Gulf Journal of Mathematics (ISSN 2309-4966) is a well-established, international, open access mathematics journal that publishes carefully refereed research articles in all mainstream branches of pure and applied mathematics that make a significant contribution to the existing literature. The current acceptance rate for publication is 14%. GJoM is indexed in Scopus, SJR, MathSciNet, Zentralblatt MATH, EBSCO, and Crossref databases.
Euler–Maruyama Schemes for G-Stochastic Volterra Equations
We study Euler–Maruyama approximations for stochastic Volterra integral equations driven by G-Brownian motion. The proposed explicit Volterra scheme discretizes the drift term, the quadratic variation term and the stochastic integral term. Under standard Lipschitz, growth and time-regularity assumptions, we prove uniform moment bounds and strong mean-square convergence under sublinear expectation. The convergence rate reflects the time regularity of the Volterra kernels and recovers the classical order one–half in the root mean square sense in the Lipschitz case. A numerical experiment under volatility uncertainty illustrates the theory.
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Adaptive Numerical Quenching for a Nonlocal Neumann Problem with Nonhomogeneous Flux
We analyze an adaptive approximation of quenching for a nonlocal parabolic problem with a nonhomogeneous Neumann-type flux. The flux is treated as a nonnegative exterior input, while the singular source is controlled by a mesh-scaled barrier time step. The discrete scheme preserves the admissible range, quenches in finite accumulated time, and satisfies upper and lower estimates for the numerical quenching time. We also prove convergence before quenching and convergence of fixed-level detection times. The numerical tests verify the invariant bound, the monotone influence of the flux intensity, and the second-order spatial behavior of the detected time.
On 2-Nil Primary Submodules of Modules Over Commutative Rings
We introduce and examine the class of 2-nil primary submodules for modules over commutative rings. This class is obtained by combining the nilpotency option occurring in 2-nil submodules with the radical condition that appears in primary-type submodule theory. Several equivalent descriptions are obtained; these descriptions are expressed by residual submodules, by ideal products, and by products of submodules. In the setting of multiplication modules, we compare the new class with primary submodules and connect it with 2-nil primary ideals for faithful finitely generated multiplication modules. We then discuss how the property is transferred through module homomorphisms, quotient modules, intersections, localization, idealization, direct products, and amalgamated algebra constructions. The examples clarify that many of the implications established here are not reversible without extra assumptions. The paper therefore gives a systematic module-theoretic treatment of a condition governed simultaneously by nilpotent and radical alternatives.
The Hilbert function of a homogeneous ideal is determined by its associated lexsegment ideal, while the Gotzmann property ensures that the ideal and its lexsegment ideal have the same number of minimal generators. For homogeneous monomial ideals with fewer minimal generators than the number of variables in the polynomial ring, the classes of critical monomial ideals and Gotzmann ideals coincide. In this article, we determine the Hilbert depth of this class of monomial ideals. As an application, we compute the Hilbert depth of the edge ideal of a star graph and propose it as an algebraic graph invariant.
Ishikawa-Mann-Fibonacci Best Proximity Points for Cyclic Relatively Nonexpansive Mappings
We study best proximity point convergence for cyclic relatively nonexpansive mappings in uniformly convex Banach spaces. We analyze two Mann-type iterations: the n-powered scheme and its Fibonacci-accelerated variant. Under suitable conditions, we establish strong and weak convergence theorems. Our main result shows that the n-powered iteration converges strongly to a best proximity point, while the Fibonacci variant achieves doubly exponential convergence. An application to optimal traffic signal timing is provided.
Blow-Up Diffusion in Singular Elliptic Problems
In this paper, we study a class of nonlinear singular elliptic problems involving a diffusion operator with blow-up behaviour as the solution approaches a finite threshold value. The combination of the singular lower-order term, the blow-up diffusion, and low-regularity data gives rise to significant analytical difficulties. Using the framework of renormalized solutions, we establish the existence of solutions under suitable assumptions on the nonlinear operator and the singular term. We also derive some regularity properties of the obtained solutions. These results contribute to the study of singular elliptic problems with blow-up diffusion and low-regularity data.
Multivariate Extreme Cvar Based on the D-Norm
Measuring systemic risk is challenging in the presence of extreme events and tail dependence. This paper proposes an extension of Conditional Value-at-Risk based on Extreme Value Theory and D-norms. A multivariate excess aggregation variable is introduced through a D-norm, and its convergence to a Generalized Pareto Distribution is established. This framework preserves dependence structures and provides an analytical decomposition of systemic risk. Finally, a simulation study illustrates the theoretical results and the practical implementation of the proposed approach.
