Axioms

20 pages, 23754 KB

Open AccessFeature PaperEditor’s ChoiceArticle

2 \frac{1}{2}

Dimensions

by Kenneth Stephenson

Axioms 2026, 15(3), 210; https://doi.org/10.3390/axioms15030210 - 12 Mar 2026

Viewed by 2994

Abstract

This paper investigates cylindrical sphere packings, that is, patterns of uniform spheres with mutually disjoint interiors which are all tangent to a common cylinder. The key unifying themes are the existence and uniqueness of hexagonal packings, in which each sphere is tangent to [...] Read more.

This paper investigates cylindrical sphere packings, that is, patterns of uniform spheres with mutually disjoint interiors which are all tangent to a common cylinder. The key unifying themes are the existence and uniqueness of hexagonal packings, in which each sphere is tangent to six others. Constructions are both intuitive and subtle, but result in the complete characterization in terms of integer parameter pairs

(m, n)

. Interesting questions in rigidity and density are encountered. Density questions arise because the packings, being of equal diameter, lie within the space between inner and outer cylinders. This density problem hovers between the 2D and 3D sphere packing cases, and though it is not solved here, it is conjectured that the hexagonal packings are densest for the countable number of cylinders which support them. Other geometric objects are along for the ride, including equilateral triangles and the packings’ dual graphs, which are associated with patterns of carbon atoms forming buckytubes. Interesting structural rigidity questions also arise. Full article

(This article belongs to the Section Geometry and Topology)

► Show Figures

Figure 1

36 pages, 432 KB

Open AccessFeature PaperEditor’s ChoiceReview

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

Viewed by 854

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

► Show Figures

Figure 1

37 pages, 2440 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Consciousness as 4-Manifold Painlevé V Dynamics: From Quantum Topology to Classical Gamma Oscillations

by Michel Planat

Axioms 2026, 15(2), 124; https://doi.org/10.3390/axioms15020124 - 6 Feb 2026

Cited by 2 | Viewed by 1080

Abstract

We propose a novel mathematical framework for understanding consciousness as a dynamical phenomenon governed by nonlinear integrable equations. The central hypothesis identifies conscious state dynamics with the Painlevé VI equation and its confluence limits, providing a unified description of stability, bifurcation, and collapse [...] Read more.

We propose a novel mathematical framework for understanding consciousness as a dynamical phenomenon governed by nonlinear integrable equations. The central hypothesis identifies conscious state dynamics with the Painlevé VI equation and its confluence limits, providing a unified description of stability, bifurcation, and collapse across cognitive regimes. In this approach, consciousness is modeled as an emergent phase sustained near criticality, where coherent quantum-like structures and classical decoherence coexist in a regulated balance. The theory is formulated in terms of isomonodromic deformations on

SL (2, C)

character varieties, allowing conscious states to be characterized by monodromy data and their controlled evolution. This geometric setting naturally encodes memory, attention, and transitions between conscious and unconscious phases, while confluence processes account for irreversible loss of coherence. A two-stage quantum-to-classical transition is identified, separating microscopic coherence from macroscopic stabilization. The framework yields universal signatures such as critical slowing down, scaling laws near transition points, and robustness under perturbations, linking consciousness dynamics to broader classes of critical phenomena observed in physics and complex systems. By replacing heuristic assumptions with a mathematically constrained dynamical structure, this work extends existing quantum consciousness models and provides a tractable platform for comparison with neural, biological, and informational data. Full article

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

► Show Figures

Figure 1

25 pages, 372 KB

Open AccessEditor’s ChoiceArticle

Recognition Geometry

by Jonathan Washburn, Milan Zlatanović and Elshad Allahyarov

Axioms 2026, 15(2), 90; https://doi.org/10.3390/axioms15020090 - 26 Jan 2026

Cited by 2 | Viewed by 1675

Abstract

We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence [...] Read more.

We introduce Recognition Geometry (RG), an axiomatic framework in which geometric structure is not assumed a priori but derived. The starting point of the theory is a configuration space together with recognizers that map configurations to observable events. Observational indistinguishability induces an equivalence relation, and the observable space is obtained as a recognition quotient. Locality is introduced through a neighborhood system, without assuming any metric or topological structure. A finite local resolution axiom formalizes the fact that any observer can distinguish only finitely many outcomes within a local region. We prove that the induced observable map

\bar{R} : C_{R} \to E

is injective, establishing that observable states are uniquely determined by measurement outcomes with no hidden structure. The framework connects deeply with existing approaches: C^*-algebraic quantum theory, information geometry, categorical physics, causal set theory, noncommutative geometry, and topos-theoretic foundations all share the measurement-first philosophy, yet RG provides a unified axiomatic foundation synthesizing these perspectives. Comparative recognizers allow us to define order-type relations based on operational comparison. Under additional assumptions, quantitative notions of distinguishability can be introduced in the form of recognition distances, defined as pseudometrics. Several examples are provided, including threshold recognizers on

R^{n}

, discrete lattice models, quantum spin measurements, and an example motivated by Recognition Science. In the last part, we develop the composition of recognizers, proving that composite recognizers refine quotient structures and increase distinguishing power. We introduce symmetries and gauge equivalence, showing that gauge-equivalent configurations are necessarily observationally indistinguishable, though the converse does not hold in general. A significant part of the axiomatic framework and the main constructions are formalized in the Lean 4 proof assistant, providing an independent verification of logical consistency. Full article

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

► Show Figures

Figure 1

9 pages, 321 KB

Open AccessEditor’s ChoiceArticle

The Complete Strong Version of Blundon’s Inequality

by Dorin Andrica, Ovidiu Bagdasar, Cătălin Barbu and Laurian-Ioan Pişcoran

Axioms 2026, 15(1), 26; https://doi.org/10.3390/axioms15010026 - 29 Dec 2025

Viewed by 2339

Abstract

In this paper we present some key results related to Blundon’s inequality, its long history, geometric interpretations and implications, as well as highlight some connections to results in other fields of mathematics. We make a case that this is a fundamental inequality in [...] Read more.

In this paper we present some key results related to Blundon’s inequality, its long history, geometric interpretations and implications, as well as highlight some connections to results in other fields of mathematics. We make a case that this is a fundamental inequality in triangle geometry. Also, we provide a new proof for the inequalities (8) and we generalize the strong version of Blundon’s inequalities presented in Theorem 1. Full article

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

► Show Figures

Figure 1

19 pages, 316 KB

Open AccessEditor’s ChoiceArticle

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 1052

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)

► Show Figures

Figure 1

66 pages, 726 KB

Open AccessEditor’s ChoiceReview

New Perspectives on Kac–Moody Algebras Associated with Higher-Dimensional Manifolds

by Rutwig Campoamor-Stursberg, Alessio Marrani and Michel Rausch de Traubenberg

Axioms 2025, 14(11), 809; https://doi.org/10.3390/axioms14110809 - 31 Oct 2025

Viewed by 1000

Abstract

In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle

S^{1}

, we extend the approach to compact and non-compact group manifolds, [...] Read more.

