José C. Pereira

Factorization Algorithm for Some Special Non-rational Matrix Functions

We construct an algorithm that allows us to determine an effective generalized factorization of a special class of matrix functions. We use the same algorithm to analyze the spectrum of a self-adjoint operator which is related to the obtained generalized factorization.

An Optimized Divide-and-Conquer Algorithm for the Closest-Pair Problem in the Planar Case

We present an engineered version of the divide-and-conquer algorithm for finding the closest pair of points, within a given set of points in the XY-plane. For this version of the algorithm we show that only two pairwise comparisons are required in the combine step, for each point that lies in the 2δ-wide vertical slab. The correctness of the algorithm is shown for all Minkowski distances with p ≥ 1. We also show empirically that, although the time complexity of the algorithm is still O(n lg n), the reduction in the total number of comparisons leads to a significant reduction in the total execution time, for inputs with size sufficiently large.

Providing an Active Learning Environment for Introducing Linear Programming

Literature on teaching and learning methods reports strong evidence of the effectiveness of using active learning techniques. This paper presents the GLP-Tool, an example of an active learning technical tool which enables an active learning environment to explore the fundamental concepts of Linear Programming (LP). The GLP-Tool is designed to solve user-defined LP problems with two variables, up to a didactical limit of five constraints (plus the non-negativity constraints). Implemented using the computer algebra system Mathematica, this interactive tool allows the user to dynamically explore different objective functions and sets of constraints. All the GLP-Tool functionalities are represented graphically and updated in real time. These interactive, dynamic, and graphical features make the GLP-Tool a powerful tool for teaching and learning LP, both in undergraduate and high school courses. The GLP-Tool is already freely available online at Sapientia—the repository of the “Universidade do Algarve” (University of the Algarve) and at one of the authors’ ResearchGate web page.

Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

Spectral theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic calculations present in many mathematical algorithms. In our work we use the CAS Mathematica to implement for the first time on a computer analytical algorithms developed by us and others within the Operator Theory. The main goal of this paper is to show how the symbolic computation capabilities of Mathematica allow us to explore the spectra of several classes of singular integral operators. For the one-dimensional case, nontrivial rational examples, computed with the automated process called [ASpecPaired-Scalar], are presented. For the matrix case, nontrivial essentially bounded and rational examples, computed with the analytical algorithms [AFact], [SInt], and [ASpecPaired-Matrix], are presented. In both cases, it is possible to check, for each considered paired singular integral operator, if a complex number (chosen arbitrarily) belongs to its spectrum.

Symbolic Computation Applied to the Study of the Kernel of a Singular Integral Operator with Non-Carleman Shift and Conjugation

On the Hilbert space

Computing some classes of Cauchy type singular integrals with Mathematica software

\widetilde{L}_{2}(\mathbb {T})

A Java Implementation of the SGA, UMDA, ECGA, and HBOA.

L~2(T) the singular integral operator with non-Carleman shift and conjugation