Axioms

21 pages, 330 KiB

Open AccessArticle

Distribution Modulo One of αp^γ + β for Special Classes of Primes

by Atanaska Georgieva and Tatiana L. Todorova

Axioms 2025, 14(7), 532; https://doi.org/10.3390/axioms14070532 (registering DOI) - 12 Jul 2025

Let

α, β \in R

with

α \neq 0

, and let

γ \in (0, 5 / 6)

. Define the set

M_{1}

to consist of primes p such that

p + 2

is almost prime, and let [...] Read more.

Let

α, β \in R

with

α \neq 0

, and let

γ \in (0, 5 / 6)

. Define the set

M_{1}

to consist of primes p such that

p + 2

is almost prime, and let

M_{2}

be the set of primes of the form

p = a^{2} + b^{2} + 1

. We study the distribution of

α p^{γ} + β

modulo one, as p ranges over the sets

M_{1}

and

M_{2}

, respectively. Full article

(This article belongs to the Special Issue Mathematical Analysis and Applications IV)

20 pages, 1791 KiB

Open AccessArticle

The Effects of the Weak Allee Effect and Disease on the Dynamics of a Predator–Prey System: Stability and Bifurcation Properties

by Yurong Dong, Hua Liu, Jianhua Ye, Gang Ma and Yumei Wei

Axioms 2025, 14(7), 531; https://doi.org/10.3390/axioms14070531 (registering DOI) - 12 Jul 2025

Abstract

In this paper, an eco-epidemiological model with a weak Allee effect and prey disease dynamics is discussed. Mathematical features such as non-negativity, boundedness of solutions, and local stability of the feasible equilibria are discussed. Additionally, the transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation [...] Read more.

In this paper, an eco-epidemiological model with a weak Allee effect and prey disease dynamics is discussed. Mathematical features such as non-negativity, boundedness of solutions, and local stability of the feasible equilibria are discussed. Additionally, the transcritical bifurcation, saddle-node bifurcation, and Hopf bifurcation are proven using Sotomayor’s theorem and Poincare–Andronov–Hopf theorems. In addition, the correctness of the theoretical analysis is verified by numerical simulation. The numerical simulation results show that the eco-epidemiological model with a weak Allee effect has complex dynamics. If the prey population is not affected by disease, the predator becomes extinct due to a lack of food. Under low infection rates, all populations are maintained in a coexistent state. The Allee effect does not influence this coexistence. At high infection rates, if the prey population is not affected by the Allee effect, the infected prey is found to coexist in an oscillatory state. The predator population and the susceptible prey population will be extinct. If the prey population is affected by the Allee effect, all species will be extinct. Full article

27 pages, 367 KiB

Open AccessArticle

Existence of Sign-Changing Solutions for a Class of p(x)-Biharmonic Kirchhoff-Type Equations

by Rui Deng and Qing Miao

Axioms 2025, 14(7), 530; https://doi.org/10.3390/axioms14070530 (registering DOI) - 12 Jul 2025

Abstract

This paper mainly studies the existence of sign-changing solutions for the following

p (x)

-biharmonic Kirchhoff-type equations: [...] Read more.

This paper mainly studies the existence of sign-changing solutions for the following

p (x)

-biharmonic Kirchhoff-type equations:

- (a + b \int_{R^{N}} \frac{1}{p (x)} {| Δ u |}^{p (x)} d x) Δ_{p (x)}^{2} u + V (x) {| u |}^{p (x) - 2} u

=

K (x) f (u), x \in R^{N}

where

Δ_{p (x)}^{2} u = Δ ({| Δ u |}^{p (x) - 2} Δ u)

is the

p (x)

biharmonic operator,

a, b > 0

are constants,

N \geq 2

,

V (x), K (x)

are positive continuous functions which vanish at infinity, and the nonlinearity f has subcritical growth. Using the Nehari manifold method, deformation lemma, and other techniques of analysis, it is demonstrated that there are precisely two nodal domains in the problem’s least energy sign-changing solution

u_{b}

. In addition, the convergence property of

u_{b}

as

b \to 0

is also established. Full article

17 pages, 255 KiB

Open AccessArticle

Wrapping Generalized Laplace and Generalized Discrete Laplace Distributions

by Barry C. Arnold and Ashis SenGupta

Axioms 2025, 14(7), 529; https://doi.org/10.3390/axioms14070529 - 11 Jul 2025

Abstract

Generalized Laplace and discrete generalized Laplace models are described and some of their properties are reviewed. For the analysis of directional data, it is natural to consider wrapped versions of these models. A well-known result for the circular and, more recently, for multivariate [...] Read more.

Generalized Laplace and discrete generalized Laplace models are described and some of their properties are reviewed. For the analysis of directional data, it is natural to consider wrapped versions of these models. A well-known result for the circular and, more recently, for multivariate circular distributions that mixing and wrapping commute, allows us to readily determine the nature of these wrapped models. Some bivariate extensions of these models are discussed, together with some consideration of the feasibility of wrapping such models. Multivariate versions of the models can be envisioned. Full article

21 pages, 1847 KiB

Open AccessArticle

Fusion of Recurrence Plots and Gramian Angular Fields with Bayesian Optimization for Enhanced Time-Series Classification

by Maria Mariani, Prince Appiah and Osei Tweneboah

Axioms 2025, 14(7), 528; https://doi.org/10.3390/axioms14070528 - 10 Jul 2025

Abstract

Time-series classification remains a critical task across various domains, demanding models that effectively capture both local recurrence structures and global temporal dependencies. We introduce a novel framework that transforms time series into image representations by fusing recurrence plots (RPs) with both Gramian Angular [...] Read more.