In this review, we present a general framework for the construction of Kac–Moody (KM) algebras associated to higher-dimensional manifolds. Starting from the classical case of loop algebras on a circle

S^{1}

, we extend the approach to compact and non-compact group manifolds, coset spaces, and soft deformations thereof. After recalling the necessary geometric background on Riemannian manifolds, Hilbert bases, and Killing vectors, we present the construction of generalized current algebras

g (M)

, their semidirect extensions with isometry algebras, and their central extensions. We show how the resulting algebras are controlled by the structure of the underlying manifold, and we illustrate the framework through explicit realizations on

S U (2)

,

S U (2) / U (1)

, and higher-dimensional spheres, highlighting their relation to Virasoro-like algebras. We also discuss the compatibility conditions for cocycles, the role of harmonic analysis, and some applications in higher-dimensional field theory and supergravity compactifications. This provides a unifying perspective on KM algebras beyond one-dimensional settings, paving the way for further exploration of their mathematical and physical implications. Full article

(This article belongs to the Special Issue New Perspectives in Lie Algebras, 2nd Edition)

24 pages, 757 KB

Open AccessEditor’s ChoiceArticle

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

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

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

Cited by 1 | Viewed by 1340

Abstract

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

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

► Show Figures

Figure 1

21 pages, 1453 KB

Open AccessEditor’s ChoiceArticle

First and Second Moments of Spherical Distributions That Are Relevant for Biological Applications

by Alexandra Shyntar and Thomas Hillen

Axioms 2025, 14(10), 743; https://doi.org/10.3390/axioms14100743 - 30 Sep 2025

Viewed by 1457

Abstract

Spherical distributions, in particular, the von Mises–Fisher distribution, are often used to analyze directional data. The first and second moments of these distributions are of central interest, as they describe mean orientations as well as anisotropic diffusion tensors. Finding these moments often requires [...] Read more.

Spherical distributions, in particular, the von Mises–Fisher distribution, are often used to analyze directional data. The first and second moments of these distributions are of central interest, as they describe mean orientations as well as anisotropic diffusion tensors. Finding these moments often requires a numerical approximation of complex trigonometric integrals. Instead, we apply the divergence theorem on suitable domains to derive explicit forms of the first and second moments for n-dimensional von Mises–Fisher and peanut distributions. Based on these new formulas, we characterize some meaningful characteristics of these distributions: fractional anisotropy and the anisotropy ratio. We find, surprisingly, that the peanut distribution has an upper bound on anisotropy, while the von-Mises Fisher distribution has no such bound. As a side benefit, we find different forms of some identities for Bessel functions. Full article

(This article belongs to the Special Issue Advances in Mathematical Modeling and Related Topics)

► Show Figures

Figure 1

15 pages, 2432 KB

Open AccessEditor’s ChoiceArticle

A Comparison Index for Costs of Interval Linear Programming Models

by Maria Letizia Guerra, Laerte Sorini and Luciano Stefanini

Axioms 2025, 14(8), 569; https://doi.org/10.3390/axioms14080569 - 24 Jul 2025

Cited by 1 | Viewed by 1242

Abstract

Interval Linear Programming (ILP) presents several compelling challenges when applied to real-world problems that cannot be easily captured by traditional robust uncertainty models. In this paper, we propose a novel solution method that employs a comparison index for interval ordering based on the [...] Read more.

Interval Linear Programming (ILP) presents several compelling challenges when applied to real-world problems that cannot be easily captured by traditional robust uncertainty models. In this paper, we propose a novel solution method that employs a comparison index for interval ordering based on the generalized Hukuhara difference. This approach proves to be highly effective in comparing solutions within ILP frameworks. Additionally, we discuss the robustness of the proposed methodology and its implications for decision-making under uncertainty. Full article

► Show Figures

Figure 1

20 pages, 7720 KB

Open AccessEditor’s ChoiceArticle

Comparative Evaluation of Nonparametric Density Estimators for Gaussian Mixture Models with Clustering Support

by Tomas Ruzgas, Gintaras Stankevičius, Birutė Narijauskaitė and Jurgita Arnastauskaitė Zencevičienė

Axioms 2025, 14(8), 551; https://doi.org/10.3390/axioms14080551 - 23 Jul 2025

Viewed by 1606

Abstract

The article investigates the accuracy of nonparametric univariate density estimation methods applied to various Gaussian mixture models. A comprehensive comparative analysis is performed for four popular estimation approaches: adaptive kernel density estimation, projection pursuit, log-spline estimation, and wavelet-based estimation. The study is extended [...] Read more.

The article investigates the accuracy of nonparametric univariate density estimation methods applied to various Gaussian mixture models. A comprehensive comparative analysis is performed for four popular estimation approaches: adaptive kernel density estimation, projection pursuit, log-spline estimation, and wavelet-based estimation. The study is extended with modified versions of these methods, where the sample is first clustered using the EM algorithm based on Gaussian mixture components prior to density estimation. Estimation accuracy is quantitatively evaluated using MAE and MAPE criteria, with simulation experiments conducted over 100,000 replications for various sample sizes. The results show that estimation accuracy strongly depends on the density structure, sample size, and degree of component overlap. Clustering before density estimation significantly improves accuracy for multimodal and asymmetric densities. Although no formal statistical tests are conducted, the performance improvement is validated through non-overlapping confidence intervals obtained from 100,000 simulation replications. In addition, several decision-making systems are compared for automatically selecting the most appropriate estimation method based on the sample’s statistical features. Among the tested systems, kernel discriminant analysis yielded the lowest error rates, while neural networks and hybrid methods showed competitive but more variable performance depending on the evaluation criterion. The findings highlight the importance of using structurally adaptive estimators and automation of method selection in nonparametric statistics. The article concludes with recommendations for method selection based on sample characteristics and outlines future research directions, including extensions to multivariate settings and real-time decision-making systems. Full article

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

► Show Figures

Figure 1

22 pages, 753 KB

Open AccessEditor’s ChoiceArticle

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

by Kuo-Shou Chiu and Fernando Córdova-Lepe

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

Cited by 2 | Viewed by 1526

Abstract

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

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

(This article belongs to the Section Mathematical Analysis)

► Show Figures

Figure 1

14 pages, 259 KB

Open AccessFeature PaperEditor’s ChoiceArticle

The Vertex-Disjoint and Edge-Disjoint Ramsey Numbers of a Set of Graphs

by Emma Jent and Ping Zhang

Axioms 2025, 14(7), 486; https://doi.org/10.3390/axioms14070486 - 21 Jun 2025

Cited by 1 | Viewed by 1612

Abstract

The Ramsey number

R (F)

of a graph F without isolated vertices is the smallest positive integer n such that every red–blue coloring of

K_{n}

produces a subgraph isomorphic to F all of whose edges are colored the same. Let [...] Read more.

The Ramsey number

R (F)

of a graph F without isolated vertices is the smallest positive integer n such that every red–blue coloring of

K_{n}

produces a subgraph isomorphic to F all of whose edges are colored the same. Let

F

be a set of graphs without isolated vertices. For a positive integer t, the vertex-disjoint Ramsey number

V R_{t} (F)

is the smallest positive integer n such that every red–blue coloring of the complete graph

K_{n}

of order n results in at least t pairwise vertex-disjoint monochromatic graphs in

F

; while the edge-disjoint Ramsey number

E R_{t} (F)

is the smallest positive integer n such that every red–blue coloring of

K_{n}

produces at least t pairwise edge-disjoint monochromatic graphs in

F

. If

t = 1

and

F

consists of a single graph F, then

V R_{1} (F) = E R_{1} (F) = R (F)

is the Ramsey number of the graph F. Thus, the concepts of vertex-disjoint and edge-disjoint Ramsey numbers provide a generalization of the standard Ramsey number. Upper and lower bounds for

V R_{t} (F)

and

E R_{t} (F)

are established for sets

F

of graphs without isolated vertices and the sharpness of these bounds is discussed. The primary goal of this paper is to investigate the values of

V R_{t} (F)

and

E R_{t} (F)

for sets

F

of graphs of size 2 or 3 without isolated vertices. The exact values of

V R_{t} (F)

are determined for all such sets

F

and all integers

t \geq 2

. The exact values of

E R_{t} (F)

of certain such sets

F

with prescribed conditions for all integers

t \geq 2

are determined. For some special sets

F

of graphs of size 2 or 3 without isolated vertices, the exact values of

E R_{t} (F)

are determined for

2 \leq t \leq 4

. Additional results, problems, and conjectures are also presented dealing with these two Ramsey concepts for graphs in general. Full article

18 pages, 251 KB

Open AccessEditor’s ChoiceArticle

Complex Riemannian Spacetime: Removal of Black Hole Singularities and Black Hole Paradoxes

by John W. Moffat

Axioms 2025, 14(6), 440; https://doi.org/10.3390/axioms14060440 - 4 Jun 2025

Cited by 3 | Viewed by 1946

Abstract

An approach is presented to resolve key paradoxes in black hole physics through the application of complex Riemannian spacetime. We extend the Schwarzschild metric into the complex domain, employing contour integration techniques to remove singularities while preserving the essential features of the original [...] Read more.

