Selecciones Matemáticas

Frobenius methods for analytic second order linear partial differential equations

Víctor Arturo Martínez León, Bruno César Azevedo Scárdua — Wed, 27 Dec 2023 00:00:00 +0000

The main subject of this text is the study of analytic second order linear partial differential equations. We aim to solve the classical equations and some more, in the real or complex analytical case. This is done by introducing methods inspired by the method of Frobenius method for second order linear ordinary differential equations. We introduce a notion of Euler type partial differential equation. To such a PDE we associate an indicial conic, which is an affine plane curve of degree two. Then comes the concept of regular singularity and finally convergence theorems, which must necessarily take into account the type of PDE (parabolic, elliptical or hyperbolic) and a nonresonance condition. This condition gives a new geometric interpretation of the original condition between the roots of the original Frobenius theorem for second order ODEs. The interpretation is something like, a certain reticulate has or not vertices on the indexical conic. Finally, we retrieve the solution of all the classical PDEs by this method (heat diffusion, wave propagation and Laplace equation), and also increase the class of those that have explicit algorithmic solution to far beyond those admitting separable variables. The last part of the text is dedicated to the construction of PDE models for the classical ODEs like Airy, Legendre, Laguerre, Hermite and Chebyshev by two different means. One model is based on the requirement that the restriction of the PDE to lines through the origin must be the classical ODE model. The second is based on the idea of having symmetries on the PDE model and imitating the ODE model. We study these PDEs and obtain their solutions, obtaining for the framework of PDEs some of the classical results, like existence of polynomial solutions (Laguerre, Hermite and Chebyshev polynomials).

A mathematical model for the formation of oscillation marks in steelmaking industry

Marcos Zambrano Fernández — Wed, 27 Dec 2023 00:00:00 +0000

Oscillation marks are defects found on the surface of the final product in the steelmaking industry. In this research, the temperature of a lubricant known as flux that lies inside of the mould is investigated due to its importance in the formation of the oscillation marks. This temperature that decreases in time is obtained numerically via the finite-volume method by using the Openfoam software.

Analysis of a delayed mathematical model for tumor-immune cell interactions with Holling type II functional response

John Medina Diaz — Wed, 27 Dec 2023 00:00:00 +0000

In this study, we analyzed a three-dimensional nonlinear differential system considering Holling type II functional response that describes the dynamics of tumor cells, cytotoxic T lymphocytes, and helper T cells, with a single interaction delay. The linear stability of the internal equilibrium point and the presence of the Hopf bifurcation are examined, with the discrete time delay serving as the bifurcation parameter. To demonstrate the rich dynamic behavior of the model, we present numerical simulations with various values of the time delay τ and the attack rate of cytotoxic T lymphocytes on tumor cells (α1). These simulations exhibit the presence of periodic oscillations and tumor death with survival of the mentioned immune cells, for all α1 greater than a fixed threshold with or without delay.

Semigroups of class Co on l2(Z)

Yolanda Santiago Ayala — Wed, 27 Dec 2023 00:00:00 +0000

In this work we begin by studying the generalized multiplication operator M on the l2(Z). We prove that this operator is not bounded, is densely defined and symmetric and therefore does not admit a symmetric linear extension to the entire space. We introduce a family of operators on the l2(Z) space with n even and demonstrate that it forms a contraction semigroup of class Co, having −M as its infinitesimal generator. We also prove that if we restrict the domains of that family of operators, they still remain a contraction semigroup. Finally, we give results of existence of solution of the associated abstract Cauchy problem and properties of continuous dependence of the solution in connection to other norms.

The region of the unit Euclidean sphere that admits a class of (r,s)-linear Weingarten hypersurfaces

Marco Antonio Lázaro Velásquez — Wed, 27 Dec 2023 00:00:00 +0000

In the unit Euclidean sphere Sn+1, we deal with a class of hypersurfaces that were characterized in [23] as the critical points of a variational problem, the so-called (r, s)-linear Weingarten hypersurfaces (0 ≤ r ≤s ≤ n−1); namely, the hypersurfaces of Sn+1 that has a linear combination arHr+1+・・・+asHs+1 of their higher order mean curvatures Hr+1 and Hs+1 being a real constant, where ar, . . . , ar are nonnegative real numbers (with at least one non zero). By assuming a geometric constraint involving the higher order mean curvatures of these hypersurfaces, we prove a uniqueness result for strongly stable (r, s)-linear Weingarten hypersurfaces immersed in a certain region determined by a geodesic sphere of Sn+1. We also establish a nonexistence result in another region of Sn+1 for strongly stable Weingarten (r, s)-linear hypersurfaces.

Solutions to the Calapso, Zoomeron, and Davey-Stewartson III Equations using Ribaucour Transformations

Marcelo Lopes Ferro, Armando M. V. Corro — Wed, 27 Dec 2023 00:00:00 +0000

In this paper, we present a method for obtaining solutions to the Calapso and Zoomeron equations using Ribaucour transformations. We provide explicit formulas for the solutions of these equations. By applying this method to the generalized cylinder, we provide new solutions of these equations, which are determined by a free function and by two other functions, each one defined in a given variable. As a consequence, we provide new solutions for the Davey-Stewartson III equation.

Dynamics of a Leslie-Gower type predation model with non-monotonic functional response and weak Allee effect on prey

Wed, 27 Dec 2023 00:00:00 +0000

This research concerns with analysis of a class of modified predator- prey type Leslie-Gower models. The model is described by an autonomous nonlinear ordinary differential equation system. The functional response of predators is Holling IV type or non-monotone, and the growth of prey is affected by the Allee effect. An important aspect is the study of the point (0, 0) since it has a strong influence on the behavior of the system being essential for the existence and extinction of both species, although the proposed system is not define there.

