Journal of the Academic department of Mathematics of the National University of Trujillo. The purpose of the journal is to publish the results of original and unpublished research in the field of Mathematics and Applied Mathematics. "Selecciones Matemáticas" is an Open Access Journal. ISSN-e: 2411 - 1783 DOI: 10.17268/Sel.mat Short title: Sel.Mat. ----------------------------------------------------------------------------------------------------------------------------------------------------------
National University of Trujillo - Academic Department of Mathematics
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Artinianness of local cohomology modules
Let R be a commutative Noetherian ring with non-zero identity and I an ideal of R. Let M be a finitely generated R-module. we some results about Artinianness of local cohomology module involving the concept of ZD-module. As applications, it is shown that the local cohomology module, under certain hypotheses, it has vanishing properties, and finite support properties.
Copyright (c) 2026 Selecciones Matemáticas https://creativecommons.org/licenses/by/4.0
<span style="font-size: 11.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-font-kerning: 0pt; mso-ligatures: none; mso-ansi-language: ES-PE; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">In this paper we introduce the notion of weakly 2-absorbing primary submodules of modules and investigate this concept when the ambient module is the edge ideal of a finite simple graph without isolated vertices. Several structural properties concerning quotients, localization and related module constructions are established.
Some identities for Fibonacci and Lucas polynomials via the Fibonacci Q(x)-matrix
In this paper, we study Fibonacci and Lucas polynomials through the Fibonacci Q(x)-matrix. By analyzing its powers and inverses, we derive several identities for both polynomial families, including extensions to negative indices. We establish that the powers of Q(x) and their inverses have identical determinants and permanents. Furthermore, we obtain an m-term Honsberger-type identity for Fibonacci and Lucas polynomials and provide a matrix-based proof.
Dynamics of a Delayed Predator–Prey Model with Fear Effect, Adaptive Refuge, and Alternative Food
In this work, a delayed predator–prey model with fear effect, adaptive refuge, and alternative food supply is investigated. Fundamental properties including positivity, boundedness, permanence, and stability are established analytically. The results show that temporal delay may destabilize the coexistence equilibrium and generate sustained oscillations through Hopf bifurcation. Numerical simulations together with global sensitivity analysis based on PRCC and Sobol indices reveal that delay and alternative food are the dominant mechanisms governing oscillatory behavior and ecological stability.
Statistical analysis of the Messi vs Cristiano Ronaldo rivalry
We will put into practice the main methods, tools, and resources for classifying statistical data and graphical representations. We will analyze a data set, how it is defined, its division, and the graphs we can use as a resource for problem-solving. We also reviewed the importance of measures of central tendency, which are used to organize statistical data presented in the various applications of probability and statistics. <span style="font-size: 11.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-font-kerning: 0pt; mso-ligatures: none; mso-ansi-language: ES-PE; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">We apply each measure of central tendency in this work, including the arithmetic mean, harmonic mean, geometric mean, median, mode, and midrange. In addition, we study the variation presented by the data associated with a statistical phenomenon, whose quantification is carried out using dispersion measures, such as range, percentiles, variance and standard deviation in the problem presented here.
On the Galois Group of Unramified Extensions of a p-adic Field
<span style="font-size: 11.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-font-kerning: 0pt; mso-ligatures: none; mso-ansi-language: ES-PE; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">In this work, we study the structure of the Galois group of finite unramified extensions of the p-adic field. Furthermore, we establish the uniqueness, up to isomorphism, of unramified extensions of a given degree, and we describe the Galois group of the maximal unramified extension of Qp.
Ab InitioWave Function Optimization for Light Atoms A PyTorch Framework for the Psiformer Architecture
The application of deep neural networks to Variational Quantum Monte Carlo (VQMC) has significantly advanced the numerical approximation of ground-state energies for quantum many-body systems. Among these advancements, the Psiformer architecture—which leverages self-attention mechanisms to model antisymmetric electron wave functions—has demonstrated state-of-the-art accuracy. However, the original implementation is built exclusively within the JAX ecosystem, presenting a barrier to entry for the broader <span style="font-size: 11.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-font-kerning: 0pt; mso-ligatures: none; mso-ansi-language: ES-PE; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">research community that predominantly utilizes PyTorch. In the interest of scientific reproducibility and computational accessibility, we present an independent open-source PyTorch implementation of the Psiformer architecture. The model design follows von Glehn, Spencer, and Pfau (2023); the code and numerical experiments reported here are the author’s own. To validate the implementation, we compute ground-state energies for helium through oxygen and compare them to a Hartree–Fock baseline. Our results demonstrate that the PyTorch Psiformer consistently recovers significant electron correlation energy across all tested atoms, confirming that the architecture is correctly implemented and behaves as expected from variational theory. By making the framework openly available, this work lowers the barrier for researchers in computational mathematics, machine learning, and quantum chemistry who wish to experiment with transformer-based wave functions. The full source code is made publicly available to facilitate future experimentation and development.
