Matemática

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

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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.
A mathematical model with nonlinear relapse: conditions for a forward-backward bifurcation
(2023) Sánchez Peña, Fabio Ariel; Arroyo Esquivel, Jorge; Calvo Alpízar, Juan Gabriel
We constructed a Susceptible-Addicted-Reformed model and explored the dynamics of nonlinear relapse in the Reformed population. The transition from susceptible considered at-risk is modeled using a strictly decreasing general function, mimicking an influential factor that reduces the flow into the addicted class. The basic reproductive number is computed, which determines the local asymptotically stability of the addicted-free equilibrium. Conditions for a forward-backward bifurcation were established using the basic reproductive number and other threshold quantities. A stochastic version of the model is presented, and some numerical examples are shown. Results showed that the influence of the temporarily reformed individuals is highly sensitive to the initial addicted population.
Adicción a Redes Sociales: Un Modelo Matemático
(2023) Garro, Valeria; Masís, Juan J.; Alarcón, Greivin J.; Campos, Camilo D.; Castro, Jorge A.; León, Juan M.; Sánchez Peña, Fabio Ariel
El artículo presenta un modelo de dinámica social para el estudio de la adicción a las redes sociales, basado en los modelos matemáticos SIR (Susceptibles-Infectados-Recuperados). El modelo considera cinco categorías de individuos: inmunes, susceptibles, adictos, controlados y recuperados. Además, se tiene en cuenta la posibilidad de generar una nueva adicción. El objetivo principal es analizar cómo se transmite y desarrolla la adicción a las redes sociales en una población determinada. Los individuos susceptibles pueden volverse adictos tras la exposición a las redes sociales e interacción con personas que ya son adictas, mientras que los adictos pueden controlarse hasta alcanzar la recuperación o experimentar nuevamente la adicción a las redes. El modelo considera factores como influencia social e interacción entre individuos. Los resultados obtenidos proporcionan una visión de cómo se propaga la adicción a las redes sociales en una población bajo ciertas condiciones. Estos hallazgos podrían ser útiles para tomar decisiones que promuevan la prevención y el tratamiento de la adicción a las redes sociales.
Sparse bounds for the discrete spherical maximal function [Presentación]
(2021-10-24) Mena Arias, Darío Alberto
We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.

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