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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

Group-theoretic generalisations of vertex and edge connectivitiesJun 19 2019Jun 25 2019Let $p$ be an odd prime. Let $P$ be a finite $p$-group of class $2$ and exponent $p$, whose commutator quotient $P/[P,P]$ is of order $p^n$. We define two parameters for $P$ related to central decompositions. The first parameter, $\kappa(P)$, is the smallest ... More

Dominator Chromatic Numbers of Orientations of TreesApr 12 2019Jun 05 2019In this paper we prove that the dominator chromatic number of every oriented tree is invariant under reversal of orientation. In addition to this marquee result, we also prove the exact dominator chromatic number for arborescences and anti-arborescences ... More

Dominator Chromatic Numbers of Orientations of TreesApr 12 2019In this paper we prove that the dominator chromatic number of every oriented tree is invariant under reversal of orientation. In addition to this marquee result, we also prove the exact dominator chromatic number for arborescences and anti-arborescences ... More

Dominator Colorings of DigraphsFeb 19 2019This paper serves as the first extension of the topic of dominator colorings of graphs to the setting of digraphs. We establish the dominator chromatic number over all possible orientations of paths and cycles. In this endeavor we discover that there ... More

The Hitchhiker guide to: Secant Varieties and Tensor DecompositionDec 26 2018We consider here the problem, which is quite classical in Algebraic geometry, of studying the secant varieties of a projective variety $X$. The case we concentrate on is when $X$ is a Veronese variety, a Grassmannian or a Segre variety. Not only these ... More

Accelerating Alternating Least Squares for Tensor Decomposition by Pairwise PerturbationNov 26 2018Jan 05 2019The alternating least squares algorithm for CP and Tucker decomposition is dominated in cost by the tensor contractions necessary to set up the quadratic optimization subproblems. We introduce a novel family of algorithms that uses perturbative corrections ... More

Representation Growth of Special Compact Linear Groups of Order TwoJul 17 2018Oct 08 2018Let $\mathfrak{o}$ be the ring of integers of a non-Archimedean local field $F$ with finite residue field of even characteristic and maximal ideal $\mathfrak{p}.$ Let $\mathrm{e}(\mathfrak{o})$ denotes the ramification index of $\mathfrak{o}$ in case ... More

Modern Regularization Methods for Inverse ProblemsJan 30 2018Regularization methods are a key tool in the solution of inverse problems. They are used to introduce prior knowledge and make the approximation of ill-posed (pseudo-)inverses feasible. In the last two decades interest has shifted from linear towards ... More

Uniform rank gradient, cost and local-global convergenceOct 28 2017Dec 01 2017We analyze the rank gradient of finitely generated groups with respect to sequences of subgroups of finite index that do not necessarily form a chain, by connecting it to the cost of p.m.p. actions. We generalize several results that were only known for ... More

On the spectrum of directed uniform and non-uniform hypergraphsOct 17 2017Aug 15 2018Here, we suggest a method to represent general directed uniform and non-uniform hypergraphs by different connectivity tensors. We show many results on spectral properties of undirected hypergraphs also hold for general directed uniform hypergraphs. Our ... More

Plethysm and fast matrix multiplicationOct 02 2017Apr 09 2018Motivated by the symmetric version of matrix multiplication we study the plethysm $S^k(\mathfrak{sl}_n)$ of the adjoint representation $\mathfrak{sl}_n$ of the Lie group $SL_n$. In particular, we describe the decomposition of this representation into ... 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

Balanced Truncation Model Order Reduction For Quadratic-Bilinear Control SystemsApr 29 2017We discuss balanced truncation model order reduction for large-scale quadratic-bilinear (QB) systems. Balanced truncation for linear systems mainly involves the computation of the Gramians of the system, namely reachability and observability Gramians. ... More

Krawtchouk-Griffiths Systems I: Matrix ApproachNov 21 2016We call \textsl{Krawtchouk-Griffiths} systems, or KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution. ... More

Krawtchouk-Griffiths Systems I: Matrix ApproachNov 21 2016Nov 23 2016We call Krawtchouk-Griffiths systems, or KG-systems, systems of multivariate polynomials orthogonal with respect to corresponding multinomial distributions. The original Krawtchouk polynomials are orthogonal with respect to a binomial distribution. Our ... More