<span style="font-size: 11.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-font-kerning: 0pt; mso-ligatures: none; mso-ansi-language: ES-PE; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">The holonomy of a leaf of a foliation was introduced by C. Ehresmann and is the version of the Poincaré first return map of a real flow in a neighborhood of a periodic orbit. In general, holonomy allows us to see the behavior of leaves near a fixed leaf. In particular, we will focus on the following problem: Let X and Y be two holomorphic vector fields defined in the three-dimensional complex space with singularity at the origin such that X is a diagonal linear vector field and Y is a quasi-homogeneous vector field, and furthermore these vector fields are related by the Lie bracket through [X, Y] = -Y. We will study the holonomy of X, Y and of the integrable 1-form determined by these vector fields.
Well-posedness for a Fourth-Order PDE with Linear Dissipation and its Generalization
<span style="font-size: 11.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-font-kerning: 0pt; mso-ligatures: none; mso-ansi-language: ES-PE; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">In this work, we demonstrate that the Cauchy problem associated to the fourth-order equation with dissipation in periodic Sobolev spaces has a unique solution and possesses the property of continuous dependence on the initial data. Moreover, compared to the third-order case, in the fourth-order case we obtain more regularity for the solution. We perform this in an intuitive manner using Fourier theory and in an abstract version using semigroup theory. Furthermore, by employing a different method, we evidence the uniqueness of the solution of this problem through its dissipative nature, motivated by the contributions of Iorio [1] and Santiago [2]. In order to increase and enrich our study, we investigate the infinite dimensional space in which differentiability occurs and its connection with the initial data. Finally, we generalize our results to the equations of nth order where n is a natural number multiple of four.
Solutions to the Calapso Equation Associated with Surfaces of Revolution
<span style="font-size: 11.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-font-kerning: 0pt; mso-ligatures: none; mso-ansi-language: ES-PE; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">We construct explicit solutions of the Calapso equation by applying Ribaucour transformations to isothermic surfaces of revolution. We derive conditions under which such transformations preserve the isothermic structure and generate new solutions of the Calapso equation. The resulting solutions depend on three functions, two depending on one variable and one depending on another variable. As a consequence, we obtain solutions of the Zoomeron and Davey–Stewartson III equations. Explicit examples associated with the cylinder, cone and catenoid are provided.
Modeling climate-mediated shifts: An almost periodic approach to exploitation and interference competition
<span style="font-size: 11.0pt; line-height: 115%; font-family: 'Calibri',sans-serif; mso-ascii-theme-font: minor-latin; mso-fareast-font-family: Calibri; mso-fareast-theme-font: minor-latin; mso-hansi-theme-font: minor-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: minor-bidi; mso-font-kerning: 0pt; mso-ligatures: none; mso-ansi-language: ES-PE; mso-fareast-language: EN-US; mso-bidi-language: AR-SA;">We consider an almost periodic competition model to analyze the simultaneous effects of exploitation and interference competition. Based on the analysis of the model, we prove the existence, uniqueness, and global stability of an almost periodic solution under certain conditions. Through numerical simulations, we illustrate the results obtained analytically. Finally, the results are discussed in detail from a biological perspective.
Centenary of the first predator-prey model. Revision, modifications and challenges of this fundamental ecological interrelationship
For many applied mathematicians, and especially for biomathematicians, the first model proposed by the Italian mathematician Vito Volterra in 1926 is well known, describing for the first time the relationship between a predator and its prey. This model coincided with a similar system, on chemical reactions, proposed by the physicist-chemist Alfred J. Lotka years earlier. Since then, and with an epidemic character, variations, modifications, and the incorporation of new phenomena or ecological principles have been formulated to ”make more realistic” the foundations and studies on this fundamental interaction between two species of living beings. In this work, we will give a brief description of the historical context of this seminal model, emphasizing its main properties; then we will add specific modifications, briefly outlining properties of some of them.
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Some Results for Generalized Local Homology Modules
We prove some results on the finiteness of co-associated primes of generalized local homology modules inspired by the conjecture of Grothendieck and the question of Huneke. We also show equivalent properties of minimax local homology modules. Here, we get applications for the generalized local homology module, in a general theory of modules.
Euler-Rodrigues Rotation for computes the tangent vector and curvature vector of the intersection curve of two surface in 3D Lorentz-Minkowski space
We present method computes the tangent and curvature vector of the intersection curve of two surface, parametric/implicit or implicit/implicit, in Lorentz-Minkowski space E3, by applying a Euler-Rodrigues rotation to a vector projected onto the tangent space. The axis of rotation is the normal vector of the surface (the surfaces can be timelike, spacelike or lightlike), therefore three types of rotations, since the normal vectors can be: spacelike, lightlike, or timelike.
