Axioms

12 pages, 282 KB

Open AccessArticle

The Cotangent Function as an Avatar of the Polylogarithm Function of Order 0 and Ramanujan’s Formula

by Ruiyang Li, Haoyang Lu and Shigeru Kanemitsu

Axioms 2025, 14(10), 774; https://doi.org/10.3390/axioms14100774 - 21 Oct 2025

Viewed by 264

In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason [...] Read more.

In this paper we will be concerned with zeta-symmetry—the functional equation for the (Riemann) zeta-function (equivalents to which are called modular relations)—and reveal the reason why so many results are intrinsic to PFE (Partial Fraction Expansion) for the cotangent function. The hidden reason is that the cotangent function (as a function in the upper half-plane, say) is the polylogarithm function of order 0 (with complex exponential argument), and therefore it shares properties intrinsic to the Lerch zeta-function of order 0. Here we view the Lerch zeta-function defined in the unit circle as a zeta-function in a wider sense, as a function defined in the upper and lower half-planes. As evidence, we give a plausibly most natural proof of Ramanujan’s formula, including the eta transformation formula as a consequence of the modular relation via the cotangent function, speculating the reason why Ramanujan had been led to such a formula. Other evidence includes the pre-Poisson summation formula as the pick-up principle (which in turn is a generalization of the argument principle). Full article

(This article belongs to the Special Issue Special Functions and Related Topics, 2nd Edition)

15 pages, 418 KB

Open AccessArticle

Threshold Dynamics of Honey Bee Colonies with General Eclosion and Social Inhibition

by Md Mostafizur Rahaman and Xiang-Sheng Wang

Axioms 2025, 14(10), 773; https://doi.org/10.3390/axioms14100773 - 21 Oct 2025

Viewed by 807

Abstract

Honey bee colonies are important to worldwide agriculture and the health of natural ecosystems. Understanding the factors behind colony persistence and failure has been a major challenge for both ecologists and mathematicians. In this paper, we present a general mathematical model to explore [...] Read more.

Honey bee colonies are important to worldwide agriculture and the health of natural ecosystems. Understanding the factors behind colony persistence and failure has been a major challenge for both ecologists and mathematicians. In this paper, we present a general mathematical model to explore the threshold dynamics of honey bee colonies. We explicitly define the basic reproduction number in terms of model parameters and demonstrate its critical role in determining whether a colony survives or collapses. We offer a biological interpretation of the basic reproduction number and prove the threshold dynamics for the general model system. Our results are established using dynamical systems techniques, including the comparison principle, persistence theory, linearization, and global analysis. For specifically chosen eclosion and social inhibition functions, we perform a sensitivity analysis to determine how the basic reproduction number depends on the model parameters. Full article

(This article belongs to the Special Issue Trends in Dynamical Systems and Applied Mathematics)

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30 pages, 31857 KB

Open AccessArticle

Fractional Solitons for Controlling Wave Dynamics in Fluids and Plasmas

by Muhammad Tehseen, Emad K. Jaradat, Elsayed M. Abo-Dahab and Hamood Ur Rehman

Axioms 2025, 14(10), 772; https://doi.org/10.3390/axioms14100772 - 20 Oct 2025

Viewed by 238

Abstract

This paper presents soliton solutions of the fractional (2+1)-dimensional Davey–Stewartson equation based on a local fractional derivative to represent wave packet propagation in dispersive media under both spatial and temporal effects. The importance of this work is in demonstrating how fractional derivatives represent [...] Read more.

This paper presents soliton solutions of the fractional (2+1)-dimensional Davey–Stewartson equation based on a local fractional derivative to represent wave packet propagation in dispersive media under both spatial and temporal effects. The importance of this work is in demonstrating how fractional derivatives represent a more capable modeling tool compared to conventional integer-order methods since they include anomalous dispersion, nonlocal interactions, and memory effects typical in most physical systems in nature. The main objective of this research is to build and examine a broad family of soliton solutions such as bright, dark, singular, bright–dark, and periodic forms, and to explore the influence of fractional orders on their amplitude, width, and dynamical stability. Specific focus is given to the comparison of the behavior of fractional-order solutions with that of traditional integer-order models so as to further the knowledge on fractional calculus and its role in governing nonlinear wave dynamics in fluids, plasmas, and other multifunctional media. Methodologically, this study uses the fractional complex transform together with a new mapping technique, which transforms the fractional Davey–Stewartson equation into solvable nonlinear ordinary differential equations. Such a systematic methodology allows one to derive various families of solitons and form a basis for investigation of nonlinear fractional systems in the general case. Numerical simulations, given in the form of three-dimensional contour maps, density plots, and two-dimensional, demonstrate stability and propagation behavior of the derived solitons. The findings not only affirm the validity of the devised analytic method but also promise possibilities of useful applications in fluid dynamics, plasma physics, and nonlinear optics, where wave structure manipulation using fractional parameters can result in increased performance and novel capabilities. Full article

(This article belongs to the Section Mathematical Analysis)

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16 pages, 305 KB

Open AccessArticle

Combinatorial Identities for the Narayana Numbers

by Ángel Plaza and Steven J. Tedford

Axioms 2025, 14(10), 771; https://doi.org/10.3390/axioms14100771 - 17 Oct 2025

Viewed by 277

Abstract

We interpret the Narayana numbers combinatorially by having them count the number of tilings of an n-strip using squares and triominoes. Using this interpretation, we develop a collection of identities satisfied by the sequence of Narayana numbers. Additionally, these techniques are used [...] Read more.

