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Axioms
Volume 14
Issue 5
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Axioms, Volume 14, Issue 5 (May 2025) – 70 articles

Cover Story (view full-size image): This Issue features a groundbreaking study that bridges kinetic theory, optimal transport, and entropy methods. Barbachoux and Kouneiher introduce a unified geometric framework that resolves degeneracies in kinetic equations using Wasserstein gradient flows and commutator structures. From the Boltzmann and Fokker–Planck equations to applications in plasma physics, astrophysics, and data science, their approach reveals the deep interplay between entropy dissipation, geometric control, and convergence to equilibrium—offering a fresh perspective on the dynamics of high-dimensional systems. View this paper
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18 pages, 998 KiB  
Article
A Novel Approach to Strengthening Cryptography Using RSA, Efficient Domination and Fuzzy Logic
by Ghulam Muhiuddin, Annamalai Meenakshi, Janusz Kacprzyk, Ganesan Ambika and Hossein Rashmamlou
Axioms 2025, 14(5), 392; https://doi.org/10.3390/axioms14050392 - 21 May 2025
Viewed by 63
Abstract
A secured communications system is a structure or infrastructure that is intended to ensure the confidentiality, integrity, and authenticity of data being exchanged between entities. Such systems use various security technologies to guarantee that communications are not tampered with, read, or accessed by [...] Read more.
A secured communications system is a structure or infrastructure that is intended to ensure the confidentiality, integrity, and authenticity of data being exchanged between entities. Such systems use various security technologies to guarantee that communications are not tampered with, read, or accessed by unauthorized parties. The intractability of factoring huge composite numbers is a prerequisite for RSA’s security. With big enough key sizes, it is still computationally infeasible for attackers to defeat RSA encryption with today’s technology. Efficient domination is an idea based on graph theory, specifically in the investigation of domination in graphs. Although it has many applications in problems in computation, it is only for cryptography in contexts involving efficient algorithms, combinatorial structures, and optimization for security that it finds uses. This idea provides minimal redundancy in domination, similar to the optimization of resources in a cryptographic scenario. In this work, we concentrate on enhancing the complexity of secure systems using a mathematical model that is based on fuzzy graph networks. The suggested model combines efficient domination, the RSA algorithm, and triangular fuzzy membership functions. By integrating these optimized parameters, we construct a very secure mathematical fuzzy graph network that can efficiently protect secret information. Full article
(This article belongs to the Special Issue Recent Advances in Fuzzy Sets and Related Topics)
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13 pages, 604 KiB  
Article
Assessing Expected Shortfall in Risk Analysis Through Generalized Autoregressive Conditional Heteroskedasticity Modeling and the Application of the Gumbel Distribution
by Bingjie Wang, Yihui Zhang, Jia Li and Tao Liu
Axioms 2025, 14(5), 391; https://doi.org/10.3390/axioms14050391 - 21 May 2025
Viewed by 43
Abstract
In this study, the Gumbel distribution is utilized to construct exact analytical representations for two pivotal measures in financial risk evaluation: Value at Risk (VaR) and Conditional Value at Risk (CVaR). These refined formulations are developed with the intention of offering resilient and [...] Read more.
In this study, the Gumbel distribution is utilized to construct exact analytical representations for two pivotal measures in financial risk evaluation: Value at Risk (VaR) and Conditional Value at Risk (CVaR). These refined formulations are developed with the intention of offering resilient and practically implementable tools to address the complexities inherent in economic risk analysis. Moreover, the newly established expressions are seamlessly integrated into the GARCH modeling framework, thereby enriching its predictive capabilities. In order to verify both the practical relevance and theoretical soundness of the presented methodology, it is systematically employed regarding the daily return series of a varied portfolio of stocks. The outcomes of the numerical experiments provide compelling evidence of the approach’s reliability and effectiveness, emphasizing its suitability for advancing contemporary risk management strategies in financial environments. Full article
(This article belongs to the Special Issue Advances in Financial Mathematics)
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21 pages, 326 KiB  
Article
Gelfand–Shilov Spaces for Extended Gevrey Regularity
by Nenad Teofanov, Filip Tomić and Milica Žigić
Axioms 2025, 14(5), 390; https://doi.org/10.3390/axioms14050390 - 21 May 2025
Viewed by 28
Abstract
We consider spaces of smooth functions obtained by relaxing Gevrey-type regularity and decay conditions. It is shown that these classes can be introduced by using the general framework of the weighted matrices approach to ultradifferentiable functions. We examine alternative descriptions of Gelfand–Shilov spaces [...] Read more.
We consider spaces of smooth functions obtained by relaxing Gevrey-type regularity and decay conditions. It is shown that these classes can be introduced by using the general framework of the weighted matrices approach to ultradifferentiable functions. We examine alternative descriptions of Gelfand–Shilov spaces related to the extended Gevrey regularity and derive their nuclearity. In addition to the Fourier transform invariance property, we present their corresponding symmetric characterizations. Finally, we consider some time–frequency representations of the introduced classes of ultradifferentiable functions. Full article
(This article belongs to the Special Issue Recent Advances in Function Spaces and Their Applications)
22 pages, 731 KiB  
Article
Measuring Semantic Stability: Statistical Estimation of Semantic Projections via Word Embeddings
by Roger Arnau, Ana Coronado Ferrer, Álvaro González Cortés, Claudia Sánchez Arnau and Enrique A. Sánchez Pérez
Axioms 2025, 14(5), 389; https://doi.org/10.3390/axioms14050389 - 21 May 2025
Viewed by 47
Abstract
We present a new framework to study the stability of semantic projections based on word embeddings. Roughly speaking, semantic projections are indices taking values in the interval [0,1] that measure how terms share contextual meaning with the words of [...] Read more.
