Boletim da Sociedade Paranaense de Matemática

Reversible cyclic Codes Over $\mathbb F_q+ u \mathbb F_q+ u^2\mathbb F_q+ u^3\mathbb F_q$ and applications to DNA codes

Muzibur Rahman Mozumder — 2025-12-29

In this work, we investigate the structure of reversible and reversible-complement cyclic codes of length $n$ over the ring $\mathbf{R}=\mathbb F_q+ u F_q+ u^2\mathbb F_q+ u^3\mathbb F_q$, $ u^4=0$, when $n\text{ and }q$ are coprime. We give the necessary and sufficient conditions for the cyclic codes of length $n$ over $\mathbf{R}$ to be reversible and reversible-compliment cyclic codes, when $(n,q)=1$ over $\mathbf{R}$. We also studied the dual cyclic codes over $\mathbf{R}$ when $(n,q)=1$ and obtain the reversibility condition for dual cyclic codes. Additionally, we explore cyclic DNA codes over nucleotide 4-base pair. First, we establish a one-to-one correspondence between $\mathbf{R_{1}}$, where $\mathbf{R_{1}}=\mathbb F_4+ u F_4+ u^2\mathbb F_4+ u^3\mathbb F_4$, $ u^4=0$ and 4-mers and then cyclic DNA codes constructed as the images of reversible-complement cyclic codes over $\mathbf{R_{1}}$.

Some Families of the Eulerian Integrals Involving Generalized Hypergeometric Functions

Shoukat Ali — 2025-12-29

In this paper, we established a family of unified Eulerian integrals involving generalized hypergeometric function by employing generalized summation theorems for the series $_{3}F_{2}$ such as generalized Watson's theorem, generalized Dixon's theorem and generalized Whipple's theorem. Various special cases are deduced as consequences of our main theorems.

Optimal Control Strategies for Mitigating Cardiovascular and Type 2 Diabetes Risks in Psoriasis and Psoriatic Arthritis : A Mathematical Model and Numerical Simulations

Oumaima Bouaoultaine — 2025-12-29

In this study, we investigate a novel mathematical model that describes the dynamics of psoriasis and psoriatic arthritis. The goal of this paper is to mitigate the risks associated with psoriasis and psoriatic arthritis on cardiovascular health and type 2 diabetes through control strategies involving three variables: awareness programs for patients with psoriasis or psoriatic arthritis, medical follow-up, and promoting a healthy and regular lifestyle. We demonstrate the existence of optimal controls and describe them using states and adjoint functions, primarily based on Pontryagin's maximum principle. Numerical simulations for various scenarios validate the effectiveness of the optimization approach.

Spray-Invariant Sets in Infinite-Dimensional Manifolds

Kaveh Eftekharinasab — 2025-12-29

We introduce the concept of spray-invariant sets on infinite-dimensional manifolds, where any geodesic of a spray starting in the set stays within it for its entire domain. These sets, possibly including singular spaces such as stratified spaces, exhibit different geometric properties depending on their regularity: Sets that are not differentiable submanifolds may show sensitive dependence, for example, on parametrization, whereas for differentiable submanifolds invariance is preserved under reparametrization.

This framework offers a broader perspective on geodesic preservation than the rigid notion of totally geodesic submanifolds, with examples arising naturally even in simple settings, such as linear spaces equipped with flat sprays.

FINITE INTEGRAL INVOLVING INCOMPLETE ALEPH-FUNCTIONS, GENERALIZED EXTENDED HURWITZ'S ZETA FUNCTION, INCOMPLETE GAMMA FUNCTION, AND ELLIPTIC INTEGRALS OF THE FIRST KIND

Dinesh Kumar — 2025-12-29

In this study, we derive a general finite integral that incorporates the generalized
Hurwitz-Lerch Zeta function of two variables, the incomplete Gamma function,
elliptic integrals of the first kind, and incomplete Aleph-functions. The paper concludes
with a discussion of several corollaries and observations.

MULTIPLICATIVE b-GENERALIZED DERIVATIONS ON NON-COMMUTATIVE RINGS

Faiza Shujat — 2025-12-29

In the current investigation, our primary objective is to find the structure of b-generalized derivations on Martindale ring of quotients. Infact, we establish that if b-generalized derivations satisfy some differential identities on non-commutative prime ring, then it will be a trivial (zero) derivation. Our proof contains an interesting approach in view of the existing classical theory about ordinary derivation presented in [12].

Employing Non-Binary Simplex Codes Over $\mathbb{F}_{p}(\mathbb{F}_{p}+v\mathbb{F}_{p})$ within Various Applications

Karima Chatouh — 2025-12-29

This article examines simplex codes categorized as types $\alpha$ and $\beta$ over $\mathbb{F}_{p}(\mathbb{F}_{p}+v\mathbb{F}_{p})$, along with exploring the $p$-array Gray representations of simplex codes of types $\alpha$ and $\beta$ over the same field. Additionally, the article investigates the covering radius of simplex codes of types $\alpha$ and $\beta$ over $\mathbb{F}_{p}(\mathbb{F}_{p}+v\mathbb{F}_{p})$. Lastly, the article delves into the construction of secret sharing schemes utilizing simplex codes over the finite field $\mathbb{F}_{p}$.

