Axioms

41 pages, 1213 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

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)

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

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 31

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, 1585 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 31

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)

33 pages, 11428 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

Viewed by 58

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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19 pages, 316 KB

Open AccessArticle

New Upper Bounds on the Number of Maximum Independent Sets in a Graph

by Vadim E. Levit and Elizabeth J. Itskovich

Axioms 2025, 14(12), 900; https://doi.org/10.3390/axioms14120900 - 5 Dec 2025

Viewed by 164

Abstract

An independent set in a graph comprises vertices that are not adjacent to one another, whereas a clique consists of vertices where all pairs are adjacent. For a given graph G, let the following notations be defined: the number of vertices in [...] Read more.

An independent set in a graph comprises vertices that are not adjacent to one another, whereas a clique consists of vertices where all pairs are adjacent. For a given graph G, let the following notations be defined: the number of vertices in G is n, the cardinality of a maximum independent set in G is

α

, the size of the largest clique in G is

ω

, the cardinality of the intersection of all maximum independent sets in G is

ξ

, and the number of maximum independent sets in G is

s_{α}

. As the main finding of this article, we present an upper bound on the number of maximum independent sets as follows:

s_{α} \leq \{\begin{matrix} ω \cdot 2^{n - α - ω + 1}, if n - α - ω + 1 \leq α - ξ - 1; \\ (\binom{n - α - ω + 1}{α - ξ}) + ω \cdot \sum_{k = 0}^{α - ξ - 1} (\binom{n - α - ω + 1}{k}), \\ if n - α - ω + 1 \geq α - ξ . \end{matrix}

. As an application of our findings, we explore a series of inequalities that connects the number of longest increasing subsequences with the number of longest decreasing subsequences in a given sequence of integers. Full article

(This article belongs to the Special Issue Advances in Combinatorial Optimization and Discrete Mathematics with Applications)

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27 pages, 2727 KB

Open AccessArticle

The Module Gradient Descent Algorithm via L₂ Regularization for Wavelet Neural Networks

by Khidir Shaib Mohamed, Ibrahim. M. A. Suliman, Abdalilah Alhalangy, Alawia Adam, Muntasir Suhail, Habeeb Ibrahim, Mona A. Mohamed, Sofian A. A. Saad and Yousif Shoaib Mohammed

Axioms 2025, 14(12), 899; https://doi.org/10.3390/axioms14120899 - 4 Dec 2025

Viewed by 81

Abstract

Although wavelet neural networks (WNNs) combine the expressive capability of neural models with multiscale localization, there are currently few theoretical guarantees for their training. We investigate the weight decay (

L_{2}

regularization) optimization dynamics of gradient descent (GD) for WNNs. Using explicit [...] Read more.

Although wavelet neural networks (WNNs) combine the expressive capability of neural models with multiscale localization, there are currently few theoretical guarantees for their training. We investigate the weight decay (

L_{2}

regularization) optimization dynamics of gradient descent (GD) for WNNs. Using explicit rates controlled by the spectrum of the regularized Gram matrix, we first demonstrate global linear convergence to the unique ridge solution for the feature regime when wavelet atoms are fixed and only the linear head is trained. Second, for fully trainable WNNs, we demonstrate linear rates in regions satisfying a Polyak–Łojasiewicz (PL) inequality and establish convergence of GD to stationary locations under standard smoothness and boundedness of wavelet parameters; weight decay enlarges these regions by suppressing flat directions. Third, we characterize the implicit bias in the over-parameterized neural tangent kernel (NTK) regime: GD converges to the minimum reproducing kernel Hilbert space (RKHS) norm interpolant associated with the WNN kernel with

L_{2}

. In addition to an assessment process on synthetic regression, denoising, and ablations across λ and stepsize, we supplement the theory with useful recommendations on initialization, stepsize schedules, and regularization scales. Together, our findings give a principled prescription for dependable training that has broad applicability to signal processing applications and shed light on when and why

L_{2}

-regularized GD is stable and quick for WNNs. Full article

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11 pages, 246 KB

Open AccessArticle

Supercyclic Weighted Translations on Quotient Spaces

by AliReza Bagheri Salec, Chung-Chuan Chen, Seyyed Mohammad Tabatabaie and Zahra Saeed Abdulazeez Alfaikhrani

Axioms 2025, 14(12), 898; https://doi.org/10.3390/axioms14120898 - 3 Dec 2025

Viewed by 85

Abstract

In this note, we give the sufficient and necessary condition for weighted translations on the Orlicz spaces of quotient spaces to be supercyclic. By applying this characterization of supercyclicity, the descriptions of hypercyclicity, topological mixing and Cesàro hypercyclicity on such spaces are obtained [...] Read more.