We present a new framework to study the stability of semantic projections based on word embeddings. Roughly speaking, semantic projections are indices taking values in the interval [0,1] that measure how terms share contextual meaning with the words of a given universe. Since there are many ways to define such projections, it is important to establish a procedure for verifying whether a group of them behaves similarly. Moreover, when fixing one particular projection, it is important to assess whether the average projections remain consistent when replacing the original universe with a similar one describing the same semantic environment. The aim of this paper is to address the lack of formal tools for assessing the stability of semantic projections (that is, their invariance under formal changes which preserve the underlying semantic context) across alternative but semantically related universes in word embedding models. To address these problems, we employ a combination of statistical and AI methods, including correlation analysis, clustering, chi-squared distance measures, weighted approximations, and Lipschitz-based estimators. The methodology provides theoretical guarantees under mild mathematical assumptions, ensuring bounded errors in projection estimations based on the assumption of Lipschitz continuity. We demonstrate the practical applicability of our approach through two case studies involving agricultural terminology across multiple data sources (DOAJ, Scholar, Google, and Arxiv). Our results show that semantic stability can be quantitatively evaluated and that the careful modeling of projection functions and universes is crucial for robust semantic analysis in NLP. Full article
(This article belongs to the Special Issue New Perspectives in Mathematical Statistics)
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26 pages, 2530 KiB  
Article
New n-Dimensional Finite Element Technique for Solving Boundary Value Problems in n-Dimensional Space
by Weam G. Alharbi, Kamal R. Raslan and Khalid K. Ali
Axioms 2025, 14(5), 388; https://doi.org/10.3390/axioms14050388 - 21 May 2025
Viewed by 29
Abstract
In this research, a groundbreaking framework for the octic B-spline collocation method in n-dimensional spaces is presented. This work is an extension of previous works that involved the creation of B-spline functions in n-dimensional space for the purpose of solving mathematical models in [...] Read more.
In this research, a groundbreaking framework for the octic B-spline collocation method in n-dimensional spaces is presented. This work is an extension of previous works that involved the creation of B-spline functions in n-dimensional space for the purpose of solving mathematical models in n-dimensions. The octic B-spline collocation approach in n-dimensional space is an extension of the standard B-spline collocation approach to higher dimensions. It involves using eighth order (octic) B-splines, which have higher smoothness and continuity properties than lower-order B-splines. To demonstrate the effectiveness and precision of the suggested method, a selection of test problems in two- and three-dimensional space is utilized. For making comparisons, we make use of a wide variety of numerical problems, which are described in this paper. Full article
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15 pages, 334 KiB  
Article
An Enhanced Method with Memory Derived from Newton’s Scheme in Solving Nonlinear Equations: Higher R-Order Convergence and Numerical Performance
by Runqi Xue, Yalin Li, Enbin Song and Tao Liu
Axioms 2025, 14(5), 387; https://doi.org/10.3390/axioms14050387 - 21 May 2025
Viewed by 55
Abstract
This article introduces a novel iterative solver with memory, derived from Newton’s scheme, for nonlinear scalar equations. The key innovation lies in integrating memory into the iterative process, enhancing the convergence rate without increasing the quantity of functional evaluations compared to conventional two-point [...] Read more.
This article introduces a novel iterative solver with memory, derived from Newton’s scheme, for nonlinear scalar equations. The key innovation lies in integrating memory into the iterative process, enhancing the convergence rate without increasing the quantity of functional evaluations compared to conventional two-point solvers. The proposed method achieves a superior R-order of convergence by adaptively utilizing past iterates to refine the current approximation. A rigorous theoretical analysis is presented to establish the convergence properties of the method. Extensive computational experiments demonstrate that the method with memory not only accelerates convergence but also enhances accuracy with fewer iterations. These findings suggest that the proposed solver has significant potential for applications in scientific computing and numerical optimization, where efficient and high-order iterative methods are crucial. Full article
(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)
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20 pages, 301 KiB  
Article
Exploring the Structural and Traversal Properties of Total Graphs over Finite Rings
by Ali Al Khabyah, Nazim and Ikram Ali
Axioms 2025, 14(5), 386; https://doi.org/10.3390/axioms14050386 - 20 May 2025
Viewed by 109
Abstract
This paper extends the concept of the total graph TΓ(R) associated with a commutative ring to the three-fold Cartesian product R=Zn×Zm×Zp, where n,m,p>1 [...] Read more.