We interpret the Narayana numbers combinatorially by having them count the number of tilings of an n-strip using squares and triominoes. Using this interpretation, we develop a collection of identities satisfied by the sequence of Narayana numbers. Additionally, these techniques are used to introduce the generalized Narayana numbers and the k-Narayana numbers and to prove corresponding identities. Full article

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19 pages, 303 KB

Open AccessArticle

About Uniqueness of Steady Ricci Schwarzschild Solitons

by Orchidea Maria Lecian

Axioms 2025, 14(10), 770; https://doi.org/10.3390/axioms14100770 - 17 Oct 2025

Viewed by 234

Abstract

In this paper, the uniqueness of steady Schwarzschild gradient Ricci solitons is studied. The role of the weight functions is analyzed. The generalized steady Schwarzschild gradient Ricci solitons are investigated; the implications of the rotational ansatz of Bryant are developed; and the new [...] Read more.

In this paper, the uniqueness of steady Schwarzschild gradient Ricci solitons is studied. The role of the weight functions is analyzed. The generalized steady Schwarzschild gradient Ricci solitons are investigated; the implications of the rotational ansatz of Bryant are developed; and the new Generalized Schwarzschildsteady gradient solitons are defined. The aspects of the weight functions of the latter type of solitons are researched as well. The new most-accurate curvature bound of the steady Ricci gradient solitons is provided. The uniqueness of the Schwarzschild solitons is discussed. The Ricci flow is reconciled with the Einstein Field Equations such that the weight functions are utilized to spell out the determinant of the metric tensor, the procedure for which is commented on following the use of the appropriate geometrical objects. The mean curvature is discussed. The configurations of the observer are issued from the geodesics spheres of the solitonic structures. Full article

17 pages, 341 KB

Open AccessArticle

Inferences for the GKME Distribution Under Progressive Type-I Interval Censoring with Random Removals and Its Application to Survival Data

by Ela Verma, Mahmoud M. Abdelwahab, Sanjay Kumar Singh and Mustafa M. Hasaballah

Axioms 2025, 14(10), 769; https://doi.org/10.3390/axioms14100769 - 17 Oct 2025

Viewed by 220

Abstract

The analysis of lifetime data under censoring schemes plays a vital role in reliability studies and survival analysis, where complete information is often difficult to obtain. This work focuses on the estimation of the parameters of the recently proposed generalized Kavya–Manoharan exponential (GKME) [...] Read more.

The analysis of lifetime data under censoring schemes plays a vital role in reliability studies and survival analysis, where complete information is often difficult to obtain. This work focuses on the estimation of the parameters of the recently proposed generalized Kavya–Manoharan exponential (GKME) distribution under progressive Type-I interval censoring, a censoring scheme that frequently arises in medical and industrial life-testing experiments. Estimation procedures are developed under both classical and Bayesian paradigms, providing a comprehensive framework for inference. In the Bayesian setting, parameter estimation is carried out using Markov Chain Monte Carlo (MCMC) techniques under two distinct loss functions: the squared error loss function (SELF) and the general entropy loss function (GELF). For interval estimation, asymptotic confidence intervals as well as highest posterior density (HPD) credible intervals are constructed. The performance of the proposed estimators is systematically evaluated through a Monte Carlo simulation study in terms of mean squared error (MSE) and the average lengths of the interval estimates. The practical usefulness of the developed methodology is further demonstrated through the analysis of a real dataset on survival times of guinea pigs exposed to virulent tubercle bacilli. The findings indicate that the proposed methods provide flexible and efficient tools for analyzing progressively interval-censored lifetime data. Full article

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32 pages, 2854 KB

Open AccessArticle

Construction of Consistent Fuzzy Competence Spaces and Learning Path Recommendation

by Ronghai Wang, Baokun Huang and Jinjin Li

Axioms 2025, 14(10), 768; https://doi.org/10.3390/axioms14100768 - 16 Oct 2025

Viewed by 224

Abstract

Artificial intelligence is playing an increasingly important role in education. Learning path recommendation is one of the key technologies in artificial intelligence education applications. This paper applies knowledge space theory and fuzzy set theory to study the construction of consistent fuzzy competence spaces [...] Read more.

