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A Cheeger type inequality in finite Cayley sum graphsJul 17 2019Let $G$ be a finite group and $S$ be a symmetric generating set of $G$ with $|S| = d$. We show that if the undirected Cayley sum graph $C_{\Sigma}(G,S)$ is an expander graph and is non-bipartite, then the spectrum of its normalised adjacency operator ... More

Independence and Matching Numbers of Unicyclic Graphs From Null SpaceJul 17 2019We characterize unicyclic graphs that are singular using the support of the null space of their pendant trees. From this, we obtain closed formulas for the independence and matching numbers of a unicyclic graph, based on the support of its subtrees. These ... More

Image-Driven Biophysical Tumor Growth Model CalibrationJul 16 2019We present a novel formulation for the calibration of a biophysical tumor growth model from a single-time snapshot, MRI scan of a glioblastoma patient. Tumor growth models are typically nonlinear parabolic partial differential equations (PDEs). Thus, ... More

Where did the tumor start? An inverse solver with sparse localization for tumor growth modelsJul 15 2019We present a numerical scheme for solving an inverse problem for parameter estimation in tumor growth models for glioblastomas, a form of aggressive primary brain tumor. The growth model is a reaction-diffusion partial differential equation (PDE) for ... More

Resolvability of Hamming GraphsJul 12 2019A subset of vertices in a graph is called resolving when the geodesic distances to those vertices uniquely distinguish every vertex in the graph. Here, we characterize the resolvability of Hamming graphs in terms of a constrained linear system and deduce ... More

A Versatile Queuing System For Sharing Economy Platform OperationsJul 12 2019The paper deals with a sharing economy system with various management factors by using a bulk input G/M/1 type queuing model. The effective management of operating costs is vital for controlling the sharing economy platform and this research builds the ... More

A survey on the classical theory for Kolmogorov equationJul 11 2019We present a survey on the regularity theory for classic solutions to subelliptic degenerate Kolmogorov equations. In the last part of this note we present a detailed proof of a Harnack inequality and a strong maximum principle.

A non commutative Kähler structure on the Poincaré disk of a C*-algebraJul 10 2019We study the Poincar\'e disk $\d=\{z\in\a: \|z\|<1\}$ of a C$^*$-algebra $\a$ as a homogeneous space under the action of an appropriate Banach-Lie group $\u(\theta)$ of $2\times 2$ matrices with entries in $\a$. We define on $\d$ a homogeneous K\"ahler ... More

The least signless Laplacian eigenvalue of the complements of bicyclic graphsJul 10 2019Suppose that $G$ is a connected simple graph with the vertex set $V(G)=\{v_1, v_2,\cdots,v_n\}$. Then the adjacency matrix of $G$ is $A(G)=(a_{ij})_{n\times n}$, where $a_{ij}=1$ if $v_i$ is adjacent to $v_j$, and otherwise $a_{ij}=0$. The degree matrix ... More

Open problems in the spectral theory of signed graphsJul 09 2019Signed graphs are graphs whose edges get a sign $+1$ or $-1$ (the signature). Signed graphs can be studied by means of graph matrices extended to signed graphs in a natural way. Recently, the spectra of signed graphs have attracted much attention from ... More

Regular Graphs with Minimum Spectral GapJul 08 2019Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with $n$ vertices is $(1+o(1)) \frac{3n^2}{2\pi^2}$. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap ... More

Tilting and Squeezing: Phase space geometry of Hamiltonian saddle-node bifurcation and its influence on chemical reaction dynamicsJul 07 2019In this article we present the influence of a Hamiltonian saddle-node bifurcation on the high-dimensional phase space structures that mediate reaction dynamics. To achieve this goal, we identify the phase space invariant manifolds using Lagrangian descriptors, ... More

An $hr$-Adaptive Method for the Cubic Nonlinear Schrödinger EquationJul 04 2019The nonlinear Schr\"{o}dinger equation (NLSE) is one of the most important equations in quantum mechanics, and appears in a wide range of applications including optical fibre communications, plasma physics and biomolecule dynamics. It is a notoriously ... More

