Axioms

15 pages, 308 KB

Open AccessArticle

Boundedness and Applications of Fractional Integral Operators in Nonlocal Problems with Fractional Laplacians

by Saba Mehmood, Dušan J. Simjanović and Branislav M. Randjelović

Axioms 2026, 15(3), 220; https://doi.org/10.3390/axioms15030220 (registering DOI) - 16 Mar 2026

In this paper, we investigate the properties of the boundedness of fractional integral operators

K_{α}

defined on general measure metric spaces. We study their action in Lebesgue spaces

L^{p} (Y)

, Morrey spaces

L_{φ}^{p} (Y)

[...] Read more.

In this paper, we investigate the properties of the boundedness of fractional integral operators

K_{α}

defined on general measure metric spaces. We study their action in Lebesgue spaces

L^{p} (Y)

, Morrey spaces

L_{φ}^{p} (Y)

, and extend our analysis to fractional Sobolev spaces

W^{α, p} (Y)

. Using classical dyadic decomposition and the Hardy–Littlewood maximal operator, we establish sharp bounds for

K_{α}

in terms of kernel parameters and the geometric structure of the space. A significant contribution of this work is the proof that

K_{α}

is bounded from

W^{α, p} (Y)

to

L^{q} (Y)

, where thus linking our operator-theoretic framework with the theory of nonlocal and fractional partial differential equations. These results provide valuable tools for studying regularity, a priori estimates, and solution mappings in nonlocal problems involving the fractional Laplacian and related operators on irregular or non- Euclidean domains. Full article

(This article belongs to the Special Issue Advancements in Applied Mathematics and Computational Physics, 2nd Edition)

21 pages, 11307 KB

Open AccessArticle

A Symmetry-Preserving Extrapolated Primal-Dual Hybrid Gradient Method for Saddle-Point Problems

by Xiayang Zhang, Wenzhuo Li, Bowen Chang, Wei Liu and Shiyu Zhang

Axioms 2026, 15(3), 219; https://doi.org/10.3390/axioms15030219 (registering DOI) - 16 Mar 2026

Abstract

The primal-dual hybrid gradient (PDHG) method is widely used for convex–concave saddle-point problems, yet its extrapolated variants are typically asymmetric because only one side is extrapolated. We propose a symmetry-preserving refinement, E-PDHG, which performs dual-side extrapolation followed by an explicit correction step. Under [...] Read more.

The primal-dual hybrid gradient (PDHG) method is widely used for convex–concave saddle-point problems, yet its extrapolated variants are typically asymmetric because only one side is extrapolated. We propose a symmetry-preserving refinement, E-PDHG, which performs dual-side extrapolation followed by an explicit correction step. Under standard step-size conditions, we establish global convergence for all

η \in (- 1, 1)

and derive a pointwise (non-ergodic)

O (1 / \sqrt{t})

rate for the last iterate. The method does not improve the asymptotic complexity order of PDHG; instead, it enlarges the practically stable parameter region while retaining the same per-iteration cost. Numerical experiments on image deblurring/inpainting and additional machine learning benchmarks (logistic regression and LASSO) demonstrate improved finite-iteration stability and efficiency. Full article

(This article belongs to the Section Mathematical Analysis)

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39 pages, 8652 KB

Open AccessArticle

The Unit Arcsine–Exponential Distribution and Its Statistical Properties with Inference and Application to Reliability Data

by Asmaa S. Al-Moisheer, Khalaf S. Sultan, Moustafa N. Mousa and Mahmoud M. M. Mansour

Axioms 2026, 15(3), 218; https://doi.org/10.3390/axioms15030218 (registering DOI) - 15 Mar 2026

Abstract

This paper presents a new continuous data model, the Unit Arcsine–Exponential distribution (UASED), a flexible data model on the unit interval. It is built up by an exponential-based arcsine-type transformation to allow it to represent a very wide range of shapes that can [...] Read more.

This paper presents a new continuous data model, the Unit Arcsine–Exponential distribution (UASED), a flexible data model on the unit interval. It is built up by an exponential-based arcsine-type transformation to allow it to represent a very wide range of shapes that can be used to model proportions and rates. A number of basic properties are obtained, such as closed-form formulas of the quantile function, moments, and entropy measures. Maximum likelihood and maximum product of spacings methods are developed to estimate parameters, and their performance is determined by Monte Carlo simulation, which shows that these methods can reasonably estimate the parameters and be stable over a variety of different parameter settings. To demonstrate that a model is practically useful, an application to real-world data on the reliability of devices in terms of failure time is discussed. The findings indicate that the UASED is a good fit to the data, in the sense that it is effective in terms of skewness and tail behavior and compares well or competes favorably with current unit distributions. All in all, the suggested model is a sparse alternative to model bounded data with sound inferential characteristics and high practical utility. Full article

13 pages, 305 KB

Open AccessArticle

Stronger Versions of Stein–Weiss Inequalities

by Youjiang Lin, Jinghong Zhou and Jiaming Lan

Axioms 2026, 15(3), 217; https://doi.org/10.3390/axioms15030217 - 13 Mar 2026

Abstract

In this paper, stronger versions of Stein–Weiss inequalities and reverse Stein–Weiss inequalities are established. Full article

24 pages, 828 KB

Open AccessArticle

Periodic Asymmetric LogGARCH Stochastic Volatility Models: Structure and Application

by Omar Alzeley and Ahmed Ghezal

Axioms 2026, 15(3), 216; https://doi.org/10.3390/axioms15030216 - 13 Mar 2026

Abstract

This paper introduces a new class of periodic volatility models, namely, the Stochastic Volatility Periodic Logarithmic Asymmetric GARCH (PlogAG-SV) model. The proposed framework extends periodic logGARCH models by incorporating a stochastic volatility component combined with a distinctive threshold mechanism, thereby significantly enhancing their [...] Read more.