Time-series classification remains a critical task across various domains, demanding models that effectively capture both local recurrence structures and global temporal dependencies. We introduce a novel framework that transforms time series into image representations by fusing recurrence plots (RPs) with both Gramian Angular Summation Fields (GASFs) and Gramian Angular Difference Fields (GADFs). This fusion enriches the structural encoding of temporal dynamics. To ensure optimal performance, Bayesian Optimization is employed to automatically select the ideal image resolution, eliminating the need for manual tuning. Unlike prior methods that rely on individual transformations, our approach concatenates RP, GASF, and GADF into a unified representation and generalizes to multivariate data by stacking transformation channels across sensor dimensions. Experiments on seven univariate datasets show that our method significantly outperforms traditional classifiers such as one-nearest neighbor with Dynamic Time Warping, Shapelet Transform, and RP-based convolutional neural networks. For multivariate tasks, the proposed fusion model achieves macro F1 scores of 91.55% on the UCI Human Activity Recognition dataset and 98.95% on the UCI Room Occupancy Estimation dataset, outperforming standard deep learning baselines. These results demonstrate the robustness and generalizability of our framework, establishing a new benchmark for image-based time-series classification through principled fusion and adaptive optimization. Full article

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26 pages, 1556 KiB

Open AccessArticle

Modified Two-Parameter Ridge Estimators for Enhanced Regression Performance in the Presence of Multicollinearity: Simulations and Medical Data Applications

by Muteb Faraj Alharthi and Nadeem Akhtar

Axioms 2025, 14(7), 527; https://doi.org/10.3390/axioms14070527 - 10 Jul 2025

Abstract

Predictive regression models often face a common challenge known as multicollinearity. This phenomenon can distort the results, causing models to overfit and produce unreliable coefficient estimates. Ridge regression is a widely used approach that incorporates a regularization term to stabilize parameter estimates and [...] Read more.

Predictive regression models often face a common challenge known as multicollinearity. This phenomenon can distort the results, causing models to overfit and produce unreliable coefficient estimates. Ridge regression is a widely used approach that incorporates a regularization term to stabilize parameter estimates and improve the prediction accuracy. In this study, we introduce four newly modified ridge estimators, referred to as RIRE1, RIRE2, RIRE3, and RIRE4, that are aimed at tackling severe multicollinearity more effectively than ordinary least squares (OLS) and other existing estimators under both normal and non-normal error distributions. The ridge estimators are biased, so their efficiency cannot be judged by variance alone; instead, we use the mean squared error (MSE) to compare their performance. Each new estimator depends on two shrinkage parameters,

k

and

d

, making the theoretical analysis complex. To address this, we employ Monte Carlo simulations to rigorously evaluate and compare these new estimators with OLS and other existing ridge estimators. Our simulations show that the proposed estimators consistently minimize the MSE better than OLS and other ridge estimators, particularly in datasets with strong multicollinearity and large error variances. We further validate their practical value through applications using two real-world datasets, demonstrating both their robustness and theoretical alignment. Full article

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

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18 pages, 300 KiB

Open AccessArticle

Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence

by Xiu-Liang Qiu, Kuldip Raj, Sanjeev Verma, Samrati Gorka, Shixiao Xiao and Qing-Bo Cai

Axioms 2025, 14(7), 526; https://doi.org/10.3390/axioms14070526 - 10 Jul 2025

Viewed by 50

Abstract

We explore the realm of uncertainty theory by investigating diverse notions of convergence and statistical convergence concerning complex uncertain sequences. Complex uncertain variables can be described as measurable functions mapping from an uncertainty space to the set of complex numbers. They are employed [...] Read more.

We explore the realm of uncertainty theory by investigating diverse notions of convergence and statistical convergence concerning complex uncertain sequences. Complex uncertain variables can be described as measurable functions mapping from an uncertainty space to the set of complex numbers. They are employed to represent and model complex uncertain quantities. We introduce the concept of lacunary almost statistical convergence of order

α

(0 < α \leq 1)

for complex uncertain sequences, examining various aspects of uncertainty such as distribution, mean, measure, uniformly almost sure convergence and almost sure convergence. Additionally, we establish connections between the constructed sequence spaces by providing illustrative instances. Importantly, lacunary almost statistical convergence provides a flexible framework for handling sequences with irregular behavior, which often arise in uncertain environments with imprecise data. This makes our approach particularly useful in practical fields such as engineering, data modeling and decision-making, where traditional deterministic methods are not always applicable. Our approach offers a more flexible and realistic framework for approximating functions in uncertain environments where classical convergence may not apply. Thus, this study contributes to approximation theory by extending its tools to settings involving imprecise or noisy data. Full article

(This article belongs to the Special Issue Numerical Computation, Approximation of Functions and Applied Mathematics, 2nd Edition)

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19 pages, 528 KiB

Open AccessArticle

Quantum-Inspired Attention-Based Semantic Dependency Fusion Model for Aspect-Based Sentiment Analysis

by Chenyang Xu, Xihan Wang, Jiacheng Tang, Yihang Wang, Lianhe Shao and Quanli Gao

Axioms 2025, 14(7), 525; https://doi.org/10.3390/axioms14070525 - 9 Jul 2025

Viewed by 127

Abstract

Aspect-Based Sentiment Analysis (ABSA) has gained significant popularity in recent years, which emphasizes the aspect-level sentiment representation of sentences. Current methods for ABSA often use pre-trained models and graph convolution to represent word dependencies. However, they struggle with long-range dependency issues in lengthy [...] Read more.