In this note, we give the sufficient and necessary condition for weighted translations on the Orlicz spaces of quotient spaces to be supercyclic. By applying this characterization of supercyclicity, the descriptions of hypercyclicity, topological mixing and Cesàro hypercyclicity on such spaces are obtained as well. Full article

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

23 pages, 910 KB

Open AccessArticle

Fractal Modeling of Generalized Weighted Pre-Invex Functions with Applications to Random Variables and Special Means

by Muhammad Muddassar, Maria Bibi, Kashif Nazar and Adil Jhangeer

Axioms 2025, 14(12), 897; https://doi.org/10.3390/axioms14120897 - 2 Dec 2025

Viewed by 110

Abstract

This article introduces certain algebraic properties of generalized

({\tilde{h}}_{1}, {\tilde{h}}_{2})

-pre-invex functions on

R^{ℵ}

(0 < ℵ \leq 1)

. A new fractal weighted integral identity is established and further employed to obtain [...] Read more.

This article introduces certain algebraic properties of generalized

({\tilde{h}}_{1}, {\tilde{h}}_{2})

-pre-invex functions on

R^{ℵ}

(0 < ℵ \leq 1)

. A new fractal weighted integral identity is established and further employed to obtain several Ostrowski-type results in the fractal setting for functions whose first derivatives in the modulus belong to the generalized

({\tilde{h}}_{1}, {\tilde{h}}_{2})

-pre-invex functions’s class. An illustrative example is presented to validate the theoretical findings. Moreover, applications of the main results are derived in connection with generalized random variables and various special means, highlighting the effectiveness and potential scope of the proposed approach. Full article

(This article belongs to the Special Issue Generalized Fractional Operators and Special Functions: Theory, Methods, and Applications)

14 pages, 289 KB

Open AccessArticle

Goedesics Completeness and Cauchy Hypersurfaces of Ricci Solitons on Pseudo-Riemannian Hypersurfaces at the Fictitious Singularity: Schwarzschild-Soliton Geometries and Generalized-Schwarzschild-Soliton Ones

by Orchidea Maria Lecian

Axioms 2025, 14(12), 896; https://doi.org/10.3390/axioms14120896 - 2 Dec 2025

Viewed by 98

Abstract

The methodology is developed here to write Ricci solitons on the newly found structure of the pseudo-spherical cylinder. The methodology is specified for Schwarzschild solitons and for Generalized-Schwarzschild solitons. Accordingly, a new classification is written for the Schwarzschild solitons and for the Generalized-Schwarzschild [...] Read more.

The methodology is developed here to write Ricci solitons on the newly found structure of the pseudo-spherical cylinder. The methodology is specified for Schwarzschild solitons and for Generalized-Schwarzschild solitons. Accordingly, a new classification is written for the Schwarzschild solitons and for the Generalized-Schwarzschild solitons. The rotational field is spelled out. The potential for a tangent vector field is used. The conditions are recalled to discriminate which submanifold of a Ricci manifold is a soliton or is an almost-Ricci soliton. It is my aim to prove that a concurrent vector field is uniquely determined after the 4-velocity vector of a Schwarzschild soliton. As a result, the analytically specified manifold, which is a spacelike submanifold of the Schwarzschild spacetime that admits Ricci solitons. The rotational killing fields are tangent to the event horizon. The conditions that are needed to match the new aspects are spelled out analytically. As a result, the two manifolds described in the work of Bardeen et al. about the requested mass of a stationary, axisymmetric solution of the Einstein Field Equations of the spacetime, which contains a blackhole surrounded with matter from the new results obtained after correcting the work of Hawking 1972 about would-be point ’beyond the conjugate point’ on the analytic continuation of the would-be geodesics: they are proven here to become the tangent manifold (which is expressed from the tangent bundle in General-Relativistic notation). The prescription here is based on one of the books of Landau et al., that the matter is not put into the metric tensor, not even in the ultra-Relativistic limit. This way, the pseudo-spherical cylinder is one implemented from the Minkowskian description and whose asymptotical limit is proven. The new methodology allows one to describe the outer region of the blackhole as one according to which the (union of the trapped) regions is one with null support. For the purpose of the present investigation, the definition of concurrent vector fields in General-Relativity is newly developed. As a further new result, the paradigm is implemented for the shrinking case, which admits as subcase the Schwarzschild manifolds and the Generalized-Schwarzschild manifolds. The Penrose 1965 Theorem is discussed for the framework outlined here; in particular, the presence of trapped hypersurfaces is discarded. The no-hair theorem can now be discussed. Full article

