Axioms

19 pages, 292 KiB

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

Viewed by 283

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 KiB

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

Viewed by 762

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)

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17 pages, 307 KiB

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

Viewed by 444

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)

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18 pages, 575 KiB

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 283

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

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13 pages, 269 KiB

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

Viewed by 426

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 KiB

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 496

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 KiB

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

Viewed by 510

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 KiB

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 489

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)

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28 pages, 399 KiB

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

Viewed by 551

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 KiB

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

Viewed by 637

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)

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14 pages, 490 KiB

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

Viewed by 466

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)

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13 pages, 517 KiB

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

Viewed by 652

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)

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17 pages, 5933 KiB

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 1 | Viewed by 513

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)

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28 pages, 2723 KiB

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

Viewed by 621

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)

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21 pages, 401 KiB

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 1 | Viewed by 742

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)

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20 pages, 338 KiB

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 816

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)

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20 pages, 465 KiB

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

Viewed by 1323

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)

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15 pages, 275 KiB

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 671

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 KiB

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 1 | Viewed by 709

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 KiB

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

Viewed by 734

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)

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10 pages, 262 KiB

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 1072

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)

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24 pages, 613 KiB

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

Viewed by 1021

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)

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17 pages, 304 KiB

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 2 | Viewed by 6162

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 KiB

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 1 | Viewed by 1251

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

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8 pages, 220 KiB

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 26 | Viewed by 1712

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 KiB

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 1 | Viewed by 967

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

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16 pages, 344 KiB

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 2 | Viewed by 7241

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)

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12 pages, 930 KiB

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

Viewed by 1483

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)

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16 pages, 284 KiB

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 3 | Viewed by 1052

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 KiB

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 1 | Viewed by 858

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)

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21 pages, 370 KiB

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 1 | Viewed by 960

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)

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38 pages, 522 KiB

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 6 | Viewed by 1412

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 KiB

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 3 | Viewed by 1352

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)

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25 pages, 10343 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Jordan-Type Inequalities and Stratification

by Miloš Mićović and Branko Malešević

Axioms 2024, 13(4), 262; https://doi.org/10.3390/axioms13040262 - 14 Apr 2024

Cited by 3 | Viewed by 1971

Abstract

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding [...] Read more.

In this paper, two double Jordan-type inequalities are introduced that generalize some previously established inequalities. As a result, some new upper and lower bounds and approximations of the sinc function are obtained. This extension of Jordan’s inequality is enabled by considering the corresponding inequalities through the concept of stratified families of functions. Based on this approach, some optimal approximations of the sinc function are derived by determining the corresponding minimax approximants. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications, 2nd Edition)

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12 pages, 310 KiB

Open AccessEditor’s ChoiceArticle

Personalized Treatment Policies with the Novel Buckley-James Q-Learning Algorithm

by Jeongjin Lee and Jong-Min Kim

Axioms 2024, 13(4), 212; https://doi.org/10.3390/axioms13040212 - 25 Mar 2024

Cited by 4 | Viewed by 1721

Abstract

This research paper presents the Buckley-James Q-learning (BJ-Q) algorithm, a cutting-edge method designed to optimize personalized treatment strategies, especially in the presence of right censoring. We critically assess the algorithm’s effectiveness in improving patient outcomes and its resilience across various scenarios. Central to [...] Read more.

This research paper presents the Buckley-James Q-learning (BJ-Q) algorithm, a cutting-edge method designed to optimize personalized treatment strategies, especially in the presence of right censoring. We critically assess the algorithm’s effectiveness in improving patient outcomes and its resilience across various scenarios. Central to our approach is the innovative use of the survival time to impute the reward in Q-learning, employing the Buckley-James method for enhanced accuracy and reliability. Our findings highlight the significant potential of personalized treatment regimens and introduce the BJ-Q learning algorithm as a viable and promising approach. This work marks a substantial advancement in our comprehension of treatment dynamics and offers valuable insights for augmenting patient care in the ever-evolving clinical landscape. Full article

(This article belongs to the Special Issue New Perspectives in Mathematical Statistics)

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23 pages, 404 KiB

Open AccessEditor’s ChoiceArticle

Optimal Construction for Decoding 2D Convolutional Codes over an Erasure Channel

by Raquel Pinto, Marcos Spreafico and Carlos Vela

Axioms 2024, 13(3), 197; https://doi.org/10.3390/axioms13030197 - 15 Mar 2024

Cited by 1 | Viewed by 1355

Abstract

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We [...] Read more.