Regular characters of groups of type A_n over discrete valuation ringsApr 04 2016Let O be a complete discrete valuation ring with finite residue field k of odd characteristic. Let G be a general or special linear group or a unitary group defined over O and let $\mathfrak{g}$ denote its Lie algebra. For every positive integer l, let ... 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

On the quasi-depth of squarefree monomial ideals and the sdepth of the monomial ideal of independent sets of a graphNov 22 2015Mar 31 2016If $J\subset I$ are two monomials ideals, we give a practical upper bound for the Stanley depth of $J/I$, which we call it the \emph{quasi-depth} of $J/I$. Also, we compute the quasi-depth of several classes of square free monomial ideals. We also study ... More

Dichromatic number and fractional chromatic numberOct 20 2015The dichromatic number of a graph $G$ is the maximum integer $k$ such that there exists an orientation of the edges of $G$ such that for every partition of the vertices into fewer than $k$ parts, at least one of the parts must contain a directed cycle ... More

On the Convergence of Alternating Least Squares Optimisation in Tensor Format RepresentationsMay 30 2015The approximation of tensors is important for the efficient numerical treatment of high dimensional problems, but it remains an extremely challenging task. One of the most popular approach to tensor approximation is the alternating least squares method. ... More

Permutability graphs of subgroups of some finite non-abelian groupsApr 02 2015In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups $D_n$, the generalized quaternion groups $Q_n$, the quasi-dihedral groups ... More

On generic identifiability of symmetric tensors of subgeneric rankApr 02 2015Jan 08 2016We prove that the general symmetric tensor in $S^d {\mathbb C}^{n+1}$ of rank r is identifiable, provided that r is smaller than the generic rank. That is, its Waring decomposition as a sum of r powers of linear forms is unique. Only three exceptional ... More

Convergence of Alternating Least Squares Optimisation for Rank-One Approximation to High Order TensorsMar 18 2015The approximation of tensors has important applications in various disciplines, but it remains an extremely challenging task. It is well known that tensors of higher order can fail to have best low-rank approximations, but with an important exception ... More

Similarity classes of integral $p$-adic matrices and representation zeta functions of groups of type $A_2$Oct 16 2014Nov 09 2015We compute explicitly Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of various $p$-adic analytic and adelic profinite groups of type $\mathsf{A}_2$. This has consequences for the representation zeta ... More

Bounds and algorithms for limited packings in graphsJul 07 2014We consider (closed neighbourhood) packings and their generalization in graphs called limited packings. A vertex set X in a graph G is a k-limited packing if for any vertex $v\in V(G)$, $\left|N[v] \cap X\right| \le k$, where $N[v]$ is the closed neighbourhood ... More

Small cancellation labellings of some infinite graphs and applicationsJun 19 2014Jun 30 2014We construct small cancellation labellings for some infinite sequences of finite graphs of bounded degree. We use them to define infinite graphical small cancellation presentations of groups. This technique allows us to provide examples of groups with ... More

The probabilistic approach to limited packings in graphsNov 07 2013We consider (closed neighbourhood) packings and their generalization in graphs. A vertex set X in a graph G is a k-limited packing if for any vertex $v\in V(G)$, $\left|N[v] \cap X\right| \le k$, where N[v] is the closed neighbourhood of v. The k-limited ... More

Geometric presentations of Lie groups and their Dehn functionsOct 20 2013We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of these groups. Our work, which also addresses algebraic groups ... More

Geometric presentations of Lie groups and their Dehn functionsOct 20 2013Nov 26 2016We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic ... More

On differences between the border rank and the smoothable rank of a polynomialMay 08 2013Mar 06 2014We consider higher secant varieties to Veronese varieties. Most points on the r-th secant variety are represented by a finite scheme of length r contained in the Veronese variety --- in fact, for generic point, it is just a union of r distinct points. ... More

Groups, Graphs, Languages, Automata, Games and Second-order Monadic LogicJan 15 2012In this paper we survey some surprising connections between group theory, the theory of automata and formal languages, the theory of ends, infinite games of perfect information, and monadic second-order logic.