(This article belongs to the Special Issue Mathematical Physics in General Relativity Theory)

15 pages, 324 KB

Open AccessArticle

Natural Representations of Black Box Groups SL₂

(F q)

by Alexandre Borovik and Şükrü Yalçınkaya

Axioms 2025, 14(12), 895; https://doi.org/10.3390/axioms14120895 - 1 Dec 2025

Viewed by 85

Abstract

In this paper, we make one step further in the recognition of black box groups of Lie type: given a black box group encrypting a special linear group of dimension 2 over a finite field of an unknown odd characteristic, we construct a [...] Read more.

In this paper, we make one step further in the recognition of black box groups of Lie type: given a black box group encrypting a special linear group of dimension 2 over a finite field of an unknown odd characteristic, we construct a black box field and a polynomial time isomorphism from the special linear group of dimension 2 over this new field to the black box, which can be made polynomial time-reversible for small characteristics at the expense of constructing a look-up table for the prime field. Our result opens a way to constructing structural proxies for black box groups of Lie type. Full article

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

13 pages, 285 KB

Open AccessArticle

Generalized Local Morrey Spaces Associated with Ball Banach Function Spaces and Their Application

by Feiyang Zhang and Jiang Zhou

Axioms 2025, 14(12), 894; https://doi.org/10.3390/axioms14120894 - 1 Dec 2025

Viewed by 96

Abstract

This paper is devoted to the analysis of boundedness for fractional integral operators, Calderón–Zygmund singular integral operators, and their corresponding commutators on generalized local Morrey spaces associated with ball Banach function spaces. These foundational results are then applied to establish the local regularity [...] Read more.

This paper is devoted to the analysis of boundedness for fractional integral operators, Calderón–Zygmund singular integral operators, and their corresponding commutators on generalized local Morrey spaces associated with ball Banach function spaces. These foundational results are then applied to establish the local regularity within the

L M_{X}^{φ}

Morrey spaces for the solution gradients of second-order elliptic equations expressed in divergence form. Full article

(This article belongs to the Special Issue Applications in Harmonic Analysis)

11 pages, 200 KB

Open AccessArticle

Approximate Common Solutions for a Family of Inverse Strongly Monotone Mappings

by Alexander J. Zaslavski

Axioms 2025, 14(12), 893; https://doi.org/10.3390/axioms14120893 - 1 Dec 2025

Viewed by 78

Abstract

In 2003 W. Takahashi and M. Toyoda showed the weak convergence of an iteration process of finding the solution of a variational inequality problem for an inverse strongly monotone mapping. In the present paper, we show that for the same process, most of [...] Read more.

In 2003 W. Takahashi and M. Toyoda showed the weak convergence of an iteration process of finding the solution of a variational inequality problem for an inverse strongly monotone mapping. In the present paper, we show that for the same process, most of its iterates are approximate common solutions for a finite family of variational inequalities induced by inverse strongly monotone mappings. Full article

3 pages, 122 KB

Open AccessEditorial

Mathematical Models and Simulations, 2nd Edition

by Giovanni Nastasi

Axioms 2025, 14(12), 892; https://doi.org/10.3390/axioms14120892 - 1 Dec 2025

Viewed by 106

Abstract

In this Editorial, we are pleased to introduce a Special Issue of the scientific journal Axioms, entitled “Mathematical Models and Simulations, 2nd Edition” [...] Full article

(This article belongs to the Special Issue Mathematical Models and Simulations, 2nd Edition)

11 pages, 278 KB

Open AccessArticle

Existence and Uniqueness of Solutions to Singular Impulsive Delay Boundary Value Problems via Paired-Chatterjea-Type Contractions

by Nicola Fabiano, Zouaoui Bekri, Amir Baklouti and Saber Mansour

Axioms 2025, 14(12), 891; https://doi.org/10.3390/axioms14120891 - 30 Nov 2025

Viewed by 113

Abstract

We establish the existence and uniqueness of solutions to a class of second-order nonlinear boundary value problems involving impulses, delay, and possible singularities. The approach leverages the recent notion of paired-Chatterjea-type contractions. Under a smallness condition ensuring the associated integral operator is a [...] Read more.

