Axioms

37 pages, 5212 KB

Open AccessArticle

A Flexible Bivariate Lifetime Model with Upper Bound: Theoretical Development and Lifetime Application

by Shuhrah Alghamdi, Tassaddaq Hussain, Hassan S. Bakouch and Maher Kachour

Axioms 2025, 14(12), 930; https://doi.org/10.3390/axioms14120930 - 18 Dec 2025

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This paper introduces the bivariate bounded Gompertz–log-logistic (BBGLL) distribution, a bounded bivariate lifetime model built by coupling two bounded Gompertz–log-logistic marginals through a Clayton copula with an independent dependence parameter. The proposed model effectively describes positively dependent lifetimes within finite support and accommodates [...] Read more.

This paper introduces the bivariate bounded Gompertz–log-logistic (BBGLL) distribution, a bounded bivariate lifetime model built by coupling two bounded Gompertz–log-logistic marginals through a Clayton copula with an independent dependence parameter. The proposed model effectively describes positively dependent lifetimes within finite support and accommodates increasing, decreasing, and bathtub-shaped hazard rates. Analytical expressions for the survival functions, hazard rate functions, and joint moments are derived, while measures of association such as Kendall’s tau, Spearman’s rho, and tail-dependence coefficients characterize the dependence structure. Parameters are estimated via maximum likelihood, inference functions for margins (IFM), and semi-parametric methods, with performance assessed through Monte Carlo simulations. A real-life data application illustrates the practical relevance of the model, showing that the BBGLL distribution achieves a superior goodness-of-fit relative to existing bivariate alternatives, highlighting its practical usefulness. Full article

(This article belongs to the Special Issue Reliability and Risk of Complex Systems: Modelling, Analysis and Optimization, 2nd Edition)

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16 pages, 291 KB

Open AccessArticle

New Generalizations of Gronwall–Bellman–Bihari-Type Integral Inequalities

by Liqiang Chen and Norazrizal Aswad Abdul Rahman

Axioms 2025, 14(12), 929; https://doi.org/10.3390/axioms14120929 - 18 Dec 2025

Viewed by 359

Abstract

This paper develops several new generalizations of Gronwall–Bellman–Bihari-type integral inequalities. We establish three novel integral inequalities that extend classical results to more complex settings, including integrals with mixed linear and nonlinear terms, delayed (retarded) arguments, and general integral kernels. In the preliminaries, we [...] Read more.

This paper develops several new generalizations of Gronwall–Bellman–Bihari-type integral inequalities. We establish three novel integral inequalities that extend classical results to more complex settings, including integrals with mixed linear and nonlinear terms, delayed (retarded) arguments, and general integral kernels. In the preliminaries, we review known Gronwall–Bellman–Bihari inequalities and useful lemmas. In the main results, we present at least three new theorems. The first theorem provides an explicit bound for solutions of an integral inequality involving a separable kernel function and a nonlinear (Bihari-type) term, significantly extending the classical Bihari inequality. The second theorem addresses integral inequalities with delayed arguments, showing that the delay does not enlarge the growth bound compared to the non-delay case. The third theorem handles inequalities with combined linear and nonlinear terms; using a monotone iterative technique, we prove the existence of a maximal solution that bounds any solution of the inequality. Rigorous proofs are given for all main results. In the Applications section, we illustrate how these inequalities can be applied to deduce qualitative properties of differential equations. As an example, we prove a uniqueness result for an initial value problem with a non-Lipschitz nonlinear term using our new inequalities. The paper concludes with a summary of results and a brief discussion of potential further generalizations. Our results provide powerful tools for researchers to obtain a priori bounds and uniqueness criteria for various differential, integral, and functional equations. It is important to note that the integral inequalities established in this work provide bounds on the solution under the assumption of its existence on the considered interval

[t_{0}, T]

. For nonlinear differential or integral equations where the nonlinearity

F

fails to be Lipschitz continuous, solutions may develop movable singularities (blow-up) in finite time. The bounds derived from our Gronwall–Bellman–Bihari-type inequalities are valid only on the maximal interval of existence of the solution. Determining the region where solutions are guaranteed to be free of such singularities is a separate and profound problem, often requiring additional techniques such as the construction of Lyapunov functions or the use of differential comparison principles. The primary contribution of this paper is to provide sharp estimates and uniqueness criteria within the domain where a solution is known to exist a priori. Full article

11 pages, 549 KB

Open AccessArticle

Several Geometric Properties in Banach Spaces and Their Further Application in Orlicz Spaces

by Xiaoxia Wang, Yunan Cui and Yaoming Niu

Axioms 2025, 14(12), 928; https://doi.org/10.3390/axioms14120928 - 17 Dec 2025

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Abstract

In this paper, locally nearly uniformly convex

(L N U C)

is studied in Banach space. Furthermore, the implication relationship between

(L N U C)

and the Kadec–Klee property

(K K)

, the fixed–point property [...] Read more.

In this paper, locally nearly uniformly convex

(L N U C)

is studied in Banach space. Furthermore, the implication relationship between

(L N U C)

and the Kadec–Klee property

(K K)

, the fixed–point property

(F P P)

are investigated in Banach space. Finally, the relationship between the uniform Kadec−Klee property

(U K K)

, the coordinate-wise uniform Kadec–Klee property

(U K K_{C})

, the coordinate-wise Kadec–Klee property

(H_{c})

and

δ_{2}

conditions are investigated in Orlicz sequence spaces equipped with the Orlicz norm, meanwhile we get a criteria that Orlicz sequence spaces equipped with the Orlicz norm are

(L N U C)

. Full article

(This article belongs to the Special Issue Advances in Functional Analysis and Banach Space)

24 pages, 344 KB

Open AccessArticle

Bayesian Updating for Stochastic Processes in Infinite-Dimensional Normed Vector Spaces

by Serena Doria

Axioms 2025, 14(12), 927; https://doi.org/10.3390/axioms14120927 - 17 Dec 2025

Viewed by 238

Abstract

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning

σ

-algebra, become inadequate when the available information is restricted to a

σ

[...] Read more.

