arXiv subjects

Fredy Vides

Publications and source records attributed to Fredy Vides.

24 records · Page 2Linked to original sources

arXiv2016

Local Matrix Homotopies and Soft Tori

We present solutions to local connectivity problems in matrix representations of the form $C([-1,1]^{N}) \to A_{n,\varepsilon} \leftarrow C_{\varepsilon}(\mathbb{T}^{2})$ for any $\varepsilon\in[0,2]$ and any integer $n\geq 1$, where $A_{n,\varepsilon}\subseteq M_n$ and where $C_{\varepsilon}(\mathbb{T}^{2})$ denotes the {\bf Soft Torus}. We solve the connectivity problems by introducing the so called toroidal matrix links, which can be interpreted as normal contractive matrix analogies of free homotopies in differential algebraic topology. In order to deal with the locality constraints, we have combined some techniques introduced in this document with some techniques from matrix geometry, combinatorial optimization, classification and representation theory of C$^*$-algebras.

math.OA↗

arXiv2013

Estimating Norms of Commutators

We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y] assuming that x and y are contractions in a C*-algebra A, with x normal and with spectrum within the domain of f. In particular we discuss [x^2, y] and [x^(1/2), y] for 0 <=, x <=, 1. For larger values of δ= \|[x; y]\| we can rigorous calculate the best possible upper bound \|[f(x), y]\| for many f. In other cases we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.

math.OA↗

arXiv2011

Approximation of Schr{ö}dinger Unitary Groups of Operators by Particular Projection Methods

In this paper we work with the approximation of unitary groups of operators of the form $e^{-itH}$ where $H\in\mathscr{L}(\mathcal{H})$ is the Hamiltonian of a given quantum dynamical system modeled in the discretizable Hilbert space $\mathcal{H}=\mathcal{H}(G)$, to perform such approximations we implement some techniques from operator theory that we name particular projection methods by compatibility with quantum theory conventions. Once particular representations are defined we study the interelation between some of them properties with the original operators that they mimic. In the end some estimates for numerical implementation are presented to verify the theoretical discussion.

math.FA↗

arXiv2011

On Matrix Schrödinger Unitary Groups in Particular Representations of Finite Dimensional Quantum Dynamical Systems

In this paper we study some particular types of matrix Schrödinger semigroups of the form $\exp(-it\mathbb{H})$ where $\mathbb{H}\in M_N(\mathbf{C})$ is the Hamiltonian of a given quantum dynamical system modeled in the finite dimensional Hilbert space $\mathcal{H}$. Once we have defined a particular matrix Schrödinger unitary group we perform some estimates for its approximation and its corresponding implementation in the numerical solution of the finite dimensional Schrödinger evolution equation to that it is related.

math.QA↗

arXiv2010

On the Approximation of Nonlinear Evolution Equations in Particular C*-Algebras of Operators

In this article we deal with the stability and convergence of numerical solutions of nonlinear evolution equations of the form $A(u(t))+f(u(t))=u'(t)$, the numerical analysis of solutions to this problems will be performed using some methods from particular algebras of operators which are sometimes represented by unital subalgebras of the unital C*-algebras of operators that are generated by some basic operators say $\mathbf{1},a,\mathcal{D}(\cdot)\in\mathcal{L}(H^m(G))$ that in some suitable sense are related to the operator $A(\cdot)\in\mathcal{L}(H^m(G))$ in the evolution equations, particular cases where the operator algebras do not verify the C*-identity with respect to the norm chosen are also studied, when applicable basic C*-algebra techniques are implemented to perform some estimates of numerical solutions to some types of problems, in all this work expressions like $H^m(G)$ will represent a prescribed discretizable Hilbert space with $G\subset\subset\mathbb{R}^n$.

math.FA↗

arXiv2010

On the Approximation of Contractive Semigroups of Operators in Discretizable Hilbert Spaces

The Computation of discrete Contractive semigroups becomes necessary when we deal with several types of evolution equations in Discretizable Hilbert spaces, in this work we study some properties of the discrete forms of the contractive semigroups induced by an approximation scheme in a prescribed Hilbert space, we also deal with the implementation of computational methods in this Hilbert Space and apply some of the results presented here in the Heisenberg representation of quantum dynamical semigroups.

math.NA↗