Axioms

11 pages, 265 KiB

Open AccessArticle

On Certain Bounds of Harmonic Univalent Functions

by Fethiye Müge Sakar, Omendra Mishra, Georgia Irina Oros and Basem Aref Frasin

Axioms 2025, 14(6), 393; https://doi.org/10.3390/axioms14060393 - 22 May 2025

Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk

U = \{z \in C : |z| < 1\}

can be written as [...] Read more.

Harmonic functions are renowned for their application in the analysis of minimal surfaces. These functions are also very important in applied mathematics. Any harmonic function in the open unit disk

U = \{z \in C : |z| < 1\}

can be written as a sum

f = h + \bar{g}

, where h and g are analytic functions in

U

and are called the analytic part and the co-analytic part of f, respectively. In this paper, the harmonic shear

f = h + \bar{g} \in S_{H}

and its rotation

f^{μ}

by

μ (μ \in C, |μ| = 1)

are considered. Bounds are established for this rotation

f^{μ}

, specific inequalities that define the Jacobian of

f^{μ}

are obtained, and the integral representation is determined. Full article

(This article belongs to the Special Issue Recent Advances in Complex Analysis and Applications, 2nd Edition)

18 pages, 837 KiB

Open AccessArticle

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

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)

13 pages, 604 KiB

Open AccessArticle

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

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

Open AccessArticle

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

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

Open AccessArticle

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

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

Open AccessArticle

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

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

Open AccessArticle

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

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

Open AccessArticle

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

Abstract

This paper extends the concept of the total graph

T_{Γ} (R)

associated with a commutative ring to the three-fold Cartesian product

R = Z_{n} \times Z_{m} \times Z_{p}

, where

n, m, p > 1

[...] Read more.

This paper extends the concept of the total graph

T_{Γ} (R)

associated with a commutative ring to the three-fold Cartesian product

R = Z_{n} \times Z_{m} \times Z_{p}

, where

n, m, p > 1

. We present complete and self-contained proofs for a wide range of graph-theoretic properties of

T_{Γ} (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

Open AccessArticle

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

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

Open AccessArticle

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

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

Open AccessArticle

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

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 (

p_{1}

and

p_{2}

) approximately at time t =

0.5

s, demonstrating its rapid convergence and robust performance. In contrast, the adaptive control method stabilized

p_{1}

at approximately

t \approx 0.5

and

p_{2}

at

t \approx 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

Open AccessArticle

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

Abstract

In this article, we will introduce a new generalized proximal

θ

-contraction for multivalued and single-valued mappings named

{(f - θ_{κ})}_{C_{P}}

-proximal contraction and

{(f - θ_{κ})}_{B_{P}}

-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 - θ_{κ})}_{C_{P}}

-proximal contraction and

{(f - θ_{κ})}_{B_{P}}

-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

{(θ_{κ})}_{C_{P}}

-proximal contraction and

{(θ_{κ})}_{B_{P}}

-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

Open AccessArticle

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

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

Open AccessArticle

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

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

Open AccessArticle

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

Abstract

In this paper, we introduce null Cartan normal helices in Minkowski space

E_{1}^{3}

. 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

E_{1}^{3}

. 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

Open AccessArticle

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

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

Open AccessArticle

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

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

Open AccessArticle

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

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

Open AccessArticle

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

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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