We establish the existence and uniqueness of solutions to a class of second-order nonlinear boundary value problems involving impulses, delay, and possible singularities. The approach leverages the recent notion of paired-Chatterjea-type contractions. Under a smallness condition ensuring the associated integral operator is a Banach contraction with constant

μ < \frac{1}{3}

, we show that it is also a Chatterjea, and hence, a paired-Chatterjea contraction. By the fixed point theorem of Chand, this guarantees at most two fixed points; a supplementary uniqueness argument then ensures a unique solution in the Banach space

P C^{1} ([a, b])

. Full article

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

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38 pages, 488 KB

Open AccessArticle

Existence and Uniqueness of Solutions for Singular Fractional Integro-Differential Equations with p-Laplacian and Two Kinds of Fractional Derivatives

by Fang Wang, Lishan Liu, Haibo Gu, Lina Ma and Yonghong Wu

Axioms 2025, 14(12), 890; https://doi.org/10.3390/axioms14120890 - 30 Nov 2025

Viewed by 109

Abstract

The paper is devoted to the study of a class of singular high-order fractional integro-differential equations with p-Laplacian operator, involving both the Riemann–Liouville fractional derivative and the Caputo fractional derivative. First, we investigate the problem by the method of reducing the order [...] Read more.

The paper is devoted to the study of a class of singular high-order fractional integro-differential equations with p-Laplacian operator, involving both the Riemann–Liouville fractional derivative and the Caputo fractional derivative. First, we investigate the problem by the method of reducing the order of fractional derivative. Then, by using the Schauder fixed point theorem, the existence of solutions is proved. The upper and lower bounds for the unique solution of the problem are established under various conditions by employing the Banach contraction mapping principle. Furthermore, four numerical examples are presented to illustrate the applications of our main results. Full article

(This article belongs to the Topic Fractional Calculus: Theory and Applications, 2nd Edition)

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17 pages, 344 KB

Open AccessArticle

Common Fixed Point Approximation for Asymptotically Nonexpansive Mapping in Hyperbolic Space with Application

by Tehreem Ishtiaq, Afshan Batool, Aftab Hussain and Hamed Alsulami

Axioms 2025, 14(12), 889; https://doi.org/10.3390/axioms14120889 - 30 Nov 2025

Viewed by 222

Abstract

This study presents a common fixed-point iteration process that includes two asymptotically nonexpansive self-mappings in a hyperbolic space and their delta convergence. To support our results, we provide an example with a comparison table and sufficient conditions for a modified iteration scheme to [...] Read more.

This study presents a common fixed-point iteration process that includes two asymptotically nonexpansive self-mappings in a hyperbolic space and their delta convergence. To support our results, we provide an example with a comparison table and sufficient conditions for a modified iteration scheme to have strong convergence to approximate the fixed point. Full article

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

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26 pages, 1043 KB

Open AccessArticle

Global Existence and Large-Time Behavior for 3D Full Compressible Magneto-Micropolar System Without Heat Conductivity

by Yuxiao Pan, Heyu Wang and Mingyu Zhang

Axioms 2025, 14(12), 888; https://doi.org/10.3390/axioms14120888 - 30 Nov 2025

Viewed by 83

Abstract

The system of full compressible magneto-micropolar flows is discussed in 3D bounded domains with slip boundary conditions. Based on the energy method, after establishing some key a priori exponential decay-in-times rates of the strong solutions, we obtain both the global existence and exponential [...] Read more.