A Robust Iterative Solver for LCPs with Singularity Detection and Performance Guarantees

Yamna ACHIK — 2025-12-29

In this study, we present a novel algorithm for solving Linear Complementarity Problems (LCPs), a class of optimization problems encountered across numerous application domains. Our approach distinguishes itself by its ability to intelligently adapt to the challenges posed by singular and ill-conditioned matrices, which often hinder the performance of traditional LCP solvers.

At the core of our algorithm is a singularity detection mechanism, analyzing matrix properties to determine the most appropriate solution method. For well-conditioned matrices, we leverage the precision of direct methods, while for singular matrices, we employ robust techniques such as the pseudo-inverse or Singular Value Decomposition (SVD).

Furthermore, we introduce an optimized preconditioning strategy, accelerating the convergence of iterative methods, particularly for large-scale LCPs. This adaptive approach, combined with intelligent handling of cases with no solution, ensures increased robustness and efficiency.

Experimental results demonstrate the superiority of our algorithm compared to conventional LCP solvers, particularly for ill-conditioned or singular problems. This advancement opens new perspectives for solving complex LCP problems in diverse application domains, including engineering, finance, and machine learning.

A Novel Computational Technique based on Python and MATLAB for Structural Analysis of Zero Divisor Graphs in Modular Rings

Nasir Ali — 2025-12-29

This paper presents advanced computational methods for visualizing zero-divisor graphs (ZDGs) of the ring of integers modulo (denoted as using MATLAB and Python. In a finite commutative ring with unity, elements and are zero divisors if The set of zero divisors forms the basis for constructing ZDGs, which are pivotal for uncovering the algebraic characteristics of . Our study focuses on developing and implementing novel algorithms in MATLAB 2020 and Python 3.12.3 for constructing and analyzing the ZDGs for efficiently. We provide a comprehensive methodology, theoretical foundation, and proofs of correctness for these algorithms. The paper also includes a detailed complexity analysis, demonstrating the computational efficiency and validity of our methods. By benchmarking our algorithms against existing approaches, we show their superior performance in terms of both speed and accuracy. This research not only enhances the understanding of algebraic structures through effective visualization but also offers a significant improvement over previous methods, paving the way for further exploration and application of ZDGs in algebraic research.

Lyapunov Stability of Singular Integral Equations via Fourier Methods

Rahim Shah — 2025-12-30

In this paper, we utilize the Fourier transform to analyze the Lyapunov stability of singular integral equations. By applying this transform, we simplify the stability analysis by converting the singular integral equation into an algebraic form. Additionally, the Fourier transform is used to derive sufficient conditions for the Lyapunov stability of singular integral equations with impulses. Our results provide enhancements over traditional methods, offering a new perspective on the stability evaluation of these equations. The approach is illustrated through several theorems, and lemmas and examples.

Almost sure linear independence of absolutely continuous Hilbert space-valued random vectors with respect to a special class of Hilbert space probability measures

Nizar El Idrissi — 2025-12-30

This note examines the implications of randomly selecting vectors from an infinite-dimensional Hilbert space on linear independence, assuming that for all k, the first k vectors follow an absolutely continuous distribution with respect to a probability measure. It demonstrates that no constraints on the random dimension of their span are necessary, provided that all strict affine subspaces are considered negligible with respect to the Hilbert space probability measure.

On Smirnov-type inequalities for polynomials

BILAL DAR — 2025-12-30

In this paper, we established certain results of Bernstein-type inequalities and provide an interesting refinement of an inequality concerning the modified version of the Smirnov operator.

Hahn Wijsman Sequence Space

Dr. Ishfaq Ahmad Malik — 2025-12-30

In this paper, we introduce and examine the Hahn-Wijsman sequence space $ h(W^u )$ , a generalized sequence space defined for a metric space $(X,d)$ and a positive sequence $\{{u_k }\}$, where a sequence $\{{x_k }\}\in h(W^u )\; if \;\sum_{k=1}|x_k | u_k< \infty$. We investigate its structural characteristics by establishing that it forms a vector space and construct an appropriate norm under which the space $h(W^u )$ becomes a normed linear space. We further demonstrate that it is complete and hence a Banach space. The dual space is characterized by bounded linear functionals, and we explore isomorphic relationships between the Hahn-Wijsman space and classical sequence spaces such as$ c_0 (p), c(p)\;and \;\ell(p)$. These results not only provide a deeper understanding of the topological and geometric aspects of the Hahn Wijsman space but also position it as a potential tool for further analysis in functional spaces and contribute to the ongoing development of sequence space theory.