Aspect-Based Sentiment Analysis (ABSA) has gained significant popularity in recent years, which emphasizes the aspect-level sentiment representation of sentences. Current methods for ABSA often use pre-trained models and graph convolution to represent word dependencies. However, they struggle with long-range dependency issues in lengthy texts, resulting in averaging and loss of contextual semantic information. In this paper, we explore how richer semantic relationships can be encoded more efficiently. Inspired by quantum theory, we construct superposition states from text sequences and utilize them with quantum measurements to explicitly capture complex semantic relationships within word sequences. Specifically, we propose an attention-based semantic dependency fusion method for ABSA, which employs a quantum embedding module to create a superposition state of real-valued word sequence features in a complex-valued Hilbert space. This approach yields a word sequence density matrix representation that enhances the handling of long-range dependencies. Furthermore, we introduce a quantum cross-attention mechanism to integrate sequence features with dependency relationships between specific word pairs, aiming to capture the associations between particular aspects and comments more comprehensively. Our experiments on the SemEval-2014 and Twitter datasets demonstrate the effectiveness of the quantum-inspired attention-based semantic dependency fusion model for the ABSA task. Full article

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30 pages, 956 KiB

Open AccessArticle

Stochastic Production Planning with Regime-Switching: Sensitivity Analysis, Optimal Control, and Numerical Implementation

by Dragos-Patru Covei

Axioms 2025, 14(7), 524; https://doi.org/10.3390/axioms14070524 - 8 Jul 2025

Viewed by 81

Abstract

This study investigates a stochastic production planning problem with regime-switching parameters, inspired by economic cycles impacting production and inventory costs. The model considers types of goods and employs a Markov chain to capture probabilistic regime transitions, coupled with a multidimensional Brownian motion representing [...] Read more.

This study investigates a stochastic production planning problem with regime-switching parameters, inspired by economic cycles impacting production and inventory costs. The model considers types of goods and employs a Markov chain to capture probabilistic regime transitions, coupled with a multidimensional Brownian motion representing stochastic demand dynamics. The production and inventory cost optimization problem is formulated as a quadratic cost functional, with the solution characterized by a regime-dependent system of elliptic partial differential equations (PDEs). Numerical solutions to the PDE system are computed using a monotone iteration algorithm, enabling quantitative analysis. Sensitivity analysis and model risk evaluation illustrate the effects of regime-dependent volatility, holding costs, and discount factors, revealing the conservative bias of regime-switching models when compared to static alternatives. Practical implications include optimizing production strategies under fluctuating economic conditions and exploring future extensions such as correlated Brownian dynamics, non-quadratic cost functions, and geometric inventory frameworks. In contrast to earlier studies that imposed static or overly simplified regime-switching assumptions, our work presents a fully integrated framework—combining optimal control theory, a regime-dependent system of elliptic PDEs, and comprehensive numerical and sensitivity analyses—to more accurately capture the complex stochastic dynamics of production planning and thereby deliver enhanced, actionable insights for modern manufacturing environments. Full article

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15 pages, 298 KiB

Open AccessArticle

Fuzzy Treatment for Meromorphic Classes of Admissible Functions Connected to Hurwitz–Lerch Zeta Function

by Ekram E. Ali, Rabha M. El-Ashwah, Abeer M. Albalahi and Rabab Sidaoui

Axioms 2025, 14(7), 523; https://doi.org/10.3390/axioms14070523 - 8 Jul 2025

Viewed by 151

Abstract

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are [...] Read more.

Fuzzy differential subordinations, a notion taken from fuzzy set theory and used in complex analysis, are the subject of this paper. In this work, we provide an operator and examine the characteristics of meromorphic functions in the punctured open unit disk that are related to a class of complex parameter operators. Complex analysis ideas from geometric function theory are used to derive fuzzy differential subordination conclusions. Due to the compositional structure of the operator, some pertinent classes of admissible functions are studied through the application of fuzzy differential subordination. Full article

(This article belongs to the Special Issue New Developments in Geometric Function Theory, 3rd Edition)

31 pages, 2227 KiB

Open AccessArticle

Observer-Linked Branching (OLB)—A Proposed Quantum-Theoretic Framework for Macroscopic Reality Selection

by Călin Gheorghe Buzea, Florin Nedeff, Valentin Nedeff, Dragos-Ioan Rusu, Maricel Agop and Decebal Vasincu

Axioms 2025, 14(7), 522; https://doi.org/10.3390/axioms14070522 - 8 Jul 2025

Viewed by 189

Abstract

We propose Observer-Linked Branching (OLB), a mathematically rigorous extension of quantum theory in which an observer’s cognitive commitment actively modulates collapse dynamics at macroscopic scales. The OLB framework rests on four axioms, employing a norm-preserving nonlinear Schrödinger evolution and Lüders-type projection triggered by [...] Read more.

We propose Observer-Linked Branching (OLB), a mathematically rigorous extension of quantum theory in which an observer’s cognitive commitment actively modulates collapse dynamics at macroscopic scales. The OLB framework rests on four axioms, employing a norm-preserving nonlinear Schrödinger evolution and Lüders-type projection triggered by crossing a cognitive commitment threshold. Our expanded formalism provides five main contributions: (1) deriving Lie symmetries of the observer–environment interaction Hamiltonian; (2) embedding OLB into the Consistent Histories and path-integral formalisms; (3) multi-agent network simulations demonstrating intentional synchronisation toward shared macroscopic outcomes; (4) detailed statistical power analyses predicting measurable biases (up to ~5%) in practical experiments involving traffic delays, quantum random number generators, and financial market sentiment; and (5) examining the conceptual, ethical, and neuromorphic implications of intent-driven reality selection. Full reproducibility is ensured via the provided code notebooks and raw data tables in the appendices. While the theoretical predictions are precisely formulated, empirical validation is ongoing, and no definitive field results are claimed at this stage. OLB thus offers a rigorous, norm-preserving and falsifiable framework to empirically test whether cognitive engagement modulates macroscopic quantum outcomes in ways consistent with—but extending—standard quantum predictions. Full article

(This article belongs to the Special Issue A Century of Quantum Mechanics: Mathematical Foundations and Computational Applications)

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25 pages, 8764 KiB

Open AccessArticle

A Comprehensive Study on the Applications of NTIM and OAFM in Analyzing Fractional Navier–Stokes Equations

by Siddiq Ur Rehman, Rashid Nawaz, Faisal Zia and Nick Fewster-Young

Axioms 2025, 14(7), 521; https://doi.org/10.3390/axioms14070521 - 7 Jul 2025

Viewed by 90

Abstract

This article introduces two enhanced techniques: the Natural Transform Iterative Method (NTIM) and the Optimal Auxiliary Function Method (OAFM). These approaches provide a close approximation for solving fractional-order Navier–Stokes equations, which are widely employed in domains such as biology, ecology, and applied sciences. [...] Read more.

