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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

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

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

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

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

Distance matrices of a tree: two more invariants, and in a unified frameworkMar 27 2019Apr 23 2019Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ depends only on ... More

Distance matrices of a tree: two more invariants, and in a unified frameworkMar 27 2019Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ depends only on ... More

Distance matrices of a tree: two more invariants, and in a unified frameworkMar 27 2019Jun 24 2019Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ depends only on ... More

Distance matrices of a tree: two more invariants, and in a unified frameworkMar 27 2019Apr 03 2019Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ depends only on ... More

On uniqueness of the foliation by comoving observers restspaces of a Generalized Robertson Walker spacetimeFeb 23 2019A characterization of the foliation by spacelike slices of an $(n+1)$-dimensional spatially closed Generalized Robertson-Walker spacetime is given by means of studying a natural mean curvature type equation on spacelike graphs. Under some natural assumptions, ... More

Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient MethodFeb 08 2019May 15 2019Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in ... More

Numerically Stable Recurrence Relations for the Communication Hiding Pipelined Conjugate Gradient MethodFeb 08 2019Pipelined Krylov subspace methods (also referred to as communication-hiding methods) have been proposed in the literature as a scalable alternative to classic Krylov subspace algorithms for iteratively computing the solution to a large linear system in ... More

Spectra of eccentricity matrices of graphsFeb 07 2019The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the least eigenvalue ... More

Complete multipartite graphs that are determined, up to switching, by their Seidel spectrumFeb 07 2019It is known that complete multipartite graphs are determined by their distance spectrum but not by their adjacency spectrum. The Seidel spectrum of a graph $G$ on more than one vertex does not determine the graph, since any graph obtained from $G$ by ... 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

Self-Polar PolytopesFeb 02 2019Self-polar polytopes are convex polytopes that are equal to an orthogonal transformation of their polar sets. These polytopes were first studied by Lov\'{a}sz as a means of establishing the chromatic number of distance graphs on spheres, and they can ... More

On the generalized distance spectral radius of graphsJan 23 2019The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_\alpha(G)=\alpha Tr(G)+(1-\alpha)D(G), \;\;0\le\alpha \le 1,$$ where $D(G)$ and $Tr(G)$ denote the distance ... More

Embedding quadratization gadgets on Chimera and Pegasus graphsJan 23 2019We group all known quadratizations of cubic and quartic terms in binary optimization problems into six and seven unique graphs respectively. We then perform a minor embedding of these graphs onto the well-known Chimera graph, and the brand new Pegasus ... More

Pegasus: The second connectivity graph for large-scale quantum annealing hardwareJan 22 2019Pegasus is a graph which offers substantially increased connectivity between the qubits of quantum annealing hardware compared to the graph Chimera. It is the first fundamental change in the connectivity graph of quantum annealers built by D-Wave since ... More

Extending partial isometries of antipodal graphsJan 14 2019We prove EPPA (extension property for partial automorphisms) for all antipodal classes from Cherlin's list of metrically homogeneous graphs, thereby answering a question of Aranda et al. This paper should be seen as the first application of a new general ... More

Quadratization in discrete optimization and quantum mechanicsJan 14 2019A book about turning high-degree optimization problems into quadratic optimization problems that maintain the same global minimum (ground state). This book explores quadratizations for pseudo-Boolean optimization, perturbative gadgets used in QMA completeness ... More

On the largest two and smallest six distance Pareto eigenvalues of a graphNov 29 2018Nov 30 2018In this article, we establish some bounds involving the largest two distance Pareto eigenvalues of a connected graph. Also we characterize all possible values for smallest six distance Pareto eigenvalues of a connected graph.