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning

σ

-algebra, become inadequate when the available information is restricted to a

σ

-algebra generated by a finite or countable family of random variables. In such settings, many events of interest are not measurable with respect to the conditioning

σ

-field, preventing the standard definition of conditional probability. To overcome this limitation, we propose an extension of the coherent conditioning model through the use of Hausdorff measures. The key idea is to exploit the non-equivalence of norms in infinite-dimensional spaces, which gives rise to distinct metric structures and corresponding Hausdorff dimensions for the same events. Conditional probabilities are then defined relative to families of Hausdorff outer measures parameterized by their dimensional exponents. This geometric reformulation allows the notion of conditionality to depend explicitly on the underlying metric and topological properties of the space. The resulting model provides a flexible and coherent framework for analyzing conditioning in infinite-dimensional stochastic systems, with potential implications for Bayesian inference in functional spaces. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

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21 pages, 342 KB

Open AccessArticle

Strongly F-Convex Functions with Structural Characterizations and Applications in Entropies

by Hasan Barsam, Slavica Ivelić Bradanović, Matea Jelić and Yamin Sayyari

Axioms 2025, 14(12), 926; https://doi.org/10.3390/axioms14120926 - 16 Dec 2025

Viewed by 257

Abstract

Strongly convex functions form a central subclass of convex functions and have gained considerable attention due to their structural advantages and broad applicability, particularly in optimization and information theory. In this paper, we investigate the class of strongly F-convex functions, which generalizes [...] Read more.

Strongly convex functions form a central subclass of convex functions and have gained considerable attention due to their structural advantages and broad applicability, particularly in optimization and information theory. In this paper, we investigate the class of strongly F-convex functions, which generalizes the classical notion of strong convexity by introducing an auxiliary convex control function F. We establish several fundamental structural characterizations of this class and provide a variety of nontrivial examples such as power, logarithmic, and exponential functions. In addition, we derive refined Jensen-type and Hermite–Hadamard-type inequalities adapted to the strongly F-convex concept, thereby extending and sharpening their classical forms. As applications, we obtain new analytical inequalities and improved error bounds for entropy-related quantities, including Shannon, Tsallis, and Rényi entropies, demonstrating that the concept of strong F-convexity naturally yields strengthened divergence and uncertainty estimates. Full article

(This article belongs to the Special Issue Advances in Functional Analysis and Banach Space)

13 pages, 279 KB

Open AccessArticle

Accelerating Propagation Induced by Slowly Decaying Initial Data for Nonlocal Reaction-Diffusion Equations in Cylinder Domains

by Ru Hou and Yu Lu

Axioms 2025, 14(12), 925; https://doi.org/10.3390/axioms14120925 - 16 Dec 2025

Viewed by 136

Abstract

This paper investigates the phenomenon of accelerating propagation for nonlocal reaction-diffusion models with spatial and trait structure in a cylinder domain

R \times Ω

. Unlike previous studies focusing on exponentially decaying or compactly supported initial data, we consider initial functions that decay [...] Read more.

This paper investigates the phenomenon of accelerating propagation for nonlocal reaction-diffusion models with spatial and trait structure in a cylinder domain

R \times Ω

. Unlike previous studies focusing on exponentially decaying or compactly supported initial data, we consider initial functions that decay more slowly than any exponential function—such as algebraic or sub-exponential decay. By constructing a pair of super- and sub-solutions via the principal eigenfunction

ψ_{0}

of the trait operator, we prove that the solution propagates with infinitely increasing speed in the spatial direction. Explicit upper and lower bounds for the locations of level sets are derived, illustrating how the decay rate of the initial data determines the acceleration profile. The results are extended to a more general model with space- and trait-dependent competition kernels under a boundedness assumption (H3). This work highlights the crucial role of slowly decaying tails in the initial distribution in driving accelerated invasion fronts, providing a theoretical foundation for assessing propagation risks in ecology and population dynamics. Full article

26 pages, 1035 KB

Open AccessArticle

Inertial Algorithm for Best Proximity Point, Split Variational Inclusion and Equilibrium Problems with Application to Image Restorations

by Mujahid Abbas, Muhammad Waseem Asghar and Ahad Hamoud Alotaibi

Axioms 2025, 14(12), 924; https://doi.org/10.3390/axioms14120924 - 16 Dec 2025

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Abstract

If S and T are two non-self-mappings, then a solution of equation

S a^{*} = T a^{*} = a^{*}

does not necessarily exist. The common best proximity point problem is to find the approximate optimal solution of such type of [...] Read more.

If S and T are two non-self-mappings, then a solution of equation

S a^{*} = T a^{*} = a^{*}

does not necessarily exist. The common best proximity point problem is to find the approximate optimal solution of such type of equation and have a key role in theory of approximation and optimization. The primary goal of this paper is to introduce an inertial-type self-adaptive algorithm for solving the common best proximity point, generalized equilibrium and split variational inclusion problems in Hilbert spaces. The strong convergence of the proposed algorithm is given under some mild conditions. It is worth mentioning that the step size in many existing algorithms requires the prior knowledge of operator norms which is difficult to compute, whereas our proposed algorithm does not require this condition. Numerical examples are given to illustrate the efficiency and applicability of the proposed approach. We further apply the proposed algorithm to an image restoration problem and show that it achieves a higher signal-to-noise ratio compared with the existing algorithms considered in this study. Full article

(This article belongs to the Section Mathematical Analysis)

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20 pages, 291 KB

Open AccessArticle

Half-Symmetric Connections of Generalized Riemannian Spaces

by Marko Stefanović, Mića S. Stanković, Ivana Djurišić and Nenad Vesić

Axioms 2025, 14(12), 923; https://doi.org/10.3390/axioms14120923 - 16 Dec 2025

Viewed by 208

Abstract

In this article, we generalize Yano’s concept of a half-symmetric affine connection. With respect to this generalization, we obtain five linearly independent curvature tensors. In the following, we examine which special kinds of affine connections may be the generalized half-symmetric affine connection. At [...] Read more.

