A GPU-based hyperbolic SVD algorithm

A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm, using a massively parallel graphics processing unit (GPU), is developed. The algorithm also serves as the final stage of solving a symmetric indefinite eigenvalue problem. Numerical testing demonstrates the gains in speed and accuracy over sequential and MPI-parallelized variants of similar Jacobi-type HSVD algorithms. Finally, possibilities of hybrid CPU--GPU parallelism are discussed.Comment: Accepted for publication in BIT Numerical Mathematic

Bethe Ansatz, Inverse Scattering Transform and Tropical Riemann Theta Function in a Periodic Soliton Cellular Automaton for A^{(1)}_n

We study an integrable vertex model with a periodic boundary condition associated with U_q(A_n^{(1)}) at the crystallizing point q=0. It is an (n+1)-state cellular automaton describing the factorized scattering of solitons. The dynamics originates in the commuting family of fusion transfer matrices and generalizes the ultradiscrete Toda/KP flow corresponding to the periodic box-ball system. Combining Bethe ansatz and crystal theory in quantum group, we develop an inverse scattering/spectral formalism and solve the initial value problem based on several conjectures. The action-angle variables are constructed representing the amplitudes and phases of solitons. By the direct and inverse scattering maps, separation of variables into solitons is achieved and nonlinear dynamics is transformed into a straight motion on a tropical analogue of the Jacobi variety. We decompose the level set into connected components under the commuting family of time evolutions and identify each of them with the set of integer points on a torus. The weight multiplicity formula derived from the q=0 Bethe equation acquires an elegant interpretation as the volume of the phase space expressed by the size and multiplicity of these tori. The dynamical period is determined as an explicit arithmetical function of the n-tuple of Young diagrams specifying the level set. The inverse map, i.e., tropical Jacobi inversion is expressed in terms of a tropical Riemann theta function associated with the Bethe ansatz data. As an application, time average of some local variable is calculated

Design of Rabin-like cryptosystem without decryption failure

In this work, we design a new, efficient and practical Rabin-like cryptosystem without using the Jacobi symbol, redundancy in the message and avoiding the demands of extra information for finding the correct plaintext. Decryption outputs a unique plaintext without any decryption failure. In addition, decryption only requires a single prime. Furthermore, the decryption procedure only computes a single modular exponentiation instead of two modular exponentiation executed by other Rabin variants. As a result, this reduces the computational effort during the decryption process. Moreover the Novak’s side channel attack is impractical over the proposed Rabin-like cryptosystem. In parallel, we prove that the Rabin-p cryptosystem is indeed as intractable as the integer factorization problem

OPUC, CMV matrices and perturbations of measures supported on the unit circle

Let us consider a Hermitian linear functional defined on the linear space of Laurent polynomials with complex coefficients. In the literature, canonical spectral transformations of this functional are studied. The aim of this research is focused on perturbations of Hermitian linear functionals associated with a positive Borel measure supported on the unit circle. Some algebraic properties of the perturbed measure are pointed out in a constructive way. We discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated Verblunsky coefficients. Then, the structure of the Theta matrices of the perturbed linear functionals, which is the main tool for the comparison of their corresponding CMV matrices, is deeply analyzed. From the comparison between different CMV matrices, other families of perturbed Verblunsky coefficients will be considered. We introduce a new matrix, named Fundamental matrix, that is a tridiagonal symmetric unitary matrix, containing basic information about the family of orthogonal polynomials. However, we show that it is connected to another family of orthogonal polynomials through the Takagi decomposition.The authors would like to thank Professor Bernhard Beckermann and Professor RogerA. Horn for valuable and insightful discussions about congruence relations. We also thank the suggestions by the referees which have contributed to improve substantially the presentation of the manuscript. The work of the first author (FM) was partially sup-ported by Dirección General de Política Científica y Tecnológica, Ministerio de Economía y Competitividad (MINECO) of Spain, under grant MTM2012-36732-C03-01. The sec-ond author (NS) thanks Alexander von Humboldt Foundation for the support and the Department of Mathematics, Universidad Carlos III de Madrid, for its constant support and friendly atmosphere during the period January–July 2014 when the manuscript was finished

Tropical R and Tau Functions

Tropical R is the birational map that intertwines products of geometric crystals and satisfies the Yang-Baxter equation. We show that the D^{(1)}_n tropical R introduced by the authors and its reduction to A^{(2)}_{2n-1} and C^{(1)}_n are equivalent to a system of bilinear difference equations of Hirota type. Associated tropical vertex models admit solutions in terms of tau functions of the BKP and DKP hierarchies.Comment: LaTeX2e, 26page