An approach is presented to resolve key paradoxes in black hole physics through the application of complex Riemannian spacetime. We extend the Schwarzschild metric into the complex domain, employing contour integration techniques to remove singularities while preserving the essential features of the original solution. A new regularized radial coordinate is introduced, leading to a singularity-free description of black hole interiors. Crucially, we demonstrate how this complex extension resolves the long-standing paradox of event horizon formation occurring only in the infinite future of distant observers. By analyzing trajectories in complex spacetime, we show that the horizon can form in finite complex time, reconciling the apparent contradiction between proper and coordinate time descriptions. This approach also provides a framework for the analytic continuation of information across event horizons, resolving the Hawking information paradox. We explore the physical interpretation of the complex extension versus its projection onto real spacetime. The gravitational collapse of a dust sphere with negligible dust is explored in the complex spacetime extension. The approach offers a mathematically rigorous framework for exploring quantum gravity effects within the context of classical general relativity. Full article

(This article belongs to the Special Issue Complex Variables in Quantum Gravity)

32 pages, 1664 KB

Open AccessEditor’s ChoiceArticle

Transfinite Elements Using Bernstein Polynomials

by Christopher Provatidis

Axioms 2025, 14(6), 433; https://doi.org/10.3390/axioms14060433 - 2 Jun 2025

Cited by 2 | Viewed by 1458

Abstract

Transfinite interpolation, originally proposed in the early 1970s as a global interpolation method, was first implemented using Lagrange polynomials and cubic Hermite splines. While initially developed for computer-aided geometric design (CAGD), the method also found application in global finite element analysis. With the [...] Read more.

Transfinite interpolation, originally proposed in the early 1970s as a global interpolation method, was first implemented using Lagrange polynomials and cubic Hermite splines. While initially developed for computer-aided geometric design (CAGD), the method also found application in global finite element analysis. With the advent of isogeometric analysis (IGA), Bernstein–Bézier polynomials have increasingly replaced Lagrange polynomials, particularly in conjunction with tensor product B-splines and non-uniform rational B-splines (NURBSs). Despite its early promise, transfinite interpolation has seen limited adoption in modern CAD/CAE workflows, primarily due to its mathematical complexity—especially when blending polynomials of different degrees. In this context, the present study revisits transfinite interpolation and demonstrates that, in four broad classes, Lagrange polynomials can be systematically replaced by Bernstein polynomials in a one-to-one manner, thus giving the same accuracy. In a fifth class, this replacement yields a robust dual set of basis functions with improved numerical properties. A key advantage of Bernstein polynomials lies in their natural compatibility with weighted formulations, enabling the accurate representation of conic sections and quadrics—scenarios where IGA methods are particularly effective. The proposed methodology is validated through its application to a boundary-value problem governed by the Laplace equation, as well as to the eigenvalue analysis of an acoustic cavity, thereby confirming its feasibility and accuracy. Full article

► Show Figures

Figure 1

25 pages, 420 KB

Open AccessFeature PaperEditor’s ChoiceArticle

An Axiomatic Approach to Mild Distributions

by Hans G. Feichtinger

Axioms 2025, 14(4), 302; https://doi.org/10.3390/axioms14040302 - 16 Apr 2025

Viewed by 3758

Abstract

The Banach Gelfand Triple

(S_{0}, L^{2}, S_{0}^{'})

consists of the Feichtinger algebra

S_{0} (R^{d})

as a space of test functions, the dual space

S_{0}^{'} (R^{d})

, [...] Read more.

The Banach Gelfand Triple

(S_{0}, L^{2}, S_{0}^{'})

consists of the Feichtinger algebra

S_{0} (R^{d})

as a space of test functions, the dual space

S_{0}^{'} (R^{d})

, known as the space of mild distributions, and the intermediate Hilbert space

L^{2} (R^{d})

. This Gelfand Triple is very useful for the description of mathematical problems in the area of time-frequency analysis, but also for classical Fourier analysis and engineering applications. Because the involved spaces are Banach spaces, we speak of a Banach Gelfand Triple, in contrast to the widespread concept of rigged Hilbert spaces, which usually involve nuclear Frechet spaces. Still, both concepts serve very similar purposes. Based on the manifold properties of

S_{0} (R^{d})

, it has found applications in the derivation of mathematical statements related to Gabor Analysis but also in providing an alternative and more lucid description of classical results, such as the Shannon sampling theory, with a potential to renew the way how Fourier and time-frequency analysis, but also signal processing courses for engineers (or physicists and mathematicians) could be taught in the future. In the present study, we will demonstrate that one could choose a relatively large variety of similar Banach Gelfand Triples, even if one wants to include key properties such as Fourier invariance (an extended version of Plancherel’s Theorem). Some of them appeared naturally in the literature. It turns out, that

S_{0} (R^{d})

is the smallest member of this family. Consequently

S_{0}^{'} (R^{d})

is the largest dual space among all these spaces, which may be one of the reasons for its universal usefulness. This article provides a study of the basic properties following from a short list of relatively simple assumptions and gives a list of non-trivial examples satisfying these basic axioms. Full article

(This article belongs to the Section Mathematical Analysis)

► Show Figures

Figure 1

18 pages, 1701 KB

Open AccessEditor’s ChoiceArticle

Furstenberg Topology and Collatz Problem

by Edward Tutaj and Halszka Tutaj-Gasinska

Axioms 2025, 14(4), 297; https://doi.org/10.3390/axioms14040297 - 15 Apr 2025

Viewed by 2820

Abstract

The aims of this paper are two-fold. First, we present the result of the decomposition on the iterations of a Collatz transform into arithmetic sequences. With this, we prove that in Furstenberg topology, the set of (odd) integers with an infinite stopping time [...] Read more.

The aims of this paper are two-fold. First, we present the result of the decomposition on the iterations of a Collatz transform into arithmetic sequences. With this, we prove that in Furstenberg topology, the set of (odd) integers with an infinite stopping time is closed and nowhere dense. Then, we move our considerations to some monoids

L

in

N

, where we define a suitably modified Collatz transform, and we present some results of numerical investigations on the behaviour of these modified transforms. Full article

► Show Figures

Figure 1

19 pages, 292 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fixed-Point Results and the Ekeland Variational Principle in Vector B-Metric Spaces

by Radu Precup and Andrei Stan

Axioms 2025, 14(4), 250; https://doi.org/10.3390/axioms14040250 - 26 Mar 2025

Cited by 1 | Viewed by 1262

Abstract

In this paper, we extend the concept of b-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in [...] Read more.