Some graft transformations and their applications on distance (signless) Laplacian spectra of graphsJul 04 2019Suppose that the vertex set of a connected graph $G$ is $V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distances between $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry ... More

On the degeneracy of integral points and entire curves in the complement of nef effective divisorsJul 01 2019As a consequence of our recently established generalized Schmidt's subspace theorem for closed subschemes in general position, we prove a degeneracy theorem for integral points on the complement of a union of nef effective divisors. A novel aspect of ... More

Isomorphism problems for tensors, groups, and cubic forms: completeness and reductionsJun 30 2019In this paper we consider the problems of testing isomorphism of tensors, $p$-groups, cubic forms, algebras, and more, which arise from a variety of areas, including machine learning, group theory, and cryptography. These problems can all be cast as orbit ... More

On nested and 2-nested graphs: two subclasses of graphs between threshold and split graphsJun 27 2019A $(0,1)$-matrix has the Consecutive Ones Property (C1P) for the rows if there is a permutation of its columns such that the ones in each row appear consecutively. We say a $(0, 1)$-matrix is nested if it has the consecutive ones property for the rows ... More

The p-norm of hypermatrices with symmetriesJun 25 2019The $p$-norm of $r$-matrices generalizes the $2$-norm of $2$-matrices. It is shown that if a nonnegative $r$-matrix is symmetric with respect to two indices $j$ and $k$, then the $p$-norm is attained for some set of vectors such that the $i$th and the ... More

Algebraic properties of perfect structuresJun 25 2019A perfect structure is a triple $(M,P,S)$ of matrices $M, P$ and $S$ of consistent sizes such that $MP = PS$. Perfect structures comprise conjugate matrices, eigenvectors, perfect colorings (equitable partitions) and graph covers. In this paper we study ... More

The strong spectral property for graphsJun 20 2019We introduce the set $\mathcal{G}^{\rm SSP}$ of all simple graphs $G$ with the property that each symmetric matrix corresponding to a graph $G \in \mathcal{G}^{\rm SSP}$ has the strong spectral property. We find several families of graphs in $\mathcal{G}^{\rm ... More

Sparse approximate matrix multiplication in a fully recursive distributed task-based parallel frameworkJun 19 2019Jun 20 2019In this paper we consider parallel implementations of approximate multiplication of large matrices with exponential decay of elements. Such matrices arise in computations related to electronic structure calculations and some other fields of science. Commonly, ... More

Sparse approximate matrix multiplication in a fully recursive distributed task-based parallel frameworkJun 19 2019Jun 26 2019In this paper we consider parallel implementations of approximate multiplication of large matrices with exponential decay of elements. Such matrices arise in computations related to electronic structure calculations and some other fields of science. Commonly, ... More

Sparse approximate matrix multiplication in a fully recursive distributed task-based parallel frameworkJun 19 2019Jun 27 2019In this paper we consider parallel implementations of approximate multiplication of large matrices with exponential decay of elements. Such matrices arise in computations related to electronic structure calculations and some other fields of science. Commonly, ... More

Sparse approximate matrix multiplication in a fully recursive distributed task-based parallel frameworkJun 19 2019In this paper we consider parallel implementations of approximate multiplication of large matrices with exponential decay of elements. Such matrices arise in computations related to electronic structure calculations and some other fields of science. Commonly, ... More

Learning Directed Graphical Models from Gaussian DataJun 19 2019In this paper, we introduce two new directed graphical models from Gaussian data: the Gaussian graphical interaction model (GGIM) and the Gaussian graphical conditional expectation model (GGCEM). The development of these models comes from considering ... More

Characterizing cospectral vertices via isospectral reductionJun 18 2019Two emerging topics in graph theory are the study of cospectral vertices of a graph, and the study of isospectral reductions of graphs. In this paper, we prove a fundamental relationship between these two areas, which is that two vertices of a graph are ... More

