Divulgaciones Matemáticas

Super quasi-topological and paratopological vector spaces versus topological vector spaces

Madhu Ram — 2024-06-10

In this paper, we introduce the idea of super quasi-topological vector space which is an extension of the concept of topological vector space and investigate some of its basic properties. We extend the existing notion of quasi-topological vector space to all complex vector spaces and investigate the relationship of super quasi-topological vector spaces with paratopological and quasi-topological vector spaces.

The Ramsey numbers for three graphs and three colors.

José Figueroa — 2024-06-10

Given a graph $\Psi$ of order $t$, simple, finite, and non-empty. An superposition of $\Psi$ will be called the complete graph $K_{t}$ defined by $\Psi\cup\overline{\Psi}$, where $\overline{\Psi}$ denotes the complement of $\Psi$. So, given two graphs $G$ and $H$, simple, connected, finite, non-empty, and two different colors. The Ramsey number $R(G, H)$, is defined as the smallest positive integer $t$ such that there exists some graph $\Psi$ that contains a monochrome copy $G'$ of $G$, or $\overline{\Psi}$ contains a monochrome copy $H'$ of $H$, and $K_{t}$ is a superposition of $\Psi$, that is, $\Psi$ is a subgraph of $K_{t}$. Similarly, given three graphs $G$, $H_{1}$, and $H_{2}$, simple, connected, finite non-empty, and three distinct colors $\{0,1,2\}$, is defined as the Ramsey number $R(G,H_{1},H_{2})$ to the smallest positive integer $t$ such that there exists a triplet of graphs $(\Psi,\Psi_{1},\Psi_{2})$ that satisfy the following: (1) $\overline{\Psi}=\Psi_{1}\cup\Psi_{2}$; (2) The complete graph $K_{t}$ is the superposition of $\Psi$; (3)$|V(K_{t})|=|V(\Psi)|=|V(\Psi_{1})|=|V(\Psi_{2})|$; (4) $E(\Psi_{1})\cap E(\Psi_{2})=\emptyset$; and (5) The graph $\Psi$ contains a monochrome copy $G'$ of $G$, or from $\Psi_{1}$ can be made a monochrome copy $H'_{1}$ of $H_{1}$, or from $\Psi_{2}$ a monochrome copy $H'_{2}$ of $H_{2}$ can be extracted. In this work it is shown that, for $n\geq 4$, given the graphs $K_{n}$ a complete graph, $W_{n}$ a roll graph, and $D_{4}$ a diamond graph. If $t=\mbox{m\'ax}\{|K_{n}|,|W_{n}|,|D_{4}|\}$, then $R(K_{n},W_{n},D_{4})=t+2$, for $t\in \mathbb{Z}^{+}\setminus\{1,2,3,4\}$.

Qualitative study of the metabolism of an ingested drug

Berónica Aguilar León — 2024-06-10

In the present paper, a qualitative study of a mathematical model for the elimination of a drug from the human body is carried out, particularly the case in which the fundamental matrix of the system has a null eigenvalue and another pair of pure imaginary eigenvalues, through examples the results obtained are verified. Additionally, a preliminary study of the metabolism of a drug in the body until its elimination, the effect it causes and its incidence
in Cuba is carried out.

Top(X) y Spec(τ ) como espacios primales

Viviana Benavides — 2024-06-10

An Alexandroff topology can be defined over a non-empty set $X$, through a function $f:X\to X$, deciding that the open sets are those subsets $A\subseteq X$ that contain their preimage, that is $\tau_f:=\{A\subset X: f^{-1}(A)\subseteq A\}$. This topology is called primal topology and the space $(X, \tau_f)$ is called primal space. In this work we explore a primal topology $\tau_\psi$ induced on $Top(X)$, through the funtion $\psi: Top(X)\to Top(X)$ defined as $\psi(\tau)=\overline{\tau}$, with $\overline{\tau}$ the closure of $\tau$ in $2^X$ with the product topology. It is shown that the set of all Alexandroff topologies in $Top(X)$ is dense in $(Top(X),\tau_{\psi^*})$, with $\tau_{\psi^*}$ the cotopology. It is also shown that the set $\phi(\tau):=\{A\in \tau_\psi :\tau\notin A\}$ is a maximal ideal of $\tau_\psi$ if and only if $\tau$ is Alexandroff. Finally we explore the primal topologies in the prime spectrum of a semiring.

Zero divisor graph of \mathbb{Z}_{2^{r} q^{s}}

Juan M. Otero Acosta — 2024-06-10

This article continues the study of zero divisor graphs, presented in 1988 by Istan Beck \cite{Beck}. There, a zero divisor graph is defined as a graph whose vertices are the elements of the set of zero divisors of a ring, where two distinct vertices $x$ and $y$ are adjacent if and only if $ x \cdot y = 0$. In this work, we present a new way to calculate the zero divisor graph of the ring $\mathbb{Z}_{2^{r}q^{s}}$ for $q$ an odd prime, with $r$ and $s$ positive integers greater than $2$, and the example of the zero divisor graph of the ring $\mathbb{Z}_{36}$ is also given.

A new method for eliminating the indeterminacy in the singularly perturbed problems with Ackerberg-O’Malley resonance

Jacques Laforgue — 2024-06-10

In the singularly perturbed problems with resonant character in the sense of Ackerberg and O’Malley, the traditional method of matched asymptotic expansions fails to determine the resonance’s amplitude. A new method is presented, based on established procedures from the theory of ordinary differential equations, for eliminating such indeterminacy taking advantage of the incomplete result of the matched asymptotic expansions and eliminating in a natural fashion the superfluous degree of freedom, through the derivation and imposition of an additional exact boundary condition that relates the slopes at both extremities of the domain. The new method is effective for the variety of problems recognized as resonant, including those exhibiting supersensitivity and also for those of a different structure but with analogous indeterminacy, for example involving a partial differential equation.

Boundary Estimation with the Fuzzy Set Regression Estimator

Jesús A. Fajardo — 2024-06-10

In order to extend the properties of the fuzzy set regression estimation method and provide new results related to the nonparametric regression estimation problems not based on kernels, this paper analyzes the possible boundary effects, if any, of the fuzzy set regression estimator and presents a criterion to remove it. Moreover, a boundary fuzzy set estimator is proposed which is defined as a particular class of fuzzy set regression estimators, where the bias, variance, mean squared error and function that minimizes the mean squared error of the proposed estimator are given. Finally, these theoretical findings are illustrated through some numerical examples, and with one real data example. Simulations show that the proposed estimator has better performance at points near zero in a spread neighborhood of the smoothing parameter, when it is compared with a general boundary kernel regression estimator for the two regression models and two density functions considered. The previously exposed represents the natural extension of the recent results to the boundary fuzzy set density estimator case.

Problems and Solutions

Tobías Rosas Soto — 2024-06-10

Appropriate problems for this section are those which may be tackled by undergraduate math students without specialized knowledge. Known open problems are not suitable. Original and interesting problems are preferred. Problem proposals and solutions should be e-mailed to the editor, in Spanish or English, to the address given above (preferably as a LATEX source file). Proposals should be accompanied by a solution or, at least, enough
information on why a solution is likely.