Investigation of Molecular Descriptors and Thermodynamic Properties of Anti-Tuberculosis Drugs Used in Tuberculosis Disease through QSPR Analysis

Muhammad Abid — 2025-12-30

Tuberculosis remains a significant public health challenge due to its widespread prevalence and severe health implications. The development of effective therapeutic agents is crucial for combating this disease. This study focuses on analyzing the structural and physicochemical characteristics of 13 key anti-tuberculosis drugs, including Isoniazid, Levofloxacin, Cycloserine, Ciprofloxacin, Pyrazinamide, Amikacin, 4-Aminosalicylic Acid, Bedaquiline, Streptomycin, Ethionamide, Ofloxacin, Ethambutol, and Kanamycin. Using a computational approach, distance-based topological descriptors, specifically the Mostar index, were investigated to explore their potential as predictors of these drugs' physicochemical properties. The methodology involved calculating the Mostar index and performing Quantitative Structure-Property Relationship (QSPR) analysis to evaluate its correlation with critical parameters such as melting point and molar mass. The results demonstrated a strong correlation (melting point $ R > 0.990 $, molar mass $ R > 0.970 $), highlighting the predictive power of the Mostar index. These findings provide valuable insights into the structural properties of anti-tuberculosis medications and support the development of novel therapeutic agents leveraging the Mostar index for enhanced drug design

Newrational-type contractions and their applications in b-metric spaces

Rahim Shah — 2025-12-30

In this paper, we analyze new rational-type contractions in the context of b-metric spaces, establish a theoretical foundation for these contractions, and explore their applications in Fredholm integral inclusions. The paper also provides examples to verify the validity of the results.

A Study on Face Index of Chemical Graph Structures and Their Derivatives

Ali Raza — 2025-12-30

A topological index is a numeric parameter that may describe the biological, physical and chemical properties which are contingent on the structural behaviour of different chemical compounds. In the vast class of topological indices: one of the index is, the face index which has been recently introduced and it assists in predicting the bond energies, the intermolecular forces, the boiling points and densities of different chemical compounds. This paper derives the formulae for the face index of the tadpole, the ladder and the wheel graphs, their subdivision graphs and the line graphs of their subdivision graphs. And briefly analysis their face indices trends.

Fixed Point Theorems for Generalized Contractions in b-Metric Spaces with Applications

Rahim Shah — 2025-12-30

In this paper, we develop new fixed point results for generalized contractions in the framework of b-metric spaces. Several illustrative examples and a concrete application to integral equations are provided to demonstrate the utility and scope of the proposed theorems.

Exploring Structural Dynamics: Characterization of the Non-Rigid Group of Diethyl Ether Compound (𝑪_𝟐 𝑯_𝟓 )_𝟐 𝑶

Enoch Suleiman — 2025-12-30

This research paper unveils the character table of the non-rigid Diethyl ether compound, . The computed group is determined to be of order 36, comprising 36 operations classified into 18 conjugacy classes and irreducible representations. This exploration is done for the first time and provides novel insights into the structural dynamics of Diethyl ether, shedding light on its non-rigid characteristics.

On p-Symmetric Rings

H M Imdadul Hoque — 2025-12-30

This article embodies the notion of p-symmetric rings using the concept of non-
zero potent elements in a ring. It is proved that R is a p-symmetric ring if and only if
pn−1Rpn−1 is a symmetric ring and pn−1 is left semicentral. Moreover, p-symmetric
rings in terms of upper triangular matrix rings and left min-p-abel rings have been
characterized. Furthermore, we introduce strongly p-symmetric ring and also pro-
vide a characterization of strongly p-symmetric rings in terms of strongly left min-
p-abel rings. In particular, it is proved that R is a strongly left min-p-abel ring if and
only if R is a strongly p-symmetric ring for each pn−1 ∈ MPl (R). Further, the notion
of p-reduced ring as a subclass of reduced ring is derived and right p-reduced rings
are p-symmetric rings have been established.

Anisotropic elliptic $\overrightarrow{p}(\cdot)-$Laplacian systems

Prof. Naceri Mokhtar — 2025-12-30

Within the framework of this paper, we aim to prove the existence of distributional solutions in the space $\mathring{W}^{1,\overrightarrow{p}(\cdot)}(\Omega,\mathbb{R}^m)$ for a new kind of nonlinear elliptic $\overrightarrow{p}(\cdot)-$anisotropic Laplace systems, such that its right-hand side is a nonlinearity connecting the solution $u$ and given functions $\psi_i\in L^{p_i(\cdot)}(\Omega,\mathbb{R}^m),\,i=1,\ldots,N.$

Security Analysis of Lightweight Block Ciphers ANU-II and LiCi

Dheeraj Singh — 2025-12-30

ANU-II is a lightweight encryption design with 64-bit block and 80/128 bits key size which is proposed
as smallest lightweight design. LiCi is a low-power lightweight block cipher with a 64-bit block and
128-bit key size. Designers have provided a proof of security for differential attack on these ciphers. In
this paper, we analyse the security of ANU-II and LiCi to provide the accurate bounds for differential
attack. We apply branch-and-bound-based algorithm to search the differential trail. We construct
the optimal differential trails for 12 rounds and RK differential trails for 13 rounds of ANU-II that
require 262 and 263 chosen plaintext pairs respectively. We also search for the differential trail of
LiCi up to 11 rounds with probability 2−67 to give accurate security bounds for this attack. Our
results are better than the security bounds claimed by the designers. Our analysis provides the best
results available so far for a differential attack on ANU-II and differential cryptanalysis of LiCi.