Boletim da Sociedade Paranaense de Matemática

On skew cyclic reversible DNA codes over F_4[v]/ < v^4 - v >

2026-01-22T14:23:11+00:00

In this paper, we study a specific class of skew cyclic codes over the ring F_4[v]/ < v^4 - v > which is suitable for describing DNA codes over this ring. Using the Gray map between F_4[v]/ < v^4 - v > and (F_4)^4 (or equivalently DNA 4-bases), we describe reversible DNA codes and reversible-complement DNA codes over this ring.

Groupoids and their Topological *-Algebras

2026-01-22T14:27:34+00:00

This study introduces the concept of atopological groupoid and some topological *-algebras are investigated, like theconvolution topological *-algebras associated with locally compact groupoids, and in particular, étale groupoids.

Spectral Properties of Identity Graph for Group of Integers Modulo using Degree-Based Matrices

2026-01-22T14:35:20+00:00

This paper investigates the spectral properties of the identity graph associated with the group $\mathbb{Z}_n$, utilizing five degree-based matrices. Specifically, the study employs the maximum and minimum degree, greatest common divisor, and first and second Zagreb matrices. For each case, the characteristic polynomial and the corresponding graph energy are derived. Furthermore, a comparative analysis is conducted between the computed energies and established results in the literature.

Difference of Convex Functions Optimization for Feature Selection in Granular Ball Support Vector Machine

2026-01-22T14:39:11+00:00

Feature selection constitutes a critical optimization problem within the domain of supervised pattern classification. It involves selecting an optimal subset of features that maximizes the retention of the data’s salient information. Granular Ball Support Vector Machine (GBSVM) has proven to be a powerful technique for enhancing the predictive accuracy and computational tractability of classification models, by exploiting the concept of granular structures in the feature space, through the generation of a set of granular balls, enabling complex decision boundary modeling and adaptability to data variability. This paper presents a novel embedded feature selection approach in the context of granular ball SVM, directly enhancing classifier performance. Our approach to the resulting optimization problem is to apply Difference of Convex (DC) functions programming to effectively handle the non-convex nature of the problem. Genetic algorithm is used to tune the model’s parameters. Experimental results on UCI datasets show the efficiency of the proposed method.

On the uniqueness of fixed points for nonlinear-linear operator sums of Krasnosel’skii type

2026-01-22T14:45:45+00:00

This paper extends Kellogg’s uniqueness fixed point theorem within the framework of Krasnosel’skii’s fixed point theorem. More precisely, we provide sufficient conditions on a linear operator B and a nonlinear mapping A to ensure the unique

ness of the fixed point of the mapping A+B. We also investigate the global asymp

totic stability of this fixed point in connection with the Belitskii-Lyubich conjecture.

An illustrative application of the main theoretical result is presented.

Asymptotics of Solutions to $p$-Laplacian Equations Involving Convection and Reaction Terms

2026-01-22T16:09:48+00:00

The purpose of this work is to investigate a nonlinear $p$-Laplacian equation that incorporates both convection and reaction effects. The model under consideration takes the form
$$
\displaystyle \mbox{div}(|\nabla U|^{p-2} \nabla U) + \lambda x\nabla(|U|^{q-1} U) + \theta U = 0 \quad \mbox{in} \quad \mathbb{R}^{N}, \\
%\displaystyle \left( |u'|^{p-2} u' \right)' + \frac{N-1}{r} |u'|^{p-2} u' + \lambda r(|u|^{q-1} u)' + \theta u = 0, \quad r > 0,
$$
with parameters $N \geq 1$, $p>2$, $q>1$, $\lambda>0$, and $\theta>0$.
Our main results concern the existence of global radial solutions, which are shown to be strictly positive under suitable assumptions. In addition, we examine the qualitative properties of these solutions and describe their asymptotic profile as $|x|\rightarrow\infty$.

Asymptotic behavior and numerical analysis for a thermoelastic-Bresse system with second sound

2026-01-22T16:29:36+00:00

In this study, we investigate the behavior of a linear one-dimensional thermoelastic Bresse system that incorporates second sound phenomena. We begin by establishing that the system is well-posed and identifying the conditions necessary for it to demonstrate exponential stability, which depend on certain parameters of the system. Our proof utilizes semigroup theory and a hybrid methodology that combines energy techniques with frequency domain analysis. Subsequently, we introduce a nite element approximation for the system and demonstrate that the associated discrete energy decreases over time. Additionally, we derive several a priori error estimates to assess the accuracy of our approximation. Finally, we validate our theoretical ndings by demonstrating that the numerical results align with our established theoretical predictions.

On the existence of renormalized solution for some nonlinear parabolic problems in Musielak-Orlicz spaces

2026-01-22T16:39:15+00:00

In this paper, we will prove in Musielak–Orlicz spaces, the existence of renormalized solution for nonlinear parabolic problems of Leray-Lions type, in the case where the Musielak–Orlicz function \varphi doesn’t satisfy the \Delta_2-condition while the right hand side f belongs to L^1(Q_{T}).

Radial Large Solutions to a Nonlinear Elliptic Problem in $\mathbb{R}^N$: On the Existence and Asymptotic Analysis.