This paper extends the concept of the total graph TΓ(R) associated with a commutative ring to the three-fold Cartesian product R=Zn×Zm×Zp, where n,m,p>1. We present complete and self-contained proofs for a wide range of graph-theoretic properties of TΓ(R), including connectivity, diameter, regularity conditions, clique and independence numbers, and exact criteria for Hamiltonicity and Eulericity. We also derive improved lower bounds for the genus and characterize the automorphism group in both general and symmetric cases. Each result is illustrated through concrete numerical examples for clarity. Beyond theoretical contributions, we discuss potential applications in cryptographic key-exchange systems, fault-tolerant network architectures, and algebraic code design. This work generalizes and deepens prior studies on two-factor total graphs, and establishes a foundational framework for future exploration of higher-dimensional total graphs over finite commutative rings. Full article
(This article belongs to the Special Issue Advances in Graph Theory with Its Applications)
36 pages, 900 KiB  
Article
Discrete Physics-Informed Training for Projection-Based Reduced-Order Models with Neural Networks
by Nicolas Sibuet, Sebastian Ares de Parga, Jose Raul Bravo and Riccardo Rossi
Axioms 2025, 14(5), 385; https://doi.org/10.3390/axioms14050385 - 20 May 2025
Viewed by 220
Abstract
This paper presents a physics-informed training framework for projection-based Reduced-Order Models (ROMs). We extend the original PROM-ANN architecture by complementing snapshot-based training with a FEM-based, discrete physics-informed residual loss, bridging the gap between traditional projection-based ROMs and physics-informed neural networks (PINNs). Unlike conventional [...] Read more.
This paper presents a physics-informed training framework for projection-based Reduced-Order Models (ROMs). We extend the original PROM-ANN architecture by complementing snapshot-based training with a FEM-based, discrete physics-informed residual loss, bridging the gap between traditional projection-based ROMs and physics-informed neural networks (PINNs). Unlike conventional PINNs that rely on analytical PDEs, our approach leverages FEM residuals to guide the learning of the ROM approximation manifold. Our key contributions include the following: (1) a parameter-agnostic, discrete residual loss applicable to nonlinear problems, (2) an architectural modification to PROM-ANN improving accuracy for fast-decaying singular values, and (3) an empirical study on the proposed physics-informed training process for ROMs. The method is demonstrated on a nonlinear hyperelasticity problem, simulating a rubber cantilever under multi-axial loads. The main accomplishment in regards to the proposed residual-based loss is its applicability on nonlinear problems by interfacing with FEM software while maintaining reasonable training times. The modified PROM-ANN outperforms POD by orders of magnitude in snapshot reconstruction accuracy, while the original formulation is not able to learn a proper mapping for this use case. Finally, the application of physics-informed training in ANN-PROM modestly narrows the gap between data reconstruction and ROM accuracy; however, it highlights the untapped potential of the proposed residual-driven optimization for future ROM development. This work underscores the critical role of FEM residuals in ROM construction and calls for further exploration on architectures beyond PROM-ANN. Full article
(This article belongs to the Section Mathematical Physics)
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11 pages, 842 KiB  
Article
Nonlinear Convection in an Inclined Porous Layer Saturated by Casson Fluid with a Magnetic Effect
by S. Suresh Kumar Raju
Axioms 2025, 14(5), 384; https://doi.org/10.3390/axioms14050384 - 20 May 2025
Viewed by 91
Abstract
The study examines the onset of magnetoconvection in a Casson fluid-saturated inclined porous layer. Oberbeck–Boussinesq approximation and Darcy law employed to characterize the fluid motion. The stability of the system is examined using both linear and nonlinear stability theories. A basic solution of [...] Read more.
The study examines the onset of magnetoconvection in a Casson fluid-saturated inclined porous layer. Oberbeck–Boussinesq approximation and Darcy law employed to characterize the fluid motion. The stability of the system is examined using both linear and nonlinear stability theories. A basic solution of the governing equation is determined. The linear instability is studied by employing disturbances to the basic flow. The nonlinear instability is analyzed utilizing the energy method. The solution to the eigenvalue problem is derived using the bvp4c routine in MATLAB R2023a. This study evaluates the influence of nondimensional parameters specifically, the Hartmann number, Casson parameter, and inclination angle on both linear and nonlinear instability. The Casson parameter destabilizes the system, whereas the Hartmann number and inclination angle stabilize it. Transverse rolls exhibit greater stability compared to longitudinal rolls. Changes in the Casson parameter significantly affect the presence or absence of transverse rolls; as its value changes, so does the disappearance of transverse rolls. Full article
(This article belongs to the Special Issue Recent Progress in Computational Fluid Dynamics)
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16 pages, 5530 KiB  
Article
Stabilization and Synchronization of a New 3D Complex Chaotic System via Adaptive and Active Control Methods
by Lamia Loudahi, Jing Yuan, Lydia Dehbi and Mawia Osman
Axioms 2025, 14(5), 383; https://doi.org/10.3390/axioms14050383 - 19 May 2025
Viewed by 154
Abstract
This paper investigates the controllability and synchronization of a newly designed three-dimensional chaotic system using active and adaptive control strategies. Although both controllers were designed with the help of a positive definite function using Lyapunov theory, for adaptive controllers, we estimated an unknown [...] Read more.
