Matemática

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https://hdl.handle.net/10669/280

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Archimedes' revenge
(2023) Villarino Bertram, Mark; Várilly Boyle, Joseph C.
We show how to compute the volume of a hyperbolic stella octangula, formed by the intersection of three solid hyperboloids centered on the coordinate axes.
Diseño y valoración de la idoneidad cognitiva de una secuencia didáctica remota para la enseñanza del concepto de derivada
(2024-09-30) Prendas Rojas, Juan Pablo; Salas Huertas, Oscar
Las problemáticas referidas al concepto de derivada de una función se han reportado en muchas investigaciones y han sido punto de partida para numerosas propuestas sobre abordaje de este tema con la intención de solventarlas. Sin embargo, no siempre está claro de donde surgen las propuestas y si, al final de la puesta en escena, estas resultan idóneas para los objetivos que se persiguen. En este proyecto de investigación se propone el uso de la teoría de Ingeniería Didáctica como guía para el proceso de diseño de una secuencia didáctica para la enseñanza del concepto de derivada de una función. Además, se muestra como los indicadores de Idoneidad Didáctica de la teoría del Enfoque Ontosemiótico de la Instrucción y el Conocimiento Matemáticos (EOS), en la faceta cognitiva, se complementan con la etapa de evaluación y reflexión docente. Se trabajó durante tres sesiones remotas con un grupo de Cálculo Diferencial e Integral del Instituto Tecnológico de Costa Rica. El proceso inició con análisis preliminares, luego análisis a priori, experimentación y finalmente reflexión sobre la labor docente desde la faceta cognitiva. Como resultado se tiene un documento con al menos tres aristas: se constituye como una guía para el diseño de secuencias didácticas con sustento en indagaciones preliminares generales en principio y locales posteriormente sobre aspectos afectivos, epistémicos y cognitivos, es una muestra sobre actividades y evaluaciones en una clase remota real y contiene una serie de indicadores para estructurar la reflexión de la labor docente. Se destaca que, desde el punto de vista teórico, se fomenta la ubicación de la Evaluación como un componente por si mismo y para éste se proponen tres indicadores de los cuales la Evaluación temporal es novedoso.
Sistema de Scoring de aseguramiento aplicado a un seguro de automóviles mediante la implementación de modelos predictivos y algoritmos de aprendizaje
(2024) González Chaves, Lauren Tatiana; Ugarte Quesada, Angie; Rodríguez Rojas, Oldemar Martín
En el presente documento se propone la implementación de un sistema de scoring de aseguramiento aplicado a un seguro de automóviles, utilizando modelos predictivos y algoritmos de aprendizaje automático. El objetivo principal es desarrollar un perfil de riesgo inicial al momento de suscripción, optimizando así la evaluación y tarificación de asegurados. Para este fin, se emplean varios modelos predictivos, incluyendo K vecinos cercanos (KNN), Arboles de Decisión, Bosques Aleatorios, AdaBoost (ADA) y Gradient Boost Tree (GBT). Se eval´uan estos modelos en términos de precisión global, precisión positiva, precisión negativa, área bajo la curva (AUC) y otras. Se parte de una cartera de aseguramiento con más de cinco años de datos suministrados por el Instituto Nacional de Seguros, complementada con un histórico de siniestros de más de diez años. A estos datos se les aplica un proceso de imputación para garantizar la calidad y completitud de la información disponible. Al tratarse de un problema desbalanceado, la metodología utilizada incluye un método de balanceo de datos para mejorar el aprendizaje y la identificación de patrones en la clase minoritaria. Los algoritmos individuales fueron ajustados mediante barrido paramétrico para identificar los hiperparámetros que optimizaran sus indicadores de desempeño. Una vez seleccionados estos hiperparámetros, se sometieron a validación cruzada. Los algoritmos con mejor desempeño avanzaron a una segunda etapa, donde se emplearon para conformar tres diferentes consensos propios. Los resultados obtenidos indican que los métodos de consenso, especialmente aquellos que incluyen técnicas de balanceo de clases, ofrecen mejores precisiones positivas y negativas, así como altos valores de AUC, en comparación con los modelos individuales. Esto demuestra la eficacia, estabilidad y adaptabilidad de los métodos de consenso en la predicción de siniestralidad con datos desbalanceados. La dinámica de evaluación incluye datos nuevos que son externos a la base de datos inicial y el cálculo de métricas de ajuste, proporcionando una visión integral sobre el desempeño del sistema de scoring propuesto. Además, se discuten las limitaciones del estudio y se sugieren futuras investigaciones y mejoras en la implementación del modelo.
