Estudio asintótico de curvas y superficies

Esta memoria se encuadra en el área de la geometría algebraica, el cálculo simbólico y sus aplicaciones en el diseño geométrico asistido por ordenador (Computer Aided Geometric Design, CAGD). Su objetivo es avanzar, desde los estudios preliminares relacionados con la caracterización de curvas algebraicas planas, hacia el análisis de las propiedades de las ramas de una curva en puntos con coordenadas "suficientemente grandes" y hacia la construcción de las "asíntotas generalizadas" de una curva, extendiendo la investigación al caso de superficies algebraicas. Se crean nuevos métodos de computación simbólica, que caracterizan a las curvas algebraicas planas y analizan el comportamiento de sus ramas infinitas, pudiendo extraer una gran parte de la información sobre el comportamiento de una curva, estudiar la topología de ciertas variedades algebraicas, curvas planas y superficies en tres dimensiones, y representarlas gráficamente en el infinito. Además, se presentan algoritmos e implementaciones que construyen de manera efectiva las asíntotas generalizadas de una curva dada, junto con un análisis de su rendimiento. Se trata, por tanto, de una investigación que involucra a dos disciplinas científicas: la matemática y la computación, haciendo que las expresiones matemáticas y los objetos matemáticos puedan manejarse, utilizando el cálculo simbólico, para resolver problemas del mundo real. Así, las principales aportaciones e innovaciones de esta tesis doctoral son: (1) el desarrollo de métodos efectivos y exactos para la construcción de las ramas infinitas y de las asíntotas de una curva algebraica plana, mediante el cálculo de límites y derivadas, (2) el diseño e implementación de algoritmos eficientes para el cálculo de las asíntotas de curvas algebraicas planas (con el software de álgebra Mapl), así como el análisis de su rendimiento computacional, (3) la determinación de ciertas propiedades obtenidas a partir de las ramas infinitas y construcción de familias de curvas a partir de ciertas asíntotas y (4) definición de conceptos de rama infinita, ramas infinitas convergentes y aproximación aplicados a superficies algebraicas. Estos resultados pueden adaptarse al espacio $n$-dimensional y a curvas definidas por parametrizaciones no necesariamente racionales. Además, se abren futuras líneas de investigación aplicada, en el área del diseño gráfico en 3-D, en el campo de la ingeniería de datos, o en el ámbito del análisis del rendimiento y eficiencia computacional de algoritmos, entre otras áreas de investigación y desarrollo. Palabras clave: Curvas algebraicas, superficies algebraicas, ramas infinitas, ramas convergentes, curvas aproximantes, asíntotas generalizadas, g-asíntotas, rendimiento computacional

A simple formula for the computation of branches and asymptotes of curves and some applications

In this paper, we obtain a simple formula based on the computation of some derivatives for determining the branches and the asymptotes of curves that are defined by a parametrization. For this purpose, we use some previous results and notions presented in Blasco and Pérez-Díaz, 2014a, Blasco and Pérez-Díaz, 2014b, Blasco and Pérez-Díaz, 2015, Blasco and Pérez-Díaz, 2020. From these results, we show how the generalized asymptotes of the input curve can be easily computed and we present some applications related to the ramification index and degree of the asymptote, the infinity form and the multiplicity of the infinity points. Furthermore, we show how to construct all the families of parametric curves having some given asymptotes. We develop this method for the plane case but it can be trivially adapted for dealing with rational curves in n-dimensional space. In addition, the formulaes presented can be similarly obtained for curves defined by a parametrization not necessarily rational.Agencia Estatal de Investigació

Asymptotic behavior of a surface implicitly defined

In this paper, we introduce the notion of infinity branches and approaching surfaces. We obtain an algorithm that compares the behavior at the infinity of two given algebraic surfaces that are defined by an irreducible polynomial. Furthermore, we show that if two surfaces have the same asymptotic behavior, the Hausdorff distance between them is finite. All these concepts are new and represent a great advance for the study of surfaces and their applications.Agencia Estatal de Investigació