Spectrum of the Laplacian on simplicial complexes by the Ricci curvatureJun 18 2019We define the Ricci curvature on simplicial complexes modifying the definition of the Ricci curvature on graphs, and prove some properties. One of our main results is an estimate of the eigenvalues of the Laplacian on simplicial complexes by the Ricci ... More

Spectrum of the Laplacian on simplicial complexes by the Ricci curvatureJun 18 2019Jun 28 2019We define the Ricci curvature on simplicial complexes modifying the definition of the Ricci curvature on graphs, and prove some properties. One of our main results is an estimate of the eigenvalues of the Laplacian on simplicial complexes by the Ricci ... More

An Efficient Structural Descriptor Sequence to Identify Graph Isomorphism and Graph AutomorphismJun 18 2019In this paper, we study the graph isomorphism and graph automorphism problems. We propose a novel technique to analyze graph isomorphism and graph automorphism. Further we handled some strongly regular datasets for prove the efficiency of our technique. ... More

Gradient Flow Finite Element Discretizations with Energy-Based Adaptivity for the Gross-Pitaevskii EquationJun 17 2019We present an effective adaptive procedure for the numerical approximation of the steady-state Gross-Pitaevskii equation. Our approach is solely based on energy minimization, and consists of a combination of gradient flow iterations and adaptive finite ... More

Wiener index and Harary index on pancyclic graphsJun 17 2019Wiener index and Harary index are two classic and well-known topological indices for the characterization of molecular graphs. Recently, Yu et al. \cite{YYSX} established some sufficient conditions for a graph to be pancyclic in terms of the edge number, ... More

PSD-throttling on TreesJun 14 2019PSD-forcing is a coloring process on a graph that colors vertices blue by starting with an initial set $B$ of blue vertices and applying a color change rule (CCR-$\Zp$). The PSD-throttling number is the minimum of the sum of the cardinality of $B$ and ... More

The Inverse Eigenvalue Problem for Linear TreesJun 14 2019We prove the sufficiency of the Linear Superposition Principle for linear trees, which characterizes the spectra achievable by a real symmetric matrix whose underlying graph is a linear tree. The necessity was previously proven in 2014. This is the most ... More

Large scale Ricci curvature on graphsJun 14 2019We define a hybrid between Ollvier and Bakry Emery curvature on graphs with dependence on a variable neighborhood. The hexagonal lattice is non-negatively curved under this new curvature notion. Bonnet-Myers diameter bounds and Lichnerowicz eigenvalue ... More

Spectra and eigenspaces from regular partitions of Cayley (di)graphs of permutation groupsJun 13 2019In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives rise to a regular ... More

Characteristic Power Series of Graph LimitsJun 13 2019Jun 16 2019In this note, we show how to obtain a ``characteristic power series'' of graphons -- infinite limits of graphs -- as the limit of normalized reciprocal characteristic polynomials. This leads to a characterization of graph quasi-randomness and another ... More

Characteristic Power Series of Graph LimitsJun 13 2019In this note, we show how to obtain a ``characteristic power series'' of graphons -- infinite limits of graphs -- as the limit of normalized reciprocal characteristic polynomials. This leads to a characterization of graph quasi-randomness and another ... More

Brouwer's conjecture holds asymptotically almost surelyJun 12 2019We show that for a sequence of random graphs Brouwer's conjecture holds true with probability tending to one as the number of vertices tends to infinity. Surprisingly, it was found that a similar statement holds true for weighted graphs with possible ... More

Distance Matrix of a Class of Completely Positive Graphs: Determinant and InverseJun 11 2019A real symmetric matrix $A$ is said to be completely positive if it can be written as $BB^t$ for some (not necessarily square) nonnegative matrix $B$. A simple graph $G$ is called a completely positive graph if every doubly nonnegative matrix realization ... More