In this article, we generalize Yano’s concept of a half-symmetric affine connection. With respect to this generalization, we obtain five linearly independent curvature tensors. In the following, we examine which special kinds of affine connections may be the generalized half-symmetric affine connection. At the end of this work, we generalize the term of Killing’s vector given by Yano to affine Killing, conformal Killing, projective Killing, harmonic, and covariant and contravariant analytic vectors. Full article

(This article belongs to the Special Issue Advances in Geometry and Its Applications)

14 pages, 259 KB

Open AccessArticle

Is the Basic Structure of the Universe Simple?

by Martin Tamm

Axioms 2025, 14(12), 922; https://doi.org/10.3390/axioms14120922 - 14 Dec 2025

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Abstract

Can our extremely complicated world be explained, starting from simple conditions and laws? Such questions easily become metaphysical, but at the same time they have historically served science well by promoting new and fruitful ideas. Of particular interest in this paper is the [...] Read more.

Can our extremely complicated world be explained, starting from simple conditions and laws? Such questions easily become metaphysical, but at the same time they have historically served science well by promoting new and fruitful ideas. Of particular interest in this paper is the unification of general relativity and quantum mechanics. It is suggested that a way to find a common basis for these theories could be to view them both as stochastic theories which, among all possible macroscopic developments, promote the simplest ones. But with the difference that general relativity uses real probabilities, whereas quantum mechanics uses complex weights. In a certain sense this approach goes back to the Principle of Least Action, although the perspective on this principle in the present paper is different from the one which is commonly used in contemporary physics. It is also suggested that a more general principle, which applies to both theories and which is beyond both stationarity and minimizing, could give us a better starting point for the unification. Full article

(This article belongs to the Special Issue Recent Advances in Quantum Mechanics and Mathematical Physics)

23 pages, 346 KB

Open AccessArticle

Fractional Stochastic Systems Driven by Fractional Brownian Motion: Existence, Uniqueness, and Approximate Controllability with Generalized Memory Effects

by Muhammad Imran Liaqat, Abdelhamid Mohammed Djaouti and Ashraf Al-Quran

Axioms 2025, 14(12), 921; https://doi.org/10.3390/axioms14120921 - 14 Dec 2025

Viewed by 294

Abstract

In this research work, we present findings on fractional stochastic systems characterized by fractional Brownian motion, which is defined by a Hurst parameter

H \in (\frac{1}{2}, 1)

. These systems are crucial for modeling complex phenomena that diverge from Markovian behavior [...] Read more.

In this research work, we present findings on fractional stochastic systems characterized by fractional Brownian motion, which is defined by a Hurst parameter

H \in (\frac{1}{2}, 1)

. These systems are crucial for modeling complex phenomena that diverge from Markovian behavior and exhibit long-range dependence, particularly in areas such as financial engineering and statistical physics. We utilize the fixed-point iteration method to demonstrate the existence and uniqueness (Ex-Un) of mild solutions. Additionally, we investigate the approximate controllability of the system. We establish all results within the framework of the

μ

-Caputo fractional derivative. This study makes a meaningful contribution to the existing body of literature by rigorously establishing the existence, uniqueness, and approximate controllability of mild solutions to generalized Caputo fractional stochastic differential equations driven by fractional Brownian motion. Full article

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

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20 pages, 2291 KB

Open AccessArticle

New Inequalities and Approximations of Cusa–Huygens Type

by Branko Malešević, Miloš Mićović and Tatjana Lutovac

Axioms 2025, 14(12), 920; https://doi.org/10.3390/axioms14120920 - 14 Dec 2025

Viewed by 220

Abstract

In this paper, we introduce a real parameter in some Cusa–Huygens-type inequalities. Using the concept of stratification, we obtain new inequalities and approximations. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

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14 pages, 2795 KB

Open AccessArticle

On the Focal Geometry of Translation Surfaces: Flatness, Minimality, and Classification Results

by Sezgin Büyükkütük, İlim Kişi, Günay Öztürk and Emre Kişi

Axioms 2025, 14(12), 919; https://doi.org/10.3390/axioms14120919 - 14 Dec 2025

Viewed by 249

Abstract

In this study, we investigate the focal surfaces associated with translation surfaces in Euclidean 3-space from the viewpoint of differential geometry. We begin by defining the translation surface generated by two planar curves and derive the corresponding focal surfaces using the framework of [...] Read more.

In this study, we investigate the focal surfaces associated with translation surfaces in Euclidean 3-space from the viewpoint of differential geometry. We begin by defining the translation surface generated by two planar curves and derive the corresponding focal surfaces using the framework of the Frenet frame. Analytical conditions are obtained under which the focal surfaces exhibit minimality or flatness. Several theorems are proven to classify the focal images, supported by illustrative examples. The results provide insights into the curvature structure of translation surfaces and contribute to the broader understanding of their geometric behavior. Full article

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

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14 pages, 263 KB

Open AccessArticle

Non-Negative Equilibrium Prices and Market Portfolio Under Minimax Diversification with Non-Homogeneous Investors

by Hongyu Yang, Jia Liu and Zijian Luo

Axioms 2025, 14(12), 918; https://doi.org/10.3390/axioms14120918 - 12 Dec 2025

Viewed by 184

Abstract

Market equilibrium is characterized by a state wherein aggregate demand equals aggregate supply for all assets, a condition arising from consumers maximizing utility within budget constraints and producers maximizing profits. This paper investigates a financial market populated by non-homogeneous investors who may employ [...] Read more.