Determining the asymptotic family of an implicit curve

In this paper we deal with the following problem: given an algebraic plane curve C, implicitly defined, we determine its “asymptotic family”, that is, the set of algebraic curves that have the same asymptotic behavior as C.Agencia Estatal de Investigació

Some New Symbolic Algorithms for the Computation of Generalized Asymptotes

We present symbolic algorithms for computing the g-asymptotes, or generalized asymptotes, of a plane algebraic curve, C, implicitly or parametrically defined. The g-asymptotes generalize the classical concept of asymptotes of a plane algebraic curve. Both notions have been previously studied for analyzing the geometry and topology of a curve at infinity points, as well as to detect the symmetries that can occur in coordinates far from the origin. Thus, based on this research, and in order to solve practical problems in the fields of science and engineering, we present the pseudocodes and implementations of algorithms based on the Puiseux series expansion to construct the g-asymptotes of a plane algebraic curve, implicitly or parametrically defined. Additionally, we propose some new symbolic methods and their corresponding implementations which improve the efficiency of the preceding. These new methods are based on the computation of limits and derivatives; they show higher computational performance, demanding fewer hardware resources and system requirements, as well as reducing computer overload. Finally, as a novelty in this research area, a comparative analysis for all the algorithms is carried out, considering the properties of the input curves and their outcomes, to analyze their efficiency and to establish comparative criteria between them.Agencia Estatal de Investigació

Cálculo de asíntotas generalizadas de curvas

XXII Encuentro de Álgebra Computacional y Aplicaciones, EACA 2022, 20-22 Junio 2022, Castellón de la Plana, España.Se presenta un algoritmo para calcular las asíntotas generalizadas o g-asíntotas de una curva algebraica plana, C, definida implícitamente en C². Las g-asíntotas generalizan el concepto clásico de asíntota de una curva definida por un polinomio de la forma yg(x) – f(x). Para ello, se definen los conceptos de ramas infinitas y ramas convergentes, y se establecen los fundamentos a partir de los cuales se definirán las g-asíntotas, es decir, las curvas aproximantes y las curvas perfectas. Estos conceptos constituyen una herramienta fundamental para analizar el comportamiento de una curva en el infinito.Agencia Estatal de Investigació

Competencias digitales de los ingresantes universitarios: acceso a la educación superior

Este artículo refleja los resultados obtenidos tras aplicar una encuesta a muestra de 741 estudiantes de nuevo ingreso en dos universidades, una europea, Universidad de Alcalá en España, y otra latinoamericana, Universidad Nacional de Córdoba en Argentina. Esta encuesta fue diseñada con el objetivo de analizar las habilidades digitales de los estudiantes que acceden a las titulaciones de ingeniería, considerando las cinco áreas clave de competencias digitales definidas en el marco europeo DigComp 2.1 para la ciudadanía, adaptándolo a las necesidades de la educación superior. Los resultados extraídos han permitido caracterizar el nivel de competencias digitales adquiridas por los estudiantes de dos regiones claramente identificadas por su diversidad económica, social y cultural. Se han observado distintos comportamientos entre los sujetos experimentales, dado que los estudiantes españoles presentan un mayor nivel de competencias digitales, frente a los argentinos. Esta tendencia se mantiene en la mayoría de las áreas particulares de las competencias evaluadas, siendo de mayor significación el área de resolución de problemas y la de creación de contenidos digitales. En todo caso, los hallazgos encontrados en esta investigación han determinado que los estudiantes ingresantes en ambas universidades, reúnen las habilidades digitales necesarias para adaptarse a las necesidades académicas, administrativas y de gestión con las que han innovado ambos sistemas universitarios, consecuencia de la pandemia de COVID-19. Como consecuencia del análisis realizado, es posible afirmar que los estudiantes de ingeniería de ambas instituciones cuentan con las competencias requeridas por la actual transformación tecnológica

Key Factors Associated With Pulmonary Sequelae in the Follow-Up of Critically Ill COVID-19 Patients

