Boletim da Sociedade Paranaense de Matemática

Conformal \eta-Ricci-Yamabe solitons on LP-Kenmotsu manifolds

ABDUL HASEEB — 2026-02-17

The aim of the present paper is to study conformal \eta-Ricci-Yamabe solitons (CERYS) on Lorentzian-para Kenmotsu n-manifolds (in brief, (LPK)_n) with certain curvature conditions. Moreover, the existence of CERYS has been proved by constructing a non-trivial example of (LPK)_3.

N-r-Ideals of Commutative Z_2-Graded Ring

Rashid Abu-Dawwas — 2026-02-17

Let $R$ be a commutative ring with nonzero unity $1$. This article introduces and investigates new classes of ideals in $\mathbb{Z}_2$-graded rings, building on the previously established notion of $r$-ideals. Using the function $N: R \to R_0$, defined by $N(x) = x_0^2 - x_1^2$ for $x = x_0 + x_1 \in R$, we define and study $N$-$r$ ideals and semi $N$-$r$-ideals. A proper ideal $I$ is $N$-$r$-ideal if $xy\in I$ implies $N(x) \in I$ or $y\in zd(R)$, while it is semi $N$-$r$-ideal if $x^2 \in I$ implies $N(x) \in I$ or $x\in zd(R)$, where $zd(R)$ is the set of zero divisors of $R$. Fundamental properties of these ideals are explored, including their relationships to existing structures in graded ring theory. These results extend the understanding of ideal theory in the context of $\mathbb{Z}_2$-graded rings and offer new perspectives for future research.

An approach to the Atom Bomb Connectivity index for graphs under transformations fact over pendent paths

Hafiz Muhammad WAQAR Ahmed — 2026-02-17

Graph theory is a dynamic tool for designing and modeling of an interconnection system by a network/graph. The processor nodes behave as the vertices and the connections between them behave as edges of such graph. The best use of system is decided by its topology. To characterize the topological aspects of underlying interconnection networks or graphs one of the most studied graph invariant is atom bomb connectivity index. To define new networks of our own choice the transformation of graph is an important tool. In this paper we will talk about the transformed family of graphs or networks. Let $\Omega$ be the connected graph of n vertices and $ \Omega_n^{k,l} $ be made up by attaching the the $k $ number of pendent paths with the fully connected vertices of the graph $\Omega$. By applying the transformations $ A_{\alpha}$ and $ A_{\alpha}^{\beta} ;$ $0\leq \alpha\leq l-2$ $ 0\leq \beta \leq k-1 $ we get the transformed graphs $ A_{\alpha}(\Omega_n^{k,l}) $ and $A_{\alpha}^{\beta}(\Omega_n^{k,l}) $ respectively. In this paper we derive new inequalities for the graph family $ \Omega_n^{k,l} $ and transformed graphs $ A_{\alpha}(\Omega_n^{k,l} )$ and $A_{\alpha}^{\beta}(\Omega_n^{k,l}) $ which involves $ ABC(\Omega) $. The existence of $ ABC(\Omega) $ made the inequalities more general than all formerly defined for $ ABC $ index. Additionally, we characterize extremal graphs which make the inequalities compact.

Unilateral Elliptic Problems with $L 1$−data in Anisotropic Weighted Sobolev Spaces

Ouidad AZRAIBI — 2026-02-17

In this note, we are interested in some results on the existence of entropy solution for quasilinear anisotropic unilateral elliptic problem of the type:

\begin{equation}\label{prbm1.0.0}
\left\{\begin{array}{ll}
-\sum_{i=1}^{N} \partial^{i} a_{i}(x, u, \nabla u)+\Phi(x, u, \nabla u)+H(x, u, \nabla u)=f \quad\text { in } \Omega \\\\
u \geq \varphi \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\qquad\qquad \text { a.e. in } \Omega,
\end{array}\right.
\end{equation}
where $ f \in L^{1}(\Omega), $ The nonlinear terms $ \Phi(x, s, \nabla u) $ satisfy the sign and growth conditions, and $ H(x, s, \nabla u) $ verifies only the growth conditions.

Newly Discovered Classes of Perfect Functions in Bitopological Spaces: Applications and Conclusions

Ali Atoom — 2026-02-17

Variable degrees of obscurity and immense quantities of information
constitute the characteristics of daily difficulties. Therefore, creating ad-
ditional mathematical methods to address problems is essential. The ideal
tool for this goal is expected to possess the perfect functions, as discussed
in this work. Consequently, in this study, we explore the use of sev-
eral set amplifiers to build perfect functions in bitopological spaces. The
associations between some kinds of pairwise perfect functions and their
traditional topologies are associated with uniformity. Alignment allows
us to investigate the characteristics and actions of traditional topological
ideas by studying sets. We present and evaluate a new class of perfect
functions in bitopological spaces, which we call P-perfect, S-perfect and
B-perfect functions, compact functions in bitoplogical spaces. We addi-
tionally identify the connections among classes of generalized functions
and this new class of perfect functions. Additionally, we demonstrate this
novel concept, explain the related connections identify the prerequisites
for their effective use, and provide instances and counter-examples while
presenting and evaluating the perfect functions that are suggested here.
We look at the images and inverse images of particular topological charac-
teristics to provide new demonstrations regarding each of these functions.
Finally, product theorems associated with these ideas have been found