Kirchhoff-Type Problems on Compact Manifolds
In this paper, we consider p-biharmonic Kirchhoff-type equations on compact manifolds with Navier boundary conditions. Using the theory of Riemannian compact Sobolev spaces and the Browder-Minty theorem, we show the existence and uniqueness of a weak solution to this problem.
Structure of Cayley Graph Over Generalized Quaternion Group
We investigate the structure of undirected Cayley graphs on generalized quaternion groups constructed via inverse-closed connection sets excluding the identity element. Through an analysis of small valencies from one to four, we generalize the structural properties to arbitrary valency. We establish a complete classification theorem, proving that all resulting Cayley graphs are isomorphic to circulant graphs, regular bipartite graphs, or the edge-disjoint union of these two structures. These findings provide a definitive characterization of Cayley graph structures on generalized quaternion groups and establish their connectivity and algebraic properties.
Simplified Lyapunov Stability for Chaotic Systems
In this paper, based on Lyapunov stability theory, sufficient conditions are established for the local stabilization of the equilibrium point of a class of chaotic nonlinear systems. The proposed conditions are expressed as explicit algebraic inequalities, making them easy to verify without computing the eigenvalues of the Jacobian matrix. The results support the design of control strategies for suppressing chaos and stabilizing system trajectories. Their effectiveness is demonstrated through several three-, four-, and five-dimensional chaotic systems, with numerical simulations confirming the theoretical results.
Difference Schemes For Convection-diffusion Equations With Generalized Solutions
High-accuracy difference schemes are proposed for a multidimensional convection-diffusion equation in classes of generalized solutions. The piecewise polynomial finite element method is used to approximate spatial variables and the resulting Cauchy problem for a system of ordinary differential equations. As a result, third-order accuracy schemes with respect to spatial variables and fourth-order accuracy schemes with respect to time are obtained. Stability conditions and a priori estimates for the solution of the scheme in classes of nonsmooth solutions are obtained. Based on a special technique, convergence theorems were established in a weaker functional norm. Convergence and accuracy theorems are proven.
Time-Optimal Control of the Heat Conduction Equation
We study a time-optimal control problem for the heat conduction equation with a general elliptic operator and Robin boundary condition. The goal is to drive the weighted average temperature to a prescribed value in minimal time using an admissible control subject to a uniform bound. Using spectral decomposition and the Green function, the problem is reduced to a Volterra integral equation. Analyzing its kernel via the operator's eigenpairs, we derive the maximal attainable average temperature. For any target below this threshold, we prove the existence of a unique minimal time and establish its asymptotic expansion, revealing logarithmic growth governed by the principal eigenvalue.
On Some Classes of Modules with Generalized Extending Condition
In this work, we present purely generalized extending (PGCS) modules. A module D is called PGCS if, for each submodule E of D, there exists a pure submodule X of D containing E for which fracXE is singular. This idea extends purely (generalized) extending modules. Moreover, PGCS modules are closed under direct summands. Furthermore, we establish the strongly purely generalized extending modules (SPGCS), where the pure submodule X is required to be fully invariant. This new structure lies properly between strongly extending modules and PGCS modules. We investigate many algebraic properties of these modules and characterize their relationships with other well-known classes of modules.
New Bounds for Gurland-Type Ratios in Terms of the Generalized Nielsen's Beta Function
We establish new bounds for Gurland-Type Ratios in terms of the generalized Nielsen's beta function βₕ(y). To achieve this, we investigate the complete monotonicity of certain functions involving βₕ(y) and the Gamma function Γ(y). Furthermore, we derive some sharp bounds that are superior to some recent results.
Bernstein and Jackson Inequalities for the Quaternionic Dunkl Transform
We study approximation theory for the quaternionic Dunkl transform (QDT) on ℝ². We establish Bernstein-type inequalities and Jackson direct theorems, relating the best approximation Q_ν(f)₂,Q to moduli of smoothness. We also prove inverse theorems showing that summability of Qₙ(f)₂,Q implies Sobolev regularity f ∈ Wₖᵐ(ℝ²,mathbbH). These results extend classical Jackson-Bernstein theory to the quaternionic Dunkl setting and provide a bridge between spectral approximation and Dunkl-Sobolev smoothness.
Wolff Potential and Birkhoff's Ergodic Theorem in Grand Variable Herz Spaces
In this article, we establish the boundedness of the Wolff potential operator and Birkhoff's ergodic theorem in grand variable Herz spaces. These concepts were previously studied in the setting of Lebesgue spaces. Our results extend and refine the existing literature by considering a more general functional framework, thereby providing broader applicability and sharper estimates than the previously known results.