Artificial intelligence is playing an increasingly important role in education. Learning path recommendation is one of the key technologies in artificial intelligence education applications. This paper applies knowledge space theory and fuzzy set theory to study the construction of consistent fuzzy competence spaces and their application to learning path recommendation. With the help of the outer fringe of fuzzy competence states, this paper proves the necessary and sufficient conditions for a fuzzy competence space to be a consistent fuzzy competence space and designs an algorithm for verifying consistent fuzzy competence spaces. It also proposes methods for constructing and reducing consistent fuzzy competence spaces, provides learning path recommendation algorithms from the competence perspective and combined with a disjunctive fuzzy skill mapping, and constructs a bottom-up gradual and effective learning path tree. Simulation experiments are carried out for the construction and reduction in consistent fuzzy competence spaces and for learning path recommendation, and the simulation studies show that the proposed methods achieve significant performance improvement compared with related research and produce a more complete recommendation of gradual and effective learning paths. The research of this paper can provide theoretical foundations and algorithmic references for the development of artificial intelligence education applications such as learning assessment systems and intelligent testing systems. Full article

(This article belongs to the Special Issue Advances in Fuzzy Preference Relations and Decision-Making Methods with Applications)

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21 pages, 629 KB

Open AccessArticle

Finite Time Stability and Optimal Control for Stochastic Dynamical Systems

by Ronit Chitre and Wassim M. Haddad

Axioms 2025, 14(10), 767; https://doi.org/10.3390/axioms14100767 - 16 Oct 2025

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Abstract

In real-world applications, finite time convergence to a desired Lyapunov stable equilibrium is often necessary. This notion of stability is known as finite time stability and refers to systems in which the state trajectory reaches an equilibrium in finite time. This paper explores [...] Read more.

In real-world applications, finite time convergence to a desired Lyapunov stable equilibrium is often necessary. This notion of stability is known as finite time stability and refers to systems in which the state trajectory reaches an equilibrium in finite time. This paper explores the notion of finite time stability in probability within the context of nonlinear stochastic dynamical systems. Specifically, we introduce sufficient conditions based on Lyapunov methods, utilizing Lyapunov functions that satisfy scalar differential inequalities involving fractional powers for guaranteeing finite time stability in probability. Then, we address the finite time optimal control problem by developing a framework for designing optimal feedback control laws that achieve finite time stochastic stability of the closed-loop system using a Lyapunov function that also serves as the solution to the steady-state stochastic Hamilton–Jacobi–Bellman equation. Full article

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18 pages, 868 KB

Open AccessArticle

Stochastic Production Planning in Manufacturing Systems

by Dragos-Patru Covei

Axioms 2025, 14(10), 766; https://doi.org/10.3390/axioms14100766 - 16 Oct 2025

Viewed by 209

Abstract

We study stochastic production planning in capacity-constrained manufacturing systems, where feasible operating states are restricted to a convex safe-operating region. The objective is to minimize the total cost that combines a quadratic production effort with an inventory holding cost, while automatically halting production [...] Read more.

We study stochastic production planning in capacity-constrained manufacturing systems, where feasible operating states are restricted to a convex safe-operating region. The objective is to minimize the total cost that combines a quadratic production effort with an inventory holding cost, while automatically halting production when the state leaves the safe region. We derive the associated Hamilton–Jacobi–Bellman (HJB) equation, establish the existence and uniqueness of the value function under broad conditions, and prove a concavity property of the transformed value function that yields a robust gradient-based optimal feedback policy. From an operations perspective, the stopping mechanism encodes hard capacity and safety limits, ensuring bounded risk and finite expected costs. We complement the analysis with numerical methods based on finite differences and illustrate how the resulting policies inform real-time decisions through two application-inspired examples: a single-product case calibrated with typical process-industry parameters and a two-dimensional example motivated by semiconductor fabrication, where interacting production variables must satisfy joint safety constraints. The results bridge rigorous stochastic control with practical production planning and provide actionable guidance for operating under uncertainty and capacity limits. Full article

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8 pages, 245 KB

Open AccessArticle

Modular Abbott Algebras

by Pavel Pták and Dominika Burešová

Axioms 2025, 14(10), 765; https://doi.org/10.3390/axioms14100765 - 15 Oct 2025

Viewed by 183

Abstract

This note adds to the investigation of Abbott algebras in relation to quantum logics (see the references below). We introduce a variety of modular Abbott algebras and show that they are isomorphic to the variety of modular quantum logics. We extend this isomorphism [...] Read more.

This note adds to the investigation of Abbott algebras in relation to quantum logics (see the references below). We introduce a variety of modular Abbott algebras and show that they are isomorphic to the variety of modular quantum logics. We extend this isomorphism for the varieties endowed with a symmetric difference. Full article

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20 pages, 347 KB

Open AccessArticle

Approximating Solutions of General Class of Variational Inclusions Involving

B^{l}

-Co-Monotone Mappings in Banach Spaces

by Sanjeev Gupta, Faizan Ahmad Khan, Reem M. Alrashidi, Maha F. Alsharari, Shurooq B. Alblawie and Mona Y. Alfefi

Axioms 2025, 14(10), 764; https://doi.org/10.3390/axioms14100764 - 15 Oct 2025

Viewed by 198

Abstract

The goal of the current study is to introduce a new class of proximal-point mappings that are associated with a new class of

B^{l}

-co-monotone mappings that are being defined. The

B^{l}

-co-monotone mapping is the sum of co-coercive and symmetric [...] Read more.