On a special class of hypersurfaces in R5
In this paper we study hypersurfaces in R5 parametrized by lines of curvature, with four distinct principal curvatures and with Laplace invariants mji = mki = mli = 0, mjik ̸= 0, mjkl ̸= 0, mljk ̸= 0, Tijkl ̸= 0 for i, j, k, l distinct fixed indices. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and four vector valued functions of one variable. Moreover, we show that these vector valued functions are invariant under inversions and homotheties. We observe that this class of hypersurfaces cannot have constant Mobius curvature.
Curves with Prescribed Curvature and Associated Surfaces
In this work, we study surfaces with a canonical principal direction (CPD surfaces) under the simultaneous prescription of two extrinsic geometric invariants: one principal curvature and the support function. Based on these prescriptions, we reformulate the first and second fundamental forms, which enables us to construct explicit examples of such surfaces. Finally, we show that when the curvature of the profile curve is constant, that is, when one of the principal curvatures is constant, the corresponding CPD surface is a tubular surface.
Numerical simulation for shell deformation using the Naghdi and Koiter models
In this research work, shell deformations generated by surface forces acting on the shell are described and simulated. The Naghdi and Koiter models are used to describe shell deformations. A weak formulation for the model is obtained, and some results on the existence of solutions are presented. The finite element method is then used to obtain the deformations. In addition, simulations of shell deformations are shown for the case of sectional curvature k = 0.
Mathematical Modeling and Numerical Simulation for the Design of Agricultural Irrigation Channels with Bend Failures
This work analyzes water flows in open irrigation channels with bend failures. To improve this failure, a mathematical model based on elliptic equations is proposed, which is solved by applying the SchwarzChristoffel transformation in complex analysis. Given its complex geometry, and with the desire to improve the structural curve of the channel, a numerical method is employed for adaptive mesh generation, based on conformal mapping via the Schwarz-Christoffel transformation. This allows for the obtaining of orthogonal meshes adapted to the geometry of the bends. The fundamental part of this study lies in the detailed simulation of the flow in the channel bends and in the evaluation of the bend’s geometric configuration and the improvement technique for hydraulic efficiency of the channel, ultimately aiming to enhance its hydraulic performance. The results obtained will be contrasted with the data found in Google Earth satellite imagery and numerical analysis, using devices such as drones to discuss the results and draw useful conclusions for the design of irrigation channels with bend failures.
Kinetic evaluation of the Arrhenius equation for artificial ageing of polymers
Artificial ageing of polymers is a crucial and complex issue, especially considering that critical infrastructures such as nuclear power plants have lifespans varying from 40 to 60 years or even longer. Controlled artificial ageing allows the evaluation of polymer lifetimes while ensuring that their properties are preserved. However, a unified and validated methodology for artificially ageing polymers is still lacking. One of the most extended methodologies for the artificial ageing of polymers is the Arrhenius methodology. This methodology is based on the application of the Arrhenius equation, which is extensively applied in the study of thermal decomposition reactions. Nevertheless, the Arrhenius methodology requires the estimation of activation energy, and the lack of a unified method introduces variability in results obtained using different methods. Furthermore, the Arrhenius ageing methodology assumes that the kinetic parameters do not change during ageing, meaning that aged and non-aged materials should exhibit the same activation energy. The present work aims to analyse the hypothesis of unvariable activation energy during ageing. This was investigated using both new and artificially aged PVC samples, evaluating the activation energy through various mathematical models based on thermogravimetric analysis.
Control and Communication in Dialogue: Controllability, Observability, and Bang–Bang Dynamics
This work interprets human conversation as a controllable and observable dynamical system. Each interlocutor influences the discursive state of the other, allowing the interaction to be modeled by differential equations. Using results from control theory, we consider a convex and compact space as admissible interactions, with extreme points corresponding to bang–bang interventions such as interruptions or emphatic statements. We validate the model through numerical simulations and synchrony metrics, showing that conversational symmetry improves stability and consensus, asymmetry produces biased equilibria, and temporal variations in coupling generate transient desynchronization followed by recovery, analogous to conflict and reconciliation phases in dialogue.
Stochastic Analysis on Efficiency Vaccination in a SIR Model
In this paper, we model the stochastic effect of vaccination on an epidemic which like epimdemic model, it can be susceptible to environmental changes. This study is carried out using a SIR model in which population remain susceptible after having been vaccinated. One important point of our research is that we apply a whit noise on recuperate rate of the infected individuals. Further, we show some stability results and some numerical simulations.
