Selecciones Matemáticas

A comprehensive review of the characterization of real numbers

2024-12-28T03:56:40+00:00

The real number system is a fundamental tool for rigorous demonstrations of the differential and integral calculus results. Even after a century of formalization on solid foundations, discussions about the construction of this field are generally omitted in advanced courses such as Real Analysis. In the present work, we present a comprehensive review on the construction and characterization of the real numbers field. The presentation focuses on the construction through Cauchy sequences of rational numbers. The notion of completeness is delimited differently from completeness when Dedekind’s cut construction is used. The results indicate Q and R Archimedean as a necessary condition for these two notions of completeness to be equivalent.

To illustrate this, inspired by the work of Leon W. Cohen and Gertrude Ehrlich, we present an example of a Cauchy-complete non-Archimedean ordered field in which the supremum axiom is not equivalent to the nested intervals principle.

Brief introduction to distributions and to Hörmander spaces Bp,k

2024-12-28T04:14:14+00:00

Partial differential equations (PDE’s) are, we believe, little known in our country, especially at a medium-advanced level, in particular, their historical roots and methods developed in their evolution. The objective of this article is to contribute to a knowledge in this direction especially to colleagues and students who are interested in studying and researching in this central Branch of mathematics. As a model we have chosen H¨ormander’s book, [1], whose study and teaching will be a sign of great progress in this direction in our country. In this article we only give an introduction to part I dedicated to functional analysis and which includes chapters I and II that deal with the theory of distributions and some special spaces of distributions, respectively.

Harmonic analysis and partial differential equations. Brief walk through these domains

2024-12-28T04:26:51+00:00

In this walk we are going to walk through some domains of harmonic analysis and partial differential equations (PDE). The objective of this article is to motivate students and colleagues to study these beautiful areas of analysis and therefore we emphasize the ideas, some mathematical results and some historical data. In this “ tourist” tour we will see a panorama of such areas in the 19th and 20 th centuries, a panorama of the Fourier series; we give a vision of the theory of distributions, of the theory of linear partial differential operators and we culminate by given a vision of harmonic analysis and its relationship we the PDE.

The Interconnection Between Calculus of Variations, Partial Differential Equations and Differential Geometry

2024-12-28T04:44:29+00:00

Calculus of variations is a fundamental mathematical discipline focused on optimizing functionals, which map sets of functions to real numbers. This field is essential for numerous applications, including the formulation and solution of partial differential equations (PDEs) and the study of differential geometry. In PDEs, calculus of variations provides methods to find functions that minimize energy functionals, leading to solutions of various physical problems. In differential geometry, it helps understand the properties of curves and surfaces, such as geodesics, by minimizing arc-length functionals. This paper explores the intrinsic connections between these areas, highlighting key principles such as the Euler-Lagrange equation, Ekeland’s variational principle, and the Mountain Pass theorem, and their applications in solving PDEs
and understanding geometrical structures.

Existence and construction of the Peano selection for a multivalued function

2024-11-28T20:56:36+00:00

In the present article, the necessary conditions are presented for a multivalued function in order to define a Peano selection. To achieve this, a bibliographic review was carried out on general results of compact topological spaces, open and closed sets and continuity. To then address the same topics, but on metric spaces. Next, the theory of multivalued functions was studied, specifically semicontinuity, both superiorly and inferiorly. Finally, using the General Theorem of multivalued functions, the necessary conditions are determined for the multivalued function, F : [0, 1]-->[0, 1] × [0, 1] to admit the construction of the selection of Peano.

A seasonal commensalism model with a weak Allee effect to describe climate-mediated shifts

2024-09-28T11:03:04+00:00

Climate change is affecting the life cycle of tight interacting species. Commonly, the seasonal population dynamics of species is analyzed through models with periodic rates; however, assuming periodicity in seasonal phenomena which depend on environmental drivers is very restrictive. In this work, we analyze seasonal commensalism between two species in which the per capita growth rate of each species is affected by a weak Allee effect and the demographic and ecological rates are assumed almost periodic. To do this, we construct and analyze an almost periodic model to describe commensalism using a wide family of functions that describe weak Allee effects and the benefits granted by the interaction. We prove that the model admits a unique almost periodic global attractor for a wide family of functions. Numerical simulations of the solutions of the model shown the result proved in this work. We show that if periodic rates are used when the phenomenon is really almost periodic, underestimation or overestimation of the population size of both species can occur, which can lead to design wrong strategies by the decision makers.