This paper introduces a new class of periodic volatility models, namely, the Stochastic Volatility Periodic Logarithmic Asymmetric GARCH (PlogAG-SV) model. The proposed framework extends periodic logGARCH models by incorporating a stochastic volatility component combined with a distinctive threshold mechanism, thereby significantly enhancing their ability to capture asymmetric and time-varying volatility dynamics. Sufficient conditions for strict stationarity, second-order stationarity, and the existence of higher-order moments are rigorously established, providing a comprehensive characterization of the model’s probabilistic properties. Parameter estimation is conducted via extensive Monte Carlo simulations, demonstrating the robustness and reliability of the proposed estimation procedure across a wide range of scenarios. Furthermore, the empirical relevance of the PlogAG-SV model is illustrated through an application to the Algerian dinar–euro exchange rate, highlighting its effectiveness in modeling real-world financial volatility. Full article

(This article belongs to the Special Issue Methods and Applications of Advanced Statistical Analysis, 3rd Edition)

16 pages, 313 KB

Open AccessArticle

Biharmonic Conformal Immersions into a 3-Dimensional Conformally Flat Space

by Ze-Ping Wang and Xue-Yi Chen

Axioms 2026, 15(3), 215; https://doi.org/10.3390/axioms15030215 - 13 Mar 2026

Viewed by 17

Abstract

This paper investigates biharmonic conformal immersions of surfaces into a conformally flat 3-space. We first establish a characterization of such immersions of totally umbilical surfaces into a generic 3-manifold. It is then proved that any biharmonic conformal immersion of a totally umbilical surface [...] Read more.

This paper investigates biharmonic conformal immersions of surfaces into a conformally flat 3-space. We first establish a characterization of such immersions of totally umbilical surfaces into a generic 3-manifold. It is then proved that any biharmonic conformal immersion of a totally umbilical surface into a nonpositively curved 3-manifold is necessarily a conformal minimal immersion. We further examine the biharmonicity of conformal immersions of totally umbilical planes into a conformally flat 3-space and construct explicit examples of such immersions from a 2-sphere (minus a point) into a conformally flat 3-sphere. Finally, the study is extended to biharmonic conformal immersions of Hopf cylinders associated with a Riemannian submersion. Full article

(This article belongs to the Section Geometry and Topology)

31 pages, 954 KB

Open AccessArticle

Poisson Mixed-Effects Count Regression Model Based on Double SCAD Penalty and Its Simulation Study

by Keqian Li, Xueni Ren, Hanfang Li and Youxi Luo

Axioms 2026, 15(3), 214; https://doi.org/10.3390/axioms15030214 - 12 Mar 2026

Viewed by 48

Abstract

This paper focuses on variable selection and parameter estimation for mixed-effects Poisson count regression models. To simultaneously select important variables in both fixed effects and random effects, we propose a double-penalized Poisson count regression model with the Smoothly Clipped Absolute Deviation (SCAD) penalty [...] Read more.

This paper focuses on variable selection and parameter estimation for mixed-effects Poisson count regression models. To simultaneously select important variables in both fixed effects and random effects, we propose a double-penalized Poisson count regression model with the Smoothly Clipped Absolute Deviation (SCAD) penalty imposed on both components. To estimate the unknown parameters, we develop a new iterative algorithm called the Double SCAD–Local Quadratic Approximation (DSCAD-LQA) algorithm. Under regularity conditions, the consistency and Oracle property of the proposed estimator are established. Simulation studies are conducted under two types of penalty parameter selection criteria: the Schwarz Information Criterion (SIC) and the Generalized Approximate Cross-Validation (GACV). We evaluate the performance of the proposed method under different levels of correlation among explanatory variables and different covariance structures of random effects. Comparisons are also carried out with the non-penalized model, the single-penalized model, and the double LASSO-penalized model. The results demonstrate that the proposed double SCAD penalty method performs better than the other three methods in terms of important variable selection and coefficient estimation, and is especially effective for sparse models. Full article

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30 pages, 1036 KB

Open AccessArticle

Classical and Bayesian Inference for the Two-Parameter Chen Distribution with Random Censored Data

by Zihan Zhao, Wenhao Gui, Minghui Liu and Lanxi Zhang

Axioms 2026, 15(3), 213; https://doi.org/10.3390/axioms15030213 - 12 Mar 2026

Viewed by 122

Abstract

This study explores classical and Bayesian estimation for the two-parameter Chen distribution with randomly censored data, where censoring times follow an independent two-parameter Chen distribution with separate shape and scale parameters. We first derive the maximum likelihood estimators of the unknown parameters, together [...] Read more.

This study explores classical and Bayesian estimation for the two-parameter Chen distribution with randomly censored data, where censoring times follow an independent two-parameter Chen distribution with separate shape and scale parameters. We first derive the maximum likelihood estimators of the unknown parameters, together with their asymptotic variances and credible intervals, and further adopt the method of moments, L-moments and least squares methods for classical estimation. Under the generalized entropy loss function and inverse gamma priors, Bayesian estimation is implemented via Gibbs sampling, with the highest posterior density credible intervals of parameters constructed accordingly. We also investigate the estimation of key reliability and lifetime characteristics of the distribution, and conduct Monte Carlo simulations to compare the performance of all aforementioned estimation methods. Finally, two real-world CMAPSS jet engine lifetime datasets from NASA are applied to validate the practical effectiveness of the proposed estimation approaches, demonstrating the enhanced flexibility of the Chen distribution compared to the exponential distribution in fitting aerospace-related censored data, given the marginal p-values in the K-S tests. Full article

(This article belongs to the Special Issue New Perspectives in Mathematical Statistics, 2nd Edition)

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21 pages, 2286 KB

Open AccessArticle

Command-Filtered Fuzzy Adaptive Output Feedback Control for Nonlinear Power Systems with Actuator Faults

by Sen Wang, Junzhe Yan, Chenxuan Sheng, Huai Liu and Guobao Liu

Axioms 2026, 15(3), 212; https://doi.org/10.3390/axioms15030212 - 12 Mar 2026

Viewed by 206

Abstract

This study presents a command-filtered fuzzy adaptive control method for nonlinear thyristor controlled series compensation (TCSC) systems subject to actuator faults, unknown nonlinearities, and unmeasurable states. To enhance applicability, the TCSC-based single-machine infinite-bus (SMIB) system is first transformed into a nonlinear form preserving [...] Read more.