Market equilibrium is characterized by a state wherein aggregate demand equals aggregate supply for all assets, a condition arising from consumers maximizing utility within budget constraints and producers maximizing profits. This paper investigates a financial market populated by non-homogeneous investors who may employ heterogeneous deviation measures to formulate their risk functions. By integrating the minimax risk diversification principle with the framework of individual utility maximization, we analytically derive the master fund for each investor. Furthermore, we establish the necessary and sufficient conditions for the existence of a unique non-negative equilibrium price system for risky assets and provide its explicit formula. A key finding is that the market portfolio is a convex combination of all investors’ master funds. Full article

(This article belongs to the Special Issue Financial Mathematics and Econophysics)

20 pages, 391 KB

Open AccessArticle

Integral Transforms in Number Theory

by Guodong Liu, Takako Kuzumaki and Shigeru Kanemitsu

Axioms 2025, 14(12), 917; https://doi.org/10.3390/axioms14120917 - 12 Dec 2025

Viewed by 337

Abstract

Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few [...] Read more.

Integral transforms play a fundamental role in science and engineering. Above all, the Fourier transform is the most vital, which has some specifications—Laplace transform, Mellin transform, etc., with their inverse transforms. In this paper, we restrict ourselves to the use of a few versions of the Mellin transform, which are best suited to the treatment of zeta functions as Dirichlet series. In particular, we shall manifest the underlying principle that automorphy (which is a modular relation, an equivalent to the functional equation) is intrinsic to lattice (or Epstein) zeta functions by considering some generalizations of the holomorphic and non-holomorphic Eisenstein series as the Epstein-type Eisenstein series, which have been treated as totally foreign subjects to each other. We restrict to the modular relations with one gamma factor and the resulting integrals reduce to a form of the modified Bessel function. In the H-function hierarchy, what we work with is the second simplest

H_{1, 1}^{1, 1} \leftrightarrow H_{0, 2}^{2, 0}

, with H denoting the Fox H-function. Full article

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

12 pages, 268 KB

Open AccessArticle

Trace of Fischer–Marsden Equation on a Riemannian Space

by Hana Al-Sodais, Sana Hamoud Alshammari and Sharief Deshmukh

Axioms 2025, 14(12), 916; https://doi.org/10.3390/axioms14120916 - 12 Dec 2025

Viewed by 234

Abstract

Among the important differential equations on a Riemannian space

(M, g)

of dimension n are the static perfect fluid equation (SPFE), namely [...] Read more.

Among the important differential equations on a Riemannian space

(M, g)

of dimension n are the static perfect fluid equation (SPFE), namely

f R i c - H e s s (f) = \frac{1}{n} (f τ - Δ f) g

, and the Fischer–Marsden equation (FME), namely

(Δ f) g + f R i c = H e s s (f)

, where

R i c

is the Ricci tensor,

τ

is the scalar curvature of

(M, g)

and

H e s s (f)

and

Δ f

are the Hessian and the Laplacian of the smooth function f. The trace of the FME is

Δ f = - \frac{τ}{n - 1} f

, which we call the TFME, and if we combine the TFME with the SPFE, we observe that it reduces to the FME. Thus, in the presence of the TFME on the Riemannian space

(M, g)

the fundamental differential equations SPFE and FME are equivalent. In this article, we consider the presence of the TFME on a Riemannian space

(M, g)

and study its impact on the Riemannian space

(M, g)

. The importance of this study follows from the fact that results obtained for Riemannian spaces admitting solutions to the TFME automatically are generalizations of corresponding results on spaces admitting solutions to the FME. First, we show that for a connected and compact Riemannian space

(M, g)

,

dim M = n > 1

, with scalar curvature

τ

that admits a nontrivial solution f to the TFME, with the Ricci operator Q satisfying

Q (\nabla f) = \frac{τ}{n} \nabla f

, and with the integral of

R i c (\nabla f, \nabla f)

having a suitable lower bound, it is necessary and sufficient that

(M, g)

is an n-sphere

S^{n} (c)

. In addition, we show that a compact and connected space

(M, g)

,

dim M = n > 1

, admits a nontrivial solution f to the TFME such that the scalar curvature

τ

satisfies

(n - 1) c < τ \leq n (n - 1) c

for some constant

c > 0

, and the Ricci curvature

R i c (\nabla f, \nabla f)

is bounded below by

(n - 1) c

, if and only if

(M, g)

is an n-sphere

S^{n} (c)

. Finally, it is shown that a connected and compact Riemannian space

(M, g)

,

dim M =

n > 1

, with constant scalar curvature

τ

admits a nontrivial solution f to the TFME, with the Ricci operator Q satisfying

Q (\nabla f) = \frac{τ}{n} \nabla f

, if and only if

(M, g)

is the sphere

S^{n} (c)

. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Singularity Theory, 2nd Edition)

16 pages, 307 KB

Open AccessArticle

Integral Inequalities for Vector (Multi)functions

by Cristina Stamate and Anca Croitoru

Axioms 2025, 14(12), 915; https://doi.org/10.3390/axioms14120915 - 12 Dec 2025

Viewed by 218

Abstract

We present some integral inequalities such as Minkowski-type and optimal bound-type for vector functions and vector multifunctions for different kinds of integrals:

G

-integral, Choquet-type integral, and Sugeno-type integral. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities, 2nd Edition)

28 pages, 2482 KB

Open AccessArticle

Research on the Flexible Job Shop Scheduling Problem with Job Priorities Considering Transportation Time and Setup Time

by Chuchu Zheng and Zhiqiang Xie

Axioms 2025, 14(12), 914; https://doi.org/10.3390/axioms14120914 - 12 Dec 2025

Viewed by 508

Abstract

This paper addresses the flexible job-shop scheduling problem with multiple time factors—namely, transportation time and setup time—as well as job priorities (referred to as FJSP-JPC-TST). An optimization model is established with the objective of minimizing the completion time. Considering the characteristics of the [...] Read more.