In general, the problem of building optimal convolutional codes under a certain criteria is hard, especially when size field restrictions are applied. In this paper, we confront the challenge of constructing an optimal 2D convolutional code when communicating over an erasure channel. We propose a general construction method for these codes. Specifically, we provide an optimal construction where the decoding method presented in the bibliography is considered. Full article

(This article belongs to the Special Issue Advances in Linear Algebra with Applications)

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15 pages, 1476 KiB

Open AccessEditor’s ChoiceArticle

A Comparison between Invariant and Equivariant Classical and Quantum Graph Neural Networks

by Roy T. Forestano, Marçal Comajoan Cara, Gopal Ramesh Dahale, Zhongtian Dong, Sergei Gleyzer, Daniel Justice, Kyoungchul Kong, Tom Magorsch, Konstantin T. Matchev, Katia Matcheva and Eyup B. Unlu

Axioms 2024, 13(3), 160; https://doi.org/10.3390/axioms13030160 - 29 Feb 2024

Cited by 3 | Viewed by 2564

Abstract

Machine learning algorithms are heavily relied on to understand the vast amounts of data from high-energy particle collisions at the CERN Large Hadron Collider (LHC). The data from such collision events can naturally be represented with graph structures. Therefore, deep geometric methods, such [...] Read more.

Machine learning algorithms are heavily relied on to understand the vast amounts of data from high-energy particle collisions at the CERN Large Hadron Collider (LHC). The data from such collision events can naturally be represented with graph structures. Therefore, deep geometric methods, such as graph neural networks (GNNs), have been leveraged for various data analysis tasks in high-energy physics. One typical task is jet tagging, where jets are viewed as point clouds with distinct features and edge connections between their constituent particles. The increasing size and complexity of the LHC particle datasets, as well as the computational models used for their analysis, have greatly motivated the development of alternative fast and efficient computational paradigms such as quantum computation. In addition, to enhance the validity and robustness of deep networks, we can leverage the fundamental symmetries present in the data through the use of invariant inputs and equivariant layers. In this paper, we provide a fair and comprehensive comparison of classical graph neural networks (GNNs) and equivariant graph neural networks (EGNNs) and their quantum counterparts: quantum graph neural networks (QGNNs) and equivariant quantum graph neural networks (EQGNN). The four architectures were benchmarked on a binary classification task to classify the parton-level particle initiating the jet. Based on their area under the curve (AUC) scores, the quantum networks were found to outperform the classical networks. However, seeing the computational advantage of quantum networks in practice may have to wait for the further development of quantum technology and its associated application programming interfaces (APIs). Full article

(This article belongs to the Special Issue Computational Aspects of Machine Learning and Quantum Computing)

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20 pages, 381 KiB

Open AccessEditor’s ChoiceArticle

The Fourier–Legendre Series of Bessel Functions of the First Kind and the Summed Series Involving ₁F₂ Hypergeometric Functions That Arise from Them

by Jack C. Straton

Axioms 2024, 13(2), 134; https://doi.org/10.3390/axioms13020134 - 19 Feb 2024

Cited by 3 | Viewed by 1645

Abstract

The Bessel function of the first kind

J_{N} (k x)

is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind

I_{N} (k x)

. The purpose of these expansions in Legendre polynomials was not an [...] Read more.

The Bessel function of the first kind

J_{N} (k x)

is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind

I_{N} (k x)

. The purpose of these expansions in Legendre polynomials was not an attempt to rival established numerical methods for calculating Bessel functions but to provide a form for

J_{N} (k x)

useful for analytical work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving

_{1} F_{2}

hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes

1 / (2^{i} 3^{j} 5^{k} 7^{l} 11^{m} 13^{n} 17^{o} 19^{p})

multiplying powers of the coefficient k. Full article

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

16 pages, 4276 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Time-Domain Fractional Behaviour Modelling with Rational Non-Singular Kernels

by Jocelyn Sabatier and Christophe Farges

Axioms 2024, 13(2), 99; https://doi.org/10.3390/axioms13020099 - 31 Jan 2024

Cited by 3 | Viewed by 1406

Abstract

This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power [...] Read more.