The goal of the current study is to introduce a new class of proximal-point mappings that are associated with a new class of

B^{l}

-co-monotone mappings that are being defined. The

B^{l}

-co-monotone mapping is the sum of co-coercive and symmetric monotone mappings and an extension of the

C_{n}

-monotone mapping. The investigation is further discussed, along with its application, which involves a variational inclusion problem (VIP) in Banach spaces. Moreover, the study proposes an iterative algorithm and systematically investigates the convergence characteristics of its generated sequences. For the purpose of illustrating our findings, a simplified numerical example is created to show the convergence graph by using the MATLAB 2015a. Full article

(This article belongs to the Section Mathematical Analysis)

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25 pages, 23378 KB

Open AccessArticle

Dispersive Soliton Solutions and Dynamical Analyses of a Nonlinear Model in Plasma Physics

by Alwaleed Kamel, Ali H. Tedjani, Shafqat Ur Rehman, Muhammad Bilal, Alawia Adam, Khaled Aldwoah and Mohammed Messaoudi

Axioms 2025, 14(10), 763; https://doi.org/10.3390/axioms14100763 - 14 Oct 2025

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Abstract

In this paper, we investigate the generalized coupled Zakharov system (GCZS), a fundamental model in plasma physics that describes the nonlinear interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves, including the influence of magnetic fields on weak ion-acoustic wave propagation. This research [...] Read more.

In this paper, we investigate the generalized coupled Zakharov system (GCZS), a fundamental model in plasma physics that describes the nonlinear interaction between high-frequency Langmuir waves and low-frequency ion-acoustic waves, including the influence of magnetic fields on weak ion-acoustic wave propagation. This research aims to achieve three main objectives. First, we uncover soliton solutions of the coupled system in hyperbolic, trigonometric, and rational forms, both in single and combined expressions. These results are obtained using the extended rational sinh-Gordon expansion method and the

(\frac{G^{'}}{G}, \frac{1}{G})

-expansion method. Second, we analyze the dynamic characteristics of the model by performing bifurcation and sensitivity analyses and identifying the corresponding Hamiltonian function. To understand the mechanisms of intricate physical phenomena and dynamical processes, we plot 2D, 3D, and contour diagrams for appropriate parameter values. We also analyze the bifurcation of phase portraits of the ordinary differential equations corresponding to the investigated partial differential equation. The novelty of this study lies in the fact that the proposed model has not been previously explored using these advanced methods and comprehensive dynamical analyses. Full article

(This article belongs to the Special Issue Trends in Dynamical Systems and Applied Mathematics)

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7 pages, 199 KB

Open AccessArticle

The Brooks–Chacon Biting Lemma, the Castaing–Saadoune Procedure, and the Baum–Katz Theorem Along Subsequences

by George Stoica, Deli Li and Liping Liu

Axioms 2025, 14(10), 762; https://doi.org/10.3390/axioms14100762 - 14 Oct 2025

Viewed by 181

Abstract

We show how the Brooks–Chacon Biting Lemma can be combined with the Castaing–Saadoune procedure to provide the complete rate of convergence along subsequences when the uniformly boundedness condition is violated. Full article

(This article belongs to the Special Issue Research on Functional Analysis and Its Applications)

22 pages, 367 KB

Open AccessArticle

Optimal Hölder Regularity for Discontinuous Sub-Elliptic Systems Structured on Hörmander’s Vector Fields

by Dongni Liao and Jialin Wang

Axioms 2025, 14(10), 761; https://doi.org/10.3390/axioms14100761 - 12 Oct 2025

Viewed by 242

Abstract

This paper studies discontinuous quasilinear sub-elliptic systems associated with Hörmander’s vector fields under controllable and natural growth conditions. By a new

A

-harmonic approximation reformulation for bilinear forms

A \in Bil (R^{k N}, R^{k N})

, we obtain [...] Read more.

This paper studies discontinuous quasilinear sub-elliptic systems associated with Hörmander’s vector fields under controllable and natural growth conditions. By a new

A

-harmonic approximation reformulation for bilinear forms

A \in Bil (R^{k N}, R^{k N})

, we obtain optimal partial Hölder continuity with exact exponents for weak solutions with vanishing mean oscillation coefficients. Full article

22 pages, 325 KB

Open AccessArticle

Global Solutions to the Vlasov–Fokker–Planck Equation with Local Alignment Forces Under Specular Reflection Boundary Condition

by Yanming Chang and Yingzhe Fan

Axioms 2025, 14(10), 760; https://doi.org/10.3390/axioms14100760 - 11 Oct 2025

Viewed by 198

Abstract

In this article, we establish the existence of global mild solutions to the Vlasov–Fokker–Planck equation with local alignment forces under specular reflection boundary conditions in the low-regularity function space

L_{k}^{1} L_{T}^{\infty} L_{v}^{2}

. A key difficulty is [...] Read more.

