Generalized companion matrix for approximate GCD

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]=(f). In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD(f; g). Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.Comment: Submitted to MEGA 201

Decay properties of spectral projectors with applications to electronic structure

Motivated by applications in quantum chemistry and solid state physics, we apply general results from approximation theory and matrix analysis to the study of the decay properties of spectral projectors associated with large and sparse Hermitian matrices. Our theory leads to a rigorous proof of the exponential off-diagonal decay ("nearsightedness") for the density matrix of gapped systems at zero electronic temperature in both orthogonal and non-orthogonal representations, thus providing a firm theoretical basis for the possibility of linear scaling methods in electronic structure calculations for non-metallic systems. We further discuss the case of density matrices for metallic systems at positive electronic temperature. A few other possible applications are also discussed.Comment: 63 pages, 13 figure

Anales del III Congreso Internacional de Vivienda y Ciudad "Debate en torno a la nueva agenda urbana"

Acta de congresoEl III Congreso Internacional de Vivienda y Ciudad “Debates en torno a la NUEVa Agenda Urbana”, ha sido una apuesta de alto compromiso por acercar los debates centrales y urgentes que tensionan el pleno ejercicio del derecho a la ciudad. Para ello las instituciones organizadoras (INVIHAB –Instituto de Investigación de Vivienda y Hábitat y MGyDH-Maestría en Gestión y Desarrollo Habitacional-1), hemos convidado un espacio que se concretó con potencia en un debate transdisciplinario. Convocó a intelectuales de prestigio internacional, investigadores, académicos y gestores estatales, y en una metodología de innovación articuló las voces académicas con las de las organizaciones sociales y/o barriales en el Foro de las Organizaciones Sociales que tuvo su espacio propio para dar voz a quienes están trabajando en los desafíos para garantizar los derechos a la vivienda y los bienes urbanos en nuestras ciudades del Siglo XXI

Decay properties of matrix functions: an application to electronic structure computation

International audienc

Structured matrix based methods for approximate polynomial GCD

Defining and computing a greatest common divisor of two polynomials with inexact coefficients is a classical problem in symbolic-numeric computation. The first part of this book reviews the main results that have been proposed so far in the literature. As usual with polynomial computations, the polynomial GCD problem can be expressed in matrix form: the second part of the book focuses on this point of view and analyses the structure of the relevant matrices, such as Toeplitz, Toepliz-block and displacement structures. New algorithms for the computation of approximate polynomial GCD are presented, along with extensive numerical tests. The use of matrix structure allows, in particular, to lower the asymptotic computational cost from cubic to quadratic order with respect to polynomial degree. 

Quantum hub and authority centrality measures for directed networks based on continuous-time quantum walks

In this article, we introduce, test and discuss three quantum methods for computing hub and authority centrality scores in directed networks. The methods are based on unitary, continuous-time quantum walks; the construction of a suitable Hermitian Hamiltonian is achieved by performing a quantum walk on the associated bipartite graph. Two methods, called CQAu and CQAw, use the same evolution operator, inspired by the classical Hyperlink-Induced Topic Search (HITS) algorithm, but with different initial states; the computation of hub and authority scores is performed simultaneously. The third method, called CQG and inspired by classical PageRank, requires instead two separate runs with different evolution operators, one for hub and one for authority scores. The methods are tested on several directed graphs with different sizes and properties; a comparison with other well-established ranking algorithms is provided. CQAw emerges as the most reliable of the three methods and yields rankings that are largely compatible with results from HITS, although CQAu and CQG also present interesting features and potential for applications