Improved global convergence results for augmented Lagrangian method with exponential penalty function

2024-09-06T20:24:34+00:00

The purpose of the present paper is to improve the global convergence results established so far concerning the Augmented Lagrangian Algorithm with exponential penalty function for solving nonlinear programming problems with equality and inequality constraints. We prove global convergence for KKT points under the PAKKT-regular constraint qualifications, which results as a consequence that accumulation points generated by the algorithm are PAKKT points. This convergence result is new for the augmented Lagrangian Method based on the exponential penalty function. An interesting consequence is that the estimates of the Lagrange multipliers computed by the method remain bounded in the presence of the quasi-normality condition. Finally we give optimality and feasibility results for the convex case.

The SIRD epidemiological model using Caputo fractional derivatives applied to study the spread of the COVID-19 in the Peruvian region of Tacna

2024-12-28T02:24:44+00:00

Mathematical models are widely used to study the spreading dynamics of infectious diseases. In particular, the “Susceptibles-Infecteds-Recovereds-Deceases”(SIRD) model provides a framework that can be adapted to describe the core spreading dynamics of several human and wildlife infectious diseases. The present work uses a SIRD model using Caputo fractional derivative. In this investigation, the existence and uniqueness of solutions for the model were established. Numerical solutions were obtained using the Adams-Bashforth method. To illustrate the model’s utility, we made forecasts for the spread of the virus SARS-Cov-2 in the region of Tacna in Perú.

It is well known that these models can help to forecast the number of infected people, understand the disease dynamics and evaluate potential control strategies.

Refuge used by prey as a function of predator numbers in a Leslie-type model

2024-12-28T02:49:29+00:00

This paper deals with a continuous-time predator-prey model of Leslie-Gower type considering the use of a physical refuge by a fraction of the prey population. The fraction of hidden prey is assumed to be dependent on the presence of predators in the environment.

The conditions for the existence of equilibrium points and their local stability are established. In particular, it is shown that the point (0; 0) has a great importance in the dynamics of the model, since it determines a separating curve Σ that divides the behavior of the trajectories.

Those trajectories that are above this curve have as their w- limit the point (0; 0), so the extinction of both populations may be possible depending on the initial conditions.

Spectral Differentiation and Mimetic Methods for Solving the Scalar Burger’s Equation

2024-12-28T03:11:56+00:00

In the present work, the spectral differentiation method was studied to solve the scalar Burger’s partial differential equation. This equation has been of considerable physical interest as it can be regarded as a simplified version of the Navier-Stokes equations. Through this study, the spectral differentiation method and its convergence were described; additionally, the mimetic method and the use of the MOLE library for numerically solving the scalar Burger’s equation were presented.

Numerical implementation of a stochastic differential equation of motion

2024-12-28T03:25:15+00:00

Using the ordinary differential equation of motion it is possible to determine the position in time of a mass that moves because it is disturbed by some deterministic action. For this work it was proposed to model a mass supporting a random disturbance. To do this, it was required to model Brownian motion since it efficiently represents the randomness of the phenomenon.

Using the fundamentals of Functional Analysis, Probability Theory and Stochastic Processes, a stochastic differential equation of motion was obtained. In order to extract solutions from this equation, the Euler-Maruyama method was used, which was implemented computationally.

The results obtained showed that the use of a non-deterministic version to model movement generates satisfactory results and of interest to science.

Normal forms of vector fields induced by holomorphic actions of the group SL(2,C) on a complex manifold

2024-12-28T03:44:24+00:00

In this work, we study actions of the Lie group SL(2,C) on a complex manifold of dimension three or higher.

It is demonstrated that these types of actions induce three complete holomorphic vector fields, one of which is periodic, and that there exists a particular relationship between them, given by the Lie bracket, which generates a singular holomorphic foliation of codimension two. Subsequently, the types of singularities are classified, and the normal forms of these vector fields are obtained in a neighborhood of each singular point of the foliation.