This article introduces two enhanced techniques: the Natural Transform Iterative Method (NTIM) and the Optimal Auxiliary Function Method (OAFM). These approaches provide a close approximation for solving fractional-order Navier–Stokes equations, which are widely employed in domains such as biology, ecology, and applied sciences. By comparing the solutions derived from these methods to exact solutions, it is clear that they provide accurate and efficient outcomes. These findings highlight the straightforward yet effective use of these methodologies in modeling engineering systems. Navier–Stokes equations have numerous practical uses, including analyzing fluid flow in pipelines and channels, predicting weather patterns, and constructing aircraft and vehicles. Full article

(This article belongs to the Special Issue Nonlinear Fractional Differential Equations: Theory and Applications)

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10 pages, 456 KiB

Open AccessArticle

A Geometric Variational Problem for Pseudo-Galilean Particles

by Ayşe Yılmaz Ceylan, Tunahan Turhan and Gözde Özkan Tükel

Axioms 2025, 14(7), 520; https://doi.org/10.3390/axioms14070520 - 7 Jul 2025

Viewed by 169

Abstract

This study explores the dynamics of particle motion in pseudo-Galilean

3 -

space

G_{3}^{1}

by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some [...] Read more.

This study explores the dynamics of particle motion in pseudo-Galilean

3 -

space

G_{3}^{1}

by considering actions that incorporate both curvature and torsion of trajectories. We consider a general energy functional and formulate Euler–Lagrange equations corresponding to this functional under some boundary conditions in

G_{3}^{1}

. By adapting the geometric tools of the Frenet frame to this setting, we analyze the resulting variational equations and provide illustrative solutions that highlight their structural properties. In particular, we examine examples derived from natural Hamiltonian trajectories in

G_{3}^{1}

and extend them to reflect the distinctive geometric features of pseudo-Galilean spaces, offering insight into their foundational behavior and theoretical implications. Full article

(This article belongs to the Section Geometry and Topology)

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11 pages, 438 KiB

Open AccessArticle

Stability Analysis of Fixed-Wing UAV Swarms Under Time-Delayed Tracking Control Law

by Ana-Maria Bordei and Andrei Halanay

Axioms 2025, 14(7), 519; https://doi.org/10.3390/axioms14070519 - 6 Jul 2025

Viewed by 125

Abstract

This paper analyzes the stability of trajectory tracking in fixed-wing UAV swarms subject to time-delayed feedback control. A delay-dependent stability criterion is established using a combination of Routh–Hurwitz analysis and a transcendental characteristic equation method. The study identifies a critical delay threshold beyond [...] Read more.

This paper analyzes the stability of trajectory tracking in fixed-wing UAV swarms subject to time-delayed feedback control. A delay-dependent stability criterion is established using a combination of Routh–Hurwitz analysis and a transcendental characteristic equation method. The study identifies a critical delay threshold beyond which the tracking objective becomes unstable. The influence of delayed feedback on the system dynamics is analyzed, showing how time delays affect the swarm’s ability to maintain formation. Numerical simulations confirm the theoretical predictions and illustrate the loss of stability as the delay increases. The findings underline the importance of accounting for delays when evaluating control performance in UAV swarm coordination. Full article

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17 pages, 343 KiB

Open AccessArticle

On the Conflation of Poisson and Logarithmic Distributions with Applications

by Abdulhamid A. Alzaid, Anfal A. Alqefari and Najla Qarmalah

Axioms 2025, 14(7), 518; https://doi.org/10.3390/axioms14070518 - 6 Jul 2025

Viewed by 126

Abstract

It is frequent for real-life count data to show inflation in lower values; however, most of the well-known count distributions cannot capture such a feature. The present paper introduces a new distribution for modeling inflated count data in small values based on a [...] Read more.

It is frequent for real-life count data to show inflation in lower values; however, most of the well-known count distributions cannot capture such a feature. The present paper introduces a new distribution for modeling inflated count data in small values based on a conflation of distributions approach. The new distribution inherits some properties from Poisson distribution (PD) and logarithmic distribution (LD), making it a powerful modeling tool. It can serve as an alternative to PD, LD, and zero-truncated distributions. The new distribution is worth considering theoretically, as it belongs to the weighted PD family. With zero as a support point, two additional models are suggested for the new distribution. These modifications yield distributions that demonstrate overdispersion models comparable to the negative binomial distribution (NBD) while retaining essential PD properties, making them suitable for accurately representing count data with frequent events of low frequency and high variance. Furthermore, we discuss the superior performance of three new distributions in modeling real count data compared to traditional count distributions such as PD and NBD, as well as other discrete distributions. This paper examines the key statistical properties of the proposed distributions. A comparison of the novel and other distributions in the literature is shown employing real-life data from some domains. All of the computations shown in this study are generated using the R programming language. Full article

(This article belongs to the Special Issue Advances in the Theory and Applications of Statistical Distributions)

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12 pages, 265 KiB

Open AccessArticle

Existence of Weakly Pareto–Nash Equilibrium for Multiobjective Games with Infinitely Many Players

by Huaxin Chen and Wensheng Jia

Axioms 2025, 14(7), 517; https://doi.org/10.3390/axioms14070517 - 4 Jul 2025

Viewed by 116

Abstract

Our work proves the existence of weakly Pareto–Nash equilibrium (PNE) in multiobjective games (MGs) with infinitely many players. First, we demonstrate the existence of weakly PNE under compactness assumptions by using the intersection theorem. Then, we extend the intersection theorem to the non-compact [...] Read more.