This study presents a command-filtered fuzzy adaptive control method for nonlinear thyristor controlled series compensation (TCSC) systems subject to actuator faults, unknown nonlinearities, and unmeasurable states. To enhance applicability, the TCSC-based single-machine infinite-bus (SMIB) system is first transformed into a nonlinear form preserving the inherent nonlinear characteristics of the power system. A state observer is then designed to estimate the unmeasurable states. Using these estimated states, a fuzzy control algorithm approximates the uncertain nonlinearities. By integrating command filtering techniques, an adaptive output feedback controller is developed, which ensures system stability and avoids the “explosion of complexity” issue. Simulation results verify the effectiveness of the proposed control approach. Full article

(This article belongs to the Topic Modeling, Stability, and Control of Dynamic Systems and Their Applications)

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18 pages, 297 KB

Open AccessArticle

Formalizing Structural Semiotics: A Category-Theoretic Approach to Actantial Systems

by Michael Fowler

Axioms 2026, 15(3), 211; https://doi.org/10.3390/axioms15030211 - 12 Mar 2026

Viewed by 70

Abstract

In this article we present a categorical reconstruction of the actantial model introduced by A. J. Greimas in structural semiotics. Using Spivak and Kent’s framework of ontological logs and schema-based categories, the actantial system is formalized as a categorical schema whose Objects and [...] Read more.

In this article we present a categorical reconstruction of the actantial model introduced by A. J. Greimas in structural semiotics. Using Spivak and Kent’s framework of ontological logs and schema-based categories, the actantial system is formalized as a categorical schema whose Objects and morphisms encode relational roles in narrative structures. Standard categorical constructions such as pullbacks, morphism images, comma categories, and functors are used both to formalize the actantial relations identified by Greimas and to reveal additional structural relations that emerge from the categorical treatment of the model. By translating the actantial model into a schema equipped with instances, the framework provides a precise account of actantial roles, their compositional properties, and their realization in discourse. Functorial and profunctorial mappings are used to model the migration of narrative instances between schemas and the correspondence between abstract roles and textual fragments, offering a formal account of Greimas’s distinction between the narrative and discursive planes. Our approach adopts a relational, extensional perspective in which actantial roles are characterized by their participation in networks of relations and instances. The results illustrate how categorical schemas can function as a unifying formalism for role-based conceptual systems, with applications in comparative narrative analysis. Full article

(This article belongs to the Special Issue Applied Mathematics and Mathematical Modeling)

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20 pages, 23754 KB

Open AccessFeature PaperArticle

Sphere Packings in

2 \frac{1}{2}

Dimensions

by Kenneth Stephenson

Axioms 2026, 15(3), 210; https://doi.org/10.3390/axioms15030210 - 12 Mar 2026

Viewed by 120

Abstract

This paper investigates cylindrical sphere packings, that is, patterns of uniform spheres with mutually disjoint interiors which are all tangent to a common cylinder. The key unifying themes are the existence and uniqueness of hexagonal packings, in which each sphere is tangent to [...] Read more.

This paper investigates cylindrical sphere packings, that is, patterns of uniform spheres with mutually disjoint interiors which are all tangent to a common cylinder. The key unifying themes are the existence and uniqueness of hexagonal packings, in which each sphere is tangent to six others. Constructions are both intuitive and subtle, but result in the complete characterization in terms of integer parameter pairs

(m, n)

. Interesting questions in rigidity and density are encountered. Density questions arise because the packings, being of equal diameter, lie within the space between inner and outer cylinders. This density problem hovers between the 2D and 3D sphere packing cases, and though it is not solved here, it is conjectured that the hexagonal packings are densest for the countable number of cylinders which support them. Other geometric objects are along for the ride, including equilateral triangles and the packings’ dual graphs, which are associated with patterns of carbon atoms forming buckytubes. Interesting structural rigidity questions also arise. Full article

(This article belongs to the Section Geometry and Topology)

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21 pages, 15067 KB

Open AccessArticle

Fixed/Predefined-Time Synchronization for Delayed Memristive Reaction-Diffusion Neural Networks Subject to Stochastic Disturbances

by Gang Wang, Ikram Mamtimin and Abdujelil Abdurahman

Axioms 2026, 15(3), 209; https://doi.org/10.3390/axioms15030209 - 12 Mar 2026

Viewed by 106

Abstract

This paper investigates the fixed-time (FXT) and predefined-time (PDT) synchronization of memristive neural networks (MNNs) subject to stochastic disturbances, reaction-diffusion terms, and time delays. First, a new PDT stability criterion is established for stochastic nonlinear systems, which permits a priori assignment of the [...] Read more.

