymptotics of the modified Bessel functions on the Stokes linesAug 31 2017Sep 02 2017We employ the exponentially improved asymptotic expansions of the confluent hypergeometric functions on the Stokes lines discussed by the author [Appl. Math. Sci. {\bf 7} (2013) 6601--6609] to give the analogous expansions of the modified Bessel functions ... More
Sampling with positive definite kernels and an associated dichotomyAug 20 2017We study classes of reproducing kernels $K$ on general domains; these are kernels which arise commonly in machine learning models; models based on certain families of reproducing kernel Hilbert spaces. They are the positive definite kernels $K$ with the ... More
Local Large deviations for empirical locality measure of typed Random Graph ModelsAug 13 2017Feb 24 2018In this article, we prove a local large deviation principle (LLDP) for the empirical locality measure of typed random networks on $n$ nodes conditioned to have a given \emph{ empirical type measure} and \emph{ empirical link measure.} From the LLDP, we ... More
Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit CircleJul 31 2017Dec 03 2017Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the ... More
Finitely dependent cycle coloringJul 28 2017We construct stationary finitely dependent colorings of the cycle which are analogous to the colorings of the integers recently constructed by Holroyd and Liggett. These colorings can be described by a simple necklace insertion procedure, and also in ... More
Reproducing kernels and choices of associated feature spaces, in the form of $L^{2}$-spacesJul 26 2017Motivated by applications to the study of stochastic processes, we introduce a new analysis of positive definite kernels $K$, their reproducing kernel Hilbert spaces (RKHS), and an associated family of feature spaces that may be chosen in the form $L^{2}\left(\mu\right)$; ... More
Mathematical aspect of the combinatorial game "Mahjong"Jul 23 2017Jan 23 2019We illustrate how one can use basic combinatorial theory and computer programming technique (Python) to analyze the combinatorial game: Mahjong. The results confirm some folklore concerning the game, and expose some unexpected results. Related results ... More
Random eigenfunctions on flat tori: universality for the number of intersectionsJul 17 2017We show that several statistics of the number of intersections between random eigenfunctions of general eigenvalues with a given smooth curve in flat tori are universal under various families of randomness.
A moment-generating formula for Erdős-Rényi component sizesJul 17 2017Mar 21 2018We derive a simple formula characterizing the distribution of the size of the connected component of a fixed vertex in the Erd\H{o}s-R\'enyi random graph which allows us to give elementary proofs of some results of Federico, van der Hofstad, den Hollander ... More
Intertwining wavelets or Multiresolution analysis on graphs through random forestsJul 14 2017May 01 2018We propose a new method for performing multiscale analysis of functions defined on the vertices of a finite connected weighted graph. Our approach relies on a random spanning forest to downsample the set of vertices, and on approximate solutions of Markov ... More
A Generating Function for the Distribution of Runs in Binary WordsJul 13 2017Let $N(n,r,k)$ denote the number of binary words of length $n$ that begin with $0$ and contain exactly $k$ runs (i.e., maximal subwords of identical consecutive symbols) of length $r$. We show that the generating function for the sequence $N(n,r,0)$, ... More
Local Large deviation: A McMillian Theorem for Coloured Random Graph ProcessesJul 06 2017For a finite typed graph on $n$ nodes and with type law $\mu,$ we define the so-called spectral potential $\rho_{\lambda}(\,\cdot,\,\mu),$ of the graph.From the $\rho_{\lambda}(\,\cdot,\,\mu)$ we obtain Kullback action or the deviation function, $\mathcal{H}_{\lambda}(\pi\,\|\,\nu),$ ... More
Metric duality between positive definite kernels and boundary processesJun 29 2017We study representations of positive definite kernels $K$ in a general setting, but with view to applications to harmonic analysis, to metric geometry, and to realizations of certain stochastic processes. Our initial results are stated for the most general ... More
Asymptotic properties of a componentwise ARH(1) plug-in predictorJun 20 2017Sep 04 2018This paper presents new results on prediction of linear processes in function spaces. The autoregressive Hilbertian process framework of order one (ARH(1) process framework) is adopted. A componentwise estimator of the autocorrelation operator is formulated, ... More
Testing Gaussian Process with Applications to Super-ResolutionJun 02 2017Jul 02 2018This article introduces exact testing procedures on the mean of a Gaussian process $X$ derived from the outcomes of $\ell_1$-minimization over the space of complex valued measures. The process $X$ can be thought as the sum of two terms: first, the convolution ... More
Local Large Deviations: McMillian Theorem for multitype Galton-Watson ProcessesMay 28 2017Jun 29 2017In this article we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential $U_{\skrik}(\,\cdot,\,\pi)$ for the Galton-Watson process, where ... More
A novel algorithmic approach to Bayesian Logic RegressionMay 22 2017Logic regression was developed more than a decade ago as a tool to construct predictors from Boolean combinations of binary covariates. It has been mainly used to model epistatic effects in genetic association studies, which is very appealing due to the ... More
