Selecciones Matemáticas

Discrete Dynamic Systems in Economics: A Study of Linear Difference Equations and Nonlinear Models with Naive and Adaptive Expectations

Jose Luis Matos Tejada — 2025-07-26

This study investigates market dynamics using a discrete economic model of supply and demand.

Initially, the linear case with naïve or static expectations is analyzed, and the stability conditions that lead to different system behaviors, such as convergence to equilibrium, periodic oscillations, or divergence, are determined analytically through the study of fixed points and cobweb dynamics.

Subsequently, the analysis is extended to the nonlinear case with adaptive expectations, where the interaction between supply nonlinearity and the expectation formation process is found to generate complex behaviors. The results show how variations in the nonlinearity parameter induce bifurcations observable in time series. The study provides a comprehensive analytical framework that links mathematical properties with observable economic phenomena, offering tools to predict and manage stability in real markets, with direct applications in economic policy and financial risk management.

Fatou’s Theorem; its contribution to harmonic analysis

Alejandro Ortiz Fernández — 2025-07-26

The objective of this article is to see how Fatou’s theorem, given at the beginning of the 20th century, motivated new developments in harmonic analysis and partial differential equations in the second half of that century. In this writing we give a brief tour of some of the contributions given by distinguished analysis and in this way our interest is to make such progress known in Peru and thus contribute to the development of this beautiful branch of mathematical analysis, which we believe is almost unknown in our country. In particular we have paid some attention to the work of Jerison - Kenig [1] because such work contains an overview that helps meet our objective.

Applied mathematics, its importance in developing it in Peru

Alejandro Ortiz Fernández — 2025-07-26

The objective of this article is to contribute with some ideas and proposals with the purpose of further developing the study and research in our country of this area of mathematics, given the historical background of how, thanks to it, many countries have achieved their technological development and in this way with the well-being of the respective society.

On pullback attractors in non-autonomous dynamical systems

Carlos Felipe Vidarte Chavez — 2025-07-26

The theory of pullback attractors constitutes an important tool to interpret the dynamics of physical, biological or engineering phenomena, as it addresses the study of the asymptotic behavior of nonautonomous dynamic systems, where differential equations explicitly depend on time. The objective of this article seeks to synthesize recent advances, present some general techniques such as the energy estimations or asymptotic compactness. This work stick out the importance of pullback attractors to model systems with variable coefficients, memory, delays or in non-cylindrical domains and also points out some challenges such as extension to stochastic systems or complex geometries.

Integration of Monomials over the Unit Sphere and Unit Ball in Rn

Calixto P. Calderón — 2025-07-26

We compute the integral of monomials of the form x^2β over the unit sphere and the unit ball in R^n where β = (β1, . . . , βn) is a multi–index with real components βk > −1/2, 1 ≤ k ≤ n, and discuss their asymptotic behavior as some, or all, βk → ∞. This allows for the evaluation of integrals involving circular and hyperbolic trigonometric functions over the unit sphere and the unit ball in Rn. We also consider the Fourier transform of monomials xα restricted to the unit sphere in Rn, where the multi–indices α have integer components, and discuss their behaviour at the origin.

Euclidean space perturbed by a constant vector field and its relation to a Zermelo navigation problem

Dik D. Lujerio Garcia — 2025-07-26

In this work, the authors perturb the Euclidean plane with a constant vector field of the form W = (0, ε) with 0 ≤ ε < 1, which can be interpreted as wind currents affecting the movement of ships in a constant unidirectional way. It is observed that the resulting perturbed norm, called the ε-Euclidean metric, which is non-reversible, is a Finsler metric. In this way, a new non-Euclidean geometry is introduced. With this, the ε-Euclidean distance is induced and defined. This new way of measuring point-to-point distances can be interpreted, physically, as optimal travel time. Due to the non-reversibility of the ε-Euclidean metric, two types of circumferences are defined and characterized. Distance formulas (or optimal travel time) from point to line, from line to point, and from line to line are obtained, as well as a geometric construction technique for obtaining the distance from a point to a parabola, which can be adapted to other curves that simulate the Edge of a beach. Examples and graphs are presented for a better understanding of the work.

Surfaces with mean of the hyperbolic curvature radii of double harmonic type

Armando M. V. Corro — 2025-07-26

In this paper, we define surfaces with mean of the hyperbolic curvature radii of double harmonic type (in short DHRMC-surfaces) in the hyperbolic space, these surfaces include the generalized Weingarten surfaces of the harmonic type (HGW-surfaces). We give a characterization of DHRMCsurfaces.

Given a real function, we will present a family of DHRMC-surfaces that depend on two holomorphic functions. Moreover, we classify the DHRMC-surfaces of rotation.

Traveling waves in a delayed reaction-diffusion SVIR epidemic model with generalized incidence function and imperfect vaccination

Rassim Darazirar — 2025-07-26

This paper is concerned with traveling wave solutions for a delayed reaction-diffusion SVIR epidemic model that includes both general incidence function and imperfect vaccination. In the model, the spread of infection in space is explicitly taken into account by using a heterogeneous environment; it takes into consideration the delay in immune response and inefficiency in vaccinations. The analysis carried out below shows that the basic reproduction number Ro will be a critical value for determining the existence of traveling waves. More precisely, when Ro > 1 there exists a minimal wave speed ρ* > 0 such that the system admits nontrivial traveling wave solutions for ρ ≥ ρ* whereas no such solutions exist for ρ < ρ*. On the other hand, if Ro ≤ 1, there are no traveling wave solutions. The introduction of delays and imperfect vaccination adds richness and complexity to the dynamics, such as possible wave speed adjustments and pattern formations, which are hallmarks of complex systems. This work develops a theoretical framework that shall guide the understanding of how delays, spatial spread, and control measures interact in epidemic systems and offers insights applicable to real-world infectious disease dynamics. Numerical simulations for some typical nonlinear incidence functions are given in the last to illustrate the existence of traveling waves.