This paper investigates the fixed-time (FXT) and predefined-time (PDT) synchronization of memristive neural networks (MNNs) subject to stochastic disturbances, reaction-diffusion terms, and time delays. First, a new PDT stability criterion is established for stochastic nonlinear systems, which permits a priori assignment of the settling time bound regardless of initial conditions, and offers a more concise form than prior results. Second, by leveraging Green’s formula, integral inequality, and stochastic analysis, some sufficient conditions are derived to guarantee FXT and PDT synchronization of introduced stochastic MNNs with reaction-diffusion terms. Finally, numerical simulations are given to validate the effectiveness of the proposed synchronization scheme. Full article

(This article belongs to the Special Issue Numerical and Analytical Methods for Partial Differential Equations with Integral Boundary Conditions)

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16 pages, 776 KB

Open AccessArticle

On the Relationship Between the Pseudospectrum and the Quadratic Numerical Range for Upper Triangular Bounded Operator Matrices

by Yang Yu and Guolin Hou

Axioms 2026, 15(3), 208; https://doi.org/10.3390/axioms15030208 - 12 Mar 2026

Viewed by 78

Abstract

This paper studies the pseudospectral inclusion property of upper triangular bounded operator matrices in Hilbert spaces. It is proven that the pseudospectra of upper triangular bounded operator matrices are contained within the closure of the quadratic numerical range. Our result extends the inclusion [...] Read more.

This paper studies the pseudospectral inclusion property of upper triangular bounded operator matrices in Hilbert spaces. It is proven that the pseudospectra of upper triangular bounded operator matrices are contained within the closure of the quadratic numerical range. Our result extends the inclusion relationship between the spectra of block operator matrices and the quadratic numerical range, as well as the inclusion relationship between the pseudospectra and the numerical range, to the pseudospectral case. Under appropriate conditions, we characterize the distribution range of the pseudospectra of upper triangular bounded operator matrices, and provide an example to illustrate the validity of the conclusions. Full article

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21 pages, 1506 KB

Open AccessArticle

A Unified Rotation-Minimizing Darboux Framework for Curves and Relativistic Ruled Surfaces in Minkowski Three-Space

by Mona Bin-Asfour, Ghaliah Alhamzi, Emad Solouma and Sayed Saber

Axioms 2026, 15(3), 207; https://doi.org/10.3390/axioms15030207 - 11 Mar 2026

Viewed by 79

Abstract

We propose a comprehensive rotation-minimizing (RM) Darboux framework for the study of curve theory and relativistic ruled surfaces in Minkowski three-space

E_{1}^{3}

. The construction merges the adaptability of the classical Darboux frame to surface geometry with the reduced rotational behavior [...] Read more.

We propose a comprehensive rotation-minimizing (RM) Darboux framework for the study of curve theory and relativistic ruled surfaces in Minkowski three-space

E_{1}^{3}

. The construction merges the adaptability of the classical Darboux frame to surface geometry with the reduced rotational behavior characteristic of RM frames, yielding a natural geometric description of curves in a Lorentzian environment. For unit speed non-null curves, the governing equations of the RM Darboux frame are derived, and precise connections between the RM curvature functions and the classical Frenet and Darboux invariants are obtained, thereby elucidating the geometric significance of RM curvatures in Lorentzian geometry. Within this setting, multiple classes of ruled surfaces are generated using RM Darboux frame vector fields. Necessary and sufficient conditions for developability, minimality, and flatness are formulated exclusively in terms of RM curvature quantities. The role of the causal character of the generating curve is analyzed in detail, revealing distinct geometric behaviors for space-like and time-like cases. These findings indicate that the RM Darboux framework constitutes a flexible and effective approach for modeling curve-induced surface geometries in Minkowski space, with potential relevance to relativistic kinematics, world sheet constructions, and geometric problems arising in mathematical physics. Full article

(This article belongs to the Special Issue Theory and Applications: Differential Geometry)

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21 pages, 1416 KB

Open AccessArticle

Mean-Variance Investment and Per-Loss Reinsurance Strategies in Contagion Financial Markets

by Xiuxian Chen and Zhongyang Sun

Axioms 2026, 15(3), 206; https://doi.org/10.3390/axioms15030206 - 11 Mar 2026

Viewed by 191

Abstract

This paper investigates the optimal investment and reinsurance problem for insurers in a financial market with contagion risk. The prices of risky assets are assumed to follow a jump–diffusion model, where the jump component is driven by a multidimensional dynamic contagion process with [...] Read more.

This paper investigates the optimal investment and reinsurance problem for insurers in a financial market with contagion risk. The prices of risky assets are assumed to follow a jump–diffusion model, where the jump component is driven by a multidimensional dynamic contagion process with diffusion (DCPD). This process simultaneously captures jumps triggered by endogenous and exogenous excitations, effectively characterizing the dynamic contagion effects arising from the joint influence of multiple factors in financial markets. The insurer aims to maximize a mean-variance (MV) utility function by purchasing per-loss reinsurance and investing the surplus in the contagion financial market. By solving the extended Hamilton–Jacobi–Bellman (HJB) equations, we derive the time-consistent equilibrium investment and reinsurance strategies, as well as explicit expressions for the equilibrium value function. These results are characterized by two nonlocal partial differential equations (PDEs), whose probabilistic solutions are obtained through the Feynman–Kac formula. Finally, numerical experiments illustrate how equilibrium strategies respond to changes in contagion intensity and confirm the effectiveness of the proposed model. Full article

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22 pages, 381 KB

Open AccessArticle

Multiplicity Result of Solutions to the Fractional Problems with (p,q)-Growth and Hardy Potentials

by Yun-Ho Kim

Axioms 2026, 15(3), 205; https://doi.org/10.3390/axioms15030205 - 10 Mar 2026

Viewed by 71

Abstract

This paper focuses on establishing the existence of infinitely many solutions for non-local fractional equations characterized by unbalanced growth and Hardy potentials. We prove that these solutions converge to zero in the

L^{\infty}

-norm, requiring conditions on the nonlinearity only near the [...] Read more.