Our work proves the existence of weakly Pareto–Nash equilibrium (PNE) in multiobjective games (MGs) with infinitely many players. First, we demonstrate the existence of weakly PNE under compactness assumptions by using the intersection theorem. Then, we extend the intersection theorem to the non-compact case and obtain a new intersection theorem. Finally, we prove the existence of weakly PNE in MGs with infinitely many players without compactness assumptions. Full article

(This article belongs to the Special Issue Mathematical Optimizations and Operations Research)

25 pages, 646 KiB

Open AccessArticle

Exponential Squared Loss-Based Robust Variable Selection with Prior Information in Linear Regression Models

by Hejun Wei, Tian Jin and Yunquan Song

Axioms 2025, 14(7), 516; https://doi.org/10.3390/axioms14070516 - 4 Jul 2025

Viewed by 118

Abstract

This paper proposes a robust variable selection method that incorporates prior information through linear constraints. For more than a decade, penalized likelihood frameworks have been the predominant approach for variable selection, where appropriate loss and penalty functions are selected to formulate unconstrained optimization [...] Read more.

This paper proposes a robust variable selection method that incorporates prior information through linear constraints. For more than a decade, penalized likelihood frameworks have been the predominant approach for variable selection, where appropriate loss and penalty functions are selected to formulate unconstrained optimization problems. However, in many specific applications, some prior information can be obtained. In this paper, we reformulate variable selection by incorporating prior knowledge as linear constraints. In addition, the loss function adopted in this paper is a robust exponential squared loss function, which ensures that the estimation of model parameter coefficient will not have a great impact when there are a few outliers in the dataset. This paper uses the designed solution algorithm to calculate the estimated values of coefficients and some other parameters, and finally conducts numerical simulations and a real-data experiment. Experimental results demonstrate that our model significantly improves estimation robustness compared to existing methods, even in outlier-contaminated scenarios. Full article

(This article belongs to the Special Issue Computational Statistics and Its Applications, 2nd Edition)

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21 pages, 1556 KiB

Open AccessArticle

Hexic-Chebyshev Collocation Method for Solving Distributed-Order Time-Space Fractional Diffusion Equations

by Afshin Babaei, Sedigheh Banihashemi, Behrouz Parsa Moghaddam and Arman Dabiri

Axioms 2025, 14(7), 515; https://doi.org/10.3390/axioms14070515 - 3 Jul 2025

Viewed by 260

Abstract

This paper presents a spectral method to solve nonlinear distributed-order diffusion equations with both time-distributed-order and two-sided space-fractional terms. These are highly challenging to solve analytically due to the interplay between nonlinearity and the fractional distributed-order nature of the time and space derivatives. [...] Read more.

This paper presents a spectral method to solve nonlinear distributed-order diffusion equations with both time-distributed-order and two-sided space-fractional terms. These are highly challenging to solve analytically due to the interplay between nonlinearity and the fractional distributed-order nature of the time and space derivatives. For this purpose, Hexic-kind Chebyshev polynomials (HCPs) are used as the backbone of the method to transform the primary problem into a set of nonlinear algebraic equations, which can be efficiently solved using numerical solvers, such as the Newton–Raphson method. The primary reason of choosing HCPs is due to their remarkable recurrence relations, facilitating their efficient computation and manipulation in mathematical analyses. A comprehensive convergence analysis was conducted to validate the robustness of the proposed method, with an error bound derived to provide theoretical guarantees for the solution’s accuracy. The method’s effectiveness is further demonstrated through two test examples, where the numerical results are compared with existing solutions, confirming the approach’s accuracy and reliability. Full article

(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inequalities)

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22 pages, 753 KiB

Open AccessArticle

Existence and Global Exponential Stability of Equilibrium for an Epidemic Model with Piecewise Constant Argument of Generalized Type

by Kuo-Shou Chiu and Fernando Córdova-Lepe

Axioms 2025, 14(7), 514; https://doi.org/10.3390/axioms14070514 - 3 Jul 2025

Viewed by 239

Abstract

The authors investigate an epidemic model described by a differential equation, which includes a piecewise constant argument of the generalized type (DEPCAG). In this work, the main goal is to find an invariant region for the system and prove the existence and uniqueness [...] Read more.

The authors investigate an epidemic model described by a differential equation, which includes a piecewise constant argument of the generalized type (DEPCAG). In this work, the main goal is to find an invariant region for the system and prove the existence and uniqueness of solutions with the defined conditions using integral equations. On top of that, an auxiliary result is established, outlining the relationship between the unknown function values in the deviation argument and the time parameter. The stability analysis is conducted using the Lyapunov–Razumikhin method, adapted for differential equations with a piecewise constant argument of the generalized type. The trivial equilibrium’s stability is examined, and the stability of the positive equilibrium is assessed by transforming it into a trivial form. Finally, sufficient conditions for the uniform asymptotic stability of both the trivial and positive equilibria are established. Full article

(This article belongs to the Section Mathematical Analysis)

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17 pages, 3336 KiB

Open AccessArticle

Modeling and Exploration of Localized Wave Phenomena in Optical Fibers Using the Generalized Kundu–Eckhaus Equation for Femtosecond Pulse Transmission

by Ejaz Hussain, Ali H. Tedjani, Khizar Farooq and Beenish

Axioms 2025, 14(7), 513; https://doi.org/10.3390/axioms14070513 - 3 Jul 2025

Viewed by 130

Abstract

This manuscript aims to explore localized waves for the nonlinear partial differential equation referred to as the

(1 + 1)

-dimensional generalized Kundu–Eckhaus equation with an additional dispersion term that describes the propagation of the ultra-short femtosecond pulses in an optical [...] Read more.