This paper addresses the flexible job-shop scheduling problem with multiple time factors—namely, transportation time and setup time—as well as job priorities (referred to as FJSP-JPC-TST). An optimization model is established with the objective of minimizing the completion time. Considering the characteristics of the FJSP-JPC-TST, we propose an improved whale optimization algorithm that incorporates multiple strategies. First, a two-layer encoding mechanism based on operations and machines is introduced. To prevent illegal solutions, a priority-based encoding repair mechanism is designed, along with an active scheduling decoding method that fully considers multiple time factors and job priorities. Subsequently, a multi-level sub-population optimization strategy, an adaptive inertia weight, and a cross-population differential evolution strategy are implemented to enhance the optimization efficiency of the algorithm. Finally, extensive simulation experiments demonstrate that the proposed algorithm offers significant advantages and exhibits high reliability in effectively solving such scheduling problems. Full article

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16 pages, 308 KB

Open AccessArticle

Generalizations of Prime Hyperideals via Hypersystems in Krasner Hyperrings

by Mehmet Bozdaş and Ummahan Acar

Axioms 2025, 14(12), 913; https://doi.org/10.3390/axioms14120913 - 12 Dec 2025

Viewed by 307

Abstract

The aim of this study is to investigate generalized prime hyperideals in the framework of Krasner hyperrings. To this end, new classes of hyperideals are introduced and analyzed based on multiplicatively closed properties. In particular, the concepts of s-m-hypersystems, f-hypersystems, and their associated [...] Read more.

The aim of this study is to investigate generalized prime hyperideals in the framework of Krasner hyperrings. To this end, new classes of hyperideals are introduced and analyzed based on multiplicatively closed properties. In particular, the concepts of s-m-hypersystems, f-hypersystems, and their associated s-prime and f-prime hyperideals are defined and examined. A subset

S \subseteq R

of a Krasner hyperring is called an s-m-hypersystem if, for every

s \in S

, there exists a multiplicatively closed subset

S^{*} \subseteq S

such that

⟨ s ⟩ \cap S^{*} \neq \emptyset

. This concept extends the classical ideal of multiplicative compatibility to the setting of hyperrings. Furthermore, for each element

a \in R

, we define a hyperideal

f (a)

satisfying the following conditions: (i)

a \in f (a)

, (ii) For any hyperideal K, if

x \in f (a) + K

, then

f (x) \subseteq f (a) + K

. Using this notion, a subset

S \subseteq R

is defined to be an f-hypersystem if there exists a multiplicatively closed subset

S^{*} \subseteq S

such that

f (a) \cap S^{*} \neq \emptyset

for every

a \in S

. We provide characterizations and original examples of these hypersystems and their corresponding prime hyperideals. The relationships and distinctions between the s-m-hypersystems and f-hypersystems are also explored. Our findings offer a refined perspective on hyperideal theory and open new pathways for the algebraic analysis of hyperstructures. Full article

(This article belongs to the Section Algebra and Number Theory)

28 pages, 4875 KB

Open AccessArticle

On Quasi-Monotone Stochastic Variational Inequalities with Applications

by Mohammad Dilshad, Ibrahim Al-Dayel, Francis O. Nwawuru and Jeremiah N. Ezeora

Axioms 2025, 14(12), 912; https://doi.org/10.3390/axioms14120912 - 11 Dec 2025

Viewed by 272

Abstract

This paper studies an efficient method for solving stochastic optimization problems formulated as stochastic variational inequalities with a quasi-monotone operator, where the cost function extends the classical monotone and pseudomonotone operators. Our proposed method iterates an adaptive stepsize that adjusts automatically without linesearch [...] Read more.

This paper studies an efficient method for solving stochastic optimization problems formulated as stochastic variational inequalities with a quasi-monotone operator, where the cost function extends the classical monotone and pseudomonotone operators. Our proposed method iterates an adaptive stepsize that adjusts automatically without linesearch and includes a momentum term to accelerate the convergence. Each iteration requires only a single projection onto the feasible set, ensuring low computational complexity. Under standard assumptions, the algorithm achieves almost sure convergence and a proven convergence rate. Furthermore, numerical experiments demonstrate its superior performance, accuracy, stability, and efficiency compared with existing stochastic approximation schemes. We also apply the method to problems such as stochastic network bandwidth allocation, stochastic complementarity problems, and the networked stochastic Nash–Cournot game, showing its strength and practical usefulness. The obtained result is an extension of existing works in the literature. Full article

(This article belongs to the Special Issue Mathematical Optimization, Variational Inequalities and Equilibrium Problems: Theory and Applications)

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20 pages, 328 KB

Open AccessArticle

Well-Posedness for a System of Generalized KdV-Type Equations Driven by White Noise

by Aissa Boukarou, Mohammadi Begum Jeelani and Nouf Abdulrahman Alqahtani

Axioms 2025, 14(12), 911; https://doi.org/10.3390/axioms14120911 - 11 Dec 2025

Viewed by 156

Abstract

In this paper, we investigate the Cauchy problem for the coupled generalized Korteweg–de Vries system driven by a cylindrical Wiener process. We prove local well-posedness for data in

H^{s} \times H^{s},

with

s > \frac{1}{2}

. The key methods [...] Read more.