This paper proposes a solution to model fractional behaviours with a convolution model involving non-singular kernels and without using fractional calculus. The non-singular kernels considered are rational functions of time. The interest of this class of kernel is demonstrated with a pure power law function that can be approximated in the time domain by a rational function whose pole and zeros are interlaced and linked by geometric laws. The Laplace transform and frequency response of this class of kernel is given and compared with an approximation found in the literature. The comparison reveals less phase oscillation with the solution proposed by the authors. A parameter estimation method is finally proposed to obtain the rational kernel model for general fractional behaviour. An application performed with this estimation method demonstrates the interest in non-singular rational kernels to model fractional behaviours. Another interest is the physical interpretation fractional behaviours that can be implemented with delay distributions. Full article

(This article belongs to the Special Issue Fractional Calculus and the Applied Analysis)

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18 pages, 338 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

A Convergence Criterion for a Class of Stationary Inclusions in Hilbert Spaces

by Mircea Sofonea and Domingo A. Tarzia

Axioms 2024, 13(1), 52; https://doi.org/10.3390/axioms13010052 - 15 Jan 2024

Cited by 2 | Viewed by 1405

Abstract

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element

f \in X

. Under appropriate assumptions on the data, the inclusion has a [...] Read more.

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set of constraints K, a nonlinear operator A, and an element

f \in X

. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence

{u_{n}} \subset X

, which guarantee its convergence to the solution u. We then present several applications that provide the continuous dependence of the solution with respect to the data K, A and f on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws. Full article

(This article belongs to the Section Hilbert’s Sixth Problem)

20 pages, 371 KiB

Open AccessEditor’s ChoiceArticle

Probability Distributions Approximation via Fractional Moments and Maximum Entropy: Theoretical and Computational Aspects

by Pier Luigi Novi Inverardi and Aldo Tagliani

Axioms 2024, 13(1), 28; https://doi.org/10.3390/axioms13010028 - 30 Dec 2023

Cited by 7 | Viewed by 1762

Abstract

In the literature, the use of fractional moments to express the available information in the framework of maximum entropy (MaxEnt) approximation of a distribution F having finite or unbounded positive support, has been essentially considered as a computational tool to improve the performance [...] Read more.

In the literature, the use of fractional moments to express the available information in the framework of maximum entropy (MaxEnt) approximation of a distribution F having finite or unbounded positive support, has been essentially considered as a computational tool to improve the performance of the analogous procedure based on integer moments. No attention has been paid to two formal aspects concerning fractional moments, such as conditions for the existence of the maximum entropy approximation based on them or convergence in entropy of this approximation to F. This paper aims to fill this gap by providing proofs of these two fundamental results. In fact, convergence in entropy can be involved in the optimal selection of the order of fractional moments for accelerating the convergence of the MaxEnt approximation to F, to clarify the entailment relationships of this type of convergence with other types of convergence useful in statistical applications, and to preserve some important prior features of the underlying F distribution. Full article

(This article belongs to the Special Issue Statistical Methods and Applications)

28 pages, 1776 KiB

Open AccessEditor’s ChoiceArticle

On the Method of Transformations: Obtaining Solutions of Nonlinear Differential Equations by Means of the Solutions of Simpler Linear or Nonlinear Differential Equations

by Nikolay K. Vitanov

Axioms 2023, 12(12), 1106; https://doi.org/10.3390/axioms12121106 - 8 Dec 2023

Cited by 2 | Viewed by 4938

Abstract

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea [...] Read more.