This paper focuses on establishing the existence of infinitely many solutions for non-local fractional equations characterized by unbalanced growth and Hardy potentials. We prove that these solutions converge to zero in the

L^{\infty}

-norm, requiring conditions on the nonlinearity only near the origin and dispensing with assumptions at infinity. As far as we are aware, results for non-local fractional

(p, q)

-Laplacian problems with singular coefficients such as Hardy potentials have not been extensively studied. To address this gap, we employ the dual fountain theorem together with the modified functional method. Full article

(This article belongs to the Special Issue Fractional Calculus—Theory and Applications, 3rd Edition)

22 pages, 2208 KB

Open AccessArticle

Analysis and Cost Optimization of a Retrial Queue with Push-Out and Feedback Using Analytical and Metaheuristic Approaches

by Suganthi Poomalai, Saeid Jafari and Jayamani V. Nanjappan

Axioms 2026, 15(3), 204; https://doi.org/10.3390/axioms15030204 - 10 Mar 2026

Viewed by 92

Abstract

The paper explores an advanced single-server M/G/1 retrial queueing model that employs a push-out service with two unique classes of customers, i.e., transient (priority) customers and recurrent customers. The arrivals of customers are Poisson process. The service time of customers and retrial time [...] Read more.

The paper explores an advanced single-server M/G/1 retrial queueing model that employs a push-out service with two unique classes of customers, i.e., transient (priority) customers and recurrent customers. The arrivals of customers are Poisson process. The service time of customers and retrial time of transit customers are follow general probability distributions. The inter-retrial time of the recurrent customer is exponentially distributed. The system also includes feedback behavior of transit customers and probabilistic push-out of repeat customers. Closed-form formulae are obtained expressing steady-state distributions of important system states using supplementary variable technique (SVT) and probability generating functions (PGFs). The impact of parameters is shown with the help of numerical experiments, and the Beetle Antennae Search (BAS) algorithm is used to optimise the performance of the system. These results are useful in designing and optimization of priority-based service systems such as cloud computing systems, communication networks, and real-time task scheduling systems. Full article

(This article belongs to the Section Mathematical Analysis)

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31 pages, 2615 KB

Open AccessArticle

Zeroth-Order Riemannian Adaptive Regularized Proximal Quasi-Newton Optimization Method

by Yinpu Ma, Cunlin Li, Zhichao Wang and Qian Li

Axioms 2026, 15(3), 203; https://doi.org/10.3390/axioms15030203 - 10 Mar 2026

Viewed by 200

Abstract

Recently, the adaptive regularized proximal quasi-Newton (ARPQN) method has demonstrated a strong performance in solving composite optimization problems over the Stiefel manifold. However, its reliance on first-order information limits its applicability to scenarios where gradient and Hessian evaluations are unavailable or costly. In [...] Read more.

Recently, the adaptive regularized proximal quasi-Newton (ARPQN) method has demonstrated a strong performance in solving composite optimization problems over the Stiefel manifold. However, its reliance on first-order information limits its applicability to scenarios where gradient and Hessian evaluations are unavailable or costly. In this paper, we propose a zeroth-order adaptive regularized proximal quasi-Newton method (ZO-ARPQN) for black-box composite optimization over Riemannian manifolds, particularly the Stiefel and symmetric positive definite (SPD) manifolds. The proposed method estimates the Riemannian gradient and curvature information through randomized one-point finite-difference approximations and adaptively updates a regularized quasi-Newton matrix to capture the local manifold geometry. Theoretically, we established global convergence and complex analyses under mild assumptions. More importantly, by incorporating curvature-aware regularization and random perturbations in the proximal quasi-Newton framework, we proved that ZO-ARPQN can escape strict saddle points with a high probability. This guarantees convergence to a stationary point, even in the absence of explicit gradients. Extensive numerical experiments were conducted on manifold-constrained problems, including sparse PCA and robot stiffness tuning. These demonstrated that ZO-ARPQN shows a competitive convergence behavior compared with other state-of-the-art Riemannian optimization methods, while requiring only function evaluations. Full article

(This article belongs to the Section Geometry and Topology)

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22 pages, 967 KB

Open AccessArticle

Solutions of a Fuzzy Difference Equation with Maximum

by Lirong Ma, Changyou Wang and Yue Sun

Axioms 2026, 15(3), 202; https://doi.org/10.3390/axioms15030202 - 9 Mar 2026

Viewed by 119

Abstract

This paper systematically investigates the dynamical properties of a class of max-type fuzzy difference equation. The study first establishes the existence and uniqueness of the solution sequence under given initial conditions with positive fuzzy numbers. Subsequently, by applying the cut-set theory, the fuzzy [...] Read more.

This paper systematically investigates the dynamical properties of a class of max-type fuzzy difference equation. The study first establishes the existence and uniqueness of the solution sequence under given initial conditions with positive fuzzy numbers. Subsequently, by applying the cut-set theory, the fuzzy equation is transformed into a system coupled by two ordinary difference equations. Through a combination of case analysis and mathematical induction, the study rigorously demonstrates that the solutions of this system exhibit global periodicity with a period of 4, while also deriving the exact closed-form expressions of the periodic solutions. Based on the periodic solutions obtained from the ordinary difference system, the research successfully reveals the periodic characteristics of the solutions to the original fuzzy difference equation and rigorously analyzes their boundedness and persistence. Finally, numerical simulations conducted with Matlab 2016 provide robust data support for the theoretical conclusions and the effectiveness of the methodology. Full article

(This article belongs to the Special Issue Delay Differential Equations: Theory, Control and Applications)

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26 pages, 760 KB

Open AccessArticle

On Contraction Principles in Product Spaces and Applications in Iterated Function Systems

by Muhammad Dabeer Mughal, Muhammad Nazam and Hami Gundogdu

Axioms 2026, 15(3), 201; https://doi.org/10.3390/axioms15030201 - 9 Mar 2026

Viewed by 146

Abstract

This paper develops an extension of the research work by Proinov to the product spaces

X^{| I |}

(I is representing an indexing set). This paper also introduces a novel class of

(L; Y)

-contractions defined on a [...] Read more.