In this paper, we extend the concept of b-metric spaces to the vectorial case, where the distance is vector-valued, and the constant in the triangle inequality axiom is replaced by a matrix. For such spaces, we establish results analogous to those in the b-metric setting: fixed-point theorems, stability results, and a variant of Ekeland’s variational principle. As a consequence, we also derive a variant of Caristi’s fixed-point theorem. Full article

(This article belongs to the Special Issue Fixed-Point Theory and Its Related Topics, 5th Edition)

26 pages, 495 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Beyond Algebraic Superstring Compactification

by Tristan Hübsch

Axioms 2025, 14(4), 236; https://doi.org/10.3390/axioms14040236 - 21 Mar 2025

Cited by 1 | Viewed by 3479

Abstract

Superstring compactifications have been vigorously studied for over four decades, and have flourished, involving an active iterative feedback between physics and (complex) algebraic geometry. This led to an unprecedented wealth of constructions, virtually all of which are “purely” algebraic. Recent developments however indicate [...] Read more.

Superstring compactifications have been vigorously studied for over four decades, and have flourished, involving an active iterative feedback between physics and (complex) algebraic geometry. This led to an unprecedented wealth of constructions, virtually all of which are “purely” algebraic. Recent developments however indicate many more possibilities to be afforded by including certain generalizations that, at first glance at least, are not algebraic—yet fit remarkably well within an overall mirror-symmetric framework and are surprisingly amenable to standard computational analysis upon certain mild but systematic modifications. Full article

(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)

► Show Figures

Figure 1

17 pages, 307 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Hammerstein Nonlinear Integral Equations and Iterative Methods for the Computation of Common Fixed Points

by María A. Navascués

Axioms 2025, 14(3), 214; https://doi.org/10.3390/axioms14030214 - 15 Mar 2025

Cited by 3 | Viewed by 2199

Abstract

In the first part of this article, a special type of Hammerstein nonlinear integral equation is studied. A theorem of the existence of solutions is given in the framework of

L^{2}

-spaces. Afterwards, an iterative method for the resolution of this kind [...] Read more.

In the first part of this article, a special type of Hammerstein nonlinear integral equation is studied. A theorem of the existence of solutions is given in the framework of

L^{2}

-spaces. Afterwards, an iterative method for the resolution of this kind of equations is considered, and the convergence of this algorithm towards a solution of the equation is proved. The rest of the paper considers two modifications of the algorithm. The first one is devoted to the sought of common fixed points of a family of nearly asymptotically nonexpansive mappings. The second variant focuses on the search of common fixed points of a finite number of nonexpansive operators. The characteristics of convergence of these methods are studied in the context of uniformly convex Banach spaces. The iterative scheme is applied to approach the common solution of three nonlinear integral equations of Hammerstein type. Full article

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

► Show Figures

Figure 1

18 pages, 575 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Some Bounds for the Fragmentation Coefficient of Random Graphs

by Katerina Adler, Reuven Cohen and Simi Haber

Axioms 2025, 14(3), 208; https://doi.org/10.3390/axioms14030208 - 12 Mar 2025

Viewed by 1199

Abstract

Graph fragmentation aims to find the smallest vertex subset whose removal breaks a graph into components of bounded size. While this problem has applications in network dismantling and combinatorics, theoretical bounds on optimal solutions remain limited. We derive rigorous bounds for several graph [...] Read more.

Graph fragmentation aims to find the smallest vertex subset whose removal breaks a graph into components of bounded size. While this problem has applications in network dismantling and combinatorics, theoretical bounds on optimal solutions remain limited. We derive rigorous bounds for several graph classes, characterize hard instances, and illuminate the relationship between graph structure and optimal fragmentation strategies. Specifically, we show that for random d-regular graphs with n vertices, the minimal size of the fragmenting subset of nodes is asymptotically almost surely

| S | \geq \frac{d - 2}{2 d - 2} n - o (n)

, and that asymptotically almost surely,

n - 2 α (G) - o (n) \leq | S | \leq n - α (G) + o (n)

, where

α (G)

is the independence number of G. For

d ≫ 1

, we prove that asymptotically almost surely,

| S | / n \approx 1 - log d / d

. However, we show that the line graphs of random regular graphs are considerably harder to fragment, with

| S | / n \geq 1 - c / d

for some constant c. Full article

► Show Figures

Figure 1

13 pages, 269 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On Equivalents of the Riemann Hypothesis Connected to the Approximation Properties of the Zeta Function

by Antanas Laurinčikas

Axioms 2025, 14(3), 169; https://doi.org/10.3390/axioms14030169 - 26 Feb 2025

Cited by 2 | Viewed by 3989

Abstract

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function

ζ (s)

(zeros different from

s = - 2 m

,

m \in N

) lie on the critical line

σ = 1 / 2

. [...] Read more.

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function

ζ (s)

(zeros different from

s = - 2 m

,

m \in N

) lie on the critical line

σ = 1 / 2

. In this paper, combining the universality property of

ζ (s)

with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts

ζ (s + i t_{τ})

approximating the function

ζ (s)

. Here,

t_{τ}

denotes the Gram function, which is a continuous extension of the Gram points. Full article

14 pages, 292 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Duality and Some Links Between Riemannian Submersion, F-Harmonicity, and Cohomology

by Bang-Yen Chen and Shihshu (Walter) Wei

Axioms 2025, 14(3), 162; https://doi.org/10.3390/axioms14030162 - 23 Feb 2025

Viewed by 1407

Abstract

Fundamentally, duality gives two different points of view of looking at the same object. It appears in many subjects in mathematics (geometry, algebra, analysis, PDEs, Geometric Measure Theory, etc.) and in physics. For example, Connections on Fiber Bundles in mathematics, and Gauge Fields [...] Read more.

Fundamentally, duality gives two different points of view of looking at the same object. It appears in many subjects in mathematics (geometry, algebra, analysis, PDEs, Geometric Measure Theory, etc.) and in physics. For example, Connections on Fiber Bundles in mathematics, and Gauge Fields in physics are exactly the same. In n-dimensional geometry, a fundamental notion is the “duality” between chains and cochains, or domains of integration and the integrands. In this paper, we extend ideas given in our earlier articles and connect seemingly unrelated areas of F-harmonic maps, f-harmonic maps, and cohomology classes via duality. By studying cohomology classes that are related with p-harmonic morphisms, F-harmonic maps, and f-harmonic maps, we extend several of our previous results on Riemannian submersions and p-harmonic morphisms to F-harmonic maps and f-harmonic maps, which are Riemannian submersions. Full article

(This article belongs to the Special Issue Trends in Differential Geometry and Algebraic Topology)

19 pages, 333 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Positive Definite Solution of System of Matrix Equations with X^−k and Y^−l via Coupled Fixed Point Theorem in Partially Ordered Spaces

by Aynur Ali, Cvetelina Dinkova, Atanas Ilchev, Hristina Kulina and Boyan Zlatanov

Axioms 2025, 14(2), 123; https://doi.org/10.3390/axioms14020123 - 7 Feb 2025

Cited by 3 | Viewed by 1411

Abstract

We establish adequate conditions for the existence and uniqueness of solutions to systems of two matrix equations when the unknown matrices are raised to a power

k \in [- 1, 1] ∖ {0}

. The findings from coupled [...] Read more.

