Axioms

15 pages, 898 KB

Open AccessArticle

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

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

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

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

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

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

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23 pages, 460 KB

Open AccessArticle

Multiary Gradings

by Steven Duplij

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

Abstract

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

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

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

18 pages, 330 KB

Open AccessArticle

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

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

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

Abstract

Provided that the initial data

(ρ_{0}, v_{0}, η_{0})

is of small energy around steady state

(ρ^{▵}, 0, 0)

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

Provided that the initial data

(ρ_{0}, v_{0}, η_{0})

is of small energy around steady state

(ρ^{▵}, 0, 0)

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

(This article belongs to the Section Mathematical Analysis)

19 pages, 18350 KB

Open AccessArticle

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

by Xiaofeng Xu and Jiajia Pan

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

Abstract

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

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

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40 pages, 1692 KB

Open AccessArticle

Fine Stability Properties of the Hankel and Wiener–Khinchin Transforms

by François Vigneron

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

Abstract

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

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

(| x |^{- α}, c_{d} | ξ |^{d - α})

on

R^{d}

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

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

27 pages, 648 KB

Open AccessArticle

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

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

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

Abstract

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

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

(This article belongs to the Section Mathematical Analysis)

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108 pages, 1969 KB

Open AccessArticle

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

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

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

Abstract

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

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

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

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10 pages, 239 KB

Open AccessArticle

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

by Chuan-Min Lee

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

Abstract

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

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

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

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36 pages, 636 KB

Open AccessArticle

Explicit Discrete Solution for Some Optimization Problems and Estimations with Respect to the Exact Solution

by Julieta Bollati, Mariela C. Olguin and Domingo A. Tarzia

Axioms 2026, 15(3), 190; https://doi.org/10.3390/axioms15030190 - 5 Mar 2026

Abstract

We consider two steady-state heat conduction systems called, S and

S_{α}

, in a multidimensional bounded domain D for the Poisson equation with source energy g. In one system, we impose mixed boundary conditions (temperature b on the boundary

Γ_{1}

[...] Read more.

We consider two steady-state heat conduction systems called, S and

S_{α}

, in a multidimensional bounded domain D for the Poisson equation with source energy g. In one system, we impose mixed boundary conditions (temperature b on the boundary

Γ_{1}

, heat flux q on

Γ_{2}

and an adiabatic condition on

Γ_{3}

). In the other system, the condition on

Γ_{1}

is replaced by a convective heat flux condition with coefficient

α

. For each of these systems, we consider three associated optimization problems

(P_{i})

and

(P_{i α})

,

i = 1, 2, 3

, where the variable is the source energy g, the heat flux q and the environmental temperature b, respectively. In the particular case where D is a rectangle, the explicit continuous optimization variables and the corresponding state of the systems are known. In the present work, by using a finite difference scheme, we obtain the discrete systems

(S^{h})

and

(S_{α}^{h})

and discrete optimization problems

(P_{i}^{h})

and

(P_{i α}^{h})

,

i = 1, 2, 3

, where h is the space step in the discretization. Explicit discrete solutions are found, and convergence and estimation errors results are proved when h goes to zero and when

α

goes to infinity. Moreover, some numerical simulations are provided in order to test theoretical results. Finally, we note that the use of a three-point finite-difference approximation for the Neumann or Robin boundary condition at the boundary improves the global order of convergence from

O (h)

to

O (h^{2})

. Full article

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16 pages, 318 KB

Open AccessArticle

Odd Right-End Numerical Semigroups

by María Ángeles Moreno-Frías and José Carlos Rosales

Axioms 2026, 15(3), 189; https://doi.org/10.3390/axioms15030189 - 5 Mar 2026

Abstract

An odd right-end semigroup (hereinafter Ore semigroup) is a numerical semigroup S verifying that

x + 1 \in S

for every

x \in S ∖ {0}

such that x is even. The introduction and study of these semigroups is the purpose [...] Read more.