Some Variants of Wayment's Mean Value Theorem for Integrals

German Lozada-Cruz — 2025-07-26

This note deals with some variants of Wayment’s Mean Value Theorem for integrals. Our approach is rather elementary and does not use advanced techniques from analysis. The simple auxiliary functions were used to prove the results.

A mathematical analysis of an eco-epidemiological model with prey-feedback effect and prey-refuge

Neisser Pino Romero — 2025-07-26

This study investigates an eco-epidemiological model incorporating feedback mechanisms and prey refuge, formulated as a system of nonlinear ordinary differential equations. The model describes interactions among susceptible prey, infected prey, and predators, including disease transmisión and nonlinear predation effects. We establish the existence, uniqueness, and positivity of solutions, and analyze their boundedness within a biologically feasible region. Local stability of equilibria is studied via linearization, while global stability is proven using Lyapunov functions. Numerical simulations complement the analytical results, demonstrating how key parameters affect the system’s long-term dynamics.

A note on Mehler's formula

Alessandri Canchoa Quispe — 2025-07-26

In this paper we propose an original proof of Mehler expansion of the Gaussian distribution in terms of probabilistic Hermite polynomials.

A note about a Morse's conjecture

Walter T. Huaraca Vargas — 2025-07-26

In dynamical systems it is known that metrically transitive systems are topologically transitive. M. Morse in 1946 ([1]) conjectured that if the dynamical system has some degree of regularity, then the converse is true. In this article, we will study the Morse conjecture for R2-actions on three-dimensional manifolds and prove the following two results: The conjecture is false if we do not impose restrictions on the singular set and we will prove that The Morse’s Conjecture is valid for locally free actions.

Tangential intersection curves of two surfaces in the three-dimensional Lorentz-Minkowski space

Osmar Alessio — 2025-07-26

We present algorithms for computing the differential geometry properties of tangential intersection curves of two surfaces in the three-dimensional Lorentz-Minkowski space E31 . We compute the tangent vector of tangential intersection curves of two surfaces parametric, where the surfaces can be: spacelike, timelike, or lightlike. The first method computed the tangent vector using the equality of the projection of the second derivative vector onto the normal vector and second method computes the tangent vector by applying a rotation to a vector projected onto the tangent space, where the axis of rotation is the normal vector of the surface. In Minkowski space, there are three types of rotations, since the normal vectors can be: spacelike, lightlike, or timelike.

On the structure of the fundamental étale group of normal schemes

Ronald Jesús Mas Huamán — 2025-07-26

The étale fundamental group is a central tool in algebraic geometry that generalizes the topological fundamental group to the context of schemes. In this article, we explore its behavior for normal schemes, highlighting its relationship to arithmetic and geometric invariants.

The energy allocation toward life-history functions: Link between the individual and population levels

Víctor Saldaña-Núñez — 2025-07-26

The population dynamics of organisms are strongly influenced by life-history strategies that result from the optimal allocation of energy to vital functions such as growth, reproduction, and survival.

These strategies, characterized by phenotypic traits, emerge as evolutionary adaptations to specific ecological conditions and define functional trade-offs relevant facing biotic and abiotic pressures. The aim of this study is to examine the link between life history and population dynamics from a bioenergetic perspective, articulating individual and population-level processes through mathematical models that capture adaptive decisions in simulated environments described in terms of constant, decreasing, and periodic resource availability over time. Using a discrete-time mathematical model with two state variables, internal energy and survival probability, energy allocation toward reproduction and foraging is incorporated in order to determine the optimal strategy that maximizes the net reproductive rate. To solve this control problem, Pontryagin’s maximum principle is applied, using the forward–backward method to obtain optimal trajectories of allocation, energy, and survival. These trajectories are analyzed with respect to relevant physiological parameters under different scenarios of resource availability, thereby allowing the exploration of how environmental conditions influence the bioenergetic decisions of organisms.

Eigen-concepts in the multiplicative linear algebra context

Fernando Córdova-Lepe — 2025-07-26

The concept of eigenvalues is associated with the linearity one, through the structure of vectorial space. The multiplicative linear algebra is a structure in which an expression such as x^3y^2 can be considered a linear combination of variables x and y. This article is reserved to show the corresponding analogues for an Eigenvalue Theory. We exemplify its applications by introducing a connection with the analysis of a nonlinear dynamical system in the standard sense, although a linear recurrence in the multiplicative one.

Tangencies for Power Functions with Integer Exponent

Cairo Henrique Vaz Cotrim — 2025-07-26

Considering a power function f(x) = x^n with exponent n as a positive integer, we show that, at each of its points, there exists a unique polynomial function of degree n − 1 that is tangent to it at that point. Similarly, we verify that every power function h(x) = x^k with exponent k as a negative integer is tangent, at each of its points, to a function of the form l(x) =Sa^t.x^t, where the exponents t are integers between k + 1 and −1.