Analyzing and improving maximal attainable accuracy in the communication hiding pipelined BiCGStab methodSep 06 2018Mar 25 2019Pipelined Krylov subspace methods avoid communication latency by reducing the number of global synchronization bottlenecks and by hiding global communication behind useful computational work. In exact arithmetic pipelined Krylov subspace algorithms are ... More

Analyzing and improving maximal attainable accuracy in the communication hiding pipelined BiCGStab methodSep 06 2018Pipelined Krylov subspace methods avoid communication latency by reducing the number of global synchronization bottlenecks and by hiding global communication behind useful computational work. In exact arithmetic pipelined Krylov subspace algorithms are ... More

An Approximation Scheme for Quasistationary Distributions of Killed DiffusionsAug 21 2018In this paper we study the asymptotic behavior of the normalized weighted empirical occupation measures of a diffusion process on a compact manifold which is also killed at a given rate and regenerated at a random location, distributed according to the ... More

Lower bound for the cost of connecting tree with given vertex degree sequenceAug 19 2018Aug 21 2018The optimal connecting network problem generalizes many models of structure optimization known from the literature, including communication and transport network topology design, graph cut and graph clustering, structure identification from data, etc. ... More

Addressing Johnson graphs, complete multipartite graphs, odd cycles and other graphsAug 14 2018Oct 29 2018Graham and Pollak showed that the vertices of any graph $G$ can be addressed with $N$-tuples of three symbols, such that the distance between any two vertices may be easily determined from their addresses. An addressing is optimal if its length $N$ is ... 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

On the distance matrices of the CP graphsMay 25 2018This paper introduces a new class of graphs, the CP graphs, and shows that their distance determinant and distance inertia are independent of their structures. The CP graphs include the family of linear $2$-trees. When a graph is attached with a CP graph, ... More

Numerical analysis of the maximal attainable accuracy in communication hiding pipelined Conjugate Gradient methodsApr 09 2018Aug 21 2018Krylov subspace methods are widely known as efficient algebraic methods for solving large scale linear systems. However, on massively parallel hardware the performance of these methods is typically limited by communication latency rather than floating ... More

A study on resistance matrix of graphsMar 24 2018In this article we consider resistance matrix of a connected graph. For unweighted graph we study some necessary and sufficient conditions for resistance regular graphs. Also we find some relationship between Laplacian matrix and resistance matrix in ... More

Lower bounds for the first eigenvalue of the Steklov problem on graphsMar 23 2018We give lower bounds for the first non-zero Steklov eigenvalue on connected graphs. These bounds depend on the extrinsic diameter of the boundary and not on the diameter of the graph. We obtain a lower bound which is sharp when the cardinal of the boundary ... More

Similarities on Graphs: Kernels versus Proximity MeasuresFeb 17 2018We analytically study proximity and distance properties of various kernels and similarity measures on graphs. This helps to understand the mathematical nature of such measures and can potentially be useful for recommending the adoption of specific similarity ... More

Matrix Group Integrals, Surfaces, and Mapping Class Groups I: $U(n)$Feb 13 2018May 03 2019Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: ... More

On Quadratic Embedding Constants of Star Product GraphsFeb 04 2018A connected graph $G$ is of QE class if it admits a quadratic embedding in a Hilbert space, or equivalently if the distance matrix is conditionally negative definite, or equivalently if the quadratic embedding constant $\mathrm{QEC}(G)$ is non-positive. ... More

A group law on the projective plane with applications in Public Key CryptographyFeb 01 2018Jun 10 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

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

Maximum principles at infinity and the Ahlfors-Khas'minskii duality: an overviewJan 16 2018This note is meant to introduce the reader to a duality principle for nonlinear equations that recently appeared in the literature. Motivations come from the desire to give a unifying potential-theoretic framework for various maximum principles at infinity ... More

The Communication-Hiding Conjugate Gradient Method with Deep PipelinesJan 15 2018Jan 28 2019Krylov subspace methods are among the most efficient solvers for large scale linear algebra problems. Nevertheless, classic Krylov subspace algorithms do not scale well on massively parallel hardware due to synchronization bottlenecks. Communication-hiding ... 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