In this paper, we investigate the Cauchy problem for the coupled generalized Korteweg–de Vries system driven by a cylindrical Wiener process. We prove local well-posedness for data in

H^{s} \times H^{s},

with

s > \frac{1}{2}

. The key methods that we used in this paper are multilinear estimates in Bourgain spaces, the Itô formula, and a fixed-point argument. Full article

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

14 pages, 267 KB

Open AccessArticle

Deriving Binomial Convolution Formulas for Horadam Sequences via Context-Free Grammars

by Jun-Ying Liu, Hai-Ling Li, Zhi-Hong Zhang and Tao Liu

Axioms 2025, 14(12), 910; https://doi.org/10.3390/axioms14120910 - 11 Dec 2025

Viewed by 230

Abstract

The Horadam sequence

H_{n} (a, b; p, q)

unifies a number of well-known sequences, such as Fibonacci and Lucas sequences. We use the context-free grammars as a new tool to study Horadam sequences. By introducing a set [...] Read more.

The Horadam sequence

H_{n} (a, b; p, q)

unifies a number of well-known sequences, such as Fibonacci and Lucas sequences. We use the context-free grammars as a new tool to study Horadam sequences. By introducing a set of auxiliary basis polynomials (

v_{1}, v_{2}, v_{3}

) and using the formal derivative associated with the Horadam grammar, we solve the convolution coefficients and provide a unified method to discover convolution formulas associated with binomial coefficients. These results are extended to subsequences with indices

k n

through a parameterized grammar

G_{k}

. Using the modified grammar

\tilde{G_{k}}

, we derive convolution formulas involving the weighting term

{(- q)}^{n - i}

. Furthermore, applying the proposed framework to

(p, q)

-Fibonacci and

(p, q)

-Lucas sequences, we derive explicit convolution formulas with parameters

(p, q)

. The framework is also applied to derive specific identities for Pell and Pell–Lucas numbers, as well as for Fermat and Fermat–Lucas numbers. Full article

(This article belongs to the Section Algebra and Number Theory)

16 pages, 501 KB

Open AccessArticle

Synchronization of Markovian Switching Stochastic Delayed Complex Dynamical Networks via Pinning Control

by Yanbo Ling and Shang Gao

Axioms 2025, 14(12), 909; https://doi.org/10.3390/axioms14120909 - 11 Dec 2025

Viewed by 264

Abstract

This study examines the synchronization of Markovian switching stochastic delayed complex dynamical networks (MSSDCDNs). MSSDCDNs have general structures and coupling forms, which are influenced by Markovian switching, random disturbances, and time delays. Simultaneously, relevant controllers are incorporated into certain nodes. Utilizing the theory [...] Read more.

This study examines the synchronization of Markovian switching stochastic delayed complex dynamical networks (MSSDCDNs). MSSDCDNs have general structures and coupling forms, which are influenced by Markovian switching, random disturbances, and time delays. Simultaneously, relevant controllers are incorporated into certain nodes. Utilizing the theory of stochastic differential equations, we establish adequate requirements to guarantee exponential synchronization in mean square inside the network by formulating suitable Lyapunov functions and employing general Itô formula and inequality approaches. Lastly, numerical examples and simulations are used to verify the validity of the derived theoretical results. Full article

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21 pages, 1271 KB

Open AccessArticle

Bell Nonlocality and EPR Steering Decay in Dephasing Hyperfine Spins

by Kamal Berrada and Smail Bougouffa

Axioms 2025, 14(12), 908; https://doi.org/10.3390/axioms14120908 - 10 Dec 2025

Viewed by 200

Abstract

This work presents a comprehensive study of quantum correlations and their degradation under environmental dephasing within the atomic hydrogen system. By analyzing the magnetic coupling between the electron and proton spins in the

1 s

hyperfine state, we elucidate how coherent spin interactions [...] Read more.

This work presents a comprehensive study of quantum correlations and their degradation under environmental dephasing within the atomic hydrogen system. By analyzing the magnetic coupling between the electron and proton spins in the

1 s

hyperfine state, we elucidate how coherent spin interactions generate entangled states and govern their temporal evolution. The investigation focuses on three key measures of quantum correlations—Bell nonlocality, Einstein–Podolsky–Rosen (EPR) steering, and quantum purity—each reflecting a different level within the hierarchy of nonclassical correlations. Analytical formulations and numerical simulations reveal that, in the absence of decay, all quantities remain steady, indicating the preservation of coherence. When dephasing is introduced, each measure decays exponentially toward a stationary lower bound, with Bell nonlocality identified as the most fragile, followed by steering and purity. A three-dimensional analysis of Werner states under dephasing further establishes the critical purity thresholds required for Bell inequality violations. The results highlight the interdependence between magnetic coupling, decoherence, and initial entanglement, providing a unified framework for understanding correlation dynamics in open quantum systems. These findings have direct implications for the development of noise-resilient quantum information protocols and spin-based quantum technologies, where preserving nonlocal correlations is essential for reliable quantum operations. Full article

(This article belongs to the Special Issue Recent Advances in Mathematical Physics with Applications in Quantum Theory)

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32 pages, 1623 KB

Open AccessArticle

Common Eigenvalues of Vertex-Decorated Regular Graphs

by Vladimir R. Rosenfeld

Axioms 2025, 14(12), 907; https://doi.org/10.3390/axioms14120907 - 10 Dec 2025

Viewed by 357

Abstract

Let

G = (V, E)

be a simple graph with the vertex set V and the edge set E

(| V | = n, | E | = m)

. An example of a vertex-decorated graph

D G

is [...] Read more.