Refuge use by prey dependent on the number of predators in a Gause-type model
This paper deals with a continuous-time predator-prey model of Gause-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. According to these results, the extinction of both species is not possible, and they coexist over the long term. We conclude that the dynamics of the studied model are very similar to the model that does not consider prey refuge. However, this cannot be stated if another function is used to describe this anti-predator behavior.
Mutualism as a stabilizing effect on the population densities of two interacting species
Mathematical models are a very useful tool to understand, describe or predict the population dynamics of species interacting. Ecologists and mathematicians have extensively studied the predator-prey, victim-exploiter, competition and mutualistic relationships. However, mutualism between species has not received the same attention as the other ecological interactions. In this work, we exclude periodic solutions of three types of systems by the construction of Dulac functions. These systems can be used to describe the population dynamics of mutualistic species. The system type I includes a wide variety of mutualistic models in which both the intrinsic rate of increase and the carrying capacity of each species increase by the interaction between species. In particular, the system type I can be applied to exclude periodic solutions of models with conditioned interactions such that mutualism occurs at low population densities and competition occurs at high population densities. The system type II includes mutualistic models that describe a consumer-resources interaction. In these models, it is assumed that the net change of benefitscosts due to the interaction depends on the densities of the recipient species and the partner one. The system type III describes mutualistic models in which the per capita growth rate of each species is affected by a weak Allee effect. We also apply the results of this work to models mentioned in a historical list of mutualistic models provided in [1]. From the results obtained, we conclude that mutualism leads to the exclusion of periodic behaviors in the population dynamics of interacting species. Therefore, the population densities of the mutualistic species converge to an equilibrium point. Then, when the population densities oscillate, the oscillatory behaviors are transient. These results are relevant since the dynamics of mutualistic species has not been deeply characterized and the discussion about the existence of sustained oscillatory behavior in mutualistic species is relevant from an ecological perspective.
Perturbation of a biological control model of malaria considering species competition
This work studies a mathematical model describing the transmission dynamics of malaria, incorporating intraspecific competition in human and mosquito populations, and biological control through larvivorous fish. We prove the existence of a compact attracting set for the proposed system of differential equations. Through stability analysis, we characterize the disease-free and endemic equilibrium points, determining epidemiologically relevant thresholds. The basic reproduction number (R0) is a key parameter: when R0 ≤ 1, the disease-free equilibrium is globally asymptotically stable, whereas for R0 > 1, a stable endemic equilibrium emerges. Numerical simulations validate the theoretical results and reveal the regulatory effect of intraspecific competition on disease prevalence. Finally, we generalize previous models by incorporating competitive interactions and vector management strategies.
Well-posedness for a Third-Order PDE with Dissipation
In this work, we prove that the Cauchy problem associated with a third-order equation with dissipation in periodic Sobolev spaces admits a unique solution. We also show that the solution depends continuously on the initial data. Our approach combines both an intuitive method, based on Fourier theory, and a more abstract framework using semigroup theory. Furthermore, by employing an alternative method, we demonstrate the uniqueness of the solution through its dissipative nature, drawing inspiration from the contributions of Iorio [1] and Santiago [2]. To deepen and enrich our study, we investigate the infinite dimensional space in which differentiability occurs and its connection to the initial data. Finally, we extend our results to equations of arbitrary nth order.
Analysis and numerical simulation of a parabolic equation with non-local terms
In this work, we investigate the existence and uniqueness of global strong solutions, as well as the exponential decay of these solutions in bounded domains, for an initial-boundary value problem associated with parabolic equations involving nonlocal terms. The theoretical results are complemented by numerical simulations obtained using the finite element method for the spatial variable and the finite difference method for the temporal variable.
Numerical Modeling and Simulations for an Ocean Circulation Model of the Southern Pacific
In this work, we consider the primitive equations of an ocean circulation model for the southern pacific, which consists of the time-dependent Navier-Stokes equations in the β-plane coupled with the temperature transport equation. Specifically, the full three-dimensional equations are adopted and formulated as a monolithic system of nonstationary, nonlinear, coupled partial differential equations. The El Nino phenomenon is simulated by the action of given wind stresses on the ocean surface. We present an approximation scheme with Crank-Nicolson finite differences in time, and in space we take inf-sup stable Galerkin finite elements for the Navier-Stokes part and bilinear elements for the temperature. We solve the resulting, nonlinear monolithic discrete system by Newton's method. Numerical experiments with realistic geometry and material data are conducted, which show the practicability of our approach.
Euclidean space perturbed by a constant vector field and its relation to a Zermelo navigation problem
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.
On pullback attractors in non-autonomous dynamical systems
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.
Applied mathematics, its importance in developing it in Peru
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.