This paper investigates the controllability and synchronization of a newly designed three-dimensional chaotic system using active and adaptive control strategies. Although both controllers were designed with the help of a positive definite function using Lyapunov theory, for adaptive controllers, we estimated an unknown parameter of the system in real time and adjusted the control signal accordingly to maintain stability. Moreover, numerical simulations demonstrated that the active control approach achieved stability for both equilibrium points (p1 and p2) approximately at time t = 0.5 s, demonstrating its rapid convergence and robust performance. In contrast, the adaptive control method stabilized p1 at approximately t≈0.5 and p2 at t≈1 s, illustrating reaching their desired conditions. Furthermore, the considered methods could effectively synchronize two identical chaotic systems, where the slave system overlapped the master system at approximately t=5 s. Apart from this, a detailed comparative analysis of the two techniques in terms of controllability and synchronization is presented. Moreover, the complementary strengths of these methods provide valuable perspectives for broader applications in chaotic system management and security-critical systems. Full article
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24 pages, 1014 KiB  
Article
A Novel Approach to Some Proximal Contractions with Examples of Its Application
by Muhammad Zahid, Fahim Ud Din, Luminiţa-Ioana Cotîrlă and Daniel Breaz
Axioms 2025, 14(5), 382; https://doi.org/10.3390/axioms14050382 - 19 May 2025
Viewed by 116
Abstract
In this article, we will introduce a new generalized proximal θ-contraction for multivalued and single-valued mappings named (f−θκ)CP-proximal contraction and (f−θκ)BP-proximal contraction. Using these newly constructed [...] Read more.
In this article, we will introduce a new generalized proximal θ-contraction for multivalued and single-valued mappings named (f−θκ)CP-proximal contraction and (f−θκ)BP-proximal contraction. Using these newly constructed proximal contractions, we will establish new results for the coincidence best proximity point, best proximity point, and fixed point for multivalued mappings in the context of rectangular metric space. Also, we will reduce these contractions for single-valued mappings, named (θκ)CP-proximal contraction and (θκ)BP-proximal contraction, to establish results for the coincidence proximity point, best proximity point, and fixed point results. We will give some illustrated examples for our newly generated results with graphical representations. In the last section, we will also find the solution to the equation of motion by using our defined results. Full article
(This article belongs to the Special Issue Numerical Methods and Approximation Theory)
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24 pages, 502 KiB  
Article
Decision-Making with Fermatean Neutrosophic Vague Soft Sets Using a Technique for Order of Preference by Similarity to Ideal Solution
by Najla Althuniyan, Abedallah Al-shboul, Sarah Aljohani, Kah Lun Wang, Kok Bin Wong, Khaleed Alhazaymeh and Suhad Subhi Aiady
Axioms 2025, 14(5), 381; https://doi.org/10.3390/axioms14050381 - 19 May 2025
Viewed by 174
Abstract
This study addresses the challenge of effectively modeling uncertainty and hesitation in complex decision-making environments, where traditional fuzzy and vague set models often fall short. To overcome these limitations, we propose the Fermatean neutrosophic vague soft set (FNVSS), an advanced extension that integrates [...] Read more.
This study addresses the challenge of effectively modeling uncertainty and hesitation in complex decision-making environments, where traditional fuzzy and vague set models often fall short. To overcome these limitations, we propose the Fermatean neutrosophic vague soft set (FNVSS), an advanced extension that integrates the concepts of neutrosophic sets with Fermatean membership functions into the framework of vague sets. The FNVSS model enhances the representation of truth, indeterminacy, and falsity degrees, providing greater flexibility and resilience in capturing ambiguous and imprecise information. We systematically develop new operations for the FNVSS, including union, intersection, complementation, the Fermatean neutrosophic vague normalized weighted average (FNVNWA) operator, the generalized Fermatean neutrosophic vague normalized weighted average (GFNVNWA) operator, and an adapted Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) method. To demonstrate the practicality of the proposed methodology, we apply it to a solar panel selection problem, where managing uncertainty is crucial. Comparative results indicate that the FNVSS significantly outperforms traditional fuzzy and vague set approaches, leading to more reliable and accurate decision outcomes. This work contributes to the advancement of predictive decision-making systems, particularly in fields requiring high precision, adaptability, and robust uncertainty modeling. Full article
(This article belongs to the Section Mathematical Analysis)
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25 pages, 383 KiB  
Article
Asymptotic Growth of Moduli of m-th Derivatives of Algebraic Polynomials in Weighted Bergman Spaces on Regions Without Zero Angles
by Uğur Değer, Meerim Imashkyzy and Fahreddin G. Abdullayev
Axioms 2025, 14(5), 380; https://doi.org/10.3390/axioms14050380 - 19 May 2025
Viewed by 136
Abstract
In this paper, we study asymptotic bounds on the m-th derivatives of general algebraic polynomials in weighted Bergman spaces. We consider regions in the complex plane defined by bounded, piecewise, asymptotically conformal curves with strictly positive interior angles. We first establish asymptotic [...] Read more.
In this paper, we study asymptotic bounds on the m-th derivatives of general algebraic polynomials in weighted Bergman spaces. We consider regions in the complex plane defined by bounded, piecewise, asymptotically conformal curves with strictly positive interior angles. We first establish asymptotic bounds on the growth in the exterior of a given unbounded region. We then extend our analysis to the closures of the region and derive the corresponding growth bounds. Combining these bounds with those for the corresponding exterior, we obtain comprehensive bounds on the growth of the m-th derivatives of arbitrary algebraic polynomials in the whole complex plane. Full article
(This article belongs to the Section Mathematical Analysis)
15 pages, 463 KiB  
Article
On Null Cartan Normal Helices in Minkowski 3-Space
by Emilija Nešović
Axioms 2025, 14(5), 379; https://doi.org/10.3390/axioms14050379 - 18 May 2025
Viewed by 95
Abstract
In this paper, we introduce null Cartan normal helices in Minkowski space E13. We obtain explicit expressions for their torsions by considering the cases when the C-constant vector field is orthogonal to their axis or not orthogonal to it. [...] Read more.