Nonlinear Dynamics of Allee Effect and Fear in a Delayed Diffusive Predator-Prey Model

Mohamed Hafdane — 2026-02-17

This paper investigates the dynamics of a predator-prey model incorporating both the Allee effect and predator-induced fear, alongside a delay representing the time required for prey to develop anti-predation defenses. The model also integrates diffusion terms to account for species movement and includes harvesting pressure on both populations. We first establish the existence and local stability of a coexistence equilibrium, then analyze the conditions under which a delay-induced Hopf bifurcation occurs. Using center manifold theory and normal form analysis, we characterize the direction, stability, and periodicity of the bifurcating solutions. Numerical simulations are performed to validate the theoretical predictions and reveal rich dynamics, including transitions from stability to periodic oscillations and the emergence of spatial patterns due to asymmetric diffusion. These results underscore the critical role of delayed behavioral responses and spatial heterogeneity in shaping ecological stability.

Semi-analytical approach to study the nonlinear oxygen absorption kinetics and oxygen diffusion in a spherical cell

Mohammad Izadi — 2026-02-17

The phenomenon of oxygen diffusion in spherical cells, governed by Michaelis-Menten uptake kinetics, gives rise to a class of challenging singular boundary value problems. A semi analytical method namely differential transform method is proposed to acquire analytical solution of aforesaid with nonlinear uptake kinetics. The proposed method is found to produce more trustworthy findings when the obtained results are compared with those available in the literature. We discuss the impact of the maximal reaction rate, Michaelis constant, and cell membrane permeability on the dimensionless oxygen concentration, supported by numerical and graphical results. The proposed method ability to handle the strongly nonlinear term make it a promising tool.

On Neutrosophic Ideals of B-algebra

Shuker Khalil — 2026-02-17

This paper presents a comprehensive study of neutrosophic concepts in B-algebras, focusing on several types of neutrosophic ideals including neutrosophic ideals, neutrosophic near ideals, neutrosophic Ns-ideals, neutrosophic power ideals, and neutrosophic near power ideals. The paper establishes and proves multiple results, such as: every neutrosophic B-algebra is both a neutrosophic near ideal and a neutrosophic Ns-ideal; every neutrosophic ideal and every neutrosophic near ideal of a B-algebra is a neutrosophic Ns-ideal; every neutrosophic regular set in a B-algebra is a neutrosophic Ns-ideal; every neutrosophic B-algebra is also a neutrosophic power ideal; every neutrosophic ideal, near ideal, and regular set in a B-algebra is a neutrosophic power ideal; every neutrosophic power ideal is a neutrosophic Ns-power ideal; and similarly, every neutrosophic B-algebra, near ideal, and regular set is a neutrosophic Ns-power ideal. In addition to these findings, the paper explores the structural relationships and properties among these different types of neutrosophic ideals, providing a deeper understanding of their significance within the framework of B-algebras. These results contribute to the theoretical development of neutrosophic algebraic structures and may serve as a foundation for future applications in logic, information systems, and decision-making models involving indeterminacy.

Invariant Muirhead Mean inequality and some Schur convexity properties

NAGARAJA K M — 2026-02-17

The aim of this paper is to define the invariant Muirhead mean, denoted by $^i{M_h}(a,b)$ and to discuss the properties of Schur, Schur geometric and Schur harmonic convexities of this mean. Also, an inequality involving the invariant Muirhead mean is constructed by applying the Newton Raphson formula. These inspections contribute to the theoretical development of generalized symmetric means in Majorization theory and Functional inequalities.

Pillai-Type Equations with Lucas Numbers and S-Unit Solutions

Abdelghani LARHLID — 2026-02-17

In this paper, we investigate the exponential Diophantine equation $
L_n - 5^x 7^y = c,$ where $L_n$ denotes the $n$-th Lucas number. The Lucas sequence is defined by the initial values
$L_0 = 2$, $L_1 = 1$, and the recurrence relation
$L_{n+2} = L_{n+1} + L_n$ for all $n \geq 0$.
We show that when $c = 0$, the equation admits exactly two distinct solutions.
Moreover, for any $c \in \mathbb{N}$, we prove that there is no integer $c$ for which the equation
has at least three distinct solutions $(n, x, y) \in \mathbb{Z}_{\geq 0}^3$.