An odd right-end semigroup (hereinafter Ore semigroup) is a numerical semigroup S verifying that

x + 1 \in S

for every

x \in S ∖ {0}

such that x is even. The introduction and study of these semigroups is the purpose of the present work. In particular, we will give some algorithms which compute all Ore semigroups with a given genus, a fixed Frobenius number and a specific multiplicity. We will see that if X is a set of positive integers, then there exists the smallest Ore semigroup, under the inclusion sets, that contains X. We will denote this semigroup by

θ [X]

and present an algorithm to calculate it. Finally, we will study the embedding dimension, the Frobenius number, and the genus of Ore semigroups of the form

θ [{m}]

, where m is a positive integer. As a consequence of this study, we will prove that this kind of semigroup satisfies Wilf’s conjecture. Full article

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

17 pages, 327 KB

Open AccessArticle

Fixed Point Approximation of Generalized α-Non-Expansive Multi-Valued Mapping in Convex Metric Space

by Tanveer Hussain, Vasile Berinde and Abdul Rahim Khan

Axioms 2026, 15(3), 188; https://doi.org/10.3390/axioms15030188 - 4 Mar 2026

Abstract

In this paper, we present approximation results for a generalized

α

-non-expansive multi-valued mapping using a four-step iteration scheme introduced in the context of a convex metric space. We extend some recent results about generalized

α

-non-expansive multi-valued mappings from the Banach space [...] Read more.

In this paper, we present approximation results for a generalized

α

-non-expansive multi-valued mapping using a four-step iteration scheme introduced in the context of a convex metric space. We extend some recent results about generalized

α

-non-expansive multi-valued mappings from the Banach space setting to a convex metric space. Two examples of generalized

α

-non-expansive multi-valued mappings are presented, and it is numerically shown that our iteration scheme enables faster convergence than other well-known schemes in the literature. To demonstrate the application of one of our results, we provide the solution of a non-linear integral equation. Full article

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

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30 pages, 1924 KB

Open AccessArticle

A Liouville–Caputo Fractional Co-Infection Model: Theoretical Analysis, Ulam-Type Stability, and Numerical Simulation

by Ghaliah Alhamzi, Mona Bin-Asfour, Najat Almutairi, Mansoor Alsulami and Sayed Saber

Axioms 2026, 15(3), 187; https://doi.org/10.3390/axioms15030187 - 4 Mar 2026

Abstract

This paper investigates a fractional-order mathematical model for the co-infection dynamics of pneumonia and typhoid fever using the Liouville–Caputo derivative. We establish the existence, uniqueness, non-negativity, and boundedness of solutions using Banach’s fixed point theorem and fractional comparison principles. The Hyers–Ulam and generalized [...] Read more.

This paper investigates a fractional-order mathematical model for the co-infection dynamics of pneumonia and typhoid fever using the Liouville–Caputo derivative. We establish the existence, uniqueness, non-negativity, and boundedness of solutions using Banach’s fixed point theorem and fractional comparison principles. The Hyers–Ulam and generalized Ulam–Hyers–Rassias stability of the system are rigorously proved; this stability analysis is epidemiologically significant because it guarantees that small perturbations in initial conditions or model parameters—inevitable in real-world data collection—do not lead to unbounded deviations in disease trajectory predictions. To approximate solutions numerically, we develop a Laplace-Based Optimized Decomposition Method (LODM) and validate its convergence against a modified predictor–corrector scheme. The LODM provides a semi-analytical series solution, while the predictor–corrector method serves as a numerical benchmark; this dual approach ensures reliability of simulations. Numerical simulations illustrate the influence of the fractional order

ξ

on system dynamics. Quantitative comparison between

ξ = 1

(integer order) and

ξ < 1

(fractional order) demonstrates that fractional modeling reduces peak infection by 12–18% and delays epidemic peaks by 15–30 days, confirming that memory effects capture long-term epidemiological dependencies that integer-order models fail to reproduce. A biological interpretation links the fractional order to immune memory, pathogen persistence, and intervention latency. This study provides both theoretical and numerical evidence supporting the use of fractional calculus in epidemiological modeling. Full article

(This article belongs to the Special Issue Fractional Calculus—Theory and Applications, 4th Edition)

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14 pages, 518 KB

Open AccessArticle

Consistency and Quantitative Backward Stability Analysis of the Two-Step Jarratt Method for Nonlinear Systems

by Vahideh Rasouli, Alicia Cordero, Taher Lotfi and Juan R. Torregrosa

Axioms 2026, 15(3), 186; https://doi.org/10.3390/axioms15030186 - 4 Mar 2026

Abstract

In this work, we revisit the two-step Jarratt method from the perspective of numerical stability. While high-order iterative schemes are often examined in terms of convergence rate and computational efficiency, their backward stability properties have received comparatively less attention. We begin by establishing [...] Read more.