We establish adequate conditions for the existence and uniqueness of solutions to systems of two matrix equations when the unknown matrices are raised to a power

k \in [- 1, 1] ∖ {0}

. The findings from coupled fixed points for ordered pairs of maps are used. Numerical examples are provided to illustrate the results shown. Some of the known results are the consequence of our acquisitions. Full article

(This article belongs to the Section Mathematical Analysis)

11 pages, 286 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Vector Meson Spectrum from Top-Down Holographic QCD

by Mohammed Mia, Keshav Dasgupta, Charles Gale, Michael Richard and Olivier Trottier

Axioms 2025, 14(1), 66; https://doi.org/10.3390/axioms14010066 - 16 Jan 2025

Viewed by 1390

Abstract

We elaborate on the brane configuration that gives rise to a QCD-like gauge theory that confines at low energies and becomes scale invariant at the highest energies. In the limit where the rank of the gauge group is large, a gravitational description emerges. [...] Read more.

We elaborate on the brane configuration that gives rise to a QCD-like gauge theory that confines at low energies and becomes scale invariant at the highest energies. In the limit where the rank of the gauge group is large, a gravitational description emerges. For the confined phase, we obtain a vector meson spectrum and demonstrate how a certain choice of parameters can lead to quantitative agreement with empirical data. Full article

(This article belongs to the Special Issue Mathematical Aspects of Quantum Field Theory and Quantization)

► Show Figures

Figure 1

28 pages, 399 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On the Work of Cartan and Münzner on Isoparametric Hypersurfaces

by Thomas E. Cecil and Patrick J. Ryan

Axioms 2025, 14(1), 56; https://doi.org/10.3390/axioms14010056 - 13 Jan 2025

Cited by 1 | Viewed by 2401

Abstract

A hypersurface

M^{n}

in a real space form

R^{n + 1}

,

S^{n + 1}

, or

H^{n + 1}

is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan [...] Read more.

A hypersurface

M^{n}

in a real space form

R^{n + 1}

,

S^{n + 1}

, or

H^{n + 1}

is isoparametric if it has constant principal curvatures. This paper is a survey of the fundamental work of Cartan and Münzner on the theory of isoparametric hypersurfaces in real space forms, in particular, spheres. This work is contained in four papers of Cartan published during the period 1938–1940 and two papers of Münzner that were published in preprint form in the early 1970s and as journal articles in 1980–1981. These papers of Cartan and Münzner have been the foundation of the extensive field of isoparametric hypersurfaces, and they have all been recently translated into English by T. Cecil. The paper concludes with a brief survey of the recently completed classification of isoparametric hypersurfaces in spheres. Full article

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

20 pages, 703 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Theoretical Basis for Classifying Hyperuniform States of Two-Component Systems

by Hiroshi Frusawa

Axioms 2025, 14(1), 39; https://doi.org/10.3390/axioms14010039 - 5 Jan 2025

Cited by 1 | Viewed by 1817

Abstract

Hyperuniform states of matter exhibit unusual suppression of density fluctuations at large scales, contrasting sharply with typical disordered configurations. Various types of hyperuniformity emerge in multicomponent disordered systems, significantly enhancing their functional properties for advanced applications. This paper focuses on developing a theoretical [...] Read more.

Hyperuniform states of matter exhibit unusual suppression of density fluctuations at large scales, contrasting sharply with typical disordered configurations. Various types of hyperuniformity emerge in multicomponent disordered systems, significantly enhancing their functional properties for advanced applications. This paper focuses on developing a theoretical framework for two-component hyperuniform systems. We provide a robust theoretical basis to identify novel conditions on structure factors for a variety of hyperuniform binary mixtures, classifying them into five distinct types with seven unique states. Our findings also offer valuable guidelines for designing multihyperuniform materials where each component preserves hyperuniformity, added to the overall hyperuniformity. Full article

(This article belongs to the Section Mathematical Physics)

► Show Figures

Figure 1

14 pages, 490 KB

Open AccessFeature PaperEditor’s ChoiceArticle

About Stabilization of the Controlled Inverted Pendulum Under Stochastic Perturbations of the Type of Poisson’s Jumps

by Leonid Shaikhet

Axioms 2025, 14(1), 29; https://doi.org/10.3390/axioms14010029 - 31 Dec 2024

Cited by 3 | Viewed by 1276

Abstract

The classical problem of stabilization of the controlled inverted pendulum is considered in the case of stochastic perturbations of the type of Poisson’s jumps. It is supposed that stabilized control depends on the entire trajectory of the pendulum. Linear and nonlinear models of [...] Read more.

The classical problem of stabilization of the controlled inverted pendulum is considered in the case of stochastic perturbations of the type of Poisson’s jumps. It is supposed that stabilized control depends on the entire trajectory of the pendulum. Linear and nonlinear models of the controlled inverted pendulum are considered, and the stability of the zero and nonzero equilibria is studied. The obtained results are illustrated by examples with numerical simulation of solutions of the equations under consideration. Full article

(This article belongs to the Special Issue Advances in Mathematical Optimal Control and Applications)

► Show Figures

Figure 1

13 pages, 517 KB

Open AccessFeature PaperEditor’s ChoiceArticle

First and Second Integrals of Hopf–Langford-Type Systems

by Vassil M. Vassilev and Svetoslav G. Nikolov

Axioms 2025, 14(1), 8; https://doi.org/10.3390/axioms14010008 - 27 Dec 2024

Cited by 1 | Viewed by 1573

Abstract

The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a [...] Read more.

The work examines a seven-parameter, three-dimensional, autonomous, cubic nonlinear differential system. This system extends and generalizes the previously studied quadratic nonlinear Hopf–Langford-type systems. First, by introducing cylindrical coordinates in its phase space, we show that the regarded system can be reduced to a two-dimensional Liénard system, which corresponds to a second-order Liénard equation. Then, we present (in explicit form) polynomial first and second integrals of Liénard systems of the considered type identifying those values of their parameters for which these integrals exist. It is also proved that a generic Liénard equation is factorizable if and only if the corresponding Liénard system admits a second integral of a special form. It is established that each Liénard system corresponding to a Hopf–Langford system of the considered type admits such a second integral, and hence, the respective Liénard equation is factorizable. Full article

(This article belongs to the Special Issue Complex Networks and Dynamical Systems)

► Show Figures

Figure 1

17 pages, 5933 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Study Using the Network Simulation Method and Nondimensionalization of the Fiber Fuse Effect

by Juan Francisco Sanchez-Pérez, Joaquín Solano-Ramírez, Fulgencio Marín-García and Enrique Castro

Axioms 2025, 14(1), 2; https://doi.org/10.3390/axioms14010002 - 26 Dec 2024

Cited by 2 | Viewed by 1419

Abstract

This paper presents an innovative approach to modelling the fiber optic fusion effect using the Network Simulation Method (NSM). An analogy between the heat conduction equations and electrical circuits is developed, allowing a complex physical problem to be transformed into an equivalent electrical [...] Read more.

This paper presents an innovative approach to modelling the fiber optic fusion effect using the Network Simulation Method (NSM). An analogy between the heat conduction equations and electrical circuits is developed, allowing a complex physical problem to be transformed into an equivalent electrical system. Using NGSpice, thermal interactions in an anisotropic optical fiber under high optical power conditions are simulated. The methodology addresses the distribution of the temperature in the system, considering thermal variations and temperature-dependent material characteristics. In an NSM equivalent circuit, the effect of applying the spark is modelled by a switch that switches the spark-generating source on and off. It can be seen that temperature variation with time, or temperature rise rate (K/s), depends on the applied power. In addition, the mathematical method of nondimensionalization is used to study the real influence of each parameter of the problem on the solution and the relationship between the variables. Four optical fiber cases are analysed, each characterised by different areas and refractive indices, revealing how these variables affect the propagation of the melting phenomenon. The results highlight the effectiveness of the NSM in solving nonlinear and coupled problems in thermal engineering, providing a solid framework for future research in the optimisation of optical communication systems. Full article

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

► Show Figures

Figure 1

28 pages, 2723 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Comprehensive Model and Numerical Study of Shear Flow in Compressible Viscous Micropolar Real Gases

by Nelida Črnjarić and Ivan Dražić

Axioms 2024, 13(12), 845; https://doi.org/10.3390/axioms13120845 - 2 Dec 2024

Cited by 1 | Viewed by 1220

Abstract

Understanding shear flow behavior in compressible, viscous, micropolar real gases is essential for both theoretical advancements and practical engineering applications. This study develops a comprehensive model that integrates micropolar fluid theory with compressible flow dynamics to accurately describe the behavior of real gases [...] Read more.