A novel algorithmic approach to Bayesian Logic RegressionMay 22 2017Jun 02 2018Logic regression was developed more than a decade ago as a tool to construct predictors from Boolean combinations of binary covariates. It has been mainly used to model epistatic effects in genetic association studies, which is very appealing due to the ... More
Analysis of Krylov Subspace Approximation to Large Scale Differential Riccati EquationsMay 21 2017Dec 29 2018We consider a Krylov subspace approximation method for the symmetric differential Riccati equation $\dot{X} = AX + XA^T + Q - XSX$, $X(0)=X_0$. The method we consider is based on projecting the large scale equation onto a Krylov subspace spanned by the ... More
Powerful sets: a generalisation of binary matroidsMay 21 2017A set $S\subseteq\{0,1\}^E$ of binary vectors, with positions indexed by $E$, is said to be a \textit{powerful code} if, for all $X\subseteq E$, the number of vectors in $S$ that are zero in the positions indexed by $X$ is a power of 2. By treating binary ... More
The spectral norm of a Horadam circulant matrixMay 09 2017Let $a$, $b$, $p$, $q$ be integers and~$(h_n)$ defined by $h_0=a$, $h_1=b$, $h_n=ph_{n-1}+qh_{n-2}$, $n=2,3,\dots$. Complementing to certain previously known results, we study the spectral norm of the circulant matrix corresponding to $h_0,\dots,h_{n-1}$. ... More
Optimal properties of the canonical tight probabilistic frameMay 09 2017A probabilistic frame is a Borel probability measure with finite second moment whose support spans $\mathbb{R}^d$. A Parseval probabilistic frame is one for which the associated matrix of the second moments is the identity matrix in $\mathbb{R}^d$. Each ... More
Xorshift random number generators from primitive polynomialsApr 28 2017Aug 06 2017A class of xorshift random number generators (RNGs) are introduced by Marsaglia. We have proposed an algorithm which constructs a full period xorshift RNG from a given primitive polynomial. It is shown there is a weakness present in those RNGs and is ... More
Wolf Barth (1942--2016)Apr 24 2017In this article we describe the life and work of Wolf Barth who died on 30th December 2016. Wolf Barth's contributions to algebraic variety span a wide range of subjects. His achievements range from what is now called the Barth-Lefschetz theorems to his ... More
Energy of commuting graph of finite groups whose centralizers are AbelianApr 21 2017Let $G$ be a finite group with centre $Z(G)$. The commuting graph of a non-Abelian group $G$, denoted by $\Gamma_G$, is a simple undirected graph whose vertex set is $G\setminus Z(G)$, and two vertices $x$ and $y$ are adjacent if and only if $xy = yx$. ... More
Conditional measure on the Brownian path and other random setsApr 19 2017Jan 12 2018Let $B$ denote the range of the Brownian motion in $\mathbb{R}^{d}$ ($d\geq3$). For a deterministic Borel measure $\nu$ on $\mathbb{R}^{d}$ we wish to find a random measure $\mu$ such that the support of $\mu$ is contained in $B$ and it is a solution ... More
Fringe Analysis of Plane Trees Related to Cutting and PruningApr 04 2017Rooted plane trees are reduced by four different operations on the fringe. The number of surviving nodes after reducing the tree repeatedly for a fixed number of times is asymptotically analyzed. The four different operations include cutting all or only ... More
Lossy Asymptotic Equipartition Property For Geometric Networked Data StructuresApr 02 2017Nov 22 2017This article extends the Generalized Asypmtotic Equipartition Property of Networked Data Structures to cover the Wireless Sensor Network modelled as coloured geometric random graph (CGRG). The main techniques used to prove this result remains large deviation ... More
Extension of Mittag-Leffler functionMar 15 2017In this paper, we present an extension of Mittag-Leffler function by using the extension of beta functions (\"{O}zergin et al. in J. Comput. Appl. Math. 235 (2011), 4601-4610) and obtain some integral representation of this newly defined function. Also, ... More
Fragmentation process, pruning poset for rooted forests, and Möbius inversionFeb 10 2017Mar 22 2018We consider a discrete-time Markov chain, called fragmentation process, that describes a specific way of successively removing objects from a linear arrangement. The process arises in population genetics and describes the ancestry of the genetic material ... More
The spectra of the unitary marix of a 2-tessellable staggered quantum walk on a graphJan 28 2017Recently, the staggered quantum walk (SQW) on a graph is discussed as a generalization of coined quantum walks on graphs and Szegedy walks. We present a formula for the time evolution matrix of a 2-tessellable SQW on a graph, and so directly give its ... More
Efficient computation of higher order cumulant tensorsJan 19 2017Apr 10 2018In this paper, we introduce a novel algorithm for calculating arbitrary order cumulants of multidimensional data. Since the $d^\text{th}$ order cumulant can be presented in the form of an $d$-dimensional tensor, the algorithm is presented using tensor ... More
Some Connections Between Cycles and Permutations that Fix a Set and Touchard Polynomials and Covers of MultisetsJan 17 2017Jan 20 2017We present a new proof of a fundamental result concerning cycles of random permutations which gives some intuition for the connection between Touchard polynomials and the Poisson distribution. We also introduce a rather novel permutation statistic and ... More