Invariant Schreier decorations of unimodular random networksJun 07 2019We prove that every $2d$-regular unimodular random network carries an invariant random Schreier decoration. Equivalently, it is the Schreier coset graph of an invariant random subgroup of the free group $F_d$. As a corollary we get that every $2d$-regular ... More

Maximum nullity and zero forcing of circulant graphsJun 07 2019It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of certain circulant graphs, including some bipartite circulants, cubic circulants, ... More

Split extensions and semidirect products of unitary magmasJun 05 2019We develop a theory of split extensions of unitary magmas, which includes defining such extensions and describing them via suitably defined semidirect product, yielding an equivalence between the categories of split extensions and of (suitably defined) ... More

Signaletic operadsJun 05 2019We introduce $k$-signaletic operads and their Koszul duals, generalizing the dendriform, diassociative and duplicial operads (which correspond to the $k=1$ case). We show that the Koszul duals of the $k$-signaletic operads act on multipermutations and ... More

On a version of the spectral excess theoremJun 04 2019Given a regular (connected) graph $\Gamma=(X,E)$ with adjacency matrix $A$, $d+1$ distinct eigenvalues, and diameter $D$, we give a characterization of when its distance matrix $A_D$ is a polynomial in $A$, in terms of the adjacency spectrum of $\Gamma$ ... More

Induction of $\mathbb{Z}^2$-actions and of partitions of the 2-torusJun 03 2019Sturmian sequences are the most simple aperiodic sequences. A result of Morse, Hedlund (1940) and Coven, Hedlund (1970) is that a biinfinite binary sequence is sturmian if and only if it is obtained as the coding of an irrational rotation on the circle ... More

Linear stability of slowly rotating Kerr black holesJun 03 2019Jul 15 2019We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equation: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in ... More

Linear stability of slowly rotating Kerr black holesJun 03 2019We prove the linear stability of slowly rotating Kerr black holes as solutions of the Einstein vacuum equation: linearized perturbations of a Kerr metric decay at an inverse polynomial rate to a linearized Kerr metric plus a pure gauge term. We work in ... More

Explicit spectral gaps for random covers of Riemann surfacesJun 03 2019We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=\Gamma\backslash\mathbb{H}$. Let $\delta$ be the Hausdorff dimension of the limit set of $\Gamma$. We say that a resonance ... More

Explicit spectral gaps for random covers of Riemann surfacesJun 03 2019Jul 04 2019We introduce a permutation model for random degree $n$ covers $X_{n}$ of a non-elementary convex-cocompact hyperbolic surface $X=\Gamma\backslash\mathbb{H}$. Let $\delta$ be the Hausdorff dimension of the limit set of $\Gamma$. We say that a resonance ... More

Planar Whitehead graphs with cyclic symmetry arising from the study of Dunwoody manifoldsMay 31 2019A fundamental theorem in the study of Dunwoody manifolds is a classification of finite graphs on $2n$ vertices that satisfy seven conditions (concerning planarity, regularity, and a cyclic automorphism of order $n$). Its significance is that if the presentation ... More

The positivity of number sequences and the Ramanujan graphsMay 31 2019Jul 01 2019Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever $$ |\lambda|\leq 2q^{\frac{1}{2}} $$ for all nontrivial eigenvalues $\lambda$ of $X$. We consider the variant $\Xi(u)$ of the Ihara ... More

The positivity of number sequences and the Ramanujan graphsMay 31 2019Let $X$ denote a connected $(q+1)$-regular undirected graph of finite order $n$. The graph $X$ is called Ramanujan whenever $$ |\lambda|\leq 2q^{\frac{1}{2}} $$ for all nontrivial eigenvalues $\lambda$ of $X$. We consider the variant $\Xi(u)$ of the Ihara ... More

The Partial differential coefficients for the second weghted Bartholdi zeta function of a graphMay 30 2019We consider the second weighted Bartholdi zeta function of a graph $G$, and present weighted versions for the result of Li and Hou's on the partial derivatives of the determinant part in the determinant expression of the Bartholdi zeta function of $G$. ... More