Quantifying uncertainty of a geometric goodness of fit measure
(2024-08-21) Hernández Ávila, Edgar Javier; Dr. Maikol Solís Chacón
Resumen El propósito de este estudio es establecer una validaci ón estadística formal de un índice de bondad de ajuste geométrico, empleando una prueba de hipótesis y una envolvente global, para determinar si una nube de puntos, que representan datos, es un proceso con un patrón completamente aleatorio (CSR por sus siglas en inglés). Utilizamos el complejo “alpha shape” de una nube de puntos de datos en R2 para generar un mapa de este índice. A continuación, establecemos una hipótesis nula que corresponde a un proceso CSR, a partir de dos diferentes estadísticos de prueba junto con ensayos Monte Carlo. Uno de estos estadísticos de prueba se emplea para construir una envolvente global que delimita una región dentro de la cual no se puede rechazar la hipótesis nula. Proporcionamos algunos ejemplos teóricos y de conjuntos de datos para ilustrar este procedimiento.
Scattering of radial solutions for quadratic-type Schrödinger systems in dimension five
(2021-08) Noguera Salgado, Norman F.; Pastor Ferreira, Ademir
In this paper we study the scattering of radial solutions to a l-component system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five. Our approach is based on the recent technique introduced by Dodson and Murphy, which relies on the radial Sobolev embedding and a Morawetz estimate.
On the dynamics of a quadratic Schrödinger system in dimension n = 5
(2018-10-02) Noguera Salgado, Norman F.; Pastor Ferreira, Ademir
In this work we give a sharp criterion for the global well-posedness, in the energy space, for a system of nonlinear Schr¨odinger equations with quadratic interaction in dimension n = 5. The criterion is given in terms of the charge and energy of the ground states associated with the system, which are obtained by minimizing a Weinstein-type functional. The main result is then obtained in view of a sharp Gagliardo-Nirenberg-type inequality.
Scattering for quadratic-type Schrödinger systems in dimension five without mass-resonance
(2021-08-11) Noguera Salgado, Norman F.; Pastor Ferreira, Ademir
In this paper we study the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our approach is based on the recent technique introduced by Dodson and Murphy (Math Res Lett 25:1805–1825, 2018), which relies on an interaction Morawetz estimate. It is proved that any solution below the ground states scatters in time.
On a system of Schrödinger equations with general quadratic-type nonlinearities
(2021-07-27) Noguera Salgado, Norman F.; Pastor Ferreira, Ademir
In this work, we study a system of Schrödinger equations involving nonlinearities with quadratic growth. We establish sharp criterion concerned with the dichotomy global existence versus blow-up in finite time. Such a criterion is given in terms of the ground state solutions associated with the corresponding elliptic system, which in turn are obtained by applying variational methods. By using the concentration-compactness method we also investigate the nonlinear stability/instability of the ground states.
Blow-up solutions for a system of Schrödinger equations with general quadratic-type nonlinearities in dimensions five and six
(2022-04-15) Noguera Salgado, Norman F.; Pastor, Ademir
This paper deals with the Cauchy problem associated with a nonlinear system of Schrödinger equations with general quadratic-type nonlinearities. The main interest is in proving the existence of blow-up solutions in dimensions five and six. We give sufficient conditions for the existence of such solutions based on the mass and the energy of the associated ground states. The existence of ground states in dimension five was already obtained in a previous paper. In the present manuscript we also establish the existence of such a special solutions in dimension six. This result can also be viewed as of independent interest. The technique we use is based on the concentration-compactness method. The blow-up solutions are obtained without the mass-resonance condition, when the initial data is radial.