Let

G = (V, E)

be a simple graph with the vertex set V and the edge set E

(| V | = n, | E | = m)

. An example of a vertex-decorated graph

D G

is a vertex-quadrangulated graph

Q G

. The vertex quadrangulation

Q G

of 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. If we contract each quadrangle of

Q G

to a point that takes over the incidence of the four edges that were previously joined to this quadrangle, then we can again get the original graph G. Any connected graph H that provides (some of) its vertices for external connections can play the role of a decorating graph, and any graph G with vertices of valency no greater than the number of contact vertices in H can be decorated with it. Herein, we consider the case when G is a regular graph. Since the decoration also depends on the way the edges are attached to the decorating graph, we clearly stipulate it. We show that all similarly decorated regular graphs

D G

that meet our conditions have at least

| V (H) |

predicted common eigenvalues. A number of related results are proven. As possible applications of these results in chemistry, cases of simplified findings of eigenvalues of a molecular graph even in the absence of the usual symmetry of the molecule may be of interest. This, in particular, can somewhat expand the possibilities of applying the simple Hückel method for large molecules. Full article

(This article belongs to the Section Algebra and Number Theory)

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20 pages, 654 KB

Open AccessArticle

Semigroups and Evolution Equations in Modular Function Spaces

by Mostafa Bachar

Axioms 2025, 14(12), 906; https://doi.org/10.3390/axioms14120906 - 10 Dec 2025

Viewed by 243

Abstract

This paper develops the theory of strongly continuous semigroups and abstract evolution equations in modular function spaces. We study the autonomous problem

\dot{u} (t) = B u (t)

with initial condition [...] Read more.

This paper develops the theory of strongly continuous semigroups and abstract evolution equations in modular function spaces. We study the autonomous problem

\dot{u} (t) = B u (t)

with initial condition

u (0) = u_{0} \in L_{ρ}

, where B is the infinitesimal generator of a strongly continuous semigroup

{(S (t))}_{t \geq 0}

on

L_{ρ}

. Within this framework, we establish modular analogues of classical results from Banach-space semigroup theory, including criteria for

ρ

-boundedness and

ρ

-continuity, a Laplace resolvent representation of the generator, and explicit resolvent bounds in terms of the modular growth function

ω_{ρ}

. Under a

Δ_{2}

-type condition on the modular, we justify Steklov regularization of semigroup orbits, obtain domain inclusion and the resolvent identity, and derive spectral consequences for classes of operators naturally acting on

L_{ρ}

. The results show that the structural features of the classical semigroup framework persist in the modular topology, providing a unified approach to linear evolution in modular function spaces. Full article

(This article belongs to the Special Issue Advances in Geometric Function Theory and Related Topics)

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41 pages, 1214 KB

Open AccessArticle

Mathematical Stability Analysis of the Full SREBP-2 Pathway Model: Insights into Cholesterol Homeostasis

by Mostafa Bachar

Axioms 2025, 14(12), 905; https://doi.org/10.3390/axioms14120905 - 9 Dec 2025

Viewed by 241

Abstract

We present a mathematical analysis of the sterol regulatory element-binding protein 2 (SREBP-2) pathway, a key regulator of intracellular cholesterol homeostasis. Using a compartment model formulated as a nonlinear system of ordinary differential equations, we investigate stability via M-matrix theory and norm-based [...] Read more.

We present a mathematical analysis of the sterol regulatory element-binding protein 2 (SREBP-2) pathway, a key regulator of intracellular cholesterol homeostasis. Using a compartment model formulated as a nonlinear system of ordinary differential equations, we investigate stability via M-matrix theory and norm-based criteria. We show that the Frobenius norm

{∥ B ∥}_{F} \geq 1

cannot ensure stability, whereas the infinity norm condition

{∥ B ∥}_{\infty} < 1

provides a practical guarantee that the spectral radius

ρ (B) < 1

. The spectral norm

{∥ B ∥}_{2}

yields sharper intermediate bounds. Numerical simulations confirm these results, highlighting parameter regions of stability and showing that the dissociation rate

k_{- 1}

has the strongest influence on system behavior. These findings demonstrate the robustness of the

ℓ_{\infty}

criterion, clarify the role of dissociation kinetics in cholesterol regulation, and provide a rigorous framework for assessing homeostatic control in the SREBP-2 pathway. Full article

(This article belongs to the Special Issue New Perspectives in Bifurcations Analysis of Dynamical Systems)

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16 pages, 331 KB

Open AccessArticle

Singularity Conditions and Probabilities of Tricyclic Graphs

by Haicheng Ma and Xianghong He

Axioms 2025, 14(12), 904; https://doi.org/10.3390/axioms14120904 - 8 Dec 2025

Viewed by 248

Abstract

A graph G is said to be singular if its adjacency matrix is a singular matrix. In this paper, by analyzing the structure of Sachs subgraphs on three types of tricyclic graphs [...] Read more.

A graph G is said to be singular if its adjacency matrix is a singular matrix. In this paper, by analyzing the structure of Sachs subgraphs on three types of tricyclic graphs

ϖ (a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2})

,

σ (a_{1}, a_{2}, b_{1}, b_{2}, c)

, and

τ (a_{1}, a_{2}, a_{3}, a_{4})

and calculating the determinants of their adjacency matrices, we investigate the necessary and sufficient conditions for the singularity of these three types of tricyclic graphs. Furthermore, we derive that the probabilities of singular graphs among tricyclic graphs

ϖ (a_{1}, a_{2}, b_{1}, b_{2}, c_{1}, c_{2})

,

σ (a_{1}, a_{2}, b_{1}, b_{2}, c)

, and

τ (a_{1}, a_{2}, a_{3}, a_{4})

are

\frac{217}{512}, \frac{113}{256}

, and

\frac{67}{128},

respectively. Full article

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20 pages, 882 KB

Open AccessArticle

Bifurcation Analysis in a Cross-Protection Model

by Yufei Wu, Zikun Han, Weixiang Wang, Yingting Yang and Qiubao Wang

Axioms 2025, 14(12), 903; https://doi.org/10.3390/axioms14120903 - 7 Dec 2025

Viewed by 186

Abstract

We analyze the population dynamics of a microbial cross-protection model and derive the exact conditions under which a Fold–Hopf bifurcation emerges. By applying center-manifold reduction and normal-form theory, we reduce the infinite-dimensional delay differential system to a finite-dimensional ordinary differential system, enabling rigorous [...] Read more.