Quantization of Polysymplectic ManifoldsMay 30 2019We adapt the framework of geometric quantization to the polysymplectic setting. Considering prequantization as the extension of symmetries from an underlying polysymplectic manifold to the space of sections of a Hermitian vector bundle, a natural definition ... More

Best Pair Formulation & Accelerated Scheme for Non-convex Principal Component PursuitMay 25 2019May 28 2019The best pair problem aims to find a pair of points that minimize the distance between two disjoint sets. In this paper, we formulate the classical robust principal component analysis (RPCA) as the best pair; which was not considered before. We design ... More

Some algebraic properties of $F(2O_k)$ graphsMay 25 2019In this research, the notation of the folded graph of $2O_k$ will be defined and denoted by $F(2O_k)$, as the graph whose vertex set is identical to the vertex set of $2O_k$, and with edge set $E_2 = E_1 \cup\{\{(v, i), (v, i^c)\} | (v, i), (v, i^c) \in ... More

Computing the Laplacian spectrum of linear octagonal-quadrilateral networks and its applicationsMay 25 2019Let Ln denote linear octagonal-quadrilateral networks. In this paper, we aim to firstly investigate the Laplacian spectrum on the basis of Laplacian polynomial of Ln. Then, by applying the relationship between the coefficients and roots of the polynomials, ... More

Convergence towards the end space for random walks on Schreier graphsMay 24 2019We consider a transitive action of a finitely generated group $G$ and the Schreier graph $\Gamma$ it defines. For a probability measure $\mu$ on $G$ with a finite first moment we show that if the induced random walk is transient, it converges towards ... More

Resistance distance-based graph invariants and the number of spanning trees of linear crossed octagonal graphsMay 23 2019Resistance distance is a novel distance function, also a new intrinsic graph metric, which makes some extensions of ordinary distance. Let On be a linear crossed octagonal graph. Recently, Pan and Li (2018) derived the closed formulas for the Kirchhoff ... More

Dual Numbers and Operational Umbral MethodsMay 22 2019Dual numbers and their higher order version are important tools for numerical computations, and in particular for finite difference calculus. Based upon the relevant algebraic rules and matrix realizations of dual numbers, we will present a novel point ... More

A basic framework for fixed point theorems: ball spaces and spherical completenessMay 22 2019We systematically develop a general framework in\linebreak which various notions of functions being contractive, as well as of spaces being complete, can be simultaneously encoded. Derived from the notions of ultrametric balls and spherical completeness, ... More

Univalent functions with quasiconformal extensions: Becker's class and estimates of the third coefficientMay 21 2019We investigate univalent functions $f(z)=z+a_2z^2+a_3z^3+\ldots$ in the unit disk $\mathbb D$ extendible to $k$-q.c.(=quasiconformal) automorphisms of $\mathbb C$. In particular, we answer a question on estimation of $|a_3|$ raised by K\"uhnau and Niske ... More

On Lipschitz-like continuity of a class of set-valued mappingsMay 20 2019We study set-valued mappings defined by solution sets of parametric systems of equalities and inequalities. We prove Lipschitz-like continuity of these mappings under relaxed constant rank constraint qualification.

Remarks on the recurrence and transience of non-backtracking random walksMay 20 2019A short proof of the equivalence of the recurrence of non-backtracking random walk and that of simple random walk on regular infinite graphs is given. It is then shown how this proof can be extended in certain cases where the graph in question is not ... More

On some properties of the $α$-spectral radius of the $k$-uniform hypergraphMay 19 2019In this paper we show how the $\alpha$-spectral radius changes under the edge grafting operations on connected $k$-uniform hypergraphs. We characterize the extremal hypertree for $\alpha$-spectral radius among $k$-uniform non-caterpillar hypergraphs with ... More