In this article, we establish the existence of global mild solutions to the Vlasov–Fokker–Planck equation with local alignment forces under specular reflection boundary conditions in the low-regularity function space

L_{k}^{1} L_{T}^{\infty} L_{v}^{2}

. A key difficulty is that the macroscopic averaged velocity u does not directly possess a dissipative structure in the equation. To overcome this, we rely on the dissipation

u - b

from the linear part, combined with the dissipation of the macroscopic component b derived from the associated macroscopic equation. Moreover, since no direct energy functional is available for u, we fully exploit the dissipative mechanisms of both

u - b

and b when handling the estimates for the nonlinear terms. Full article

(This article belongs to the Special Issue Recent Advances in Differential Equations and Related Topics)

14 pages, 290 KB

Open AccessArticle

Z-Solitons and Gradient Z-Solitons on α-Cosymplectic Manifolds

by Mustafa Yildirim, Mehmet Akif Akyol, Majid Ali Choudhary and Foued Aloui

Axioms 2025, 14(10), 759; https://doi.org/10.3390/axioms14100759 - 10 Oct 2025

Viewed by 311

Abstract

In this paper, we study Z-solitons and gradient Z-solitons on

α

-cosymplectic manifolds. The soliton structure is defined by the generalized tensor

Z = S + β g

, where S denotes the Ricci tensor, g the metric tensor, and

β

[...] Read more.

In this paper, we study Z-solitons and gradient Z-solitons on

α

-cosymplectic manifolds. The soliton structure is defined by the generalized tensor

Z = S + β g

, where S denotes the Ricci tensor, g the metric tensor, and

β

a smooth function. We investigate the geometric implications of Z-solitons under various curvature conditions, with a focus on the interplay between the Z-tensor and the Q-curvature tensor, as well as the case of Z-recurrent

α

-cosymplectic manifolds. Our classification results establish that such manifolds can be Einstein,

η

-Einstein, or of constant curvature. Finally, we construct a concrete five-dimensional example of an

α

-cosymplectic manifold that admits a Z-soliton structure, thereby illustrating the theoretical framework. Full article

(This article belongs to the Special Issue Differential Geometry and Its Application, 3rd Edition)

16 pages, 374 KB

Open AccessArticle

An Extended Complex Method to Solve the Predator–Prey Model

by Hongqiang Tu and Guoqiang Dang

Axioms 2025, 14(10), 758; https://doi.org/10.3390/axioms14100758 - 10 Oct 2025

Viewed by 301

Abstract

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predator–prey model. These meromorphic solutions contain the rational solutions, exponential solutions, elliptic solutions, and [...] Read more.

Through transformation and utilizing a novel extended complex method combining with the Weierstrass factorization theorem, Wiman–Valiron theory and the Painlevé test, new non-constant meromorphic solutions were constructed for the predator–prey model. These meromorphic solutions contain the rational solutions, exponential solutions, elliptic solutions, and transcendental entire function solutions of infinite order in the complex plane. The exact solutions contribute to understanding the predator–prey model from the perspective of complex differential equations. In fact, the presented synthesis method provides a new technology for studying some systems of partial differential equations. Full article

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24 pages, 757 KB

Open AccessArticle

A One-Phase Fractional Spatial Stefan Problem with Convective Specification at the Fixed Boundary

by Diego E. Guevara, Sabrina D. Roscani, Domingo A. Tarzia and Lucas D. Venturato

Axioms 2025, 14(10), 757; https://doi.org/10.3390/axioms14100757 - 8 Oct 2025

Viewed by 336

Abstract

We address a fractional spatial Stefan problem derived from a non-Fourier heat flux model with a convective boundary condition at the fixed boundary. An explicit solution is obtained in terms of a three-parameter Mittag–Leffler function. A dimensionless formulation is used to derive a [...] Read more.

We address a fractional spatial Stefan problem derived from a non-Fourier heat flux model with a convective boundary condition at the fixed boundary. An explicit solution is obtained in terms of a three-parameter Mittag–Leffler function. A dimensionless formulation is used to derive a family of fractional spatial Stefan problems that depend on the Biot and Stefan numbers. Finally, a straightforward numerical method for approximating the solutions is presented, along with numerical experiments analyzing the influence of the physical parameters and the order of fractional differentiation. Full article

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24 pages, 421 KB

Open AccessArticle

Non-Uniformly Multidimensional Moran Random Walk with Resets

by Mohamed Abdelkader

Axioms 2025, 14(10), 756; https://doi.org/10.3390/axioms14100756 - 7 Oct 2025

Viewed by 237

Abstract

In this paper, we investigate the non-uniform m-dimensional Moran walk

(Z_{n}^{(1)}, \dots, Z_{n}^{(m)})

, where each component process

{(Z_{n}^{(j)})}_{1 \leq j \leq m}

, [...] Read more.