In this work, we revisit the two-step Jarratt method from the perspective of numerical stability. While high-order iterative schemes are often examined in terms of convergence rate and computational efficiency, their backward stability properties have received comparatively less attention. We begin by establishing the method’s strong consistency. Next, we provide a quantitative backward stability assessment within the standard floating-point arithmetic framework, deriving explicit perturbation bounds that show that the iteration errors remain proportional to machine precision. To support the theoretical findings, we present numerical experiments—including tests under finite-precision perturbations—as well as Python implementations and visualizations of the numerical examples. The results illustrate that the two-step Jarratt method not only achieves a high convergence order but also remains numerically robust for well-conditioned nonlinear systems. Full article

(This article belongs to the Section Mathematical Analysis)

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17 pages, 319 KB

Open AccessArticle

Approximate Synchronization of Memristive Hopfield Neural Networks

by Yuncheng You

Axioms 2026, 15(3), 185; https://doi.org/10.3390/axioms15030185 - 4 Mar 2026

Abstract

Asymptotic synchronization is one of the essential differences between artificial neural networks and biologically inspired neural networks due to mismatches from the dynamical update of weight parameters and heterogeneous activations. In this paper, a new concept of approximate synchronization is proposed and investigated [...] Read more.

Asymptotic synchronization is one of the essential differences between artificial neural networks and biologically inspired neural networks due to mismatches from the dynamical update of weight parameters and heterogeneous activations. In this paper, a new concept of approximate synchronization is proposed and investigated for Hopfield neural networks coupled with nonlinear memristors. It is proved that global solution dynamics are robustly dissipative and a sharp ultimate bound is acquired. Through a priori uniform estimates on the interneuron differencing equations, it is rigorously and analytically shown that approximate synchronization to any prescribed small gap at an exponential convergence rate of the memristive Hopfield neural networks occurs if an explicitly computable threshold condition is satisfied by the interneuron coupling strength parameter. The main result is also extended to Hopfield neural networks with Hebbian learning rules for a broad range of applications in unsupervised learning. The contribution of this approximate synchronization framework and the analytic methodology in this work advance the exploration of asymptotic dynamics for more AI mathematical models. Full article

36 pages, 432 KB

Open AccessFeature PaperReview

Classical Entanglement: Parametric Geometry and Non-Parametric Synthesis of Asymptotic Laws

by Simon Gluzman

Axioms 2026, 15(3), 184; https://doi.org/10.3390/axioms15030184 - 3 Mar 2026

Abstract

This review develops a unified geometric framework for synthesizing global asymptotic laws, termed classical entanglement. The central tool is the entanglement operator, a Minkowski–

L_{a}

metric blend that couples asymptotic regimes through an index

a > 1

, producing a nonlinear global [...] Read more.

This review develops a unified geometric framework for synthesizing global asymptotic laws, termed classical entanglement. The central tool is the entanglement operator, a Minkowski–

L_{a}

metric blend that couples asymptotic regimes through an index

a > 1

, producing a nonlinear global state whose intermediate region is metrically non-separable and cannot be written as a linear combination of its limits. The framework reveals a universal transition knee whose curvature scales linearly with a, independent of amplitudes or local scales. We show that this geometric mechanism encompasses Orlicz norms, weighted Hölder metrics, and iterated Hölder constructions, the latter being structurally isomorphic to self-similar root approximants. A conceptual “Rosetta Stone” links practitioner terminology, geometric meta-language, and functional-analytic structures, clarifying how classical entanglement unifies empirical blending, metric curvature, and Calderón-type interpolation. Applications to turbulence (Darcy friction factor), fractional dynamics, and scale-dependent diffusion illustrate how classical entanglement provides stable, asymptotically consistent global states across multi-scale systems. Full article

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19 pages, 325 KB

Open AccessArticle

Strongly Exposed Points of Orlicz Sequence Spaces Equipped with the p-Amemiya Norm

by Xiaoyan Li and Yunan Cui

Axioms 2026, 15(3), 183; https://doi.org/10.3390/axioms15030183 - 2 Mar 2026

Abstract

Using some new techniques, criteria for strongly exposed points of Orlicz sequence spaces generated by arbitrary Orlicz function and equipped with the p-Amemiya

(1 < p < \infty)

norm are obtained. Next, strongly exposed property of [...] Read more.