The uniform perfectness of diffeomorphism groups of open manifoldsMay 19 2019In this paper we study the uniform perfectness, boundedness and uniform simplicity of diffeomorphism groups of compact manifolds with boundary and open manifolds and obtain some upper bounds of their diameters with respect to commutator length, those ... More

Fourier series in orthogonal polynomials on a cone of revolutionMay 18 2019Orthogonal polynomials and the Fourier orthogonal series on a cone of revolution in $\mathbb{R}^{d+1}$ are studied. It is shown that orthogonal polynomials with respect to the weight function $(1-t)^\gamma (t^2-\|x\|^2)^{\mu-\frac12}$ on the cone $\mathbb{V}^{d+1} ... More

Improving strong scaling of the Conjugate Gradient method for solving large linear systems using global reduction pipeliningMay 15 2019This paper presents performance results comparing MPI-based implementations of the popular Conjugate Gradient (CG) method and several of its communication hiding (or 'pipelined') variants. Pipelined CG methods are designed to efficiently solve SPD linear ... More

Vaught's Conjecture for Almost Chainable TheoriesMay 14 2019A structure ${\mathbb Y}$ of a relational language $L$ is called almost chainable iff there are a finite set $F \subset Y$ and a linear order $<$ on the set $Y\setminus F$ such that for each partial automorphism $\varphi$ (i.e., local automorphism, in ... More

Spectrum of some arrow-bordered circulant matrixMay 12 2019Given a circulant matrix $\mathrm{circ}(c,a,0,0,...,0,a)$, $a\ne 0$, of order~$n$, we ``border'' it from left and from above by constant column and row, respectively, and we set the left top entry to be $-nc$. This way we get a~particular title object, ... More

The compound product distribution; a solution to the distributional equation X=AX+1May 12 2019The solution of $ X=AX+1 $ is analyzed for a discrete variable $ A $ with $ \mathbb{P}\left[A=0\right]>0 $. Accordingly, a fast algorithm is presented to calculate the obtained heavy tail density. To exemplify, the compound product distribution is studied ... More

A construction for clique-free pseudorandom graphsMay 12 2019A construction of Alon and Krivelevich gives highly pseudorandom $K_k$-free graphs on $n$ vertices with edge density equal to $\Theta(n^{-1/(k -2)})$. In this short note we improve their result by constructing an infinite family of highly pseudorandom ... More

The nef cone of the Hilbert scheme of points on rational elliptic surfaces and the cone conjectureMay 11 2019We compute the nef cone of the Hilbert scheme of points on a general rational elliptic surface. As a consequence of our computation, we show that the Morrison-Kawamata cone conjecture holds for these nef cones.

Orthogonal sequences constructed from quasi-orthogonal ultraspherical polynomialsMay 10 2019Let $\displaystyle \{x_{k,n-1}\} _{k=1}^{n-1}$ and $\displaystyle \{x_{k,n}\} _{k=1}^{n},$ $n \in \mathbb{N}$, be two sets of real, distinct points satisfying the interlacing property $ x_{i,n}<x_{i,n-1}< x_{i+1,n}, \, \, \, i = 1,2,\dots,n-1.$ Wendroff ... More

Inverse sum indeg energy of graphsMay 10 2019Suppose G is an n-vertex simple graph with vertex set {v1,..., vn} and d(i), i = 1,..., n, is the degree of vertex vi in G. The ISI matrix S(G) = [sij] of G is a square matrix of order n and is defined by sij = d(i)d(j)/d(i)+d(j) if the vertices vi and ... More

Some Remarks on Systems of Equiangular LinesMay 09 2019In this note, we study the maximum number $N_\alpha(d)$ of a system of equiangular lines in $\mathbb{R}^d$ with cosine $\alpha$, where $\frac{1}{\alpha}$ is not an odd positive integer. This note is motivated by a remark in a $2018$ paper by Balla, Dr\"{a}xler, ... More

Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systemsMay 08 2019This work consists an introduction to the classical and quantum information theory of geometric flows of (relativistic) Lagrange--Hamilton mechanical systems. Basic geometric and physical properties of the canonical nonholonomic deformations of G. Perelman ... More