Cameron-Liebler sets of generators in finite classical polar spacesDec 17 2017Cameron-Liebler sets were originally defined as collections of lines ("line classes") in $\mathrm{PG}(3,q)$ sharing certain properties with line classes of symmetric tactical decompositions. While there are many equivalent characterisations, these objects ... More

Cameron-Liebler sets of generators in finite classical polar spacesDec 17 2017May 07 2019Cameron-Liebler sets were originally defined as collections of lines (`line classes') in $\mathrm{PG}(3,q)$ sharing certain properties with line classes of symmetric tactical decompositions. While there are many equivalent characterisations, these objects ... More

Inverse Perron values and connectivity of a uniform hypergraphNov 27 2017In this paper, we show that a uniform hypergraph $\mathcal{G}$ is connected if and only if one of its inverse Perron values is larger than $0$. We give some bounds on the bipartition width, isoperimetric number and eccentricities of $\mathcal{G}$ in terms ... 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

A Characterization of Effective Resistance MetricsOct 04 2017Jun 11 2018We produce a characterization of finite metric spaces which are given by the effective resistance of a graph. This characterization is applied to the more general context of resistance metrics defined by Kigami. A countably infinite resistance metric ... More

Linear Computer-Music through Sequences over Galois FieldsSep 19 2017It is shown how binary sequences can be associated with automatic composition of monophonic pieces. We are concerned with the composition of e-music from finite field structures. The information at the input may be either random or information from a ... More

Some improved bounds on two energy-like invariants of some derived graphsSep 16 2017Given a simple graph $G$, its Laplacian-energy-like invariant $LEL(G)$ and incidence energy $IE(G)$ are the sum of square root of its all Laplacian eigenvalues and signless Laplacian eigenvalues, respectively. Applying the Cauchy-Schwarz inequality and ... More

Embedded Picard-Vessiot extensionsAug 31 2017We prove that if T is a theory of large, bounded, fields of characteristic zero, with almost quantifier elimination, and T_D is the model companion of T + "D is a derivation", then for any model U of T_D, and differential subfield K of U whose field of ... More

Numerical properties of Koszul connectionsAug 03 2017We use the notation EX(S>M), EXF(S>M) and DL(S>M), where M is a smooth manifold and S is a geometric structure. EX(S>M) is the question whether S exists in M. EXF(S>M) is the question whether M admits S-foliations. DL(S>M) is the search of an invariant ... More

Rational invariants of even ternary forms under the orthogonal groupJul 31 2017Nov 13 2018In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction relies on two ... More

Eigenvalues and Wiener index of the Zero Divisor graph $Γ[\mathbb {Z}_n]$Jul 17 2017The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma[R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. In this paper, we consider the zero divisor graph $\Gamma[\mathbb{Z}_n]$ ... 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

There are no cycles in the $3n+1$ sequenceJun 20 2017Jun 27 2017In 1937, Lothar Collatz conjectured that the sequence generated by the rule $f(n)=3n+1$ for $n\in\mathbb{N}$ odd, $f(n)=n/2$ for $n\in\mathbb{N}$ even, starting in any positive integer $n$ produces $1$. This is equivalent to (1) there are no cycles except ... More

More Circulant Graphs exhibiting Pretty Good State TransferMay 24 2017The transition matrix of a graph $G$ corresponding to the adjacency matrix $A$ is defined by $H(t):=\exp{\left(-itA\right)},$ where $t\in\mathbb{R}$. The graph is said to exhibit pretty good state transfer between a pair of vertices $u$ and $v$ if there ... More

Adaptive aggregation on graphsApr 29 2017Nov 16 2017We generalize some of the functional (hyper-circle) a posteriori estimates from finite element settings to general graphs or Hilbert space settings. We provide several theoretical results in regard to the generalized a posteriori error estimators. We ... More