Understanding shear flow behavior in compressible, viscous, micropolar real gases is essential for both theoretical advancements and practical engineering applications. This study develops a comprehensive model that integrates micropolar fluid theory with compressible flow dynamics to accurately describe the behavior of real gases under shear stress. We formulate the governing equations by incorporating viscosity and micropolar effects and transform the obtained system into the mass Lagrangian coordinates. Two numerical methods, Faedo–Galerkin approximation and finite-difference methods, are used to solve it. These methods are compared using several benchmark examples to assess their accuracy and computational efficiency. Both methods demonstrate good performance, achieving equally precise results in capturing essential flow characteristics. However, the finite-difference method offers advantages in speed, stability, and lower computational demands. This research bridges gaps in existing models and establishes a foundation for further investigations into complex flow phenomena in micropolar real gases. Full article

(This article belongs to the Special Issue Recent Progress in Computational Fluid Dynamics)

► Show Figures

Figure 1

21 pages, 401 KB

Open AccessFeature PaperEditor’s ChoiceArticle

From Uncertainty Relations to Quantum Acceleration Limits

by Carlo Cafaro, Christian Corda, Newshaw Bahreyni and Abeer Alanazi

Axioms 2024, 13(12), 817; https://doi.org/10.3390/axioms13120817 - 22 Nov 2024

Cited by 4 | Viewed by 1627

Abstract

The concept of quantum acceleration limit has been recently introduced for any unitary time evolution of quantum systems under arbitrary nonstationary Hamiltonians. While Alsing and Cafaro used the Robertson uncertainty relation in their derivation, employed the Robertson–Schrödinger uncertainty relation to find the upper [...] Read more.

The concept of quantum acceleration limit has been recently introduced for any unitary time evolution of quantum systems under arbitrary nonstationary Hamiltonians. While Alsing and Cafaro used the Robertson uncertainty relation in their derivation, employed the Robertson–Schrödinger uncertainty relation to find the upper bound on the temporal rate of change of the speed of quantum evolutions. In this paper, we provide a comparative analysis of these two alternative derivations for quantum systems specified by an arbitrary finite-dimensional projective Hilbert space. Furthermore, focusing on a geometric description of the quantum evolution of two-level quantum systems on a Bloch sphere under general time-dependent Hamiltonians, we find the most general conditions needed to attain the maximal upper bounds on the acceleration of the quantum evolution. In particular, these conditions are expressed explicitly in terms of two three-dimensional real vectors, the Bloch vector that corresponds to the evolving quantum state and the magnetic field vector that specifies the Hermitian Hamiltonian of the system. For pedagogical reasons, we illustrate our general findings for two-level quantum systems in explicit physical examples characterized by specific time-varying magnetic field configurations. Finally, we briefly comment on the extension of our considerations to higher-dimensional physical systems in both pure and mixed quantum states. Full article

(This article belongs to the Special Issue Mathematical Aspects of Quantum Field Theory and Quantization)

► Show Figures

Figure 1

20 pages, 338 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Eccentric p-Summing Lipschitz Operators and Integral Inequalities on Metric Spaces and Graphs

by Roger Arnau, Enrique A. Sánchez Pérez and Sergi Sanjuan

Axioms 2024, 13(11), 760; https://doi.org/10.3390/axioms13110760 - 2 Nov 2024

Viewed by 1389

Abstract

The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens–Eells [...] Read more.

The extension of the concept of p-summability for linear operators to the context of Lipschitz operators on metric spaces has been extensively studied in recent years. This research primarily uses the linearization of the metric space M afforded by the associated Arens–Eells space, along with the duality between M and the metric dual space

M^{#}

defined by the real-valued Lipschitz functions on

M .

However, alternative approaches to measuring distances between sequences of elements of metric spaces (essentially involved in the definition of p-summability) exist. One approach involves considering specific subsets of the unit ball of

M^{#}

for computing the distances between sequences, such as the real Lipschitz functions derived from evaluating the difference in the values of the metric from two points to a fixed point. We introduce new notions of summability for Lipschitz operators involving such functions, which are characterized by integral dominations for those operators. To show the applicability of our results, in the last part of this paper, we use the theoretical tools obtained in the first part to analyze metric graphs. In particular, we show new results on the behavior of numerical indices defined on these graphs satisfying certain conditions of summability and symmetry. Full article

(This article belongs to the Special Issue Theory and Application of Integral Inequalities)

► Show Figures

Figure 1

20 pages, 465 KB

Open AccessFeature PaperEditor’s ChoiceArticle

A Global Method for Approximating Caputo Fractional Derivatives—An Application to the Bagley–Torvik Equation

by Maria Carmela De Bonis and Donatella Occorsio

Axioms 2024, 13(11), 750; https://doi.org/10.3390/axioms13110750 - 30 Oct 2024

Cited by 4 | Viewed by 3177

Abstract

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order

α

[...] Read more.

In this paper, we propose a global numerical method for approximating Caputo fractional derivatives of order

α

(D^{α} f) (y) = \frac{1}{Γ (m - α)} \int_{0}^{y} {(y - x)}^{m - α - 1} f^{(m)} (x) d x, y > 0

, with

m - 1 < α \leq m, m \in N .

The numerical procedure is based on approximating

f^{(m)}

by the m-th derivative of a Lagrange polynomial, interpolating f at Jacobi zeros and some additional nodes suitably chosen to have corresponding logarithmically diverging Lebsegue constants. Error estimates in a uniform norm are provided, showing that the rate of convergence is related to the smoothness of the function f according to the best polynomial approximation error and depending on order

α

. As an application, we approximate the solution of a Volterra integral equation, which is equivalent in some sense to the Bagley–Torvik initial value problem, using a Nyström-type method. Finally, some numerical tests are presented to assess the performance of the proposed procedure. Full article

(This article belongs to the Special Issue Orthogonal Polynomials, Special Functions and Applications: 2nd Edition)

► Show Figures

Figure 1

15 pages, 275 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fredholm Determinant and Wronskian Representations of the Solutions to the Schrödinger Equation with a KdV-Potential

by Pierre Gaillard

Axioms 2024, 13(10), 712; https://doi.org/10.3390/axioms13100712 - 15 Oct 2024

Viewed by 1295

Abstract

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and [...] Read more.

From the finite gap solutions of the KdV equation expressed in terms of abelian functions we construct solutions to the Schrödinger equation with a KdV potential in terms of fourfold Fredholm determinants. For this we establish a connection between Riemann theta functions and Fredholm determinants and we obtain multi-parametric solutions to this equation. As a consequence, a double Wronskian representation of the solutions to this equation is constructed. We also give quasi-rational solutions to this Schrödinger equation with rational KdV potentials. Full article

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

13 pages, 315 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Blow Up of Solutions to Wave Equations with Combined Logarithmic and Power-Type Nonlinearities

by Milena Dimova, Natalia Kolkovska and Nikolai Kutev

Axioms 2024, 13(10), 709; https://doi.org/10.3390/axioms13100709 - 14 Oct 2024

Cited by 2 | Viewed by 1782

Abstract

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we [...] Read more.