In this paper, we introduce null Cartan normal helices in Minkowski space E13. We obtain explicit expressions for their torsions by considering the cases when the C-constant vector field is orthogonal to their axis or not orthogonal to it. We find that the tangent vector field of a null Cartan normal helix satisfies the third-order linear homogeneous differential equation and obtain its general solution in a special case. We prove that null Cartan helices are the only normal helices having two axes and, in a particular case, three axes. Finally, we provide the necessary and sufficient conditions for null Cartan normal helices lying on a timelike surface to be isophotic curves, silhouettes, normal isophotic curves and normal silhouettes with respect to the same axis and provide some examples. Full article
(This article belongs to the Section Geometry and Topology)
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23 pages, 340 KiB  
Article
Third-Order Fuzzy Subordination and Superordination on Analytic Functions on Punctured Unit Disk
by Ekram E. Ali, Georgia Irina Oros, Rabha M. El-Ashwah and Abeer M. Albalahi
Axioms 2025, 14(5), 378; https://doi.org/10.3390/axioms14050378 - 17 May 2025
Viewed by 143
Abstract
This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best [...] Read more.
This work’s theorems and corollaries present new third-order fuzzy differential subordination and superordination results developed by using a novel convolution linear operator involving the Gaussian hypergeometric function and a previously studied operator. The paper reveals methods for finding the best dominant and best subordinant for the third-order fuzzy differential subordinations and superordinations, respectively. The investigation concludes with the assertion of sandwich-type theorems connecting the conclusions of the studies conducted using the particular methods of the theories of the third-order fuzzy differential subordination and superordination, respectively. Full article
(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)
26 pages, 332 KiB  
Article
Uniqueness Methods and Stability Analysis for Coupled Fractional Integro-Differential Equations via Fixed Point Theorems on Product Space
by Nan Zhang, Emmanuel Addai and Hui Wang
Axioms 2025, 14(5), 377; https://doi.org/10.3390/axioms14050377 - 16 May 2025
Viewed by 91
Abstract
In this paper, we obtain unique solution and stability results for coupled fractional differential equations with p-Laplacian operator and Riemann–Stieltjes integral conditions that expand and improve the works of some of the literature. In order to obtain the existence and uniqueness of solutions [...] Read more.
In this paper, we obtain unique solution and stability results for coupled fractional differential equations with p-Laplacian operator and Riemann–Stieltjes integral conditions that expand and improve the works of some of the literature. In order to obtain the existence and uniqueness of solutions for coupled systems, several fixed point theorems for operators in ordered product spaces are given without requiring the existence conditions of upper–lower solutions or the compactness and continuity of operators. By applying the conclusions of the operator theorem studied, sufficient conditions for the unique solution of coupled fractional integro-differential equations and approximate iterative sequences for uniformly approximating unique solutions were obtained. In addition, the Hyers–Ulam stability of the coupled system is discussed. As applications, the corresponding results obtained are well demonstrated through some concrete examples. Full article
8 pages, 235 KiB  
Article
Variational Principle of the Unstable Nonlinear Schrödinger Equation with Fractal Derivatives
by Kang-Jia Wang and Ming Li
Axioms 2025, 14(5), 376; https://doi.org/10.3390/axioms14050376 - 16 May 2025
Viewed by 95
Abstract
The well-known nonlinear Schrödinger equation (NLSE) plays a crucial role in describing the temporal evolution of disturbances in marginally stable or unstable media. However, when the media is a fractal form, it becomes ineffective. Thus, the fractal modification to the NLSE is presented [...] Read more.
The well-known nonlinear Schrödinger equation (NLSE) plays a crucial role in describing the temporal evolution of disturbances in marginally stable or unstable media. However, when the media is a fractal form, it becomes ineffective. Thus, the fractal modification to the NLSE is presented based on the fractal derivative in this work for the first time. The semi-inverse method is employed to establish the fractal variational principle. The entire process of deriving the fractal variational principle is presented in detail. To our knowledge, the fractal variational principle mentioned in this article is the first exploration and report to date. The fractal variational principle established in this paper is expected to deepen our understanding of the essence of physical phenomena in the fractal space and offer new ideas for the application and exploration of the variational approaches. Full article
(This article belongs to the Special Issue Principles of Variational Methods in Mathematical Physics)
21 pages, 320 KiB  
Article
Combined Matrix of a Tridiagonal Toeplitz Matrix
by Begoña Cantó, Rafael Cantó and Ana Maria Urbano
Axioms 2025, 14(5), 375; https://doi.org/10.3390/axioms14050375 - 16 May 2025
Viewed by 76
Abstract
In this work, combined matrices of tridiagonal Toeplitz matrices are studied. The combined matrix is known as the Relative Gain Array in control theory. In particular, given a real tridiagonal Toeplitz matrix of order n, the characterization of its combined matrix as [...] Read more.