Transformations are much used to connect complicated nonlinear differential equations to simple equations with known exact solutions. Two examples of this are the Hopf–Cole transformation and the simple equations method. In this article, we follow an idea that is opposite to the idea of Hopf and Cole: we use transformations in order to transform simpler linear or nonlinear differential equations (with known solutions) to more complicated nonlinear differential equations. In such a way, we can obtain numerous exact solutions of nonlinear differential equations. We apply this methodology to the classical parabolic differential equation (the wave equation), to the classical hyperbolic differential equation (the heat equation), and to the classical elliptic differential equation (Laplace equation). In addition, we use the methodology to obtain exact solutions of nonlinear ordinary differential equations by means of the solutions of linear differential equations and by means of the solutions of the nonlinear differential equations of Bernoulli and Riccati. Finally, we demonstrate the capacity of the methodology to lead to exact solutions of nonlinear partial differential equations on the basis of known solutions of other nonlinear partial differential equations. As an example of this, we use the Korteweg–de Vries equation and its solutions. Traveling wave solutions of nonlinear differential equations are of special interest in this article. We demonstrate the existence of the following phenomena described by some of the obtained solutions: (i) occurrence of the solitary wave–solitary antiwave from the solution, which is zero at the initial moment (analogy of an occurrence of particle and antiparticle from the vacuum); (ii) splitting of a nonlinear solitary wave into two solitary waves (analogy of splitting of a particle into two particles); (iii) soliton behavior of some of the obtained waves; (iv) existence of solitons which move with the same velocity despite the different shape and amplitude of the solitons. Full article

(This article belongs to the Topic Mathematical Modeling)

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12 pages, 311 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

Ramsey Chains in Linear Forests

by Gary Chartrand, Ritabrato Chatterjee and Ping Zhang

Axioms 2023, 12(11), 1019; https://doi.org/10.3390/axioms12111019 - 29 Oct 2023

Viewed by 1330

Abstract

Every red–blue coloring of the edges of a graph G results in a sequence

G_{1}

,

G_{2}

, …,

G_{ℓ}

of pairwise edge-disjoint monochromatic subgraphs

G_{i}

(

1 \leq i \leq ℓ

) of size i, such [...] Read more.

Every red–blue coloring of the edges of a graph G results in a sequence

G_{1}

,

G_{2}

, …,

G_{ℓ}

of pairwise edge-disjoint monochromatic subgraphs

G_{i}

(

1 \leq i \leq ℓ

) of size i, such that

G_{i}

is isomorphic to a subgraph of

G_{i + 1}

for

1 \leq i \leq ℓ - 1

. Such a sequence is called a Ramsey chain in G, and

A R_{c} (G)

is the maximum length of a Ramsey chain in G, with respect to a red–blue coloring c. The Ramsey index

A R (G)

of G is the minimum value of

A R_{c} (G)

among all the red–blue colorings c of G. If G has size m, then

(\binom{k + 1}{2}) \leq m < (\binom{k + 2}{2})

for some positive integer k. It has been shown that there are infinite classes S of graphs, such that for every graph G of size m in S,

A R (G) = k

if and only if

(\binom{k + 1}{2}) \leq m < (\binom{k + 2}{2})

. Two of these classes are the matchings

m K_{2}

and paths

P_{m + 1}

of size m. These are both subclasses of linear forests (a forest of which each of the components is a path). It is shown that if F is any linear forest of size m with

(\binom{k + 1}{2}) < m < (\binom{k + 2}{2})

, then

A R (F) = k

. Furthermore, if F is a linear forest of size

(\binom{k + 1}{2})

, where

k \geq 4

, that has at most

(\binom{k - 1}{2})

components, then

A R (F) = k

, while for each integer t with

(\binom{k - 1}{2}) < t < (\binom{k + 1}{2})

there is a linear forest F of size

(\binom{k + 1}{2})

with t components, such that

A R (F) = k - 1

. Full article

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

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12 pages, 6794 KiB

Open AccessEditor’s ChoiceCommunication

A Comparative Study of the Explicit Finite Difference Method and Physics-Informed Neural Networks for Solving the Burgers’ Equation

by Svetislav Savović, Miloš Ivanović and Rui Min

Axioms 2023, 12(10), 982; https://doi.org/10.3390/axioms12100982 - 18 Oct 2023

Cited by 14 | Viewed by 3291

Abstract

The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical [...] Read more.