Introduction: Critical COVID-19 survivors have a high risk of respiratory sequelae. Therefore, we aimed to identify key factors associated with altered lung function and CT scan abnormalities at a follow-up visit in a cohort of critical COVID-19 survivors. Methods: Multicenter ambispective observational study in 52 Spanish intensive care units. Up to 1327 PCR-confirmed critical COVID-19 patients had sociodemographic, anthropometric, comorbidity and lifestyle characteristics collected at hospital admission; clinical and biological parameters throughout hospital stay; and, lung function and CT scan at a follow-up visit. Results: The median [p25–p75] time from discharge to follow-up was 3.57 [2.77–4.92] months. Median age was 60 [53–67] years, 27.8% women. The mean (SD) percentage of predicted diffusing lung capacity for carbon monoxide (DLCO) at follow-up was 72.02 (18.33)% predicted, with 66% of patients having DLCO < 80% and 24% having DLCO < 60%. CT scan showed persistent pulmonary infiltrates, fibrotic lesions, and emphysema in 33%, 25% and 6% of patients, respectively. Key variables associated with DLCO < 60% were chronic lung disease (CLD) (OR: 1.86 (1.18–2.92)), duration of invasive mechanical ventilation (IMV) (OR: 1.56 (1.37–1.77)), age (OR [per-1-SD] (95%CI): 1.39 (1.18–1.63)), urea (OR: 1.16 (0.97–1.39)) and estimated glomerular filtration rate at ICU admission (OR: 0.88 (0.73–1.06)). Bacterial pneumonia (1.62 (1.11–2.35)) and duration of ventilation (NIMV (1.23 (1.06–1.42), IMV (1.21 (1.01–1.45)) and prone positioning (1.17 (0.98–1.39)) were associated with fibrotic lesions. Conclusion: Age and CLD, reflecting patients’ baseline vulnerability, and markers of COVID-19 severity, such as duration of IMV and renal failure, were key factors associated with impaired DLCO and CT abnormalities

Clustering COVID-19 ARDS patients through the first days of ICU admission. An analysis of the CIBERESUCICOVID Cohort

Background Acute respiratory distress syndrome (ARDS) can be classified into sub-phenotypes according to different inflammatory/clinical status. Prognostic enrichment was achieved by grouping patients into hypoinflammatory or hyperinflammatory sub-phenotypes, even though the time of analysis may change the classification according to treatment response or disease evolution. We aimed to evaluate when patients can be clustered in more than 1 group, and how they may change the clustering of patients using data of baseline or day 3, and the prognosis of patients according to their evolution by changing or not the cluster.Methods Multicenter, observational prospective, and retrospective study of patients admitted due to ARDS related to COVID-19 infection in Spain. Patients were grouped according to a clustering mixed-type data algorithm (k-prototypes) using continuous and categorical readily available variables at baseline and day 3.Results Of 6205 patients, 3743 (60%) were included in the study. According to silhouette analysis, patients were grouped in two clusters. At baseline, 1402 (37%) patients were included in cluster 1 and 2341(63%) in cluster 2. On day 3, 1557(42%) patients were included in cluster 1 and 2086 (57%) in cluster 2. The patients included in cluster 2 were older and more frequently hypertensive and had a higher prevalence of shock, organ dysfunction, inflammatory biomarkers, and worst respiratory indexes at both time points. The 90-day mortality was higher in cluster 2 at both clustering processes (43.8% [n = 1025] versus 27.3% [n = 383] at baseline, and 49% [n = 1023] versus 20.6% [n = 321] on day 3). Four hundred and fifty-eight (33%) patients clustered in the first group were clustered in the second group on day 3. In contrast, 638 (27%) patients clustered in the second group were clustered in the first group on day 3.Conclusions During the first days, patients can be clustered into two groups and the process of clustering patients may change as they continue to evolve. This means that despite a vast majority of patients remaining in the same cluster, a minority reaching 33% of patients analyzed may be re-categorized into different clusters based on their progress. Such changes can significantly impact their prognosis