Enumerative Combinatorics: Recent Advances and Conjectures

Sarita Agarwal — 2026-02-17

We have studied a branch of mathematics primarily concerned with
counting, arrangement, and combination of elements within sets, and their
properties with it's structures.This paper has a good collection of such Com-
binatorics, It deals with discrete objects and has connections with the areas
of mathematics and science.
We have collected all the conjectures and properties related with numbers
and the sets.Further we have discussed Kruskal -Katona and Alon- Tarsi con-
jecture connected with Graph Theory and their interdisciplinary approach.

Stability of the sine addition-subtraction law

Karim Farhat — 2026-02-18

In this paper, we investigate the stability of the functional equation

f(xy) = f(x)g(y) + \beta\, g(x)f(y) + \gamma\, f(x)f(y), \qquad x, y \in S,

where $S$ is a semigroup, $f, g : S \to \mathbb{C}$ are two unknown functions, $\beta \in \mathbb{C} \setminus \{0\}$ and $\gamma \in \mathbb{C}$ are fixed constants. We extend our analysis to the functional equation

f(x\sigma(y)) = f(x)g(y) + \beta\, g(x)f(y) + \gamma\, f(x)f(y), \qquad x, y \in S,

where $\sigma : S \to S$ is an involutive automorphism.

Partial Prime Exposure Attack on the Cubic Pell RSA Cryptosystem.

Mostafa CHAKER — 2026-02-18

A recent contribution by Rahmani and Nitaj (AfricaCrypt 2025) investigates the cryptanalysis of an RSA-inspired scheme derived from the cubic Pell curve $t_1^3 + f t_2^3 + f^2 t_3^3 - 3 f t_1 t_2 t_3 \equiv 1 \pmod{\mathtt{N}}$, where $\mathtt{N} = \mathtt{p}\mathtt{q}$ is a standard RSA modulus and the public–private exponent pair satisfies $ed-1 \equiv 0 \pmod{(\mathtt{p}-1)^2 (\mathtt{q}-1)^2}$. In this paper, we revisit their attack showing that when an approximation of one prime factor is known, the scheme becomes significantly more vulnerable. Using a variant of Coppersmith's method, one can factor $\mathtt{N}$ in polynomial time under explicit bounds, which improve previous results.

INFINITE FAMILIES OF SEXTIC NUMBER FIELDS WITH ALL POSSIBLE INDICES

Hamid Bouaouina — 2026-02-18

For each rational prime $p\in\{2,3,5\}$, we construct infinite families of sextic number fields $K$ such that the $p$-adic valuation of the index $i(K)$ satisfies $\nu_p(i(K))=\nu_p$, for every possible positive integer $\nu_p$. We illustrate our results by some computational examples.

ANALYTICAL AND GEOMETRICAL PROPERTIES OF NEW CLASS OF UNIVALENT FUNCTIONS ASSOCIATED WITH THE FRACTIONAL DERIVATIVE OPERATOR

Kirti Pal — 2026-02-18

In this work, we introduce and analyze a new subclass $\mathcal{F}_{0,z}^\varrho(\varepsilon, \lambda,\eta,\mu)$ of analytic univalent functions related to the fractional derivative operator within the open unit disk
$ \mathbb{U}=\{z:z\in \mathbb{C},|z|<1\}$. We investigate coefficient estimates, distortion bounds and growth theorems, convex set, radius of convexity, radius of stralikeness, arithmetic mean, weighted mean, and also we establish some basic results like extreme points, Hadamard product, closure theorem for the functions in the class.

Contribution to the Study of Linear Cryptosystems: An Analysis of Non-Invertible Matrix-Based Techniques Beyond the Hill Cipherer

Najat RAFI — 2026-02-18

Linear cryptosystem, such as the Hill cipher, are foundational in symmetric-key encryption but are limited by the requirement of invertible key matrices, reducing key space and security. This study investigates the use of non-invertible matrices to enhance cryptographic complexity and resilience. We analyze the mathematical principles, design optimized encryption and decryption algorithms, and evaluate their performance against known attacks. Experimental results show that non-invertible matrix-based methods provide stronger data protection than conventional approaches while remaining practically feasible. This proposed symmetric encryption algorithm advances matrix-based cryptography, offering a robust framework for secure communication and guiding future cryptosystem development.

Integral Kannappan-cosine addition law on semigroups

Omar Ajebbar — 2026-02-18

Let $S$ be a semigroup, $\sigma:S \longrightarrow S$ be an involutive automorphism, $\mu$ be a complex measure that is a linear combination of Dirac measures and $\alpha \in \mathbb{C}$.

We determine the complex-valued solutions of the following integral Kannappan-cosine addition law with an additional term $$\int_{S}g(x\sigma(y)t) d\mu(t)=g(x)g(y)-f(x)f(y)+\alpha \int_{S}f(x\sigma(y)t) d\mu(t) ,\; x,y \in S.$$

As application we solve two functional equations that have not been studied until now.

The continuous solutions on topological semigroups are found.