This manuscript aims to explore localized waves for the nonlinear partial differential equation referred to as the

(1 + 1)

-dimensional generalized Kundu–Eckhaus equation with an additional dispersion term that describes the propagation of the ultra-short femtosecond pulses in an optical fiber. This research delves deep into the characteristics, behaviors, and localized waves of the

(1 + 1)

-dimensional generalized Kundu–Eckhaus equation. We utilize the multivariate generalized exponential rational integral function method (MGERIFM) to derive localized waves, examining their properties, including propagation behaviors and interactions. Motivated by the generalized exponential rational integral function method, it proves to be a powerful tool for finding solutions involving the exponential, trigonometric, and hyperbolic functions. The solutions we found using the MGERIF method have important applications in different scientific domains, including nonlinear optics, plasma physics, fluid dynamics, mathematical physics, and condensed matter physics. We apply the three-dimensional (3D) and contour plots to illuminate the physical significance of the derived solution, exploring the various parameter choices. The proposed approaches are significant and applicable to various nonlinear evolutionary equations used to model nonlinear physical systems in the field of nonlinear sciences. Full article

(This article belongs to the Special Issue Applied Nonlinear Dynamical Systems in Mathematical Physics)

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11 pages, 260 KiB

Open AccessArticle

Averaging of Linear Quadratic Parabolic Optimal Control Problem

by Olena Kapustian, Oleksandr Laptiev and Adalbert Makarovych

Axioms 2025, 14(7), 512; https://doi.org/10.3390/axioms14070512 - 2 Jul 2025

Viewed by 139

Abstract

This paper studies an averaged Linear Quadratic Regulator (LQR) problem for a parabolic partial differential equation (PDE), where the system dynamics are affected by uncertain parameters. Instead of assuming a deterministic operator, we model the uncertainty using a probability distribution over a set [...] Read more.

This paper studies an averaged Linear Quadratic Regulator (LQR) problem for a parabolic partial differential equation (PDE), where the system dynamics are affected by uncertain parameters. Instead of assuming a deterministic operator, we model the uncertainty using a probability distribution over a set of possible system dynamics. This approach extends classical optimal control theory by incorporating an averaging framework to account for parameter uncertainty. We establish the existence and uniqueness of the optimal control solution and analyze its convergence as the probability distribution governing the system parameters changes. These results provide a rigorous foundation for solving optimal control problems in the presence of parameter uncertainty. Our findings lay the groundwork for further studies on optimal control in dynamic systems with uncertainty. Full article

(This article belongs to the Special Issue Modern Trends and Application of Decision-Making Theory, Stability and Control)

20 pages, 1115 KiB

Open AccessArticle

A Novel Computational Framework for Time-Fractional Higher-Order KdV Models: CLADM-Based Solutions and Comparative Analysis

by Priti V. Tandel, Anant Patel and Trushitkumar Patel

Axioms 2025, 14(7), 511; https://doi.org/10.3390/axioms14070511 - 1 Jul 2025

Viewed by 144

Abstract

This study applies the Conformable Laplace Adomian Decomposition Method (CLADM) to solve generalized time-fractional Korteweg–de Vries (KdV) models, including seventh- and fifth-order models. CLADM combines the conformable fractional derivative and Laplace transform with the Adomian decomposition technique, offering analytic approximate solutions. Numerical and [...] Read more.

This study applies the Conformable Laplace Adomian Decomposition Method (CLADM) to solve generalized time-fractional Korteweg–de Vries (KdV) models, including seventh- and fifth-order models. CLADM combines the conformable fractional derivative and Laplace transform with the Adomian decomposition technique, offering analytic approximate solutions. Numerical and graphical results, generated using MATLAB R2020a 9.8.0.1323502, validate the method’s efficiency and precision in capturing fractional-order dynamics. Fractional parameters

ϱ

significantly influence wave behavior, with higher orders yielding smoother profiles and reduced oscillations. Comparative analysis confirms CLADM’s superiority over existing methods in minimizing errors. The versatility of CLADM highlights its potential for studying nonlinear wave phenomena in diverse applications. Full article

(This article belongs to the Special Issue Fractional Calculus and Applied Analysis, 2nd Edition)

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59 pages, 1417 KiB

Open AccessArticle

Symmetrized Neural Network Operators in Fractional Calculus: Caputo Derivatives, Asymptotic Analysis, and the Voronovskaya–Santos–Sales Theorem

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

Axioms 2025, 14(7), 510; https://doi.org/10.3390/axioms14070510 - 30 Jun 2025

Viewed by 198

Abstract

This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for [...] Read more.

This work presents a comprehensive mathematical framework for symmetrized neural network operators operating under the paradigm of fractional calculus. By introducing a perturbed hyperbolic tangent activation, we construct a family of localized, symmetric, and positive kernel-like densities, which form the analytical backbone for three classes of multivariate operators: quasi-interpolation, Kantorovich-type, and quadrature-type. A central theoretical contribution is the derivation of the Voronovskaya–Santos–Sales Theorem, which extends classical asymptotic expansions to the fractional domain, providing rigorous error bounds and normalized remainder terms governed by Caputo derivatives. The operators exhibit key properties such as partition of unity, exponential decay, and scaling invariance, which are essential for stable and accurate approximations in high-dimensional settings and systems governed by nonlocal dynamics. The theoretical framework is thoroughly validated through applications in signal processing and fractional fluid dynamics, including the formulation of nonlocal viscous models and fractional Navier–Stokes equations with memory effects. Numerical experiments demonstrate a relative error reduction of up to 92.5% when compared to classical quasi-interpolation operators, with observed convergence rates reaching

O (n^{- 1.5})

under Caputo derivatives, using parameters

λ = 3.5

,

q = 1.8

, and

n = 100

. This synergy between neural operator theory, asymptotic analysis, and fractional calculus not only advances the theoretical landscape of function approximation but also provides practical computational tools for addressing complex physical systems characterized by long-range interactions and anomalous diffusion. Full article

(This article belongs to the Special Issue Advances in Fuzzy Logic and Computational Intelligence)

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18 pages, 271 KiB

Open AccessArticle

New Results on Idempotent Operators in Hilbert Spaces

by Salma Aljawi, Cristian Conde, Kais Feki and Shigeru Furuichi

Axioms 2025, 14(7), 509; https://doi.org/10.3390/axioms14070509 - 30 Jun 2025

Viewed by 252

Abstract

This paper provides a new proof of the operator norm identity

∥ Q ∥ = ∥ I - Q ∥

, where Q is a bounded idempotent operator on a complex Hilbert space, and I is the identity operator. We also [...] Read more.