In this paper, we investigate the non-uniform m-dimensional Moran walk

(Z_{n}^{(1)}, \dots, Z_{n}^{(m)})

, where each component process

{(Z_{n}^{(j)})}_{1 \leq j \leq m}

, either increases by one unit or resets to zero at each step. Using probability generating functions, we analyze key statistical properties of the walk, with particular emphasis on the mean and variance of its final altitude. We further establish closed-form expressions for the limiting distribution of the process, as well as for the mean and variance of each component. These results extend classical findings on one- and two-dimensional Moran models to the general m-dimensional setting, thereby providing new insights into the asymptotic behavior of multi-component random walks with resets. Full article

(This article belongs to the Section Mathematical Analysis)

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11 pages, 254 KB

Open AccessArticle

Construction Theorem and Application of a Semi-Discrete Hilbert-Type Inequality Involving Partial Sums and Variable Upper Limit Integral Functions

by Yong Hong, Bing He and Qian Zhao

Axioms 2025, 14(10), 755; https://doi.org/10.3390/axioms14100755 - 7 Oct 2025

Viewed by 232

Abstract

By using the construction theorem of semi-discrete Hilbert-type inequalities with quasi-homogeneous kernels and real analysis techniques, this paper establishes a semi-discrete Hilbert-type inequality involving partial sums and variable upper limit integral functions, obtains the necessary and sufficient condition for constructing such an inequality, [...] Read more.

By using the construction theorem of semi-discrete Hilbert-type inequalities with quasi-homogeneous kernels and real analysis techniques, this paper establishes a semi-discrete Hilbert-type inequality involving partial sums and variable upper limit integral functions, obtains the necessary and sufficient condition for constructing such an inequality, and, under certain conditions, derives the computational expression of the best constant factor. Finally, we discuss the boundedness and operator norm of the corresponding operator using the obtained results. Full article

15 pages, 597 KB

Open AccessArticle

Developments of Semi-Type-2 Interval Approach with Mathematics and Order Relation: A New Uncertainty Tackling Technique

by Rukhsar Khatun, Sadiah Aljeddani, Shuhrah Alghamdi, Md Sadikur Rahman and Asoke Kumar Bhunia

Axioms 2025, 14(10), 754; https://doi.org/10.3390/axioms14100754 - 6 Oct 2025

Viewed by 318

Abstract

This paper aims to introduce a new interval approach called the Semi-Type-2 interval to represent imprecise parameters in uncertain decision-making. The proposed work establishes arithmetic operations of Semi-Type-2 intervals with algebraic properties. Additionally, a new interval ranking is proposed in order to compare [...] Read more.

This paper aims to introduce a new interval approach called the Semi-Type-2 interval to represent imprecise parameters in uncertain decision-making. The proposed work establishes arithmetic operations of Semi-Type-2 intervals with algebraic properties. Additionally, a new interval ranking is proposed in order to compare Semi-Type-2 interval numbers, and the corresponding properties of total order relations are also derived. All the definitions and properties related to Semi-Type-2 intervals are illustrated with the help of numerical examples. Numerical illustrations confirm the consistency of the framework and its effectiveness in extending classical interval mathematics. Finally, some probable applications of the Semi-Type-2 interval approach are demonstrated for future implementation. Full article

(This article belongs to the Special Issue Recent Advances in Fuzzy Sets and Related Topics, 2nd Edition)

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39 pages, 1781 KB

Open AccessArticle

An Exponentiated Inverse Exponential Distribution Properties and Applications

by Aroosa Mushtaq, Tassaddaq Hussain, Mohammad Shakil, Mohammad Ahsanullah and Bhuiyan Mohammad Golam Kibria

Axioms 2025, 14(10), 753; https://doi.org/10.3390/axioms14100753 - 3 Oct 2025

Viewed by 351

Abstract

This paper introduces Exponentiated Inverse Exponential Distribution (EIED), a novel probability model developed within the power inverse exponential distribution framework. A distinctive feature of EIED is its highly flexible hazard rate function, which can exhibit increasing, decreasing, and reverse bathtub (upside-down bathtub) shapes, [...] Read more.

This paper introduces Exponentiated Inverse Exponential Distribution (EIED), a novel probability model developed within the power inverse exponential distribution framework. A distinctive feature of EIED is its highly flexible hazard rate function, which can exhibit increasing, decreasing, and reverse bathtub (upside-down bathtub) shapes, making it suitable for modeling diverse lifetime phenomena in reliability engineering, survival analysis, and risk assessment. We derived comprehensive statistical properties of the distribution, including the reliability and hazard functions, moments, characteristic and quantile functions, moment generating function, mean deviations, Lorenz and Bonferroni curves, and various entropy measures. The identifiability of the model parameters was rigorously established, and maximum likelihood estimation was employed for parameter inference. Through extensive simulation studies, we demonstrate the robustness of the estimation procedure across different parameter configurations. The practical utility of EIED was validated through applications to real-world datasets, where it showed superior performance compared to existing distributions. The proposed model offers enhanced flexibility for modeling complex lifetime data with varying hazard patterns, particularly in scenarios involving early failure periods, wear-in phases, and wear-out behaviors. Full article

(This article belongs to the Special Issue Probability, Statistics and Estimations, 2nd Edition)