In this work, combined matrices of tridiagonal Toeplitz matrices are studied. The combined matrix is known as the Relative Gain Array in control theory. In particular, given a real tridiagonal Toeplitz matrix of order n, the characterization of its combined matrix as a bisymmetric and doubly quasi-stochastic matrix is studied. Furthermore, this paper addresses the inverse problem, that is, given a bisymmetric, doubly quasi-stochastic tridiagonal Jacobi matrix U of order n, determine under what conditions there exists a real tridiagonal Toeplitz matrix A such that its combined matrix is U. Full article
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21 pages, 300 KiB  
Article
Existence and Uniqueness of Solutions to SDEs with Jumps and Irregular Drifts
by Mhamed Eddahbi
Axioms 2025, 14(5), 374; https://doi.org/10.3390/axioms14050374 - 16 May 2025
Viewed by 77
Abstract
We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages a space transformation and Itô-Krylov’s formula to effectively eliminate the singular component of the drift, allowing us to obtain a [...] Read more.
We focus on solving stochastic differential equations driven by jump processes (SDEJs) with measurable drifts that may exhibit quadratic growth. Our approach leverages a space transformation and Itô-Krylov’s formula to effectively eliminate the singular component of the drift, allowing us to obtain a transformed SDEJ that satisfies classical solvability conditions. By applying the inverse transformation proven to be a one-to-one mapping, we retrieve the solution to the original equation. This methodology offers several key advantages. First, it extends the well-known result of Le Gall (1984) from Brownian-driven SDEs to the jump process setting, broadening the range of applicable stochastic models. Second, it provides a robust framework for handling singular drifts, enabling the resolution of equations that would otherwise be intractable. Third, the approach accommodates drifts with quadratic growth, making it particularly relevant for financial modeling, insurance risk assessment, and other applications where such growth behavior is common. Finally, the inclusion of multiple examples illustrates the practical effectiveness of our method, demonstrating its flexibility and applicability to real-world problems. Full article
(This article belongs to the Section Mathematical Analysis)
20 pages, 762 KiB  
Article
Hybrid Inertial Self-Adaptive Iterative Methods for Split Variational Inclusion Problems
by Doaa Filali, Mohammad Dilshad, Atiaf Farhan Yahya Alfaifi and Mohammad Akram
Axioms 2025, 14(5), 373; https://doi.org/10.3390/axioms14050373 - 15 May 2025
Viewed by 259
Abstract
Herein, we present two hybrid inertial self-adaptive iterative methods for determining the combined solution of the split variational inclusions and fixed-point problems. Our methods include viscosity approximation, fixed-point iteration, and inertial extrapolation in the initial step of each iteration. We employ two self-adaptive [...] Read more.
Herein, we present two hybrid inertial self-adaptive iterative methods for determining the combined solution of the split variational inclusions and fixed-point problems. Our methods include viscosity approximation, fixed-point iteration, and inertial extrapolation in the initial step of each iteration. We employ two self-adaptive step sizes to compute the iterative sequence, which do not require the pre-calculated norm of a bounded linear operator. We prove strong convergence theorems to approximate the common solution of the split variational inclusions and fixed-point problems. Further, we implement our methods and results to examine split variational inequality and split common fixed-point problems. Finally, we illustrate our methods and compare them with some known methods existing in the literature. Full article
(This article belongs to the Section Mathematical Analysis)
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7 pages, 207 KiB  
Article
Polygonal Quasiconformality and Grunsky’s Operator
by Samuel L. Krushkal
Axioms 2025, 14(5), 372; https://doi.org/10.3390/axioms14050372 - 15 May 2025
Viewed by 75
Abstract
This paper concerns the old problem of the connection between the dilatations of a given quasisymmetric homeomorphism h of a circle and the associated polygonal quasiconformal maps with a fixed finite number of boundary points, namely whether [...] Read more.
This paper concerns the old problem of the connection between the dilatations of a given quasisymmetric homeomorphism h of a circle and the associated polygonal quasiconformal maps with a fixed finite number of boundary points, namely whether k(h)=supkn, where the supremum is taken over all possible n-gons formed by the disk with n distinguished boundary points. A still open question is whether such equality is valid under the additional assumption that the naturally related univalent functions with quasiconformal extensions have equal Grunsky and Teichmüller norms. We solved this problem in the negative for n≥4. Full article
(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)
36 pages, 637 KiB  
Article
On the Dynamics of Some Three-Dimensional Systems of Difference Equations
by Turki D. Alharbi and Jawharah G. AL-Juaid
Axioms 2025, 14(5), 371; https://doi.org/10.3390/axioms14050371 - 15 May 2025
Viewed by 121
Abstract
This paper looks into the dynamics of nonlinear systems of difference equations, with particular emphasis on fourth-order cases. Analytical solutions are derived for some cases of systems, a tedious task due to the lack of explicit mathematical techniques for their solution. In addition, [...] Read more.
This paper looks into the dynamics of nonlinear systems of difference equations, with particular emphasis on fourth-order cases. Analytical solutions are derived for some cases of systems, a tedious task due to the lack of explicit mathematical techniques for their solution. In addition, the qualitative properties of the solutions, such as boundedness and periodicity, are analyzed through theoretical methods and numerical simulations. The results advance our understanding of nonlinear systems, providing important implications for their use in various scientific fields. Full article
(This article belongs to the Section Mathematical Analysis)
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22 pages, 945 KiB  
Article
Opposition and Implication in Aristotelian Diagrams
by Alexander De Klerck
Axioms 2025, 14(5), 370; https://doi.org/10.3390/axioms14050370 - 15 May 2025
Viewed by 197
Abstract
In logical geometry, Aristotelian diagrams are studied in a systematic fashion. Recent developments in this field have shown that the square of opposition generalizes in two ways, which correspond precisely to the theory of opposition (leading to α-structures) and the theory of [...] Read more.