This paper provides a new proof of the operator norm identity

∥ Q ∥ = ∥ I - Q ∥

, where Q is a bounded idempotent operator on a complex Hilbert space, and I is the identity operator. We also derive explicit lower and upper bounds for the distance from an arbitrary idempotent operator to the set of orthogonal projections. Our approach simplifies existing proofs. Full article

(This article belongs to the Section Mathematical Analysis)

18 pages, 361 KiB

Open AccessArticle

Analyzing Competing Risks with Progressively Type-II Censored Data in Dagum Distributions

by Raghd Badwan and Reza Pakyari

Axioms 2025, 14(7), 508; https://doi.org/10.3390/axioms14070508 - 30 Jun 2025

Viewed by 133

Abstract

Competing risk models are essential in survival analysis for studying systems with multiple mutually exclusive failure events. This study investigates the application of competing risk models in the presence of progressively Type-II censored data for the Dagum distribution, a flexible distribution suited for [...] Read more.

Competing risk models are essential in survival analysis for studying systems with multiple mutually exclusive failure events. This study investigates the application of competing risk models in the presence of progressively Type-II censored data for the Dagum distribution, a flexible distribution suited for modeling data with heavy tails and varying skewness and kurtosis. The methodology includes maximum likelihood estimation of the unknown parameters, with a focus on the special case of a common shape parameter, which allows for a closed-form expression of the relative risks. A hypothesis test is developed to assess the validity of this assumption, and both asymptotic and bootstrap confidence intervals are constructed. The performance of the proposed methods is evaluated through Monte Carlo simulations, and their applicability is demonstrated with a real-world example. Full article

(This article belongs to the Special Issue Recent Stochastic and Statistical Approaches for Modeling Complex Systems and Dependent Variables)

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21 pages, 565 KiB

Open AccessArticle

Transfer Learning of High-Dimensional Stochastic Frontier Model via Elastic Net

by Jiahao Chen, Wenjun Chen and Yunquan Song

Axioms 2025, 14(7), 507; https://doi.org/10.3390/axioms14070507 - 28 Jun 2025

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Abstract

In this paper, the high-dimensional stochastic frontier model problem is explored via Elastic Net under the transfer learning framework. When the target data is limited, transfer learning improves the accuracy of model estimation and prediction by transferring the source data. When the transferable [...] Read more.

In this paper, the high-dimensional stochastic frontier model problem is explored via Elastic Net under the transfer learning framework. When the target data is limited, transfer learning improves the accuracy of model estimation and prediction by transferring the source data. When the transferable source is known, a transfer learning algorithm for a high-dimensional stochastic frontier model is proposed based on Elastic Net. In addition, based on the prior knowledge of the parameters, this paper introduces linear constraints to improve the estimation accuracy in transfer learning. When the transferable source is unknown, this paper designs a corresponding algorithm to detect the transferable source. Finally, the effectiveness of the method is proved by simulation experiments and actual cases. Full article

(This article belongs to the Special Issue Methods and Applications of Advanced Statistical Analysis, 2nd Edition)

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17 pages, 322 KiB

Open AccessArticle

A New Class of (α,η,(Q,h),L)-Contractions in Triple Controlled Metric-Type Spaces with Application to Polynomial Sine-Type Equations

by Fatima M. Azmi

Axioms 2025, 14(7), 506; https://doi.org/10.3390/axioms14070506 - 27 Jun 2025

Viewed by 158

Abstract

This paper introduces a novel class of generalized contractions, termed

(α, η, (Q, h), L)

-contraction mapping, within the context of triple controlled metric-type spaces, extending the framework of fixed point theory in controlled structures. [...] Read more.

This paper introduces a novel class of generalized contractions, termed

(α, η, (Q, h), L)

-contraction mapping, within the context of triple controlled metric-type spaces, extending the framework of fixed point theory in controlled structures. The proposed mapping is defined using

α

-admissible and

η

-subadmissible functions, in conjunction with a control pair

(Q, h)

of upper class of type I, and incorporates Wardowski’s function

L

-contraction condition. Under suitable hypotheses, we establish both the existence and uniqueness of fixed points for this class of mappings. Several corollaries are derived as special cases of the main result. Moreover, we provide a nontrivial application by analyzing the solvability of a nonlinear equation involving powers of the sine function, thereby illustrating the utility of the developed theory. Full article

(This article belongs to the Section Mathematical Analysis)

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21 pages, 1675 KiB

Open AccessArticle

H_∞ Preview Tracking Control of Time-Delay Discrete Systems and Its Application in Nuclear Reactor Problems

by Fucheng Liao, Hao Xie, Xianchun Meng, Jiang Wu, Yucheng Wei and Jiamei Deng

Axioms 2025, 14(7), 505; https://doi.org/10.3390/axioms14070505 - 27 Jun 2025

Viewed by 167

Abstract

Improving the tracking accuracy and effectiveness of the pressurizer control system with respect to the reference signal is an effective method to enhance the safe and stable operation of nuclear reactors. This paper applies preview tracking control to the pressurizer control system. For [...] Read more.