Weak Continuity of the Cartan Structural System on Semi-Riemannian Manifolds with Lower RegularityMay 07 2019We are concerned with the global weak continuity of the Cartan structural system - or equivalently, the Gauss-Codazzi-Ricci system - on semi-Riemannian manifolds. We prove the $W^{2,p}$ weak continuity of the Cartan structural system for $p>2$: For a ... More

Selection properties of the split interval and the Continuum HypothesisMay 06 2019We prove that every usco multimap $\Phi:X\to Y$ from a metrizable separable space $X$ to a GO-space $Y$ has an $F_\sigma$-measurable selection. On the other hand, for the split interval $\ddot{\mathbb I}$ and the projection $P:\ddot{\mathbb I}^2\to{\mathbb ... More

Orthonormal representations of $H$-free graphsMay 04 2019Let $x_1, \ldots, x_n \in \mathbb{R}^d$ be unit vectors such that among any three there is an orthogonal pair. How large can $n$ be as a function of $d$, and how large can the length of $x_1 + \ldots + x_n$ be? The answers to these two celebrated questions, ... More

Quantification of thermally-driven flows in microsystems using Boltzmann equation in deterministic and stochastic contextsMay 03 2019When the flow is sufficiently rarefied, a temperature gradient, for example, between two walls separated by a few mean free paths, induces a gas flow---an observation attributed to the thermo-stress convection effects at microscale. The dynamics of the ... More

Digit expansions of numbers in different basesMay 02 2019A folklore conjecture in number theory states that the only integers whose expansions in base $3,4$ and $5$ contain only binary digits are $0, 1$ and $82000$. In this paper we present the first significant progress on this conjecture. Furthermore, we ... More

The asymptotic value of graph energy for random graphs with degree-based weightsApr 30 2019May 14 2019In this paper, we investigate the energy of weighted random graphs $G_{n,p}(f)$, in which each edge $ij$ takes the weight $f(d_i,d_j)$, where $d_v$ is a random variable, the degree of vertex $v$ in the random graph $G_{n,p}$ of Erd\"{o}s--R\'{e}nyi model, ... More

The asymptotic value of graph energy for random graphs with degree-based weightsApr 30 2019In this paper, we investigate the energy of weighted random graphs $G_{n,p}(f)$, in which each edge $ij$ takes the weight $f(d_i,d_j)$, where $d_v$ is a random variable, the degree of vertex $v$ in the random graph $G_{n,p}$ of Erd\"{o}s--R\'{e}nyi model, ... More

Signless Laplacian eigenvalue problems of Nordhaus-Gaddum typeApr 30 2019Let $G$ be a graph of order $n$, and let $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)$ denote the signless Laplacian eigenvalues of $G$. Ashraf and Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) \#P3.6] showed that $q_1(G)+q_1(\overline{G})\leq 3n-4$, with ... More

Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$Apr 30 2019Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\mathrm{O}(n)$ gives a function ${\cal T}r_{w}^{\mathrm{O}}(n)$ of $n$. We show that ${\cal T}r_{w}^{\mathrm{O}}(n)$ ... More

Matrix Group Integrals, Surfaces, and Mapping Class Groups II: $\mathrm{O}\left(n\right)$ and $\mathrm{Sp}\left(n\right)$Apr 30 2019Jun 24 2019Let $w$ be a word in the free group on $r$ generators. The expected value of the trace of the word in $r$ independent Haar elements of $\mathrm{O}(n)$ gives a function ${\cal T}r_{w}^{\mathrm{O}}(n)$ of $n$. We show that ${\cal T}r_{w}^{\mathrm{O}}(n)$ ... More

Tracelets and Tracelet Analysis Of Compositional Rewriting SystemsApr 29 2019Taking advantage of a recently discovered associativity property of rule compositions, we extend the classical concurrency theory for rewriting systems over adhesive categories. We introduce the notion of tracelets, which are defined as minimal derivation ... More