Using some new techniques, criteria for strongly exposed points of Orlicz sequence spaces generated by arbitrary Orlicz function and equipped with the p-Amemiya

(1 < p < \infty)

norm are obtained. Next, strongly exposed property of these spaces is deduced. The obtained results broaden the knowledge about these notions in Orlicz sequence spaces equipped with the p-Amemiya norm. Full article

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

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16 pages, 4951 KB

Open AccessArticle

Structural Robustness of Consensus Models with Layered Correlated Graphs

by Zhi Song, Jian Zhu, Da Huang, Xing Chen and Zhongming Hu

Axioms 2026, 15(3), 182; https://doi.org/10.3390/axioms15030182 - 2 Mar 2026

Abstract

This study analyzes network coherence in two-layer and three-layer networks with positive and negative inter-layer correlation patterns. Using algebraic graph theory, we construct the Laplacian matrices for different correlated graph structures and compute their Laplacian spectra. The impact of correlation patterns on network [...] Read more.

This study analyzes network coherence in two-layer and three-layer networks with positive and negative inter-layer correlation patterns. Using algebraic graph theory, we construct the Laplacian matrices for different correlated graph structures and compute their Laplacian spectra. The impact of correlation patterns on network coherence is investigated for two-layer and three-layer structures, and numerical evaluations are performed. The results show that negative-correlation patterns yield better network coherence than positive ones. This work provides fundamental insights into the structural robustness of layered networks and offers theoretical guidance for the design of robust networked systems. Full article

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18 pages, 1400 KB

Open AccessArticle

A Structure-Preserving Scheme for the Space-Fractional Klein-Gordon-Schrödinger System with the Invariant Energy Quadratization Method

by Wenye Jiang, Yu Li and Yan Fan

Axioms 2026, 15(3), 181; https://doi.org/10.3390/axioms15030181 - 1 Mar 2026

Abstract

This paper investigates the conservation of mass and energy in the space-fractional Klein-Gordon-Schrödinger system with fractional Laplacian operators. Firstly, the invariant energy quadratization method is applied to transform the original system into an equivalent form. For spatial discretization, Fourier spectral methods are employed, [...] Read more.

This paper investigates the conservation of mass and energy in the space-fractional Klein-Gordon-Schrödinger system with fractional Laplacian operators. Firstly, the invariant energy quadratization method is applied to transform the original system into an equivalent form. For spatial discretization, Fourier spectral methods are employed, yielding a semi-discrete scheme. Subsequently, an invariant energy quadratization Runge-Kutta approach is used for temporal discretization, resulting in a fully discrete scheme. Owing to its diagonally implicit structure, the proposed scheme is both highly accurate and efficient while preserving mass and energy exactly. Numerical experiments are conducted to verify the accuracy and conservation properties of the method. Full article

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

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16 pages, 468 KB

Open AccessArticle

From Anticommutative Algebras to Combinatorial Structures: A Graph-Theoretic Approach

by Jesús Baena, Manuel Ceballos and Desamparados Fernández-Ternero

Axioms 2026, 15(3), 180; https://doi.org/10.3390/axioms15030180 - 28 Feb 2026

Abstract

This article develops a combinatorial and graph-theoretic framework for the study of finite-dimensional anticommutative algebras. Given a fixed basis, we associate to each algebra a directed graph that may contain filled oriented triangles, encoding the pattern of nonzero products and the constraints imposed [...] Read more.

This article develops a combinatorial and graph-theoretic framework for the study of finite-dimensional anticommutative algebras. Given a fixed basis, we associate to each algebra a directed graph that may contain filled oriented triangles, encoding the pattern of nonzero products and the constraints imposed by anticommutativity. Focusing on three-dimensional algebras, we provide the complete list of combinatorial structures associated with each algebra in the known classification. Moreover, we develop two algorithmic procedures: the first determines whether a given anticommutative algebra satisfies the Tortkara identity, and the second identifies the additional conditions a given combinatorial structure must satisfy to correspond to a Tortkara algebra. Finally, we carry out a computational study of these algorithms. Our approach establishes a clear link between algebraic identities and discrete combinatorial objects, opening new avenues for the graphical analysis and classification of anticommutative algebras. Full article

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

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