On graphs with $m(\partial^L_1)=n-3$Apr 11 2017Let $\partial^L_1\ge\partial^L_2\ge\cdots\ge\partial^L_n$ be the distance Laplacian eigenvalues of a connected graph $G$ and $m(\partial^L_i)$ the multiplicity of $\partial^L_i$. It is well known that the graphs with $m(\partial^L_1)=n-1$ are complete ... More

Positive definite kernels and boundary spacesNov 13 2016We consider a kernel based harmonic analysis of "boundary," and boundary representations. Our setting is general: certain classes of positive definite kernels. Our theorems extend (and are motivated by) results and notions from classical harmonic analysis ... More

On the Wiener index, distance cospectrality and transmission regular graphsSep 22 2016In this paper we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral graphs with ... More

On the Wiener index, distance cospectrality and transmission regular graphsSep 22 2016Jul 03 2017In this paper, we investigate various algebraic and graph theoretic properties of the distance matrix of a graph. Two graphs are $D$-cospectral if their distance matrices have the same spectrum. We construct infinite pairs of $D$-cospectral graphs with ... More

Cauchy and signaling problems for the time-fractional diffusion-wave equationSep 18 2016In this paper, some known and novel properties of the Cauchy and signaling problems for the one-dimensional time-fractional diffusion-wave equation with the Caputo fractional derivative of order $\beta,\ 1 \le \beta \le 2$ are investigated. In particular, ... More

Constructing Frequency Domains on Graphs in Near-Linear TimeSep 14 2016Apr 05 2018Analysis of big data has become an increasingly relevant area of research, with data often represented on discrete networks both constructed and organic. While for structured domains, there exist intuitive definitions of signals and frequencies, the definitions ... More

Constructing Frequency Domains on Graphs in Near-Linear TimeSep 14 2016Analysis of big data has become an increasingly relevant area of research, with data often represented on discrete networks both constructed and organic. While for structured domains, there exist intuitive definitions of signals and frequencies, the definitions ... More

The XML and Semantic Web Worlds: Technologies, Interoperability and Integration. A Survey of the State of the ArtAug 11 2016In the context of the emergent Web of Data, a large number of organizations, institutes and companies (e.g., DBpedia, Geonames, PubMed ACM, IEEE, NASA, BBC) adopt the Linked Data practices and publish their data utilizing Semantic Web (SW) technologies. ... More

Pretty Good State Transfer on Circulant GraphsJul 13 2016Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)},\;t\in\Rl$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there ... More

Do logarithmic proximity measures outperform plain ones in graph clustering?May 03 2016Feb 18 2017We consider a number of graph kernels and proximity measures including commute time kernel, regularized Laplacian kernel, heat kernel, exponential diffusion kernel (also called "communicability"), etc., and the corresponding distances as applied to clustering ... More

Torsion of the Graph Laplacian: Sandpiles, Electrical Networks, and Homological AlgebraApr 24 2016Aug 15 2016An important technique in the electrical inverse problem is to "layer-strip" a graph with boundary by iteratively contracting boundary spikes and deleting boundary edges. We construct an algebraic invariant $\Upsilon$ to test whether a graph can be completely ... More

Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological AlgebraApr 24 2016Jan 10 2017We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with a generalized ... More

Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equationsMar 30 2016In this paper, we investigate under which conditions a differential inequality F on a manifold X satisfies a Liouville-type property. This question, arising in nonlinear potential theory, is related to the validity of an appropriate maximum principle ... More

Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equationsMar 30 2016Apr 23 2018In this paper, we investigate under which conditions a differential inequality F on a manifold X satisfies a Liouville-type property. This question, arising in nonlinear potential theory, is related to the validity of an appropriate maximum principle ... More

Convex spaces, affine spaces, and commutants for algebraic theoriesMar 10 2016Certain axiomatic notions of $\textit{affine space}$ over a ring and $\textit{convex space}$ over a preordered ring are examples of the notion of $\mathcal{T}$-algebra for an algebraic theory $\mathcal{T}$ in the sense of Lawvere. Herein we study the ... More