The system of full compressible magneto-micropolar flows is discussed in 3D bounded domains with slip boundary conditions. Based on the energy method, after establishing some key a priori exponential decay-in-times rates of the strong solutions, we obtain both the global existence and exponential stability of strong solutions. In particular, it should be pointed out that the estimates of

{∥ (curl u, curl w) ∥}_{L^{2}}

and

{∥ (div u, div w) ∥}_{L^{2}}

are established separately, which implies that the growth rate of

(div u, div w)

in

L^{2}

are faster than that of

(curl u, curl w)

under the condition that the diameter of the domain is suitably large. Compared with previous works, we no longer consider the pressure P as

ρ θ

, but as variable in

(x, t)

, and directly deal with

{∥ \nabla P ∥}_{L^{2}}

. Based on slip boundary conditions, we established the

L^{p}

-norm for the gradient of effective viscous flux, and the term

{∥ \nabla P ∥}_{L^{2}}

can be controlled by

∥ (\nabla u_{t}, \nabla w_{t}, \nabla b_{t}) ∥_{L^{2}}

. Through precise calculations, we found that

∥ (\nabla u_{t}, \nabla w_{t}, \nabla b_{t}) ∥_{L^{2}}

is dependent on

{∥ \nabla P ∥}_{L^{2}}

. Therefore, the smallness condition we propose does not depend on the

L^{r}

-norm of the density gradient, which means that density can contain large oscillations. Full article

(This article belongs to the Section Mathematical Physics)

20 pages, 1296 KB

Open AccessArticle

GrImp: Granular Imputation of Missing Data for Interpretable Fuzzy Models

by Krzysztof Siminski and Konrad Wnuk

Axioms 2025, 14(12), 887; https://doi.org/10.3390/axioms14120887 - 30 Nov 2025

Viewed by 117

Abstract

Data incompleteness is a common problem in real-life datasets. This is caused by acquisition problems, sensor failures, human errors, and so on. Missing values and their subsequent imputation can significantly affect the performance of data-driven models and can also distort the interpretability of [...] Read more.

Data incompleteness is a common problem in real-life datasets. This is caused by acquisition problems, sensor failures, human errors, and so on. Missing values and their subsequent imputation can significantly affect the performance of data-driven models and can also distort the interpretability of explainable artificial intelligence (XAI) models, such as fuzzy models. This paper presents a novel imputation algorithm based on granular computing. This method benefits from the local structure of the dataset, explored using the granular approach. The method elaborates a set of granules that are then used to impute missing values in the dataset. The method is evaluated on several datasets and compared with several state-of-the-art imputation methods, both directly and indirectly. The direct evaluation compares the imputed values with the original data. The indirect evaluation compares the performance of fuzzy models built with TSK and ANNBFIS neuro-fuzzy systems. This enables not only the evaluation of the quality of numerically imputed values but also their impact on the interpretability of the constructed fuzzy models. This paper is accompanied by numerical experiments. The implementation of the method is available in a public GitHub repository. Full article

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

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29 pages, 383 KB

Open AccessArticle

New Approach for Closure Spaces on Graphs Based on Relations and Graph Ideals

by Rehab Alharbi, Salah El Deen Abbas, Hossam Mahmoud Omar Khiamy and Ismail Ibedou

Axioms 2025, 14(12), 886; https://doi.org/10.3390/axioms14120886 - 29 Nov 2025

Viewed by 132

Abstract

The main aim of this paper is to combine the connections between graph theory and rough set theory. We created graph ideal ASs by proposing the interior and closure operators, utilizing the concept of graph ideals. With the help of various graphical examples, [...] Read more.

The main aim of this paper is to combine the connections between graph theory and rough set theory. We created graph ideal ASs by proposing the interior and closure operators, utilizing the concept of graph ideals. With the help of various graphical examples, we applied some topological concepts to the induced graph ideal ASs, including the subspace, continuous functions, lower separation axioms, and connectedness. However, simple directed graphs with or without loops are the ones that are discussed throughout the study. The obtained results are valid for any type of graph: multi-graphs or simple graphs, connected or disconnected graphs, with loops or without loops, and undirected or directed graphs. Full article

(This article belongs to the Section Geometry and Topology)

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19 pages, 1125 KB

Open AccessArticle

Synchronization of Networked Reaction-Diffusion Nonlinear Systems via Hybrid Control

by Dongfang Mao, Guoping Jiang, Qian Ye and Meilin Chen

Axioms 2025, 14(12), 885; https://doi.org/10.3390/axioms14120885 - 29 Nov 2025

Viewed by 106

Abstract

This paper addresses the asymptotic synchronization problem for networked nonlinear reaction–diffusion systems under a novel hybrid control strategy. The hybrid controller consists of two components: impulsive control and continuous feedback control. By combining the comparison principle of impulsive systems with the introduction of [...] Read more.