In this paper, we study the initial boundary value problem for wave equations with combined logarithmic and power-type nonlinearities. For arbitrary initial energy, we prove a necessary and sufficient condition for blow up at infinity of the global weak solutions. In addition, we derive a growth estimate for the blowing up global solutions. Full article

(This article belongs to the Special Issue Advances in Nonlinear Analysis and Boundary Value Problems)

15 pages, 326 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Non-Fragile Sampled Control Design for an Interconnected Large-Scale System via Wirtinger Inequality

by Volodymyr Lynnyk and Branislav Rehák

Axioms 2024, 13(10), 702; https://doi.org/10.3390/axioms13100702 - 10 Oct 2024

Cited by 5 | Viewed by 1589

Abstract

A control design for a linear large-scale interconnected system composed of identical subsystems is presented in this paper. The control signal of all subsystems is sampled. For different subsystems, the sampling times are not identical. Nonetheless, it is assumed that a bound exists [...] Read more.

A control design for a linear large-scale interconnected system composed of identical subsystems is presented in this paper. The control signal of all subsystems is sampled. For different subsystems, the sampling times are not identical. Nonetheless, it is assumed that a bound exists for the maximal sampling time. The control algorithm is designed using the Wirtinger inequality, and the non-fragile control law is proposed. The size of the linear matrix inequalities to be solved by the proposed control algorithm is independent of the number of subsystems composing the overall system. Hence, the algorithm is computationally effective. The results are illustrated by two examples. The first example graphically illustrates the function of the proposed algorithm while the second one compares with a method for stabilizing a large-scale system obtained earlier, thus illustrating the improved capabilities of the presented algorithm. Full article

(This article belongs to the Special Issue Advances in Mathematical Methods in Optimal Control and Applications)

► Show Figures

Figure 1

10 pages, 262 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Greedoids and Violator Spaces

by Yulia Kempner and Vadim E. Levit

Axioms 2024, 13(9), 633; https://doi.org/10.3390/axioms13090633 - 17 Sep 2024

Viewed by 1727

Abstract

This research explores the interplay between violator spaces and greedoids—two distinct theoretical frameworks developed independently. Violator spaces were introduced as a generalization of linear programming, while greedoids were designed to characterize combinatorial structures where greedy algorithms yield optimal solutions. These frameworks have, until [...] Read more.

This research explores the interplay between violator spaces and greedoids—two distinct theoretical frameworks developed independently. Violator spaces were introduced as a generalization of linear programming, while greedoids were designed to characterize combinatorial structures where greedy algorithms yield optimal solutions. These frameworks have, until now, existed in isolation. This paper bridges the gap by showing that greedoids can be defined using a modified violator operator. The established connections not only deepen our understanding of these theories but also provide a new characterization of antimatroids. Full article

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

► Show Figures

Figure 1

24 pages, 613 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Round-Off Error Suppression by Statistical Averaging

by Andrej Liptaj

Axioms 2024, 13(9), 615; https://doi.org/10.3390/axioms13090615 - 11 Sep 2024

Cited by 1 | Viewed by 1684

Abstract

Regarding round-off errors as random is often a necessary simplification to describe their behavior. Assuming, in addition, the symmetry of their distributions, we show that one can, in unstable (ill-conditioned) computer calculations, suppress their effect by statistical averaging. For this, one slightly perturbs [...] Read more.

Regarding round-off errors as random is often a necessary simplification to describe their behavior. Assuming, in addition, the symmetry of their distributions, we show that one can, in unstable (ill-conditioned) computer calculations, suppress their effect by statistical averaging. For this, one slightly perturbs the argument of

f (x_{0})

many times and averages the resulting function values. In this text, we forward arguments to support the assumed properties of round-off errors and critically evaluate the validity of the averaging approach in several numerical experiments. Full article

(This article belongs to the Special Issue Numerical Analysis and Applied Mathematics)

► Show Figures

Figure 1

17 pages, 304 KB

Open AccessFeature PaperEditor’s ChoiceArticle

On a Generic Fractional Derivative Associated with the Riemann–Liouville Fractional Integral

by Yuri Luchko

Axioms 2024, 13(9), 604; https://doi.org/10.3390/axioms13090604 - 4 Sep 2024

Cited by 8 | Viewed by 8170

Abstract

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its [...] Read more.

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed-form formula for its unique solution, and study its complete monotonicity. Full article

(This article belongs to the Section Mathematical Analysis)

14 pages, 673 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Fox’s H-Functions: A Gentle Introduction to Astrophysical Thermonuclear Functions

by Hans J. Haubold, Dilip Kumar and Ashik A. Kabeer

Axioms 2024, 13(8), 532; https://doi.org/10.3390/axioms13080532 - 6 Aug 2024

Cited by 2 | Viewed by 3090

Abstract

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, [...] Read more.

Needed for cosmological and stellar nucleosynthesis, we are studying the closed-form analytic evaluation of thermonuclear reaction rates. In this context, we undertake a comprehensive analysis of three largely distinct velocity distributions, namely the Maxwell–Boltzmann distribution, the pathway distribution, and the Mittag-Leffler distribution. Moreover, a natural generalization of the Maxwell–Boltzmann velocity distribution is discussed. Furthermore, an explicit evaluation of the reaction rate integral in the high-energy cut-off case is carried out. Generalized special functions of mathematical physics like Meijer’s G-function and Fox’s H-functions and their utilization in mathematical physics are the prime focus of this paper. Full article

► Show Figures

Figure 1

8 pages, 220 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Integrable Couplings and Two-Dimensional Unital Algebras

by Wen-Xiu Ma

Axioms 2024, 13(7), 481; https://doi.org/10.3390/axioms13070481 - 18 Jul 2024

Cited by 35 | Viewed by 3417

Abstract

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs [...] Read more.

The paper aims to demonstrate that a linear expansion in a unital two-dimensional algebra can generate integrable couplings, proposing a novel approach for their construction. The integrable couplings presented encompass a range of perturbation equations and nonlinear integrable couplings. Their corresponding Lax pairs and hereditary recursion operators are explicitly detailed. Concrete applications to the KdV equation and the AKNS system of nonlinear Schrödinger equations are extensively explored. Full article

21 pages, 747 KB

Open AccessEditor’s ChoiceArticle

A Reduced-Dimension Weighted Explicit Finite Difference Method Based on the Proper Orthogonal Decomposition Technique for the Space-Fractional Diffusion Equation

by Xuehui Ren and Hong Li

Axioms 2024, 13(7), 461; https://doi.org/10.3390/axioms13070461 - 8 Jul 2024

Cited by 3 | Viewed by 2036

Abstract

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure [...] Read more.

A kind of reduced-dimension method based on a weighted explicit finite difference scheme and the proper orthogonal decomposition (POD) technique for diffusion equations with Riemann–Liouville fractional derivatives in space are discussed. The constructed approximation method written in matrix form can not only ensure a sufficient accuracy order but also reduce the degrees of freedom, decrease storage requirements, and accelerate the computation rate. Uniqueness, stabilization, and error estimation are demonstrated by matrix analysis. The procedural steps of the POD algorithm, which reduces dimensionality, are outlined. Numerical simulations to assess the viability and effectiveness of the reduced-dimension weighted explicit finite difference method are given. A comparison between the reduced-dimension method and the classical weighted explicit finite difference scheme is presented, including the error in the

L_{2}

norm, the accuracy order, and the CPU time. Full article

► Show Figures

Figure 1

16 pages, 344 KB

Open AccessEditor’s ChoiceReview

Monogenity and Power Integral Bases: Recent Developments

by István Gaál

Axioms 2024, 13(7), 429; https://doi.org/10.3390/axioms13070429 - 26 Jun 2024

Cited by 10 | Viewed by 10234

Abstract

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled [...] Read more.