A non-parametric dynamic model for measuring efficiency

Kelly Patricia Murillo — Wed, 27 Dec 2023 00:00:00 +0000

Conventional efficiency evaluation systems present little diversity in the type of variables considered. This aspect generally leads to errors in the application of the models and in the corresponding interpretation of results. This study proposes a dynamic model to measure the efficiency of decision-making units, based on non-parametric Multidirectional Efficiency Analysis (MEA). The model presented here considers a complete structure, which includes intertemporal variables (desirable/undesirable intermediate inputs and outputs), discretionary/non-discretionary inputs; desirable/undesirable inputs and outputs. Dynamic score is defined first in a particular version and then in a more general version, considering two optimization problems.

The directional contribution of each variable is examined, showing excess inputs and deficit outputs. In addition, a dynamic inefficiency index to measure the number of times each input was used inefficiently, is presented for both desirable and undesirable cases.

The Frobenius endomorphism to Elliptic Curves and same properties

Jaime Edmundo Apaza Rodriguez — Wed, 27 Dec 2023 00:00:00 +0000

In this work we present the Frobenius endomorphism for Elliptical Curves defined over finite fields and some of their properties relative to their kernel and the condition of separability.

Relationship Between the Cantor-Bendixson Derivative and the Algebra of Sets

Andres Merino, Sebastián Heredia F. — Wed, 27 Dec 2023 00:00:00 +0000

This article provides a detailed analysis of the relationship between the Cantor-Bendixson derivative and set containment, as well as the standard set operations of union and intersection. In particular, it is shown that the Cantor-Bendixson derivative is increasing with respect to set containment and, under suitable hypotheses, generates a decreasing family of sets. On the other hand, we study both the derivative of a union and the derivative of an intersection, under different restrictions on the cardinality of the family of sets being operated on, while taking into account the order of derivative being performed.

Dolph Algebras and Dolph Groups

Felipe Clímaco Ccolque Taipe — Wed, 27 Dec 2023 00:00:00 +0000

A finite Hopf crossed product whose base ring is a finite field will be called a Dolph algebra, and the corresponding group of units will be called a Dolph group. Assuming known the crossed product of a ring and a group under a crossed mapping [1], the units of the cross products Z2[X] ⟨X2⟩ ⋊f C2 and Z3[X] ⟨X2⟩ ⋊f C2 are calculated. Furthermore, we give concrete examples of 4 classes of Hopf crossed products.

Modeling and calibration for a simple pendulum

Daiana Soares Barreiro, Frank Henry Acasiete Quispe, Laura Costa Pereira Miranda — Wed, 27 Dec 2023 00:00:00 +0000

This work aims to present a report on the analysis and calibration of uncertainties of the model that describes the movement of a simple pendulum. The results will be presented through different figures generated by numerical methods and tables whose data are the result of simulation.

A characterization of the natural isomorphism of bifunctors

Wed, 27 Dec 2023 00:00:00 +0000

We are going to characterize the natural isomorphism between the bifunctors of the examples 2.13 and 2.14, for that it is enough to show that η and ζ of the lemma 3.1 are natural isomorphisms.

Hopf bifurcation in an autonomous system with logistic growth and Holling type II functional response

Danny Estefany Paz Vidal, Joan Esteban Salazar Gordillo — Wed, 27 Dec 2023 00:00:00 +0000

The autonomous prey-predator system with logistic growth and Holling type II functional response, which describes the population dynamics of two species. In this study, the equilibrium points of the system (1.1) were identified. Two saddle-node points and one non-trivial P3 equilibrium point were found. For this latter point, the conditions were determined for the Jacobian matrix of (1.1), evaluated at P3, to have a pair of purely complex eigenvalues (necessary condition for the Hopf bifurcation). Through this analysis,

values c0, δ0, k0 were found that satisfied these conditions. Subsequently, each of these values is considered as a bifurcation parameter value, and the remaining two are considered as control parameters, under the assumptions of the normal form theorem for the Hopf bifurcation, it’s concluded that by varying these values slightly, the system undergoes the Hopf bifurcation. Finally, the first Lyapunov coefficient was calculated

to determine the conditions under which the system exhibits supercritical, subcritical, and degenerate Hopf bifurcation.

The analysis was supported by using MAPLE and MATLAB software, which enabled graphical visualization of the obtained results.

Computational methods in algebraic geometry

Joe Palacios, Ruth Cabanillas — Wed, 27 Dec 2023 00:00:00 +0000

In this article we give a general description of the field of algebraic geometry and its computational aspects using Macaulay 2. In particular, we present an application to the bidimensional robotic arm of n spans for any natural n.

The integral, a vision of its evolution through time II

Alejandro Ortiz Fernández — Wed, 27 Dec 2023 00:00:00 +0000

In this part II we intend to give an overview of the different ways in which the idea of integral has evolved over time; in particular the Riemann integral has been the motivation for further research up to the present.

We only want to motivate this part of the history of mathematical analysis in our environment with the aim that student and teachers have a guide for more detailed studies and thus contribute to the development of matematics in our region.

The integral, a vision of its evolution through time III

Alejandro Ortiz Fernández — Wed, 27 Dec 2023 00:00:00 +0000

In this opportunity we present some areas of the evolution of the integral, which complement what was deal with in parts I and II. Thus we give a conceptual overview of the McShane integral, of the C-integral, of the Bochner integral, of the Lm HK integral , of the integral on manifolds and on a modern point of view of the Riemann integral.