Existence Of Solutions For A Class Of Multidimensional Nonlinear Integral Equations
In this paper, we construct a measure of noncompactness adapted to multidimensional Lebesgue spaces over product domains. Using this measure together with a Darbo-type fixed point theorem, we prove the existence of at least one solution in Lebesgue spaces for a broad class of multidimensional nonlinear integral equations. This operational framework includes the classical mixed Volterra-Fredholm integral equation as a special case.
Existence Theory for Nonlinear Langevin-Type Fractional Differential Equations
This paper is devoted to establishing the solvability of a class of nonlinear Langevin fractional differential equations subject to boundary conditions and formulated in terms of the generalized Caputo proportional fractional derivative. The study is conducted in a general Banach space setting. By combining the Kuratowski measure of noncompactness with Mönch’s fixed point theorem, we derive sufficient criteria ensuring the existence of solutions. A concrete example is presented at the end to confirm the effectiveness and practical relevance of the obtained results.
generalized Caputo proportional fractional derivative
Robust Event-Triggered Control for Networked Fuzzy Systems
This paper presents colorbluea Neural Gradient Descent-based Event-Triggered (NGDET) control design colorbluefor Networked Control Systems (NCS) based on Takagi-Sugeno (T-S) fuzzy systems affected by measurement noise, external disturbances, and network-induced delays. The main objective is to improve communication efficiency while preserving closed-loop stability and robust performance. To this end, an adaptive event-triggering mechanism is proposed, where the triggering threshold is updated online through a colorblueneural gradient descent technique according to the system behavior and communication network constraints. Then, colorblue by constructing an appropriate Lyapunov–Krasovskii Functional (LKF), colorblue relaxed Linear Matrix Inequality (LMI) conditions are developed for the design of a networked controller involving a sampled data Parallel Distributed Compensation (PDC) approach ensuring asymptotic stability colorblue and robustness of NCS closed-loop dynamic. The simulation results demonstrate its effectiveness by enlarging the admissible delay bound, reducing packet transmissions, and achieving less conservative performance compared to existing methods from the literature, while preserving robustness against disturbances and measurement noise. newline noindent Keywords: Networked control systems, Takagi–Sugeno fuzzy systems, Event-triggered control, colorblue Neural gradient descent, measurement noise.
Sharp Threshold Dynamics for a Stochastic SIRS Model with General Nonlinear Incidence
This paper studies the dynamics of a stochastic SIRS epidemic model with a general nonlinear incidence rate. The model incorporates environmental noise through white noise perturbations. We first establish basic properties of the total population, including boundedness, recurrence, and the existence of a stationary distribution. Then, under suitable hypotheses on the incidence function, we derive conditions for disease extinction and persistence. The theoretical results are illustrated by a corollary providing explicit lower bounds for the time averages of the infected, recovered, and susceptible compartments. Numerical simulations are used to confirm the theoretical threshold dynamics and demonstrate the practical applicability of the model.
A Study of (α,β)-Intuitionistic Fuzzy Orders
In this work, we establish some properties of (α, β)-intuitionistic fuzzy order relations. Firstly, we prove that the infimum of any family of (mathbbα, mathbbβ)-intuitionistic fuzzy orders defined on a non-empty set mathcal X is also an (α, β)-intuitionistic fuzzy order on it. Secondly, we present a characterization of a class of total (α, β)-intuitionistic fuzzy order relations compatibles with the usual algebraic structures on the real line ℝ.
On Some Cohomological Invariants Associated to a Reduced Normal Crossing Divisor in the Affine Plane
This work focuses on the study of cohomological invariants associated with a reduced normal crossing divisor in the affine plane. After determining the classical Poisson cohomology groups and the logarithmic Poisson cohomology groups along the associated ideal, we establish that, for a Poisson bivector attached to the log-symplectic structure under consideration, there exists an isomorphism of vector bundles between the module of logarithmic derivations and the module of logarithmic differential 1-forms relative to the ideal. This allows us to construct a cochain morphism related to the de Rham complex and subsequently compute the de Rham cohomology groups in order to carry out a comparison between the resulting cohomology groups.
