Selecciones Matemáticas

Quadratic Fractionally Integrated Moving Average Processes with Long-Range Dependence

Jonas F. de Medeiros — 2024-07-29

Stochastic processes with the long-range dependency (LRD) property are fundamental to modeling data that exhibit slow power decay of the covariance function. Such behavior often appears in the analysis of financial data, telecommunications, and various natural phenomena. Thus, introducing new stochastic models and statistical methods that take the LRD into account is of great interest. Based on previous work, we introduce a new stochastic process called quadratic fractionally integrated moving average, that arises from the Quadratic Ornstein-Uhlenbeck Type (QOUT) process, proposed in the literature. We consider Lévy noises of finite second-order moments and use a construction based on a moving average stochastic process whose kernel is that of a QOUT process. Then, using Riemann-Liouville fractional integrals, we propose a fractionally integrated moving average process, for which we highlight some results, including the LRD. We also propose the estimation of the parameters for this process for the case of fractional Brownian noise, showing its efficiency through a Monte Carlo simulation. By an application based on Brazil’s stock market prices, we illustrate how this process can be used in practice with the Sao Paulo’s Stock Exchange Index data set, also known as the BOVESPA Index.

Surfaces with quadratic support function

Armando M. V. Corro — 2024-07-29

In this paper, we study oriented surfaces S in R3, called surfaces with quadratic support function (in short QSF-surfaces). We obtain a Weierstrass type representation for the QSF-surfaces which depends on two holomorphic functions. Moreover, classify the QSF-surfaces of rotation. Also, we give some explicit examples of this class of surfaces.

Prime submodules and maximal submodules in the idealization of a module

Alex Molina — 2024-07-29

This paper describes certain types of prime submodules and maximal submodules that are found within the idealization of a module. As a consequence of this, it is concluded that every module over a commutative ring with identity element has an extension whose prime spectrum is nonempty.

Extensions in affine spaces of bilinear applications, differentiable actions, and tensors

Benito Leonardo Ostos Cordero — 2024-07-29

In this article, several generalizations in affine spaces are studied. First, the notion of affine mappings to bilinear mappings defined in affine spaces is explored, referred to as affine bilinear mappings. Subsequently, differentiable actions of a Lie group on affine spaces are defined, and their isotropy group, orbit space, and set of fixed points are examined. Finally, the concept of tensor product between vector spaces is extended to the tensor product between affine spaces.

Influence of the prey refuge use on the Volterra predation model

Eduardo González-Olivares — 2024-07-29

In various previous works, different predation models have been analyzed considering the use of refuge by the prey population, without carrying out an exhaustive analysis of its dynamics.

In some of them it is stated that the use of shelters or dens by a fraction of the prey population has a stabilizing effect on predator-prey interaction. One of the objectives of this work is to show that some of these new systems considering refuge may have the same topological portrait in the phase plane as the original; but in others the dynamics change strongly.

In our research we will introduce modifications to the well-known Volterra model, considering various ways to express the number of prey in refuge. In several of them, local dynamics equivalent to the original model are obtained, confirming that results reported as new were previously known based on the original system, without taking into account the refuge of the prey. We conclude that the behavior of the systems depends on the mathematical expression to describe the number of sheltered prey.

Mathematical modeling to evaluate potato cultivation in high-andean soils using data analysis and the Runge Kutta method

Irla Mantilla Núñez — 2024-07-29

According to data collected by the International Potato Center (CIP) and the Departmental Federation of Peasant Communities (FEDECH), regarding the Huancavelica region. One of the most devastating diseases in the potato crop is ”Rancha”; scientifically known as Phytophtora infestan. Frog can be controlled by the application of fungicides, but due to the excessive growth of the population density of this pathogen produced by the frog fungus, it would be necessary to apply more fungicides, which poses a high environmental and human health risk to those who consume this type of tuber in their diet. To measure the population growth of both species, a database will be built related to the percentage of potato crops per hectare that are attacked by this type of fungus. In this sense, a numerical mathematical model will be developed based on the Lotka-Volterra type differential equation system, which is solved with the Runge-Kutta RK2 and RK4 method, in order to contribute, in a non-invasive way, with an alternative for the prevention and control of this disease in potato crops in high Andean soils.

Well-Possednes of a predator-prey reaction-diffusion model with functional response dependent on species

Neisser Pino-Romero — 2024-07-29

In this work, the well-possednes of a predator-prey model with reaction-diffusion will be studied where the Allee effect in prey reproduction will be included and predation dynamics will be considered a functional response dependent on the species (prey and predator) with the incorporation of the refuge in the dam. In this way, the existence, uniqueness and positivity of the system's solutions as the main result will be guaranteed.

Automatic placement of elements of up to three sets in Venn Diagram operations using a Wolfram-Mathematica command

Jorge L. Viera-Jimenez — 2024-07-29

The representation of set algebra operations using Venn diagrams has been widely adopted in various fields. However, it is possible to create these diagrams manually in programs such as Microsoft Word, Microsoft PowerPoint, Microsoft Visio, etc., and symbolic calculus systems such as Wolfram-Mathematica, Maxima, Maple, MatLab, etc. The real innovation lies in the automation offered by specialized software such as Wolfram-Mathematica. This paper presents a new Wolfram-Mathematica command that allows operations with up to three sets, offering unprecedented versatility. The objective of this paper is to explore how Wolfram-Mathematica facilitates these operations automatically, highlighting its superiority in terms of efficiency and precision. n compared to manual methods.

The Integral, a Vision of its Evolution Through Time IV. Harmonic Analysis, Stochastic Processes. Stochastic Integration

Alejandro Ortiz Fernández — 2024-07-29

The objective of this article is to give an overview of the relationship between harmonic analysis and stochastic processes, a relationship that has motivated many investigations and applications to specific problems. We will also emphasize stochastic integration.

Some Aspects in the PDE. The Dirichlet and the Cauchy Problems

Alejandro Ortiz Fernández — 2024-07-29

In this article we present two classic problems in PDE: the Dirichlet problem and the Cauchy problem. Some ideas are previously discussed, such as functions and identities’s Green, and “bad positions” problems. The work of L.Nirenberg [1] is discussed in detail where some results of L.Hörmander [2] are used.

Fourier analysis with R

Mississippi Valenzuela — 2024-07-29

Fourier analysis is currently one of the important parts of mathematical analysis. Fourier analysis methods are used in physics, electrical engineering, and other sciences to solve various problems. In this article we will give the main concepts of Fourier analysis and its applications in spectral analysis in solving problems using R, as shown in the examples.