In logical geometry, Aristotelian diagrams are studied in a systematic fashion. Recent developments in this field have shown that the square of opposition generalizes in two ways, which correspond precisely to the theory of opposition (leading to α-structures) and the theory of implication (leading to ladders) it exhibits. These two kinds of Aristotelian diagrams are dual to each other, in the sense that they are the oppositional and implicative counterpart of the same construction. This paper formalizes this duality as OI-companionship, explores its properties, and applies it to various σ-diagrams. This investigation shows that OI-companionship has some interesting, but unusual behaviors. While it is symmetric, and works well on the level of Aristotelian families, it lacks (ir)reflexivity, transitivity, functionality, and seriality. However, we show that all important Aristotelian families from the literature do have a unique OI-companion. These findings explore the limits that arise when extending the duality between opposition and implication beyond the limits of α-structures and ladders. Full article
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21 pages, 5020 KiB  
Article
Influence of Heat Transfer on Stress Components in Metallic Plates Weakened by Multi-Curved Holes
by Faizah M. Alharbi and Nafeesa G. Alhendi
Axioms 2025, 14(5), 369; https://doi.org/10.3390/axioms14050369 - 14 May 2025
Viewed by 175
Abstract
This manuscript addresses an application study by employing a mathematical model of a thermoelastic plate weakened by multi-curved holes under the effect of stress forces in the presence of heat conduction. When the initial heat flow is directed to the plate system, complex [...] Read more.
This manuscript addresses an application study by employing a mathematical model of a thermoelastic plate weakened by multi-curved holes under the effect of stress forces in the presence of heat conduction. When the initial heat flow is directed to the plate system, complex variable procedures are used to compute the basic Goursat functions, taking into account the time-dependent variables through conformal mapping, which transfers the domain to the exterior of a unit circle. The problem reduces to a general form of a contact problem in two dimensions, which is called an integrodifferential equation of the second type with the Cauchy kernel. Additionally, different hole shapes are generated using Maple 2023. Computational simulations are performed to determine the normal and shear stress components in the presence and absence of heat effects at various times. Furthermore, numerical calculations of Goursat functions are carried out and graphically displayed for some specific materials. This investigation provides valuable information about industries, such as those regarding ceramic tile, glass, rubber, paint, ceramic pigment, and metal alloys. Full article
(This article belongs to the Special Issue Mathematical Methods in the Applied Sciences, 2nd Edition)
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10 pages, 464 KiB  
Article
Optimizing Parameter Estimation Precision in Open Quantum Systems
by Kamal Berrada
Axioms 2025, 14(5), 368; https://doi.org/10.3390/axioms14050368 - 13 May 2025
Viewed by 163
Abstract
In the present manuscript, we demonstrate the potential to control and enhance the accuracy of parameter estimation (P-E) in a two-level atom (TLA) immersed in a cavity field that interacts with another cavity. We investigate the dynamics of quantum Fisher information (FI), considering [...] Read more.
In the present manuscript, we demonstrate the potential to control and enhance the accuracy of parameter estimation (P-E) in a two-level atom (TLA) immersed in a cavity field that interacts with another cavity. We investigate the dynamics of quantum Fisher information (FI), considering the influence of coupling strength between the two cavities and the detuning parameter. Our findings reveal that, in the case of a perfect cavity, a high quantum FI value can be maintained during the dynamics concerning the detuning and coupling strength parameters. The results indicate that with a proper choice of quantum model parameters, long-term protection of the FI can be achieved without being affected by decoherence. Full article
(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)
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27 pages, 631 KiB  
Article
Kaczmarz-Type Methods for Solving Matrix Equation AXB = C
by Wei Zheng, Lili Xing, Wendi Bao and Weiguo Li
Axioms 2025, 14(5), 367; https://doi.org/10.3390/axioms14050367 - 13 May 2025
Viewed by 139
Abstract
This paper proposes a class of randomized Kaczmarz and Gauss–Seidel-type methods for solving the matrix equation AXB=C, where the matrices A and B may be either full-rank or rank deficient and the system may be consistent or inconsistent. [...] Read more.
This paper proposes a class of randomized Kaczmarz and Gauss–Seidel-type methods for solving the matrix equation AXB=C, where the matrices A and B may be either full-rank or rank deficient and the system may be consistent or inconsistent. These iterative methods offer high computational efficiency and low memory requirements, as they avoid costly matrix–matrix multiplications. We rigorously establish theoretical convergence guarantees, proving that the generated sequences converge to the minimal Frobenius-norm solution (for consistent systems) or the minimal Frobenius-norm least squares solution (for inconsistent systems). Numerical experiments demonstrate the superiority of these methods over conventional matrix multiplication-based iterative approaches, particularly for high-dimensional problems. Full article
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26 pages, 789 KiB  
Article
Stability and Hopf Bifurcation of Fractional-Order Quaternary Numerical Three-Neuron Neural Networks with Different Types of Delays
by Qiankun Wang, Tianzeng Li, Yu Wang and Xiaowen Tan
Axioms 2025, 14(5), 366; https://doi.org/10.3390/axioms14050366 - 13 May 2025
Viewed by 133
Abstract
In this paper, the stability and Hopf bifurcation of fractional-order quaternion-valued neural networks (FOQVNNs) with various types of time delays are studied. The fractional-order quaternion neural networks with time delays are decomposed into an equivalent complex-valued system through the Cayley–Dickson construction. The existence [...] Read more.
