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Strong cliques and forbidden cyclesMar 14 2019Given a graph $G$, the strong clique number $\omega_2'(G)$ of $G$ is the cardinality of a largest collection of edges every pair of which are incident or connected by an edge in $G$. We study the strong clique number of graphs missing some set of cycle ... 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
Fair splittings by independent sets in sparse graphsSep 10 2018Given a partition $V_1 \sqcup V_2 \sqcup \dots \sqcup V_m$ of the vertex set of a graph, we are interested in finding multiple disjoint independent sets that contain the correct fraction of vertices of each $V_j$. We give conditions for the existence ... More
Motion Feasibility Conditions for Multi-Agent Control Systems on Lie GroupsAug 14 2018We study motion feasibility conditions of decentralized multi-agent control systems on Lie groups with collision avoidance constraints, modeled by an undirected graph. We first consider agents modeled by a kinematic left invariant control systems (single ... More
Variational Principles for Optimal Control of Left-Invariant Multi-Agent Systems with Asymmetric Formation ConstraintsFeb 05 2018We study an optimal control problem for a multi-agent system modeled by an undirected formation graph with nodes describing the kinematics of each agent, given by a left invariant control system on a Lie group. The agents should avoid collision between ... 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
Extremal solutions to some art gallery and terminal-pairability problemsAug 29 2017Nov 07 2017The chosen tool of this thesis is an extremal type approach. The lesson drawn by the theorems proved in the thesis is that surprisingly small compromise is necessary on the efficacy of the solutions to make the approach work. The problems studied have ... More
Layered graphs: a class that admits polynomial time solutions for some hard problemsMay 18 2017May 25 2017The independent set on a graph $G=(V,E)$ is a subset of $V$ such that no two vertices in the subset have an edge between them. The MIS problem on $G$ seeks to identify an independent set with maximum cardinality, i.e. maximum independent set or MIS. $V* ... 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
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
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
The (p,q)-extremal problem and the fractional chromatic number of Kneser hypergraphsAug 14 2014The problem of computing the chromatic number of Kneser hypergraphs has been extensively studied over the last 40 years and the fractional version of the chromatic number of Kneser hypergraphs is only solved for particular cases. The \emph{$(p,q)$-extremal ... More
Convex Hull of Face Vectors of Colored ComplexesOct 31 2012In this paper we verify a conjecture by Kozlov (Discrete Comput Geom 18 (1997) 421--431), which describes the convex hull of the set of face vectors of $r$-colorable complexes on $n$ vertices. As part of the proof we derive a generalization of Tur\'{a}n's ... More
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
Conflict-free coloring of graphsNov 23 2011Sep 19 2013We study the conflict-free chromatic number chi_{CF} of graphs from extremal and probabilistic point of view. We resolve a question of Pach and Tardos about the maximum conflict-free chromatic number an n-vertex graph can have. Our construction is randomized. ... More
On Krawtchouk TransformsJun 24 2011Krawtchouk polynomials appear in a variety of contexts, most notably as orthogonal polynomials and in coding theory via the Krawtchouk transform. We present an operator calculus formulation of the Krawtchouk transform that is suitable for computer implementation. ... More
Graph products of spheres, associative graded algebras and Hilbert seriesJan 28 2009Mar 08 2010Given a finite, simple, vertex-weighted graph, we construct a graded associative (non-commutative) algebra, whose generators correspond to vertices and whose ideal of relations has generators that are graded commutators corresponding to edges. We show ... More
Face vectors of flag complexesMay 26 2006A conjecture of Kalai and Eckhoff that the face vector of an arbitrary flag complex is also the face vector of some particular balanced complex is verified.
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