Monogenity is a classical area of algebraic number theory that continues to be actively researched. This paper collects the results obtained over the past few years in this area. Several of the listed results were presented at a series of online conferences titled “Monogenity and Power Integral Bases”. We also give a collection of the most important methods used in several of these papers. A list of open problems for further research is also given. Full article

(This article belongs to the Special Issue Advances in Generalized Hypergeometric Functions, Integral Transforms and Number Theory)

► Show Figures

Figure 1

12 pages, 930 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Constructing Approximations to Bivariate Piecewise-Smooth Functions

by David Levin

Axioms 2024, 13(7), 428; https://doi.org/10.3390/axioms13070428 - 26 Jun 2024

Cited by 1 | Viewed by 2413

Abstract

This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal [...] Read more.

This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal discontinuities across curves, and even across more involved geometries such as a three-corner discontinuity. The provided data may be uniform or non-uniform, and noisy, and the approximation procedure involves non-linear least-squares minimization. Also included is a basic approximation theorem for functions with jump discontinuity across a smooth curve. Full article

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

► Show Figures

Figure 1

16 pages, 284 KB

Open AccessEditor’s ChoiceArticle

On the Generalized Stabilities of Functional Equations via Isometries

by Muhammad Sarfraz, Jiang Zhou, Yongjin Li and John Michael Rassias

Axioms 2024, 13(6), 403; https://doi.org/10.3390/axioms13060403 - 14 Jun 2024

Cited by 4 | Viewed by 1831

Abstract

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large [...] Read more.

The main goal of this research article is to investigate the stability of generalized norm-additive functional equations. This study demonstrates that these equations are Hyers-Ulam stable for surjective functions from an arbitrary group G to a real Banach space B using the large perturbation method. Furthermore, hyperstability results are investigated for a generalized Cauchy equation. Full article

(This article belongs to the Special Issue New Theory and Applications of Nonlinear Analysis, Fractional Calculus and Optimization, 2nd Edition)

22 pages, 533 KB

Open AccessEditor’s ChoiceArticle

Fixed Time Synchronization of Stochastic Takagi–Sugeno Fuzzy Recurrent Neural Networks with Distributed Delay under Feedback and Adaptive Controls

by Yiran Niu, Xiaofeng Xu and Ming Liu

Axioms 2024, 13(6), 391; https://doi.org/10.3390/axioms13060391 - 11 Jun 2024

Cited by 4 | Viewed by 1408

Abstract

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a [...] Read more.

In this paper, the stochastic Takagi–Sugeno fuzzy recurrent neural networks (STSFRNNS) with distributed delay is established based on the Takagi–Sugeno (TS) model and the fixed time synchronization problem is investigated. In order to synchronize the networks, we design two kinds of controllers: a feedback controller and an adaptive controller. Then, we obtain the synchronization criteria in a fixed time by combining the Lyapunov method and the related inequality theory of the stochastic differential equation and calculate the stabilization time for the STSFRNNS. In addition, to verify the authenticity of the theoretical results, we use MATLABR2023A to carry out numerical simulation. Full article

(This article belongs to the Special Issue Recent Advances in Applied Mathematics and Artificial Intelligence)

► Show Figures

Figure 1

21 pages, 370 KB

Open AccessEditor’s ChoiceArticle

Exploring Clique Transversal Problems for d-degenerate Graphs with Fixed d: From Polynomial-Time Solvability to Parameterized Complexity

by Chuan-Min Lee

Axioms 2024, 13(6), 382; https://doi.org/10.3390/axioms13060382 - 4 Jun 2024

Cited by 2 | Viewed by 1904

Abstract

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations [...] Read more.

This paper explores the computational challenges of clique transversal problems in d-degenerate graphs, which are commonly encountered across theoretical computer science and various network applications. We examine d-degenerate graphs to highlight their utility in representing sparse structures and assess several variations of clique transversal problems, including the b-fold and

{b}

-clique transversal problems, focusing on their computational complexities for different graph categories. Our analysis identifies that certain instances of these problems are polynomial-time solvable in specific graph classes, such as 1-degenerate or 2-degenerate graphs. However, for d-degenerate graphs where

d \geq 2

, these problems generally escalate to NP-completeness. We also explore the parameterized complexity, pinpointing specific conditions that render these problems fixed-parameter tractable. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Combinatorial Optimization)

► Show Figures

Figure 1

38 pages, 522 KB

Open AccessFeature PaperEditor’s ChoiceArticle

Static Spherically Symmetric Perfect Fluid Solutions in Teleparallel F(T) Gravity

by Alexandre Landry

Axioms 2024, 13(5), 333; https://doi.org/10.3390/axioms13050333 - 17 May 2024

Cited by 14 | Viewed by 2389

Abstract

In this paper, we investigate static spherically symmetric teleparallel

F (T)

gravity containing a perfect isotropic fluid. We first write the field equations and proceed to find new teleparallel

F (T)

solutions for perfect isotropic and linear fluids. By [...] Read more.

In this paper, we investigate static spherically symmetric teleparallel

F (T)

gravity containing a perfect isotropic fluid. We first write the field equations and proceed to find new teleparallel

F (T)

solutions for perfect isotropic and linear fluids. By using a power-law ansatz for the coframe components, we find several classes of new non-trivial teleparallel

F (T)

solutions. We also find a new class of teleparallel

F (T)

solutions for a matter dust fluid. After, we solve the field equations for a non-linear perfect fluid. Once again, there are several new exact teleparallel

F (T)

solutions and also some approximated teleparallel

F (T)

solutions. All these classes of new solutions may be relevant for future cosmological and astrophysical applications. Full article

(This article belongs to the Special Issue Advances in Differential Geometry and Mathematical Physics)

20 pages, 383 KB

Open AccessEditor’s ChoiceArticle

Ideals and Filters on Neutrosophic Topologies Generated by Neutrosophic Relations

by Ravi P. Agarwal, Soheyb Milles, Brahim Ziane, Abdelaziz Mennouni and Lemnaouar Zedam

Axioms 2024, 13(5), 292; https://doi.org/10.3390/axioms13050292 - 25 Apr 2024

Cited by 4 | Viewed by 2775

Abstract

Recently, Milles and Hammami presented and studied the concept of a neutrosophic topology generated by a neutrosophic relation. As a continuation in the same direction, this paper studies the concepts of neutrosophic ideals and neutrosophic filters on that topology. More precisely, we offer [...] Read more.

Recently, Milles and Hammami presented and studied the concept of a neutrosophic topology generated by a neutrosophic relation. As a continuation in the same direction, this paper studies the concepts of neutrosophic ideals and neutrosophic filters on that topology. More precisely, we offer the lattice structure of neutrosophic open sets of a neutrosophic topology generated via a neutrosophic relation and examine its different characteristics. Furthermore, we enlarge to this lattice structure the notions of ideals (respectively, filters) and characterize them with regard to the lattice operations. We end this work by studying the prime neutrosophic ideal and prime neutrosophic filter as interesting types of neutrosophic ideals and neutrosophic filters. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Editor’s Choice Articles

Further Information

Guidelines

MDPI Initiatives

Follow MDPI