This paper addresses the asymptotic synchronization problem for networked nonlinear reaction–diffusion systems under a novel hybrid control strategy. The hybrid controller consists of two components: impulsive control and continuous feedback control. By combining the comparison principle of impulsive systems with the introduction of a time-varying function and a power exponent to flexibly adjust the system, sufficient conditions for synchronization of networked nonlinear reaction–diffusion systems are derived, ensuring that the error dynamics between the network nodes converge to zero. Numerical simulations of a representative example are presented to demonstrate the practical validity and effectiveness of the proposed theoretical control scheme, confirming that the hybrid controller successfully achieves synchronization. Full article

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20 pages, 685 KB

Open AccessArticle

Parameter Estimation for Stochastic Korteweg–de Vries Equations

by Zhenyu Lang, Xiuling Yin, Yanqin Liu and Yaru Wang

Axioms 2025, 14(12), 884; https://doi.org/10.3390/axioms14120884 - 29 Nov 2025

Viewed by 103

Abstract

In this paper, we propose two methods for parameter estimation in stochastic Korteweg–de Vries (KdV) equations with unknown parameters. Both methods are based on the numerical discretization of the stochastic KdV equation. Moreover, we further propose an extrapolation-based approach to improve the accuracy [...] Read more.

In this paper, we propose two methods for parameter estimation in stochastic Korteweg–de Vries (KdV) equations with unknown parameters. Both methods are based on the numerical discretization of the stochastic KdV equation. Moreover, we further propose an extrapolation-based approach to improve the accuracy of parameter estimation. In addition, for the deterministic case, the convergence and conservation of the fully discrete schemes are analyzed. Both our theoretical analysis and numerical tests indicate the efficiency of the proposed methods for the KdV equations considered. Full article

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

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20 pages, 609 KB

Open AccessArticle

Geometric Fusion Mechanism for Reliable Central Measure Construction Amid Partial and Distorted Information

by Mohammed Ahmed Alomair and Muhammad Raza

Axioms 2025, 14(12), 883; https://doi.org/10.3390/axioms14120883 - 29 Nov 2025

Viewed by 91

Abstract

Biased estimates and fluctuating measures of central tendency are significant impediments to statistical inference and computational data analysis and are often caused by partial and distorted observations. The imputed least-squares-based estimators are very sensitive to non-normality, outliers, and missing data; thus, they cannot [...] Read more.

Biased estimates and fluctuating measures of central tendency are significant impediments to statistical inference and computational data analysis and are often caused by partial and distorted observations. The imputed least-squares-based estimators are very sensitive to non-normality, outliers, and missing data; thus, they cannot guarantee reliability in the presence of anomalous data. In an attempt to overcome these inadequacies, this paper utilizes a geometric fusion scheme, the Minimum Regularized Covariance Determinant (MRCD), to construct high-quality central measures. The suggested mechanism incorporates the concept of geometric dispersion and resistance-based principles of covariance to form stable dispersion structures, irrespective of data contamination and incompleteness. In this computational scheme, three estimators are developed, all of which use adaptive logarithmic transformations to boost efficiency and robustness. The theoretical argument can be characterized by analytical derivations of bias and Mean Squared Error (MSE) in large-sample situations, and empirical gains were verified by large-scale Monte Carlo experiments using synthetic populations and real-world datasets. The proposed MRCDdriven estimators are known to have a lower MSE as well as higher percentage relative efficiency (PRE) as compared to classical estimators. Overall, the findings indicate that the geometric fusion mechanism (MRCD) is a powerful, self-scaling, and statistically sound way of computing central measures in a situation in which information is incomplete and distorted. Full article

(This article belongs to the Section Mathematical Analysis)

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11 pages, 1298 KB

Open AccessArticle

A Modified Auxiliary Method for Efficient Solutions to the (2+1)-Dimensional Variable-Coefficient Burgers’ Equation

by Yiman Han and Yanni Zhang

Axioms 2025, 14(12), 882; https://doi.org/10.3390/axioms14120882 - 29 Nov 2025

Viewed by 110

Abstract

This paper explores an innovative expansion method for solving variable-coefficient partial differential equations. Combining specific auxiliary equations with the aid of mathematical software, our method achieves notable perspectives for understanding and solving related physical problems. The validity of this method was verified through [...] Read more.