2026-01-22T16:45:03+00:00

We consider the following elliptic equation
\[
\Delta_p u = g(x) h(u) \quad \text{in } \mathbb{R}^N,
\]
where \,
$
\Delta_p u \text{ is the } p-\mbox{Laplacian } \, \text{ with \,} N > p > 2,\quad
$
and the functions $h$ and $g$ satisfy appropriate assumptions. We establish existence results for large solutions and describe their asymptotic behavior as $|x| \to \infty$.

A UML-Driven Framework for Intelligent Collaborative E-Learning

2026-01-22T16:49:39+00:00

The growth of e-learning has introduced significant challenges related to maintaining learner engagement, facilitating effective collaboration, and offering personalized learning paths. Exciting opportunities are presented by artificial intelligence (AI), but its implementation is frequently not based on any solid foundations of pedagogy. This paper addresses this issue by formally describing a pedagogical scenario using the Unified Modeling Language (UML). The UML model not only describes what happened but also suggests a scenario in blueprint format. This framework synergizes human teamwork with a multi-layered substructure of AI-based support, including conversational agents and intelligent tutors. Using class, use, activity, and sequence diagrams, we specify the system's architecture, actors, and their dynamic interactions. The paper proposes that developing a scenario through formal modeling is an important approach that can bridge the gap between pedagogical intentions and technical realization in developing learning environments that are technologically sophisticated and fundamentally based on solid educational principles.

Optimal Strategies for a Controlled SIR Model with Dynamic Reproduction Number and Economic Feedback

2026-01-22T16:54:00+00:00

Mathematical models play a critical role in analyzing infectious disease dynamics, yet existing frameworks often overlook the interplay between epidemiological and socio-economic factors. This study develops the SIRK$\rho$ model, a novel mathematical framework that integrates time-varying transmission dynamics with economic feedback mechanisms. The model incorporates optimal control theory to determine vaccination v and public health intervention c strategies that simultaneously minimize disease prevalence and economic losses while maintaining the effective reproduction number below unity. Through analytical derivation using Pontryagin's Maximum Principle and numerical validation with Hepatitis B (HBV) parameters, we demonstrate the model's effectiveness in outbreak control. Simulation results show that optimized intervention strategies can reduce HBV infections while supporting economic recovery. The SIRK$\rho$ framework provides a comprehensive approach for public health decision-making that balances epidemiological control with economic considerations.

Numerical Approximation of the Timoshenko System with Temperature and Microtemperature Effects in the Absence of Thermal Conductivity

2026-01-22T16:57:12+00:00

This study presents a numerical investigation of a thermoelastic Timoshenko system where dissipation arises exclusively from microtemperature effects, with thermal diffusion neglected. The primary objective is to analyze the system’s energy evolution and exponential decay properties. We start by formulating the problem variationally, employing transformed derivatives to derive a coupled system of four first-order variational equations. A fully discrete numerical scheme is then proposed, and its discrete stability is rigorously established. We also derive a priori error estimates for the method. To support our theoretical analysis, numerical experiments are carried out, confirming the expected decay behavior and accuracy of the solution.

Asymptotic analysis of the processor sharing multi-queue

2026-01-22T17:03:31+00:00

Queueing theory is a key tool for analyzing complex systems like cloud computing and networks. It helps understand how delays, congestion, and resource sharing behave under different regimes. This paper studies the asymptotic behavior of the fluid model solution associated with a network of processor sharing multi-queues. This model is particularly relevant to modern applications where multiple tasks share processing resources. The network consists of $J$ queues, each with a single server, an infinite waiting room and arbitrary interarrival and service time distributions. Under the processor-sharing discipline, all customers present in a queue are served simultaneously. In this system, customers may arrive at a queue either from outside the system or from the previous queue.

Upon completing service at one queue, customers proceed to the next. Our results show that, as time approaches infinity, the fluid model solution converges in the critical regime and grows asymptotically linearly with time in the supercritical regime.

Identification of Initial Condition in Parabolic Convection Diffusion Equation Using Radial Basis Function Partition of Unity

2026-01-22T17:06:46+00:00

The present paper deals with the identification problem of an unknown initial condition in convection diffusion equation. This inverse problem is transformed into a constrained optimization problem. We prove its solution existence. The radial basis function based partition of unity is considered as a discretization method. The obtained matrix system is solved via a robust approach based on preconditioning techniques. Using the quasi Newton algorithm we approach the solution of the optimization problem. At the end, we establish several numerical examples in order to illustrate our theoretical results and the validity of the constructed numerical scheme.

Variational and Numerical Study for Eigenvalues Double-Phase Elliptic Problems with Robin Boundary Conditions

2026-01-22T17:14:25+00:00

In this paper, we study the numerical approximation of the first eigenvalue in double-phase equation subject to Robin boundary conditions. When the parameters of this problem satisfy certain assumptions, particularly regarding the right-hand side function, we establish the existence of the first eigenvalue with its corresponding eigenfunction. The Physics-Informed Neural Networks approach is considered to approximate this solution. At the end, we conduct several numerical test for different exponents and analytical solution.

Eta Quotients of Level $18$ and Weight $1$: Classification and Applications

2026-01-22T17:20:41+00:00

We classify all eta quotients in the space $M_{1}\left(\Gamma_{0}(18), \left(\frac{-3}{*}\right)\right)$ of modular forms 
and explicitly compute their Fourier coefficients, 
where $\left(\frac{d}{*}\right)$ denotes the Legendre–Jacobi–Kronecker symbol, 
viewed as a Dirichlet character modulo $18$ taking values in $\mathbb{Q}$.