This paper develops an extension of the research work by Proinov to the product spaces

X^{| I |}

(I is representing an indexing set). This paper also introduces a novel class of

(L; Y)

-contractions defined on a supremum metric

(X^{| I |}, d^{'})

. In supremum metric

(X^{| I |}, d^{'})

, several new fixed point theorems for

(L; Y)

-contractions have been established that generalize well-known ideas like the Banach, Geraghty, Boyd–Wong, and Wardowski principles. This paper also contributes a new iterated function system (IFS) built on the family of

(L; Y)

-contractions and demonstrates the existence and uniqueness of fractals (in other words, compact attractors) in a complete supremum metric

(X^{| I |}, d^{'})

. Theoretical work is illustrated with examples and graphs. Full article

(This article belongs to the Section Mathematical Analysis)

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13 pages, 277 KB

Open AccessArticle

Multivariate Approximation by Overactivated and Spiked Multivariate Convolutions as Positive Linear Operators

by George A. Anastassiou

Axioms 2026, 15(3), 200; https://doi.org/10.3390/axioms15030200 - 8 Mar 2026

Viewed by 183

Abstract

This study quantitatively approximates the unit operator using three types of multivariate, overactivated, and spiked convolution operators. The core of these operators is a multivariate “cusp” kernel, which acts as a novel, compact-support activation function derived from a constructed S-shaped finite-length arc. This [...] Read more.

This study quantitatively approximates the unit operator using three types of multivariate, overactivated, and spiked convolution operators. The core of these operators is a multivariate “cusp” kernel, which acts as a novel, compact-support activation function derived from a constructed S-shaped finite-length arc. This arc is itself formed by combining two general sigmoid activation functions. The operators are multivariate positive linear ones. The research initially establishes their basic convergence properties. It then explores simultaneous and iterated approximations, utilizing inequalities and the multivariate modulus of continuity of the function being approximated. Full article

(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)

27 pages, 425 KB

Open AccessArticle

Fractional Kirchhoff-Hardy Problem: Breaking of Resonance and General Existence Results

by Boumediene Abdellaoui, Abdelhalim Azzouz, Ahmed Bensedik and Rachid Bentifour

Axioms 2026, 15(3), 199; https://doi.org/10.3390/axioms15030199 - 7 Mar 2026

Viewed by 204

Abstract

In this paper, we study a fractional Kirchhoff problem with a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient: [...] Read more.

In this paper, we study a fractional Kirchhoff problem with a Hardy-type singular potential and general nonlinearities depending on the solution and its gradient:

M ({\int \int}_{R^{N} \times R^{N}} \frac{{| u (x) - u (y) |}^{q}}{{| x - y |}^{N + q s}} d x d y) {(- Δ)}^{s} u = λ \frac{u}{{| x |}^{2 s}} + f (x, u, \nabla u) in Ω,

where

Ω \subset R^{N}

is a bounded domain containing the origin,

s \in (0, 1)

,

q \in (1, 2]

with

N > 2 s

,

λ > 0

, and f is a measurable non-negative function satisfying suitable hypotheses. The main objective is to establish the existence of positive solutions for the largest possible class of nonlinearities f without imposing restrictions on

λ

. Two main cases areconsidered:

(I) - f (x, u, \nabla u) = u^{p} + μ, and (I I) - f (x, u, \nabla u) = {| \nabla u |}^{p} + μ g .

Existence is proved under suitable hypotheses on

q, p

and the data

g, μ

. The results are new, including for the local case

s = 1

. Full article

(This article belongs to the Section Mathematical Analysis)

15 pages, 898 KB

Open AccessArticle

Exploring Nonlinear Dynamics of the (3+1)-Dimensional Boussinesq-Type Equation: Wave Patterns and Sensitivity Insight

by Ejaz Hussain, Ali H. Tedjani and Muhammad Amin S. Murad

Axioms 2026, 15(3), 198; https://doi.org/10.3390/axioms15030198 - 6 Mar 2026

Viewed by 208

Abstract

This study examines a nonlinear partial differential equation, namely the (3+1)-dimensional Boussinesq-type equation. To explore this model, three versatile analytical approaches are applied: the Exp-function method, the Kudryashov method, and the Riccati equation method. Using these techniques, a range of exact analytical solutions [...] Read more.

This study examines a nonlinear partial differential equation, namely the (3+1)-dimensional Boussinesq-type equation. To explore this model, three versatile analytical approaches are applied: the Exp-function method, the Kudryashov method, and the Riccati equation method. Using these techniques, a range of exact analytical solutions is derived, exhibiting diverse structural forms such as periodic, kink-type, rational, and trigonometric solutions. The analysis reveals the rich dynamical behavior of the equation and demonstrates its effectiveness in modeling a variety of nonlinear wave phenomena across different physical contexts. Several of the obtained solutions are illustrated through graphical representations for better interpretation. The results include hyperbolic, trigonometric, and rational function solutions, along with a sensitivity analysis. To highlight the physical relevance of the findings, suitable parameter values are selected, and the corresponding wave behaviors are visualized using three-dimensional and contour plots generated with Maple 2024. Overall, the study provides valuable insights into the mechanisms underlying the generation and propagation of complex nonlinear phenomena in fields such as fluid dynamics, optical fiber systems, plasma physics, and ocean wave transmission. Full article

(This article belongs to the Special Issue New Perspectives in Applied Mathematics with Nonlinear Equations and Dynamical Systems)

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23 pages, 434 KB

Open AccessArticle

Multiary Gradings

by Steven Duplij

Axioms 2026, 15(3), 197; https://doi.org/10.3390/axioms15030197 - 6 Mar 2026

Viewed by 226

Abstract

This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate various compatibility conditions between the arity of algebra operations and grading [...] Read more.