The Burgers’ equation is solved using the explicit finite difference method (EFDM) and physics-informed neural networks (PINN). We compare our numerical results, obtained using the EFDM and PINN for three test problems with various initial conditions and Dirichlet boundary conditions, with the analytical solutions, and, while both approaches yield very good agreement, the EFDM results are more closely aligned with the analytical solutions. Since there is good agreement between all of the numerical findings from the EFDM, PINN, and analytical solutions, both approaches are competitive and deserving of recommendation. The conclusions that are provided are significant for simulating a variety of nonlinear physical phenomena, such as those that occur in flood waves in rivers, chromatography, gas dynamics, and traffic flow. Additionally, the concepts of the solution techniques used in this study may be applied to the development of numerical models for this class of nonlinear partial differential equations by present and future model developers of a wide range of diverse nonlinear physical processes. Full article

(This article belongs to the Special Issue Axioms and Methods for Handling Differential Equations and Inverse Problems)

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23 pages, 9565 KiB

Open AccessEditor’s ChoiceArticle

Color Image Recovery Using Generalized Matrix Completion over Higher-Order Finite Dimensional Algebra

by Liang Liao, Zhuang Guo, Qi Gao, Yan Wang, Fajun Yu, Qifeng Zhao, Stephen John Maybank, Zhoufeng Liu, Chunlei Li and Lun Li

Axioms 2023, 12(10), 954; https://doi.org/10.3390/axioms12100954 - 10 Oct 2023

Cited by 59 | Viewed by 2140

Abstract

To improve the accuracy of color image completion with missing entries, we present a recovery method based on generalized higher-order scalars. We extend the traditional second-order matrix model to a more comprehensive higher-order matrix equivalent, called the “t-matrix” model, which incorporates a pixel [...] Read more.

To improve the accuracy of color image completion with missing entries, we present a recovery method based on generalized higher-order scalars. We extend the traditional second-order matrix model to a more comprehensive higher-order matrix equivalent, called the “t-matrix” model, which incorporates a pixel neighborhood expansion strategy to characterize the local pixel constraints. This “t-matrix” model is then used to extend some commonly used matrix and tensor completion algorithms to their higher-order versions. We perform extensive experiments on various algorithms using simulated data and publicly available images. The results show that our generalized matrix completion model and the corresponding algorithm compare favorably with their lower-order tensor and conventional matrix counterparts. Full article

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18 pages, 346 KiB

Open AccessEditor’s ChoiceArticle

Positive Solutions for a System of Hadamard Fractional Boundary Value Problems on an Infinite Interval

by Alexandru Tudorache and Rodica Luca

Axioms 2023, 12(8), 793; https://doi.org/10.3390/axioms12080793 - 16 Aug 2023

Cited by 6 | Viewed by 1595

Abstract

Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both [...] Read more.

Our investigation is devoted to examining the existence, uniqueness, and multiplicity of positive solutions for a system of Hadamard fractional differential equations. This system is defined on an infinite interval and is subject to coupled nonlocal boundary conditions. These boundary conditions encompass both Hadamard fractional derivatives and Riemann–Stieltjes integrals, and the nonlinearities within the system are non-negative functions that may not be bounded. To establish the main results, we rely on the utilization of mathematical theorems such as the Schauder fixed-point theorem, the Banach contraction mapping principle, and the Avery–Peterson fixed-point theorem. Full article

(This article belongs to the Special Issue Research on Fixed Point Theory and Application)

13 pages, 353 KiB

Open AccessEditor’s ChoiceArticle

Solvability, Approximation and Stability of Periodic Boundary Value Problem for a Nonlinear Hadamard Fractional Differential Equation with

p

-Laplacian

by Kaihong Zhao

Axioms 2023, 12(8), 733; https://doi.org/10.3390/axioms12080733 (registering DOI) - 27 Jul 2023

Cited by 18 | Viewed by 1570

Abstract

The fractional order

p

-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary [...] Read more.