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10 pages, 226 KB

Open AccessArticle

Completeness Theorems for Impulsive Dirac Operator with Discontinuity

by Kai Wang, Murat Sat, Xin-Jian Xu and Ran Zhang

Axioms 2025, 14(10), 752; https://doi.org/10.3390/axioms14100752 - 3 Oct 2025

Viewed by 292

Abstract

In this work, the discontinuous Dirac operator with weight is studied. We prove the completeness theorems of the system of eigenfunctions for the discontinuous Dirac operator. Full article

17 pages, 304 KB

Open AccessArticle

Quasisymmetric Minimality on Packing Dimension for Homogeneous Perfect Sets

by Shishuang Liu, Yanzhe Li and Jiaojiao Yang

Axioms 2025, 14(10), 751; https://doi.org/10.3390/axioms14100751 - 2 Oct 2025

Viewed by 241

Abstract

The quasisymmetric minimality for fractal sets is a hot research topic for scholars focused on the fractal geometry and quasisymmetric mappings. In this paper, we study the quasisymmetric minimality on packing dimension for homogeneous perfect sets. By using some mathematical tools such as [...] Read more.

The quasisymmetric minimality for fractal sets is a hot research topic for scholars focused on the fractal geometry and quasisymmetric mappings. In this paper, we study the quasisymmetric minimality on packing dimension for homogeneous perfect sets. By using some mathematical tools such as the mass distribution principle, we find that a special class of homogeneous perfect sets with packing dimension 1 is quasisymmetrically packing minimal. Our result generalizes the results in the references. Full article

20 pages, 345 KB

Open AccessArticle

A Novel Approach to Polynomial Approximation in Multidimensional Cylindrical Domains via Generalized Kronecker Product Bases

by Mohra Zayed

Axioms 2025, 14(10), 750; https://doi.org/10.3390/axioms14100750 - 2 Oct 2025

Viewed by 365

Abstract

The Kronecker product has been commonly seen in various scientific fields to formulate higher-dimensional spaces from lower-dimensional ones. This paper presents a generalization of the Cannon–Kronecker product bases by introducing generalized Kronecker product bases of polynomials within an analytic framework. It investigates the [...] Read more.

The Kronecker product has been commonly seen in various scientific fields to formulate higher-dimensional spaces from lower-dimensional ones. This paper presents a generalization of the Cannon–Kronecker product bases by introducing generalized Kronecker product bases of polynomials within an analytic framework. It investigates the convergence behavior of infinite series formed by these generalized products in various polycylindrical domains, including both open and closed configurations. The paper also delves into essential analytic properties such as order, type, and the

T_{ρ}

-property to analyze the growth and structural characteristics of these bases. Moreover, the theoretical insights are applied to a range of classical special functions, notably Bernoulli, Euler, Gontcharoff, Bessel, and Chebyshev polynomials. Full article

31 pages, 399 KB

Open AccessArticle

Weakly B-Symmetric Warped Product Manifolds with Applications

by Bang-Yen Chen, Sameh Shenawy, Uday Chand De, Safaa Ahmed and Hanan Alohali

Axioms 2025, 14(10), 749; https://doi.org/10.3390/axioms14100749 - 2 Oct 2025

Viewed by 257

Abstract

This work presents a comprehensive study of weakly B-symmetric warped product manifolds

{(W B S)}_{n}

, a natural extension of several classical curvature-restricted geometries including B-flat, B-parallel, and B-recurrent manifolds. We begin by formulating the fundamental [...] Read more.

This work presents a comprehensive study of weakly B-symmetric warped product manifolds

{(W B S)}_{n}

, a natural extension of several classical curvature-restricted geometries including B-flat, B-parallel, and B-recurrent manifolds. We begin by formulating the fundamental properties of the B-tensor

B (X, Y) = a S (X, Y) + b r g (X, Y)

, where S is the Ricci tensor, r the scalar curvature, and

a, b

are smooth non-vanishing functions. The warped product structure is then exploited to obtain explicit curvature identities for base and fiber manifolds under various geometric constraints. Detailed characterizations are established for Einstein conditions, Codazzi-type tensors, cyclic parallel tensors, and the behavior of geodesic vector fields. The weakly B-symmetric condition is analyzed through all possible projections of vector fields, leading to sharp criteria describing the interaction between the warping function and curvature. Several applications are discussed in the context of Lorentzian geometry, including perfect fluid and generalized Robertson–Walker spacetimes in general relativity. These results not only unify different curvature-restricted frameworks but also reveal new geometric and physical implications of warped product manifolds endowed with weak B-symmetry. Full article

(This article belongs to the Section Mathematical Physics)

13 pages, 290 KB

Open AccessArticle

The Existence of Fixed Points for Generalized

ω_{b}^{φ}

-Contractions and Applications

by Ahad Hamoud Alotaibi and Maha Noorwali

Axioms 2025, 14(10), 748; https://doi.org/10.3390/axioms14100748 - 1 Oct 2025

Viewed by 205

Abstract

This article introduces a new type of contractions via

φ

-admissibility and

ω_{b}

-distance called generalized

ω_{b}^{φ}

-contractions. We prove the existence of fixed points for this type of contractions under some conditions. Moreover, we give an example to demonstrate [...] Read more.