Convex spaces, affine spaces, and commutants for algebraic theoriesMar 10 2016May 14 2017Certain axiomatic notions of $\textit{affine space}$ over a ring and $\textit{convex space}$ over a preordered ring are examples of the notion of $\mathcal{T}$-algebra for an algebraic theory $\mathcal{T}$ in the sense of Lawvere. Herein we study the ... More

Fast calculation of correlations in recognition systemsMar 06 2016Computationally efficient classification system architecture is proposed. It utilizes fast tensor-vector multiplication algorithm to apply linear operators upon input signals . The approach is applicable to wide variety of recognition system architectures ... More

Beyond CCA: Moment Matching for Multi-View ModelsFeb 29 2016Jun 03 2016We introduce three novel semi-parametric extensions of probabilistic canonical correlation analysis with identifiability guarantees. We consider moment matching techniques for estimation in these models. For that, by drawing explicit links between the ... More

Ordering connected graphs by their Kirchhoff indicesFeb 21 2016The Kirchhoff index $Kf(G)$ of a graph $G$ is the sum of resistance distances between all unordered pairs of vertices, which was introduced by Klein and Randi\'c. In this paper we characterized all extremal graphs with Kirchhoff index among all graphs ... More

Nonuniform sampling, reproducing kernels, and the associated Hilbert spacesJan 27 2016In a general context of positive definite kernels $k$, we develop tools and algorithms for sampling in reproducing kernel Hilbert space $\mathscr{H}$ (RKHS). With reference to these RKHSs, our results allow inference from samples; more precisely, reconstruction ... More

Atiyah classes and dg-Lie algebroids for matched pairsJan 23 2016Apr 01 2016For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal{M}=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \mathcal{M}$ and the projection $p:\mathcal{M}\to L[1]$ are morphisms ... More

Atiyah classes and dg-Lie algebroids for matched pairsJan 23 2016Sep 21 2017For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal M=L[1]\oplus L/A$ such that the inclusion $\iota: A[1] \to \mathcal M$ and the projection $p:\mathcal M\to L[1]$ are morphisms ... More

On the spectral distributions of distance-k graph of free product graphsJan 18 2016We calculate the distribution with respect to the vacuum state of the distance-$k$ graph of a $d$-regular tree. From this result we show that the distance-$k$ graph of a $d$-regular graphs converges to the distribution of the distance-$k$ graph of a regular ... More

Signal Flow Graph Approach to Efficient DST I-IV AlgorithmsJan 18 2016In this paper, fast and efficient discrete sine transformation (DST) algorithms are presented based on the factorization of sparse, scaled orthogonal, rotation, rotation-reflection, and butterfly matrices. These algorithms are completely recursive and ... More

Non-archimedean valuations of eigenvalues of matrix polynomialsJan 04 2016We establish general weak majorization inequalities, relating the leading exponents of the eigenvalues of matrices or matrix polynomials over the field of Puiseux series with the tropical analogues of eigenvalues. We also show that these inequalities ... More

Crushing runtimes in adiabatic quantum computation with Energy Landscape Manipulation (ELM): Application to Quantum FactoringOct 26 2015We introduce two methods for speeding up adiabatic quantum computations by increasing the energy between the ground and first excited states. Our methods are even more general. They can be used to shift a Hamiltonian's density of states away from the ... More

Ordered Fields, the Purge of Infinitesimals from Mathematics and the Rigorousness of Infinitesimal CalculusSep 13 2015We present a characterization of the completeness of the field of real numbers in the form of a \emph{collection of several equivalent statements} borrowed from algebra, real analysis, general topology, and non-standard analysis. We also discuss the completeness ... More

On the distance spectra of graphsSep 03 2015Oct 01 2015The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the distance spectra ... More

