Fredy Vides

Fredy Vides, Idelfonso B. R. Nogueira, Gabriela Lopez Gutierrez et al.

In this document, we present key findings in structured matrix approximation theory, with applications to the regressive representation of dynamic financial processes. Initially, we explore a comprehensive approach involving generic nonlinear time delay embedding for time series data extracted from a financial or economic system under examination. Subsequently, we employ sparse least-squares and structured matrix approximation methods to discern approximate representations of the output coupling matrices. These representations play a pivotal role in establishing the regressive models corresponding to the recursive structures inherent in a given financial system. The document further introduces prototypical algorithms that leverage the aforementioned techniques. These algorithms are demonstrated through applications in approximate identification and predictive simulation of dynamic financial and economic processes, encompassing scenarios that may or may not exhibit chaotic behavior.

Fredy Vides

In this document we study the uniform local path connectivity of sets of $m$-tuples of pairwise commuting normal matrices with some additional constraints. More specifically, given given $\varepsilon>0$, a fixed metric $ð$ in ${M_n(\mathbb{C})}^m$ induced by the operator norm $\|\cdot\|$, any collection of $r$ non-constant multivariable polynomials $p_1(x_1,\ldots,x_m),\ldots,p_r(x_1,\ldots,x_m)$ over $\mathbb{C}$ with finite zero set $\mathbf{Z}(p_1,\ldots,p_r)\subset \mathbb{C}^m$, and any $m$-tuple $\mathbf{X}=(X_1,\ldots,X_m)$ in the set $\mathbb{ZD}_n^m(p_1,\ldots,p_r)\subseteq M_n^m(\mathbb{C})$, of pairwise commuting normal matrix contractions such that, $\|p_j(Y_1,\ldots,Y_m)\|=0$ for each $(Y_1,\ldots,Y_m)\in \mathbb{ZD}_n^m(p_1,\ldots,p_r)$ and each $1\leq j\leq r$. We prove the existence of paths between arbitrary $m$-tuples, that lie in the intersection of $\mathbb{ZD}_n^m(p_1,\ldots,p_r)$, and the $δ$-ball $B_ð(\mathbf{X},δ)$ centered at $\mathbf{X}$ for some $δ>0$, with respect to $ð$. Two of the key features of these matrix paths is that $δ$ can be chosen independent of $n$, and that they are contained in the intersection of $B_ð(\mathbf{X},\varepsilon)$ and $\mathbb{ZD}_n^m(p_1,\ldots,p_r)$. Some connections with the approximation theory for matrix functions of several matrix variables, are studied as well.

Fredy Vides

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.

Fredy Vides

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.

Fredy Vides, Idelfonso B. R. Nogueira, Gabriela Lopez Gutierrez et al.

The investigation reported in this document focuses on identifying systems with symmetries using equivariant autoregressive reservoir computers. General results in structured matrix approximation theory are presented, exploring a two-fold approach. Firstly, a comprehensive examination of generic symmetry-preserving nonlinear time delay embedding is conducted. This involves analyzing time series data sampled from an equivariant system under study. Secondly, sparse least-squares methods are applied to discern approximate representations of the output coupling matrices. These matrices play a critical role in determining the nonlinear autoregressive representation of an equivariant system. The structural characteristics of these matrices are dictated by the set of symmetries inherent in the system. The document outlines prototypical algorithms derived from the described techniques, offering insight into their practical applications. Emphasis is placed on the significant improvement on structured identification precision when compared to classical reservoir computing methods for the simulation of equivariant dynamical systems.

Fredy Vides

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.

Fredy Vides

Some elements of the theory and algorithmics corresponding to the computation of semilinear sparse models for discrete-time signals are presented. In this study, we will focus on approximately eventually periodic discrete-time signals, that is, signals that can exhibit an aperiodic behavior for an initial amount of time, and then become approximately periodic afterwards. The semilinear models considered in this study are obtained by combining sparse representation methods, linear autoregressive models and GRU neural network models, initially fitting each block model independently using some reference data corresponding to some signal under consideration, and then fitting some mixing parameters that are used to obtain a signal model consisting of a linear combination of the previously fitted blocks using the aforementioned reference data, computing sparse representations of some of the matrix parameters of the resulting model along the process. Some prototypical computational implementations are presented as well.