This paper explores an innovative expansion method for solving variable-coefficient partial differential equations. Combining specific auxiliary equations with the aid of mathematical software, our method achieves notable perspectives for understanding and solving related physical problems. The validity of this method was verified through the (2+1)-dimensional variable-coefficient Burgers’ equation, and the results were visualized using three-dimensional surface plots. This study proposes an effective method for solving partial differential equations that holds broad application prospects in the field of fluid physics. Full article

(This article belongs to the Topic Functional Equations: Methods and Applications)

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21 pages, 1292 KB

Open AccessArticle

Modified Tseng’s Extragradient Method for Solving Variational Inequality Problems and Fixed Point Problems with Applications in Optimal Control Problems

by Yaling Bai, Guolin Yu, Linqi Sun and Shengquan Weng

Axioms 2025, 14(12), 881; https://doi.org/10.3390/axioms14120881 - 28 Nov 2025

Viewed by 147

Abstract

This paper presents an enhanced inertial Tseng’s extragradient method designed to address variational inequality problems involving pseudomonotone operators, along with fixed point problems governed by quasi-nonexpansive operators in real Hilbert spaces. Provided that the parameters satisfy appropriate conditions, the proposed method is shown [...] Read more.

This paper presents an enhanced inertial Tseng’s extragradient method designed to address variational inequality problems involving pseudomonotone operators, along with fixed point problems governed by quasi-nonexpansive operators in real Hilbert spaces. Provided that the parameters satisfy appropriate conditions, the proposed method is shown to converge strongly. Finally, we provide computational results and illustrate their utility through optimal control applications. These aim to show the efficacy and superiority of the proposed algorithm compared with some existing algorithms. Full article

(This article belongs to the Special Issue Mathematics and Its Applications in Other Disciplines)

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32 pages, 2764 KB

Open AccessArticle

Convergence-Enhanced and ANN-Accelerated Solvers for Absolute Value Problems

by Mudassir Shams and Bruno Carpentieri

Axioms 2025, 14(12), 880; https://doi.org/10.3390/axioms14120880 - 28 Nov 2025

Viewed by 116

Abstract

Absolute value problems of the form

A x - | x | = b

, where

x \in R^{n}

is the unknown vector,

b \in R^{n}

is a given vector, and

A \in R^{n \times n}

is a matrix, arise [...] Read more.

Absolute value problems of the form

A x - | x | = b

, where

x \in R^{n}

is the unknown vector,

b \in R^{n}

is a given vector, and

A \in R^{n \times n}

is a matrix, arise in a wide range of scientific and engineering applications. Their solution is challenging due to the non-differentiability of the absolute value operator and the possible existence of multiple solutions. Classical iterative techniques often suffer from slow convergence, strong sensitivity to the choice of initial vectors, and limited global convergence guarantees. In this study, we introduce a novel two-step iterative scheme that incorporates an adaptive initialization strategy enhanced by artificial neural networks (ANNs). The proposed method attains global linear convergence and local third-order convergence, thereby combining robustness with high accuracy. Numerical experiments on a range of benchmark problems—including cases with both unique and multiple solutions—demonstrate that the ANN-assisted initialization substantially accelerates convergence. In particular, it reduces the number of iterations, computational time, and residual errors across multiple norms, including both the Euclidean and infinity norms. These findings demonstrate that coupling a high-order two-step solver with ANN-based adaptive initialization yields a reliable and efficient framework for solving absolute value problems in both theoretical analysis and practical large-scale applications. Full article

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25 pages, 540 KB

Open AccessArticle

A Higher-Order Ising Model with Gradient-Free Update

by Gengsheng L. Zeng

Axioms 2025, 14(12), 879; https://doi.org/10.3390/axioms14120879 - 28 Nov 2025

Viewed by 143

Abstract

The Ising model is able to memorize some patterns or solutions as stable states. An Ising network may automatically converge to a pre-stored solution for a random input. However, in many cases, the Ising model cannot perform this task. The gap is that [...] Read more.