Matrices in ${\cal A}(R,S)$ with minimum $t$-term ranksApr 24 2019Let $R$ and $S$ be two sequences of nonnegative integers in nonincreasing order and with the same sum, and let ${\cal A}(R,S)$ be the class of all $(0,1)$-matrices having row sum $R$ and column sum $S$. For a positive integer $t$, the $t$-term rank of ... More

Solution of Dirichlet and Riquier-type problems for $λ$-polyharmonic functions on regular treesApr 23 2019This paper studies the boundary behaviour of $\lambda$-polyharmonic functions for the simple random walk operator on a regular tree, where $\lambda$ is complex and $|\lambda|> \rho$, the $\ell^2$-spectral radius of the random walk. In particular, subject ... More

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

Compositionality of Rewriting Rules with ConditionsApr 19 2019We extend the notion of compositional associative rewriting as recently studied in the rule algebra framework literature to the setting of rewriting rules with conditions. Our methodology is category-theoretical in nature, where the definition of rule ... More

Critical Robertson-Walker universesApr 18 2019The integral of the energy density function $\mathfrak m$ of a closed Robertson-Walker (RW) spacetime with source a perfect fluid and cosmological constant $\Lambda$ gives rise to an action functional on the space of scale functions of RW spacetime metrics. ... More

Sesqui-Pushout Rewriting: Concurrency, Associativity and Rule Algebra FrameworkApr 17 2019Sesqui-pushout (SqPO) rewriting provides a variant of transformations of graph-like and other types of structures that fit into the framework of adhesive categories where deletion in unknown context may be implemented. We provide the first account of ... More

Determining Finite Connected Graphs Along the Quadratic Embedding Constants of PathsApr 17 2019The QE constant of a finite connected graph $G$, denoted by $\mathrm{QEC}(G)$, is by definition the maximum of the quadratic function associated to the distance matrix on a certain sphere of codimension two. We prove that the QE constants of paths $P_n$ ... More

Generic identities for finite group actionsApr 16 2019Let $G$ be a finite group of order $n$, and $Z_G=\mathbb{Z}\langle\zeta_{i,g}\mid g\in G,\ i=1,2,\dots,n\rangle$ be the free generic algebra, with canonical action of $G$ according to $(\zeta_{i,g})^x=\zeta_{i,x^{-1}g}$. It is proved that there exists ... More

Generic identities for finite group actionsApr 16 2019May 20 2019Let $G$ be a finite group of order $n$, and $Z_G=\mathbb{Z}\langle\zeta_{i,g}\mid g\in G,\ i=1,2,\dots,n\rangle$ be the free generic algebra, with canonical action of $G$ according to $(\zeta_{i,g})^x=\zeta_{i,x^{-1}g}$. It is proved that there exists ... More

Generic identities for finite group actionsApr 16 2019Apr 24 2019Let $G$ be a finite group of order $n$, and $Z_G=\mathbb{Z}\langle\zeta_{i,g}\mid g\in G,\ i=1,2,\dots,n\rangle$ be the free generic algebra, with canonical action of $G$ according to $(\zeta_{i,g})^x=\zeta_{i,x^{-1}g}$. It is proved that there exists ... More

Combinatorial Conversion and Moment Bisimulation for Stochastic Rewriting SystemsApr 15 2019We develop a novel method to analyze the dynamics of stochastic rewriting systems evolving over finitary adhesive, extensive categories. Our formalism is based on the so-called rule algebra framework and exhibits an intimate relationship between the combinatorics ... More

The Numerical Stability of Regularized Barycentric Interpolation Formulae for Interpolation and ExtrapolationApr 15 2019Jun 25 2019The $\ell_2-$ and $\ell_1-$regularized modified Lagrange interpolation formulae over $[-1,1]$ are deduced in this paper. This paper mainly analyzes the numerical characteristics of regularized barycentric interpolation formulae, which are presented in ... More