This article develops a comprehensive theory of multiary graded polyadic algebras, extending the classical concept of group-graded algebras to higher-arity structures. We introduce the notion of grading by multiary groups and investigate various compatibility conditions between the arity of algebra operations and grading group operations. Key results include quantization rules connecting arities, classification of graded homomorphisms, the First Isomorphism Theorem for graded polyadic algebras and concrete examples including ternary superalgebras and polynomial algebras over n-ary matrices. The theory reveals fundamentally new phenomena not present in the binary case, such as the existence of higher power gradings and nontrivial constraints on arity compatibility. Full article

(This article belongs to the Special Issue Computational Algebra, Coding Theory and Cryptography: Theory and Applications, 3rd Edition)

18 pages, 335 KB

Open AccessArticle

Global Low-Energy Weak Solutions of a Fluid–Particle Interaction Model with Vacuum in ℝ³

by Bingyuan Huang, Jinrui Huang, Zonghao Lin and Yongtong Liu

Axioms 2026, 15(3), 196; https://doi.org/10.3390/axioms15030196 - 6 Mar 2026

Viewed by 182

Abstract

Provided that the initial data

(ρ_{0}, v_{0}, η_{0})

is of small energy around steady state

(ρ^{▵}, 0, 0)

, in this work we obtain the global-in-time existence of weak solutions to [...] Read more.

Provided that the initial data

(ρ_{0}, v_{0}, η_{0})

is of small energy around steady state

(ρ^{▵}, 0, 0)

, in this work we obtain the global-in-time existence of weak solutions to a fluid particle interaction system. It should be pointed out that vacuum is allowed in this work. Full article

(This article belongs to the Section Mathematical Analysis)

19 pages, 18350 KB

Open AccessArticle

Upper and Lower Bounds for Eigenvalues of the Elliptic Operator by Weak Galerkin Quadrilateral Spectral Element Methods

by Xiaofeng Xu and Jiajia Pan

Axioms 2026, 15(3), 195; https://doi.org/10.3390/axioms15030195 - 6 Mar 2026

Viewed by 209

Abstract

In this study, we investigate the upper- and lower-bound approximations of numerical eigenvalues derived by weak Galerkin spectral element methods on arbitrary convex quadrilateral meshes for the Laplace eigenvalue problem. Firstly, the Piola transformation is employed to construct the approximation space for weak [...] Read more.

In this study, we investigate the upper- and lower-bound approximations of numerical eigenvalues derived by weak Galerkin spectral element methods on arbitrary convex quadrilateral meshes for the Laplace eigenvalue problem. Firstly, the Piola transformation is employed to construct the approximation space for weak gradients on each convex quadrilateral element, while a one-to-one mapping is used to establish the approximation space for weak functions. Subsequently, based on the weak Galerkin spectral element approximation space defined on convex quadrilateral meshes, a Galerkin approximation scheme is formulated, and its well-posedness is then analyzed. Furthermore, numerical experiments are performed on arbitrary convex quadrilateral meshes of the square and L-shaped domains to explore the upper- and lower-bound approximations of numerical eigenvalues. Numerical findings indicate that the presented method not only obtains optimal orders of convergence with respect to both the mesh size and the polynomial degree, but also provides upper- and lower-bound approximations for the reference eigenvalues by proper choices of polynomial degrees in approximation spaces and parameters of the approximation scheme in both h-version and p-version weak Galerkin spectral element methods. This study offers new perspectives and methodologies for the high-precision numerical solution of eigenvalue problems in elliptic equations. Full article

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40 pages, 1733 KB

Open AccessArticle

Fine Stability Properties of the Hankel and Wiener–Khinchin Transforms

by François Vigneron

Axioms 2026, 15(3), 194; https://doi.org/10.3390/axioms15030194 - 6 Mar 2026

Viewed by 142

Abstract

The Fourier transform is continuous in the weak sense of tempered distribution; this ensures the weak stability of Fourier pairs. This article investigates a stronger form of stability of the pair of homogeneous profiles [...] Read more.

The Fourier transform is continuous in the weak sense of tempered distribution; this ensures the weak stability of Fourier pairs. This article investigates a stronger form of stability of the pair of homogeneous profiles

(| x |^{- α}, c_{d} | ξ |^{d - α})

on

R^{d}

that encompasses the case where the homogeneous profiles exist only on a large but finite range. In this case, largely overlooked in the literature, we provide precise error estimates in terms of the size of the tails outside the homogeneous range. We also prove a series of refined properties of the Fourier transform on related questions including criteria that ensure an approximate homogeneous behavior asymptotically near the origin or at infinity. The sharpness of our results is checked with numerical simulations. We also investigate briefly how these results consolidate the mathematical foundations of turbulence theory. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics, 2nd Edition)

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27 pages, 648 KB

Open AccessArticle

Synergistic Evolutionary Optimization with Reinforcement Learning for Multi-Objective Energy-Efficient Hybrid Flow Shop Scheduling

by Yuchen Liu, Ting Shu, Xuesong Yin and Jinsong Xia

Axioms 2026, 15(3), 193; https://doi.org/10.3390/axioms15030193 - 6 Mar 2026

Viewed by 248

Abstract

The Energy-Efficient Hybrid Flow Shop Scheduling Problem poses a significant multi-objective optimization challenge, necessitating the simultaneous minimization of conflicting objectives: Total Tardiness, Total Energy Cost, and Carbon Trading Cost. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is a classic algorithm in the field [...] Read more.