Hermite Structure And Spectral Properties Of Symmetric Gated Activation Functions
We study a family of self-gated functions where the gate satisfies a reflection symmetry, and this forces a rigid Hermite structure. We then specialize the abstract theory to the Swish and GELU families by proving strict monotonicity of the spectral ratio, and deriving both the small- and large-parameter asymptotic expansions. For GELU specifically, we also derive the exact closed form for the spectral ratio. We then enrich the symmetric family by adding odd perturbations and show that the leading Hermite mode becomes continuously controllable.
Coupled Mkdv with Memory: Existence and Exponential Decay
We investigate a coupled system of modified Korteweg-de Vries (MKdV) equations with memory effects through convolution integrals with relaxation kernels. Under suitable assumptions on the memory kernel positivity, monotonicity, and exponential decay we establish two main results. First, using Faedo-Galerkin approximations together with a priori estimates and compactness arguments, we prove local existence of weak solutions. Second, we construct an appropriate Lyapunov functional to exploit the dissipative nature of the memory terms, from which we derive an exponential decay estimate for the total energy. Our analysis shows how memory influences both solvability and long-term dynamics in this coupled dispersive system.
Opial Inequalities on Timescales Via Fractional Conformable Calculus
In this work, we prove a new class of weighted inequalities of Opial type via conformable fractional calculus on timescales. The inequalities obtained unify and generalize a number of results previously established in the literature, while also provide significant improvements. Furthermore, we apply our findings to determine the distance between successive zeros of solutions to conformable fractional dynamic equations with damping terms. In addition, we obtain lower bounds for the distance between a zero of a solution and a zero of its derivative in the conformable fractional sense. our results provide suitable conditions for the right and left disfocality, as well as the disconjugacy, of such conformable fractional dynamic equations on compact intervals timescales.
Location and Perturbation Analysis of Quaternionic Matrix Polynomials
Localization theorems for the right eigenvalues of quaternionic matrix polynomials are given. A Bauer–Fike–type theorem for the right eigenvalues of diagonalizable quaternionic matrices is derived using the real adjoint matrix representation. A perturbation analysis for the right eigenvalues of quaternionic matrix polynomials is presented. The condition number of a simple right eigenvalue of a scalar polynomial is obtained. Numerical examples are provided to illustrate the results.
A Doubly Critical Elliptic Problem with Submanifold Singularities
Let N ≥ 4 and Ω be a bounded domain in ℝᴺ containing a smooth closed submanifold Σ of dimension k with 2 ≤ k ≤ N-2. We study the existence of positive solutions to a doubly critical Hardy–Sobolev elliptic equation whose singular weights are powers of the distance to Σ, with two distinct critical exponents 2s₁^* and 2s₂^* corresponding to the parameters s₁ and s₂ with 0 ≤ s₂ < s₁ < 2. Using the mountain pass lemma and asymptotic expansions of the energy functional in Fermi coordinates, we prove that a positive solution exists whenever a geometric quantity involving the mean curvature H, the scalar curvature Rg of Σ, and the potential h satisfies a negativity condition at some point of Σ.
Time-Fractional Wave-Diffusion-Wave Equation: Inverse Problem in a Cylindrical Domain
This paper addresses the inverse source problem for a mixed fractional wave-diffusion-wave equation posed in a cylindrical domain. The governing equation involves a time-dependent variable-order fractional derivative, which enables the model to effectively capture temporal transitions between wave-like and diffusive behaviors. The solution is constructed in the form of a Fourier-Bessel series. By employing the method of separation of variables together with fundamental properties of Bessel functions, we analyze the uniform convergence of the resulting infinite series. This analysis ultimately leads to a rigorous proof of the existence of a solution
Regularized Prabhakar fractional derivative
Rotation-Coordinate Density Estimation on Spheres
We study a coordinate-based kernel density estimator for spherical directional data. The construction represents each point of the sphere by a selected rotation and performs smoothing with a distance inherited from the rotation group. The method is formulated as a section-based procedure, not as a fully intrinsic rotation-invariant estimator. We derive local bias, variance, mean squared error and consistency properties away from the polar coordinate singularities. To address numerical instability near the poles, we add a locally normalized pole-corrected version inspired by boundary correction techniques. Simulations show that this correction substantially reduces polar errors.
Square-Free Harmonic Numbers: Basic Properties
We study a variant of harmonic numbers defined by restricting the summation to square?free indices, denoted by Hₙˢq. Using the Mobius function to detect square?free integers, we establish a relation between Hₙˢq and the ordinary harmonic numbers Hₙ. We derive an asymptotic expansion for Hₙˢq with an explicit constant involving the Riemann zeta function. Moreover, we obtain the ordinary generating function, a convolution identity, and the arithmetic mean of these numbers.