On genuine infinite algebraic tensor productsDec 14 2011A genuine infinite tensor product of complex vector spaces is a vector space ${\bigotimes}_{i\in I} X_i$ whose linear maps coincide with multilinear maps on an infinite family $\{X_i\}_{i\in I}$ of vector spaces. We give a direct sum decomposition of ... More

Legendre Duality Between Lagrangian and Hamiltonian MechanicsAug 29 2011In some previous papers, a Legendre duality between Lagrangian and Hamiltonian Mechanics has been developed. The (\rho,\eta)-tangent application of the Legendre bundle morphism associated to a Lagrangian L or Hamiltonian H is presented. Using that, a ... More

Families of twisted tensor product codesJul 06 2011Using geometric properties of the variety $\cV_{r,t}$, the image under the Grassmannian map of a Desarguesian $(t-1)$-spread of $\PG(rt-1,q)$, we introduce error correcting codes related to the twisted tensor product construction, producing several families ... More

The generalized Lie algebroids and their applicationsJul 09 2010Aug 10 2010In this paper we introduce the notion of generalized Lie algebroid and we develop a new formalism necessary to obtain a new solution for the Weistein's Problem. Many applications emphasize the importance and the utility of this new framework determined ... More

Gallai colorings and domination in multipartite digraphsJun 10 2010Assume that D is a digraph without cyclic triangles and its vertices are partitioned into classes A_1,...,A_t of independent vertices. A set $U=\cup_{i\in S} A_i$ is called a dominating set of size |S| if for any vertex $v\in \cup_{i\notin S} A_i$ there ... More

A Simple Proof of an Inequality Connecting the Alternating Number of Independent Sets and the Decycling NumberMay 21 2009If alpha=alpha(G) is the maximum size of an independent set and s_{k} equals the number of stable sets of cardinality k in graph G, then I(G;x)=s_{0}+s_{1}x+...+s_{alpha}x^{alpha} is the independence polynomial of G. In this paper we provide an elementary ... More

Upper bounds for alpha-domination parametersMay 05 2008In this paper, we provide a new upper bound for the alpha-domination number. This result generalises the well-known Caro-Roditty bound for the domination number of a graph. The same probabilistic construction is used to generalise another well-known upper ... More

Divergence in lattices in semisimple Lie groups and graphs of groupsJan 27 2008May 01 2009Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $\le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having ... More

Divergence in lattices in semisimple Lie groups and graphs of groupsJan 27 2008Jun 12 2017Divergence functions of a metric space estimate the length of a path connecting two points $A$, $B$ at distance $\le n$ avoiding a large enough ball around a third point $C$. We characterize groups with non-linear divergence functions as groups having ... More

Offensive k-alliances in graphsMar 20 2007Let $G=(V,E)$ be a simple graph. For a nonempty set $X\subset V,$ and a vertex $v\in V,$ $\delta_{X}(v)$ denotes the number of neighbors $v$ has in $X.$ A nonempty set $S\subset V$ is an \emph{offensive $r$-alliance} in $G$ if $\delta_S(v)\ge \delta_{\bar{S}}(v)+r,$ ... More

Groups acting on tree-graded spaces and splittings of relatively hyperbolic groupJan 13 2006Jan 31 2006Tree-graded spaces are generalizations of R-trees. They appear as asymptotic cones of groups (when the cones have cut points). Since many questions about endomorphisms and automorphisms of groups, solving equations over groups, studying embeddings of ... More

Ranking Participants in Tournaments by means of Rating FunctionsJan 03 2006Dec 02 2006In this paper we bring a novel approach to the theory of tournament rankings. We combine two different theories that are widely used to establish rankings of populations after a given tournament. First, we use the statistical approach of paired comparison ... More

Symmetric Groups and Expander GraphsMay 28 2005We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This answers affirmatively ... More

Symmetric Groups and ExpandersMar 10 2005We construct an explicit generating sets $F_n$ and $\tilde F_n$ of the alternating and the symmetric groups, which make the Cayley graphs $C(Alt(n), F_n)$ and $C(Sym(n), \tilde F_n)$ a family of bounded degree expanders for all sufficiently large $n$. ... More

Universal lattices and unbounded rank expandersFeb 11 2005We study the representations of non-commutative universal lattices and use them to compute lower bounds for the \TauC for the commutative universal lattices $G_{d,k}= \SL_d(\Z[x_1,...,x_k])$ with respect to several generating sets. As an application of ... More

Symmetric groups and the cup product on the cohomology of Hilbert schemesSep 13 2000Dec 12 2000The integral cohomology ring of the Hilbert scheme of n-tuples on the affine plane is shown to be isomorphic to the graded ring associated to a filtration of the ring of integral class functions on the symmetric group.