Improving the tracking accuracy and effectiveness of the pressurizer control system with respect to the reference signal is an effective method to enhance the safe and stable operation of nuclear reactors. This paper applies preview tracking control to the pressurizer control system. For the simplified control system model of the pressurizer, we first study its general structure, which can be characterized as a discrete-time system with state delay. Unlike conventional control systems, the system considered in this study features control inputs that are represented as cumulative sums of historical inputs. In order to design a preview tracking controller for such systems, we adopt the difference method and state augmentation technique and introduce an equality containing the reference signal and a discrete integrator to construct an augmented error system. Simultaneously, a performance signal is defined to evaluate the impact of external disturbances on system performance. Thus, the preview tracking control problem of the original system is reformulated as an

H_{\infty}

control problem for the augmented error system. Subsequently, a memory-based state feedback controller is designed for the augmented error system. Then, by employing the Lyapunov function and linear matrix inequality (LMI), the

H_{\infty}

preview tracking controller for the original system is derived. Finally, the proposed control strategy is applied to a pressurizer control system model, and numerical simulations are conducted to validate the effectiveness of the proposed controller by using MATLAB (R2023a, MathWorks, Natick, MA, USA). Full article

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22 pages, 670 KiB

Open AccessArticle

LDC-GAT: A Lyapunov-Stable Graph Attention Network with Dynamic Filtering and Constraint-Aware Optimization

by Liping Chen, Hongji Zhu and Shuguang Han

Axioms 2025, 14(7), 504; https://doi.org/10.3390/axioms14070504 - 27 Jun 2025

Viewed by 156

Abstract

Graph attention networks are pivotal for modeling non-Euclidean data, yet they face dual challenges: training oscillations induced by projection-based high-dimensional constraints and gradient anomalies due to poor adaptation to heterophilic structure. To address these issues, we propose LDC-GAT (Lyapunov-Stable Graph Attention Network with [...] Read more.

Graph attention networks are pivotal for modeling non-Euclidean data, yet they face dual challenges: training oscillations induced by projection-based high-dimensional constraints and gradient anomalies due to poor adaptation to heterophilic structure. To address these issues, we propose LDC-GAT (Lyapunov-Stable Graph Attention Network with Dynamic Filtering and Constraint-Aware Optimization), which jointly optimizes both forward and backward propagation processes. In the forward path, we introduce Dynamic Residual Graph Filtering, which integrates a tunable self-loop coefficient to balance neighborhood aggregation and self-feature retention. This filtering mechanism, constrained by a lower bound on Dirichlet energy, improves multi-head attention via multi-scale fusion and mitigates overfitting. In the backward path, we design the Fro-FWNAdam, a gradient descent algorithm guided by a learning-rate-aware perceptron. An explicit Frobenius norm bound on weights is derived from Lyapunov theory to form the basis of the perceptron. This stability-aware optimizer is embedded within a Frank–Wolfe framework with Nesterov acceleration, yielding a projection-free constrained optimization strategy that stabilizes training dynamics. Experiments on six benchmark datasets show that LDC-GAT outperforms GAT by 10.54% in classification accuracy, which demonstrates strong robustness on heterophilic graphs. Full article

(This article belongs to the Section Mathematical Analysis)

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13 pages, 281 KiB

Open AccessArticle

m-Isometric Operators with Null Symbol and Elementary Operator Entries

by Bhagwati Prashad Duggal

Axioms 2025, 14(7), 503; https://doi.org/10.3390/axioms14070503 - 27 Jun 2025

Viewed by 131

Abstract

A pair

(A, B)

of Banach space operators is strict

(m, X)

-isometric for a Banach space operator

X \in B (X)

and a positive integer m if [...] Read more.

A pair

(A, B)

of Banach space operators is strict

(m, X)

-isometric for a Banach space operator

X \in B (X)

and a positive integer m if

▵_{A, B}^{m} (X) = (\sum_{j = 0}^{m} (\begin{matrix} m \\ j \end{matrix}) L_{A}^{j} R_{B}^{j}) (X) = 0

and

▵_{A, B}^{m - 1} (X) \neq 0

, where

L_{A}

and

R_{B} \in B (B (X))

are, respectively, the operators of left multiplication by A and right multiplication by B. Define operators

E_{A, B}

and

E_{A, B} (X)

by

E_{A, B} = L_{A} R_{B}

and

{(E_{A, B} (X))}^{n} = E_{A, B}^{n} (X)

for all non-negative integers n. Using little more than an algebraic argument, the following generalised version of a result relating

(m, X)

-isometric properties of pairs

(A_{1}, A_{2})

and

(B_{1}, B_{2})

to pairs

(E_{A_{1}, A_{2}} (S_{1}), E_{B_{1}, B_{2}} (S_{2}))

and

(E_{A_{1}, A_{2}}, E_{B_{1}, B_{2}})

is proved: if

A_{i}, B_{i}, S_{i}, X

are operators in

B (X)

,

1 \leq i \leq 2

and X a quasi-affinity, then the pair

(E_{A_{1}, A_{2}} (S_{1}), E_{B_{1}, B_{2}} (S_{2}))

(resp., the pair

(E_{A_{1}, A_{2}}, E_{B_{1}, B_{2}})

) is strict

(m, X)

-isometric for all

X \in B (X)

if and only if there exist positive integers

m_{i} \leq m

,

1 \leq i \leq 2

and

m = m_{1} + m_{2} - 1

, and a non-zero scalar

β

such that

I - E_{β A_{1}, A_{2}} (S_{1})

is (strict)

m_{1}

-nilpotent and

I - E_{\frac{1}{β} B_{1}, B_{2}} (S_{2})

is (strict)

m_{2}

-nilpotent (resp.,

(β A_{1}, B_{1})

is strict

(m_{1}, I)

-isometric and

(\frac{1}{β} B_{2}, A_{2})

is strict

(m_{2}, I)

-isometric). Full article

(This article belongs to the Special Issue Trends in Fixed Point Theory and Fractional Calculus)

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Axioms, Volume 14, Issue 7 (July 2025) – 54 articles

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