The Energy-Efficient Hybrid Flow Shop Scheduling Problem poses a significant multi-objective optimization challenge, necessitating the simultaneous minimization of conflicting objectives: Total Tardiness, Total Energy Cost, and Carbon Trading Cost. The Non-dominated Sorting Genetic Algorithm II (NSGA-II) is a classic algorithm in the field of multi-objective optimization. However, this algorithm frequently lacks the adaptive capability required to navigate high-dimensional solution spaces, often trapping the search in local optima, particularly when constrained by practical energy states of heterogeneous machines. To address these complexities, this study proposes a hybrid algorithm, named QGN, integrating Q-learning, the Grey Wolf Optimizer (GWO), and the NSGA-II. Specifically, QGN algorithm integrates NSGA-II for robust diversity maintenance with GWO for high-precision intensification. Unlike static hybrid methods, QGN employs a Q-learning agent as an adaptive controller to dynamically balance global exploration and local refinement, providing a theoretically grounded response to the rugged search landscape created by machine heterogeneity. Comprehensive experimental validation across diverse production scenarios confirms that QGN significantly outperforms baselines, including NSGA-II, Jaya, and Multi-Objective Evolutionary Algorithm based on Decomposition (MOEA/D), as well as the state-of-the-art Q-learning- and GVNS-driven NSGA-II (QVNS) algorithm, in terms of both convergence and diversity. The results indicate that the proposed algorithm yields superior solution dominance, generates a substantially larger set of non-dominated solutions, and maintains a more uniform distribution along the Pareto front. Full article

(This article belongs to the Section Mathematical Analysis)

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108 pages, 1969 KB

Open AccessArticle

Ramanujan–Santos–Sales Hypermodular Operator Theorem and Spectral Kernels for Geometry-Adaptive Neural Operators in Anisotropic Besov Spaces

by Rômulo Damasclin Chaves dos Santos and Jorge Henrique de Oliveira Sales

Axioms 2026, 15(3), 192; https://doi.org/10.3390/axioms15030192 - 6 Mar 2026

Viewed by 161

Abstract

We present Hyperbolic Symmetric Hypermodular Neural Operators (ONHSH), a novel operator learning framework for solving partial differential equations (PDEs) in curved, anisotropic, and modularly structured domains. The architecture integrates three components: hyperbolic-symmetric activation kernels that adapt to non-Euclidean geometries, modular spectral smoothing informed [...] Read more.

We present Hyperbolic Symmetric Hypermodular Neural Operators (ONHSH), a novel operator learning framework for solving partial differential equations (PDEs) in curved, anisotropic, and modularly structured domains. The architecture integrates three components: hyperbolic-symmetric activation kernels that adapt to non-Euclidean geometries, modular spectral smoothing informed by arithmetic regularity, and curvature-sensitive kernels based on anisotropic Besov theory. In its theoretical foundation, the Ramanujan–Santos–Sales Hypermodular Operator Theorem establishes minimax-optimal approximation rates and provides a spectral-topological interpretation through noncommutative Chern characters. These contributions unify harmonic analysis, approximation theory, and arithmetic topology into a single operator learning paradigm. In addition to theoretical advances, ONHSH achieves robust empirical results. Numerical experiments on thermal diffusion problems demonstrate superior accuracy and stability compared to Fourier Neural Operators and Geo-FNO. The method consistently resolves high-frequency modes, preserves geometric fidelity in curved domains, and maintains robust convergence in anisotropic regimes. Error decay rates closely match theoretical minimax predictions, while Voronovskaya-type expansions capture the tradeoffs between bias and spectral variance observed in practice. Notably, ONHSH kernels preserve Lorentz invariance, enabling accurate modeling of relativistic PDE dynamics. Overall, ONHSH combines rigorous theoretical guarantees with practical performance improvements, making it a versatile and geometry-adaptable framework for operator learning. By connecting harmonic analysis, spectral geometry, and machine learning, this work advances both the mathematical foundations and the empirical scope of PDE-based modeling in structured, curved, and arithmetically. Full article

(This article belongs to the Special Issue Fractional Differential Equation and Its Applications)

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10 pages, 239 KB

Open AccessArticle

Efficient Minus and Signed Domination in Proper Interval Graphs with a Totally Unimodular Structure

by Chuan-Min Lee

Axioms 2026, 15(3), 191; https://doi.org/10.3390/axioms15030191 - 5 Mar 2026

Viewed by 166

Abstract

The efficient minus domination problem (EMDP) and the efficient signed domination problem (ESDP) are domination-type problems in graphs. These problems are known to be NP-complete on chordal graphs and polynomially solvable on chain interval graphs, while the complexity on proper interval graphs remained [...] Read more.

The efficient minus domination problem (EMDP) and the efficient signed domination problem (ESDP) are domination-type problems in graphs. These problems are known to be NP-complete on chordal graphs and polynomially solvable on chain interval graphs, while the complexity on proper interval graphs remained open. By exploiting the totally unimodular structure of the closed-neighborhood matrix induced by a proper interval ordering, we obtain linear programming formulations under which both the EMDP and ESDP become polynomially solvable. The same perspective naturally extends to vertex-weighted settings and to other domination variants defined by similar neighborhood constraints. Full article

(This article belongs to the Special Issue Advances in Graph Theory with Its Applications)

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Axioms, Volume 15, Issue 3 (March 2026) – 64 articles

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