Relación entre los indicadores de progreso socioeconómico y el Índice de Felicidad de los países en el periodo 2014-2018
(2024) Amey Apuy, Luis Fernando; Jiménez Navarro, Anthony Mauricio; Hernández Navarro, Javier Antonio; Venegas Espinoza, Erick Javier
El estudio se centra principalmente en examinar la relación entre diversos marcadores socioeconómicos y el Índice de Felicidad de diferentes países en el periodo del 2014-2018. Para llevar a cabo esto, se utilizaron datos recolectados de The World Bank (2018), World Happiness Report (2018) y Transparency International (2018) en conjunto con metodologías y pruebas estadísticas relevantes como el coeficiente de correlación de Pearson, el cual se utilizó para obtener correlaciones significativas entre el Índice de Felicidad y variables socioeconómicas relevantes como el PIB per cápita, el acceso a electricidad y agua potable, la clase social y el Índice de Percepción de la Corrupción. A su vez, se utilizó la Transformación Z de Fisher con el fin de obtener intervalos de confianza asegurando en un 95 % la probabilidad de que los coeficientes de dichas correlaciones sean los correctos. Los resultados muestran que en efecto existe una correlación positiva entre las variables seleccionadas y el Índice de Felicidad, sugiriendo que mejorar estos marcadores podrían influir en los niveles de felicidad percibidos por la sociedad. Finalmente, las pruebas realizadas a la variable del Índice de Felicidad mostraron tiene un comportamiento normal que, en conjunto con las correlaciones observadas, ofrece una base sólida para futuros análisis para decisiones políticas.
A two level overlapping Schwarz preconditioner for discontinuous Galerkin methods
(2024) Calvo Alpízar, Juan Gabriel; Solano Córdoba, Moisés Eduardo
This article presents a two-level overlapping additive Schwarz algorithm designed to solve an elliptic problem discretized with the symmetric discontinuous Galerkin method. The algorithm allows for the use of irregular subdomains, overcoming limitations of other approaches where the coarse mesh was based on triangular elements. Additionally, a brief description of the numerical implementation of the Galerkin method is included. Numerical results validating the relevance of our algorithm are also presented, including cases where the coefficient of the differential equation is discontinuous, which is relevant in different applications.
A posteriori error analysis of a semi-augmented finite element method for double-diffusive natural convection in porous media
(2024) Álvarez Guadamuz, Mario Andrés; Colmenares García, Eligio Antonio; Sequeira Chavarría, Filander A.
This paper presents our contribution to the a posteriori error analysis in 2D and 3D of a semi-augmented mixed-primal finite element method previously developed by us to numerically solve double-diffusive natural convection problem in porous media. The model combines Brinkman-Navier-Stokes equations for velocity and pressure coupled to a vector advection-diffusion equation, representing heat and concentration of a certain substance in a viscous fluid within a porous medium. Strain and pseudo-stress tensors were introduced to establish scheme solvability and provide a priori error estimates using Raviart-Thomas elements, piecewise polynomials and Lagrange finite elements. In this work, we derive two reliable residual-based a posteriori error estimators. The first estimator leverages ellipticity properties, Helmholtz decomposition as well as Clément interpolant and Raviart-Thomas operator properties for showing reliability; efficiency is guaranteed by inverse inequalities and localization strategies. An alternative estimator is also derived and analyzed for reliability without Helmholtz decomposition. Numerical tests are presented to confirm estimator properties and demonstrate adaptive scheme performance.
Characterization of principal bundles: the noncommutative algebraic case
(2024) Ugalde Gómez, William Javier
We review Hopf–Galois extensions, in particular faithfully flat ones, accepted to be the noncommutative algebraic dual of a principal bundle. We also make a short digression into how quantum groups relate to Hopf—Galois extensions. Several examples are given, in order to provide a satisfactory understanding of each topic.
Characterization of principal bundles: the commutative case
(2023-10-04) Ugalde Gómez, William Javier
A review of the characterization of principal bundles, through the different properties of the action of a group and its related canonical and translation maps, is presented. The work is divided in three stages: a topological group acting on a topological space, a discrete group acting on a smooth manifold, and a Lie group acting on a smooth manifold.
Improved Epstein-Glaser renormalization in x-space versus differential renormalization
(2014-09) Gracia Bondía, José M.; Gutiérrez Garro, Heidy; Várilly Boyle, Joseph C.
