Algorithms for curve design and accurate computations with totally positive matrices

Esta tesis doctoral se enmarca dentro de la teoría de la Positividad Total. Las matrices totalmente positivas han aparecido en aplicaciones de campos tan diversos como la Teoría de la Aproximación, la Biología, la Economía, la Combinatoria, la Estadística, las Ecuaciones Diferenciales, la Mecánica, el Diseño Geométrico Asistido por Ordenador o el Álgebra Numérica Lineal. En esta tesis nos centraremos en dos de los campos que están relacionados con matrices totalmente positivas.This doctoral thesis is framed within the theory of Total Positivity. Totally positive matrices have appeared in applications from fields as diverse as Approximation Theory, Biology, Economics, Combinatorics, Statistics, Differential Equations, Mechanics, Computer Aided Geometric Design or Linear Numerical Algebra. In this thesis, we will focus on two of the fields that are related to totally positive matrices.<br /

Accurate and efficient computations with Wronskian matrices of Bernstein and related bases

In this article, we provide a bidiagonal decomposition of the Wronskian matrices of Bernstein bases of polynomials and other related bases such as the Bernstein basis of negative degree or the negative binomial basis. The mentioned bidiagonal decompositions are used to achieve algebraic computations with high relative accuracy for these Wronskian matrices. The numerical experiments illustrate the accuracy obtained using the proposed decomposition when computing inverse matrices, eigenvalues or singular values, and the solution of some related linear systems. © 2021 John Wiley & Sons Ltd

A new class of trigonometric B-Spline Curves

We construct one-frequency trigonometric spline curves with a de Boor-like algorithm for evaluation and analyze their shape-preserving properties. The convergence to quadratic B-spline curves is also analyzed. A fundamental tool is the concept of the normalized B-basis, which has optimal shape-preserving properties and good symmetric properties

Implementation of an efficient strategy to analyze the mathematical training required in undergraduate degrees in engineering and architecture

Engineering and Architecture studies aim to develop professionals capable of facing and solving complex multidisciplinary problems. This requires abilities that cannot be acquired without a comprehensive training model, addressing real-world problems to provide technical solutions. Furthermore, it is necessary to ensure the acquisition of skills to generalise and think abstractly about reality. For this purpose, fundamental disciplines training, such as mathematics, play a relevant role in technical undergraduate studies and their corresponding contents and temporal order traineeship should be properly analysed when designing their curricula. This work describes an active and collaborative methodology used to analyse the mathematical concepts and tools that should be introduced in the undergraduate degrees in the School of Engineering and Architecture at the University of Zaragoza. The methodology applied is based on the activation of communication mechanisms between the teaching staff of mathematical subjects and those of higher courses, in which the students acquire the specific skills of each degree. Moreover, it has motivated interesting discussions between mathematics teaching staff. As a result, a great deal of information has been gathered about the mathematical knowledge required, and its appropriate scheduling, in all degree programs. Also, some deficiencies that should be addressed in the initial training have been identified. Finally, a strategy has been planned for contextualising the mathematical training in different disciplines to help students to understand the relevance of mathematics formation and motivate them in their study

Trajectory definition with high relative accuracy (HRA) by parametric representation of curves in nano-positioning systems

Nanotechnology applications demand high accuracy positioning systems. Therefore, in order to achieve sub-micrometer accuracy, positioning uncertainty contributions must be minimized by implementing precision positioning control strategies. The positioning control system accuracy must be analyzed and optimized, especially when the system is required to follow a predefined trajectory. In this line of research, this work studies the contribution of the trajectory definition errors to the final positioning uncertainty of a large-range 2D nanopositioning stage. The curve trajectory is defined by curve fitting using two methods: traditional CAD/CAM systems and novel algorithms for accurate curve fitting. This novel method has an interest in computer-aided geometric design and approximation theory, and allows high relative accuracy (HRA) in the computation of the representations of parametric curves while minimizing the numerical errors. It is verified that the HRA method offers better positioning accuracy than commonly used CAD/CAM methods when defining a trajectory by curve fitting: When fitting a curve by interpolation with the HRA method, fewer data points are required to achieve the precision requirements. Similarly, when fitting a curve by a least-squares approximation, for the same set of given data points, the HRA method is capable of obtaining an accurate approximation curve with fewer control points

Problemas resueltos de álgebra, cálculo y ecuaciones diferenciales relacionados con ODS

La consecución global de los Objetivos de Desarrollo Sostenible (ODS) y la Agenda 2030 es un propósito que solo se alcanzará en la medida en que todos los agentes sociales se involucren. Es evidente que la Universidad debe ser uno de los instrumentos que promueva la concienciación de la ciudadanía en los problemas de sostenibilidad y la formación necesaria para aportar las soluciones que nos lleven hacia ese nuevo horizonte. Los contenidos evaluables de las asignaturas básicas de formación matemática que se imparten en los grados de la EINA y la EUPT por sí solos no capacitan en principio al alumnado para aportar a la consecución de la Agenda 2030; aunque son imprescindibles para fundamentar los conocimientos posteriores del resto de la titulación correspondiente, que sí se relacionan más directamente con los ODS y por lo tanto con la Agenda2030. Además, en literatura reciente, aparecen ejemplos contextualizados en los diferentes ODS que pueden trabajarse con herramientas matemáticas básicas incluidas en las asignaturas de los grados

A Shape Preserving Class of Two-Frequency Trigonometric B-Spline Curves

This paper proposes a new approach to define two frequency trigonometric spline curves with interesting shape preserving properties. This construction requires the normalized B-basis of the space U4(Iα)=span{1,cost,sint,cos2t,sin2t} defined on compact intervals Iα=[0,α], where α is a global shape parameter. It will be shown that the normalized B-basis can be regarded as the equivalent in the trigonometric space U4(Iα) to the Bernstein polynomial basis and shares its well-known symmetry properties. In fact, the normalized B-basis functions converge to the Bernstein polynomials as α→0. As a consequence, the convergence of the obtained piecewise trigonometric curves to uniform quartic B-Spline curves will be also shown. The proposed trigonometric spline curves can be used for CAM design, trajectory-generation, data fitting on the sphere and even to define new algebraic-trigonometric Pythagorean-Hodograph curves and their piecewise counterparts allowing the resolution of C(3 Hermite interpolation problems