In this paper, the stability and Hopf bifurcation of fractional-order quaternion-valued neural networks (FOQVNNs) with various types of time delays are studied. The fractional-order quaternion neural networks with time delays are decomposed into an equivalent complex-valued system through the Cayley–Dickson construction. The existence and uniqueness of the solution for the considered fractional-order delayed quaternion neural networks are proven by using the compression mapping theorem. It is demonstrated that the solutions of the involved fractional delayed quaternion neural networks are bounded by constructing appropriate functions. Some sufficient conditions for the stability and Hopf bifurcation of the considered fractional-order delayed quaternion neural networks are established by utilizing the stability theory of fractional differential equations and basic bifurcation knowledge. To validate the rationality of the theoretical results, corresponding simulation results and bifurcation diagrams are provided. The relationship between the order of appearance of bifurcation phenomena and the order is also studied, revealing that bifurcation phenomena occur later as the order increases. The theoretical results established in this paper are of significant guidance for the design and improvement of neural networks. Full article
(This article belongs to the Special Issue Complex Networks and Dynamical Systems)
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12 pages, 244 KiB  
Article
Graded 1-Absorbing Prime Ideals over Non-Commutative Graded Rings
by Azzh Saad Alshehry, Rashid Abu-Dawwas and Rahaf Abudalo
Axioms 2025, 14(5), 365; https://doi.org/10.3390/axioms14050365 - 13 May 2025
Viewed by 118
Abstract
In this article, we define and study graded 1-absorbing prime ideals and graded weakly 1-absorbing prime ideals in non-commutative graded rings as a new class of graded ideals that lies between graded prime ideals (graded weakly prime ideals) and graded 2-absorbing ideals (graded [...] Read more.
In this article, we define and study graded 1-absorbing prime ideals and graded weakly 1-absorbing prime ideals in non-commutative graded rings as a new class of graded ideals that lies between graded prime ideals (graded weakly prime ideals) and graded 2-absorbing ideals (graded weakly 2-absorbing ideals). Let G be a group and let R be a non-commutative G-graded ring with nonzero unity. Let P be a proper graded ideal of R. We then say that P is a graded 1-absorbing prime ideal (a graded weakly 1-absorbing prime ideal) of R if, for each nonunit homogeneous element r,s,t∈R with rRsRt⊆P ({0}≠rRsRt⊆P), either rs∈P or t∈P. We present a number of properties and characterizations of these graded ideals. Full article
14 pages, 366 KiB  
Article
Superconvergence of a Nonconforming Interface Penalty Finite Element Method for Elliptic Interface Problems
by Xiaoxiao He
Axioms 2025, 14(5), 364; https://doi.org/10.3390/axioms14050364 - 12 May 2025
Viewed by 128
Abstract
In our previous works, we developed the superconvergence of a nonconforming finite element method based on unfitted meshes for an elliptic interface problem and elliptic problem, respectively. In this paper, a nonconforming interface penalty finite element method (NIPFEM) based on body-fitted meshes is [...] Read more.
In our previous works, we developed the superconvergence of a nonconforming finite element method based on unfitted meshes for an elliptic interface problem and elliptic problem, respectively. In this paper, a nonconforming interface penalty finite element method (NIPFEM) based on body-fitted meshes is explored for elliptic interface problems, which allows us to use different meshes in different sub-domains separated by the interface. A nonconforming finite element method based on rectangular meshes is studied and the supercloseness property between the gradient of the numerical solution and the gradient of the interpolation of the exact solution is proven for both symmetric NIPFEM and nonsymmetric NIPFEM. Then, the global superconvergence rate O(hi32) between the postprocessed numerical solution of NIPFEM and the exact solution is derived by using an interpolation postprocessing technique. Numerical examples are carried out to demonstrate the theoretical results. Full article
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13 pages, 1850 KiB  
Article
An Innovative Analytical Approach for the Solution of Fractional Differential Equations Using the Integral Transform
by Eltaib M. Abd Elmohmoud and Tarig M. Elzaki
Axioms 2025, 14(5), 363; https://doi.org/10.3390/axioms14050363 - 12 May 2025
Viewed by 186
Abstract
In this study, we suggest a straightforward analytical/semi-analytical method based on the Elzaki transform (ET) method to find the solution to a number of differential fractional boundary value problems with initial conditions (ICs). The suggested approach not only resolves the issue of some [...] Read more.
In this study, we suggest a straightforward analytical/semi-analytical method based on the Elzaki transform (ET) method to find the solution to a number of differential fractional boundary value problems with initial conditions (ICs). The suggested approach not only resolves the issue of some equation nonlinearity but also transforms the issue into a simpler algebraic recurrence problem. In science and engineering, fractional differential equations (FDEs) can be solved with the help of this basic but effective concept. Some illustrative cases are used to demonstrate the efficacy and value of the suggested technique. Full article
(This article belongs to the Special Issue Current Perspectives in Fractional Calculus: Theory, Methods, and Applications)
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