Reducing multi-qubit interactions in adiabatic quantum computation without adding auxiliary qubits. Part 2: The "split-reduc" method and its application to quantum determination of Ramsey numbersAug 28 2015Quantum annealing has recently been used to determine the Ramsey numbers R(m,2) for 3 < m < 9 and R(3,3) [Bian et al. (2013) PRL 111, 130505]. This was greatly celebrated as the largest experimental implementation of an adiabatic evolution algorithm to ... More

Reducing multi-qubit interactions in adiabatic quantum computation without adding auxiliary qubits. Part 1: The "deduc-reduc" method and its application to quantum factorization of numbersAug 19 2015Oct 01 2015Adiabatic quantum computing has recently been used to factor 56153 [Dattani & Bryans, arXiv:1411.6758] at room temperature, which is orders of magnitude larger than any number attempted yet using Shor's algorithm (circuit-based quantum computation). However, ... More

Limit theorems for random walksApr 07 2015Feb 20 2017We consider a random walk $S_{\tau}$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner's subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau}$ ... More

Limit theorems for random walksApr 07 2015Dec 31 2015We consider a random walk $S_{\tau}$ which is obtained from the simple random walk $S$ by a discrete time version of Bochner's subordination. We prove that under certain conditions on the subordinator $\tau$ appropriately scaled random walk $S_{\tau}$ ... More

A Survey of Manoeuvring Target Tracking MethodsMar 06 2015A comprehensive review of the literature on manoeuvring target tracking for both uncluttered and cluttered measurements is presented. Various discrete-time dynamical models including non-random input, random-input and switching or hybrid system manoeuvre ... More

A degenerating Cahn-Hilliard system coupled with complete damage processesFeb 20 2015Sep 15 2016In this work, we analytically investigate a degenerating PDE system for phase separation and complete damage processes considered on a nonsmooth time-dependent domain with mixed boundary conditions. The evolution of the system is described by a \textit{degenerating} ... More

Expanders are order diameter non-hyperbolicJan 30 2015Feb 25 2015We show that expander graphs must have Gromov-hyperbolicity at least proportional to their diameter, with a constant of proportionality depending only on the expansion constant and maximal degree. In other words, expanders contain geodesic triangles which ... More

The signless Laplacian Estrada index of tricyclic graphsDec 06 2014Oct 28 2015The signless Laplacian Estrada index of a graph $G$ is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q_i}$ where $q_1, q_2, \ldots, q_n$ are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we show that there are exactly two tricyclic graphs ... More

The Signless Laplacian Estrada Index of Unicyclic GraphsDec 06 2014For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$,where $q_1, q_2, \dots, q_n$are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with ... More

Quantum factorization of 56153 with only 4 qubitsNov 25 2014Nov 27 2014The largest number factored on a quantum device reported until now was 143. That quantum computation, which used only 4 qubits at 300K, actually also factored much larger numbers such as 3599, 11663, and 56153, without the awareness of the authors of ... More

On the symmetry of the Laplacian spectra of signed graphsNov 22 2014We study the symmetry properties of the spectra of normalized Laplacians on signed graphs. We find a new machinery that generates symmetric spectra for signed graphs, which includes bipartiteness of unsigned graphs as a special case. Moreover, we prove ... More

Numerical method of characteristics for one-dimensional blood flowNov 20 2014Mar 27 2015Mathematical modeling at the level of the full cardiovascular system requires the numerical approximation of solutions to a one-dimensional nonlinear hyperbolic system describing flow in a single vessel. This model is often simulated by computationally ... More

Cheeger constants, structural balance, and spectral clustering analysis for signed graphsNov 13 2014Dec 08 2017We introduce a family of multi-way Cheeger-type constants $\{h_k^{\sigma}, k=1,2,\ldots, n\}$ on a signed graph $\Gamma=(G,\sigma)$ such that $h_k^{\sigma}=0$ if and only if $\Gamma$ has $k$ balanced connected components. These constants are switching ... More