Renormalization of massless Feynman amplitudes in x-space is reexamined here, using almost exclusively real-variable methods. We compute a wealth of concrete examples by means of recursive extension of distributions. This allows us to show perturbative expansions for the four-point and two-point functions at several loop order. To deal with internal vertices, we expound and expand on convolution theory for log-homogeneous distributions. The approach has much in common with differential renormalization as given by Freedman, Johnson and Latorre; but differs in important details.
Metric properties of the fuzzy sphere
(2013-02) D'Andrea, Francesco; Lizzi, Fedele; Várilly Boyle, Joseph C.
The fuzzy sphere, as a quantum metric space, carries a sequence of metrics which we describe in detail. We show that the Bloch coherent states, with these spectral distances, form a sequence of metric spaces that converge to the round sphere in the high-spin limit.
Testing one-body density functionals on a solvable model
(2012-10) Benavides Riveros, Carlos L.; Várilly Boyle, Joseph C.
There are several physically motivated density matrix functionals in the literature, built from the knowledge of the natural orbitals and the occupation numbers of the one-body reduced density matrix. With the help of the equivalent phase-space formalism, we thoroughly test some of the most popular of those functionals on a completely solvable model.
The lowest excited configuration of harmonium
(2012) Benavides Riveros, Carlos L.; Gracia Bondía, José M.; Várilly Boyle, Joseph C.
The harmonium model has long been regarded as an exactly solvable laboratory bench for quantum chemistry [W. Heisenberg, Z. Phys. 38, 411 (1926)]. For studying correlation energy, only the ground state of the system has received consideration heretofore. This is a spin singlet state. In this work we exhaustively study the lowest excited (spin triplet) harmonium state, with the main purpose of revisiting the relation between entanglement measures and correlation energy for this quite different species. The task is made easier by working with Wigner quasiprobabilities on phase space.
Density functional theory on phase space
(2012) Blanchard, Philippe; Gracia Bondía, José M.; Várilly Boyle, Joseph C.
Forty-five years after the point de départ [Hohenberg and Kohn, Phys. Rev. 1964, B864, 136] of density functional theory, its applications in chemistry and the study of electronic structures keep steadily growing. However, the precise form of the energy functional in terms of the electron density still eludes us - and possibly will do so forever [Schuch and Verstraete, Nat. Phys. 2009, 5, 732]. In what follows we examine a formulation in the same spirit with phase space variables. The validity of Hohenberg-Kohn-Levy-type theorems on phase space is recalled. We study the representability problem for reduced Wigner functions, and proceed to analyze properties of the new functional. Along the way, new results on states in the phase space formalism of quantum mechanics are established. Natural Wigner orbital theory is developed in depth, with the final aim of constructing accurate correlation-exchange functionals on phase space. A new proof of the overbinding property of the Müller functional is given. This exact theory supplies its home at long last to that illustrious ancestor, the Thomas-Fermi model.
Wavelet analysis of dengue incidence and its correlation with weather and vegetation variables in Costa Rica
(2021) García Puerta, Yury Elena; Barboza Chinchilla, Luis Alberto; Sánchez Peña, Fabio Ariel; Vásquez Brenes, Paola Andrea; Calvo Alpízar, Juan Gabriel
Dengue represents a serious public health problem in tropical and subtropical regions worldwide. The number of dengue cases and its geographical expansion has increased in recent decades, driven mostly after by social and environmental factors. In Costa Rica, it has been endemic since it was first introduced in 1993. In this article, wavelet analyzes (wavelet power spectrum and wavelet coherence) were performed to detect and quantify dengue periodicity and describe patterns of synchrony between dengue incidence and climatic and environmental factors: Normalized Difference Water Index, Enhanced Vegetation Index, Normalized Difference Vegetation Index, Tropical North Atlantic indices, Land Surface Temperature, and El Niño Southern Oscillation indices in 32 different cantons, using dengue surveillance from 2000 to 2019. Results showed that the dengue dominant cycles are in periods of 1, 2, and 3 years. The wavelet coherence analysis showed that the vegetation indices are correlated with dengue incidence in places located in the central and Northern Pacific of the country in the period of 1 year. Climatic variables such as El Niño 3, 3.4, 4, showed a strong correlation with dengue incidence in the period of 3 years and the Tropical North Atlantic is correlated with dengue incidence in the period of 1 year. Land Surface Temperature showed a strong correlation with dengue time series in the 32 cantons.

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