The Ising model is able to memorize some patterns or solutions as stable states. An Ising network may automatically converge to a pre-stored solution for a random input. However, in many cases, the Ising model cannot perform this task. The gap is that for a set of desired patterns, one may not be able to construct an Ising model such that the desired patterns are the stable solutions of the Ising model. The Ising model has limited power, because its energy function is limited to a second-order polynomial. Our research outline is as follows. This paper extends the conventional Ising model so that it has wider applications, where the Hebbian rule no longer works. The extended model does not have a limit on the order of the energy function. The extended Ising is defined by combining all desired patterns in a product. Our findings are that the extended Ising model has explicit closed-from update formulas, which do not require the evaluation of gradients. Thus, no network training is necessary. The update algorithm takes finite steps to reach a local minimum. Full article

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18 pages, 302 KB

Open AccessArticle

On Non-Archimedean Fuzzy Metric Free Topological Groups

by Cristina Bors and Manuel Sanchis

Axioms 2025, 14(12), 878; https://doi.org/10.3390/axioms14120878 - 28 Nov 2025

Viewed by 111

Abstract

We construct the free group over a non-Archimedean fuzzy metric space

(X, M, \land)

in the sense of George and Veeramani where ∧ is the minimum t-norm. The two main tools used are the concept of a scheme [...] Read more.

We construct the free group over a non-Archimedean fuzzy metric space

(X, M, \land)

in the sense of George and Veeramani where ∧ is the minimum t-norm. The two main tools used are the concept of a scheme (for every non-empty subset S of

N

of even cardinal, a permutation

φ

on S is a scheme for S if it is idempotent, with no fixed points and, additionally,

i < j < φ (i) < φ (j)

does not hold for every

i, j \in S

), and the notion of a fuzzy prenorm on a fuzzy topological group. As a consequence of our results, we prove that every non-Archimedean fuzzy metric space

(X, M, \land)

in the sense of George and Veeramani is isometric to a closed subspace of a non-Archimedean fuzzy metric free (Abelian) group and also that every metric space

(X, d)

is uniformly isomorphic to a closed subspace of a non-Archimedean fuzzy metric free (Abelian) group. Our results also apply to non-Archimedean fuzzy metric spaces in the sense of Kramosil and Michálek. Full article

18 pages, 1074 KB

Open AccessArticle

Third-Order Functional Differential Equations with Damping Term: Oscillatory Behavior of Solutions

by Asma Al-Jaser, Eman Alluqmani, Belgees Qaraad and Higinio Ramos

Axioms 2025, 14(12), 877; https://doi.org/10.3390/axioms14120877 - 28 Nov 2025

Viewed by 138

Abstract

In this paper, we investigate the asymptotic and oscillatory behavior of a specific class of third-order functional differential equations with damping terms and deviating arguments. By employing the comparison principle, Riccati transformation, and the integral averaging technique, we derive new criteria that guarantee [...] Read more.

In this paper, we investigate the asymptotic and oscillatory behavior of a specific class of third-order functional differential equations with damping terms and deviating arguments. By employing the comparison principle, Riccati transformation, and the integral averaging technique, we derive new criteria that guarantee all solutions to the studied equation oscillate when

\int_{⊤_{0}}^{\infty} 1 / γ^{1 / α} (⊤) d ⊤ = \infty

and

ϱ (⊤) \leq ϱ_{0} < \infty

. This study introduces novel conditions and effective analytical tools, which enhance our understanding of such equations and broaden their range of applications. Illustrative examples are provided to demonstrate the applicability of the results. Full article

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

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17 pages, 289 KB

Open AccessArticle

Additive Derivations of Incidence Modules

by Naseer Ullah, Hailou Yao, Dalal Alhwikem and Imran Shabir Chuhan

Axioms 2025, 14(12), 876; https://doi.org/10.3390/axioms14120876 - 28 Nov 2025

Viewed by 139

Abstract

Let R be an associative ring and M a left R-module. This paper examines the structural properties of the incidence module

I (P, M)

, associated with a module M over a ring R and a locally finite poset [...] Read more.

Let R be an associative ring and M a left R-module. This paper examines the structural properties of the incidence module

I (P, M)

, associated with a module M over a ring R and a locally finite poset

P

. We provide a complete characterization of when an additive derivation on

I (P, M)

is inner, for the case where

P

is a finite and connected poset. These criteria are then generalized to arbitrary posets, revealing a profound connection between the algebraic properties of the module and the graph-theoretic structure of

P

as a directed graph. Full article

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