We analyze the population dynamics of a microbial cross-protection model and derive the exact conditions under which a Fold–Hopf bifurcation emerges. By applying center-manifold reduction and normal-form theory, we reduce the infinite-dimensional delay differential system to a finite-dimensional ordinary differential system, enabling rigorous bifurcation analysis. Numerical simulations reveal a rich repertoire of dynamical behaviors, including stable equilibria, sustained oscillations, and noise-induced irregularities. Our findings identify time-delay-induced Fold–Hopf bifurcation as a fundamental mechanism driving oscillatory coexistence in cross-protection mutualisms, for previously reported experimental observations. Full article

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32 pages, 1584 KB

Open AccessArticle

Adaptive Sparse Clustering of Mixed Data Using Azzalini-Encoded Ordinal Variables

by Ismail Arjdal, Mohamed Alahiane, Echarif Elharfaoui and Mustapha Rachdi

Axioms 2025, 14(12), 902; https://doi.org/10.3390/axioms14120902 - 7 Dec 2025

Viewed by 205

Abstract

In this paper, we propose a novel sparse clustering method designed for high-dimensional mixed-type data, integrating Azzalini’s score-based encoding for ordinal variables. Our approach aims to retain the inherent nature of each variable type—continuous, ordinal, and nominal—while enhancing clustering quality and interpretability. To [...] Read more.

In this paper, we propose a novel sparse clustering method designed for high-dimensional mixed-type data, integrating Azzalini’s score-based encoding for ordinal variables. Our approach aims to retain the inherent nature of each variable type—continuous, ordinal, and nominal—while enhancing clustering quality and interpretability. To this end, we extend classical distance metrics and adapt the Davies–Bouldin Index (DBI) to better reflect the structure of mixed data. We also introduce a weighted formulation that accounts for the distinct contributions of variable types in the clustering process. Empirical results on simulated and real-world datasets demonstrate that our method consistently achieves better separation and coherence of clusters compared to traditional techniques, while effectively identifying the most informative variables. This work opens promising directions for clustering in complex, high-dimensional settings such as marketing analytics and customer segmentation. Full article

(This article belongs to the Special Issue Stochastic Modeling and Optimization Techniques)

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33 pages, 11429 KB

Open AccessArticle

Two-Dimensional Coupling-Enhanced Cubic Hyperchaotic Map with Exponential Parameters: Construction, Analysis, and Application in Hierarchical Significance-Aware Multi-Image Encryption

by Wei Feng, Zixian Tang, Xiangyu Zhao, Zhentao Qin, Yao Chen, Bo Cai, Zhengguo Zhu, Kun Qian and Heping Wen

Axioms 2025, 14(12), 901; https://doi.org/10.3390/axioms14120901 - 6 Dec 2025

Cited by 2 | Viewed by 281

Abstract

As digital images proliferate across open networks, securing them against unauthorized access has become imperative. However, many recent image encryption algorithms are limited by weak chaotic dynamics and inadequate cryptographic design. To overcome these, we propose a new 2D coupling-enhanced cubic hyperchaotic map [...] Read more.

As digital images proliferate across open networks, securing them against unauthorized access has become imperative. However, many recent image encryption algorithms are limited by weak chaotic dynamics and inadequate cryptographic design. To overcome these, we propose a new 2D coupling-enhanced cubic hyperchaotic map with exponential parameters (2D-CCHM-EP). By incorporating exponential terms and strengthening interdependence among state variables, the 2D-CCHM-EP exhibits strict local expansiveness, effectively suppresses periodic windows, and achieves robust hyperchaotic behavior, validated both theoretically and numerically. It outperforms several recent chaotic maps in key metrics, yielding significantly higher Lyapunov exponents and Kolmogorov–Sinai entropy, and passes all NIST SP 800-22 randomness tests. Leveraging the 2D-CCHM-EP, we further develop a hierarchical significance-aware multi-image encryption algorithm (MIEA-CPHS). The core of MIEA-CPHS is a hierarchical significance-aware encryption strategy that decomposes input images into high-, medium-, and low-significance layers, which undergo three, two, and one round of vector-level adaptive encryption operations. An SHA-384-based hash of the fused data dynamically generates a 48-bit adaptive control parameter, enhancing plaintext sensitivity and enabling integrity verification. Comprehensive security analyses confirm the exceptional performance of MIEA-CPHS: near-zero inter-pixel correlation (<0.0016), near-ideal Shannon entropy (>7.999), and superior plaintext sensitivity (NPCR

\approx 99.61 %

, UACI

\approx 33.46 %

). Remarkably, the hierarchical design and vectorized operations achieve an average encryption throughput of 87.6152 Mbps, striking an outstanding balance between high security and computational efficiency. This makes MIEA-CPHS highly suitable for modern high-throughput applications such as secure cloud storage and real-time media transmission. Full article

(This article belongs to the Special Issue Nonlinear Dynamical System and Its Applications)

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Axioms, Volume 14, Issue 12 (December 2025) – 75 articles

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