This article introduces a new type of contractions via

φ

-admissibility and

ω_{b}

-distance called generalized

ω_{b}^{φ}

-contractions. We prove the existence of fixed points for this type of contractions under some conditions. Moreover, we give an example to demonstrate the applications of our results. Full article

(This article belongs to the Special Issue Advances in Fixed Point Theory with Applications)

21 pages, 441 KB

Open AccessArticle

Discovering New Recurrence Relations for Stirling Numbers: Leveraging a Poisson Expectation Identity for Higher-Order Moments

by Ying-Ying Zhang and Dong-Dong Pan

Axioms 2025, 14(10), 747; https://doi.org/10.3390/axioms14100747 - 1 Oct 2025

Viewed by 282

Abstract

This paper establishes two novel recurrence relations for Stirling numbers of the second kind—an L recurrence and a vertical recurrence—discovered through a probabilistic analysis of Poisson higher-order origin moments. While the link between these moments and Stirling numbers is known, our derivation via [...] Read more.

This paper establishes two novel recurrence relations for Stirling numbers of the second kind—an L recurrence and a vertical recurrence—discovered through a probabilistic analysis of Poisson higher-order origin moments. While the link between these moments and Stirling numbers is known, our derivation via a specific expectation identity provides a clear and efficient pathway to their computation, circumventing the need for infinite series. The primary theoretical contribution is the proof of these previously undocumented combinatorial recurrences, which are of independent mathematical interest. Furthermore, we demonstrate the severe practical inadequacy of high-order sample moments as estimators, highlighting the necessity of our analytical approach to obtaining reliable estimates in applied fields. Full article

(This article belongs to the Section Mathematical Analysis)

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36 pages, 437 KB

Open AccessArticle

Formulas Involving Cauchy Polynomials, Bernoulli Polynomials, and Generalized Stirling Numbers of Both Kinds

by José L. Cereceda

Axioms 2025, 14(10), 746; https://doi.org/10.3390/axioms14100746 - 1 Oct 2025

Viewed by 285

Abstract

In this paper, we derive novel formulas and identities connecting Cauchy numbers and polynomials with both ordinary and generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler polynomials, r-Whitney numbers, and hyperharmonic polynomials, as well as Bernoulli numbers and polynomials. We also [...] Read more.

In this paper, we derive novel formulas and identities connecting Cauchy numbers and polynomials with both ordinary and generalized Stirling numbers, binomial coefficients, central factorial numbers, Euler polynomials, r-Whitney numbers, and hyperharmonic polynomials, as well as Bernoulli numbers and polynomials. We also provide formulas for the higher-order derivatives of Cauchy polynomials and obtain corresponding formulas and identities for poly-Cauchy polynomials. Furthermore, we introduce a multiparameter framework for poly-Cauchy polynomials, unifying earlier generalizations like shifted poly-Cauchy numbers and polynomials with a q parameter. Full article

17 pages, 1214 KB

Open AccessArticle

Fusion Maximal Information Coefficient-Based Quality-Related Kernel Component Analysis: Mathematical Formulation and an Application for Nonlinear Fault Detection

by Jie Yuan, Hao Ma and Yan Wang

Axioms 2025, 14(10), 745; https://doi.org/10.3390/axioms14100745 - 30 Sep 2025

Viewed by 189

Abstract

Amid intensifying global competition, industrial product quality has become a critical determinant of competitive advantage. However, persistent quality-related faults in production environments threaten product integrity. To address this challenge, a Fusion Maximal Information Coefficient-based Quality-Related Kernel Component Analysis (FMIC-QRKCA) methodology is proposed in [...] Read more.

Amid intensifying global competition, industrial product quality has become a critical determinant of competitive advantage. However, persistent quality-related faults in production environments threaten product integrity. To address this challenge, a Fusion Maximal Information Coefficient-based Quality-Related Kernel Component Analysis (FMIC-QRKCA) methodology is proposed in this paper by capitalizing on information fusion principles and statistical metric theory. Based on information fusion principles, a Fusion Maximal Information Coefficient (FMIC) strategy is first studied to quantify correlations between process variables and multivariate quality indicators. Subsequently, by integrating the proposed FMIC method with Kernel Principal Component Analysis (KPCA), a Quality-Related Kernel Component Analysis (QRKCA) method is proposed. In the proposed QRKCA strategy, the complete latent variable space is first obtained; on this basis, FMIC is further applied to quantify the correlation between each latent variable and quality variables, thereby completing the screening of quality-related latent variables. Additionally, the

T^{2}

and squared prediction error monitoring statistics are used as the key indices to determine the occurrence of faults. This integration overcomes the limitation of conventional KPCA, which does not explicitly consider quality indicators during the principal component extraction, thereby enabling precise isolation of quality-related fault features. Validation through the numerical case and the industrial process case demonstrates that FMIC-QRKCA significantly outperforms established methods in detection accuracy for quality-related faults. Full article

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