The fractional order

p

-Laplacian differential equation model is a powerful tool for describing turbulent problems in porous viscoelastic media. The study of such models helps to reveal the dynamic behavior of turbulence. Therefore, this article is mainly concerned with the periodic boundary value problem (BVP) for a class of nonlinear Hadamard fractional differential equation with

p

-Laplacian operator. By virtue of an important fixed point theorem on a complete metric space with two distances, we study the solvability and approximation of this BVP. Based on nonlinear analysis methods, we further discuss the generalized Ulam-Hyers (GUH) stability of this problem. Eventually, we supply two example and simulations to verify the correctness and availability of our main results. Compared to many previous studies, our approach enables the solution of the system to exist in metric space rather than normed space. In summary, we obtain some sufficient conditions for the existence, uniqueness, and stability of solutions in the metric space. Full article

(This article belongs to the Special Issue Differential Equations in Applied Mathematics)

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25 pages, 3239 KiB

Open AccessEditor’s ChoiceArticle

Fuzzy Logic and Decision Making Applied to Customer Service Optimization

by Gabriel Marín Díaz and Ramón Alberto Carrasco González

Axioms 2023, 12(5), 448; https://doi.org/10.3390/axioms12050448 - 30 Apr 2023

Cited by 12 | Viewed by 3592

Abstract

In the literature, the Information Technology Infrastructure Library (ITIL) methodology recommends determining the priority of incident resolution based on the impact and urgency of interactions. The RFID model, based on the parameters of Recency, Frequency, Importance and Duration in the resolution of incidents, [...] Read more.

In the literature, the Information Technology Infrastructure Library (ITIL) methodology recommends determining the priority of incident resolution based on the impact and urgency of interactions. The RFID model, based on the parameters of Recency, Frequency, Importance and Duration in the resolution of incidents, provides an individual assessment and a clustering of customers based on these factors. We can improve the traditional concept of waiting queues for customer service management by using a procedure that adds to the evaluation provided by RFID such additional factors as Impact, Urgency and Emotional character of each interaction. If we also include aspects such as Waiting Time and Contact Center Workload, we have a procedure that allows prioritizing interactions between the customer and the Contact Center dynamically and in real time. In this paper we propose to apply a model of unification of heterogeneous information in 2-tuple linguistic evaluations, to obtain a global evaluation of each interaction by applying the Analytic Hierarchy Process (AHP), and in this way be able to have a dynamic process of prioritization of interactions. Full article

(This article belongs to the Special Issue Fuzzy Set Theory and Its Applications in Decision Making)

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21 pages, 1852 KiB

Open AccessEditor’s ChoiceArticle

An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

by Heng-Pin Hsu, Te-Wen Tu and Jer-Rong Chang

Axioms 2023, 12(5), 416; https://doi.org/10.3390/axioms12050416 - 24 Apr 2023

Cited by 3 | Viewed by 5252

Abstract

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case [...] Read more.

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type. Full article

(This article belongs to the Special Issue Applied Mathematics in Energy and Mechanical Engineering)

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22 pages, 4652 KiB

Open AccessEditor’s ChoiceArticle

A Coupled PDE-ODE Model for Nonlinear Transient Heat Transfer with Convection Heating at the Boundary: Numerical Solution by Implicit Time Discretization and Sequential Decoupling

by Stefan M. Filipov, Jordan Hristov, Ana Avdzhieva and István Faragó

Axioms 2023, 12(4), 323; https://doi.org/10.3390/axioms12040323 - 24 Mar 2023

Cited by 2 | Viewed by 3036

Abstract

This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower [...] Read more.

This article considers heat transfer in a solid body with temperature-dependent thermal conductivity that is in contact with a tank filled with liquid. The liquid in the tank is heated by hot liquid entering the tank through a pipe. Liquid at a lower temperature leaves the tank through another pipe. We propose a one-dimensional mathematical model that consists of a nonlinear PDE for the temperature along the solid body, coupled to a linear ODE for the temperature in the tank, the boundary and the initial conditions. All equations are converted into a dimensionless form reducing the input parameters to three dimensionless numbers and a dimensionless function. A steady-state analysis is performed. To solve the transient problem, a nontrivial numerical approach is proposed whereby the differential equations are first discretized in time. This reduces the problem to a sequence of nonlinear two-point boundary value problems (TPBVP) and a sequence of linear algebraic equations coupled to it. We show that knowing the temperature in the system at time level n − 1 allows us to decouple the TPBVP and the corresponding algebraic equation at time level n. Thus, starting from the initial conditions, the equations are decoupled and solved sequentially. The TPBVPs are solved by FDM with the Newtonian method. Full article

(This article belongs to the Special Issue Computational Heat Transfer and Fluid Dynamics)

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